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The location this year is Gilley's Honky Tonk, that features historical memorabilia, including the mechanical bull from 1980's Urban Cowboy staring John Travolta and Debra Winger.
Another highlight is a panel session – Get Real or Go Home: Marketing Therapy for SMBs & Freelancers – that will feature Duane Forrester, Wil Reynolds and Zeph Snapp from Altura Interactive; with Sean Dolan from Pushfire as the moderator.
There is still time to book yourself a ticket and flight to "the premiere digital marketing conference in the South." More information, and the registration form can be found by clicking here.
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{"url":"https:\/\/possiblywrong.wordpress.com\/2020\/02\/26\/fair-dice-some-isohedra-are-less-fair-than-others\/","text":"## Fair dice: some isohedra are less fair than\u00a0others\n\nIntroduction\n\nWhat makes dice fair? Intuitively, when we roll a fair die with $n$ sides, we expect each of its possible outcomes to have the same probability $1\/n$ of occurring. What shapes have this property?\n\nI think this is an interesting problem, in part because it seems like there should be an elegant mathematical solution, but some unpleasantly complicated physics gets in the way. For example, even a standard six-sided die can be vulnerable to manipulation by a skilled cheat. The difficulty is that the fairness of a particular shape of die may depend on assumptions about how the die is rolled\u2013 what is the probability distribution from which we randomly draw the die\u2019s initial position, velocity, and angular momentum? What surface does the die land on\u2013 can it slide and\/or bounce, and if so, with what coefficients of friction and restitution, etc.?\n\n(For this discussion, we will focus on dice that are convex polyhedra, having flat polygonal faces with no holes or indentations. There are other interesting possibilities, though. For example, consider flipping a cylinder as a \u201cthree-sided coin,\u201d which may land on heads, tails, or on its curved \u201cedge.\u201d)\n\nSymmetry\n\nOne way to get around this dependence on physics modeling assumptions is to appeal to symmetry: if there is any shape of die that could possibly be considered fair, then a sufficient (but perhaps not necessary) condition for fairness is face-transitivity\u2014 the group of symmetries of the die should act transitively on its faces. That is, given any pair of faces $f_1$ and $f_2$, there must be a rigid transformation of the die into itself that maps $f_1$ to $f_2$.\n\nTo see why this is the criteria that we want, suppose that Charlie is a skilled-but-myopic cheat with the ability to roll the die biased toward any face $f_1$ that he desires. However, he can only see the resting position and shape of the rolled die, not the labels on its faces. So, after the roll, but before the outcome is announced, an objective third party, Oscar, selects a face $f_2$ uniformly at random, and has an opportunity to secretly rotate the rolled die, preserving its original location, to show the randomly selected face $f_2$, instead of the face $f_1$ that Charlie intended to roll.\n\nIntuitively, if the die is fair, then Oscar should be able to prevent Charlie from having undue advantage, without Charlie being aware that the die was moved. No matter what face Charlie tries to roll, and no matter what truly random face Oscar selects, it should be possible to rotate one to the other without changing the \u201cspace taken up\u201d by the die.\n\nRotations vs. reflections\n\nA convex polyhedron that is face-transitive is called an isohedron. Wolfram\u2019s MathWorld page (as well as a great Numberphile video with Persi Diaconis) describes 30 types of isohedra, noting that they \u201cmake fair dice\u201d \u2026 but face-transitivity is a property that requires some qualification. These 30 types of dice all have the property that their full symmetry group acts transitively on their faces, where by \u201cfull\u201d we mean that not just rotations but also reflections\u2013 think turning the die \u201cinside out\u201d\u2013 are allowed.\n\nSince we can\u2019t expect Oscar to secretly turn a physical die inside out, if we more realistically constrain the allowed transformations to include rotations only, then we can ask whether this proper symmetry subgroup also acts transitively on the faces of a die. It turns out that there are six types of isohedra\u2013 including two infinite classes of dipyramids\u2013 where this is not the case, i.e., the proper symmetry group does not act transitively on the faces. Instead, these \u201cless fair\u201d dice each have two distinct orbits of faces, where it is impossible to rotate a face from one orbit into a face from another orbit while preserving the overall space taken up by the die.\n\nThe figure below shows all 30 isohedra, with the six less fair dice shown with their two orbits of faces in red and green.\n\nIsohedra as classified on MathWorld, with six classes whose proper (rotational) symmetry groups do not act transitively on faces. Orbits are shown in red\/green.\n\nImplementation notes\n\nModels of these isohedra and Python code to compute their symmetry groups and face orbits are available on GitHub. I started with the models on the MathWorld page, but modified them to provide exact coordinates for all vertices, scaled to have unit minimum edge length. Computing the symmetries mapping one face to another is very similar to the telescope registration problem described here, although there are some interesting additional wrinkles due to working in limited floating-point precision.\n\nReferences:\n\n1. Diaconis, P. and Keller, J., Fair Dice, The American Mathematical Monthly, 96(4) April 1989, p. 337-339 [PDF]\n2. Weisstein, Eric W., Isohedron, MathWorld\u2013A Wolfram Web Resource\u00a0[HTML]\nThis entry was posted in Uncategorized. Bookmark the permalink.\n\n### 3 Responses to Fair dice: some isohedra are less fair than\u00a0others\n\n1. Regarding your code: The Wavefront .obj format is really convenient. It\u2019s like the Netpbm of 3D models \u2014 including irritating minor limitations \u2014 and it\u2019s always the first format I go to when I need to handle model data, especially since Blender supports it.\n\n\u2022 Yeah, I chose it since Mathematica supports it, but mostly because it looked really easy to parse, as in my too-quick-and-dirty Python loader.\n\nSpeaking of irritating minor limitations\u2013 I didn\u2019t see a similarly \u201cminimal\u201d way to specify per-face colors (e.g., the separate face orbits in red\/green) without resorting to a separate materials file.\n\nThis site uses Akismet to reduce spam. Learn how your comment data is processed.","date":"2021-06-14 18:52:21","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\": 10, \"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\": 0.6271210312843323, \"perplexity\": 897.8813297291332}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"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-2021-25\/segments\/1623487613380.12\/warc\/CC-MAIN-20210614170602-20210614200602-00294.warc.gz\"}"}
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{"url":"https:\/\/www.terrychan.org\/2021\/08\/linux-%E7%8E%AF%E5%A2%83%E5%8F%98%E9%87%8F\/","text":"# linux \u73af\u5883\u53d8\u91cf\n\nlinux \u73af\u5883\u53d8\u91cf\n\nimport os\n\nos.environ['ENV_VAR']=\"True\"\n\nmy_env =None\n\nif bool(my_env):\n<h1>\/\/ if my_env: gives the same result<\/h1>\n<h1>Env exists<\/h1>\nprint(\"ok\")\n\nelse:\n\npass\n<h1>Env not set<\/h1>\n\n\n## #Dcoker \u9650\u5236\u5bb9\u5668\u7684\u5185\u5b58\n\nDocker can enforce hard memory limits, which allow the container to use no more than a given amount of user or system memory, or soft limits, which allow the container to use as much memory as it needs unless certain conditions are met, such as when the kernel detects low memory or contention on the host machine. Some of these options have different effects when used alone or when more than one option is set.\n\nMost of these options take a positive integer, followed by a suffix of b, k, m, g, to indicate bytes, kilobytes, megabytes, or gigabytes.\n\nOption\n\nDescription\n\n-m or --memory=\n\nThe maximum amount of memory the container can use. If you set this option, the minimum allowed value is 4m (4 megabyte).\n\n--memory-swap*\n\nThe amount of memory this container is allowed to swap to disk. See --memory-swap details.\n\n--memory-swappiness\n\nBy default, the host kernel can swap out a percentage of anonymous pages used by a container. You can set --memory-swappiness to a value between 0 and 100, to tune this percentage. See --memory-swappiness details.\n\n--memory-reservation\n\nAllows you to specify a soft limit smaller than --memory which is activated when Docker detects contention or low memory on the host machine. If you use --memory-reservation, it must be set lower than --memory for it to take precedence. Because it is a soft limit, it does not guarantee that the container doesn\u2019t exceed the limit.\n\n--kernel-memory\n\nThe maximum amount of kernel memory the container can use. The minimum allowed value is 4m. Because kernel memory cannot be swapped out, a container which is starved of kernel memory may block host machine resources, which can have side effects on the host machine and on other containers. See --kernel-memory details.\n\n--oom-kill-disable\n\nBy default, if an out-of-memory (OOM) error occurs, the kernel kills processes in a container. To change this behavior, use the --oom-kill-disable option. Only disable the OOM killer on containers where you have also set the -m\/--memory option. If the -m flag is not set, the host can run out of memory and the kernel may need to kill the host system\u2019s processes to free memory.","date":"2023-03-24 22:31:26","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\": 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\": 0.31786441802978516, \"perplexity\": 2298.7015479105676}, \"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\/1679296945289.9\/warc\/CC-MAIN-20230324211121-20230325001121-00041.warc.gz\"}"}
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{"url":"https:\/\/mathematica.stackexchange.com\/questions\/241797\/stringcases-and-shortest-from-right-to-left","text":"# StringCases and Shortest from right to left\n\nAs explained in https:\/\/mathematica.stackexchange.com\/a\/72293\/45020, using Shortest in StringCases does not actually return the shortest matching string. Instead it looks until some part matches (from left to right) and then it looks ahead and stops at the first complete match from that position. Of course this misses the true shortest string whenever this first match is not part of the shortest string. See for example\n\nStringCases[\"This is not what I'm looking for. This is part of the\nshortest match\", Shortest[\"This is\" ~~ any__ ~~ \"match\"]]\n\n\nwhere the whole string gets matched while a stronger string match is available.\n\nI need the reverse behavior. Where one searches from right to left (Alternatively I need to find the \"true\" Shortest substring, that would also work for my application).\n\nI have a StringExpression that describes some pattern, say \"out [\"~~x: NumberString~~\"]\", which at some point is preceded by some strong \"some_string\" from which I want to extract the string between the end of the regular pattern and the closest preceding instance of \"some_string\". So for example\n\nreverseShortestCase[\"some_string other stuff I don't want. some_string, the part I do want, out [3.1]\", \"some_string\", \"out [\"~~x: NumberString~~\"]\"]\n\n\nshould return \"some_string, the part I do want, out [3.1]\".\n\nHow should I go about doing this? I guess I can try to reverse the string and the patterns in order to search right to left? But this seems complicated if you want to be able to use things like NumberString.\n\nThe solutions proposed in the linked question do not seem to offer a solution for this case where there is no fixed begin and end string and instead a StringExpression needs to be used to describe these.\n\n\u2022 In your first example, what do you mean by \"misses the true shortest string\"? That StringCases call returns two matches to the input, which seems like the correct behavior. \u2013\u00a0Jason B. Mar 15 at 17:55\n\u2022 @Jason B. I wonder if the OP has seen that the StringCases returns a list of 2 strings, (and not the whole initial string) \u2013\u00a0andre314 Mar 15 at 17:58\n\u2022 @andre314 - or maybe a better example of their problem is StringCases[\"This is not what I'm looking for. This is part of the shortest match\", Shortest[\"This is\" ~~ any__ ~~ \"match\"]] which only has one match, and is not the intended match. \u2013\u00a0Jason B. Mar 15 at 18:00\n\u2022 So sorry, thoughtlessly altered the example for an aesthetics reason after writing (and testing it) accidentally destroying the whole point. \u2013\u00a0Kvothe Mar 15 at 18:34\n\nI would generally prefer an approach that retains the ability to apply replacement rules (e.g. Jason B's ) but... another approach would be to find all matches using Overlaps->True and to then pick out the shortest one afterwards:\n\n$$string = \"some_string other stuff I don't want. some_string, the part I do want, out [3.1]\";$$pattern = \"some_string\"~~___~~\"out [\"~~x:NumberString~~\"]\";\n\nStringCases[$$string,$$pattern, Overlaps -> True] \/\/ MinimalBy[StringLength]\n\n(* {\"some_string, the part I do want, out [3.1]\"} *)\n\n\nThis brute-force approach might have some appeal if, for example, there were the possibility of multiple shortest matches and we wanted to apply complex logic pick one out or otherwise process the hits.\n\nYou can specify that the trigger \"some_string\" only occurs at the beginning of the pattern, and not in the middle:\n\nIn[49]:=\npattern =\n\"some_string\" ~~ Shortest[in__ \/; StringFreeQ[in, \"some_string\"]] ~~\n\"out [\" ~~ NumberString ~~ \"]\";\n\nIn[50]:= StringCases[\"some_string other stuff I don't want. some_string, the part I do want, out [3.1]\", pattern]\n\nOut[50]= {\"some_string, the part I do want, out [3.1]\"}\n$$$$\n`","date":"2021-05-11 09:55:56","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"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\": 3, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.2620733976364136, \"perplexity\": 1882.4911171161796}, \"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-2021-21\/segments\/1620243991982.8\/warc\/CC-MAIN-20210511092245-20210511122245-00283.warc.gz\"}"}
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Webster's Dictionary: any extended period of leave from one's customary work, especially for rest, to acquire new skills or training, etc.
Urban Dictionary: The time a businessman spends passed out on the hotel lobby floor after consuming too much alcohol at a business event.
Whatever your definition, that's where I am. I quit. Resigned. Retired. Left the Company. Removed myself from the situation. I'm on sabbatical!
Wait, you left your job without having another? Uh oh, did you feel that? That shaking of the earth was my father rolling over in his grave. Okay, let me explain.
For the past two years I've worked between 75 and 82 hours per week. No days (or nights) off for 3-4 months at a time. Still, I believed I could make a difference. Make magic. So why leave, you say? Well, even codeine has an expiration date. After a while, its magic gets to be more like a parlor trick and it gives you less of a high.
What did I love about my job? Oh lots. I loved seeing new places (well, as much as I could see in the 2-3 hours off I had. And new places? Not so many new ones unless you consider that sparkly slot machine at Atlantis a new place), meeting new people, learning more about leadership, dreaming with Imagineers (I used to have Imaginary friends, now I have Imagineer friends), having Captain Hook wave to me as he walked past my office, knowing princesses (Cinderella's my favorite and I always wanted to go shoe shopping with her) and generally, making magic for children. Oh, and sparkles! I loved taking a break from reports and E-mails to do arts and crafts where I would invariably end up covered in sparkles.
Why did I leave, you ask? Oh, those reasons are more complicated. You see, this was never meant to be long term. If you recall, in the year preceding these last two, things were, shall we say, a bit tough financially. Oh, who am I kidding, I was so broke I couldn't even pay attention (any further questions on this please refer to, 'I'm From the Government and I'm Here to Help' or, 'Yes Virginia, There is a Mortgage Modification.').
This wasn't just about the money (okay, at the time it was mostly about the money). I never had any intention of going back to my previous life working on board cruise ships (add over five years to these past two). I was truly ready to leave the past in the past. It's a bit like going to live in the college dorm, only I was in my 40's. But the job presented itself and, as my house was due to go on the auction block in a month, I wouldn't have anywhere to live anyway.
So now, here I am. The situation is much different now. I was able to pay off some bills, save some money and even have a lovely vacation in England (yes, you can read about the unplanned adventures beginning with "On a Wing and a Prayer"). And another benefit, I have The Walt Disney Company on my resume (or my c/v if you're from Europe).
The decision to leave certainly wasn't like buying that Snickers bar at the grocery checkout. It was definitely not an impulse buy. Last year I spoke with my manager about transitioning off ships and into a land based position with the company. One big challenge, I won't live in Florida. Nothing against Florida but. . . okay, sorry but I do have something against Florida. Tried it, hated it. In order not to outrage any of my Florida followers I will leave it at that (so unlike me).
So, you want to work for Disney and not live in Florida? Believe it or not, it can be done. California, England, Tokyo, Paris, Shanghai, the list goes on. Still, I was told that it may need to be a lateral move (fine by me) or even a downgrade (not so fine). I'm not twenty-two years old anymore (I know from my photo you find that hard to believe). I have made certain advances in my career by gaining knowledge and working really hard. Going backwards is not in the plan. I did a lot of networking (much with the PR and Marketing departments) but I was also open to outside opportunities. After all, I keep getting told how great it is to have Disney on a resume. Well, let's just see if Uncle Walt has some influence outside his own company.
I should also mention something else that happened this past summer. My friend BreAnn took a break in her chemo. and cruised with me in Alaska. At the end of the trip she commented, "They don't pay you enough. I don't know what they pay you but it's not enough." When I arrived home after this latest contract and received the call that Bre wasn't doing well, I went over to her house. I climbed into bed with her, held her hand and told her I was quitting. At this point she wasn't responding to much. She opened her eyes and looked at me and said, "Good, you need to." Well, there you go. Promise your dying friend something and changing your mind is not an option.
it's time to enjoy it for a bit. Okay, still setting up my life again. Getting my taxes done, securing health insurance, getting internet for my house and attempting to kill my newly acquired weed garden (it turns out that, as much as they may look like herbs, you cannot cook any sort of edible chicken with them). Also, a little time to enjoy riding my new bike, doing yoga, reestablishing friendships that I have (unwillingly) had to ignore for the past two years, a bit of traveling (beautiful, exotic Chicago) and hot bubble baths with scotch and candles (wait, not bathing in scotch – water in the tub, scotch in a glass).
Oh, and another thing I get to do on a regular basis now – write. And while I enjoy being a bit cliché and sitting at the coffee bar drinking a latte or espresso while writing, it turns out I am just as inspired sitting in the park drinking a bottle of wine.
Stand by while I find some new adventures or perhaps, just make fun of the world in general. And, if you're looking to hire me, well, I'm not cheap but I can be had.
Thanks for the laugh, Carole. Enjoy the sabbatical. Oh, and Florida doesn't want you, anyway.
Thanks Laura. I am enjoying it. But unless I win the lottery I will need to start thinking about a job soon. In the meantime, more wine.
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"redpajama_set_name": "RedPajamaC4"
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\section{Introduction}
Programming is the core of software development, but it is also an inherently error-prone activity. The likelihood of errors will even be significantly higher when programming with a quantum computer, as the techniques used for classical programming are, unfortunately, hard to apply to quantum computers because quantum systems are essentially different from classical ones. Thus, there is a pressing need to provide verification and analysis techniques for reasoning about the correctness of quantum programs. Furthermore, these techniques would also be very useful for compiling and optimising quantum programs.
Among other techniques, Hoare logic~\cite{hoare1969axiomatic} provides a syntax-oriented proof system to reason about program correctness. After decades of development, Hoare logic has been successfully applied in analysis of programs with non-determinism, recursion, parallel execution, etc~\cite{apt2019fifty,apt2010verification}. It was also extended to programming languages with probabilistic features. Remarkably, as the program states for probabilistic languages are (sub)distributions over evaluations of program variables, the extension naturally follows two different approaches, depending on how assertions of probabilistic states are defined. The first one
takes subsets of distributions as (qualitative) assertions, similar to the non-probabilistic case, and the \emph{satisfaction} relation between distributions and assertions is then just the ordinary membership~\cite{ramshaw1979formalizing,den2002verifying,chadha2007reasoning,BEGGHS18}. In contrast, the other approach takes
non-negative functions on evaluations as (quantitative) assertions. Consequently, one is concerned with the \emph{expectation} of a distribution satisfying an assertion~\cite{morgan1996probabilistic,mciver2005abstraction,olmedo2016reasoning,kozen1981semantics,kozen1985probabilistic}.
In the past two decades, several Hoare-type logic systems for quantum programs (QHL) have been proposed, also following the two approaches as in the probabilistic setting.
\textbf{Expectation-based QHLs}. In the logic systems proposed in~\cite{Yin12,ying2016foundations,ying2018reasoning,li2019quantum} for purely quantum programs, the assertions $P$ and $Q$ in a Hoare triple $\{P\} S \{Q\}$ are both positive operators with the eigenvalues lying in $[0,1]$, and such a triple is valid in the sense of total correctness if for any initial quantum state $\rho$, ${\rm tr}(P\rho) \leq {\rm tr}(Q\rho')$ where $\rho'$ is the final state obtained by executing $S$ on $\rho$, and ${\rm tr}(P\rho)$ denotes the expectation/degree of $\rho$ satisfying $P$ (or physically, the average outcome when measuring $\rho$ according to the projective measurement determined by $P$). This definition captures the idea that the precondition $P$ (on the initial state) provides a lower bound on the degree of satisfaction of the postcondition $Q$ (on the finial state).
More recently, this type of expectation-based QHL has been extended to quantum programs with classical variables~\cite{feng2020quantum}
as well as distributed quantum programs with classical communication~\cite{feng2021verification}.
These logic systems have proven to be useful in describing and verifying correctness of a wide range of quantum algorithms such as Shor's algorithm~\cite{Sho94}, Grover's algorithm~\cite{Gro96}, etc. Moreover, they are theoretically elegant: all of these systems are (relatively) complete in the sense that every semantically valid Hoare triple can be deduced from the corresponding proof system.
\textbf{Limit of the expectation-based approach}. However, the expectation-based quantum Hoare logic systems proposed in the literature all suffer from the following \emph{expressiveness} problems.
\begin{itemize}
\item Unlike the classical boolean-valued assertions, the positive operator assertions cannot exclude undesirable quantum inputs. Note that the correctness of many algorithms such as quantum teleportation (an EPR pair is assumed as part of the input) and phase estimation (the corresponding eigenstate is given) all have some restrictions on the input. The expressiveness of expectation-based QHL might be limited, as the following artificial example suggests.
Let
$$S\equiv x := M_{\pm}[q]$$
where $M_{\pm}$ is the measurement according to the $|\pm\>$ basis.
Obviously, starting with $|0\>$, the program ends at $|+\>$ with probability 0.5. However, expectation-based QHL cannot describe this property: the natural candidate
\begin{equation}\label{eq:cexample}
\{0.5 |0\>\<0|\}\ S\ \{|+\>\<+|\}
\end{equation}
is actually not valid. The reason is, it somehow \emph{over-specifies} the correctness: in addition to the above desirable property, it also puts certain requirement for \emph{all other} possible input states. To see this, take $\rho = |-\>\<-|$. Then $\Denote{S}(\rho) = |-\>\<-|$. Thus $${\rm tr}(0.5 |0\>\<0|\rho) = 0.25 > {\rm tr}(|+\>\<+|\Denote{S}(\rho)) =0,$$
making the Hoare triple in Eq.~\eqref{eq:cexample} invalid.
\item As logic operations such as conjunction and disjunction are difficult, if at all possible, to define for positive operators, complicated properties can only be analysed separately, making the verification process cumbersome. This has been pointed out in~\cite{BEGGHS18} for expectation-based probabilistic Hoare logics. The same is obviously true for expectation-based QHLs as well.
\end{itemize}
\textbf{Satisfaction-based QHLs}. An Ensemble Exogenous Quantum Propositional Logic (EEQPL) was proposed in~\cite{chadha2006reasoning} for a simple quantum language with bounded iteration. The assertions in EEQPL can access amplitudes of quantum states, which makes it very strong in expressiveness but also hinders its use in applications such as debugging, as amplitudes are not physically accessible through measurements. The completeness of EEQPL is only proven in a special case where all real and complex values involved range over a finite set. In contrast, the QHL proposed in~\cite{Kakutani:2009} takes as the assertion language an extended first-order logic with the primitives of applying a matrix on a set of qubits and computing the probability that a classical predicate is satisfied by the outcome of a quantum measurement. The proof system is shown to be sound, but no completeness result was established.
Another way of defining satisfaction-based QHLs proposed in~\cite{zhou2019applied,unruh2019quantum} regard subspaces of the Hilbert space as assertions, and a quantum state $\rho$ satisfies an assertion $P$ iff the support (the image space of linear operators) of $\rho$ is included in $P$. The subspace assertion
makes it easy to describe and determine properties of quantum programs, but the expressive power of the assertions is limited: they only assert if a given quantum state lies completely within a subspace. Consequently, quantum algorithms which succeed with certain probability cannot be verified in their logic systems.
\textbf{Contribution of the current paper}. In this paper, motivated by~\cite{BEGGHS18}, we propose two Hoare-style proof systems: an abstract one and a concrete one.
It is worth noting that the imperative language we consider here involves both classical and quantum constructs.
Our work distinguishes itself from the works on QHLs mentioned above in the following aspects:
\begin{itemize}
\item \emph{Assertion language}. The assertions used in our logic systems are boolean-typed, so that they can be easily combined using logic operations such as disjunction and conjunction. On the other hand, all information used in the assertions are physically accessible: they can be obtained through quantum measurement applying on the program states.
For example, consider the protocol of superdense coding (see Section~\ref{sec:example}). In order to send a message of two bits stored in two variables $x_0x_1$, Alice actually sends a qubit to Bob. From the received quantum information Bob can recover the message $y_0y_1$. The property we would expect is $x_0=y_0 \wedge x_1=y_1$. Indeed, by letting $SC$ be a quantum program to implement the protocol, we can prove that the following judgement is derivable:
\[\triple{{\textbf{true}}} {SC}{\Box(x_0=y_0 \wedge x_1=y_1)}\]
where the precondition ${\textbf{true}}$ is satisfied by any program state. Intuitively, the judgement says that the message received by Bob is always the same as that sent by Alice, no matter what the initial program state is. There is no need to mention the concrete values of $x_0$ and $x_1$. This is a natural and concise way of specifying the correctness of the program $SC$.
\item \emph{Satisfaction-based complete QHL}. The existing satisfaction-based QHLs proposed in the literature all lack of completeness; the only exception is~\cite{zhou2019applied}, but as mentioned above, the assertions there are not expressive enough to verify probabilistic correctness.
The abstract proof system proposed in the current work is shown to be sound and relatively complete, while the concrete proof system is sound only. By soundness, we mean that if the Hoare triple $\triple{P}{c}{Q}$ is derivable, then for any program state $\mu$ that satisfies $P$, the program state $\Denote{c}_\mu$ after the execution of command $c$ always satisfies $Q$. Completeness means the converse. We only have relative completeness because our assertion language allows for first-order logic operators such as implication. Note that in the presence of both classical and quantum variables, we represent each program state $\mu$ as a partial density operator valued distribution (POVD), and interpret a program as a transformer of POVDs. We establish a consistence result between the denotational semantics and the small-step operational semantics based on POVDs.
\end{itemize}
The rest of the paper is structured as follows. In Section~\ref{sec:pre} we recall some basic notations from linear algebra and quantum mechanics. In Section~\ref{sec:qimp} we define the syntax and operational semantics of a simple classical-quantum imperative language. In Section~\ref{sec:abs} we present an abstract proof system and show its soundness and relative completeness. In Section~\ref{sec:conc} we provide a concrete proof system. In Section~\ref{sec:example} we use the example of superdense coding to illustrate the concrete proof system. Finally, we conclude in Section~\ref{sec:clu}.
\section{Preliminaries}\label{sec:pre}
We briefly recall some basic notations
from linear algebra and quantum mechanics which are needed in this paper.
For more details, we refer to \cite{NC00}.
A {\it Hilbert space} $\ensuremath{\mathcal{H}}$ is a complete vector space with an inner
product $\langle\cdot|\cdot\rangle:\ensuremath{\mathcal{H}}\times \ensuremath{\mathcal{H}}\rightarrow \mathbf{C}$
such that
\begin{enumerate}
\item
$\langle\psi|\psi\rangle\geq 0$ for any $|\psi\>\in\ensuremath{\mathcal{H}}$, with
equality if and only if $|\psi\rangle =0$;
\item
$\langle\phi|\psi\rangle=\langle\psi|\phi\rangle^{\ast}$;
\item
$\langle\phi|\sum_i c_i|\psi_i\rangle=
\sum_i c_i\langle\phi|\psi_i\rangle$,
\end{enumerate}
where $\mathbf{C}$ is the set of complex numbers, and for each
$c\in \mathbf{C}$, $c^{\ast}$ stands for the complex
conjugate of $c$. For any vector $|\psi\rangle\in\ensuremath{\mathcal{H}}$, its
length $|||\psi\rangle||$ is defined to be
$\sqrt{\langle\psi|\psi\rangle}$, and it is said to be {\it normalised} if
$|||\psi\rangle||=1$. Two vectors $|\psi\>$ and $|\phi\>$ are
{\it orthogonal} if $\<\psi|\phi\>=0$. An {\it orthonormal basis} of a Hilbert
space $\ensuremath{\mathcal{H}}$ is a basis $\{|i\rangle\}$ where each $|i\>$ is
normalised and any pair of them are orthogonal.
Let $\ensuremath{\mathcal{L(H)}}$ be the set of linear operators on $\ensuremath{\mathcal{H}}$.
For any $A\in \ensuremath{\mathcal{L(H)}}$, $A$ is {\it Hermitian} if $A^\dag=A$ where $A^\dag$ is the adjoint operator of $A$ such that
$\<\psi|A^\dag|\phi\>=\<\phi|A|\psi\>^*$ for any
$|\psi\>,|\phi\>\in\ensuremath{\mathcal{H}}$.
A linear operator $A\in \ensuremath{\mathcal{L(H)}}$ is {\it unitary} if $A^\dag A=A A^\dag=I_\ensuremath{\mathcal{H}}$ where $I_\ensuremath{\mathcal{H}}$ is the
identity operator on $\ensuremath{\mathcal{H}}$.
The {\it trace} of $A$ is defined as ${\rm tr}(A)=\sum_i \<i|A|i\>$ for some
given orthonormal basis $\{|i\>\}$ of $\ensuremath{\mathcal{H}}$.
A linear operator $A\in \ensuremath{\mathcal{L(H)}}$ is {\it positive} if $\<\phi|A|\phi\> \geq 0$ for any state $|\phi\> \in\ensuremath{\mathcal{H}}$.
The \emph{L\"{o}wner order} $\sqsubseteq$ on the set of Hermitian operators on $\ensuremath{\mathcal{H}}$ is defined by letting $A\sqsubseteq B$ iff $B-A$ is positive.
Let $\ensuremath{\mathcal{H}}_1$ and $\ensuremath{\mathcal{H}}_2$ be two Hilbert spaces. Their {\it tensor product} $\ensuremath{\mathcal{H}}_1\otimes \ensuremath{\mathcal{H}}_2$ is
defined as a vector space consisting of
linear combinations of the vectors
$|\psi_1\psi_2\rangle=|\psi_1\>|\psi_2\rangle =|\psi_1\>\otimes
|\psi_2\>$ with $|\psi_1\rangle\in \ensuremath{\mathcal{H}}_1$ and $|\psi_2\rangle\in
\ensuremath{\mathcal{H}}_2$. Here the tensor product of two vectors is defined by a new
vector such that
$$\left(\sum_i \lambda_i |\psi_i\>\right)\otimes
\left(\sum_j\mu_j|\phi_j\>\right)=\sum_{i,j} \lambda_i\mu_j
|\psi_i\>\otimes |\phi_j\>.$$ Then $\ensuremath{\mathcal{H}}_1\otimes \ensuremath{\mathcal{H}}_2$ is also a
Hilbert space where the inner product is defined as the following:
for any $|\psi_1\>,|\phi_1\>\in\ensuremath{\mathcal{H}}_1$ and $|\psi_2\>,|\phi_2\>\in
\ensuremath{\mathcal{H}}_2$,
$$\<\psi_1\otimes \psi_2|\phi_1\otimes\phi_2\>=\<\psi_1|\phi_1\>_{\ensuremath{\mathcal{H}}_1}\<
\psi_2|\phi_2\>_{\ensuremath{\mathcal{H}}_2}$$ where $\<\cdot|\cdot\>_{\ensuremath{\mathcal{H}}_i}$ is the inner
product of $\ensuremath{\mathcal{H}}_i$.
By applying quantum gates to qubits, we can change their
states. For example, the Hadamard gate (H gate) can be applied
on a single qubit, while the CNOT gate can be applied on two
qubits. Some commonly used gates and their
representation in terms of matrices are as follows.
$${\textit{CNOT}}=\left(%
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0
\end{array}%
\right),$$
\[
H=\frac{1}{\sqrt{2}}\left(%
\begin{array}{cc}
1 & 1 \\
1 & -1 \\
\end{array}%
\right),\ \ I_2=\left(%
\begin{array}{cc}
1 & 0 \\
0 & 1 \\
\end{array}%
\right),\ \
X=\left(%
\begin{array}{cc}
0 & 1 \\
1 & 0 \\
\end{array}%
\right),\ \ Z=\left(%
\begin{array}{cc}
1 & 0 \\
0 & -1 \\
\end{array}%
\right).
\]
According to von Neumann's formalism of quantum mechanics
\cite{vN55}, an isolated physical system is associated with a
Hilbert space which is called the {\it state space} of the system. A {\it pure state} of a
quantum system is a normalised vector in its state space, and a
{\it mixed state} is represented by a density operator on the state
space. Here a \emph{density operator} $\rho$ on Hilbert space $\ensuremath{\mathcal{H}}$ is a
positive linear operator such that ${\rm tr}(\rho)= 1$. A \emph{partial density operator} $\rho$ is a positive linear operator with ${\rm tr}(\rho)\leq 1$.
The evolution of a closed quantum system is described by a unitary
operator on its state space: if the states of the system at times
$t_1$ and $t_2$ are $\rho_1$ and $\rho_2$, respectively, then
$\rho_2=U\rho_1U^{\dag}$ for some unitary operator $U$ which
depends only on $t_1$ and $t_2$.
A quantum {\it measurement} is described by a
collection $\{M_m\}$ of measurement operators, where the indices
$m$ refer to the measurement outcomes. It is required that the
measurement operators satisfy the completeness equation
$\sum_{m}M_m^{\dag}M_m=I_\ensuremath{\mathcal{H}}$. If the system is in state $\rho$, then the probability
that measurement result $m$ occurs is given by
$$p(m)={\rm tr}(M_m^{\dag}M_m\rho),$$ and the state of the post-measurement system
is $M_m\rho M_m^{\dag}/p(m).$
\section{{\textbf{QIMP}}}\label{sec:qimp}
We define the syntax and operational semantics of a simple classical-quantum imperative language called {\textbf{QIMP}}. The language is essentially extended from \textbf{IMP} \cite{Win93} by adding quantum data and a few operations for manipulating quantum data.
\subsection{Syntax }
We assume three types of data in our language: {\tt Bool} for booleans, {\tt Int} for integers, and qubits {\tt Qbt} for quantum data. Let $\mathbb{Z}$ be the set of constant integer numbers, ranged over by $n$. Let ${\textbf{Cvar}}$, ranged over by $x,y,...$, be the set of classical variables, and ${\textbf{Qvar}}$, ranged over by $q,q',...$, the set of quantum variables. It is assumed that both ${\textbf{Cvar}}$ and ${\textbf{Qvar}}$ are countably infinite. We assume a set {{\textbf{Aexp}}} of arithmetic expressions over {\tt Int}, which includes ${\textbf{Cvar}}$ as a subset and is ranged over by $a, a',...$, and a set of boolean-valued expressions ${\textbf{Bexp}}$, ranged over by $b,b',...$, with the usual boolean constants {{\textbf{true}}, {\textbf{false}}} and boolean operators $\neg, \wedge,\vee$. In particular, we let $a=a'$ and $a\leq a'$ be boolean expressions for any $a,a'\in{\textbf{Aexp}}$. We further assume that only classical variables can occur free in both arithmetic and boolean expressions.
We let $U$ range over unitary operators, which can be user-defined matrices or built in if the language is implemented. For example, a concrete $U$ could be the $1$-qubit Hadamard operator $H$, or the 2-qubit controlled-NOT operator ${\textit{CNOT}}$, etc. Similarly, we write $M$ for the measurement described by a collection $\{M_i\}$ of measurement operators, with each index $i$ representing a measurement outcome. For example, to describe the measurement of the qubit referred to by variable $q$ in the computational basis, we can write $M:=\{M_0, M_1\}$, where $M_0=|0\>_q\<0|$ and $M_1=|1\>_q\<1|$.
\leaveout{
The following list summarizes the syntactic sets associated with \Imp.
\begin{itemize}
\item real constants $\mathbb{R}$, consisting of all real constant numbers, ranged over by \emph{metavariables} \Math{n,m}
\item truth values {\textbf{T}}=$\sset{{\textbf{true}},{\textbf{false}}}$,
\item classical variables {\textbf{Cvar}}, ranged over by \Math{x, y}
\item quantum variables {\textbf{Qvar}}, ranged over by \Math{q, r}
\item arithmetic expressions {\textbf{Aexp}}, ranged over by \Math{a}
\item boolean expressions {\textbf{Bexp}}, ranged over by \Math{b}
\item commands {\textbf{Com}}, ranged over by \Math{c}
\end{itemize}
}
Sometimes we use metavariables which are primed or subscripted, e.g. $x', x_0$ for classical variables. We abbreviate a tuple of quantum variables $\pair{q_1,...,q_n}$ as $\bar{q}$ if the length $n$ of the tuple is not important.
The formation rules for arithmetic and boolean expressions as well as commands are defined by the following grammar.
\begin{itemize}
\item For {\textbf{Aexp}}: \quad \Math{a ::= n \mid x \mid a_0+a_1 \mid a_0 - a_1 \mid a_0\times a_1}
\item For {\textbf{Bexp}}: \quad \Math{b ::={\textbf{true}} \mid {\textbf{false}} \mid a_0=a_1 \mid a_0 \leq a_1 \mid \neg b \mid b_0\wedge b_1 \mid b_0 \vee b_1}
\item For {{\textbf{Com}}}: \[\begin{array}{rl}
c ::= & {\textbf{skip}} \mid x:=a \mid c_0;c_1 \mid \textbf{if}~ b ~\textbf{then}~ c_0 ~\textbf{else}~ c_1 \mid \textbf{while}~ b ~\textbf{do}~ c \\
& \mid q := |0\> \mid { U}[\bar{q}] \mid x := { M}[\bar{q}]
\end{array}\]
\end{itemize}
An arithmetic expression can be an integer, a variable, or built from other arithmetic expressions by addition, subtraction, or multiplication. A boolean expression can be formed by comparing arithmetic expressions or by using the usual boolean operators. A command can be a skip statement, a classical assignment, a conditional statement, or a while-loop, as in many classical imperative languages. In addition, there are three commands that involve quantum data. The command $q := |0\>$ initialises the qubit referred to by variable $q$ to be the basis state $|0\>$. The command ${ U}[\bar{q}]$ applies the unitary operator ${ U}$ to the quantum system referred to by $\bar{q}$. The command $x := { M}[\bar{q}]$ performs a measurement $M$ on $\bar{q}$ and assigns the measurement outcome to $x$. It differs from a classical assignment because the measurement $M$ may change the quantum state of $\bar{q}$, besides the fact that the value of $x$ is updated.
\subsection{Operational Semantics}
Since the execution of a {{\textbf{QIMP}}} program may involve both classical and quantum data, we
consider the setting where the CPU that processes the program has two registers: one stores classical data and the other quantum data. Therefore, we will model a machine state as a pair composed of a classical state and a quantum state.
The notion of classical state is standard. Formally, a \emph{classical state} is a function \Math{\sigma: {\textbf{Cvar}}\rightarrow\mathbb{Z}} from classical variables to integers. Thus $\sigma(x)$ is the value of variable $x$ in state $\sigma$. The notion of quantum state is slightly more complicated. For each quantum variable $q\in {\textbf{Qvar}}$, we assume a $2$-dimensional Hilbert space ${\cal H}_q$ to be the state space of the $q$-system. For any finite subset $V$ of ${\textbf{Qvar}}$, we denote
\[ {\cal H}_V \ = \ \bigotimes_{q\in V}{\cal H}_q.\]
That is, ${\cal H}_V$ is the tensor product of the individual state spaces of all the quantum variables in $V$. Throughout the paper, when we refer to a subset of ${\textbf{Qvar}}$, it is always assumed to be finite.
Given $V\subseteq{\textbf{Qvar}}$,
the set of \emph{quantum states} consists of all partial density operators in the space ${\cal H}_V$, denoted by $\padist{{\cal H}_V}$.
A \emph{machine state} is a pair $\pair{\sigma,\rho}$ where $\sigma$ is a classical state and $\rho$ a quantum state. In the presence of measurements, we often need to consider an ensemble of states. For that purpose, we introduce a notion of distribution.
\begin{definition}
Suppose $V\subseteq{\textbf{Qvar}}$ and $\Sigma$ is the set of classical states, i.e., the set of functions of type ${\textbf{Cvar}}\rightarrow\mathbb{Z}$.
A \emph{partial density operator valued distribution (POVD)} is a function $\mu: \Sigma\rightarrow\padist{{\cal H}_V}$ with $\sum_{\sigma\in\Sigma}{\rm tr}(\mu(\sigma))\leq 1$.
\end{definition}
Intuitively, a POVD $\mu$ represents a collection of machine states where each classical state $\sigma$ is associated with a quantum state $\mu(\sigma)$.
The notation of POVD is called classical-quantum state in \cite{feng2020quantum}.
If the collection has only one element $\sigma$, we explicitly write $(\sigma,\mu(\sigma))$ for $\mu$.
The support of $\mu$, written $\support{\mu}$, is the set $\{\sigma\in\Sigma \mid \mu(\sigma)\not=0\}$.
We can also define the addition of two distributions by letting
$(\mu_1+\mu_2)(\sigma) = \mu_1(\sigma)+\mu_2(\sigma)$.
A \emph{configuration} is a pair $\pair{e,\sigma,\rho}$, where $e$ is an expression and $(\sigma,\rho)$ is a POVD.
We define the small-step operational semantics of arithmetic and boolean expressions as well as commands in a syntax-directed way by using an evaluation relation $\hookrightarrow$ between configurations. In Figure~\ref{fig:evalab} we list the rules for evaluating integer variables, sums, and expressions of the form $a_0\leq a_1$; the rules for other arithmetic and boolean expressions are similar.
When evaluating an arithmetic or boolean expression, we only rely on the information from the given classical state, therefore we omit the quantum state in the configuration. This is not the case when we execute commands.
\begin{figure}
\[\begin{array}{l}
\slinfer[\Rlts{Evaluation of local variables}]{\pair{x,\sigma}\hookrightarrow \pair{\sigma(x), \sigma}}\vspace{2mm}\\
\prooftree
\pair{a_0,\sigma}\hookrightarrow \pair{a_0',\sigma}
\justifies
\pair{a_0+a_1,\sigma}\hookrightarrow \pair{a_0'+a_1,\sigma}
\endprooftree\qquad
\prooftree
\pair{a_1,\sigma}\hookrightarrow \pair{a_1',\sigma}
\justifies
\pair{n+a_1,\sigma}\hookrightarrow \pair{n+a'_1,\sigma}
\endprooftree
\vspace{2mm}\\
\pair{n+m,\sigma}\hookrightarrow\pair{p,\sigma} \qquad \mbox{if $p$ is the sum of $n$ and $m$}\vspace{2mm}\\
\prooftree
\pair{a_0,\sigma}\hookrightarrow \pair{a'_0,\sigma}
\justifies
\pair{a_0\leq a_1,\sigma}\hookrightarrow \pair{a'_0\leq a_1,\sigma}
\endprooftree
\qquad
\prooftree
\pair{a_1,\sigma}\hookrightarrow \pair{a'_1,\sigma}
\justifies
\pair{n\leq a_1,\sigma}\hookrightarrow \pair{n\leq a'_1,\sigma}
\endprooftree
\vspace{2mm}\\
\pair{n\leq m,\sigma}\hookrightarrow\pair{{\textbf{true}},\sigma} \qquad \mbox{if $n$ is less than or equal to $m$.}
\vspace{2mm}\\
\pair{n\leq m,\sigma}\hookrightarrow\pair{{\textbf{false}},\sigma} \qquad \mbox{if $n$ is greater than $m$.}
\end{array}\]
\caption{Evaluation of arithmetic and boolean expressions (selected rules)}\label{fig:evalab}
\end{figure}
Let $\sigma$ be a classical state and $n\in \mathbb{Z}$. We write $\sigma[n/x]$ for the updated state satisfying
\[
\sigma[n/x](y)=\left\{\begin{array}{ll}
n & \mbox{if $y=x$,}\\
\sigma(y) & \mbox{if $y\not=x$.}
\end{array}\right.
\]
We are going to write $\rightarrow$ for the execution of commands.
The transition rules are given in Figure~\ref{fig:exec}. Here we introduce a special command {\bf nil} that stands for a successful termination of programs.
We follow \cite{Yin12} to define the operational semantics of quantum measurements in a non-deterministic way, and the probabilities of different branches
are encoded in the quantum part of the configurations. For that reason we need to take partial density
operators instead of the normalised density operators to represent quantum states.
After the measurement $M$ defined by some measurement operators $M_i$, the original state $(\sigma,\rho)$ may evolve into a new state whose classical part is the updated state $\sigma[i/x]$ and the quantum part is the new quantum state $M_i\rho M_i^\dag$. In all other rules, the execution of a command changes a configuration to another one. Among them, the rules for initialising qubits and unitary transformations only affect the quantum part of the original machine state. On the contrary, the commands for manipulating classical data only update the classical part of a state.
\begin{figure}
\[\begin{array}{l}
\pair{{\textbf{skip}},\sigma,\rho}\rightarrow \pair{\textbf{nil},\sigma,\rho}\vspace{2mm}\\
\prooftree
\pair{a,\sigma}\hookrightarrow \pair{a',\sigma'}
\justifies
\pair{x:=a,\sigma,\rho}\rightarrow \pair{x:=a',\sigma',\rho}
\endprooftree
\qquad
\pair{x:=n,\sigma,\rho}\rightarrow \pair{\textbf{nil},\sigma[n/x],\rho} \vspace{2mm}\\
\prooftree\pair{c_0,\sigma,\rho}\rightarrow \pair{c'_0,\sigma',\rho'}
\justifies
\pair{c_0;c_1,\sigma,\rho}\rightarrow \pair{c'_0;c_1,\sigma',\rho'}
\endprooftree
\qquad
\prooftree
\pair{c_1,\sigma,\rho}\rightarrow \pair{c'_1,\sigma',\rho'}
\justifies
\pair{\textbf{nil};c_1,\sigma,\rho}\rightarrow \pair{c'_1,\sigma',\rho'}
\endprooftree
\vspace{2mm}\\
\prooftree
\pair{b,\sigma}\hookrightarrow \pair{b',\sigma'}
\justifies
\pair{\textbf{if}~ b ~\textbf{then}~ c_0 ~\textbf{else}~ c_1,\sigma,\rho}\rightarrow \pair{\textbf{if}~ b' ~\textbf{then}~ c_0 ~\textbf{else}~ c_1,\sigma',\rho'}
\endprooftree
\vspace{2mm}\\
\pair{\textbf{if}~ {\textbf{true}} ~\textbf{then}~ c_0 ~\textbf{else}~ c_1,\sigma,\rho}\rightarrow \pair{ c_0,\sigma,\rho}
\qquad
\pair{\textbf{if}~ {\textbf{false}} ~\textbf{then}~ c_0 ~\textbf{else}~ c_1,\sigma,\rho}\rightarrow \pair{ c_1,\sigma,\rho}
\vspace{2mm}\\
\pair{\textbf{while}~ b ~\textbf{do}~ c,\sigma,\rho} \rightarrow \pair{\textbf{if}~ b ~\textbf{then}~ (c;\textbf{while}~ b~\textbf{do}~ c)~\textbf{else}~ {\textbf{skip}},\sigma,\rho}
\vspace{2mm}\\
\pair{q:=|0\>,\sigma,\rho}\rightarrow \pair{\textbf{nil},\sigma,\rho'}
\qquad\mbox{ with } \rho'= |0\>_q\<0|\rho |0\>_q\<0| + |0\>_q\<1|\rho |1\>_q\<0|
\vspace{2mm}\\
\pair{{ U}[\bar{q}],\sigma,\rho}\rightarrow \pair{\textbf{nil},\sigma,{ U}\rho{ U}^\dag}
\vspace{2mm}\\
\prooftree
M:=\{M_i\}_{i\in I}
\justifies
\pair{x:={ M}[\bar{q}],\sigma,\rho}\rightarrow \pair{\textbf{nil}, \sigma[i/x],M_i \rho M_i^\dag}
\endprooftree
\end{array}\]
\caption{Execution of commands}\label{fig:exec}
\end{figure}
\subsection{Denotational Semantics}
For the purpose of presenting the denotational semantics, we add an {\textbf{abort}} command that halts the computation with no result.
We interpret programs as POVD transformers.
We write
${\textbf{POVD}}$ for the set of POVDs called distribution states.
\begin{lemma}
We impose an order between POVDs by letting $\mu_1\leq\mu_2$ if for any classical state $\sigma$ we have $\mu_1(\sigma)\sqsubseteq\mu_2(\sigma)$, where $\sqsubseteq$ is the L\"{o}wner order.
Let $(\mu_n)_{n\in{\mathbb N}}\in{\textbf{POVD}}$ be an increasing sequence of POVDs. This sequence converges to some POVD $\mu_\infty$ and $\mu_n\leq \mu_\infty$ for any $n\in {\mathbb N}$.
\end{lemma}
Given an expression $e$, we denote its interpretation with respect to machine state $(\sigma,\rho)$ by $\Denote{e}_{(\sigma,\rho)}$.
The denotational semantics of commands is displayed in Figure~\ref{fig:de}, where we omit the denotational semantics of arithmetic and boolean expressions such as $\Denote{a}_\sigma$ and $\Denote{b}_\sigma$, which is almost the same as in the classical setting because the quantum part plays no role for those expressions. This is an extension of the semantics for probabilistic programs presented in \cite{BEGGHS18}. Instead of probabilistic assignments are measurements of quantum systems. A state evolves into a POVD after some quantum qubits are measured, with the measurement outcomes assigned to a classical variable. Two other quantum commands, initialisation of qubits and unitary operations, are deterministic and only affect the quantum part of a state. As usual, we define the semantics of a loop ($\textbf{while}~ b ~\textbf{do}~ c$) as the limit of its lower approximations, where the $n$-th lower approximation of $\Denote{\textbf{while}~ b ~\textbf{do}~ c}_{(\sigma,\rho)} $ is $\Denote{(\textbf{if}~ b ~\textbf{then}~ c)^n; \textbf{if}~ b ~\textbf{then}~ \textbf{abort}}_{(\sigma,\rho)} $, where ($\textbf{if}~ b ~\textbf{then}~ c$) is shorthand for ($\textbf{if}~ b ~\textbf{then}~ c ~\textbf{else}~ {\textbf{skip}}$) and $c^n$ is the command $c$ iterated $n$ times with $c^0\equiv{\textbf{skip}}$. The limit exists because the sequence $(\Denote{(\textbf{if}~ b ~\textbf{then}~ c)^n; \textbf{if}~ b ~\textbf{then}~ \textbf{abort}}_{(\sigma,\rho)} )_{n\in{\mathbb N}}$ is increasing and bounded. We write $\varepsilon$ for the special POVD whose support is the empty set.
\begin{proposition}
The semantics $\Denote{c}_{(\sigma,\rho)} $ of a command $c$ in initial state ${(\sigma,\rho)} $ is a POVD. The lifted semantics $\Denote{c}_\mu$ of a command $c$ in initial POVD $\mu$ is a POVD.
\end{proposition}
\begin{figure}
\[\begin{array}{rcl}
\Denote{{\textbf{skip}}}_{(\sigma,\rho)} & = & (\sigma,\rho) \vspace{2mm}\\
\Denote{\textbf{abort}}_{(\sigma,\rho)} & = & \varepsilon \vspace{2mm}\\
\Denote{x := a}_{(\sigma,\rho)} & = & (\sigma[\Denote{a}_\sigma/x], \rho) \vspace{2mm}\\
\Denote{c_0;c_1}_{(\sigma,\rho)} & = & \Denote{c_1}_{\Denote{c_0}_{(\sigma,\rho)} }\vspace{2mm}\\
\Denote{\textbf{if}~ b ~\textbf{then}~ c_0 ~\textbf{else}~ c_1}_{(\sigma,\rho)} & = & \left\{\begin{array}{ll}
\Denote{c_0}_{(\sigma,\rho)} & \mbox{if $\Denote{b}_\sigma={\textbf{true}}$}\\
\Denote{c_1}_{(\sigma,\rho)} & \mbox{if $\Denote{b}_\sigma={\textbf{false}}$}
\end{array}\right.\vspace{2mm}\\
\Denote{\textbf{while}~ b ~\textbf{do}~ c}_{(\sigma,\rho)} & = & \lim_{n\rightarrow\infty}
\Denote{(\textbf{if}~ b ~\textbf{then}~ c)^n; \textbf{if}~ b ~\textbf{then}~ \textbf{abort}}_{(\sigma,\rho)} \vspace{2mm}\\
\Denote{q := |0\>}_{(\sigma,\rho)} & = & \pair{\sigma,\rho'}\\
& & \mbox{where $\rho' := |0\>_q\<0|\rho |0\>_q\<0| + |0\>_q\<1|\rho |1\>_q\<0| $} \vspace{2mm}\\
\Denote{{ U}[\bar{q}] }_{(\sigma,\rho)} & = & \pair{\sigma, { U}\rho{ U}^\dag} \vspace{2mm}\\
\Denote{x := { M}[\bar{q}]}_{(\sigma,\rho)} & = & \mu \vspace{2mm}\\
& & \mbox{where $M=\sset{M_i}_{i\in I}$ and $\mu(\sigma')=\sum_i\{M_i \rho M_i^\dag \mid \sigma[i/x]=\sigma'\}$}
\vspace{4mm}\\
\Denote{c}_{\mu} & = & \sum_{\sigma\in\support{\mu}} \Denote{c}_{(\sigma,\mu(\sigma))}.
\end{array}\]
\caption{Denotational semantics of commands}\label{fig:de}
\end{figure}
The operational and denotational semantics are related by the following theorem.
\begin{theorem}
For any command $c$ and state $(\sigma,\rho)$, we have
\[\Denote{c}_{(\sigma,\rho)} = \sum_{i}\{(\sigma_i,\rho_i) \mid \pair{c,\sigma,\rho}\rightarrow ^*\pair{\textbf{nil},\sigma_i,\rho_i}\}\ .\]
\end{theorem}
\begin{proof}
We proceed by induction on the structure of $c$. The most difficult case is when $c\equiv \textbf{while}~ b ~\textbf{do}~ c'$ for some command $c'$. Below we consider this case.
Let $\textbf{While}^n = (\textbf{if}~ b ~\textbf{then}~ c')^n; \textbf{if}~ b ~\textbf{then}~ \textbf{abort}$ and
$\pair{c,\sigma,\rho} \rightarrow^{n}\pair{\textbf{nil},\sigma',\rho'}$ be the sequence of maximal transitions from $\pair{c,\sigma,\rho}$ such that the unfolding rule
\[\pair{\textbf{while}~ b ~\textbf{do}~ c',\sigma'',\rho''} \rightarrow \pair{\textbf{if}~ b ~\textbf{then}~ (c';\textbf{while}~ b~\textbf{do}~ c')~\textbf{else}~ {\textbf{skip}},\sigma'',\rho''}\ ,\] for any $\sigma''$ and $\rho''$, has been applied at most $n$ times.
\[\mbox{\bf Claim: }\hspace{2cm} \Denote{\textbf{While}^n}_{(\sigma,\rho)} = \sum_i\{(\sigma_i,\rho_i)\mid
\pair{c,\sigma,\rho} \rightarrow^{n+1}\pair{\textbf{nil},\sigma_i,\rho_i}\}\ .\]
We prove the above claim by induction on $n$.
\begin{itemize}
\item $n=0$. On the left hand side, we have
$$\Denote{\textbf{While}^0}_{(\sigma,\rho)} = \Denote{\textbf{if}~ b ~\textbf{then}~ \textbf{abort}}_{(\sigma,\rho)}
= \left\{\begin{array}{ll}
\varepsilon\ & \mbox{if $\Denote{b}_\sigma={\textbf{true}}$}\\
(\sigma,\rho) & \mbox{if $\Denote{b}_\sigma={\textbf{false}}$}
\end{array}\right. . $$
On the right hand side, we observe that
\begin{equation}\label{eq:while}
\begin{array}{rcl}
\pair{\textbf{while}~ b ~\textbf{do}~ c',\sigma,\rho} & \rightarrow & \pair{\textbf{if}~ b ~\textbf{then}~ (c';\textbf{while}~ b~\textbf{do}~ c')~\textbf{else}~ {\textbf{skip}},\sigma,\rho} \\
& \rightarrow^* & \left\{\begin{array}{ll}
\pair{(c';\textbf{while}~ b~\textbf{do}~ c'),\sigma,\rho}\ & \mbox{if $\Denote{b}_\sigma={\textbf{true}}$}\\
\pair{{\textbf{skip}}, \sigma,\rho} & \mbox{if $\Denote{b}_\sigma={\textbf{false}}$}
\end{array}\right.
\end{array}\end{equation}
The unfolding rule has been used in the first reduction step in (\ref{eq:while}).
If $\Denote{b}_\sigma={\textbf{false}}$ then the claim clearly holds. If $\Denote{b}_\sigma={\textbf{true}}$ then the configuration $\pair{(c';\textbf{while}~ b~\textbf{do}~ c'),\sigma,\rho}$ cannot reduce to any $\pair{\textbf{nil}, \sigma'',\rho''}$ without using the unfolding rule again, which means that there is no maximal transition from $\pair{c,\sigma,\rho}$ that uses the unfolding rule at most once. It follows that the claim also holds in this case.
\item Suppose $n=k+1$ and the claim holds for some $k$. On the left hand side, we have
\begin{equation}\label{eq:m0}
\begin{array}{rcl}
\Denote{\textbf{While}^{k+1}}_{(\sigma,\rho)} & = & \Denote{\textbf{if}~ b ~\textbf{then}~ c'; \textbf{While}^k}_{(\sigma,\rho)}\\
& = & \left\{\begin{array}{ll}
\Denote{\textbf{While}^k}_{\Denote{c'}_{(\sigma,\rho)}} & \mbox{if $\Denote{b}_\sigma={\textbf{true}}$}\\
(\sigma,\rho) & \mbox{if $\Denote{b}_\sigma={\textbf{false}}$}
\end{array}\right.
\end{array}
\end{equation}
On the right hand side, we have the same transitions as in (\ref{eq:while}). If $\Denote{b}_\sigma={\textbf{false}}$ then the claim clearly holds. If $\Denote{b}_\sigma={\textbf{true}}$ then we infer as follows.
Since $c'$ is a subterm of $c$, we know from the hypothesis of the structural induction that \begin{equation}\label{eq:0}
\Denote{c'}_{(\sigma,\rho)}
= \sum_{j\in J}\{(\sigma_j,\rho_j) \mid \pair{c',\sigma,\rho}\rightarrow ^*\pair{\textbf{nil},\sigma_j,\rho_j}\}
\end{equation}
for some set $J$.
It follows that
\begin{equation}\label{eq:aa}\Denote{\textbf{While}^k}_{\Denote{c'}_{(\sigma,\rho)}} = \sum_{j\in J}\Denote{\textbf{While}^k}_{(\sigma_j,\rho_j)} \ .
\end{equation} By induction hypothesis on $k$,
\begin{equation}\label{eq:bb}
\Denote{\textbf{While}^k}_{(\sigma_j,\rho_j)} = \sum_{i\in I_j}\{(\sigma_i,\rho_i)\mid
\pair{c,\sigma_j,\rho_j} \rightarrow^{k+1}\pair{\textbf{nil},\sigma_i,\rho_i}\}
\end{equation}
for some index set $I_j$.
As a result, when $\Denote{b}_\sigma={\textbf{true}}$, we have
\begin{equation}\label{eq:c}
\begin{array}{rll}
\pair{c,\sigma,\rho} & \rightarrow & \pair{\textbf{if}~ b ~\textbf{then}~ (c';c)~\textbf{else}~ {\textbf{skip}},\sigma,\rho} \\
& \rightarrow^* & \pair{(c';c),\sigma,\rho} \\
& \rightarrow^* & \pair{c, \sigma_j,\rho_j} \qquad\mbox{by (\ref{eq:0})} \\
& \rightarrow^{k+1} & \pair{\textbf{nil},\sigma_i,\rho_i} \qquad\mbox{by (\ref{eq:bb})}
\end{array}
\end{equation}
for each $j\in J$ and $i\in I_j$. This means that
\begin{equation}\label{eq:d}
\pair{c,\sigma,\rho} \rightarrow^{k+2} \pair{\textbf{nil},\sigma_i,\rho_i}
\end{equation}
for each $j\in J$ and $i\in I_j$. Thus, we rewrite (\ref{eq:bb}) as follows.
\begin{equation}\label{eq:e}
\Denote{\textbf{While}^k}_{(\sigma_j,\rho_j)} = \sum_{i\in I_j}\{(\sigma_i,\rho_i)\mid
\pair{c,\sigma,\rho} \rightarrow^{k+2}\pair{\textbf{nil},\sigma_i,\rho_i}\}
\end{equation}
Combining (\ref{eq:m0}), (\ref{eq:aa}) and (\ref{eq:e}), we obtain the desired result that
\[\Denote{\textbf{While}^{k+1}}_{(\sigma,\rho)} = \sum_{j\in J}\sum_{i\in I_j} \{(\sigma_i,\rho_i)\mid
\pair{c,\sigma,\rho} \rightarrow^{k+2}\pair{\textbf{nil},\sigma_i,\rho_i}\}\]
\end{itemize}
So far we have proved the claim. Then by taking the limit on both sides of the claim, we see that $\Denote{c}_{(\sigma,\rho)} = \sum_{i}\{(\sigma_i,\rho_i) \mid \pair{c,\sigma,\rho}\rightarrow ^*\pair{\textbf{nil},\sigma_i,\rho_i}\}$.
\end{proof}
\section{An Abstract Proof System}\label{sec:abs}
In this section, we present an abstract proof system, where assertions are arbitrary predicates on POVDs. We show that the proof system is sound and relatively complete.
\begin{definition}
The set ${\textbf{Assn}}$ of assertions is defined as $\cal{P}({\textbf{POVD}})$, the powerset of ${\textbf{POVD}}$. Each assertion $P$ can be constructed by the following grammar.
\[P ~:=~ {\bf 1}_{\mu} \mid S \mid \neg P \mid P_1\wedge P_2 \mid \Box\psi \mid P_1\oplus P_2 \mid P[f]\]
where $\mu\in{\textbf{POVD}}$, $S\subseteq {\textbf{POVD}}$, $\psi$ is a predicate over states and $f$ is a function from ${\textbf{POVD}}$ to ${\textbf{POVD}}$.
\end{definition}
Here ${\bf 1}_\mu$ is also called the characteristic function of the
POVD $\mu$, which is a predicate requiring that
${\bf 1}_\mu$ holds on $\mu'$ if and only if $\mu'=\mu$, for any distribution state $\mu'$.
The satisfaction relation $\models$ between a POVD and an assertion is defined as follows.
\[\begin{array}{rcl}
\mu \models {\bf 1}_{\mu'} & \mbox{iff} & \mu=\mu' \\
\mu \models S & \mbox{iff} & \mu\in S \\
\mu \models \neg P & \mbox{iff} &
\mbox{not } \mu\models P \\
\mu \models P_1\wedge P_2 & \mbox{iff} &
\mu \models P_1 \wedge \mu\models P_2\\
\mu \models \Box\psi & \mbox{iff} & \forall \sigma.\ \sigma\in\support{\mu} \Rightarrow \Denote{\psi}_\sigma ={\textbf{true}}\\
\mu \models P_1\oplus P_2 & \mbox{iff} & \exists \mu_1, \mu_2.\ \mu=\mu_1 + \mu_2 \wedge \mu_1\models P_1 \wedge \mu_2\models P_2\\
\mu \models P[f] & \mbox{iff} & f(\mu)\models P
\end{array}\]
Let $\Denote{P}:=\{\mu \mid \mu\models P\}$ be the semantic interpretation of assertion $P$. We see that boolean operations of assertions are represented by set operations. For example, we have $\Denote{\neg P}=\mathcal{P}({\textbf{POVD}})\backslash \Denote{P}$ and $\Denote{P_1\wedge P_2}=\Denote{P_1}\cap\Denote{P_2}$.
The predicate $\Box\psi$ is lifted from a state predicate by requiring that $\Box\psi$ holds on the POVD $\mu$ when $\psi$ holds on all the states in the support of $\mu$. For example, a particular predicate over states is a boolean expression $b$ with $\sigma\models b$ iff $\Denote{b}_\sigma={\textbf{true}}$. Therefore, the predicate $\Box b$ holds on the POVD $\mu$ when $b$ evaluates to be true under any state $\sigma$ in the support of $\mu$.
The assertion $P_1\oplus P_2$ holds on the POVD $\mu$ if we can split $\mu$ into the sum of two POVDs such that $P_1$ and $P_2$ hold on each of them. Lastly, $P[f]$ holds on a POVD $\mu$ only when $P$ holds on the image of $\mu$ under $f$.
\leaveout{
We define boolean operations of assertions by set operations. For example, $P\wedge P' := P\cap P'$ and $\neg P := {\textbf{POVD}}\backslash P$.
Given a predicate $\psi$ over states, we lift it to be a predicate $\Box\psi$ over subdistributions by letting
\[ \Box\psi (\Delta) ~:=~ \forall \sigma. \sigma\in\support{\Delta} \Rightarrow \psi(\sigma) .\]
Intuitively, $\Box\psi$ holds on the subdistribution $\Delta$ when $\psi$ holds on all the states in the support of $\Delta$. For example, a particular predicate over states is a boolean expression $b$ with $m\models b$ iff $\Denote{b}_m={\textbf{true}}$. Therefore, the predicate $\Box b$ holds on the subdistribution $\Delta$ when $b$ evaluates to be true under any state $m$ in the support of $\Delta$.
To reason about branching commands, we combine two assertions $P_1$ and $P_2$ into a new assertion $P_1\oplus P_2$ by letting
\[(P_1\oplus P_2)(\Delta) ~:=~ \exists \Delta_1, \Delta_2. \Delta=\Delta_1 + \Delta_2 \wedge P_1(\Delta_1) \wedge P_2(\Delta_2) .\]
In other words, the assertion $P_1\oplus P_2$ holds on the subdistribution $\Delta$ if we can split $\Delta$ into the sum of two subdistributions such that $P_1$ and $P_2$ holds on each of them.
Given an assertion $P$ and a function $f$ from ${\textbf{POVD}}$ to ${\textbf{POVD}}$, we define $$P[f] := \lambda \Delta. P(f(\Delta)) .$$ Therefore, $P[f]$ holds on a subdistribution $\Delta$ only when $P$ holds on the image of $\Delta$ under $f$.
}
\begin{definition}
A sequence of assertions $(P_n)_{n\in{\mathbb N}^\infty}$ is \emph{u-closed}, if for each increasing sequence of POVDs $(\mu_n)_{n\in{\mathbb N}}$ such that $\mu_n\models P_n$ for all $n\in{\mathbb N}$, we have $\lim_{n\rightarrow\infty}\mu_n\models P_\infty$.
\end{definition}
\begin{definition}
A judgement is a triple in the form $\triple{P}{c}{P'}$, where $c$ is a command, $P$ and $P'$ are assertions. It is valid, written $\models\triple{P}{c}{P'}$, if
\[\forall \mu.\ \mu\models P \ \Rightarrow \ \Denote{c}_\mu \models P' .\]
\end{definition}
In Figure~\ref{fig:proofrules} we give the rules for an abstract proof system denoted by
$\mathcal{S}_a$. It extends the system in \cite{BEGGHS18} with the last three rules to handle the manipulations of quantum systems. In order to show the soundness of $\mathcal{S}_a$, we need a few technical lemmas.
\begin{figure}
\[\begin{array}{c}
\linfer{}{\triple{P}{{\textbf{skip}}}{P}}{[\sf Skip]}
\qquad
\linfer{}{\triple{P}{\textbf{abort}}{\Box{\textbf{false}}}}{[\sf Abort] }
\qquad
\linfer{}{\triple{P[\Denote{x:=a}]}{x : = a }{P}}{[\sf Assgn]}
\vspace{2mm}\\
\prooftree
\triple{P_0}{c_0}{P_1} \quad
\triple{P_1}{c_1}{P_2}
\justifies
\triple{P_0}{c_0;c_1}{P_2}
\using
[\sf Seq]
\endprooftree
\qquad
\prooftree
\triple{P_0}{c}{P'_0}
\quad
\triple{P_1}{c}{P'_1}
\justifies
\triple{P_0\oplus P_1}{c}{P'_0\oplus P'_1}
\using
[\sf Split]
\endprooftree
\vspace{2mm}\\
\prooftree
\triple{P_0\wedge\Box b}{c_0}{P'_0}
\quad
\triple{P_1\wedge\Box \neg b}{c_1}{P'_1}
\justifies
\triple{(P_0\wedge\Box b)\oplus (P_1 \wedge \Box\neg b)}
{\textbf{if}~ b ~\textbf{then}~ c_0 ~\textbf{else}~ c_1}{P'_0\oplus P'_1}
\using
[\sf Cond]
\endprooftree
\vspace{2mm}\\
\prooftree
\justifies
\triple{{\textbf{false}}}{c}{P}
\using
[\sf Absurd]
\endprooftree
\qquad
\prooftree
P_0 \Rightarrow P_1 \quad
\triple{P_1}{c}{P_2} \quad
P_2 \Rightarrow P_3
\justifies
\triple{P_0}{c}{P_3}
\using
[\sf Conseq]
\endprooftree
\qquad
\prooftree
\forall \mu.\ \triple{{\bf 1}_\mu\wedge P}{c}{P'}
\justifies
\triple{P}{c}{P'}
\using
[\sf All]
\endprooftree
\vspace{2mm}\\
\prooftree
\begin{array}{c}
\sf uclosed((P'_n)_{n\in{\mathbb N}^\infty})\\
\forall n.\
\triple{P_n}{\textbf{if}~ b ~\textbf{then}~ c}{P_{n+1}}
\quad
\forall n.\ \triple{P_n}{\textbf{if}~ b ~\textbf{then}~ \textbf{abort}}{P'_n}
\end{array}
\justifies
\triple{P_0}{\textbf{while}~ b ~\textbf{do}~ c}{P'_\infty \wedge \Box \neg b}
\using
[\sf While]
\endprooftree
\vspace{2mm}\\
\linfer{}{\triple{P[\Denote{q := |0\>}]}{q := |0\> }{P}}{[\sf QInit]}
\qquad
\linfer{}{\triple{P[\Denote{{ U}[\bar{q}]}]}{{ U}[\bar{q}]}{P}}{[\sf QUnit]}
\vspace{2mm}\\
\linfer{}{\triple{P[\Denote{x := { M}[\bar{q}]}]}{x := { M}[\bar{q}]}{P}}{[\sf QMeas]}
\end{array}\]
\caption{Proof rules for $\mathcal{S}_a$}\label{fig:proofrules}
\end{figure}
\begin{lemma}\label{lem:assgn1}
Let $P$ be an assertion and $c$ a command. Then
$\models \triple{P[\Denote{c}]}{c}{P}$.
\end{lemma}
\begin{proof}
Suppose $\mu$ is a distribution state and $\mu\models P[\Denote{c}]$. By the definition of $P[\Denote{c}]$, this means that $\Denote{c}_\mu \models P$, which is the desired result.
\end{proof}
\begin{lemma}\label{lem:split}
Let $\sigma$ be a classical state, $\rho_1, \rho_2$ be two quantum states, and $\mu_1, \mu_2$ be two POVDs. For any command $c$, we have
\begin{enumerate}
\item $\Denote{c}_{(\sigma,\rho_1+\rho_2)} = \Denote{c}_{(\sigma,\rho_1)} + \Denote{c}_{(\sigma,\rho_2)}$;
\item $\Denote{c}_{(\mu_1+\mu_2)} = \Denote{c}_{\mu_1} + \Denote{c}_{\mu_2}$.
\end{enumerate}
\end{lemma}
\begin{proof}
The two clauses can be proved by a simultaneous induction on the structure of command $c$.
\end{proof}
\begin{lemma}\label{lem:seq1}
For any commands $c_0, c_1$ and distribution state $\mu$, we have $\Denote{c_0;c_1}_\mu = \Denote{c_1}_{\Denote{c_0}_\mu}$.
\end{lemma}
\begin{proof}
\[\begin{array}{rcl}
\Denote{c_1}_{\Denote{c_0}_\mu}
& = & \sum_\sigma \Denote{c_1}_{(\sigma,\Denote{c_0}_\mu(\sigma))}\\
& = & \sum_\sigma \Denote{c_1}_{(\sigma,\sum_{\sigma'}\Denote{c_0}_{(\sigma',\mu(\sigma'))}(\sigma))}\\
& = & \sum_\sigma\sum_{\sigma'}\Denote{c_1}_{(\sigma,\Denote{c_0}_{(\sigma',\mu(\sigma'))}(\sigma))}\qquad\mbox{by Lemma~\ref{lem:split}(1)}\\
& = & \sum_{\sigma'}\sum_{\sigma}\Denote{c_1}_{(\sigma,\Denote{c_0}_{(\sigma',\mu(\sigma'))}(\sigma))}\\
& = & \sum_{\sigma'}\Denote{c_1}_{\Denote{c_0}_{(\sigma',\mu(\sigma'))}}\\
& = & \sum_{\sigma'}\Denote{c_0;c_1}_{(\sigma',\mu(\sigma'))} \\
& = & \Denote{c_0;c_1}_{\mu}
\end{array}\]
\end{proof}
\begin{theorem}\label{thm:sounda}
{\bf (Soundness)}
Every judgement provable using the proof system $\mathcal{S}_a$ is valid.
\end{theorem}
\begin{proof}
We analyze the cases one by one.
\begin{itemize}
\item Rule {[\sf Skip]}. Suppose $\mu\models P$ for some distribution state $\mu$. Then we have $\Denote{{\textbf{skip}}}_\mu = \mu$ and thus
$\Denote{{\textbf{skip}}}_\mu\models P$ as required.
\item Rule {[\sf Abort]}. This case is easy by noting that $\Denote{\textbf{abort}}_\mu=\varepsilon$ and $\varepsilon\models\Box{\textbf{false}}$ for any $\mu$.
\item The cases for rules {[\sf Assgn], [\sf QInit], [\sf QUnit]}, and {[\sf QMeas]} follow from Lemma~\ref{lem:assgn1}.
\item Rule {[\sf Seq]}. Suppose $\mu\models P_0$ for some distribution state $\mu$. By the premises, both $\triple{P_0}{c_0}{P_1}$ and $ \triple{P_1}{c_1}{P_2}$ are valid. It follows that $\Denote{c_0}_\mu \models P_1$ and then $\Denote{c_1}_{\Denote{c_0}_\mu}\models P_2$, which is $\Denote{c_0;c_1}_\mu\models P_2$ by Lemma~\ref{lem:seq1} as required.
\item Rule {[\sf Split]}. Suppose $\mu \models P_0\oplus P_1$ for some distribution state $\mu$. Then there exist $\mu_0$ and $\mu_1$ such that $\mu=\mu_0+\mu_1$, $\mu_0\models P_0$ and $\mu_1\models P_1$. By the premises, both $\triple{P_0}{c_0}{P'_0}$ and $ \triple{P_1}{c_1}{P'_1}$ are valid. Therefore, we have that $\Denote{c}_{\mu_0}\models P'_0$ and $\Denote{c}_{\mu_1}\models P'_1$. By Lemma~\ref{lem:split} we obtain $\Denote{c}_\mu = \Denote{c}_{\mu_0} + \Denote{c}_{\mu_1}$. It follows that $\Denote{c}_\mu\models P'_0\oplus P'_1$ as required.
\item Rule {[\sf Cond]}. We first claim that $\triple{P_0\wedge\Box b}{\textbf{if}~ b ~\textbf{then}~ c_0 ~\textbf{else}~ c_1}{P'_0}$ is valid. To see this, suppose $\mu$ is a POVD with $\mu\models P_0\wedge\Box b$. Obviously, we have $\mu\models\Box b$ and thus $\Denote{b}_\sigma={\textbf{true}}$ for each $\sigma\in\support{\mu}$. It follows that
\[\begin{array}{rcl}
\Denote{\textbf{if}~ b ~\textbf{then}~ c_0 ~\textbf{else}~ c_1}_\mu & = & \sum_\sigma \Denote{\textbf{if}~ b ~\textbf{then}~ c_0 ~\textbf{else}~ c_1}_{(\sigma,\mu(\sigma))} \\
& = & \sum_\sigma \Denote{c_0}_{(\sigma,\mu(\sigma))} \\
& = & \Denote{c_0}_\mu .
\end{array}\]
By the first premise, $\triple{P_0\wedge\Box b}{c_0}{P'_0}$ is valid. Therefore,
we have $\Denote{c_0}_\mu\models P'_0$, and thus
$$\Denote{\textbf{if}~ b ~\textbf{then}~ c_0 ~\textbf{else}~ c_1}_\mu \models P'_0 $$
and the above claim is proved. Similarly, we can prove that
$\triple{P_1\wedge\Box \neg b}{\textbf{if}~ b ~\textbf{then}~ c_0 ~\textbf{else}~ c_1}{P'_1}$ is valid.
By the soundness of {[\sf Split]}, it follows that $$\triple{(P_0\wedge\Box b)\oplus (P_1 \wedge \Box\neg b)}
{\textbf{if}~ b ~\textbf{then}~ c_0 ~\textbf{else}~ c_1}{P'_0\oplus P'_1}$$ is also valid.
\item Rule {[\sf Absurd]}. There exists no $\mu$ with $\mu\models {\textbf{false}}$. Thus, we always have $\forall \mu.\ \mu\models {\textbf{false}} \Rightarrow \Denote{c}_\mu\models P$.
\item Rule {[\sf Conseq]}. Let $\mu$ be a distribution state and $\mu\models P_0$. The first premise gives $\mu\models P_1$. The second premise tells us that $\Denote{c}_\mu\models P_2$. By the third premise, we derive that $\Denote{c}_\mu\models P_3$. It follows that $\triple{P_0}{c}{P_3}$ is valid.
\item Rule {[\sf All]}. Let $\mu$ be a POVD and $\mu\models P$. It is clear that $\mu\models {\bf 1}_\mu \wedge P$. By the premise, $\triple{{\bf 1}_\mu\wedge P}{c}{P'}$ is valid. Therefore, we have $\Denote{c}_\mu \models P'$, and thus $\triple{P}{c}{P'}$ is valid.
\item Rule {[\sf While]}. We first observe that, for any state $(\sigma,\rho)$,
\[ \Denote{\textbf{if}~ b ~\textbf{then}~ \textbf{abort}}_{(\sigma,\rho)} = \left\{\begin{array}{ll}
\varepsilon & \mbox{if $\Denote{b}_{(\sigma,\rho)}={\textbf{true}}$}\\
{(\sigma,\rho)} & \mbox{if $\Denote{b}_{(\sigma,\rho)}={\textbf{false}}$} \ .
\end{array}
\right. \]
Thus, if a state $\sigma'$ is in the support of $\Denote{\textbf{if}~ b ~\textbf{then}~ \textbf{abort}}_{(\sigma,\rho)} $, it must be the case that $\sigma'\models \neg b$. Furthermore, for any distribution state $\mu$, if a state $\sigma'$ is in the support of $\Denote{\textbf{if}~ b ~\textbf{then}~ \textbf{abort}}_\mu$ then $\sigma'\models \neg b$. It follows that, for any command $c'$ and distribution state $\mu$, we have
\[\Denote{c'; \textbf{if}~ b ~\textbf{then}~ \textbf{abort}}_\mu\models \Box \neg b. \]
By definition, $\Denote{\textbf{while}~ b ~\textbf{do}~ c}_\mu$ is the limit of the sequence
\[(\Denote{(\textbf{if}~ b ~\textbf{then}~ c)^n; \textbf{if}~ b ~\textbf{then}~ \textbf{abort}}_\mu)_{n\in{\mathbb N}}\]
and so we have that
\begin{equation}\label{e:1}
\Denote{\textbf{while}~ b ~\textbf{do}~ c}_\mu\models \Box \neg b.
\end{equation}
By the first premise and the soundness of {[\sf Seq]}, it is easy to show by induction that
\[\forall n.\
\triple{P_0}{(\textbf{if}~ b ~\textbf{then}~ c)^n}{P_{n}}\]
is valid. By the second premise and {[\sf Seg]} again, the following judgement
\[\forall n.\
\triple{P_0}{(\textbf{if}~ b ~\textbf{then}~ c)^n; \textbf{if}~ b ~\textbf{then}~ \textbf{abort}}{P'_{n}}\]
is valid. Let $\mu$ be any POVD with $\mu\models P_0$. Then
\[\forall n.\
\Denote{(\textbf{if}~ b ~\textbf{then}~ c)^n; \textbf{if}~ b ~\textbf{then}~ \textbf{abort}}_\mu\models P'_{n} .\]
By assumption, the sequence of assertions $(P'_n)_{n\in{\mathbb N}^\infty}$ is u-closed. Hence, we can infer that $$\Denote{\textbf{while}~ b ~\textbf{do}~ c}_\mu \models P'_\infty, $$
which means that the judgement
\begin{equation}\label{e:2}
\triple{P_0}{\textbf{while}~ b ~\textbf{do}~ c}{P'_\infty}
\end{equation} is valid.
Combining (\ref{e:1}) and (\ref{e:2}), we finally obtain that
$ \triple{P_0}{\textbf{while}~ b ~\textbf{do}~ c}{P'_\infty \wedge \Box \neg b}$ is valid.
\end{itemize}
\end{proof}
Now we turn to the relative completeness of the proof system $\mathcal{S}_a$. Formulas of the form ${\bf 1}_\mu$ will be helpful for that purpose.
\begin{lemma}\label{lem:seq}
For any distribution state $\mu$ and command $c$,
\[{\bf 1}_\mu \Rightarrow {\bf 1}_{\Denote{c}_\mu}[\Denote{c}] .\]
\end{lemma}
\begin{proof}
Let $\mu'$ be any distribution state.
\[\begin{array}{rcl}
\mu'\models {\bf 1}_\mu & \Leftrightarrow & \mu'=\mu \\
& \Rightarrow & \Denote{c}_{\mu'} = \Denote{c}_\mu \\
& \Leftrightarrow & \Denote{c}_{\mu'} \models {\bf 1}_{ \Denote{c}_{\mu} } \\
& \Leftrightarrow & \mu' \models {\bf 1}_{ \Denote{c}_{\mu}}[\Denote{c}]
\end{array}\]
\end{proof}
\begin{definition}
Let $\mu$ be a distribution state and $b$ a boolean expression. The restriction $\mu_{|b}$ of $\mu$ to $b$ is the distribution state such that $\mu_{|b}(\sigma)=\mu(\sigma)$ if $\Denote{b}_\sigma={\textbf{true}}$ and $0$ otherwise.
\end{definition}
According to the definition above, it is easy to see that we can split any $\mu$ into two parts w.r.t. a boolean expression.
\begin{lemma}\label{lem:decomp}
For any distribution state $\mu$ and boolean expression b, we have
$\mu = \mu_{|b} + \mu_{|\neg b}$.
\end{lemma}
\begin{proof}
This is straightforward because, at each state $\sigma$ in the support of $\mu$, the boolean expression $b$ evaluates to either ${\textbf{true}}$ or ${\textbf{false}}$.
\end{proof}
With Lemmas~\ref{lem:decomp} and \ref{lem:split}, it is easy to see that the denotational semantics of conditional commands can be rewritten as follows.
\begin{equation}\label{eq:cond}
\Denote{\textbf{if}~ b ~\textbf{then}~ c_0 ~\textbf{else}~ c_1}_\mu ~=~ \Denote{c_0}_{\mu_{|b}} + \Denote{c_1}_{\mu_{|\neg b}}
\end{equation}
The following facts are also easy to show.
\begin{equation}\label{eq:Char}
\begin{array}{rcl}
{\bf 1}_{\mu | b} & \Leftrightarrow &{\bf 1}_{\mu | b} \wedge \Box b \\
{\bf 1}_{\mu} & \Leftrightarrow &{\bf 1}_{\mu} \wedge P \qquad\mbox{if $\mu\models P$}\\
{\bf 1}_{\mu} & \Leftrightarrow & {\bf 1}_{\mu|b} \oplus {\bf 1}_{\mu|\neg b}\\
{\bf 1}_{\mu_1+\mu_2} & \Leftrightarrow & {\bf 1}_{\mu_1} \oplus {\bf 1}_{\mu_2}
\end{array}
\end{equation}
\begin{lemma}\label{lem:CharP}
For any POVD $\mu$, the following judgement is provable:
\[\triple{{\bf 1}_{\mu}}{c}{{\bf 1}_{\Denote{c}_\mu}} . \]
\end{lemma}
\begin{proof}
We proceed by induction on the structure of $c$.
\begin{itemize}
\item $c\equiv {\textbf{skip}}$. This case is immediate as $\Denote{{\textbf{skip}}}_\mu=\mu$ and by {[\sf Skip]} we have $\vdash \triple{{\bf 1}_{\mu}}{c}{{\bf 1}_{\mu}} . $
\item $c\equiv\textbf{abort}$. Then $\Denote{c}_\mu=\varepsilon$. For any POVD $\mu'$, we note that \[\mu'\models\Box {\textbf{false}} ~\Leftrightarrow~ \mu'=\varepsilon ~\Leftrightarrow~ \mu'\models{\bf 1}_{\varepsilon} . \]
By rules {[\sf Abort]} and {[\sf Conseq]} we can infer
$\vdash \triple{{\bf 1}_{\mu}}{c}{{\bf 1}_{\varepsilon}} $.
\item $c\equiv x:=a,\ q := |0\>,\ { U}[\bar{q}]$ or $x := { M}[\bar{q}]$. By the corresponding rules
{[\sf Assgn]}, {[\sf QInit]}, {[\sf QUnit]} or {[\sf QMeas]}, we have
$$\vdash \triple{{\bf 1}_{\Denote{c}_\mu} [\Denote{c}]}{c}{{\bf 1}_{\Denote{c}_\mu}} . $$
By Lemma~\ref{lem:seq} and rule {[\sf Conseq]}, we obtain that
$\vdash \triple{{\bf 1}_{\mu}} {c}{{\bf 1}_{\Denote{c}_\mu}} . $
\item $c\equiv c_0;c_1$. By induction, we have $\vdash\triple{{\bf 1}_{\mu}}{c_0}{{\bf 1}_{\Denote{c_0}_\mu}}$ and
$\vdash\triple{{\bf 1}_{\Denote{c_0}_\mu}}{c_1}{{\bf 1}_{\Denote{c_1}_{\Denote{c_0}_\mu}}}$. Using the rule {[\sf Seq]}, we obtain that
$\vdash\triple{{\bf 1}_{\mu}}{c}{{\bf 1}_{\Denote{c_1}_{\Denote{c}_\mu}}}$.
\item $c\equiv \textbf{if}~ b ~\textbf{then}~ c_0 ~\textbf{else}~ c_1$. By induction, we have
$\vdash \triple{{\bf 1}_{\mu_{|b}}}{c_0}{{\bf 1}_{\Denote{c_0}_{\mu_{|b}}}}$. By the first clause in (\ref{eq:Char}) and rule {[\sf Conseq]}, we have
$\vdash \triple{{\bf 1}_{\mu_{|b}}\wedge \Box b}{c_0}{{\bf 1}_{\Denote{c_0}_{\mu_{|b}}}}$. Similarly,
$\vdash \triple{{\bf 1}_{\mu_{|\neg b}}\wedge \Box \neg b}{c_1}{{\bf 1}_{\Denote{c_1}_{\mu_{|\neg b}}}}$. Using rule {[\sf Cond]}, we infer
$$\vdash \triple{({\bf 1}_{\mu_{|b}} \wedge \Box b) \oplus ({\bf 1}_{\mu_{|\neg b}}\wedge \Box \neg b)}{\textbf{if}~ b ~\textbf{then}~ c_0 ~\textbf{else}~ c_1}{{\bf 1}_{\Denote{c_0}_{\mu_{|b}}} \oplus {\bf 1}_{\Denote{c_1}_{\mu_{|\neg b}}}}. $$
Using (\ref{eq:cond}), (\ref{eq:Char}) and rule {[\sf Conseq]}, we finally obtain that
$$\vdash \triple{{\bf 1}_{\mu}}{\textbf{if}~ b ~\textbf{then}~ c_0 ~\textbf{else}~ c_1}{{\bf 1}_{\Denote{\textbf{if}~ b ~\textbf{then}~ c_0 ~\textbf{else}~ c_1}_\mu}}.$$
\item $c\equiv \textbf{while}~ b ~\textbf{do}~ c'$. For each $n\in{\mathbb N}$, let
\[\begin{array}{rcl}
P_n & = & {\bf 1}_{\Denote{(\textbf{if}~ b ~\textbf{then}~ c')^n}_\mu} \\
P'_n & = & {\bf 1}_{\Denote{(\textbf{if}~ b ~\textbf{then}~ c')^n; \textbf{if}~ b ~\textbf{then}~ \textbf{abort}}_\mu} \\
P'_\infty & = & {\bf 1}_{\lim_{n\rightarrow\infty}\Denote{(\textbf{if}~ b ~\textbf{then}~ c')^n; \textbf{if}~ b ~\textbf{then}~ \textbf{abort}}_\mu}
\end{array}\]
Obviously, the sequence of assertions $(P'_n)_{n\in{\mathbb N}^\infty}$ is u-closed.
As in the last case, we can show that $\vdash \triple{P_n}{\textbf{if}~ b ~\textbf{then}~ c'}{P_{n+1}}$ by induction hypothesis and rules {[\sf Conseq]}, {[\sf Skip]} and {[\sf Cond]}. It is also easy to see that
$\vdash\triple{P_n}{\textbf{if}~ b ~\textbf{then}~ \textbf{abort}}{P'_n}$ for each $n\in {\mathbb N}$. Therefore, we can use rule {[\sf While]} to infer that
$\vdash \triple{P_0}{\textbf{while}~ b ~\textbf{do}~ c}{P'_\infty \wedge \Box \neg b}$. Using (\ref{e:1}), the second clause of (\ref{eq:Char}), and rule {[\sf Conseq]}, we obtain that $\vdash \triple{P_0}{\textbf{while}~ b ~\textbf{do}~ c}{P'_\infty}$, which is exactly $\vdash\triple{{\bf 1}_\mu}{\textbf{while}~ b ~\textbf{do}~ c'}{{\bf 1}_{\Denote{\textbf{while}~ b ~\textbf{do}~ c'}_\mu}}$.
\end{itemize}
\end{proof}
With the preparations above, we are in the position to show that the proof system ${\cal{S}}_a$ is relatively complete.
\begin{theorem}
{\bf (Relative completeness)}
Every valid judgement is derivable in ${\cal{S}}_a$.
\end{theorem}
\begin{proof}
Let $\triple{P}{c}{P'}$ be a valid judgement. Suppose $\mu$ be any POVD. There are two possibilities:
\begin{itemize}
\item $\mu\models P$. The validity of the judgement says that $\Denote{c}_\mu\models P'$.
By Lemma~\ref{lem:CharP}, we have that
$\vdash \triple{{\bf 1}_\mu}{c}{{\bf 1}_{\Denote{c}_\mu}}$.
By the second clause of (\ref{eq:Char}) and rule {[\sf Conseq]}, we can obtain that
$\vdash \triple{{\bf 1}_\mu\wedge P}{c}{{\bf 1}_{\Denote{c}_\mu} \wedge P'}$.
Using {[\sf Conseq]} again gives $\vdash \triple{{\bf 1}_\mu\wedge P}{c}{ P'}$.
\item $\mu\not\models P$. Then it is obvious that ${\bf 1}_\mu\wedge P \Leftrightarrow {\textbf{false}}$. By rules {[\sf Absurd]} and {[\sf Conseq]}, we also obtain
$\vdash \triple{{\bf 1}_\mu\wedge P}{c}{ P'}$.
\end{itemize}
Since $\mu$ is arbitrarily chosen, the premise of rule {[\sf All]} is derivable. Therefore, we can use that rule to obtain $\vdash\triple{P}{c}{P'}$.
\end{proof}
\section{A Concrete Program Logic}\label{sec:conc}
In this section, we present a concrete program logic. We first define the concrete syntax of assertions. Following \cite{BEGGHS18}, we define a two-level assertion language in Figure~\ref{fig:syn}. Formally, assertions are divided into two categories: \emph{state assertions} are formulas that describe the properties of machine states and \emph{distribution assertions} are used to describe the properties of POVDs. Distribution assertions are based on comparison of distribution expressions, or built with first-order quantifiers and connectives, as well as the connective $\oplus$ mentioned in Section~\ref{sec:abs}.
A \emph{distribution expression} is either the expectation ${\mathbb E}[e]$ of a state expression $e$, the expectation ${\mathbb E}_{\bar{x}\sim M[\bar{q}]}[e]$ of state expression $e$ w.r.t. the measurement $M$, or an operator applied to distribution expressions. A \emph{state expression} is either a classical variable, the characteristic function ${\bf 1}_\psi$ of a state assertion $\psi$, or an operator applied to state expressions.
Finally, a \emph{state assertion} is either a comparison of state expressions, or a first-order formula over state assertions. In particular, the boolean expressions in \textbf{Bexp} are included as state assertions.
Note that the set of operators is left unspecified but we assume that some basic operators such as addition, subtraction, scalar multiplication for both arithmetic expressions and matrix representation of partial density operators are included. With a slight abuse of notation, when $\leq$ is used to compare matrices, we essentially mean $\sqsubseteq$. Similarly for $<$ and $=$.
For convenience of presentation,
in this section we consider a general form of quantum measurement.
\begin{definition}
A \emph{general measurement} $M$ is a pair $\pair{\{M_i\}_{i\in I}, {l}}$, where each $M_i$ is a measurement operator as usual, and ${l}: I\mapsto J$ is a labelling function that maps each measurement outcome $i$ to some some label ${l}(i)$.
If the state of a quantum system is specified by density operator $\rho$ immediately before the measurement $M$, then the probability with which those results with label $j$ occur is given by
\[p(j)=\sum_{i:{l}(i)=j} {\rm tr}(M^\dag_iM_i\rho),\]
and the state of the system after the measurement is
\[\\\frac{\sum_{i:{l}(i)=j} M_i\rho M^\dag_i}{p(j)} .\]
\end{definition}
General measurements are convenient to describe the situation where we would like to group some measurement outcomes. For example, if $i_1, i_2 \in I$ are two different outcomes, but for some reasons we would not like to distinguish them, then we simply give them the same label by letting ${l}(l_1)={l}(l_2)$. In the special case that $l$ is the identity function ${\it Id}$, then the labelling function has no effect and we degenerate to the usual notion of measurements.
\begin{figure}
\[\begin{array}{rcll}
e & ::= & x
\mid {\bf 1}_{\psi} \mid o(\mathbf{e})
\qquad
& \mbox{(State expressions)} \\
\psi & ::= & e \rhd\!\!\!\lhd e \mid FO(\psi) & \mbox{(State assertions)} \\
r & ::= & {\mathbb E}[e] \mid {\mathbb E}_{\bar{x}\sim M[\bar{q}]}[e] \mid o(\mathbf{r}) & \mbox{(Distribution expressions)} \\
P & ::= & r \rhd\!\!\!\lhd r \mid P \oplus P \mid FO(P) & \mbox{(Distribution assertions)} \\
\rhd\!\!\lhd & \in & \{=,\ <,\ \leq\} \qquad o\in Ops & \mbox{(Operations)}
\end{array}\]
\caption{Syntax of assertions
}\label{fig:syn}
\end{figure}
\begin{figure}
\[\begin{array}{rcl}
\Denote{x}_\sigma & := & \sigma(x) \\
\Denote{{\bf 1}_{\psi}}_\sigma & := & {\bf 1}_{\Denote{\psi}_{\sigma}} \\
\Denote{o(e)}_\sigma & := & o(\Denote{e}_\sigma)\\
\hline \\
\Denote{e_1 \rhd\!\!\!\lhd e_2}_\sigma & := & \Denote{e_1}_\sigma \rhd\!\!\!\lhd \Denote{e_2}_\sigma \\
\Denote{FO(\psi)}_\sigma & := & FO(\Denote{\psi}_\sigma) \\
\hline\\
\Denote{{\mathbb E}[e]}_\mu & := & \sum_{\sigma}\mu(\sigma)\cdot \Denote{e}_\sigma\\
\Denote{{\mathbb E}_{\bar{x}\sim M[\bar{q}]}[e]}_\mu & := & \sum_\sigma \sum_{i} M_i \mu(\sigma) M_i^\dag \cdot \Denote{e}_{\sigma[l(i)/\bar{x}]} \\
& & \mbox{where $M=\pair{\sset{M_i}_{i\in I},{l}}$}\\
\Denote{o(r)}_\mu & := & o(\Denote{r}_\mu)\\
\hline \\
\Denote{r_1 \rhd\!\!\!\lhd r_2}_\mu & := & \Denote{r_1}_\mu \rhd\!\!\!\lhd \Denote{r_2}_\mu \\
\Denote{P_1\oplus P_2}_\mu & := & \exists \mu_1, \mu_2.\ \mu = \mu_1 + \mu_2 \wedge \Denote{P_1}_{\mu_1}
\wedge\Denote{P_2}_{\mu_2} \\
\Denote{FO(P)}_\mu & := & FO(\Denote{P}_\mu)
\end{array}\]
\caption{Semantics of assertions}\label{fig:sem}
\end{figure}
The interpretation of assertions is given in Figure~\ref{fig:sem}. Comparing the interpretation with that in \cite{BEGGHS18}, we see that the main difference is the introduction of a distribution expression related to a quantum measurement. The meaning of
${\mathbb E}_{\bar{x}\sim M[\bar{q}]}[e]$ is the expected Hermitian operator weighted by the value of $e$ after a measurement entailed by $M$.
Note that the formula $\Box\psi$, where $\psi$ is a state assertion, can now be viewed as a syntactic sugar in view of the following lemma.
\begin{lemma}\label{lem:box}
\begin{enumerate}
\item $\Box\psi \ \Leftrightarrow\ {\mathbb E}[{\bf 1}_{\psi}] = {\mathbb E}[{\bf 1}_{{\textbf{true}}}]$
\item $\Box\psi \ \Leftrightarrow\ {\mathbb E}_{\bar{x}\sim M[\bar{q}]}[{\bf 1}_{\psi}] = {\mathbb E}_{\bar{x}\sim M[\bar{q}]}[{\bf 1}_{{\textbf{true}}}]$
\item $\Box\psi\ \Leftrightarrow\ \Box(\psi\wedge b) \oplus \Box(\psi\wedge\neg b) $
\end{enumerate}
\end{lemma}
\begin{proof}
Let us consider the first clause; the second one is similar and the third one is easier.
\[\begin{array}{lcl}
\mu\models\Box\psi &
\mbox{iff} & \forall \sigma\in\support{\mu}. \Denote{\psi}_\sigma={\textbf{true}}\\
& \mbox{iff} & \sum_{\sigma}\mu(\sigma)\cdot\Denote{{\bf 1}_{\psi}}_\sigma = \sum_{\sigma}\mu(\sigma)\cdot\Denote{{\bf 1}_{{\textbf{true}}}}_\sigma\\
& \mbox{iff} & \Denote{{\mathbb E}[{\bf 1}_{\psi}]}_\mu = \Denote{{\mathbb E}[{\bf 1}_{\textbf{true}}]}_\mu \\
& \mbox{iff} & \mu\models ({\mathbb E}[{\bf 1}_{\psi}] = {\mathbb E}[{\bf 1}_{{\textbf{true}}}])
\end{array}\]
\end{proof}
Using the concrete syntax for assertions, we propose a syntactic version of the existing proof rules by avoiding the semantics of commands. We call the concrete proof system $\mathcal{S}_c$. Specifically, we keep all proof rules in Figure~\ref{fig:proofrules} but replace [{\sf Assgn}], [{\sf QInit}], [{\sf QUnit}], and [{\sf QMeas}] with the four rules in Figure~\ref{fig:syntacticproofrules}.
\begin{figure}
\[\begin{array}{c}
\linfer{}{\triple{P[a/x]}{x : = a }{P}}{[\sf Assgn']}
\qquad
\prooftree
\justifies
\triple{h(P)}{q:=|0\>}{P}
\using
[\sf QInit']
\endprooftree
\vspace{2mm}\\
\prooftree
\justifies
\triple{g^U(P)}{U[\bar{q}]}{P}
\using
[\sf QUnit']
\endprooftree
\qquad
\prooftree
\justifies
\triple{f_{x,\bar{q}}^M(P)}{x :=M[\bar{q}]}{P}
\using
[\sf QMeas']
\endprooftree
\end{array}\]
\caption{Selected syntactic proof rules}\label{fig:syntacticproofrules}
\end{figure}
In rule [{\sf QInit'}] we use the notation $h(P)$ for a syntactic substitution. It changes all
${\mathbb E}_{\bar{x}\sim M[\bar{q}]}[e]$ in $P$ into ${\mathbb E}_{\bar{x}\sim M'[\bar{q}]}[e]$ and distributes over most other syntactic constructors of assertions, where
$M'$ is obtained from $M=\pair{\{M_i\}_i, l}$ by constructing two measurement operators $M_{i0}, M_{i1}$ for each $M_i$ in $M$ with the mapping ${l}'$ given by ${l}'(i0)={l}'(i1)=l(i)$. A formal definition is given below.
\[\begin{array}{rcl}
h(o(\mathbf{r})) & := & o(h(\mathbf{r})) \qquad\mbox{where $o\in Ops$} \\
h({\mathbb E}[e]) & := & {\mathbb E}_{x\sim M[q]}[e] \qquad\mbox{where } M=\pair{\{M_{0}, M_{1}\}, Id} \mbox{ with }
M_{0}=|0\>\<0|,\ M_{1}=|0\>\<1|, \\
& & \hfill \ x \mbox{ is fresh} \\
h({\mathbb E}_{\bar{x}\sim M[\bar{q}]}[e]) & := & {\mathbb E}_{\bar{x}\sim M'[\bar{q}]}[e] \quad \mbox{where } M'=\pair{\{M_{i0}, M_{i1}\}_i, {l}'}\\
& & \hfill \mbox{ with }
M_{i0}=M_i|0\>\<0|,\ M_{i1}=M_i|0\>\<1|,
{l}'(i0)={l}'(i1)=l(i) \\
h(r_1 \rhd\!\!\!\lhd r_2) & := & h(r_1) \rhd\!\!\!\lhd h(r_2)\\
h(FO(P)) & := & FO(h(P)) \\
h(P_1\oplus P_2) & := & h(P_1) \oplus h(P_2)
\end{array}\]
To ensure the freshness requirement on $x$ in $h({\mathbb E}[e])$, we assume an enumeration of all the variables in ${\textbf{Cvar}}$. Each time a fresh variable is needed, we take the next one which has not appeared in all the programs under consideration.
In rule [{\sf QUnit'}] we use the notation $g^U(P)$ for a syntactic substitution. It changes all
${\mathbb E}_{\bar{x}\sim M[\bar{q}]}[e]$ in $P$ into ${\mathbb E}_{\bar{x}\sim M'[\bar{q}]}[e]$, where $M=\pair{\{M_i \}_{i\in I},{l}}$, $M'=\pair{\{M_i U\}_{i\in I},{l}}$, and distributes over most other syntactic constructors of assertions. A formal definition is given below.
\[\begin{array}{rcl}
g^U(o(r)) & := & o(g^U(r)) \qquad\mbox{where $o\in Ops$} \\
g^U({\mathbb E}[e]) & := & {\mathbb E}_{x\sim M[\bar{q}]}[e] \qquad \mbox{where } M=\pair{\{M_0\}, Id} \mbox{ with } M_0=U \mbox{ and $x$ is fresh}\\
g^U({\mathbb E}_{\bar{x}\sim M[\bar{q}]}[e]) & := & {\mathbb E}_{\bar{x}\sim M'[\bar{q}]}[e] \qquad \mbox{where } M'=\pair{\{M_iU\}_{i\in I},{l}} \\
g^U(r_1 \rhd\!\!\!\lhd r_2) & := & g^U(r_1) \rhd\!\!\!\lhd g^U(r_2)\\
g^U(FO(P)) & := & FO(g^U(P)) \\
g^U(P_1\oplus P_2) & := & g^U(P_1) \oplus g^U(P_2)
\end{array}\]
In rule [{\sf QMeas'}] we use the notation $f_{x,\bar{q}}^M(P)$ for a syntactic substitution. It changes all ${\mathbb E}[e]$ in $P$ into ${\mathbb E}_{x\sim M[\bar{q}]}[e]$. For the distribution expression ${\mathbb E}_{\bar{y}\sim N[\bar{q}]}[e]$, it adds an outer layer of measurement to $N$. A formal definition is given below.
\[\begin{array}{rcl}
f_{x,\bar{q}}^M(o(r)) & := & o(f_{x,\bar{q}}^M(r)) \qquad\mbox{where $o\in Ops$} \\
f_{x,\bar{q}}^M({\mathbb E}[e]) & := & {\mathbb E}_{x\sim M[\bar{q}]}[e]\\
f_{x,\bar{q}}^M({\mathbb E}_{\bar{y}\sim N[\bar{q'}]}[e]) & := & \left\{\begin{array}{ll}
{\mathbb E}_{x\bar{y}\sim M'[\bar{q}\cup\bar{q'}]}[e] \mbox{ with } M'=\pair{\{N_j M_i\}_{ij}, k'} \mbox{ and } k'(ij)=(k(i),l(j)) & \mbox{if } x\not\in\bar{y}\\
{\mathbb E}_{\bar{y}\sim M'[\bar{q}\cup\bar{q'}]}[e] \mbox{ with } M'=\pair{\{N_j M_i\}_{ij}, k'} \mbox{ and } k'(ij)=l(j) & \mbox{if } x\in\bar{y}
\end{array}\right.\\
f_{x,\bar{q}}^M(r_1 \rhd\!\!\!\lhd r_2) & := & f_{x,\bar{q}}^M(r_1) \rhd\!\!\!\lhd f_{x,\bar{q}}^M(r_2)\\
f_{x,\bar{q}}^M(FO(P)) & := & FO(f_{x,\bar{q}}^M(P)) \\
f_{x,\bar{q}}^M(P_1\oplus P_2) & := & f_{x,\bar{q}}^M(P_1) \oplus f_{x,\bar{q}}^M(P_2)
\end{array}\]
We are going to show that the three functions $h(\cdot)$, $g^U(\cdot)$ and $f^M_{x,\bar{q}}(\cdot)$ behave well as they help to transform postconditions into preconditions for three kinds of commands: initialisation, applications of unitary operations, and measurements of quantum systems.
\begin{lemma}\label{lem:qinit}
The following two clauses hold.
\begin{enumerate}
\item[(i)] $\Denote{h(r)}_\mu = \Denote{r}_{\Denote{q:=|0\>}_\mu}$.
\item[(ii)] $\Denote{h(P)}_\mu \Rightarrow \Denote{P}_{\Denote{q:=|0\>}_\mu}$.
\end{enumerate}
\end{lemma}
\begin{proof}
We prove the two statements by structural induction.
\begin{enumerate}
\item[(i)] There are three cases for the structure of $r$.
\begin{itemize}
\item $r\equiv{\mathbb E}[e]$. We note that $\Denote{e}_\sigma = \Denote{e}_{\sigma[n/x]}$ for any number $n$ and fresh variable $x$ in the sense that $x$ does not appear in $e$. Then we reason as follows.
\[\begin{array}{rcl}
\Denote{h(r)}_\mu & = & \Denote{{\mathbb E}_{x\sim M}[e]}_\mu \qquad\mbox{where } M=\pair{\{M_{0}, M_{1}\}, Id} \mbox{ with }
M_{0}=|0\>\<0|,\ M_{1}=|0\>\<1|, \\
& & \hfill \ x \mbox{ is fresh} \\
& = & \sum_\sigma (|0\>\<0|\mu(\sigma)|0\>\<0| \cdot \Denote{e}_{\sigma[0/x]} + |0\>\<1|\mu(\sigma) |1\>\<0|\cdot \Denote{e}_{\sigma[1/x]} ) \\
& = & \sum_\sigma (|0\>\<0|\mu(\sigma)|0\>\<0| \cdot \Denote{e}_{\sigma} + |0\>\<1|\mu(\sigma) |1\>\<0|\cdot \Denote{e}_{\sigma} ) \qquad x \mbox{ is fresh}\\
& = & \sum_\sigma (|0\>\<0|\mu(\sigma)|0\>\<0| + |0\>\<1|\mu(\sigma) |1\>\<0|)\cdot \Denote{e}_{\sigma} \\
& = & \sum_\sigma \Denote{q:=|0\>}_\mu(\sigma)\cdot \Denote{e}_{\sigma} \\
& = & \Denote{{\mathbb E}[e]}_{\Denote{q:=|0\>}_\mu} \\
& = & \Denote{r}_{\Denote{q:=|0\>}_\mu}
\end{array}\]
\item $r\equiv {\mathbb E}_{\bar{x}\sim M[\bar{q}]}[e]$ for some $M=\pair{\{M_i\}_i, {l}}$. Then
\[\begin{array}{rcl}
\Denote{h(r)}_\mu & = &\Denote{{\mathbb E}_{\bar{x}\sim M'[\bar{q}]}[e]}_\mu \quad \mbox{where } M'=\pair{\{M_{i0}, M_{i1}\}_i, {l}'} \\
& & \hspace{2.5cm} \mbox{ with }
M_{i0}=M_i|0\>\<0|,\ M_{i1}=M_i|0\>\<1|,
{l}'(i0)={l}'(i1)=l(i) \\
& = & \sum_\sigma \sum_i (M_{i}|0\>\<0|\mu(\sigma)|0\>\<0|M^\dag_{i} + M_{i} |0\>\<1|\mu(\sigma) |1\>\<0| M^\dag_{i})\cdot \Denote{e}_{\sigma[l(i)/\bar{x}]} \\
& = & \sum_\sigma \sum_i (M_{i}(|0\>\<0|\mu(\sigma)|0\>\<0| + |0\>\<1|\mu(\sigma) |1\>\<0|) M^\dag_{i}\cdot \Denote{e}_{\sigma[l(i)/\bar{x}]} \\
& = & \sum_\sigma \sum_i M_{i} \Denote{q:=|0\>}_\mu(\sigma) M^\dag_{i}\cdot \Denote{e}_{\sigma[l(i)/\bar{x}]} \\
& = & \Denote{{\mathbb E}_{\bar{x}\sim M[\bar{q}]}[e]}_{\Denote{q:=|0\>}_\mu} \\
& = & \Denote{r}_{\Denote{q:=|0\>}_\mu}
\end{array}\]
\item $r\equiv o(r_1,...,r_k)$. The case can be proved by induction.
\[\begin{array}{rcl}
\Denote{h(r)}_\mu & = & \Denote{o(h(r_1),...,h(r_k)}_\mu \\
& = & o(\Denote{h(r_1)}_\mu,...,\Denote{h(r_k)}_\mu) \\
& = & o(\Denote{r_1}_{\Denote{q:=|0\>}_\mu},..., \Denote{r_k}_{\Denote{q:=|0\>}_\mu}) \\
& = & \Denote{o(r_1,...,r_k)}_{\Denote{q:=|0\>}_\mu} \\
& = & \Denote{r}_{\Denote{q:=|0\>}_\mu}
\end{array}\]
\end{itemize}
\item[(ii)] There are three cases for the structure of $P$.
\begin{itemize}
\item $P\equiv r_1\rhd\!\!\!\lhd r_2$. In this case, we need to use statement (iii).
\[\begin{array}{rcl}
\Denote{h(P)}_\mu & = & \Denote{h(r_1) \rhd\!\!\!\lhd h(r_2)}_\mu \\
& = & \Denote{h(r_1)}_\mu \rhd\!\!\!\lhd \Denote{h(r_2)}_\mu \\
& = & \Denote{r_1}_{\Denote{q:=|0\>}_\mu} \rhd\!\!\!\lhd \Denote{r_2}_{\Denote{q:=|0\>}_\mu}) \\
& = & \Denote{r_1 \rhd\!\!\!\lhd r_2}_{\Denote{q:=|0\>}_\mu} \\
& = & \Denote{r}_{\Denote{q:=|0\>}_\mu}
\end{array}\]
\item $P\equiv P_1\oplus P_2$. This case is proved by induction.
\[\begin{array}{rcl}
\Denote{h(P)}_\mu & = & \Denote{h(P_1) \oplus h(P_2)}_\mu \\
& = & \exists \mu_1, \mu_2.\ \mu=\mu_1 + \mu_2 \wedge
\Denote{h(P_1)}_{\mu_1} \wedge \Denote{h(P_2)}_{\mu_2} \\
& \Rightarrow & \exists \mu_1, \mu_2.\ \Denote{q:=|0\>}_\mu=\Denote{q:=|0\>}_{\mu_1} + \Denote{q:=|0\>}_{\mu_2} \\& & \wedge
\Denote{P_1}_{\Denote{q:=|0\>}_{\mu_1}} \wedge \Denote{P_2}_{\Denote{q:=|0\>}_{\mu_2}} \qquad\mbox{by Lemma~\ref{lem:split}}\\
& = & \Denote{P_1\oplus P_2}_{\Denote{q:=|0\>}_\mu} \\
& = & \Denote{r}_{\Denote{q:=|0\>}_\mu}
\end{array}\]
\item $P\equiv FO(P_1,...,P_k)$. Again, this case is proved by induction.
\[\begin{array}{rcl}
\Denote{h(P)}_\mu & = & \Denote{FO(h(P_1),..., h(P_k)}_\mu \\
& = & FO(\Denote{h(P_1)}_\mu,..., \Denote{h(P_k)}_\mu) \\
& = & FO(\Denote{P_1}_{\Denote{q:=|0\>}_\mu},..., \Denote{P_k}_{\Denote{q:=|0\>}_\mu} )\\
& = & \Denote{FO(P_1,...,P_k)}_{\Denote{q:=|0\>}_\mu} \\
& = & \Denote{P}_{\Denote{q:=|0\>}_\mu}
\end{array}\]
\end{itemize}
\end{enumerate}
\end{proof}
\begin{lemma}\label{lem:qunit}
The following two clauses hold.
\begin{enumerate}
\item[(i)] $\Denote{g^U(r)}_\mu = \Denote{r}_{\Denote{U[\bar{q}]}_\mu}$.
\item[(ii)] $\Denote{g^U(P)}_\mu \Rightarrow \Denote{P}_{\Denote{U[\bar{q}]}_\mu}$.
\end{enumerate}
\end{lemma}
\begin{proof}
The proof is similar to that of Lemma~\ref{lem:qinit} except for the treatment of two cases for statement (i).
\begin{itemize}
\item $r\equiv {\mathbb E}[e]$. We infer that
\[\begin{array}{rcl}
\Denote{g^U(r)}_\mu & = & \Denote{{\mathbb E}_{x\sim M[\bar{q}]}[e]}_\mu \qquad \mbox{where } M=\pair{\{M_0\}, Id} \mbox{ with } M_0=U \mbox{ and $x$ is fresh}\\
& = & \sum_\sigma U\mu(\sigma)U^\dag \cdot\Denote{e}_{\sigma[0/x]}\\
& = & \sum_\sigma U\mu(\sigma)U^\dag \cdot\Denote{e}_{\sigma}\\
& = & \sum_\sigma \Denote{U[\bar{q}]}_\mu(\sigma) \cdot\Denote{e}_{\sigma}\\
& = & \Denote{{\mathbb E}[e]}_{\Denote{U[\bar{q}]}_\mu}\\
& = & \Denote{r}_{\Denote{U[\bar{q}]}_\mu}
\end{array}\]
\item $r\equiv{\mathbb E}_{\bar{x}\sim M[\bar{q}]}[e]$. Suppose $M=\pair{\sset{M_i}_{i\in I},{l}}$.
We reason as follows.
\[\begin{array}{rcl}
\Denote{g^U(r)}_\mu & = & \Denote{{\mathbb E}_{\bar{x}\sim M'[\bar{q}]}[e]}_\mu \quad\mbox{where } M'=\pair{\{M_i U\}_{i\in I},{l}} \\
& = & \sum_\sigma\sum_{i}M_iU\mu(\sigma)U^\dag M^\dag_i\cdot\Denote{e}_{\sigma[l(i)/\bar{x}]} \\
& = & \sum_\sigma\sum_{i}M_i\Denote{U[\bar{q}]}_\mu(\sigma) M^\dag_i\cdot\Denote{e}_{\sigma[l(i)/\bar{x}]} \\
& = & \Denote{{\mathbb E}_{\bar{x}\sim M[\bar{q}]}[e]}_{\Denote{U[\bar{q}]}_\mu}\\
& = & \Denote{r}_{\Denote{U[\bar{q}]}_\mu}
\end{array}\]
\end{itemize}
\end{proof}
\begin{lemma}\label{lem:assgn}
Let $a$ be an arithmetic expression, $\sigma$ be any state and $\sigma':=\sigma[\Denote{a}_\sigma/x]$.
The following four clauses hold, where $x$ is not a bound variable in $e$, $\psi$, $r$ and $P$.
\begin{enumerate}
\item[(i)] $\Denote{e[a/x]}_\sigma =\Denote{e}_{\sigma'}$
\item[(ii)] $\Denote{\psi[a/x]}_\sigma = \Denote{\psi}_{\sigma'}$
\item[(iii)] $\Denote{r[a/x]}_\mu = \Denote{r}_{\Denote{x:=a}_\mu}$.
\item[(iv)] $\Denote{P[a/x]}_\mu \Rightarrow \Denote{P}_{\Denote{x:=a}_\mu}$.
\end{enumerate}
\end{lemma}
\begin{proof}
The proof is similar to that of Lemma~\ref{lem:qinit}. As an example, we only consider one case for statement (iii).
Suppose $r\equiv{\mathbb E}_{\bar{y}\sim M[\bar{q}]}[e]$ with $M=\pair{\{M_i\}_i,l}$. There are two possibilities.
\begin{itemize}
\item $x\in \bar{y}$. In this case, $x$ is a bound variable in $r$, which contradicts our assumption.
\item $x\not\in \bar{y}$. Notice that
\begin{equation}\label{eq:a}
\Denote{x:=a}_\mu(\sigma')\ =\ \sum_\sigma\{\mu(\sigma)\mid \sigma[\Denote{a}_\sigma/x]=\sigma'\}
\end{equation}
holds for any $\mu$ and $\sigma'$.
We reason as follows.
\[\begin{array}{rcl}
\Denote{r[a/x]}_\mu & = & \Denote{{\mathbb E}_{\bar{y}\sim M[\bar{q}]}[e[a/x]]}_\mu \\
& = & \sum_\sigma\sum_iM_i\mu(\sigma)M^\dag_i\Denote{e[a/x]}_{\sigma[l(i)/\bar{y}]}\\
& = & \sum_\sigma\sum_iM_i\mu(\sigma)M^\dag_i\Denote{e}_{\sigma[l(i)/\bar{y}][\Denote{a}_{\sigma[l(i)/\bar{y}]}/x]}\qquad\mbox{by statement (i)}\\
& = & \sum_iM_i \sum_\sigma\mu(\sigma)M^\dag_i\Denote{e}_{\sigma[\Denote{a}_\sigma/x][l(i)/\bar{y}]}\\
& = & \sum_iM_i \sum_{\sigma'} \sum_\sigma\{\mu(\sigma)\mid \sigma[\Denote{a}_\sigma/x]=\sigma'\}M^\dag_i\Denote{e}_{\sigma'[l(i)/\bar{y}]}\\
& = & \sum_iM_i \sum_{\sigma'} \Denote{x:=a}_\mu(\sigma')M^\dag_i\Denote{e}_{\sigma'[l(i)/\bar{y}]}\qquad\mbox{by }(\ref{eq:a})\\
& = & \sum_{\sigma'} \sum_iM_i \Denote{x:=a}_\mu(\sigma')M^\dag_i\Denote{e}_{\sigma'[l(i)/\bar{y}]}\\
& = & \Denote{{\mathbb E}_{\bar{y}\sim M[\bar{q}]}[e]}_{\Denote{x:=a}_\mu}\\
& = & \Denote{r}_{\Denote{x:=a}_\mu}
\end{array}\]
\end{itemize}
\end{proof}
\begin{lemma}\label{lem:meas}
\begin{enumerate}
\item[(i)] $\Denote{f_{x,\bar{q}}^M(r)}_\mu = \Denote{r}_{\Denote{x:=M[\bar{q}]}_\mu}$.
\item[(ii)] $\Denote{f_{x,\bar{q}}^M(P)}_\mu \Rightarrow \Denote{P}_{\Denote{x:=M[\bar{q}]}_\mu}$.
\end{enumerate}
\end{lemma}
\begin{proof}
We consider two cases for statement (i); the other cases are easier. Assume that $M=\pair{\{M_i\}_i, k}$.
\begin{itemize}
\item $r\equiv {\mathbb E}[e]$.
We reason as follows.
\[\begin{array}{rcl}
\Denote{f_{x,\bar{q}}^M(r)}_\mu & = & \Denote{{\mathbb E}_{x\sim M[\bar{q}]}[e]}_\mu \\
& = & \sum_\sigma\sum_i M_i\mu(\sigma)M_i^\dag\cdot\Denote{e}_{\sigma[l(i)/x]} \\
& = & \sum_{\sigma'}\sum_\sigma\sum_i \{M_i\mu(\sigma)M_i^\dag\mid \sigma[l(i)/x]=\sigma'\}\cdot\Denote{e}_{\sigma'} \\
& = & \sum_{\sigma'}\sum_\sigma\mu_\sigma(\sigma')\cdot\Denote{e}_{\sigma'} \qquad\mbox{where } \mu_\sigma(\sigma')=\sum_i \{M_i\mu(\sigma)M_i^\dag\mid \sigma[l(i)/x]=\sigma'\}\\
& = &\Denote{{\mathbb E}[e]}_{\sum_\sigma\mu_\sigma}\\
& = & \Denote{{\mathbb E}[e]}_{\Denote{x:=M[\bar{q}]}_\mu} \\
& = & \Denote{r}_{\Denote{x:=M[\bar{q}]}_\mu}
\end{array}\]
The second last equality holds because $\Denote{x:=M[\bar{q}]}_{(\sigma,\mu(\sigma))}=\mu_\sigma$.
\item $r\equiv {\mathbb E}_{\bar{y}\sim N[\bar{q'}]}[e]$. There are two possibilities. Let us first assume that $x\not\in\bar{y}$ and $N=\pair{\{N_j\}_j, l}$.
\[\begin{array}{rcl}
\Denote{f_{x,\bar{q}}^M(r)}_\mu & = & \Denote{{\mathbb E}_{x\bar{y}\sim M'[\bar{q}\cup\bar{q'}]}[e]}_\mu \mbox{ with } M'=\pair{\{N_j M_i\}_{ij}, k'} \mbox{ and } k'(ij)=(k(i),l(j)) \\
& = & \sum_\sigma\sum_{ij} N_jM_i\mu(\sigma)M_i^\dag N_j^\dag \cdot\Denote{e}_{\sigma[k(i),l(j)/x\bar{y}]} \\
& = & \sum_\sigma\sum_{j} N_j(\sum_i M_i\mu(\sigma)M_i^\dag) N_j^\dag \cdot\Denote{e}_{\sigma[k(i)/x][l(j)/\bar{y}]} \\
& = & \sum_{\sigma'}\sum_\sigma\sum_j N_j(\sum_i\{M_i\mu(\sigma)M_i^\dag \mid \sigma[k(i)/x]=\sigma'\})N_j^\dag\cdot\Denote{e}_{\sigma'[l(j)/\bar{y}]}\\
& = & \sum_{\sigma'}\sum_\sigma\sum_{j} N_j\mu_\sigma(\sigma') N_j^\dag \cdot\Denote{e}_{\sigma'[l(j)/\bar{y}]} \\
& & \qquad\mbox{where } \mu_\sigma(\sigma')=\sum_i \{M_i\mu(\sigma)M_i^\dag\mid \sigma[k(i)/x]=\sigma'\}\\
& = & \sum_{\sigma'}\sum_{j} N_j\sum_\sigma\mu_\sigma(\sigma') N_j^\dag \cdot\Denote{e}_{\sigma'[l(j)/\bar{y}]} \\
& = &\Denote{{\mathbb E}_{\bar{y}\sim N[\bar{q'}]}[e]}_{\sum_\sigma\mu_\sigma}\\
& = & \Denote{{\mathbb E}_{\bar{y}\sim N[\bar{q'}]}[e]}_{\Denote{x:=M[\bar{q}]}_\mu} \\
& = & \Denote{r}_{\Denote{x:=M[\bar{q}]}_\mu}
\end{array}\]
If $x\in\bar{y}$, the proof is similar by noting that $\sigma[k(i)/x][l(j)/\bar{y}]=\sigma[l(j)/\bar{y}]$.
\end{itemize}
\end{proof}
The next theorem states that the concrete proof system is sound.
\begin{theorem}
Every judgement provable in $\mathcal{S}_c$ is valid.
\end{theorem}
\begin{proof}
We only need to prove that the four new rules [{\sf Assgn'}], [{\sf QInit'}], [{\sf QUnit'}] and [{\sf QMeas'}] are sound, which follows from Lemmas~\ref{lem:qinit} - \ref{lem:meas}; the soundness of all other rules are already shown in Theorem~\ref{thm:sounda}.
\end{proof}
We can define a precondition calculus to help with syntactic reasoning. Given an assertion $P$ as a postcondition and a loop-free command $c$, we construct an assertion as a precondition for $c$, written as
$pc(c, P)$.
The computation rules for preconditions are given in Figure~\ref{fig:pc}.
\begin{figure}
\[\begin{array}{rcl}
pc({\textbf{skip}}, P) & := & P \\
pc(x := a, P) & := & P[a/x] \\
pc(c_0; c_1, P) & := & pc(c_0, pc(c_1, P))\\
pc(\textbf{if}~ b ~\textbf{then}~ c_0 ~\textbf{else}~ c_1, P) & := &
(pc(c_0,P)\wedge\Box b)\oplus (pc(c_1,P)\wedge\Box \neg b)
\\
pc(\textbf{abort}, P) & := & \left\{\begin{array}{ll}
\top & \mbox{if } P=\Box{\textbf{false}}\\
\mbox{undefined} & \mbox{otherwise}
\end{array}\right.
\\
pc(q:=|0\>, P) & := & h(P)\\
pc(U[\bar{q}], P) &:=& g^U(P)\\
pc(x:=M[\bar{q}], P) & := & f_{x,\bar{q}}^M(P)
\end{array}\]
\caption{Precondition calculus
}\label{fig:pc}
\end{figure}
\begin{theorem}
Let $c$ be a non-looping command. The following rule is derivable.
\[
\prooftree
\justifies
\triple{pc(c, P)}{c}{P}
\using
[\sf PC]
\endprooftree
\]
\end{theorem}
\begin{proof}
We proceed by induction on the structure of $c$.
\begin{itemize}
\item $c\equiv{\textbf{skip}}$. Then $pc(c,P)=P$ and we have
$\vdash \triple{P}{c}{P}$ by rule {\sf [Skip]}.
\item $c\equiv\textbf{abort}$. Then $pc(c, P)$ is only defined for $P=\Box{\textbf{false}}$. In this case, we can infer that $\vdash \triple{\top}{\textbf{abort}}{\Box{\textbf{false}}}$ by rule {\sf [Abort]}.
\item $c\equiv x:=a$. Then $pc(c,P) = P[a/x]$ and we have $\vdash\triple{P[a/x]}{c}{P}$ by rule {\sf [Assgn']}.
\item $c\equiv c_0;c_1$. Then $pc(c,P) = pc(c_0, pc(c_1,P))$. By induction, we have $\vdash\triple{pc(c_1,P)}{c_1}{P}$ and $\vdash\triple{pc(c_0,pc(c_1,P))}{c_0}{pc(c_1,P)}$. By using rule {\sf [Seq]}, we obtain $\vdash \triple{pc(c_0,pc(c_1,P))}{c}{P}$.
\item $c\equiv \textbf{if}~ b ~\textbf{then}~ c_0 ~\textbf{else}~ c_1$. Then $pc(c,P) = (pc(c_0,P)\wedge\Box b)\oplus (pc(c_1,P)\wedge\Box \neg b)$. By induction, we have that $\vdash \triple{pc(c_0,P)}{c_0}{P}$ and $\vdash \triple{pc(c_1,P)}{c_1}{P}$. It is obvious that $pc(c_0,P)\wedge \Box b\Rightarrow pc(c_0,P)$. We can use rule {\sf [Conseq]} to infer that $\vdash\triple{pc(c_0,P)\wedge \Box b}{c_0}{P}$. Similarly, we have $\vdash\triple{pc(c_1,P)\wedge \Box \neg b}{c_1}{P}$.
By applying rule {\sf [Cond]}, we have that
$\vdash\triple{pc(c, P\oplus P)}{c}{P\oplus P}$.
Since $P\oplus P\Leftrightarrow P$, we use rule {\sf [Conseq]} again to infer the required result that $\vdash\triple{pc(c, P)}{c}{P}$.
\item $c\equiv q:=|0\>$. A direct consequence of rule {\sf [QInit']}.
\item $c\equiv U[\bar{q}]$. By using rule {\sf [QUnit']}.
\item $c\equiv M[\bar{q}]$. By using rule {\sf [QMeas']}.
\end{itemize}
\end{proof}
\section{Example: superdense coding}\label{sec:example}
In this section, we illustrate the use of the proof system $\mathcal{S}_c$ via the example of superdense coding.
Superdense coding was proposed by Bennett and Wiesner in 1992~\cite{BW92}. It is a quantum communication protocol allowing two classical bits to be encoded in one qubit during a transmission, so it needs only one quantum channel. Such advantage is based on the use of a maximally
entangled state, EPR state. An EPR state can be transformed into all the four kinds of EPR states through 1-qubit operations, and these EPR states are mutually orthogonal.
\paragraph*{Protocol.}
We suppose the sender and the receiver of the communication are Alice and Bob, then the protocol goes as follows:
\begin{enumerate}
\item Alice and Bob prepare an EPR state $\frac{|00\>+|11\>}{\sqrt{2}}$ together. Then they share the qubits, Alice holding $q_0$ and Bob holding $q_1$.
\item Depending on the message Alice wants to send, she applies a gate to her qubit. If Alice wants to send $00$, she does nothing. If Alice wants to send $01$, she applies the X gate. To send $10$, she applies the Z gate. To send $11$, she applies both $X$ and $Z$.
\item Then Alice sends the qubit $q_0$ to Bob.
\item Bob applies a CNOT operation on $q_0,q_1$ and a Hadamard operation on $q_0$ to remove the entanglement.
\item Bob measures $q_0$ and $q_1$ to get the message.
\end{enumerate}
After the execution of the protocol above, Bob gets the value that Alice wants to send. The protocol exactly transmits two classical bits of information by sending one qubit from Alice to Bob. A quantum circuit implementing the protocol is illustrated in Figure~\ref{fig:sc}.
\begin{figure}[t]
\[
\Qcircuit @C=1.2em @R=2em {
\lstick{x_0} & \cw & \cw & \cw & \cw & \cw & \cw & \cw & \cctrl{2} & & & & & & & & & \\
\lstick{x_1} & \cw & \cw & \cw & \cw & \cw & \cctrl{1} & & & & & & & & & & & \\
\lstick{q_0} & \qw & \gate{H} & \qw & \ctrl{1} & \qw & \gate{X} & \qw & \gate{Z} & \qw & \ctrl{1} & \qw & \gate{H} & \qw & \meter & \cw & \cw & y_0 \\
\lstick{q_1} & \qw & \qw & \qw & \targ & \qw & \qw & \qw & \qw & \qw & \targ & \qw & \qw & \qw & \meter & \cw & \cw & y_1 \\
}
\]
\caption{Superdense coding}\label{fig:sc}
\end{figure}
The protocol can also be described by the quantum program $SC$ given in Figure~\ref{fig:prog}, where for any pure state $|\varphi\>$, we write $[|\varphi\>]$ for its density operator $|\varphi\>\<\varphi|$.
\begin{figure}
\[\begin{array}{cl}
SC \equiv & \\
1: & q_0 := |0\>; \\
2: & q_1 := |0\>; \\
3: & H[q_0]; \\
4: & CNOT[q_0 q_1]; \\
5: & \textbf{if}~ x_1=1 ~\textbf{then}~ X[q_0]; \\
6: & \textbf{if}~ x_0=1 ~\textbf{then}~ Z[q_0]; \\
7: & CNOT[q_0 q_1]; \\
8: & H[q_0]; \\
9: & y_0 :=M[q_0]; \\
10: & y_1 :=M[q_1] \\
& \quad\mbox{where } M=\pair{\{M_0, M_1\}, Id}, M_0=[|0\>], M_1=[|1\>]
\end{array}\]
\caption{The quantum program of implementing superdense coding}\label{fig:prog}
\end{figure}
According to the operational rules in Figure~\ref{fig:exec}, we can derive the following sequence of transitions, where the initial values of the four classical variables in the first configuration can be arbitrary and we use $*$ to stand for unimportant commands or the values of variables.
\[\begin{array}{ll}
& (SC, x_0x_1y_0y_1, [|00\>])\vspace{2mm}\\
\rightarrow & (*, *, [\frac{|0\>+|1\>}{\sqrt{2}}|0\>]) \vspace{2mm}\\
\rightarrow & (*, *, [\frac{|00\>+|11\>}{\sqrt{2}}]) \vspace{2mm}\\
\rightarrow & (*, *, [X_0^{x_1}\frac{|00\>+|11\>}{\sqrt{2}}]) \vspace{2mm}\\
\rightarrow & (*, *, [Z_0^{x_0}X_0^{x_1}\frac{|00\>+|11\>}{\sqrt{2}}]) \vspace{2mm}\\
\equiv & \left\{\begin{array}{ll}
(*, 00y_0y_1, [\frac{|00\>+|11\>}{\sqrt{2}}]) & \mbox{ if } x_0=x_1=0 \\
(*, 01y_0y_1, [\frac{|10\>+|01\>}{\sqrt{2}}]) & \mbox{ if } x_0=0, x_1=1 \\
(*, 10y_0y_1, [\frac{|00\>-|11\>}{\sqrt{2}}]) & \mbox{ if } x_0=1, x_1=0 \\
(*, 11y_0y_1, [\frac{|10\>-|01\>}{\sqrt{2}}]) & \mbox{ if } x_0=x_1=1
\end{array} \right.\vspace{2mm}\\
\rightarrow & \left\{\begin{array}{l}
(*, 00y_0y_1, [\frac{|00\>+|10\>}{\sqrt{2}}]) \\
(*, 01y_0y_1, [\frac{|11\>+|01\>}{\sqrt{2}}]) \\
(*, 10y_0y_1, [\frac{|00\>-|10\>}{\sqrt{2}}]) \\
(*, 11y_0y_1, [\frac{|11\>-|01\>}{\sqrt{2}}])
\end{array} \right.\vspace{2mm}\\
\rightarrow & \left\{\begin{array}{l}
(*, 00y_0y_1, [|00\>]) \\
(*, 01y_0y_1, [|01\>]) \\
(*, 10y_0y_1, [|10\>]) \\
(*, 11y_0y_1, [|11\>])
\end{array} \right.\vspace{2mm}\\
\rightarrow & \left\{\begin{array}{l}
(*, 000y_1, [|00\>]) \\
(*, 010y_1, [|01\>]) \\
(*, 101y_1, [|10\>]) \\
(*, 111y_1, [|11\>])
\end{array} \right.\vspace{2mm}\\
\rightarrow & \left\{\begin{array}{l}
(\textbf{nil}, 0000, [|00\>]) \\
(\textbf{nil}, 0101, [|01\>]) \\
(\textbf{nil}, 1010, [|10\>]) \\
(\textbf{nil}, 1111, [|11\>])
\end{array} \right.\\
\end{array}\]
We observe that in each case of the four last configurations, we always have the value of $x_0x_1$ coincide with $y_0y_1$ as expected. Indeed, we would like to show that the judgement
\begin{equation}\label{eq:correct}
\triple{{\textbf{true}}} {SC}{\Box(x_0=y_0 \wedge x_1=y_1)}
\end{equation}
is provable in our concrete proof system. This can be accomplished by a sequence of derivations; for every line of command in Figure~\ref{fig:prog} we need to prove a Hoare triple. We start from line 10 and proceed backwards. The first six steps can be derived by using the rules {\sf[QMeas']}, {\sf [QUnit']}, {\sf [Cond]} and {\sf [Split]}, as shown in Figure~\ref{fig:cond}. Continue the reasoning until line 1, we obtain the following precondition for SC.
\[\begin{array}{l}
\{({\mathbb E}_{y_0y_1\sim M_{10}[q_0 q_1]}[{\bf 1}_{\psi\wedge x_0=1\wedge x_1=1}]= {\mathbb E}_{y_0y_1\sim M_{10}[q_0 q_1]}[{\bf 1}_{{\textbf{true}}}]) \\
\oplus ({\mathbb E}_{y_0y_1\sim M_9[q_0 q_1]}[{\bf 1}_{\psi\wedge x_0=1\wedge x_1=0}]= {\mathbb E}_{y_0y_1\sim M_9[q_0 q_1]}[{\bf 1}_{{\textbf{true}}}]) \qquad\qquad (\dag)\\
\oplus ({\mathbb E}_{y_0y_1\sim M_8[q_0 q_1]}[{\bf 1}_{\psi\wedge x_0=0\wedge x_1=1}]= {\mathbb E}_{y_0y_1\sim M_8[q_0 q_1]}[{\bf 1}_{{\textbf{true}}}]) \\
\oplus ({\mathbb E}_{y_0y_1\sim M_7[q_0 q_1]}[{\bf 1}_{\psi\wedge x_0=0\wedge x_1=0}]= {\mathbb E}_{y_0y_1\sim M_7[q_0 q_1]}[{\bf 1}_{{\textbf{true}}}]) \},\\
\mbox{where } M_{10}\equiv \pair{\{E_{0000}, E_{0001},..., E_{1111}\}, f}\mbox{ with}\\
\hspace{2.3cm} E_{0000}\equiv E_{00}H_{q_0}CNOT_{q_0q_1}Z_{q_0}X_{q_0}CNOT_{q_0q_1}H_{q_0}|0\>_{q_1}\<0|\cdot |0\>_{q_0}\<0|, \\
\hspace{2.3cm} E_{0001}\equiv E_{00}H_{q_0}CNOT_{q_0q_1}Z_{q_0}X_{q_0}CNOT_{q_0q_1}H_{q_0}|0\>_{q_1}\<0|\cdot |0\>_{q_0}\<1|, \\
\hspace{2.3cm} E_{0010}\equiv E_{00}H_{q_0}CNOT_{q_0q_1}Z_{q_0}X_{q_0}CNOT_{q_0q_1}H_{q_0}|0\>_{q_1}\<1|\cdot |0\>_{q_0}\<0|,\\ \hspace{2.3cm} E_{0011}\equiv E_{00}H_{q_0}CNOT_{q_0q_1}Z_{q_0}X_{q_0}CNOT_{q_0q_1}H_{q_0}|0\>_{q_1}\<1|\cdot |0\>_{q_0}\<1|, \\
\hspace{2.3cm} E_{0100}\equiv E_{01}H_{q_0}CNOT_{q_0q_1}Z_{q_0}X_{q_0}CNOT_{q_0q_1}H_{q_0}|0\>_{q_1}\<0|\cdot |0\>_{q_0}\<0|, \\
\hspace{3.3cm} \vdots \\
\hspace{2.3cm} E_{1111}\equiv E_{11}H_{q_0}CNOT_{q_0q_1}Z_{q_0}X_{q_0}CNOT_{q_0q_1}H_{q_0}|0\>_{q_1}\<1|\cdot |0\>_{q_0}\<1|, \\
\hspace{2.3cm} f(00**)=00,\ f(01**)=01,\ f(10**)=10,\ f(11**)=11\\
M_{9}\equiv \pair{\{E'_{0000}, E'_{0001},..., E'_{1111}\}, f}\mbox{ with}\\
\hspace{1.3cm} E'_{0000}\equiv E_{00}H_{q_0}CNOT_{q_0q_1}Z_{q_0}CNOT_{q_0q_1}H_{q_0}|0\>_{q_1}\<0|\cdot |0\>_{q_0}\<0|, \\
\hspace{2.3cm} \vdots \\
\hspace{1.3cm} E'_{1111}\equiv E_{11}H_{q_0}CNOT_{q_0q_1}Z_{q_0}CNOT_{q_0q_1}H_{q_0}|0\>_{q_1}\<1|\cdot |0\>_{q_0}\<1| \\
M_{8}\equiv \pair{\{E''_{0000}, E''_{0001},..., E''_{1111}\}, f}\mbox{ with}\\
\hspace{1.3cm} E''_{0000}\equiv E_{00}H_{q_0}CNOT_{q_0q_1}X_{q_0}CNOT_{q_0q_1}H_{q_0}|0\>_{q_1}\<0|\cdot |0\>_{q_0}\<0|, \\
\hspace{2.3cm} \vdots \\
\hspace{1.3cm} E''_{1111}\equiv E_{11}H_{q_0}CNOT_{q_0q_1}X_{q_0}CNOT_{q_0q_1}H_{q_0}|0\>_{q_1}\<1|\cdot |0\>_{q_0}\<1| \\
M_{7}\equiv \pair{\{E'''_{0000}, E'''_{0001},..., E'''_{1111}\}, f}\mbox{ with}\\
\hspace{1.3cm} E'''_{0000}\equiv E_{00}H_{q_0}CNOT_{q_0q_1}CNOT_{q_0q_1}H_{q_0}|0\>_{q_1}\<0|\cdot |0\>_{q_0}\<0|, \\
\hspace{2.3cm} \vdots\\
\hspace{1.3cm} E'''_{1111}\equiv E_{11}H_{q_0}CNOT_{q_0q_1}CNOT_{q_0q_1}H_{q_0}|0\>_{q_1}\<1|\cdot |0\>_{q_0}\<1| \\
\end{array}\]
The measurement operators in $M_{10}$ look complicated. However, a simple calculation shows that among the 16 operators only four of them are non-zero. Indeed, the simplified form of $M_{10}$ is
\[M_{10}=\pair{\{E_{1100}, E_{1101}, E_{1110}, E_{1111}\}, f}\]
where
\[E_{1100}=\begin{bmatrix}
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
1 & 0 & 0 & 0
\end{bmatrix}
\quad E_{1101}=\begin{bmatrix}
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 1 & 0
\end{bmatrix}
\quad E_{1110}=\begin{bmatrix}
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 1 & 0 & 0
\end{bmatrix}
\quad E_{1111}=\begin{bmatrix}
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 1
\end{bmatrix}
\]
Similarly, the simplified form of $M_{9}$ is
$M_{9}=\pair{\{E'_{1000}, E'_{1001}, E'_{1010}, E'_{1011}\}, f}$,
where
\[E'_{1000}=\begin{bmatrix}
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
1 & 0 & 0 & 0\\
0 & 0 & 0 & 0
\end{bmatrix}
\quad E'_{1001}=\begin{bmatrix}
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 0
\end{bmatrix}
\quad E'_{1010}=\begin{bmatrix}
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 0 & 0
\end{bmatrix}
\quad E'_{1011}=\begin{bmatrix}
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 1\\
0 & 0 & 0 & 0
\end{bmatrix}
\]
The simplified form of $M_{8}$ is
$M_{8}=\pair{\{E''_{0100}, E''_{0101}, E''_{0110}, E''_{0111}\}, f}$,
where
\[E''_{0100}=\begin{bmatrix}
0 & 0 & 0 & 0\\
1 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0
\end{bmatrix}
\quad E''_{0101}=\begin{bmatrix}
0 & 0 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0
\end{bmatrix}
\quad E''_{0110}=\begin{bmatrix}
0 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0
\end{bmatrix}
\quad E''_{0111}=\begin{bmatrix}
0 & 0 & 0 & 0\\
0 & 0 & 0 & 1\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0
\end{bmatrix}
\]
Finally, the simplified form of $M_{7}$ is
$M_{7}=\pair{\{E'''_{0000}, E'''_{0001}, E'''_{0010}, E'''_{0011}\}, f}$,
where
\[E'''_{0000}=\begin{bmatrix}
1 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0
\end{bmatrix}
\quad E'''_{0001}=\begin{bmatrix}
0 & 0 & 1 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0
\end{bmatrix}
\quad E'''_{0010}=\begin{bmatrix}
0 & 1 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0
\end{bmatrix}
\quad E'''_{0011}=\begin{bmatrix}
0 & 0 & 0 & 1\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0
\end{bmatrix}
\]
\begin{figure}[t]
\[\begin{array}{cl}
& \cond{\{{\textbf{true}}\}} \\
SC \equiv & \\
1: & q_0 := |0\>; \\
2: & q_1 := |0\>; \\
3: & H[q_0]; \\
4: & CNOT[q_0 q_1]; \\
& \cond{\{({\mathbb E}_{y_0y_1\sim M_6[q_0 q_1]}[{\bf 1}_{\psi\wedge x_0=1\wedge x_1=1}]= {\mathbb E}_{y_0y_1\sim M_6[q_0 q_1]}[{\bf 1}_{{\textbf{true}}}])} \\
& \cond{ \oplus ({\mathbb E}_{y_0y_1\sim M_4[q_0 q_1]}[{\bf 1}_{\psi\wedge x_0=1\wedge x_1=0}]= {\mathbb E}_{y_0y_1\sim M_4[q_0 q_1]}[{\bf 1}_{{\textbf{true}}}]) }\\
& \cond{ \oplus ({\mathbb E}_{y_0y_1\sim M_5[q_0 q_1]}[{\bf 1}_{\psi\wedge x_0=0\wedge x_1=1}]= {\mathbb E}_{y_0y_1\sim M_5[q_0 q_1]}[{\bf 1}_{{\textbf{true}}}]) }\\
& \cond{ \oplus ({\mathbb E}_{y_0y_1\sim M_3[q_0 q_1]}[{\bf 1}_{\psi\wedge x_0=0\wedge x_1=0}]= {\mathbb E}_{y_0y_1\sim M_3[q_0 q_1]}[{\bf 1}_{{\textbf{true}}}]) \},}\\
& \cond{\mbox{where } M_6\equiv\langle\{E_{00}H_{q_0}CNOT_{q_0q_1}Z_{q_0}X_{q_0}, \ E_{01}H_{q_0}CNOT_{q_0q_1}Z_{q_0}X_{q_0},} \\
& \hspace{2.3cm} \cond{E_{10}H_{q_0}CNOT_{q_0q_1}Z_{q_0}X_{q_0},\ E_{11}H_{q_0}CNOT_{q_0q_1}Z_{q_0}X_{q_0}\}, Id\rangle ,} \\
& \cond{M_5\equiv\langle\{E_{00}H_{q_0}CNOT_{q_0q_1}X_{q_0}, \ E_{01}H_{q_0}CNOT_{q_0q_1}X_{q_0},} \\
& \hspace{2.3cm} \cond{E_{10}H_{q_0}CNOT_{q_0q_1}X_{q_0},\ E_{11}H_{q_0}CNOT_{q_0q_1}X_{q_0}\}, Id\rangle} \\
5: & \textbf{if}~ x_1=1 ~\textbf{then}~ X[q_0]; \\
& \cond{\{({\mathbb E}_{y_0y_1\sim M_4[q_0 q_1]}[{\bf 1}_{\psi\wedge x_0=1}]= {\mathbb E}_{y_0y_1\sim M_4[q_0 q_1]}[{\bf 1}_{{\textbf{true}}}])} \\
& \cond{ \oplus ({\mathbb E}_{y_0y_1\sim M_3[q_0 q_1]}[{\bf 1}_{\psi\wedge x_0=0}]= {\mathbb E}_{y_0y_1\sim M_3[q_0 q_1]}[{\bf 1}_{{\textbf{true}}}]) \},}\\
& \cond{\mbox{where } M_4\equiv\langle\{E_{00}H_{q_0}CNOT_{q_0q_1}Z_{q_0}, \ E_{01}H_{q_0}CNOT_{q_0q_1}Z_{q_0},} \\
& \hspace{2.3cm} \cond{E_{10}H_{q_0}CNOT_{q_0q_1}Z_{q_0},\ E_{11}H_{q_0}CNOT_{q_0q_1}Z_{q_0}\}, Id\rangle} \\
6: & \textbf{if}~ x_0=1 ~\textbf{then}~ Z[q_0]; \\
& \cond{\{{\mathbb E}_{y_0y_1\sim M_3[q_0 q_1]}[{\bf 1}_{\psi}]= {\mathbb E}_{y_0y_1\sim M_3[q_0 q_1]}[{\bf 1}_{{\textbf{true}}}] \}, }\\
& \cond{\mbox{where } M_3\equiv\pair{\{E_{00}H_{q_0}CNOT_{q_0q_1}, E_{01}H_{q_0}CNOT_{q_0q_1}, E_{10}H_{q_0}CNOT_{q_0q_1}, E_{11}H_{q_0}CNOT_{q_0q_1}\}, Id}} \\
7: & CNOT[q_0 q_1]; \\
& \cond{\{{\mathbb E}_{y_0y_1\sim M_2[q_0 q_1]}[{\bf 1}_{\psi}]= {\mathbb E}_{y_0y_1\sim M_2[q_0 q_1]}[{\bf 1}_{{\textbf{true}}}] \},}\\
& \cond{\mbox{where } M_2\equiv\pair{\{E_{00}H_{q_0}, E_{01}H_{q_0}, E_{10}H_{q_0}, E_{11}H_{q_0}\}, Id} } \\
8: & H[q_0]; \\
& \cond{\{{\mathbb E}_{y_0y_1\sim M_1[q_0 q_1]}[{\bf 1}_{\psi}]= {\mathbb E}_{y_0y_1\sim M_1[q_0 q_1]}[{\bf 1}_{{\textbf{true}}}] \}}\\
& \cond{\mbox{where } M_1\equiv\pair{\{E_{00}, E_{01}, E_{10}, E_{11}\}, Id}, } \\
& \cond{ E_{00}\equiv [|0\>]_{q_1}\cdot [|0\>]_{q_0}, \
E_{01}\equiv [|1\>]_{q_1}\cdot [|0\>]_{q_0}, \
E_{10}\equiv [|0\>]_{q_1}\cdot [|1\>]_{q_0}, \
E_{11}\equiv [|1\>]_{q_1}\cdot [|1\>]_{q_0}
} \\
9: & y_0 :=M[q_0]; \\
& \cond{\{{\mathbb E}_{y_1\sim M[q_1]}[{\bf 1}_{\psi}]= {\mathbb E}_{y_1\sim M[q_1]}[{\bf 1}_{{\textbf{true}}}] \}}\\
10: & y_1 :=M[q_1] \\
& \quad\mbox{where } M=\pair{\{E_0, E_1\}, Id}, E_0=[|0\>], E_1=[|1\>]\\
& \cond{\{\Box\psi\}, \mbox{ where }\psi\equiv x_0=y_0 \wedge x_1=y_1 }
\end{array}\]
\caption{The quantum program with pre- and postconditions}\label{fig:cond}
\end{figure}
Write $P$ for the assertion in ($\dag$). We have seen that
\begin{equation}\label{eq:b}
\triple{P}{SC}{\Box\psi} .
\end{equation}
We observe that ${\textbf{true}} \Leftrightarrow P$. This can be seen as follows. Let
\[\begin{array}{rcl}
P_{11} & \equiv & ({\mathbb E}_{y_0y_1\sim M_{10}[q_0 q_1]}[{\bf 1}_{\psi\wedge x_0=1\wedge x_1=1}]= {\mathbb E}_{y_0y_1\sim M_{10}[q_0 q_1]}[{\bf 1}_{{\textbf{true}}}]) \\
P_{10} & \equiv & ({\mathbb E}_{y_0y_1\sim M_9[q_0 q_1]}[{\bf 1}_{\psi\wedge x_0=1\wedge x_1=0}]= {\mathbb E}_{y_0y_1\sim M_9[q_0 q_1]}[{\bf 1}_{{\textbf{true}}}]) \\
P_{01} & \equiv & ({\mathbb E}_{y_0y_1\sim M_8[q_0 q_1]}[{\bf 1}_{\psi\wedge x_0=0\wedge x_1=1}]= {\mathbb E}_{y_0y_1\sim M_8[q_0 q_1]}[{\bf 1}_{{\textbf{true}}}]) \\
P_{00} & \equiv & ({\mathbb E}_{y_0y_1\sim M_7[q_0 q_1]}[{\bf 1}_{\psi\wedge x_0=0\wedge x_1=0}]= {\mathbb E}_{y_0y_1\sim M_7[q_0 q_1]}[{\bf 1}_{{\textbf{true}}}])\\
b_{11} & \equiv & x_0=1\wedge x_1=1\\
b_{10} & \equiv & x_0=1\wedge x_1=0\\
b_{01} & \equiv & x_0=0\wedge x_1=1\\
b_{00} & \equiv & x_0=0\wedge x_1=0
\end{array}\]
We have $P = P_{11}\oplus P_{10}\oplus P_{10}\oplus P_{00}$. For any POVD $\mu$, it is easy to see that
\[\mu = \mu_{|b_{11}} + \mu_{|b_{10}} + \mu_{|b_{01}} + \mu_{|b_{00}} \]
We have that $\mu_{|b_{11}} \models P_{11}$ because
\[\begin{array}{ll}
& \Denote{{\mathbb E}_{y_0y_1\sim M_{10}[q_0 q_1]}[{\bf 1}_{\psi\wedge x_0=1\wedge x_1=1}]}_{\mu_{|b_{11}} } \\
= & \sum_{\sigma}\sum_i E_i\mu_{|b_{11}}(\sigma)M^\dag_i\cdot {\bf 1}_{\Denote{\psi\wedge x_0=1\wedge x_1=1}_{\sigma[f(i)/y_0y_1]}}\\
& \mbox{where } i\in\{1100, 1101, 1110, 1100\} \\
= & \sum_{\sigma}\sum_i E_i\mu_{|b_{11}}(\sigma)M^\dag_i\cdot {\bf 1}_{\Denote{\psi\wedge x_0=1\wedge x_1=1}_{\sigma[11/y_0y_1]}}\\
= & \sum_{\sigma}\sum_i E_i\mu_{|b_{11}}(\sigma)M^\dag_i\cdot 1 \\
= & \sum_{\sigma}\sum_i E_i\mu_{|b_{11}}(\sigma)M^\dag_i\cdot {\bf 1}_{\Denote{{\textbf{true}}}_{\sigma[11/y_0y_1]}}\\
= & \Denote{{\mathbb E}_{y_0y_1\sim M_{10}[q_0 q_1]}[{\bf 1}_{{\textbf{true}}}]}_{\mu_{|b_{11}} }
\end{array}\]
which implies $\Denote{P_{11}}_{\mu_{|b_{11}} } ={\textbf{true}}$.
Similarly, we can check that $\mu_{|b_{10}} \models P_{10}$, etc. Therefore, we obtain that $\mu \models P$. As $\mu$ is arbitrarily chosen, we have verified that ${\textbf{true}}\Leftrightarrow P$. By (\ref{eq:b}) and rule {\sf [Conseq]}, we finally see that the triple $\triple{{\textbf{true}}}{SC}{\Box\psi}$ is provable.
As we can see in (\ref{eq:correct}), the postcondition of that Hoare triple is an assertion about classical variables, even though quantum computation takes place during the execution of the program $SC$. For such scenario, it is very natural to prove the correctness of programs via an satisfaction-based proof system, rather than an expectation-based one.
\section{Conclusion and future work}\label{sec:clu}
We have introduced a simple quantum imperative language that has both classical and quantum constructs by extending the language \textbf{IMP} studied in depth by Winskel. We have investigated its formal semantics by providing a small-step operational semantics, a denotational semantics and two Hoare-style proof systems: an abstract one and a concrete one. In order to define the semantics, we have used the notion of POVDs to represent the states of programs. Therefore, a program can be considered as a transformer of POVDs. Following the work of Barthe et al, we have designed
two satisfaction-based proof systems, as opposed to the usual expectation-based systems. The abstract proof system turns out to be sound and relatively complete, while the concrete one is sound.
As to the future work, at least two immediate improvements are interesting and worth being considered.
\begin{itemize}
\item The proof rule {\sf [While]} is not satisfactory because it involves two sequences of assertions $(P_n)$ and $(P'_n)$. In either the purely classical \cite{hoare1969axiomatic} or purely quantum setting \cite{Yin12}, the rule can be elegantly formulated. However, in the presence of both classical and quantum variables, it remains a challenge to find a more concise formulation of the rule.
\item The reasoning in Section~\ref{sec:example} about the example of superdense coding was done manually. In the future, we would like to embed our program logic in a proof assistant so as to facilitate the reasoning.
\end{itemize}
\bibliographystyle{abbrv}
|
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"redpajama_set_name": "RedPajamaArXiv"
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| 4,884
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{"url":"https:\/\/ai.stackexchange.com\/questions\/27772\/is-there-literature-on-neural-network-with-activation-functions-of-bounded-domai","text":"Is there literature on Neural Network with activation functions of bounded domain?\n\nI think to have found a somewhat interesting connection between neural networks and another area of mathematics. However, it requires the activation functions in the network to have a bounded - ideally small - domain. For the sake of simplicity, I am restricting this to feedforward networks.\n\nMy approach has been the following: Assuming bounded input and weights, a maximal input can be derived. Before each application of an activation function, I thus just scale down the range from the maximal one to the one permitted.\n\nThis however causes nearly all weights that are not close (in absolute terms) to the maximal ones to lead to very small outputs of the network, meaning nearly all weight combinations lead to outputs near zero. The network thus has issues learning even simple tasks.\n\nMy question, therefore, is: Has anyone ever studied these issues and maybe found a network architecture that works well with this? Or another solution for bounded domains?\n\n\u2022 Might not be what you look for but I remember the paper \"Neural Arithmetic Logic Units\" bounding weights in the architecture with activation functions. arxiv.org\/pdf\/1808.00508.pdf May 14 at 22:04","date":"2021-11-27 14:22:26","metadata":"{\"extraction_info\": {\"found_math\": false, \"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\": 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\": 0.8253741264343262, \"perplexity\": 558.2684359958915}, \"config\": {\"markdown_headings\": false, \"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-2021-49\/segments\/1637964358189.36\/warc\/CC-MAIN-20211127133237-20211127163237-00096.warc.gz\"}"}
| null | null |
https://www.concordia.ca/content/concordia/en/finearts/art-history/faculty.html
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Nicola Tullio Pezolet, PhD
Associate Professor, Art History
Graduate Program Director, Art History
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Office: S-EV 3765
Engineering, Computer Science and Visual Arts Integrated Complex,
1515 St. Catherine W.
Phone: (514) 848-2424 ext. 5449
Email: nicola.pezolet@concordia.ca
Availability: Office hours on Mondays (by email appointment). Please do not call or leave a voicemail message. Instead, contact via email to schedule a phone or Zoom call.
Nicola Pezolet's Explore Concordia page
PhD (2013) Massachusetts Institute of Technology, Cambridge (MIT School of Architecture and Planning: History, Theory and Criticism of Architecture and Art)
MA (2008) Université Laval, Québec (Département des sciences historiques: Histoire de l'art)
BA (2005) Université Laval, Québec (Département des sciences historiques: Histoire de l'art)
Research & Teaching Interests
History and Theories of Architecture and the Built Environment in Europe and North America (19th-20th Century)
Modernism, Public Space and the 'Synthesis of the Arts'
Critical Historiography and Social History of Art
Cold War Cultural Politics
Christian Architecture, Art and Liturgy
Theological Aesthetics
Utopian Thought
Distinctions & Awards
Gail and Stephen A. Jarislowsky for Studies in Canadian Art collaboration grant (2019)
Social Sciences and Humanities Research Council of Canada, Insight Development Grant (2016-2018)
Christopher Jackson Teaching Award, Department of Art History, Concordia University
Concordia University, Aid to Research-Related Events
Fonds de Recherche Société et Culture du Québec, Établissement de nouveaux professeurs-chercheurs (2014-2017)
Concordia University Undergraduate Student Research Award (supervisor)
Concordia University Faculty Research and Development Program Award
Social Science Research Council, International Dissertation Research Fellowship, residency at the Maison des Étudiants Canadiens / Cité internationale universitaire de Paris (Paris, 2011-2012)
Canadian Centre for Architecture collection research grant (Montreal, 2009)
Social Sciences and Humanities Research Council of Canada doctoral fellowship (Cambridge, 2007- 2011)
Fonds Québécois de Recherche sur la Société et la Culture doctoral scholarship (2007- 2011) (declined)
MIT presidential fellowship (Cambridge, 2007-2008)
Fonds Québécois de Recherche sur la Société et la Culture master's scholarship (2006-2007)
Centre Interuniversitaire d'Études sur les Lettres, les Arts et les Traditions (CELAT) travel grant (Vancouver, 2006)
Social Sciences and Humanities Research Council of Canada master's scholarship (Quebec/Montreal, 2005- 2006)
Canada Millennium scholarship (Quebec, 2005)
Professional Memberships & Research Teams
Society for the Study of Architecture in Canada
Universities Art Association of Canada
Gail and Stephen A. Jarislowsky Institute for Studies in Canadian Art
Les lieux des savoirs photographiques: Archives, ateliers, récits, réseaux (FRQSC)
Material Religion Initiative (Concordia/Centre for Interdisciplinary Studies in Society and Culture)
ARTH 270 Icons of Architectural History
ARTH 283 The Life and Work of Le Corbusier
ARTH 362 Studies in Early Christian and Byzantine Art and Architecture
ARTH 371 Studies in Canadian Architecture
ARTH 374 Architecture and Urbanism in Montreal
ARTH 400 Architecture and the "Synthesis of the Arts" (advanced seminar)
ARTH 450 Architecture and Utopia (advanced seminar)
Graduate Seminars
ARTH 633 Image Building: Art, Architecture, and Print Media
ARTH 633 Of Utopian Space
ARTH 639 The Cold War: Art, Architecture, and Global Politics (1945-1989)
ARTH 655 Art History MA Thesis Seminar
ARTH 810 Art History and its Methodologies/Séminaire Intégrateur (Interuniversity Art History PhD program, co-taught w/ Dr. Christina Contandriopoulos, UQÀM)
ARTH 804 The Utopian Imagination in Art and Architecture / L'imaginaire utopique en art et architecture (Interuniversity Art History PhD program)
MA in Art History
Joia Duskic, L'oeuvre Maman de Louise Bourgeois: Vecteur de compréhension de la relation entre l'institution muséale et l'espace public (in progress)
Agnieszka Frasunkiewicz, Drawing the Curtain: (Re)discovering the Women Artists of the Polish People's Republic (in progress)
Jacqueline Grassi, Illuminating the Role of Play in teamLab's Immersive Digital Installations (in progress)
Nina Chabelnik, Modernist Architecture and Religion in the Soviet Union: The Case of the Palace of the Soviets and the Cathedral of Christ the Saviour (2021)
Lucile Cordonnier, Maurice Denis (1870-1943) and the Sacred Grove: Temporality in Fin-de-Siècle France (2021)
Christa Nemnon, Losing Sight to Gain Vision: The Eye in European Surrealist Painting (2021)
Patricia Pérez Rabelo, Collaborative Matter: Agency and Materiality in Barragán's Casa Estudio (2021)
Benjamin Peterson, Modernizing a Marginal Maritime Metropolis: The Emergence of Canada's Postwar Planning Practice in Saint John, New Brunswick (2020)
Gabrielle Montpetit, Douglas Cardinal's Circle of Life Thunderbird House: Lessons in Indigenous Planning and Architecture in Winnipeg's North End (2019)
Lindsay Leblanc, Nicolas Schöffer and the Scattered Origins of Cybernetic Art History (2019)
Emily Bergsma, "Canadian Reject": Melvin Charney's Design for the Canadian Pavilion at Expo 70 (2017)
Gillian MacCormack, The Art of Nation-Building: Two Murals by Charles Comfort (1936-37) (2016)
Nancy Webb, In Search of the New Sensibility: Susan Sontag Writing on Art in the Sixties (2015)
PhD in Art History / Humanities
Stephanie Weber
Interuniversity PhD in Art History program
Patricia Pérez Rabelo
Rebecca Lemire, Decolonizing Design History in the Southwest: Indigenous Architectures, Cross-Cultural-Knowledge, and the Formation of American Modernism (ABD)
Melanie Schnidrig, Constructed Atmospheres: Synesthesia and the Senses in Contemporary Art (ABD)
PhD in Humanities
Material Religion Initiative
(former working group funded by Concordia's Centre for Interdisciplinary Studies in Society and Culture, with Drs. Jeremy Stolow and Hillary Kaell)
"In recent years, the study of religion has undergone an important shift, as a growing number of scholars foreground the role of diverse forms of media, technology, material culture, and embodiment in research on religious experiences, practices, affiliations, and forms of public-making. Such research has generated new bases for studying key dimensions of religious life -- such as ritual practice, communal identity, knowledge formation, or ecstatic experience - in local settings, both historically and today. This 'material turn' in the study of religion has also facilitated the re-evaluation of a range of phenomena not always classified as religious but that merit comparison, from magic and occultism to politics, theatre, art, science, consumer culture and popular entertainment. Bringing together Concordia and Montreal-based experts in a variety of disciplines, this working group will explore how 'material' approaches to the study of religion (broadly defined) inform individual research projects and also provide new opportunities for comparison and debate."
(forthcoming) Sites of Photographic Knowledge: The Book, co-editor with Eduardo Ralickas (Montreal: Artexte, 2022).
Reconstruction and the Synthesis of the Arts in France, 1944-1962 (Abingdon, Oxon/New York, NY: Routledge/Ashgate Studies in Architecture, 2017).
Volumes edited
(co-editor) Thresholds 41: Revolution! (Cambridge, Mass.: MIT School of Architecture and Planning, 2013).
Peer-Reviewed Journal Articles
(in print) "From Automatisme to Automation: Jean-Paul Mousseau's 1962 Lumino-Kinetic Mural for Hydro-Québec," Journal of Canadian Art History / Annales d'histoire de l'art Canadien, vol. 37, no. 2/vol. 38, no.1 (2022): 100-131.
"God and his World: The Architecture of the Christian Pavilion at Expo 67," Research Notebooks/Laboratoire d'Étude de l'Architecture Potentielle, no. 3 (2019): 104-111.
"Art in Transnational Architecture: Paul F. Damaz's Popularization of the Synthesis of the Arts, Between Europe and Latin America," special issue on Contemporary Latino and Latin American Art, ed. Olga U. Herrera and María C. Gaztambide, Diálogo: An Interdisciplinary Studies Journal (University of Texas Press/Center for Latino Research, Chicago), vol. 20, no. 1 (Spring 2017): 21-36.
"Etienne Gaboury, Vatican II and Catholic Liturgical Renewal in Postwar Canada," special issue Apocryphal and Apostolic Modernism: Connections between Religion and Architectural Modernism, 1945-1970, ed. Rajesh Heynickx and Stéphane Symons, The European Legacy (Routledge), vol. 22, no. 3 (Winter 2017): 293-317.
"Spiritual Matter: Universal And Regional Forms In Étienne Gaboury's Précieux-Sang Church," special issue Kingdoms of God, Manifest: A Journal of American Architecture and Urbanism, no. 2 (December 2015): 90-101.
"Bauhaus Ideas: Jorn, Max Bill and Reconstruction Culture," October (The MIT Press), no. 141 (Summer 2012): 86-110.
"Primitivism, Humanism, and Ambivalence: Cobra and Post-Cobra," Res: Anthropology and Aesthetics (Harvard University Press), no. 59-60 (Spring/Autumn 2011): 282-302. Co-authored with Karen Kurczynski, University of Massachusetts (Amherst).
"The Cavern of Anti-Matter: Giuseppe Gallizio and the Technological Imaginary of the Early Situationist International," Grey Room (The MIT Press), no. 38 (March 2010): 62-89.
"Signs of Inhabitation: The Critical Legacies of Patio and Pavilion," Thresholds (MIT School of Architecture), no. 35 (Spring 2009): 44-49.
Book Chapters / Exhibition Catalogue Essays
"Dionysus in the Democracy: Jorn Contra Le Corbusier," in What Moves Us? Le Corbusier and Asger Jorn in Art and Architecture, ed. Ruth Baumeister (Zurich/Silkeborg: Scheidegger and Spiess/Museum Jorn, 2015), 76-81. Translated in Danish as "Dionysos i demokratiet: Jorn Contra Le Corbusier".
"'Le Peintre Asgaer', the Popular Front and the Synthesis of the Arts," in Expo Jorn: Art is a Festival, ed. Karen Kurczynski (Silkeborg: Museum Jorn, 2014), 168-171. Translated in Danish as "'Maleren Asgaer', Folkefronten og syntesen af kunstarterne".
"The Village Polychrome, 1952-53," Canadian Centre for Architecture website, December 2009.
Book and Exhibition Reviews
"The Social Project: Housing Postwar France, by Kenny Cupers (University of Minnesota Press, 2014)," Journal of the Society of Architectural Historians, vol. 76, no. 1 (March 2017): 115-117.
"The SAAL Process: Housing in Portugal, 1974–76, exhibition at the Canadian Centre for Architecture (2 May to 4 October 2015)," Journal of Architectural Education, vol. 70, no. 1 (2016): 180-181.
"The Allied Arts: Architecture and Craft in Postwar Canada, by Sandra Alfoldy (McGill-Queen's University Press, 2012)," RACAR, vol. 40, no. 2 (2015): 161-163.
"On Architecture: Melvin Charney - A Critical Anthology, ed. Louis Martin (McGill-Queen's University Press, 2013)," Future Anterior, vol. 9, no. 1 (2014): 99-105.
Participation activities
"Modernisme, oecuménisme et humanisme à Expo 67," doctoral seminar HAR 7005: Constructing Sacred Space / Construire l'espace sacré, Université de Montréal, November 15, 2021
"'The World Around Us': Photography and the Humanistic Theology of the Christian Pavilion at Expo 67," panel Photographic Narrative and Counter-Narrative in Architectural Histories, Society for the Study of Architecture in Canada conference, Halifax, May 31, 2019
"La Construction des Églises: Réflexions sur le Directoire Pastoral de la Commission Diocésaine de Liturgie de Montréal (1965)," (Re)connaissance de l'architecture religieuse au Québec, journée d'étude du CELAT, Université Laval, Québec, April 18, 2019
"Terre Nouvelle: Modernism, Humanistic Theology and the Didactic Publications of the Centre d'Animation Pastorale at Expo 67," Laboratoire d'Étude de l'Architecture Potentielle (LEAP), Montreal, April 7, 2018
"Lumino-Kineticism and the Postwar Years," Centre for Expanded Poetics, Concordia University, Montreal, January 26, 2018
"Plastique armé: À propos des vitraux et fresques lumineuses de Jean-Paul Mousseau," panel Art sacré et avant-garde: De l'art religieux à l'art civique, Expo 67 et les métiers d'art colloquium, Musée des maîtres et artisans du Québec, Montreal, October 22, 2017
"Secular and Sacred Light in the Work of Jean-Paul Mousseau," panel Craft and Public Art, Canadian Craft Biennial conference, OCADU, Toronto, September 16, 2017
"St. Ignatius of Loyola Parish at 100 (1917-2017)," panel Religious Architecture in Canada, Society for the Study of Architecture in Canada conference, Niagara-on-the-Lake, May 26, 2017
"Le Corbusier: Architecture, Art et Politique Modernes," undergraduate course HAR 1425: L'architecture depuis les Lumières (1750-1950) (Prof. Christina Contandriopoulos), Université du Québec à Montréal, April 4, 2017
Panel "Utopia, Territory and Media Cultures / Utopie, Territoire et Cultures Médiatiques," Universities Art Association of Canada conference, Université du Québec à Montréal, October 27-30, 2016 (co-chair with Dr. Christina Contandriopoulos)
"Paul Damaz' Transatlantic Network and the Historiography of Midcentury Modernism," Networked Art Histories, 1960s to the Present Day (in Canada and Elsewhere) conference, Concordia University, October 22, 2016
"Hydro-Quebec and the Cultural Legacies of the 'Quiet Revolution': On Photography and the Restoration of Jean-Paul Mousseau's Lumière et mouvement dans la couleur (1962-2002)," panel What does Photography Preserve? Reification and Ruin in the Photographic Heritage of a Place Called Montreal, Association of Critical Heritage Studies conference, Université du Québec à Montréal, June 7-10, 2016
"The Spirit of the Matter: Materiality and Metamorphosis in the Early Work of Étienne Gaboury," panel Churches: Past, Present, and Future, Society for the Study of Architecture in Canada annual conference, Regina, Saskatchewan, June 1-4, 2016
"The Spirit of the Matter: Materiality and Metamorphosis in the Early Work of Étienne Gaboury," conference The Sacred and the Profane, CUJAH, Concordia University, February 13, 2016
"Colorful Montreal: Modern Architecture, Urban Life, and the 1960s Abstract Murals of Jean-Paul Mousseau," panel Geometric Abstraction, Op, and Kinetic Art in Transnational Perspective, College Art Association annual conference, Washington DC, February 4, 2016
"'La Culture Humaine': Utopianism, Teleology and Synthesis in Jorn's Theories of Art and Architecture of the 1940s," Le Corbusier: What Moves Us? conference, Aarhus School of Architecture, Denmark, 19 November, 2015
"French Modern Architecture and Cold War Diplomacy, circa 1950," panel Reassessing the Cold War in Architecture and Planning, Society of Architectural Historians annual conference, Chicago, April 17, 2015
"Art et mobilier modernes à l'ère industrielle," Montreal Museum of Fine Arts, Montreal, March 2, 2015
"Modern Art and Furniture Design in the First Machine Age," Montreal Museum of Fine Arts, Montreal, March 2, 2015
"Edgar Pillet, Abstract Art and the Postwar Catholic Left: The Mame Factories in Tours," Forum des Professeurs, Université Laval, Québec, January 30, 2015
"Photography and the Documentation of Modernist Church Architecture in Canada," panel The Arc of Photographic Knowledge: From Representation to Experience, Universities Art Association of Canada, OCAD University, Toronto, October 23-26, 2014
"Le Corbusier et la 'synthèse des arts majeurs' comme fiction architecturale," doctoral seminar HAR 8001: La structure fictionnelle de l'œuvre architecturale (Prof. Marc Grignon), Université du Québec à Montréal, October 17, 2014
"Art and the Urban Environment, 1950s-1960s," undergraduate course ARTH 375: Issues in the Montreal Art Milieu (Prof. Johanne Sloan), Concordia University, September 24, 2014
"Modernism and Catholic Architecture, Before and After Vatican II," panel Apocryphal and Apostolic Modernism: Forgotten Connections Between Religion and Architecture, International Society for Intellectual History annual conference, University of Toronto, June 27, 2014
"The Spaces of Religion and Identity," Afternoons at the Institute, Conversation with Dr. Tania Martin, Gail and Stephen A. Jarislowsky Institute for Studies in Canadian Art, Concordia University, February 5, 2014
"Art on the Assembly Line," colloquium BOOM! Growth, Form, Sustainable Bodies 1950-1967, University City London, London (UK), April 5, 2014
"The Postwar Territorialization of Neo-Plasticism," panel Referential Meaning in Abstraction, Universities Art Association of Canada, Banff Centre, October 19, 2013
"Couleur dans l'espace: Art mural, abstraction, biopolitique," followed by a public conversation with Dr. Yve-Alain Bois, Dr. Jaleh Mansoor and Melanie Gilligan, International Colloquium Max et Iris Stern 7, Musée d'Art Contemporain de Montréal, October 6, 2013
"Polychrome Architecture: Groupe Espace and the Renault Industrial Complex in Flins," MIT, Department of Architecture, April 2013
"Jorn, Second Futurism and the Italian Neo-Avant-Garde," Museum Jorn, Silkeborg, Denmark, March 11, 2012
"Rational/Irrational Production: The Bauhaus Idea," MIT, Department of Architecture, April 17, 2011
"Giuseppe Gallizio et la synthèse des arts: Entre utopie technologique et nostalgie préhistorique," Colloquium Les Apories de l'Art Total, Institut National d'Histoire de l'Art, Paris, France, December 15-17, 2011
"Modernist Polychromy in France," MIT, Department of Architecture, March 10, 2009
Research in Progress 2009, MIT, School of Architecture and Planning, March 13-14, 2009 (organizing committee member)
"Primitives of a New Sensibility: Futurism, Popular Art and the European Neo-Avant- Garde," International symposium Futurism at 100: To Measure a Century, Harvard University, April 16-17 2009
"Design, Education and Nueva Vision: On Tomás Maldonado, Max Bill and the 'Bauhaus idea' at Mid-Century," Annual meeting of the American Comparative Literature Association, Harvard University, March 29, 2009
"Hopes in Design: Tomás Maldonado's 'Scientific Operationalism' and the Architectural Culture of the Postwar Years," Study Centre lecture series, Canadian Centre for Architecture, Montreal, July 24, 2009
"The Cavern of Anti-Matter: The Techno-Primitivist Imaginary of the Situationist International, circa 1959," Universities Art Association of Canada, University of Alberta, October 23, 2009
"Signs of Inhabitation: Patio and Pavilion, London, 1956," Graduate colloquium Research in Progress, MIT School of Architecture and Planning, April 5, 2008
"Surrealism and Situationism's Cartographic Practices," Harvard University, Graduate School of Design, April 15, 2008
"Partez sur les Routes: Situationist Nomadism and the Anxious Legacies of Surrealism," Symposium Anxious Mobilities, Yale School of Architecture, April 12, 2008
"Le nomadisme et son rôle critique dans les publications architecturales d'après-guerre," International conference Bohemia Without Borders: The Production and Internationalization of a Literary Attitude, University of Toronto, 12 December, 2008
"Guy Debord and the Art of Cartography," Stanford University, French Department, November 13, 2007
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DRCNet Library | Schaffer Library | Major Studies
Drug Addiction, Crime or Disease?
Interim and Final Reports of the Joint Committee of the American Bar Association and the American Medical Association on Narcotic Drugs.
Some Basic Problems in Drug Addiction and Suggestions for Research*
by MORRIS PLOSCOWE
*The author wishes to express his deep appreciation to Miss Marge Moraitis for her intelligent and painstaking assistance in the preparation of this report. The author also wishes to express his gratitude to the staff of the Russell Sage Foundation for their unfailing cooperation.
I. INTRODUCTION-SEVERITY OF PUNISHMENT AS DETERRENCE TO DRUG ADDICTION
In 1914 Congress, with the passage of the Harrison Act, embarked upon a policy of prohibiting legal access to narcotic drugs on the part of those addicted to such drugs. This prohibitory policy has been strengthened by subsequent legislation. It has been implemented with considerable vigor by the Narcotics Bureau of the Treasury Department and by other state and local enforcement agencies throughout the country, acting under the authority of state and local statutes. Despite this effort, a Senate Committee recently came to the conclusion that, "The United States has more narcotic addicts, both in total numbers and population-wise, than any other country of the Western World." Such a finding, that we have more drug addicts than any other Western country, despite forty years of enforcement of prohibitory laws, raises doubts concerning the wisdom of the prohibitory approach to problems of drug addiction. It would seem to require a re-examination of our narcotics policy. Nevertheless, the two Congressional Committees which recently conducted nationwide inquiries into problems of drug addiction and the drug traffic appeared to be oblivious to doubts concerning the wisdom of the current policy toward narcotic drugs. Both Committees took the basic position that even stronger prohibitions were required if our narcotic addiction problems were to be satisfactorily controlled.
(The first inquiry was conducted by a House Committee under the Chairmanship of Hale Boggs; the second by a Senate Committee under the Chairmanship of Price Daniel. Both Committees made determined efforts to get at the facts surrounding drug addiction and the drug traffic in this country. The printed records of the testimony taken run into several thousand pages.) Three basic concepts run through the recommendations of both Congressional Committees: (1) more stringent narcotic law enforcement; (2) severer penalties for offenders against the narcotic laws; (3) the permanent isolation of incurable drug addicts. The thinking of the Committees is contained in the following extracts from their reports: "Effective control of the vicious narcotic traffic requires not only vigorous enforcement, but also certainty of punishment. Conclusive evidence was presented during your sub-committee's investigation that the imposition of heavier penalties was the strongest deterrent to narcotic addiction and narcotic traffic." ...
"Unless immediate action is taken to prohibit probation or suspension of sentence, it is the sub-committee's considered opinion that the first offender peddler problem will become eventually worse and eventually lead to the large scale recruiting of our youth by the upper echelon of traffickers." ... "Some testimony received by the sub-committee that ... a distinction should be made between the non-addict trafficker and the addict trafficker, with the latter group being dealt with less severely. It is the view of your sub-committee that the addict trafficker is just as vicious a person as the non-addict trafficker." ...
"It is urged ... that the minimum and maximum penalties applicable to conviction for violations of the narcotic laws be increased on both the federal and state levels."3 "Criminal laws and procedures are insufficient to insure the apprehension and punishment of narcotics offenders." ...
"Penalties for narcotic violations are neither commensurate with the seriousness of the crime nor sufficient to remove the profits." ...
"The minimum and maximum penalties be increased for all violations of the narcotics law, with greatly increased penalties for sales to juveniles." ...4 "The Committee has found that whenever and wherever penalties are severe and strictly enforced drug addiction and narcotic trafficking have decreased proportionately." ...5 "That habitual narcotic addicts be committed to 'an indeterminable quarantine type of confinement on a suitable narcotics farm'." ...6
The Report to the President of the Inter-Departmental Committee On Narcotics' also stresses the vital importance of severe punishment as a basic means of controlling drug addiction and the drug problem.
"The Committee has arrived at the conclusion that there is need for a continuation of the policy of punishment of a severe character as a deterrent to narcotic law violations. It therefore recommends an increase of maximum sentences for first as well as subsequent offenses. With respect to the mandatory minimum features of such penalties and prohibitions of suspended sentences or probation, the Committee fully recognizes the objections in principle. It feels however that in order to define the gravity of this class of crime and the assured penalty to follow, these features of the law must be regarded as essential elements of the desired deterrents, although some difference of opinion still exists regarding their application to first offenses of certain types."8 These predilections for stringent law enforcement and severer penalties as answers to the problems of drug addiction reflect the philosophy and the teachings of the Bureau of Narcotics. For years the Bureau has supported the doctrine that if penalties for narcotic drug violations were severe enough and if they could be enforced strictly enough, drug addiction and the drug traffic would largely disappear from the American scene. This approach to problems of narcotics has resulted in spectacular modifications of our narcotic drug laws on both the state and federal level. The 84th Congress passed legislation which provided that whoever "receives, conceals, buys or sells" heroin, etc., shall be punished by 5 to 10 years' imprisonment for a first offense. The giving, selling or furnishing of heroin to a person under 18 years of age was made punishable by sentences of 10 years to life or the death sentence if directed by the jury. Legal provisions permitting suspended sentence and probation for violations of the drug laws were struck from the federal statutes.
The states have followed the lead of the Federal Government in strengthening penalties for violations of the drug laws. In California, unlawful possession of narcotics was formerly punishable by a maximum of 6 years in the State prison. A 1953 amendment increased the maximum to 10 years and to 20 years for a second offense. In Illinois, illicit possession of a narcotic drug used to be punished by a maximum of one year in the County jail. It is now punishable by 2 to 10 years in the penitentiary for a first offense, and 5 years to life for subsequent offenses. In Michigan, unlawful possession of narcotic drugs was punishable by a maximum of 4 years imprisonment. At present, such possession is punishable by a maximum of to years for a first offense, 20 years for a second offense, and 29 to 40 years for a third offense. In Ohio, unlawful possession of drugs was punishable by a maximum of 5 years imprisonment. Today, the penalties for unlawful possession as a first offense are 2 to 15 years, for a second offense, 5 to 20 years, and for a third offense, 10 to 30 years. Stringent law enforcement has its place in any system of controlling narcotic drugs. However, it is by no means the complete answer to American problems of drug addiction. In the first place it is doubtful whether drug addicts can be deterred from using drugs by threats of jail or prison sentences. The belief that fear of punishment is a vital factor in deterring an addict from using drugs rests upon a superficial view of the drug addiction process and the nature of drug addiction. This will be apparent from the discussion of the nature and mechanics of drug addiction (see infra). It is also doubtful whether it will be possible to incarcerate indefinitely relapsing, uncured drug addicts as recommended by the Senate Committee. The Committee urged this step because of the fear that incurable drug addicts carry the contagion of drug addiction to others. In order to prevent such contagion, incurable drug addicts must be permanently incarcerated and permanently isolated from the community. There are thousands of men and women in this country who are confirmed drug addicts and who are incurable by present methods and techniques. If the Senate Committee recommendation is to be acted upon, places of detention will have to be set up for these thousands of men and women, by Congress and state legislatures. There is little likelihood that federal and state legislation will provide new places of detention for large numbers of confirmed drug addicts. Men and women may jam our prisons and penitentiaries for alleged violations of the drug laws. But it is not likely that in the foreseeable future there will be any wholesale round-up of chronic and incurable drug addicts for more or less permanent isolation.
Since all confirmed addicts cannot be incarcerated, permanently, there will always be addicts at liberty to serve as customers for an illicit drug traffic. Even where drug addicts are sentenced to penal or correctional institutions, they eventually come out. They may be off the drug when in the institution but they usually relapse to the use of drugs shortly after they are released from institutional confinement. Severe penalties and strict enforcement may deter or discourage some drug peddlers. But there will always be others attracted by the lure of the large profits to be made in the drug traffic. The very severity of law enforcement tends to increase the price of drugs on the illicit market and the profits to be made therefrom. The lure of profits and the risks of the traffic simply challenge the ingenuity of the underworld peddlers to find new channels of distribution and new customers, so that profits can be maintained despite the risks involved. So long as a non-addict peddler is willing to take the risk of serving as a wholesaler of drugs, he can always find addict pushers or peddlers to handle the retail aspects of the business in return for a supply of the drugs for themselves.* Thus, it is the belief of the author of this report that no matter how severe law enforcement may be, the drug traffic cannot be eliminated under present prohibitory repressive statutes.
*It should be noted that on occasion, law enforcement agencies themselves may act as suppliers of drugs to addicts. The greater the pressure upon law enforcement agencies, the greater the necessity of producing arrests in drug cases. Arrests in drug cases cannot be made without information. Stool pigeons or informers are vital suppliers of information. Nobody is better equipped to provide information concerning violations of the narcotic drug laws than the narcotic addict himself. One pays off the stool pigeon in money, in winking at his illegal activity, and in the case of the addict, sometimes in seeing that he obtains his drugs. Thus it has been alleged that the law enforcement agencies that are engaged in enforcing the narcotic laws may themselves see that drugs are supplied to addicts.
Moreover, even if it were [theoretically] possible to eliminate the drug traffic through strict and uniform enforcement of narcotic laws, this objective is practically unrealistic. In the first place, inefficiency in law enforcement is endemic in this country. The causes are many and varied.
Among such causes are inadequate recruiting and training of police officials, lack of specialized expert direction of police departments, political selection of police chiefs and district attorneys, part time and amateur administration in attorney's offices and courts, political selection of judges, lack of coordination between law enforcement agencies, lack of State supervision of local law enforcement, conflicts between uncoordinated law enforcement agencies, inadequacies in the law of arrest, search and seizure, and other branches of procedural law, etc.
Any particular community can overcome the factors contributing to inefficient law enforcement and stage a concerted drive against drug addicts and drug peddlers. Such a drive can result in imprisoning many individuals. But it will also bring about an exodus of drug addicts and drug peddlers to communities where the "heat" is not on, and the law enforcement is a little more lax and lenient. So long as our law enforcement agencies consist of thousands of independent units, there will always be communities where the enforcement of the drug laws will be viewed with relative indifference and where drug addicts and drug peddlers can wait out a flurry of law enforcement in their own communities.
Strict law enforcement and severe penalties are therefore not the easy answers to problems of drug addiction. We must look elsewhere for a rational drug control program for this country. Any such program must be based on a thorough understanding of the phenomenon that we are seeking to control. Failure to understand the nature of the phenomenon of drug addiction and the practical problems involved in controlling it are responsible for the fact that drug addiction has such serious consequences in this country.
II. THE DEFINITION OF DRUG ADDICTION
An authoritative definition of drug addiction is that propounded by the World Health Organization: "Drug addiction is a state of periodic and chronic intoxication detrimental to the individual and to society, produced by the repeated consumption of a drug (natural or synthetic). Its characteristics include: (1) An overpowering desire or need (compulsion) to continue taking the drug and to obtain it by any means; (2) A tendency to increase the dose; (3) A psychic (psychological) and sometimes a physical dependence on the effects of the drug." This definition of drug addiction includes many drugs which are not within the scope of our study, such as hypnotic and sedative drugs (barbiturates, etc.) alcohol, amphetamine, mescaline (peyote).9 We are interested primarily in the abuse of the opiate drugs and the synthetic-like opiates, such as heroin, morphine, opium, laudanum dilaudid, codeine, demerol, etc.
We are not engaged in anthropological investigation. Accordingly, we shall not study the abuse of mescaline or peyote, which is of little practical importance in this country and is used primarily by Indians in the Southwest for religious rites. We shall, however, pay some attention to cocaine and marihuana, which are included within the above definition, even though the effects of cocaine or marihuana differ from the opiate drugs. Cocaine is sometimes used alone. It is, however, frequently used as a concomitant of opiate addiction to obtain a special kind of thrill (speedball). The use of marihuana frequently precedes experimentation with the more powerful drugs, such as heroin. Neither cocaine nor marihuana, however, produces the characteristic withdrawal syndrome resulting from physical dependence on the opiates.
We shall not deal with the abuse of alcohol, even though there are many more alcoholics in the United States than opiate addicts. Nor shall we deal with the barbiturates or the amphetamine Problem, even though two Congressional Committees were concerned with their abuse. The legal, though regulated, distribution of alcohol, the barbiturates and amphetamine drugs presents a different set of problems from the complete prohibition of the non medical use and sale of the opiate drugs.* There are those who believe that the legal attitude of strict prohibition of the non medical use of opiate drugs is largely responsible for the character of the drug addiction problems in this country.
*We should note, however, that addiction to and intoxication with alcohol or the barbiturates may produce withdrawal symptoms or an abstinence syndrome which is the characteristic of opiate addiction. Addiction to barbiturates, moreover, may be even more dangerous and harmful than addiction to morphine or to an opium derivative. (See for example Nyswander, The Drug Addict As A Patient, Grune & Stratton, p. 126) The prime drug of addiction in this country is heroin. There is a great deal of use and experimentation with heroin, which does not quite fall within the above definition of the World Health Organization. There are many persons, particularly in the slum areas of our large cities, who have the drug habit--who use drugs more or less regularly, but who have not become addicted. While they may have become psychologically dependent upon heroin, they are not physically dependent upon it and deprivation of heroin may not, in these individuals, produce the characteristic withdrawal symptoms which appear whenever an addict to an opiate drug fails to obtain his usual "fix".
This is illustrated by the following comments of the N. Y. U. study, "Heroin Use and Street Gangs": "Heroin addiction is typified by regular use, increased tolerance and physical dependence. An addict uses at least one dose of heroin (or another drug) every day and his intake increases with time. Yet we find that not all of the 94 heroin users studied are seriously dependent upon the drug, even though most of them have been taking heroin for 2-3 years. For one thing, only 43% take one or more doses of heroin daily; only these can be presumed addicted. The rest take the drug two or three times a week or even less often and many of them remain on this non-addictive level, even though some of them inject directly into a vein. Furthermore, only about half of them (54%) Use the drug intravenously. Such casual or weekend use represents a type that is not usually encountered in the medical literature because such users do not show the typical characteristics of addiction, tolerance and physical dependence.
For this group, heroin use may be largely a social activity, the drug being taken as part of the leisure time patterns the boys have adopted." The terrific adulteration of the drugs sold may explain this phenomenon of use and experimentation with heroin without addiction. A Chicago police officer testified before the Senate Committee: "You see now there is something else. When we test the stuff in our crime laboratory, the quality is over 2%, what they are getting is all milk sugar. I remember years ago, back in 1928 and 1929 an addict would get a cap and it would last him 2 days because it was 50%, or 60% pure . . .
"Here is something else that is very important We have these addicts every day in our bureau and very seldom do we get an addict that is sick. They are all needle addicts. It is just a rare case of where we have an addict that is really sick and going through a withdrawal period"l0 A similar phenomenon was noted for Detroit: "I have tested 1,492 addicts.. . I would assume that there are at least half as many addicts unknown to us ... and when we refer to addicts . . . we are covering marihuana smokers, occasional and the regular type, people who are not really addicted. They are occasional users, what we call 'joy poppers' - lightly addicted people. And the drugs in Michigan . . . are terrifically adulterated The average capsule of heroin on the street is almost 11/2 to 2%,. . . In other words, a lot of addicts are taking voluntary cures in this city." . . .11 Were it not for the aforementioned adulteration, our drug addict problem would be much more serious than it is at present. The greed of peddlers of narcotics has saved many from a full blown addiction. Nevertheless, there can be little doubt that much addiction results from the occasional or weekend use of drugs like heroin, even where the drugs are greatly adulterated.
III. THE EXTENT OF DRUG ADDICTION
It is impossible to give any exact estimates of the number of drug addicts in this country. Nor can one with full confidence determine the basic question as to whether drug addiction is increasing or decreasing. The strong social disapproval of the use of opiate drugs and the police pressures against drug users and traffickers necessarily cause drug addicts and drug takers to conceal themselves from strange or unfriendly eyes. They do not come into the open to be counted. Any statistics with respect to the extent of drug addiction must, therefore, usually be based on apprehended addicts or apprehended users of drugs. If we knew what proportion of drug addicts or users are arrested every year, we would have a reasonable basis for estimating the extent of drug addiction in this country. Unfortunately, we do not know the proportion that arrested drug addicts or drug users bear to the total addict and user population. Moreover, increases or decreases in the number of arrests are just as likely to reflect increases or decreases in police activity rather than an increase or decrease in drug use or drug addiction. A further complication is the fact that a person arrested as a drug user may not necessarily be an addict, even though he may be so classified by the police. Thus, any statements with respect to the extent of drug addiction and its changes from year to year must be viewed with a considerable degree of reserve and caution.
In 1924, Dr. Lawrence Kolb and A. G. DuMetz12 conducted a survey of the extent of narcotic addiction for the Public Health Service. They came to the conclusion, on the basis of an analysis of surveys of narcotics use, reports on narcotics clinics, statistics on narcotics imported into this country, interviews with physicians and other data that the maximum number of addicts in the U. S. was 150,000. However, Kolb and DuMez believed that the figure of 110,000 addicts for the country in 1924 was more "nearly correct." They also believed that the number of addicts had decreased steadily since l900. Before this decrease set in there may have been 264,000 addicts in this country. The careful analysis of Kolb and DuMez was criticized by Terry and Pellens, who believed that this estimate of 110,000 addicts for the country was too low: "We cannot agree that the ultimate estimate of 110,000 is warranted. While we on the one hand deplore sensational exaggerations, on the other hand we recognize the danger of basing maximal estimates on selected data."13 Until the second World War, the reports of the Bureau of Narcotics were full of statements concerning the reduction in drug addiction. For example, the 1935 Report states: "This recent survey shows that the total number of non medical addicts in the U. S. has decreased to the extent that there is now less than one addict known to the authorities in every thousand of the population"14 A similar comment is found in the 1937 Report: "From the present study it is evident that addiction has decreased to the extent that there are now less than two addicts known to the authorities in every 10,000 of the population."l5 Shortly after the second World War, the use of narcotic drugs appeared to have spread with epidemic force in the slum areas of our large cities, particularly among minority groups of the population. Negroes and Puerto Ricans were especially involved in this increasing use of narcotic drugs, particularly heroin. We do not have any specific statistics on the Puerto Ricans, but the fact that large numbers of Negroes were arrested in recent years for violations of the narcotics laws is apparent from the following figures. In Chicago in 1954, there was a total of 7,639 narcotics arrests; 6,601 of these were Negro arrests, 752 were White arrests and 286 were classified as "other races." In Detroit in the year 1955, Of a total of 1,812 arrests, for violations of the narcotics laws, ,593 were classified as Negroes; 184 were classified as White; 12 were classified as "Yellow"; and 23 were classified as Mexican. The Bureau of Narcotics made a survey of addicts in the United States in the year 1953-1954. Its report noted a total of 6,957 addicts in the Illinois, Indiana and Wisconsin areas; of this number 6,057 were Negro; 916 were listed as Caucasian and 2 were listed as Oriental. In the New York, New Jersey area, of a total of 7,931 addicts, 4,740 were listed as Negro; 3,037 were listed as White, and 160 were listed as Orientals. It should be noted that the ratio of male to female among the persons arrested for violation of the drug laws is approximately 5 to 1.* *For example, in Chicago in the year of 1954, of the 7,699 arrests for violation of the narcotic laws 6,182 were male and 1,457 were female. In New York in 1956, of a total of 6,098 arrests, 5,082 were male and 1,061 were female. In Detroit in the year 1955~ there were 1,812 arrests of which 1, 454 were male and 318 were female.
The most disturbing feature of the increasing resort to the use of narcotic drugs in the post-war period was the apparently increasing use of heroin by adolescents and "teenagers." As the Boggs Committee put it: "In 1948 an upsurge in addiction and an outbreak of teen-age use of narcotic drugs occurred. By 1950, narcotic addiction approached grave proportions in certain metropolitan areas of the country."16 A similar conclusion was reached by the New York Attorney General's Survey, 1952: "The investigation revealed with disturbing clarity that (a) Narcotic use and addiction ...has increased in tremendous fashion since World War II and particularly in the last two years.
(b) The disease has spread with alarming rapidity through the ranks of our adolescent society."17 A Chicago study of drug addiction made in 1952 determined that there were 5,310 individual drug addicts in the City of Chicago, "slightly more than 1/l0th of 1% of Chicago's 1950 population." Upwards of 90% of these persons acquired records as drug addicts during the five years 1947-1952. Males made up 83% of this drug user and addict population. Moreover, more than 4/5ths (84%) were non-white.
Public concern with the apparent increase in the use of drugs after World War II led to a spectacular rise in arrests for violations of the narcotic drug laws. Arrests for violations of narcotic laws appear in the table on pages 32-33.
It is apparent from the table below that arrests for narcotics violations of all types in New York were 712 in 1946 as against 5,965 in 1956; in Chicago the comparable figures were 424 and 9,011; in Los Angeles, 1,166 and 5,091; in Detroit, 339 and 2,646.
There appears to be little doubt that drugs like heroin were readily available in many of the slum areas of our larger cities in the post-war years, despite considerable pressure from law enforcement. This availability of heroin, together with a social and cultural climate in the areas which favored drug use, undoubtedly encouraged many teen-agers and many young adults to try heroin. This experimentation with drugs like heroin must have inevitably increased the drug addict population in this country. Just how much of an increase has resulted from the increased availability and the increased experimentation with heroin, it is difficult to say.
Despite the difficulties in determining the exact number of addicts in the country, the need for data on the extent of the problem has brought about a considerable acceptance of the Bureau of Narcotics' statement that there are almost 60,000 addicts in the country.l8 This estimate was accepted as reasonably accurate in the report to the President of the Inter-Departmental Committee on Narcotics.l9 "Many and varied estimates have been made as to the number of persons in the U. S. addicted to narcotic drugs. The Committee regards the current estimate of the Bureau of Narcotics as the most accurate available. This estimate of 60,000 is based on the records of its own agents and cooperating state and municipal authorities." "While there are far less drug addicts in the Nation today than there were before the Harrison Narcotics Act was passed and before the Federal Bureau of Narcotics was created, the figure of 60,000 addicts today is far more than the number reported by other western nations." There are indications that this estimate of 60,000 narcotics addicts for this country is too low. For example, a recent California report to the Attorney General on Narcotic Addiction stated: "What is the extent of addiction in California? No one knows with any degree of accuracy. It' s known that we have in our State medical files some 32,000 persons who are legally using narcotics medicinally, although a certain percentage of them may be using it illegally because they are going to several different doctors concurrently The state criminal files reflect that there are approximately 10,000 additional illegal traffickers or users of narcotics in California. It is believed that 10,000 represents approximately one-half of the total illegal addicts in this State. Our estimated total, therefore, would be 32,000 medical or legal users and probably 20,000 illegal, a total of 52,000 persons.-If it is true that there are at least 20,000 illegal users of opiate drugs in California alone, then it is questionable that there are only 40,000 more addicts in the rest of the United States. It appears to be obvious that the exact number of drug addicts in this country is unknown. However, it is apparent that whatever the extent of drug addiction in this country may be, as Terry and Pellens pointed out almost 30 years ago: "The surveys and estimates indicate sufficiently the existence of a major medical-social problem.... As a matter of fact it is not necessary to know the exact number of users or even the minimal extent to realize that there are a large number in the country and that the problem is serious."21
IV. THE NATURE AND CHARACTERISTICS OF DRUG ADDICTION
The law has largely acted on the premise, which is supported by some of the earlier writers, that drug addiction was largely a vice, which an effort of the will could conquer. Severe penalties were necessary to compel the will to make the effort to conquer the vice. Medical writers, on the other hand, have taken the view that drug addiction was a disease and that the drug addict was a sick person. For example, Emest S. Bishop wrote many years ago: "The fundamental truth which applies to all cases of narcotic drug addiction is this--whatever may have been the circumstances of the primary administration of narcotic drugs, or whatever the physical, ethical or personal status of the person addicted... Continued administration of the drug creates within the body of the person to whom the drug is administered a physical disease process. A demonstration of material cause and effect in obvious symptomatology, in physical suffering, and in nerve strain and exhaustion, unless there is applied to that person in sufficient amounts the drug of his addiction. Every addict is sick of a disease condition... insufficiently recognized and insufficiently studied."22 Or as Dr. William G. Somerville put it: "Drug addiction is a disease, a pathological condition just as much as the psychoneuroses of any of the various toxic states."23 If the physiological and psychological need for the drug inherent in drug addiction is a disease, then it will be apparent from our discussion of relapse that it is a disease which is largely incurable by present methods and techniques. The course of the disease can only be controlled by the continued administration of the drug of addiction or some similar drug.
There are, however, many who do not regard drug addiction as a disease entity. Maurer and Vogel for example have pointed out that: "All the research done on drug addiction within the past two generations indicates that addiction is not a disease, rather a symptom of personality difficulties, which if they did not lead to drug addiction would lead to difficulties of other types."24
Maurer and Vogel would say that drug addicts are sick, unbalanced, disturbed, abnormal individuals. Unfortunately as we shall see in our discussion of the personality types of drug addicts (infra), it is easier to attach a psycho-pathological label to the drug addict than to explain how or why he became addicted or why he continues his addiction.
Many with similar psychological difficulties do not become addicted to drugs. Some become alcoholics rather than drug addicts. The mere designation of a drug addict as a sick, unbalanced, disturbed or abnormal individual conceals more than it reveals. This is clearly indicated by the comments of Dr. H. Isbell: "Addiction is a complex process in which pharmacological, psychological, socio-economic and legal factors play interdependent roles. It is viewed in two ways: (1) as a distinct disease; (2) as a symptom of an underlying personality disorder.
Both views can be supported by evidence established so far. Studies have shown that the majority of addicts have personality disorders which antedate drug use. Also, addicts use many drugs and change from one to another especially when their favorite drug is difficult to obtain. Drugs used by addicts also have diverse actions; they not only use drugs that cause 'depression' but also stimulants. The only common denominator among drugs abused by addicts seems to be that they all are compounds which exert powerful effects on the central nervous system. These facts suggest that there is nothing specific about the drugs that addicts take and, therefore, addiction is nothing more than a symptom of the personality disorder. This view cannot be accepted without reservation. The theories of personality that are used to explain addiction are the same theories that are used to explain neurosis, psychoses, character disorders, etc. Since it is known that many persons with personality characteristics similar to those of addicts never abuse drugs, it is apparent that factors other than personality must be operating. Furthermore, under conditions of equal drug exposure, one individual may choose opiates, another alcohol. This implies some sort of specificity in the choice of the drug of addiction."25
Whether addiction to drugs be viewed as a disease or as a symptom of personality disorder it usually involves the three characteristically related phenomena, noted in the definition of the World Health Organization, namely, (1) tolerance, (2) physical dependence and (3) emotional dependence. These phenomena have been described by Isbell and White as follows: "By tolerance is meant a decreasing effect on repetition of the same dose of a drug. This particular characteristic is very marked in addiction to the opiates and synthetic analgesics.
Patients with well developed tolerance have injected as much as 5 gm. (78 gr.) of morphine sulfate intravenously in less than twenty-four hours without developing significant toxic symptoms. Tolerance to the various effects of morphine and related drugs develops, however, at different rates and in different degrees. For example, tolerance to the toxic, sedative, emetic, analgesic and respiratory-depressant effects of morphine develops very rapidly and becomes marked, whereas tolerance to the miotic effects and to the spasmogenic effects on gastro-intestinal smooth muscle, if developed at all, is never complete. "Physical dependence refers to the development of an altered physiologic state which requires continued administration of a drug to prevent the appearance of a characteristic illness, termed an 'abstinence syndrome.' Physical dependence is an extremely important characteristic of addiction to morphine and similar drugs, since it leads to continuity of intoxication with resultant subservience of all phases of the addict's life to the one aim of obtaining and maintaining a constant supply of the drug.
"Emotional dependence is defined as a substitution of the use of the drug for other types of adaptive behavior. In other words, use of the drug becomes the answer to all of life's problems. Instead of taking constructive action about his difficulties, regardless of their type, the addict seeks refuge in his drug."26 It is simpler to describe the phenomenon of tolerance to and physical dependence upon opiate drugs, as Isbell and White have done, than to explain the exact mechanics of their action upon the human organism. A great deal of research has been done on both phenomena in the attempt to find such explanations. Much of value has been uncovered by this research. Nevertheless, the fundamental effects of narcotic drugs upon the human system are still obscure. As Maurer and Vogel have noted: "The action of the opiate drugs and their synthetic equivalents upon human beings is still imperfectly understood. This fact is striking when we consider that opium has been used generally for thousands of years, and that no single medicine is more useful or more generally used by the physician than the modern opium derivatives and opium-like synthetics .
Certain fundamental questions are still unanswered; many peripheral or incidental problems remain to be solved. With some of the basic reactions of opiates upon the human physiology and neurology still obscure, it is not surprising that the nature of addiction to drugs of the opiate series ...should still be a controversial matter. The nature of narcotic addiction is still not yet fully understood."27* *"An examination of Nathan A. Eddy's classic chapter on tolerance and addiction, which summarizes the studies up to 1940, indicates the soundness of the above observation. Dr. Eddy wrote as follows: "The last word has not been said by any means on the mechanism of tolerance and addiction to morphine. Evidence is accumulating that morphine is handled differently in the tolerant animal. In addition, the phasic character of morphine action (excitation on the one hand, and apparent depression on the other hand), seems to be intimately concerned in the tolerance development and addiction, whether it is a question of the time relations of the two effects or of an alteration of the biologic substrate. The disturbed autonomic and hormone balance in addiction and withdrawal needs further careful thorough study."28 Commenting on tolerance, Dr. Eddy stated: "The evidence as a whole points to a change in the cells of the nervous system as the important factor, but the exact nature of the change and its fundamental mechanism are still unknown." In the same volume as Dr. Eddy's study, Margaret Sumwalt analyzed the studies which attempted to answer the question of what the organism does to morphine. She pointed out that man disposed of between 1/3 to 1/10 of his intake of morphine in his urine and feces. Sweat and saliva carry trivial amounts. Milk perhaps more.
"The remaining 65 to 85% is got rid of rather promptly by unknown chemical processes ... The chemistry of morphine metabolism is unknown."29 The present status of research on tolerance and dependence is clearly summarized by Isbell, in his authoritative article, "Trends Research On Opiate Addiction." "Most physiological research in addiction has been concerned with tolerance and physical dependence. Two major hypotheses have been developed. The first is that of Tatum, Seevers and Collins and is based on the dual character of the effects of morphine on the central nervous system. Morphine has both excitant and depressant effects within the central nervous system. In animals, the excitant effects appear to outlast the depressant effects. Therefore, as morphine is repeatedly administered, the excitant effects constantly increase. This excess excitation requires more and more of the drug in order to obtain the excitant effects, which are still present, and unopposed by depressant effects; hence, abstinence symptoms occur. Recently this hypothesis has been expanded. It is conceived that morphine exerts its depressant effects by attachment to receptors within the cells. The drug at receptor sites on the cell surface is in equilibrium with drugs in body fluids, is easily detached and swiftly metabolized. Morphine hypothetically diffuses into and out of cells quite slowly, so that degradation of drug attached at this site is slow. Since drugs on the cell surface are more rapidly dissipated than are drugs within the cells, the excitant effects outlast the depressant. Unfortunately this concept is completely untestable with present technics, and there are also objections to the 'depressant-excitant' formulation. In the lower animals, codeine is a more excitant drug than is morphine. One would therefore, assume that symptoms of abstinence from codeine would be more severe than symptoms of abstinence from morphine; actually, the reverse is the case. Also, in the chronic spinal dog, morphine markedly enhances-'stimulates'--the ipsilateral extensor thrust reflex.
"If this is regarded as an excitant action of morphine, the reflex should be even more hyperactive following withdrawal of the drug. Actually, the extensor thrust reflex disappears during abstinence. "The other theory of tolerance and physical dependence was first formulated by Joel and Ettinger and has been further developed by Himmelsbach. In this formulation, it is hypothesized that the administration of morphine calls into play homeostatic responses which oppose the effects--chiefly the depressant effects--of the morphine. These homeostatic responses are gradually strengthened by repeated administration of the drug and, therefore, more drug is required to induce the initial degree of effect. When morphine is removed, the enhanced homeostatic responses still remain and are released from the brake imposed on them by the continued presence of morphine in the organism, thus accounting for symptoms of abstinence. This formulation seems to fit the facts. For example, morphine constricts the pupils and depresses respiratory rate in minute volume. When morphine is withdrawn, the pupils dilate and hyperpnea ensues. Many other examples could be stated. In fact, the development of counter responses which oppose the main effects of drugs may be a general pharmacological phenomenon... It is apparent, however, that the homeostatic theory is more descriptive than explanatory. It tells us what happens but not really how. We have very little knowledge of the mechanisms of the homeostatic responses that supposedly oppose the actions of morphine. Due to the researches of Wikler, we can describe them in neurophysiological terms. In the non-addicted chronic spinal dog, morphine enhances the extensor reflexes, and has little effect on the knee jerk. It can be inferred from these facts that morphine depresses reflexes which are mediated through multi- neuron arcs (the flexor) and has little effect on reflex arcs that are mediated through a single synapse (the patellar). As tolerance develops, multineuron arcs become hyperexcitable and, on withdrawal of the morphine, excitability in these arcs is unmasked and accounts for withdrawal symptoms. Similar phenomena have been observed in spinal man. It is also known that chronic decorticated dogs develop tolerance and symptoms of abstinence on withdrawal of morphine. One may infer therefore, that the cerebral cortex is not necessary for the development of physical dependence. Although these facts give us some concept of the neurophysiological changes associated with addiction we know little about the nature of the changes at levels between the cord and the cortex. Technical difficulties in studying the activity of these levels in chronically intoxicated animals have not yet been solved, but the current trend in neurophysiological research in addiction consists partly of attempts to develop such methods.
"Biochemical studies have shown that tolerance and physical dependence are not related to changes in excretion or distribution of morphine within the organism, and are also not due to any known degradation product of morphine. Eisenman and Fraser have shown that maintained addiction causes a decrease in the urinary excretion of 17-ketosteroids, 17-hydroxycorticoids, and of pituitary gonadotropin although the adrenal and the gonads remain responsive to ACTH and chronic gonadotropin. On withdrawal of morphine, excretion of 17-ketosteroids is increased, serum corticoid levels rise and eosinophile counts decrease. These results indicate depression of the adrenals, the gonads or both by morphine during maintained addiction probably because of depression of the pituitary through unknown central mechanisms. During abstinence, there is a marked adrenal discharge. These findings are very important since they explain the decreased libido and sexual activity present during opiate addiction. The psychiatric significance of this effect requires no comment. "Efforts to elucidate the biochemical mechanisms underlying dependence and tolerance have not yet proved fruitful. Though the technical difficulties are great, studies of this sort are now being pushed. Obviously, we cannot explain all the phenomena of addiction by physiological data alone. Physiological data, though very useful in understanding symptoms and in managing them, contribute to total understanding of the problem only insofar as correlation of physiological mechanisms with drug induced changes in behavior are possible. "30 Whatever the mechanics of tolerance and dependence, if the addict has reached the stage of physical dependence upon a drug, he must obtain the drug regularly if he is to avoid the distressing experience of the withdrawal syndrome. How much of the drug he will use, will depend in the first instance on how much he can get. If the drug is available, despite the mechanism of tolerance, each addict eventually tends to find a level or a physical plateau in the use of the drugs. He tends to stop increasing the dosage at a point where he feels right physically and psychologically or where the drug will give him the euphoria that he is looking for. But whatever his level or plateau, the addict must obtain enough drugs to ward off the withdrawal symptoms which inevitably follow any failure to obtain the drug.
The withdrawal syndrome or sickness is no mere figment of the addict's imagination, but an illness which constitutes one of the most stereotyped syndromes in clinical medicine.
Wikler has demonstrated that physical dependence is a real physiological entity and is not psychic in origin. He has distinguished the purposive from the non purposive features of the withdrawal syndrome as follows: "The train of events which follow abrupt cessation of morphine in the tolerant addict varies within limits in different individuals, and is related to previous dosage, duration of addiction and the degree of tolerance which had been developed. However, for any given dose level and period of addiction, the morphine abstinence syndrome is remarkably reproducible in any given individual. The significance of the morphine abstinence syndrome to the individual is also highly individualized and is partly determined by particular situations. Thus, it may serve as a means for expressing hostility, expiating guilt and even justifying relapse. When observed in a hospital situation after abrupt and complete withdrawal of the drug, the fully developed morphine and abstinence syndrome is characterized by the following changes, which may be separated into two groups: "
(a) NON-PURPOSIVE. These consist of yawning, lachrymation, rhinorrhea, mydriasis, gooseflesh (piloerection) tremors, muscle twitches (particularly in the lower extremities), restlessness, hot and cold flashes, nausea, vomiting, diarrhea, anorexia, weight loss, ejaculations in men and orgasms in women, elevation of body temperature, cardiac and respiratory rates and blood pressure, leucocytosis, hemoconcentration, elevation of blood sugar and a precipitous drop in circulating eosinophile count. In addition, the subject often exhibits a rather typical facies suggestive of an individual with an acute febrile infectious disease. Often the patient 'curls up' in the lateral recumbent position with a blanket drawn over his head, preferably on a hard cold surface, such as the floor. Curiously, alpha activity in the electroencephalogram if present in prewithdrawal records, continues during the abstinence period in spite of manifest 'anxiety', although an increase in slow activity may be observed during abstinence following periods of addiction to other morphine-like drugs.
(b) PURPOSIVE. This group of morphine abstinence changes refers to such behavior as appears to be directed toward obtaining the drug. It is expressed verbally in terms of 'craving' and demanding drugs. Also, the subject may exhibit patterns of behavior which are highly individualized-threatening suicide, or violence, assuming bizarre postures and exaggerating his distress in dramatic ways.
"The non purposive abstinence changes reach peak intensity about 48-72 hours after abrupt withdrawal of morphine and subside gradually over a period of about one week, although some physiological variables do not return to control levels for as long as six months, while the 'purposive' abstinence changes may continue indefinitely."31 According to Lindesmith, the necessity of avoiding withdrawal distress provides the basic explanation of the nature and the processes of drug addiction. "Addiction to opiates," he points out: "... is determined by the individual's reaction to the withdrawal symptoms which occur when the drug's effects are beginning to wear off, rather than upon positive euphoric effects often erroneously attributed to its continued use. More specifically, the complex of attitudes which constitute addiction is built up in the process of conscious use of the drug to alleviate or avoid withdrawal distress. This theory, though simple in form, has considerable explanatory value, and offers a means of accounting for varied and paradoxical aspects of the habit, such as the addict's claim that he feels normal under the drug's influence, as well as his tendency to increase the dose to a point where its use becomes much more unpleasant and burdensome than it need be. The hypothesis presented makes intelligible the constant preoccupation of the addict with the drug, and explains how the unpleasant and unwelcome appellation 'dope fiend' is forced upon him."32 "Addiction occurs only when opiates are used, to alleviate withdrawal distress, after this distress has been properly understood or interpreted, that is to say, after it has been represented to the individual in terms of the linguistic symbols and cultural patterns which have grown up around the opiate habit. If the individual fails to conceive of his distress as withdrawal distress brought about by the absence of opiates he cannot become addicted, but if he does, addiction is quickly and permanently established through further use of the drug. All of the evidence unequivocally supports this conclusion. "This theory furnishes a simple but effective explanation, not only of the manner in which addiction becomes established, but also of the essential features of addiction behavior, those features which are found in addiction in all parts of the world and which are common to all cases."33 Whatever the truth of Lindesmith's theory, there can be little doubt that once a user of drugs realizes that he has become addicted his entire life becomes centered around the search for the drug. He must obtain the drug in order to be comfortable and to be able to function. He may also want the drug in order to obtain an ever elusive euphoria. The drive and compulsion for the drug is such that family, friends, property, profession may all be sacrificed to feed it. The compulsion to take the drug cannot be stopped by a threat of jail or prison sentences.
V. THE EFFECTS OF DRUG ADDICTION
The compulsion to take the drug is one of the components of the "drug fiend" myth which has been propagated by irresponsible journalism and irresponsible law enforcement. Another vital aspect to this myth is the misconception concerning what narcotic drugs do to human beings and the kind of behavior that such drugs foster. It is alleged that drugs like heroin and morphine have devastating effects on the persons who use them. Murder on the installment plan is a phrase frequently used to describe heroin addiction. It is charged, moreover, that the use of narcotic drugs leads to the commission of all kinds of serious crime, particularly crimes of violence. The printed proceedings of the House and the Senate Committees are full of such charges concerning the use of narcotic drugs. The pernicious effects of narcotic drugs on human beings were the justification for the severe penalties that were recommended as a means of dealing with the drug traffic and problems of drug addiction.
Unfortunately, the facts concerning the effects of such drugs as morphine and heroin on human beings differ considerably from these misconceptions. The facts tend to indicate that the use of drugs like heroin and morphine is consistent both with a reasonable state of health and with a reasonable degree of efficiency on the part of the individual user.
Over thirty years ago, Dr. Kolb pointed out that there was no evidence that the use of a narcotic drug made one less efficient. It was lack of the drug and the constant pre-occupation with obtaining it which led to a loss of efficiency on the part of the individual. Thus, the drug addict is not, by virtue of the fact that he takes a drug, necessarily a parasite, who is unable to function in any productive capacity. Nor is he necessarily a degenerate human being who, because he takes drugs is sliding rapidly towards the grave. This is apparent from the comments of Dr. Nathan B. Eddy, who analyzed the world literature on morphine in 1940 and who observed: "Given an addict who is receiving morphine in amount and at intervals adequate to keep the withdrawal symptoms completely in abeyance, the deviations from normal physiological behavior are minor for the most part within the range of normal variations."34 Professor W. G. Karr, a biochemist of the University of Pennsylvania, wrote in a similar vein: "The addict under his normal tolerance of morphine is medically a well man. Careful studies of all known medical tests for pathological variation indicated, with a few minor exceptions, that the addict is a well individual when receiving satisfying quantities of a drug. He responds to work in the normal manner. He is as agreeable a patient, even more so, than other hospital cases. When he is abruptly withdrawn from the drug he is most decidedly a sick individual."35 The feeling of normality and well being which an addict has, when he is using the drug was observable by Dr. Marie Nyswander, when she tested a group of patients at Lexington by means of Rorschach tests, both before and after using morphine. She writes: "With the administration of morphine a striking change is observed in the Rorschach-a change which corresponds to the addict's subjective feeling that he has attained normalcy. The responses begin to fall into more normal categories; the constriction is lessened, and movement response and fantasy appear."36 Dr. Lawrence Kolb noted that many prominent people who led socially useful lives have been addicted to narcotic drugs, yet were able to function effectively in their business and profession.37 As a matter of fact some who at one time were gutter alcoholics have improved themselves and their social functioning by shifting to morphine. This notion that the use of an opiate drug may actually improve the functioning of a particular individual is clearly presented by Wikler and Rasor: "On further interrogation, the majority of such individuals explain that in ordinary life situations opiates (usually heroin or morphine) reduce appetite, pain and erotic urges of all sorts, heterosexual, homosexual or autoerotic. In addition, intravenous injection of these agents produces a transient 'thrill' akin to sexual orgasm, except that it is centered in the abdomen. After these effects have developed a sense of gratification or satisfaction is achieved and they feel more 'at ease' and free to do what they 'want to do.' In some situations they may 'want' to doze peacefully and enjoy daydreams of wealth, power or social prestige. In other situations they may want to socialize, and they feel more comfortable to the presence of women. Furthermore, some opiate users state that these agents do not impair, others state that they actually improve, their ability to do useful work and that under the influence of opiates, they are less aggressive and 'keep out of trouble.' "It is difficult, of course, to verify statements such as these relative to the contrasting effects of drugs in actual life situations. However, observations made under experimental conditions are in substantial agreement with them. Thus, as long as adequate amounts of opiates are administered, aggressive, antisocial behavior is practically never observed, personal hygiene is maintained, assigned responsibilities are discharged satisfactorily, psychologic tests of performance reveal little or no impairment, and the sensorium remains quite clear, while anxiety associated with anticipation of pain is reduced."38 Opiate drugs like morphine do have certain effects upon the individual. The use of the drug causes a loss of appetite. Thus there may be a failure to maintain a proper intake of foods. This may affect health. However, as Maurer and Vogel point out: "... it has not been possible to demonstrate that opiate drugs in themselves actually destroy tissue or are directly the cause of tissue deterioration."39 The existence of emaciation and anemia in drug addicts, "...may be due to the unhygienic and impoverished life of the addict rather than to the direct effect of the drug."40 The drug addict simply does not eat enough, because on the one hand the drug he uses reduces appetite and on the other hand, costs so much that he has no money left over for food. Another effect of drug addiction is in reducing the urge to sex.
"The reproductive system generally tends to become inactive. ... In both males and females, opiates have a general tendency to reduce or obliterate sexual desire, although there may be individual exceptions to this."41 This lowering of sexual desire resulting from narcotic drugs would cause one to be skeptical of the claim that heroin and morphine incite to violent sexual crimes. Drug addiction may result in moral and character deterioration. But here the legal and social policy concerning drug addiction may be at fault rather than the use of the drug. An addict can only obtain the drug from underworld sources. He is cut off from any legitimate supply. The underworld will supply him at a price. The price is high and most addicts do not have the kind of money necessary to feed a habit. The obvious alternative is to raise the money by theft or if the addict is a woman, by prostitution.
Once the addict is started on a criminal or prostitutional career, his moral deterioration becomes almost inevitable. But the question may well be raised whether it is the drug or the short sighted social policy which utterly fails to take into account the desperate need of the addict for his drugs which causes the breakdown in character.
As Lindesmith notes: "Addicts escape most of the alleged degenerate results of the drug if they are sufficiently well-to-do, and many addicts suffer serious 'character deterioration' only after the narcotic agents catch up with them. In other countries ... addicts do not suffer evil effects ... forced upon the American users. They do not steal, lie, engage in prostitution, or become derelicts to the extent that our addicts do. If the toxic effect of the drug on the central nervous system promotes degeneration, or if addiction is a bio-chemical affair,... why do not similar conditions result in other countries or in our own upper class."42
VI. PSYCHIATRIC AND PSYCHOLOGICAL FACTORS IN DRUG ADDICTION
As we have seen, medical men have tended to regard drug addiction either as a disease or as a symptom of a disturbed or abnormal personality that requires drugs in order to be able to cope with life's problems. Drug addiction may be considered a disease if the focus of attention is the pathologic process in the human organism created by addiction. A healthy human organism does not need morphine or heroin to ward off withdrawal symptoms. The diseased body of an addict, however, requires its daily dosage of drug for the addict to be comfortable. On the other hand, drug addiction is not an accidental process. Individual factors are at work in the determination of who will and who will not become addicted, even in those areas of our cities, where the incidence of drug use is high. There are individuals who are exposed to drug use, who through an effort of will, strength of character or force of personality reject all contact with narcotic drugs. These individuals will never become drug addicts. There are also some persons who although once addicted, through will power, or force of personality and character manage to stay off drugs. It is obvious that character and personality factors are at work in the selection of addicts and in determining which addicts will relapse to the use of drugs, once they have been taken off drugs.
Who, then, are the individuals who succumb to drug addiction? What factors of personality, of character, of psychological organization or disorganization distinguish the drug addict from the non-addict? Can the phenomenon of drug addiction be explained by the disciplines of psychology and psychiatry? Even the most casual reading of the psychiatric and psychological literature on drug addiction indicates that psychology and psychiatry are still far from satisfactory explanations as to why specific individuals take to drugs, and why others who may be similarly exposed do not take to drugs to resolve their personal problems. Over and over again, one reads that drug addiction is an expression of personality disturbance or maladjustment. An individual takes drugs to overcome the shortcomings of personality which make it difficult for him to cope with the world in which he lives. He needs drugs to enable him to deal with the anxieties and tensions arising from familial conflicts, sexual difficulties and the necessity of growing up and taking one's place in an adult society. A vast majority of drug addict patients, write Vogel, Isbell and Chapman, "... are fundamentally emotionally immature children like persons who have never made a proper adaptation to the problems of living."43 Not all drug addicts, however, fit into a single psychiatric classification or diagnosis. The personality disorders of drug addicts, "... run the gamut of the standard psychiatric nomenclature from the simple anxiety states to the major psychoses."44 Thus, all kinds of people, both normal and abnormal, become drug addicts. This can be seen from the summary by Vogel, Isbell and Chapman of the pioneering work on the classification of drug addicts done by Kolb* and Felix.** *In 1925 Dr. Lawrence Kolb made his pioneer study of 230 drug addicts recruited from prisons, a municipal hospital, a clinic "and other addicts in good social standing in various parts of the country." This fell into the following general classifications: 1. People of normal nervous constitution necessarily or accidentally addicted through medication in course of illness. This group constituted 14% of the total; 9% being necessary addicts and 5% were accidental cases.
2. Care-free individuals, devoted to pleasure, seeking new excitements and sensations, and usually having some ill-defined instability of personality that often expresses itself in mild infractions of social customs. This group constituted 38% of the total.
3. Cases with definite neuroses not falling into classes 2, 4 or 5. This group constituted 13.5% of the total.
4. Habitual criminals, always psychopathic. This group constituted l3% of the total.
5. Inebriates. (Only those who had a definite history of periodic drinking with sprees were considered for this study.) This group constituted 21.5% of the total.
In 1937. Dr. Kolb and Dr. Ossenfort attempted to refine the classification in this earlier study, based upon an analysis of the first 1,750 admissions at Lexington. The addicts were classified as follows: 1. Normal individuals accidentally addicted. This group includes persons of normal nervous constitution accidentally or necessarily addicted through medication in the course of illness.
2. Psychopathic Diathesis. This group includes individuals who show psychopathic dispositions or tendencies characterized by behavior resulting from misinterpretations of environmental settings or situations, but not a well-crystalized personality defect.
3. Psychoneurosis. This group includes individuals suffering with ordinary types of psychoneurosis.
4. Psychopathic personality without psychosis. This group is composed of persons who show deviation of personality usually expressed as constitutional psychopathic inferiority, psychopathic personality or constitutional states, where volitional and emotional control are gravely distorted from the normal.
5. Inebriate. This group includes individuals in whom alcoholic indulgence, either periodic or more or less continuous, played an important role as a precipitating factor in their addiction. They apparently have a so-called inebriate impulse.
6. Drug addiction associated with psychosis. This group includes addicts suffering with frank psychosis, organic, toxic or functional. In 1939. D'. Michael Pescor made an analysis of the personalities of 1036 addicts at Lexington, Kentucky, based upon the aforementioned psychiatric classifications. Dr. Pescor came to the conclusion that the 1036 addicts studied by him fell into the following:
1. Normal individuals, accidentally addicted--3.8%
2. Psychopathic Diathesis--54.5%
3. Psychoneurosis (ordinary type)—6.3%
4. Inebriate -Inebriate Impulse—21.9%
5. Psychopathic Personality Without Psychosis--11.7%
6. Drug Addiction Associated With Psychosis—11.7%
7. Psychosis Caused by Opiates--None.
**Dr. Robert H. Felix, in 1939, attempted to further define three categories in the above Kolb, Ossenfort, Pescor classification, namely the psychoneurotic, the psychopathic personality and the psychopathic diathesis.
His difficulties with these elusive categories are apparent from the following extracts of his article: "The concept of the psychopathic-diathesis group may not be as dear as that of the other two, but probably can best be described as a state in which, because of some ill-defined instability of personality, no better than a border-line adjustment is made. The individual is not fundamentally anti-social and, with some artificial assistance, can make an acceptable adjustment. The most striking characteristic of this group is the fact that, as a whole, they were adjusting marginally before they became acquainted with narcotics. After their first few experiences with the drug, they felt an exhilaration and a sense of relief comparable to the solution of a difficult problem or the shaking off of a heavy responsibility. Many of them also felt an increase in efficiency which, in some cases, at least, appears to have been an actual improvement. Having once found this new world of greater happiness and efficiency, they attempted to regain it and to live therein for all time.
"This phenomenon is not so prominent in the other two groups. The psychoneurotic takes his drugs to relieve himself of whatever type of symptom he may have. The psychopath uses narcotics rather as an aggressive behavior reaction--that is, he feels a desire to be more important or prominent among his associates. He wishes to excel in deeds of daring, to be more clever than his fellows, or to stand out as an object of admiration. Under narcotics, he feels that he has more nearly accomplished these ends. As Kolb has put it, his use of drugs is 'comparable to the compensation of little men who endeavor to lift themselves to greatness.' In other cases, he uses this means to gain an experience of pleasure over and beyond the requirements for comfortable living. He is a hedonist. What he desires to do, he does for the pure pleasure to be derived from it. He is morally defective and hence does not consider social or ethical standards a check upon its activity. The only restraint he recognizes is painful or physical in nature. The patient with a psychopathic predisposition, however, takes his opium as a medicine which he believes-- sometimes with good reason-helps him to make a more satisfactory adjustment to life as he finds it, and without which he feels inadequate to meet many of life's problems. "The same fundamental drive, then, is present in all cases--namely, the desire to derive from life more pleasure and satisfaction, which, after all is a striving present in all mankind. The differentiations made above are probably of theoretical rather than principal importance, but it is felt that they help to clarify the problem." "The kinds of personality disorders which underlie drug addiction have been well described by Kolb and Felix, who list four general personality types.
"The first group is made up of normal persons accidentally addicted. It consists of patients who in the course of an illness have received drugs over an extended period of time and, following relief of their ailments, have continued the use of drugs. These persons are frequently termed 'accidental' or 'medical' addicts. Such persons are regarded by some authors as constituting a special group of addicts who are different from those persons who began the use of drugs as a result of association with persons who were already addicted. In our experience, all 'medical' addicts have some fundamental emotional problem which causes them to continue the use of drugs beyond the period of medical need. There is, then, no basic difference between 'medical' and 'non-medical' addicts except in the mode of the original contact with drugs. In persons with stable personalities, social pressure, conscience and well balanced emotional makeup negate the pleasure produced by drugs sufficiently to prevent their continued use.
"The second group consists of persons with all kinds of psychoneurotic disorders who, as Felix said, take drugs to relieve whatever symptoms they may have. The manifestation of the neurosis may be anxiety, an obsession or compulsion or any of the great group of psychosomatic disorders.
"The third and largest group consists of psychopathic persons, who ordinarily become addicted through contact and association with persons already addicted. They are generally emotionally undeveloped aggressive hostile persons who take drugs merely for pleasure arising from the unconscious relief of inner tension, as shown by this statement of an addict: I was always getting into trouble before I got on drugs-never could seem to get comfortable; I had to go somewhere and do something all the time. I was always in trouble with the law. Some fellows told me about drugs and how good they made you feel, and I tried them. From then on, I was content, as long as I had my drugs--I didn't care to do anything, but to sit around, talk to my friends occasionally, listen to the radio, and only be concerned with the problem of getting money for drugs. This I usually did by picking pockets or other such petty stuff.
"The fourth and smallest group is characterized by drug addiction with psychosis. The persons in this group, many of whom have borderline mental illness and sometimes frank mental illness, are seemingly able to make a better adjustment while taking drugs. Sometimes it is difficult to establish the diagnosis and not until drugs are withheld, does the psychosis become apparent.
"There is a category of patients not included in the aforementioned groups. Kolb originally listed these as patients with psychopathic diathesis. While it is true that some of these exhibit much of the overt behavior pattern of psychopathic persons, when studied carefully, they usually fall into a milder behavior or character disorder group, which has characteristics of both the psychoneurotic and the psychopathic groups. Included are persons with severe dependency problems, withdrawn schizoid types, emotionally immature adults, as well as those suffering with the milder degrees of maladjustment and inadaptiveness to the complications of living. Felix stated that most of the persons falling into this group were making a marginal adjustment to life before becoming acquainted with narcotics. After their first few experiences with narcotics, they felt an exhilaration and a sense of relief comparable to the solution of a difficult problem or the shaking off of a heavy responsibility. Many of them also felt an increase in efficiency which, in some cases, appeared to have been actual improvement. "In general, persons who never have been able to make a satisfactory adjustment to life, whose adaptive patterns of behavior have been inadequate, frequently find in morphine, much as the tired business man finds in the preprandial cocktail, a means of return to 'normal.' This is a false situation which may be recognized by the tired business man but is not recognized by the drug addict. Our studies indicate that patients who have made a marginal degree of emotional adjustment to life, and then have begun to use drugs, lose some of their normal adaptive patterns of adjustment. This regression in personality represents the greatest danger of drug addiction."45 A consideration of the aforementioned classifications makes it obvious that none of the classifications provide specific explanations for drug addiction. Large numbers of individuals fitting into the categories of psychopathic diathesis, psychopathic personality or psychoneurosis, never take drugs as a means of resolving their personality difficulties or emotional problems. One begins to see the wisdom of Dr. Wikler's observation: "The attractiveness of morphine for certain individuals seems to be related to some of its remarkable pharmacologic properties, namely, its effectiveness in reducing such anxiety as is associated with fear of pain, anger and sexual urges, without seriously impairing the sensorium or the effectiveness of internalized controls on behavior. The intensity of this attraction is enhanced greatly for such individuals as have been unable to gratify these needs by other means, be they 'normal,' neurotic or psychopathic..." "... the degree of attractiveness of morphine is related to 'personality structure' but not necessarily to 'neurosis' or psychopathy as such ..."46 This notion that the use of opiates is a highly individualized process and is not necessarily related to mental pathology is also expressed by Gerard and Kornetsky in their study on "Adolescent Opiate Addiction." They diagnosed 30 narcotic addicts and 30 adolescent non-addicts of roughly similar background and status.
The writers conclude as follows : "... The psychologic and psychiatric data of the study indicated that the addicts exceeded the controls in personality malfunction to a statistically significant and clinically impressive extent. These findings support the hypothesis that youths living in urban areas where illicit opiate use is widespread do not become addicted independently of psychiatric pathology. The data also indicate that the converse need not be true; as youths who exhibit personality malfunction similar to that of addicts need not become addicted. As the writers pointed out previously, becoming an opiate addict is a highly individualized process which can be understood only in the context of the individual's personality structure, past life situation and present interactions with the significant figures of his familial and peer groups."47 The addict as Winick points out: "... is responding to personality problems of great complexity. The drug addict is a person with certain personality characteristics who happens to have selected this way of coping with his problems for a variety of reasons, of which he is usually unaware. Not the least of these reasons is his access to a social group in which drug use was both practised and valued. He takes one drug rather than another because it provides satisfaction for him. Other people with exactly the same kind of personality substratum never become addicts and select other means of expression for their basic conflicts."48
VII. SOCIAL FACTORS IN ADDICTION
Psychopaths, psychopathic diathetics, psychoneurotics, emotionally disturbed persons, etc. would not use narcotics as a solution for their personal problems, unless such drugs were available. If such individuals happen to live in the slum areas of our cities, this offers no problem, for one of the facts of life in connection with narcotics is that illicit drugs can be purchased most readily in the slum neighborhoods of our large cities. Police officers from many different cities testified before the Congressional Committees that most drug arrests and violations of the drug laws occurred in certain limited areas of their cities, usually the areas of greatest social disorganization. In these neighborhoods live the most economically deprived groups of our population; the racial and religious minorities, and the recent immigrants into the cities. These are the areas of poor and squalid housing, of overcrowding, of a shifting disorganized family life. They are the areas with the largest number of relief cases; the highest rates of juvenile delinquency, adolescent and adult crime. They are also areas with high rates of mental disturbance and psychological bnormality.49 It is these disorganized slum neighborhoods, whether they exist in New York, Chicago, Los Angeles, Detroit or Washington, D. C., which develop a special cultural climate which is favorable to drug use and experimentation particularly by juveniles and adolescents. Two major studies in recent years (The N. Y. U. Study and the Chicago Study) were concerned with an analysis of the factors in such neighborhoods which were conducive to a high degree of drug use. The Chicago Study pointed out that the social environment for young males in these areas comes to be dominated by a "street corner society" and that such societies flourish in communities where the traditional influences and controls over youth tend to be weak and uncertain. The central feature of this society or culture is the support that it gives to behavior which is inconsistent with the norms of conventional society and often openly hostile to many of its expectations. This orientation on the part of the street boys is expressed in a variety of ways, particularly by delinquency and crime "and in the search for and exploitation of kicks." Success in the exploitation of "kicks" entails willingness to experiment with new drugs whose effects and properties are not precisely known to the user. However, the street corner groups appear to vary in the degree to which they court the double interest in delinquency and "kicks." There are, as an N. Y. U. study pointed out, street gangs with a high degree of narcotic use, a low degree of narcotic use and gangs which do not permit their members to use narcotics at all. Nevertheless, as the Chicago study notes, the introduction of heroin to street groups in Chicago, "...was facilitated by an established and pre-existing interest in the use of stimulants and intoxicants and by the tendency to experiment freely with new drugs."50 Heroin was "pushed" vigorously by the frenetic search of the street corner boy for newer, stranger and more status giving intoxicants, and after heroin use had been defined as "desirable and valuable by intimate associates whose views are meaningful to the potential user." The New York University studies came to similar conclusions : "We have learned that the social pattern of using narcotics is highly concentrated in the most deprived areas of the city; that it is associated with the type of delinquency that produces ready cash; that the pattern of using drugs spreads within the peer-group and apparently is meaningful in the context of the social reality in which the boys live; that the users (and nonusing delinquents) live in a special defiant and escapist subculture side by side with the other subculture of 'squares' who want to life themselves out of their depriving environment."51 Obviously not all the boys who participate in the activities of street corner society or of street gangs wind up as habitual or professional criminals or drug addicts. The wider, conventional society exercises its pressures for conformity even against the members of the delinquent and deviant sub-culture. As the boys grow older, the youthful preoccupations with delinquency, kicks, hell-raising, gang fights, etc., give way for many of the boys to a concern about the future, a steady girl, a job, a home, etc. The problem of who will and who will not become a professional criminal or a drug addict is dependent upon the personality of the individual boy. This is noted by both the Chicago and the N. Y. U. studies: "... Most likely to become extreme delinquents or drug users are those who by virtue of their personal histories are least responsive to the expectations of conventional society. Thus, the problem of differences between those who do and those who do not become drug addicts in the world of the street boy may be regarded as a problem of the difference in life history among individuals, with each life history constituting a unique equation of forces."52
"But as the group grows older, two things happen. Sport, hell-raising and gang fights become 'kid stuff' and are given up. In the normal course of events, the youthful preoccupations are replaced by more individual concerns about work, future, a 'steady' girl and the like. If most of the gang members are sufficiently healthy to face these new personal needs and societal demands and engage in the new activities appropriate for their age, the availability of drugs will not attract their interest. But for those gang members who are too disturbed emotionally to face the future as adults, the passing of adolescent hell-raising leaves emptiness, boredom, apathy and restless anxiety. In a gang where there are many such disturbed members, experimentation with drugs for 'kicks' will soon lead to frequent and, later, habitual use; cliques of users will grow quickly. Enmeshed in the patterns of activities revolving around the purchase, sale and use of drugs and the delinquent efforts to get money to meet the exorbitant cost of heroin, the young users can comfortably forget about girls, careers, status and recognition in the society at large. Their sexual drive is diminished, they maintain a sense of belonging in the limited world of the addict, they remain children forever. They may give up all sense of personal responsibility for their lives and conveniently project the blame for their shiftless existence on the 'habit'."53 It is obvious that in the production of a drug addict, just as in the production of a delinquent or a criminal, there is an interaction of personality and environmental factors. But there is also a shaping of personality by environmental factors, cultural attitudes, and interpersonal relationships. Nowhere is this more true than in the intimate confines of family life. The N. Y. U. group compared the family background of 30 White, Negro, and Puerto Rican families with a non addict boy and 30 such families with a boy who was an addict. All the families lived in a high drug use neighborhood. Almost all of the 30 addicts came from families where there was a disturbed relationship between the parents as evidenced by separation, divorce, overt hostility or lack of warmth and mutual interest.
The addicts experienced much more frequently than the controls, "... cool or hostile parent figures, weak parent-child relationships, lack of clarity as to the way in which disciplinary policies were established and vague or inconsistent parental standards for the boy."54 As a result of these findings, this study came to the conclusion: " ... that the pathologic personality characteristics of the juvenile heroin addict are consistent outgrowths of the disturbed pattern of family relationships to which he has been exposed."55
VIII. DRUG ADDICTION IN RELATION TO CRIME
One of the compelling reasons why more rational methods of dealing with drug addicts must be devised is the close relationship between drug addiction and crime. The compulsion for the drug makes every drug addict a law violator and a criminal. Mere possession of a narcotic drug which the addict must have to ward off withdrawal distress is a violation of the narcotic laws. Thus, every drug addict is subject to arrest by the police, and as we have seen, the arrests of addicts and of narcotic law violators have gone up by leaps and bounds. Addicts guilty of no other crime than illegal possession of narcotics are filling the jails, prisons and penitentiaries of the country.
However, this is only a part of the distressing picture of the relationship between narcotic addiction and criminality. For most narcotic addicts, predatory crime (larceny, shoplifting, sneak thievery, burglary, embezzlement, robbery, etc.), is a necessary way of life. This was clearly recognized by the law enforcement officials who appeared before the Congressional Committees and gave testimony concerning the close relationship between property crime and drug addiction in their communities. These officials were convinced that property crimes could be reduced materially if all drug addicts could be incarcerated.
The New York University and the Chicago studies on drug addiction support the notion that drug addiction necessarily leads to predatory crime as a way of life. For example, Chein and Rosenfeld make the following comments based on their studies of juvenile addicts: "Drug use leads to a criminal way of life. The illegality of purchase and possession of opiates and similar drugs makes a drug user a delinquent ipso facto. The high cost of heroin, the drug generally used by juvenile users, also forces specific delinquency against property for cash returns. The average addicted youngster spends almost forty dollars a week on drugs, often as much as seventy dollars. He is too young and unskilled to be able to support his habit by his earnings. The connection between drug use and delinquency for profit has been established beyond any doubt."56 A Chicago study comes to a similar conclusion: "...Almost without exception addicts resort to theft to obtain money for the purchase of the drugs. The compulsion of the addiction itself coupled with the astronomically high cost of heroin leads the addict inescapably to crime. For the addict there is very simply no alternative."57 There has been considerable debate as to whether the criminality of the addict preceded or is merely a consequence of the drug addiction. Studies like those of Pescor can be cited for the proposition that most narcotic addicts became delinquents and criminals after the onset of their addiction. Pescor found in 1943. that of the 1,036 patients at Lexington, studied by him, 75.3% had no history of delinquency prior to addiction.58 Anslinger, however, has the always taken the view that the drug addict was usually a criminal first before becoming addicted.59 The answer to the question of whether the addict was a delinquent or criminal prior to addiction largely depends upon the particular groups of addicts studied. For example, Kolb60, in 1928, studied a group of 119 so called "medical addicts", persons who became addicted to drugs as a result of the prescription of narcotics for ailments other than addiction. Kolb found that of these 119 addicts, 90 had never previously been arrested. However, the studies conducted in New York and Chicago present a different picture. These studies of drug addiction were conducted in areas with high rates of delinquency and crime. They were also concerned with youthful and adolescent offenders. The conclusion from the Chicago and New York studies is inescapable that "delinquency both preceded and followed addiction to heroin."61 "Persons who became users," stated the Chicago report, "were found to have engaged in delinquency in a group habitual form either prior to their use of drugs or simultaneously with their developing interest in drugs. There was little evidence of a consistent sequence from drug use without delinquency to drug use with delinquency."62 Nevertheless, even in the delinquency areas of our large cities, there are persons who become addicted to drugs without a prior career of delinquency and crime. After addiction, however, they will usually turn to delinquency and crime "often after overcoming severe psychological conflict occasioned by their repugnance to theft."63 Moreover, the addict who had previously been a delinquent loses all chance of shaking off habits of delinquency and crime as he grows older. Not all non addicted delinquents and adolescent offenders living in the delinquency areas of our large cities grow up to be habitual and professional criminals. Many abandon their delinquent and criminal pursuits when they reach early adulthood. They find jobs, marry and settle down to productive lives. But if the delinquent or adolescent offender adds narcotic addiction to his patterns of behavior, ". ..All possible future retreat from a delinquent mode of life is cut off regardless of whatever later impulses they may have to reject a criminal career in favor of a conventional one. They are constrained by their unremitting need and the high cost of heroin to continue in crime. This interpretation supports the conclusion that drug addiction results in a large and permanent increase in the volume of crime."64 Thus, the realities of the relationship between narcotic addiction and crime appear to be much more somber than the romantic myth, "that hold-up men, murderers, rapists and other violent criminals take drugs to give them courage or stamina to go through with acts which they might not commit when not drugged."65 Dr. Kolb has labeled this notion an "absurd fallacy." The crimes committed by opiate addicts are generally of a parasitic, predatory nonviolent character. Drug addicts may, on occasion, commit violent crimes. This is hardly surprising since so many are classified as psychopaths. A psychopath tends towards serious criminality with or without drug addiction. Generally, however, the use of opiate drugs (whatever may be the case with marihuana and cocaine) tends to discourage violent crime. As Maurer and Vogel point out: "The sense of well-being and satisfaction with the world are so strong that, coupled with the depressant action of the drug, the individual is unlikely to commit aggressive or violent crime after he is addicted, even though he habitually or professionally did so previous to addiction. In the words of Kolb, 'Both heroin and morphine in large doses change drunken fighting psychopaths into sober, cowardly, non-aggressive idlers ...' "...To date, there has been no evidence collected to show that any significant percentage of opiate addicts commit violent crimes either professionally or casually while under the influence of these drugs ... the reduction or elimination of sexual desire tends to remove the opiate addict from the category of psychopathic sex offenders, even though he might have a tendency to commit sex crimes when not addicted ..."66 Since opiate drugs do not act as a stimulant for the commission of violent crime, should not confirmed addicts have a means of obtaining such drugs legally, so that they will not have to engage in crime in order to raise the money necessary for their needs? This basic question goes to the heart of our present policy in dealing with drugs addiction.
IX. METHODS OF TREATMENT OF DRUG ADDICTION
1. The Doctor and the Drug Addict In Western Europe, and in England, the treatment of drug addiction and drug addicts is primarily a matter for the physician. (See Appendix B, appended hereto.) Physicians may prescribe drugs to addicts either in the attempt to cure them of their addiction or to keep them in a state of comfort so that they can function without fear of the dreaded withdrawal symptoms. In this country, on the other hand, the physician has largely been deprived of an appropriate role in the treatment of drug addicts. There are many who believe that the physician must be substituted for the jailer in dealing with drug addicts, before fundamental progress can be made in controlling addiction.
This requires a review of the development of the laws in this country which has to a considerable degree resulted in the exclusion of doctors from the field of drug addiction.
Prior to 1915 physicians were permitted to treat addicts as they saw fit, and opiates were available to the general public. But Congress, pressured by the public's concern over the growing number of addicts in the country, enacted the Harrison Narcotic Law67 which was designed to control the domestic manufacture, sale and distribution of narcotic drugs. The Act requires importers and manufacturers to purchase and affix stamps to all opiates and cocaine packages. In addition, importers, manufacturers, wholesalers, retailers, and doctors must register and pay a graduated tax for the use of narcotics. Narcotics can only be legally transferred under the Act by registered persons through the use of special order forms. The Act does not seek to interfere with the legitimate practices of medicine, nor with the medical treatment of addicts, for it provides that: "Nothing contained in this chapter shall apply to the dispensing or distribution of any of the drugs ... to a patient by a physician, dentist, or veterinary surgeon registered ... in the course of his professional practice only." If an addict is a patient of a doctor, narcotic drugs can be dispensed to him, if it is done in the course of the "professional practice" of the doctor. The Harrison Act did not seek to regulate the practice of medicine nor impinge upon a doctor's relationship to his patient. Nevertheless, despite the exception in favor of physicians many doctors were subjected to criminal prosecution because of the charge that their treatment of and prescription for drug addicts was not legitimate "professional practice" within the meaning of the Act. Targets of the initial prosecution were doctors who had many addict patients for whom they prescribed large amounts of drugs. Such doctors were charged with the illegal sale of narcotics in violation of the Act.
In the first Supreme Court case under the Act (United States v. Doremus),68 the defendant, a doctor, had dispensed 500 one-sixth grain tablets of heroin to addicts, and was convicted of a violation of the Act. He contended that the Act was unconstitutional because it sought the control of the distribution of narcotic drugs through the device of taxing- such drugs. It was contended that Congress could not constitutionally control the distribution of narcotic drugs. However, the Supreme Court, in a 5-4 decision, upheld the constitutionality of the Act, stating in the course of its opinion: "... the Act may not be declared unconstitutional because its effects may be to accomplish another purpose as well as the raising of revenue. If the legislation is within the taxing authority of Congress, that is sufficient to sustain it."69 This case did not directly pass upon the question of what a doctor may or may not do in the treatment of a drug addict. In the case of Webb v. United States70 however, which came before the Court on the same day, the Narcotics Bureau was able to persuade the Supreme Court to adopt its views concerning the treatment of drug addicts by physicians. Dr. Webb had been indicted and convicted for selling at 50 cents apiece, over 1,000 prescriptions for narcotic drugs, indiscriminately to anyone, and occasionally using fictitious names on the prescriptions. It was obvious that the defendant was a mere prescription peddler, who was neither treating patients nor practising medicine. His conviction, therefore, should have been affirmed since his activity in relation to drugs was not covered by the exception in the Act in favor of physicians. The Narcotics Bureau, however, apparently wanted more from the Supreme Court than the affirmance of a conviction. It wanted an authoritative expression of opinion from the Court as to what was and what was not, the legitimate practice of medicine in dealing with narcotic addicts. It therefore had a question certified to the Court for its answer, which went far beyond the facts of this case and which seems to impinge upon the domain of medical practice. The certified question reads as follows: "If a practicing and registered physician issues an order for morphine to an habitual user thereof, the order not being issued by him in the course of professional treatment in the attempted cure of the habit, but being issued for the purpose of providing the user with morphine sufficient to Keep him comfortable by maintaining his customary use, is such order a physician's prescription under exception (b) of section 2 (of the Harrison Act)?"71 A majority of the Supreme Court (5-4) answered this question as follows: "to call such an order for the use of morphine a physician's prescription would be so plain a perversion of meaning that no discussion is required."72 Under this decision, it became possible for the Narcotics Bureau to warn doctors against prescribing drugs to addicts for the purpose of avoiding withdrawal distress or keeping the addicts comfortable. The position of the Narcotic Bureau was strengthened by another flagrant case the following year,'" in which the physician had prescribed 8 to 16 drams of morphine at a time, indiscriminately to anyone, for $1 a dram. In dismissing the appeal from the conviction the Supreme Court observed: "Manifestly the phrase 'to a patient' and 'in the course of his professional practice only' are intended to confine the immunity of a registered physician, in dispensing the narcotic drugs mentioned in the Act, strictly within the appropriate bounds of a physician's professional practice, and not to extend it to include a sale to a dealer or a distribution intended to cater to the appetite or satisfy the craving of one addicted to the use of the drug. A 'prescription' issued for either of the latter purposes protects neither the physician who issues it nor the dealer who knowingly accepts and fills it."74 In the Behrman case75 two years later, the Supreme Court began to realize that the earlier cases may have trespassed upon the domain of medical practice in attempting to dictate what a doctor could or could not do in relation to a drug addict. The Court dismissed the demurrer to the indictment of Dr. Behrman, who had prescribed 150 grains of heroin, 360 grains of morphine and 910 grains of cocaine to an addict, at one time. But it observed in the course of its opinion that: "It may be admitted that to prescribe a single dose or even a number of doses, may not bring a physician within the penalties of the Act."76 It should be noted that the indictment in the aforementioned case did not allege bad faith on the part of the physician-defendant. Nevertheless the Court held that such wholesale prescribing of drugs to an addict regardless of good or bad faith of the doctor was a violation of the Act.
The aforementioned precedents enabled the Narcotics Bureau to prosecute many physicians, and unquestionably resulted in most doctors leaving the narcotic addict severely alone. However, a few physicians continued to treat and prescribe drugs for addicts. One such man was Dr. C. O.
Linder,77 who was charged with the unlawful sale to an addict "stoolie" of one tablet of morphine and three tablets of cocaine for self-administration in divided doses over a period of time. The Linder indictment, like the Behrman indictment, did not question the physician's good faith. But the Court sustained the demurrer to this indictment and observed: "Obviously, direct control of medical practice in the states is beyond the power of the federal government. Incidental regulation of such practice by Congress through a taxing act cannot extend to matters plainly inappropriate and unnecessary to reasonable enforcement of a revenue measure. The enactment under consideration levies a tax, upheld by the court ... and may regulate medical practice in the states only so far as reasonably appropriate for or merely incidental to its enforcement. It says nothing of 'addicts' and does not undertake to prescribe methods for their medical treatment. They are diseased and proper subjects for such treatment, and we cannot possibly conclude that a physician acted improperly or unwisely or for other than medical purposes solely because he has dispensed to one of them in the ordinary course and in good faith four small tablets of morphine or cocaine for relief of conditions incident to addiction. What constitutes bona fide medical practice must be determined upon consideration of evidence and attending circumstances. Mere pretense of such practice, of course, cannot legalize forbidden sales, or otherwise nullify valid provisions of the statute, or defeat such regulations as may be fairly appropriate to its enforcement within the proper limitations of a revenue measure."77 The Court refused to adopt the interpretation placed upon the Webb Case (supra) that no prescription to an addict which sought to keep him comfortable or ward off withdrawal distress could be justified under the Act: "The question (in the Webb Case) specified no definite quantity of drugs, nor the time intended for their use. The narrated facts show, plainly enough, that physician and druggist conspired to sell large quantities of morphine to addicts under the guise of issuing and filling orders. The so-called prescriptions were issued without consideration of individual cases and for the quantities of the drugs which applicants desired for the continuation of customary use. The answer thus given must not be construed as forbidding every prescription for drugs, irrespective of quantity, when designed temporarily to alleviate an addict's pains, although it may have been issued in good faith and without design to defeat the revenues."78 In commenting on the Behrman Case (supra), the Court stated: "This opinion related to definitely alleged facts and must be so understood. The enormous quantity of drugs ordered, considered in connection with the recipient's character, without explanation, seemed enough to show prohibited sales and to exclude the idea of bona fide professional action in the ordinary course. The opinion cannot be accepted as authority for holding that a physician who acts bona fide and according to fair medical standards, may never give an addict moderate amounts of drugs for self-administration in order to relieve conditions incident to addiction. Enforcement of the tax demands no such drastic rule, and if the Act had such scope it would certainly encounter grave constitutional difficulties."80* *In a subsequent case, Nigro v. United States, 276 U. S. 332 (1928), the case involved a layman who was accused of selling one ounce of morphine not in pursuance of a written order form, and he argued that the act only applied to professionals. The Court said: "In interpreting the act, we must assume that it is a taxing measure, for otherwise, it would be no law at all. If it is a mere act for the purpose of regulating and restraining the purchase of the opiate and other drugs, it is beyond the power of Congress, and must be regarded as invalid ...
Everything in the construction of section 2 must be regarded as directed toward the collection of the taxes imposed in section 1 and the prevention of evasion by persons subject to the tax. If the words cannot be read as reasonably, serving such purposes, section 2 cannot be supported."81 Thus, the Linder Case lays down the rule that a doctor acting in good faith and guided by proper standards of medical practice may give an addict moderate amounts of drugs "in order to relieve conditions incident to addiction." The Harrison Act does not regulate how much a physician may or may not prescribe to an addict nor delimit either the quantity or frequency with which a physician may prescribe for an addict in his practice. This is illustrated by the case of Boyd v. United States,82 where a physician had been convicted of unlawful sale of 30 to 98 grams of morphine, by means of prescriptions, issued to two known, confirmed addicts.
The trial court had charged the jury that: "... it was not admissible to issue prescriptions to a known addict for an amount of morphine for a greater number of doses than was sufficient for the necessity of any particular administration of it." The Supreme Court pointed out that this statement was: ... ambiguous and might be regarded as meaning that it never is admissible for a physician, in treating an addict, to give him a prescription for a greater quantity than is reasonably appropriate for a single dose or administration. So understood, the statement would be plainly in conflict with what this court said in the Linder case."83 The rule of the Linder Case was also applied by the Circuit Court of Appeals of the 10th Circuit in the case of Strader v. United States.84 There the trial judge had charged the jury that a prescription for morphine to an addict is a violation of the law, and that it may not be given merely for the purpose of relieving pain incident to addiction. The court in reversing the conviction stated: "We think the court incorrectly stated the law and unduly circumscribed the testimony. The statute does not prescribe the diseases for which morphine may be supplied. Regulation 85 (of the Narcotics Bureau) issued under its provisions forbids the giving of a prescription to an addict or habitual user of narcotics, not in the course of professional treatment, but for the purpose of providing him with a sufficient quantity to keep him comfortable by maintaining his customary use. Neither the statute nor the regulation precludes a physician from giving an addict a moderate amount of drugs in order to relieve a condition incident to addiction, if the physician acts in good faith and in accord with fair medical standards."85 Under these decisions, the exception in the Harrison Act in favor of physicians still has vitality. The Act does not purport to regulate medical practice, nor determine what drugs a physician may prescribe to an addict; nor indicate the quantity or frequency of the prescriptions.
The responsibility for prescribing rests upon the physician in charge of any given case, and the courts have been clear in holding that if he acts in good faith and prescribes a narcotic drug in the course of his professional practice, he is entitled to the benefit of the exception under the Act." As the court put it in the case of Bush v. United States:86 "A physician may give an addict moderate amounts of drugs for self administration, if he does so in good faith and according to fair medical standards." In the Strader Case (supra) the court ruled: "In Mitchell v. United States, 3 Fed 516 (6th Cir.,1925) the physician was indicted for dispensing drugs unlawfully. The defense argued that no offense was alleged because the indictment did not allege that the disposition of narcotics was not made to a patient in the course of the physician's professional practice. The court held that the indictment was defective because it did not negative the exceptions specified in the act.
* The Bush case involved a physician who was indicted for violating the act by issuing prescriptions for morphine, varying from 10o to 16 grains, to known addicts who pretended to be suffering with painful diseases. The indictment further charged that the quantities prescribed were enough to last more than one day. The defense relied upon the rule laid down in the Linder case that a physician is within his rights when he prescribes morphine to an habitual user as he sees fit.
"A physician issuing morphine prescriptions in good faith to a federal narcotics agent, whom he believes to be a bona fide patient, for the purpose of curing a disease or relieving suffering would not be guilty of violating the Harrison Act." But while the present law permits a physician to treat an addict in good faith and in the course of his professional practice, doctors are still reluctant to treat or prescribe for addict patients. A physician who treats and/or prescribes drugs for an addict patient in good faith according to medical standards will be protected from a conviction. But his good faith and adherence to medical standards can only be determined after a trial. The issue of whether the doctor acted in good faith and adhered to proper medical standards must be decided by a judge or a jury. If the judge or jury decide against the physician, the latter may be sent to prison or deprived of his license to practice medicine.
The physician has no way of knowing before he attempts to treat, and/or prescribe drugs to an addict, whether his activities will be condemned or condoned. He does not have any criteria or standards to guide him in dealing with drug addicts, since what constitutes bona fide medical practice and good faith depends upon the facts and circumstances of each case. (See Bush Case supra.) The physician's dilemma in treating drug addicts is illustrated by the case of Teter v. United States,"87 where the physician dispensed nine one-quarter grain tablets of morphine over a two week period to an addict who was used as an informer by the Narcotics Bureau. The defense argued that the indictment was insufficient because of the small amount of drugs dispensed. In sustaining the indictment, the court said: "While the quantity was not large, nevertheless there was evidence tending to indicate that the sales were not in good faith from a physician's standpoint, and were for no other purpose than to enable this addict to further indulge her unfortunate propensities ... Notwithstanding two other physicians testified that in the treatment of addicts, it was not improper to give them doses such as appear to have been given to the complaining witness, we are satisfied that under all the circumstances, it was for the jury to say whether or not these sales of drugs to the complaining witness were in good faith, or were solely for the purpose of pandering to the habit of a drug addict, and selling the drug."88* * In Hawkins v. United States, 90 F. 2nd 551 (5th Cir., 1987), the physician was convicted for prescribing 15 grains of morphine to three known addicts, who he claimed were suffering from serious pulmonary conditions. The government had one witness who testified that he examined the addicts and found that none of them were suffering from such a condition. After the trial, one of these addicts died from a pulmonary condition. The defense argued that the amount prescribed was small and therefore it comes within the Linder Case because it was not large enough to put it within the power of the addict to sell part of the drug and thereby violate the act. The court, in sustaining the conviction, held: ". . .15 grains of morphine was enough to present a question of fact as to the good faith of the doctor to be decided by the jury." In United States v. Brandenburg, 155 F. 2nd 110 (3rd Cir., 1946) the physician was convicted for prescribing drugs to a narcotics officer who was introduced to him by an addict "stoolie," as a "tubercular brother-in-law." The physician also prescribed drugs for the addict who claimed that he had serious gall bladder trouble and that his doctor who was out of town prescribed morphine. The defendant was treating this doctor's patients while he was away so that when the addict walked into the defendant's office, there was no reason to suspect him. Subsequently, the addict and the agent received additional prescriptions. In sustaining the conviction, the court said: "The frequency of the issuing of the prescriptions and the quantities prescribed were factors which made the question of good faith one for the jury." In the recent case of McBride v. United States, 2229 F. 2nd 249 (5th Cir., 1955) an " osteopath was convicted for falsifying his records and illegally dispensing codeine. The facts show that the Chief of Police of Houston, Texas, who was a personal friend of the defendant, had suffered a back injury and he was in constant pain. The defendant had given him a shot of codeine on one occasion in order to relieve a severe pain and when the Chief discovered that his pain could be relieved and that he was able to work, he asked the defendant to give him more; the defendant was reluctant to administer more of the drug because he feared the narcotics regulations. The Chief assured him that he would be within his rights if he dispensed the drug while treating him, and he brought in the regulations so that the defendant would be assured. Each time he prescribed the drug, he gave him an osteopathic treatment. The Chief had asked him not to use his name on the records which he kept because he feared losing his job if he was discovered; thus, he convinced him to use the name of an incurable cancer patient, again showing him the regulations, which he interpreted as being complied with so long as the dispensing of the drug was recorded. In sustaining. the conviction, the court observed that none of the expert witnesses (4 were called by the government, one being an osteopath and two were called by the defense) would say that the dispensing of codeine in quantities given by the defendant was standard medical practice, and it further said: "Evidence of the failure to follow standard medical practice shows a lack of good faith. So also as bearing on good faith is evidence of appellant's unorthodox attitude toward narcotics and addiction." The case of United States v. Anthonys89 crystalizes the problem which the physician faces in dealing with drug addicts. There, the defendant was approached by the City of Los Angeles to take over the treatment of addicts who were former patients at the City's narcotics clinic, before it was closed. These patients were confirmed addicts who were thoroughly examined by the defendant before he prescribed drugs for them, At the trial, three doctors testified that such prescription was good professional practice. Two other doctors testified that the ambulatory treatment of drug addicts was not proper medical practice under any circumstances. In acquitting the defendant, the court said:
"Good faith must be determined on the basis of evidence and expert testimony. The courts cannot arbitrarily say that, irrespective of the beliefs of the physician that he is effecting a cure or properly prescribing narcotics, the amount is excessive and ipso facto a violation of the law."
"There is no dogmatic rule which the courts have laid down for the purpose of determining what is good or bad professional practice."
"Ultimately, the question to determine is not whether the judgment used was good or bad, but whether the defendant believed... that the treatment he administered was proper by ordinary medical standards."90
This state of the law offers a challenge to the medical profession. It may question the somewhat misleading Regulation No. 5, Art. 167, of the Narcotics Bureau founded on the too sweeping language of the Webb and Behrman Cases to the effect that:
"...An order purporting to be a prescription issued to an addict or habitual user of narcotics, not in the course of professional treatment but for the purpose of providing the user with narcotics sufficient to keep him comfortable by maintaining his customary use, is not a prescription within the meaning and intent of the act; and the person filling such an order, as well as the person issuing it, may be charged with violation of the law."
Despite this regulation, physicians may legally treat addicts. They may prescribe narcotic drugs to addicts. But they must act in good faith and according to proper medical standards. However, the medical profession should not leave the task of determining good faith and proper medical standards to an ex post facto judgment made by twelve laymen on a jury. It should not be left to the conflicting opinions of so-called experts, who may have differing views on how to treat narcotic addiction. The profession itself, through its authoritative body, the American Medical Association, should lay down the criteria by which a physician's treatment of an addict can be judged.
The A. M. A. itself should determine the standards of good faith and the limits of proper medical practice in the treatment of addicts. If the A. M. A. were to lay down standards, then the physician will know what is proper medical practice in dealing with addicts before he acts. A physician will also know that he need not fear criminal prosecution if he adheres to standards laid down by his profession. He will not be at the mercy of the stool pigeon and the informer. He will not tend to divorce himself entirely from the treatment of one group of unfortunate individual s, whose troubles lie legitimately within the domain of medicine. In laying down standards for the treatment of addicts, the American Medical Association may have to reconsider its resolution of 1924 condemning all "so called ambulatory methods of treating narcotic addiction." Thus, the present law provides the framework within which the medical profession, acting through the American Medical Association, can authoritatively determine what the role of the doctor should be in the treatment of addicts and in the treatment of problems of addiction.
2. Outpatient Clinics Medical counseling outpatient clinics for drug addicts have been set up particularly for adolescents in such cities as Chicago, Detroit and Los Angeles.91 These clinics offer some social case work and psychotherapy as well as some medical help for the addict. None of these clinics supply drugs to their patients. These clinics were established as a result of the concern with narcotic addiction immediately after the war. They were established in various communities under pressure to do something about the narcotics problem. A clinic is cheaper and easier to operate than a hospital dedicated to the rehabilitation of drug addicts. Unfortunately, the founders of outpatient clinics were not fully aware of the difficulties involved in attempting to treat drug addicts. This awareness and understanding came as the clinic obtained actual experience. As the report of the Medical Counseling Clinic of Chicago pointed out: "The treatment of addicts is an extremely difficult problem, in large part due to the inadequate motivation of the person and to his instability and unpredictability, which results in sudden breaks of contact with treatment and a lack of noticeable progress over a long period of time. When the individual is able to continue in treatment over a sufficiently long period, we do observe movement and progress in adjustment, both in personal and social levels. It would then seem that successful treatment of the person with a history of narcotic addiction is a very slow gradual process taking place over a long period of treatment contacts, and fraught with difficulties created by outside social and legal pressures, as well as by the extremely inadequate and weak personality that we have to deal with."92 One hopeful development in connection with the outpatient clinics has been the establishment of agencies where the person who has been a patient at the federal narcotics hospital at Lexington or New York City's Riverside Hospital may come for advice, counsel, guidance and help.
For years, officials at the Lexington Hospital deplored the necessity of sending the released addict back to his community, where he had no one to turn to in case he needed help with his personal problems. Similarly, the officials at Riverside Hospital felt that contact must be maintained with the young addict after his discharge from this institution. A beginning has been made in New York City in providing after-care facilities for drug addicts discharged from Riverside Hospital. Similar facilities have been provided for Lexington graduates, in connection with a follow-up study of persons released from the latter institution. The clinics serve only a small part of the drug user or drug addict population in their cities. Confirmed addicts do not willingly attend outpatient clinics if they cannot obtain drugs there. Where they do attend such clinics, it is usually under pressure of official agencies such as courts, parole or probation officers or under pressure from parents or relatives. Contacts under these circumstances are restricted and are broken off at the earliest possible opportunity.
Many persons coming to the clinics may have been helped by their contacts with these agencies. Some may have been persuaded to stay off drugs. Many addicts may have been induced, by contact with these clinics, to take the more drastic institutional treatment at Lexington. One can, however, be skeptical as to whether outpatient clinics have any kind of decisive impact on the confirmed addicts living in the communities they serve.
3. Institutional Treatment of Drug Addicts Jail or prison is the usual method of treating drug addicts in this country. Drug addicts are incarcerated by the thousands all over the country for violations of the drug laws, or for thefts and other offenses committed in order to obtain money for drugs. Drug addicts also surrender themselves voluntarily for self incarceration under the laws providing for self commitment of drug addicts. The only value of jail or prison for the treatment of drug addiction is that the addict may be temporarily withdrawn from drugs during the period of incarceration. Even this objective may not be achieved if the jail or prison is one into which drugs may be smuggled. There are practically no facilities for treatment of drug addicts in jails or penal institutions, beyond the forcible withdrawal of drugs. As a result, the drug addict comes out of jail or prison with his basic problems unresolved. The tensions, anxieties, pressures and personality problems which caused him to take drugs in the first instance are still with him. He usually goes back to the same environmental setting which facilitated his use of drugs in the first instance. There he also finds the same friends and acquaintances who have the same basic interest in drugs as himself. Under these circumstances, relapse to drugs is almost inevitable. The only value of prison or jail incarceration is in diminishing the dose of heroin or morphine necessary to keep the addict comfortable. But once the addict takes his first shot or "fix," after leaving jail or prison, he starts on the inevitable treadmill of tolerance and dependence, requiring greater and greater doses to obtain the elusive euphoria. His capacity will be limited only by the amount of money that he can borrow or steal in order to obtain the drugs necessary for his physical needs. If the addict was released from prison on parole (as he may well have been) he is usually an extremely unsatisfactory parolee. No threat of reincarceration prevents an addict from continuing to use the drug. Parole officers cannot prevent continued use of the drug or association with other addicts from whom parolees can obtain drugs, when they need them. Beyond jails and prisons and occasional addicts who may be accommodated at mental hospitals or private institutions, the only other institutional facilities for large scale treatment of drug addicts are the two federal installations at Lexington and Fort Worth, and the Riverside Hospital in New York City. These institutions were set up because of the belief that it is only possible to treat drug addicts in an institutional setting; that treatment of a drug addict is impossible unless he is first hospitalized in a drug free environment. The advocates of the hospitalization of drug addicts feel that only in a hospital setting can the addict be withdrawn from drugs and given the supportive psychological, medical, vocational and educational therapy necessary to enable him to cope with life without the use of drugs.
The author does not wish to minimize the great contributions that institutions like Lexington, Fort Worth and Riverside Hospital have made to an understanding of problems of drug addiction. Nevertheless, the limitations on the scope of their operations and their impact on the control of drug addiction in this country must be clearly understood.
The capacity of Lexington is 1280, of Fort Worth 1,053 . Riverside, the only narcotics hospital in New York City, is open only to adolescents under 21. It has a capacity of approximately 180.
It is obvious that these institutions of limited capacity can accommodate only a small fraction of the drug addict population of this country. Lexington and Fort Worth take federal prisoners who are drug addicts and who are permitted to serve their sentences in these institutions.
No technique has yet been worked out whereby drug addicts who have offended against state laws can be committed directly to Lexington or Fort Worth. Within the limit of the capacity of these institutions such offenders may be admitted as voluntary patients at Lexington or Fort Worth for the q-6 months believed necessary for rehabilitation. Most voluntary patients at Lexington and Fort Worth leave long before it is thought advisable that they should do so.
But even if all patients at Fort Worth or Lexington stayed for the 4-6 months believed desirable, it is unlikely that a permanent rehabilitation would result from such a stay. The programs of the institutions like Lexington, Fort Worth and Riverside are directed towards: (1) successfully withdrawing the patients from drugs; (2) building them up physically; (3) strengthening their vocational skills so that they can become productive members of the community; (4) eliminating gaps in their educational background; (5) attempting to give them understanding as to why they have had to resort to drugs in order to cope with life's problems, and (6) enabling them to resist the compulsion to use drugs as a means of resolving their difficulties.
There can be no doubt that institutions like Lexington, Fort Worth and Riverside have been a great deal more successful in the first four aspects of their program than in giving addicts a thorough understanding of why they use drugs and a resolve to resist the compulsion of drugs in the future. Addicts undoubtedly benefit considerably from their stays in Lexington, Forth Worth and Riverside.
Their systems are cleared of drugs, they become physically healthier and stronger. They are taught habits of regular work and may learn some academic subjects. But the exposure of a few months to a minimum amount of psychiatry, social case work, educational and vocational-activity, cannot eradicate the deep seated necessity and compulsion for drugs which most addicts seem to have.
There are no magic cures at narcotics hospitals. We simply do not know enough about the processes of drug- addiction to produce such cures.
The statistics on relapse to addiction after attempted cures at narcotics hospitals like Lexington, Fort Worth or Riverside tell the stark story of the basic failure of the hospital centered approach in dealing with problems of drug addiction.
X. THE RELAPSE AND REHABILITATION OF DRUG ADDICTS
One of the foundations of the present policy for dealing with narcotic addiction, which frowns upon ambulatory treatment by physicians or anyone else, is that narcotic addiction must be treated in an institutional setting. The preference for treatment of drug addiction in hospitals led to the establishment of the Lexington, Kentucky and the Fort Worth, Texas instaIlations maintained by the federal government. It also led to the foundation of the Riverside Hospital for adolescent narcotic addicts in New York City.
Those who established these institutions were undoubtedly buoyed by the hope that the institutional treatment would result in the rehabilitation of a substantial percentage of addicts. The founders of Lexington, Fort Worth and Riverside must have held the opinion that the combination of medical and psychiatric treatment, and social work, educational and vocational treatment could eliminate the curse of addiction to narcotic drugs from the patients admitted to their institutions. We have already noted the limited facilities for drug addicts provided by the aforementioned hospitals. If sound statistical studies were available on the relapse of drug addiction and they showed considerable percentages of success secured by institutional treatment, then the conclusion might well be drawn that better and more extensive hospital facilities were necessary for any rational drug control program. Unfortunately, sound, carefully conceived studies on a broad scale of the success or failure of hospital treatment of drug addicts have simply not been made. Whatever studies have been made present a somewhat discouraging picture of the possibility of preventing narcotic drug use by chronic addicts through current methods of hospital treatment.
In 1941, M. J. Pescor93 made a follow-up study of 4,766 patients released from the Lexington narcotics institution.
He was attempting to determine the present addiction status of these former patients of Lexington. Pescor tried to obtain this information from the F. B. I. which reported the re-arrests of these patients, from their probation officers who may have had them under supervision and from the patients themselves, primarily through letters. Pescor did not make any field studies nor did he have any means for determining the truth of a report by a patient or by a probation officer, that the patient was abstinent and no longer used drugs. Thus, on the one hand, Pescor was unable to obtain replies from a considerable percentage of the patients and, on the other hand, where the reply was favorable, there was some question as to whether it was reliable.
Pescor's conclusion was that the present addiction status could not be determined in 39.6% of the cases; 7% had died following release from the hospital; 39.9% relapsed to the use of drugs; and, 13.5% were reported as still abstinent. Even if we accept Pescor's figures as reliable, it is apparent that of the cases on which Pescor had data (39.9% relapsed and 13.5% abstinent), relapse had occurred in 3/4ths of the cases and only 1/4th of the patients could be deemed cured. More accurate tracing of the unreporting patients and more careful checking of those who reported themselves abstinent (through such devices as the use of Nallymorphine), would undoubtedly show that considerably less than 1/4th of the patients that pass through Lexington remain abstinent for any considerable period of time.
The high percentage of relapse after hospital treatment is experienced by other institutions besides Lexington. A study was made in California on 584 cases treated at the state narcotics hospital between the years 1928-1936. This study showed that only 15% of the cases could be deemed abstinent as of the date of the study and that 85% had relapsed; most of them during the 16 month parole period after release from the hospital.
In 1930, Dr. Alexander Lambert94 reported on 318 addicts voluntarily committed to Bellevue Hospital by a City Magistrate. Of these addicts, 279 or 87%, admitted prior cures as against 39 or 13% who denied any prior cures.
More than half of these 279 addicts had "taken the cure" four or more times .
In 1951, Knight and Prout95 reported on a follow up study of 75 private patients treated at the New York Hospital, Westchester Division. These were, obviously, luxury patients who could pay for hospital treatment. Data as to addiction were available on only 38 of these 75 patients.
Less than half of these 38 patients (15) were reported as "abstaining" from drugs.
One might expect a considerably higher percentage of cures from an institution like Riverside, which attempts to select only promising cases for treatment and which takes only adolescent drug addicts to start with. Yet the N. Y.U.96 study made on the post hospitalization adjustment of 30 adolescent opiate addicts is quite discouraging. Only two of the patients remained abstinent during the period of one year after discharge from the hospital. Twenty-two of the thirty patients were reinstitutionalized one year after discharge, either in a jail or in a hospital.
In the light of the data, it is obvious that more careful studies of the effect of hospital treatment of drug addicts must be made. If it is true that most chronic addicts cannot be cured by present hospital methods, this fact should be known as quickly as possible. It may necessitate the complete revision of present methods of dealing with addicts.
XI. THE SUGGESTED CLINICS FOR LEGAL NARCOTICS DISTRIBUTION
The dissatisfaction with present methods of controlling narcotic addiction has led to numerous proposals for the legal distribution of narcotic drugs to addicts. These proposals are not made by sensation seekers or by subversives who desire to undermine our society by spreading addiction to narcotic drugs. Proposals for the legalization of narcotics have been seriously advanced by conservative physicians, medical societies, lawyers, judges and responsible community groups. These individuals and organizations have been concerned with the fact that present methods condemn the addict to a life of parasitism and crime, while they also fail to control the illicit traffic or halt the spread of narcotic addiction.
Dr. Lawrence Kolb is one of many doctors who have strongly urged the legalization of narcotics under appropriate safeguards. In a recent Saturday Evening Post Article,97 he has written as follows: "A major move in the right direction would be to stop the false propaganda about the nature of drug addicts and present it for what it is--a health problem which needs some police measure for adequate control..."
"'We need an increase in treatment facilities and recognition that some opiate addicts, having reached the stage they have, should be given opiates for their own welfare and the public welfare, too..."
". . . A workable solution would be to have the medical societies or health department appoint competent physicians to decide which patients should be carried on an opiate while being prepared for treatment and which ones should be given opiates indefinitely..."
"The details of a scheme of operation should be worked out by a committee of physicians and law-enforcement officers, with the physicians predominant in authority." Other physicians and medical societies have favored the establishment of narcotic clinics, where drugs could be legally dispensed to addicts. Dr. Andrew E. Eggston of the New York State delegation submitted a resolution to the American Medical Association in 1954, which proposed that the American Medical Association go on record as favoring:
(1) The establishment of narcotics clinics under the aegis of the Federal Bureau of Narcotics.
(1) Registration and fingerprinting of narcotic addicts.
(3) Keeping of accurate records.
(4) Administering. optimal doses to addicts at regular intervals at cost or free.
(5) Prevention of self-administration.
(6) Attempt cures through voluntary hospitalization, if possible.
(7) Avoidance of forcible confinement.
The New York Academy of Medicine has proposed a more detailed plan which differs in many particulars from the aforementioned Eggston resolution. The Academy proposed a six point program to stamp out drug addiction by:
1. Changing the attitudes towards the addict. He should be treated as a "sick person, not a criminal."
2. Taking the profit out of the illicit traffic by furnishing drugs to addicts at low cost under federal control.
3. Medical supervision of existing addicts with vigorous efforts towards their rehabilitation through: (a) persuasion (b) appraisal of methods of treatment and their success, and (c) supervision of addicts resistant to treatment.
4. Continued efforts to suppress the illicit traffic in drugs.
5. Formulation of an education program on the dangers of drugs for adults as well as adolescents.
6. Obtaining an accurate count of addicts and knowledge of the success or failure of treatment, so that epidemiology and parthenogenesis of drug addiction can be properly studied.
To implement this program, the Academy proposed that:
1. Clinics be attached to general hospitals, whether federal, municipal or voluntary, dispensing narcotics to addicts, open 24 hours daily, 7 days a week.
2. No person be given drugs at such clinic unless he is willing to enter a hospital for evaluation of his drug needs. After a medical evaluation, he should receive at cost from the clinic the amount of the drug which he requires medically.
3. Safeguards against the addict registering in more than one clinic.
4. Drugs could be given to the addict for self-administration, but no more than two days supply would be furnished at any one time.
5. The addict be re-admitted to the hospital for reevaluation of drug needs so that the factor of tolerance can be handled.
6. Addicts detected giving away or selling any or all of their supplies be liable to commitment to a hospital for attempted rehabilitation.
7.. Current enforcement machinery be maintained to continue suppression of the illicit market in drugs.*
* See also clinic plans advocated by Richmond County Medical Society which proposed the following: 1. Establishment of narcotic clinics in large centers where the problem is acute. Suitable private physicians can care for the occasional addict in isolated areas.
2. Fingerprinting, photographing, and registering the addict to be sure that the addict uses no more than one such facility.
3. The addict will receive his narcotics only at the clinic, hospital, or doctor's office so that he cannot resell them elsewhere.
4. Examination of the individual to determine whether or not he is actually an addict.
5. Attempt to permanently withdraw the individual from the drug.
Compare with plan advocated by Dr. Hubert S. Howe:
1. Establishing narcotic hospital facilities under federal, state or municipal auspices in cities which are centers of addiction.
2. Equipping these hospitals to examine, classify, hospitalize, and treat addicted persons on their premises for necessary periods, after which the appropriate cases would be referred to specifically commissioned physicians who would be appointed by the hospital staff.
3. Treatment of addict patients in the offices of the physicians under strict supervision of hospitals.
4. Returning patients to the hospital for final cure after achievement of an adequate social and economic adjustment.
5. Upon release from the hospital prevention of relapse through care of a commissioned physician, during the critical period when the patient is becoming adjusted to his resurgent sexual and other emotions.
An interesting plan for the dispensation of drugs legally to chronic addicts has been formulated by the Citizens Advisory Committee to the Attorney General of California, on Crime Prevention.98 This Committee consisting of eminent representative Californians proposed the following:
1. Upon a medical determination that a person is an addict, he shall be institutionalized for a period of at least 90 days, during which time the patient will be withdrawn from narcotics and exposed to an over-all educational and psychiatric program.
2. On release from this institutional treatment, the patient will be assisted by outpatient supervision It will include psychological, sociological, economic, cultural and other elements in an effort to determine the narcotic-proneness of the individual.
3. Treatment should be on either a voluntary self-commitment basis or involuntary. The patient must be required legally to continue the treatment supervision in the outpatient clinic. This phase shall be known as Treatment supervision.
4. It is recommended that a Disposition Board be established consisting of individuals experienced in the field of human relations who shall evaluate the disposition of cases and the duration of treatment and control.
5. If during outpatient treatment, it is ascertained by administration of Nalline or other means that the patient is again using narcotics as indicated by his withdrawal symptoms, the Disposition Board would have the responsibility of determining the further disposition of the case.
6. Should the disposition Board conclude, after repeated failures, that the patient is "incurable," he might then be certified or registered so that thereafter, he shall receive indicated dosages of narcotic drugs from a determined governmental agency and thereby remove said addict as a potential market for criminally or illegally secured narcotics. The establishment of this phase of the program should be deferred until two years after the institution of the over-all management program.
It is obvious that various clinic plans and plans for dispensing legal drugs to addicts differ in important particulars. For example, the New York Academy plan would provide morphine to addicts for self-administration, but no more than a two day supply at a time. Other plans do not envisage furnishing drugs to addicts for self-administration because of the fear that the addict would peddle the drugs given to him and thus help create new addicts. But if an addict does not have the drug for self-administration, then he must come to a clinic or hospital to get the drug several times a day, so that he can avoid withdrawal distress.
It is difficult for an addict to work on a job in any productive capacity if he must visit a clinic several times a day. To obviate this Dr. Eggston and Dr. Berger envision the use of a "depot morphine", a slow acting morphine whose effect would last at least 24 hours. Unfortunately, there is no such drug on the market today and none which has yet been devised, which does not have some rather bad side effects.
The aforementioned is an illustration of the practical difficulties which abound in all plans for the legal distribution of narcotics. The New York Academy plan envisioned hospital clinics open 24 hours a day, 7 days a week, where an addict could come and obtain his drug. This is a very expensive way of attempting to meet the needs of the addicts and would hardly find favor with hospital trustees.
Other examples may be cited. The plans envisage the distribution of drugs to confirmed addicts after a study of the addict and his needs to determine whether it may be possible to rehabilitate him so that he may care to live without drugs. Unfortunately, except for hospitals like Lexington, Fort Worth and Riverside, facilities do not exist for such study. New facilities will have to be provided in communities with a large addict population. The plans are vague and indefinite as to the nature of such facilities. Moreover, the criteria for distinguishing chronic unrehabilitable addicts who must be furnished drugs in order to lead a normal life and addicts who may be reclaimed from the curse of addiction are not sufficiently precise nor are they sufficiently well known to the medical profession generally.
Any adequate hospital treatment or study program concerning addicts requires follow-up facilities in the community to assist addicts in the process of rehabilitation.
The test of rehabilitation is not whether an addict can exist without drugs in an institution, but whether he can live, function and work without drugs in the community.
Before a decision can be made as to whether to supply an addict with legal drugs, he should be observed in the community, and helped in any resolve that he may have to live without drugs. Unfortunately, follow-up facilities for drug addicts do not exist at the present time. Nor do we have a blueprint as to the kind of facilities which are necessary.
An underlying assumption of all the plans is that addicts will not patronize illicit peddlers if they receive a sufficient dose of drugs to keep themselves comfortable. Unfortunately, this expectation does not sufficiently take into account the mechanism of tolerance, and the increasing need or desire for drugs on the part of the addict. None of the plans suggest how this matter of tolerance can be handled so that an addict will be satisfied with his legal supply of drugs and stay away from peddlers for additional supplies. Nor do the plans take any account of an addict's desire for drugs like cocaine, which will not be supplied by the clinic.
Finally, there is an insufficient realization in the various clinic plans that large numbers of addicts have serious personality difficulties even without the problem of drug addiction and that a mere supplying of drugs to such individuals will not solve such difficulties. A criminal psychopath drug addict is likely to continue his criminality despite the fact that he may be supplied with drugs legally. If clinics are to have success in rehabilitating drug addicts they may have to do a great deal more than merely serve as dispensaries for drugs. They obviously need social work and psychiatric facilities to deal with the personality problems of the addict. Unfortunately, even if such facilities were provided, successes in dealing with addicts are not assured. As the Council on Mental Health Report points out:
"... Psychiatrists, experienced in managing addicts, doubt that there would be any great success in persuading addicts to undergo withdrawal and to engage in psychotherapy as long as drugs are supplied to them. A large percentage of addicts are poorly motivated for treatment. They feel that, in the drug, they have the answer to their symptoms. They do not regard themselves as being psychiatrically abnormal and, therefore, are resistant to psychotherapeutic measures. All psychiatrists are familiar with the difficulties in treating psychopaths of the kind that constitute a large proportion of addicts. It would seem unwise in the light of lack of knowledge of the etiology and treatment of character disorders for the medical profession to promise good results in managing such persons by purely medical means alone."99
There has been a violent opposition to the plans for legalizing the distribution of narcotics to confirmed addicts and to plans for narcotic clinics. Dr. George H. Stevenson conducted a study of drug addiction problems in Vancouver and British Columbia. He examined the arguments for and against the legal sale of narcotics and came to the conclusion that: " ... the proposal for legal sale of narcotics if adopted would not only fail to solve the addiction problems but would actually make them more serious than they are at present."100
The U. S. Senate Committee also considered the legalization of narcotics and concluded:
"The sub-committee is unalterably opposed to and rejects the clinic plan proposed for supplying narcotic addicts with free or low cost drugs. We are opposed to all types of so-called ambulatory treatment... Finally, we believe the thought of permanently maintaining drug addiction with 'sustaining' doses of narcotic drugs to be utterly repugnant to the moral principles inherent in our law and the character of our people."101
The spearhead of the opposition to legal narcotics clinics has been the present Bureau of Narcotics. For years it has opposed legal clinics and dispensaries for the treatment of drug addicts. Its main weapon against the establishment of present day clinics was the alleged failure of the approximately 44 earlier clinics,* established between 1919-1923 by state and municipal health officials throughout the country to meet a purported emergency resulting from the Supreme Court decisions which prevented doctors from prescribing for drug addicts. The author cannot enter the debate as to whether these early clinics did or did not produce detrimental results.** There is too little objective data concerning the operation of these clinics. Most of the clinics operated for too short a time for any results to be evaluated. Many of the clinics were hastily organized to meet a threatened emergency without any serious planning or precise knowledge of the problems that they were intended to meet.
*See publication, Narcotics Clinics In The United States, U. S. Government Printing Office, 1955.
** See the discussion in Report on Narcotic Addiction of the Council on Mental Health, American Medical Association, page 3 et seq.
The ambulatory treatment aspect of these clinics evoked considerable criticism and was in part responsible for the resolution adopted by the American Medical Association in 1924, urging:
"Both Federal and State governments to put an end to all manner of so called ambulatory methods of treatment of narcotic drug addiction whether practiced by the private physician or by the so called narcotics clinic or dispensary."102
Incidentally, it should be noted that the condemnation of any system of treatment which places opiates in the hands of addicts for self administration is still the official policy of the American Medical Association. The Council on Mental Health of the American Medical Association reported after examining the arguments for and against narcotics clinics that: governments to put an end to all ". . . The Council does not feel that the American Medical Association should approve proposals for establishment of clinics which would dispense drugs to addicts at this time ..." However the Council did suggest: ". .. the possibility of devising a limited experiment which would test directly the hypothesis that clinics would eliminate the illicit traffic and reduce addiction."103
The author of this report tends to agree with the Council's recommendation that we should go slow in establishing narcotics clinics. He would like to see the various problems involved in the establishment of clinics carefully tested in a research setting. Clinics cannot be established on the basis of broad general principles alone. We need to know what facilities are necessary for the successful operation of any clinic. We also must be clear concerning the techniques and procedures that should be used by such clinics.
Careful research and planning may make a modern clinic a success, and avoid the mistakes which bedeviled their 1920 counterparts.
XII. PROPOSALS FOR RESEARCH
It is obvious from the preceding pages that a fundamental attack on drug addiction requires some basic research.
The author would 1ike to recommend the following research projects:
An Outpatient Experimental Clinic for the Treatment of Drug Addicts
The Council on Mental Health of the American Medical Association has suggested, as we have seen, that a limited experiment be devised which would test directly the hypothesis that clinics would eliminate the illicit traffic and reduce drug addiction. But an experimental clinic can do more than this. It can also aid in the determination of whether it is possible to rehabilitate addicts, in a noninstitutional setting, so that they can live and function without drugs. Heretofore, the opinion has been that this could not be done outside institutional walls. An experimental clinic can also try out varied techniques in the rehabilitation of addicts and decide which are most useful.
Finally, an experimental clinic can resolve a basic problem in dealing with addiction; whether confirmed, unrehabilitable addicts can be transformed into productive members of the community if their drug needs are met.
The Committee should sponsor an experimental clinic for the outpatient treatment of drug addicts with the broad objectives outlined above. The clinic should be organized in a metropolitan center like Washington, D. C., Chicago or New York, where hospital facilities can be made available for the use of the clinic. It should be restricted to dealing with only a limited number of drug addicts; one hundred may be found to be sufficient, for experimental purposes. The clinic should provide facilities for a thoroughgoing study and diagnosis of each addict. The disciplines of medicine, psychiatry, psychology, social casework and education should all be used in making such a diagnosis and study. After the diagnosis and study, the attempt should be made to take the addicts off drugs and keep them off drugs through the use of all the techniques available in the disciplines aforementioned.
For the purpose of diagnosis, study and withdrawing the addict from drugs, a short stay in a hospital may be necessary. The clinic should have access to hospital facilities for its addict patients. It is desirable that the personnel of the clinic use every device and every technique presently available to try and keep the addict patients off drugs. Success or failure in this connection can be determined through a process of periodic reporting to the clinic and periodic follow-up in the field. Besides concentrating on the drug problems of the addict, the clinic should offer psychotherapeutic guidance to the patient in dealing with his personality problems; social work help with his family difficulties; vocational guidance on how to find and keep a job; and help in overcoming his educational shortcomings.
Every effort should be directed toward enabling the addict to become a productive individual without drugs.
If the clinic does not succeed in taking and keeping the addict patient off drugs after a period of intensive treatment, its personnel then should consider supplying the addict with sufficient drugs for his needs, so that he does not have to patronize the illicit peddler. The clinic will obviously have to determine what the minimum needs of the addict are for drugs. It will also have to struggle with the problem of tolerance and the demand of the addict for increasing doses of the drug. It will need to determine whether the addict is supplementing the supply of drugs from the clinic with "junk" obtained from illicit peddlers.
The clinic should also continue to provide every patient with vocational, educational, social work and psychotherapeutic guidance in the effort to make him a productive individual and wean him away from drugs.
It is obvious that many coming to the clinic will need to be supplied with drugs and will be unable to function without drugs. This will give the clinic the opportunity of testing the hypothesis that an addict is in a state of apparent normality when his drug needs are satisfied, and can function productively, with relative efficiency. But can he do so in the community in which he must live, work and bring up a family, and where he is subjected to the strains and stresses of everyday living? No demonstration of the relative normality of addicts under drugs in the controlled conditions at Lexington will be as effective as a similar demonstration made by addicts whose drug needs are satisfied, yet who are able to live and function in the outside world, hold down jobs and meet their family responsibilities.
In connection with the confirmed addicts who may be unable to function without drugs for a considerable period, the clinic will have to determine the best means of supplying such drugs to such patients with the least danger to the community. Since ambulatory treatment of drug addicts is frowned upon, the clinic may decide to dispense the drugs only on the premises of the clinic itself. In that connection, the clinic will have to determine whether an addict can hold down a job if he must come to the clinic for his supply of drugs several times daily. The clinic may also experiment with the so-called "depot morphine," if such a drug can be made available in order to obviate the necessity of repeated trips to the clinic daily. On the other hand, the clinic may decide that some form of ambulatory dispensation of drugs is a necessity in dealing with confirmed addicts. In that event, it is to be expected that the clinic will work out the best means of supplying drugs to addicts on an ambulatory basis.
It will be relatively easy to get the addict patients to attend the clinic if they are to receive drugs. It will be far more difficult to get them to attend in the earlier stage when the clinic is trying to take them off drugs. In that event, some degree of official control over the addict patients may be necessary. This may be provided by taking the addict patients from the courts of the city in which the clinic is located and working out cooperative agreements with the courts and their probation departments concerning the patients that the clinic will handle.
The experimental clinic can be envisaged as primarily a research enterprise which will provide data on the best methods of dealing with narcotic addicts outside institutional walls. Such data are indispensable for the establishment of public health clinics and will contribute greatly to their ultimate success. The experimental clinic will have to keep thorough records on what it does to and for its addict patients. Only through such records will it be possible to evaluate the possibility of success or failure in the use of specific techniques for dealing with addicts. The thorough case studies made by the clinic should also throw considerable light on causative factors in drug addiction and help in the formulation of prevention programs. It could also provide indispensable data on the procedures and techniques for dealing with addicts which could be used by the individual physician in the smaller communities which cannot support public health clinics.
In proposing the experimental clinic, the author is not unaware of the legal problems involved in supplying drugs to addicts. However, it is his belief that the operation of the experimental clinics proposed herein will not violate present federal statutes on narcotics, as interpreted by our courts. This is apparent from our discussion concerning the doctor and the drug addict. (See supra.)
The addicts coming to the experimental clinic will be treated as patients in the effort to overcome their addiction, and will only be supplied with drugs when it is determined that such drugs are absolutely necessary to their health and well-being and their ability to function as productive individuals in the community. If they are supplied with drugs it will be in a desire to avoid the discomforts and physical difficulties arising out of addiction. The good faith of the doctors supplying the drugs can hardly be questioned within the experimental clinic setting. In our view supplying drugs to a confirmed addict in the research setting of an out-patient clinic is no more illegal than the similar supply of drugs to an addict at Lexington in connection with studies made there. Both efforts advance materially the frontiers of our medical knowledge of how to deal with the problem of addiction.
2. A Study of Relapse and Causative Factors in Addiction and Rehabilitation
We have indicated that our present social policy demands that an addict be treated only in a hospital, in a drug free environment. We have expressed doubts as to whether hospital treatment, given our present knowledge, is very effective and can effectuate many cures or rehabilitate many drug addicts. We have noted the lack of good studies of the relapse of addicts after treatment at such institutions as Lexington, Fort Worth and Riverside. The author would like to propose a thorough research study of the after careers of graduates of Lexington, Fort Worth and Riverside. A study of addicts should be made with the same thoroughness as that used by the Gluecks in their classic studies of the after careers of delinquents and criminals.'" This will necessitate careful detective work to locate the subjects studied and thorough case work to determine what they have been doing since they left the institution in which they were originally treated.
Through such detective work and case work analysis we should be able to determine the relative effectiveness or ineffectiveness of hospital treatment. We should also be able to determine periods of abstinence and periods of relapse after leaving the institutions in which they were treated.
If a relapse study is undertaken, a thorough psychological, psychiatric and sociological study of the same addicts would be desirable. It should be able to determine why these individuals became addicted to drugs, while others who grew up with them in the same neighborhoods with roughly the same background did not become addicted. The basic concern will be causative factors in addiction based on a comparison of drug addicts and control groups. At the same time, a study of cured and rehabilitated addicts should be made in order to determine how and why men and women conquer the drug habit as well as how and why they became addicted in the first instance.
Success in dealing with addiction has as much to teach us as failure. A comparison is therefore desirable of those addicts who succeeded with those who have failed.
Educational and Preventative Research
There is need for sound, authoritative, educational materials that could be used in campaigns for the prevention of narcotic addiction. Materials are required on both the adult as well as the adolescent level. The preparation of such materials and the planning of campaigns for the prevention of narcotic addiction will require the collaboration of the disciplines of public health, mental hygiene and education. The challenge to be met by such collaboration is not unfamiliar. It is similar to the challenges which were met and the campaigns undertaken in connection with the prevention of venereal disease, heart disease and the dissemination of better information concerning problems of mental illness and emotional disturbance.
It is the author's belief that it is possible to prevent narcotic addiction through a dissemination of knowledge concerning the nature and effects of narcotics and understanding concerning the methods whereby narcotic patterns of behavior are transmitted from one individual to another. Preventative materials on narcotics, developed by the disciplines of public health, mental hygiene, and education are particularly necessary for use in those areas of our large cities, where narcotic addiction is a common phenomenon.
Sound educational and preventative materials on narcotic addiction have not been developed largely because of the fear that the dissemination of information about narcotics would lead adolescents to experiment with drugs.
It is the author's belief that ignorance is more to be feared than knowledge and that knowledge of the nature and effects of narcotic drugs would prevent experimentation rather than lead to it. As Justice Singer observed, sound education on narcotics would no more stimulate increased narcotics use "than education on fire prevention leads to more fires by stimulating people to become pyromaniacs."105 Prevention through education, however, is but one facet of a narcotics addiction prevention program. There is at the present time little available information on what community and neighborhood techniques are effective for preventing experimentation with drugs by juveniles and adolescents. This problem is closely related to the prevention of juvenile delinquency and crime, since many of those who experiment with narcotics engage in such anti-social activities. There are many programs for the prevention of juvenile delinquency and crime, but such programs are generally not geared to the prevention of drug addiction.
The author would like to propose a research project which would attempt to formulate community preventative techniques for dealing specifically with narcotic experimentation and narcotic addiction. If sound preventative techniques could be devised and applied in our cities, they might considerably cut down the incidence of narcotics addiction.
4. Research in the Administration of Present Laws
A careful, authoritative analysis should be made of the administration of recently enacted narcotics legislation.
These laws have provided increasingly severe penalties in narcotics cases. They have eliminated judicial discretion with respect to sentences. They have provided fixed minimum sentences which the judges were required to impose on convicted offenders. They have eliminated the use of probation and parole in narcotics cases, thus eliminating the possibility of controlling narcotics offenders outside prison walls.
Many believe that these changes in our laws have done more harm than good and that they have not advanced the goal of effective control over narcotic drugs. Some also feel that the new legislation has created many more problems than it solved. Accordingly, we should like to find out what has happened in the enforcement of the new state and federal narcotics statutes.
Are they providing a greater degree of control over the violation of the narcotics law? Have they contributed to any decline in narcotics addiction? Are judges and prosecutors enforcing the laws as they are written? Are mandatory minimum sentences being imposed by judges or are they ignoring the provisions of the new laws in cases where they feel that statutory penalties are too severe? Has severity of sentence had any effect on convictions under the new statutes, or are the new statutes self-defeating because juries will not convict and expose offenders to severe punishments in relatively minor cases? Are the sentencing? provisions of the new statutes being evaded by permitting pleas of guilty to other types of offenses which provide for lesser penalties and the possibility of probation and parole? Has the new legislation made it possible to reach the upper echelons in the illicit drug traffic or are our prisons and penitentiaries being filled by run-of-the-mill drug addicts? Have the new laws changed the relationship between state and federal law enforcement in connection with narcotics by having more cases brought into the federal courts because of the severity of federal legislation? Are state prosecutors refusing to press narcotics prosecutions because of the severity of the sentences involved? These are a few of the many questions which should be answered by an analysis of the administration of existing narcotics laws. Such an analysis will throw considerable light on the success or failure of present narcotics legislation and the utility of our present narcotics policy. It will also serve as a fundamental guide in the formulation of the new narcotics legislation, projected by the legal research that the author recommends in the next section.
5. Legal Research
Many defects in our present narcotic laws will be disclosed by the analysis of their administration. This will require the formulation of amendments to the narcotics laws. However, the author feels that more basic legal research must be done.
At the present time the addict is treated by our statutes like a criminal. If he is found in possession of even the minutest portion of narcotics for his own use, or a hypodermic needle, he may be jailed. He may even be jailed under some state statutes because of the mere fact that he is an addict. If the addict is, as the author believes him to be, a sick, maladjusted individual driven by a compulsion, then these statutes are wrong. They must be replaced by statutes which in the first instance require the treatment of addicts and not their incarceration in jails or prisons.
If such treatment is ineffective and the addict cannot be rehabilitated, the law should not prevent the addict from obtaining a legal supply of narcotics.
In the author's opinion, a new Uniform or Model Narcotics Act must be formulated, which can be recommended to the various states, and which must be based on the enlightened premises of a new social policy towards addiction.
Before such a statue is formulated, a careful survey and study must be made of existing state statutes and state court decisions concerning the control of narcotic drugs and methods provided for dealing with addicts.
If a new statute concerning addiction is to be drafted, then a new look should also be taken at the problem of the non-addict peddler. Even if chronic addicts are to be given a legal supply of narcotics, the illicit peddler of narcotics must still be jailed. We should like to strengthen law enforcement in dealing with the non-addict peddler. There are many devices which can be used in this connection, some of which have already been included in particular state statutes. New devices for strengthening law enforcement can also be devised in the course of legal research.
Thus, the legal research recommended herein would have two major functions:
(1) Provide the legal framework for a new orientation in dealing with narcotic addicts.
(2) Provide the most effective controls and sanctions for dealing with drug peddlers and the illicit drug traffic.
6. The Preparation of a Volume of Readings on Narcotic Addiction
In the 1920s, Dr. Charles E. Terry and Mildred Pellens undertook a critical review of the literature on what was then called chronic opium intoxication. This work was done for a Committee on Drug Addiction, which sought answers to fundamental questions concerning the extent, etiology, nature and treatment of drug addiction. Unfortunately, as Terry and Pellens pointed out (The Opium Problem, xiii-xiv), such answers were generally unavailable because of the lack of unanimity of opinion on almost every phase of the drug problem.
Thirty years have elapsed since the publication of the Terry-Pellens volume. During these thirty years, a great deal of research has been done and considerable literature has appeared on problems of drug addiction. Drug addiction has been the subject of at least two nationwide Congressional investigations and innumerable state and local investigations. The material concerning drug addiction is spread through hundreds of pages of official reports, legal opinions, articles in all kinds of technical journals, publications of medical societies, the U. S. Public Health Service, Narcotics Bureau and other agencies. Particular phases of what we may know about drug addiction are found in various textbooks. Some aspects of the drug addiction problem have become clearer since the Terry-Pellens volume. Others have changed considerably since the 1920's.
There still is considerable divergence of opinion and controversy over many phases of the problem of narcotic addiction.
The author believes that there would be considerable advantage in the preparation and publication of a critical selection of the materials on drug addiction which have appeared since the Terry-Pellens volume. This would help to determine the state of our knowledge of drug addiction.
It would serve to illustrate and point up the various phases of the drug addiction problem in this country. The analysis of the literature would also demonstrate whether there are still any answers of fundamental questions concerning the extent, nature and effects of narcotic addiction, the social, psychological and familial factors involved, the relationship between narcotic addiction and crime, and the best methods of treating narcotic addicts. In addition the preparation of a critical volume of readings on drug addiction would help define areas where further research would be useful in helping to control the scourge of narcotic addiction.
1. Subcommittee on Improvements In The Federal Criminal Code of The Committee On The Judiciary, United States Senate, Eighty-Fourth Congress. First Session On: The Causes, Treatment, And Rehabilitation Of Drug Addicts.
Price Daniel, Chairman Joseph O'Mahoney James O. Eastland Herman Welker John Marshall Butler C. Aubrey Gasque, General Counsel W. Lee Speer, Chief Investigator
2. Subcommittee of The Committee On Ways And Means, House of Representatives, Eighty-Fourth Congress. On: Traffic In, And Control Of, Narcotics, Barbiturates, And Amphetamines.
Hale Boggs, Chairman Frank M. Karsten Eugene J. McCarthy Frank Ikard John W. Byrnes Antoni N. Sadlak Howard H. Baker Henry L. Giordano, Chief Investigator
3. Report to the House Committee on Ways and Means, from the SubCommittee on Narcotics, May 10, 1956 (PP. 13-14).
4. Report of the Committee of the Judiciary, U. S. Senate Summary and Preliminary Findings and Recommendations Sub-Committee on Improving the Federal Criminal Code, January 16, 1956.
5. Ibid. Report #1997. May 14, 1956 (P. 5).
6. Ibid. April 24, 1956 (p. 2).
7. Consisting of representatives of the Department of State, Defense, Health, Education and Welfare, Treasury, and Justice.
8. Report of the Interdepartmental Committee on Narcotics To The President, Washington, D. C., February, 1956 (P. 16).
9. H. Isbell and W. White, "Clinical Characteristics of Addictions," American Journal of Medicine, May 1, 1955 (P. 558)
10. Senate Committee Report (p. 4198).
11. Ibid. (p. 4624).
12. L. Kolb and A. G. DuMez, "The Prevalence And Trend Of Drug Addiction in the U. S. and Factors Influencing It," Public Health Reports, Reprint #924 May, 1924
13. C. Terry and M. Pellens, The Opium Problem, New York: The Committee On Drug Addiction with the Bureau of Social Hygiene, Inc., 1928 (P. 48).
14. Bureau of Narcotics, Traffic In Opium And Other Dangerous Drugs, 1935.
16. Op. Cit. (p. 8).
17. New York Attorney General's Survey, 1952 (p. 9).
18. See testimony Senate Committee 1441; House Committee, 9188.
20. Report on Narcotic Addiction to the Attorney General by the Citizen's Advisory Committee To The Attorney General On Crime Prevention. March 26, 1954 (p.17).
21. Op. Cit. (p. 52).
22. Ernest S. Bishop, Medical Times, May, 1916.
23. See #13 supra (p. 72).
24. D. Mauer and V. Vogel, Narcotics and Narcotics Addiction, Springfield, Illinois: C. C. Thomas, 1954 (P. 72).
25. H. Isbell, "Trends in Research On Opiate Addiction," Transactions and Studies of The College Of Physicians Of Philadelphia, Vol. 24 No. 1, June, 1956.
26. H. Isbell and W. W. White, "Clinical Characteristics of Addictions," American Journal of Medicine, Vol. 14, No. 5, May, 1953 (P. 558).
27. See #24, supra (p. 20).
28. H. Kreuger, N. Eddy, and M. Sumwalt, "The Pharmacology of the Opium Alkaloids: Parts 1 and 2." Public Health Reports, Supplement No. 165, 1943 (P. 758).
29. Ibid. (p. 808) .
30. Op. Cit. (pp. 5-6).
31. A. Wikler, Opiate Addiction, Springfield, Illinois: C. C. Thomas, 1953 (PP. 36-37)
32. A. Lindesmith, Opiate Addiction, Evanston, Illinois: Principia Press, 1947 (PP. 87-88).
33. Ibid. (p. 165)
34. Op. Cit. (p. 729)
35. Documentation of the Fifth Annual Conference of Committees of the World Narcotic Defense Association and International Narcotic Education Association, New York, 1992.
36. M. Nyswander, The Drug Addict As A Patient, New York: Grune & Stratton, 1956 (P. 61).
37. L. Kolb, "Drug Addiction As A Public Health Menace," Scientific Monthly, May, '939 (P. 4).
38. A. Wikler and R. Rasor, Psychiatric Aspects of Drug Addiction, American Journal of Medicine, Vol. 14, No. 5, May, 1953 (PP. 567-568) ss. Op. Cit. (p. 61).
40 See #28, supra (p. 709).
43. V. Vogel, H. Isbell, and K. Chapman, "Present Status of Narcotic Addiction," Journal of American Medical Association, Vol. 188, December 4, 1948 (P. 1997) .
45. Ibid. (pp. 1997-1999).
47. D. Gerard and C. Kornetsky, Adolescent Opiate Addiction: A Study Of Control And Addict Subjects, Research Center For Human Relations New York University, 1955.
48. C. Winick, "Narcotics Addiction And Its Treatment," Law And Contemporary Problems, Duke University School of Law, Vol. 42, W. 1957 (pp. 9-34).
49. See, for example, R. F. Faris and H. W. Dunham, Mental Disorders In Urban Areas, Chicago, 1999.
50. Illinois Institute for Juvenile Research and Chicago Area Project, Drug Addiction Among Young Persons In Chicago, October, 1953.
51. Research Center for Human Relations, New York University, Studies On Narcotics Use Among Juveniles, September,1955 (P. 12).
52. See #50. supra (p. 16).
53. Research Center for Human Relations, New York University, Heroin Use and Street Gangs, March, 1956 (pp. 12-13).
54. Research Center for Human Relations, New York University, Family Background As An Etiologic Factor In Personality Predisposition To Heroin Addiction, 1956 (P. 8).
55. Ibid. (p. 9).
56. Law and Contemporary Problems, Duke University School of Law, Vol. 22, No. 1, 1957 (PP. 53-54)
57. Special Report on Heroin Addiction in Chicago, 1957 (P. 43).
58. M. Pescor, "A Study Of Drug Addicts," Public Health Reports, Supplement No. 143. 1943.
59. H. J. Anslinger and W. F. Tompkins, The Traffic in Narcotics, New York: Funk & Wagnalls Co., 1953 (P. 170).
60. L. Kolb, "Drug Addiction, A Study Of Some Medical Cases," Archives of Neurology and Psychiatry, Vol. 20, 1928 (pp. 171-188).
61. See #50, supra (p. 6).
62. See #57. Supra.
64. Ibid. (p. 44).
65. See #24, supra (p. 211).
66. Ibid. (pp. 216-217).
67. 38 Stat. 785 (1914). 26 U. S. C. 2550 (1959), 16 U. S. C. 4701 (1954).
68. 249 U. S. 86 (1919).
71. (Italics mine).
72. 249 U. S. 96, 100.
73. Jin Fuey Moy v. United States, 254 U. S. 189 (1920).
74. Ibid. (p. 194) (Italics mine).
75. United States v. Behrman, 258 U. S. 280 (1922).
76. Ibid. (p. P89).
77. United States v. Linder, 268 U. S. 5 (1925).
78. Ibid. (p. 18) (Italics mine).
81. 276 U. S. 332, 342 (1928).
82. 271 U. S. 104 (1926) .
84. 72 F. 2d 589 (10th Cir., 1994).
86. 16 F. 2nd 709 (5th Cir., 1927).
88. Ibid. (p. 245).
89. 15 F. Supp. 558. Dist. Ct., S. D. Calif. (1936).
91. Department of Public Health, Illinois, Medical Counselling For Drug Addicts, 1953.
93. M. J. Pescor, "Follow-Up Study of Treated Narcotic Drug Addicts," Public Health Reports, Supplement No.170, 1941 (PP. 1-18).
94. A. Lambert, "Narcotic Addiction, Report Of The Mayor's Committee To Honorable Richard C. Patterson, Jr., Commissioner Of Correction," Journal Of American Medical Association, Vol. 98, October 26, 1929 pp 1297-1801).
95. R. Knight and C. Prout, "A Study Of Results in Hospital Treatment of Drug Addictions," American Journal of Psychiatry, Vol. 108, 1951 (P. 303)
96. Research Center for Human Relations, New York University, Post Hospitalization Adjustment: A Follow-up Study Of Adolescent Opiate Addicts, October, 1956 (P. 48).
97. L. Kolb, "Let's Stop This Narcotic Hysteria," Saturday Evening Post, 829: July 28, 1956
99. Report of the Council on Mental Health, of the American Medical Association, on Narcotic Addiction (p. 86).
100. G. H. Stevenson, "Arguments For And Against The Legal Sale of Narcotics," Bulletin of the Vancouver Medical Association, Vol. 31, No. 4.
101. Op. Cit.
102. See #99. Supra (pp. 7-8).
103 Ibid. (p. 48).
104 Sheldon and Eleanor Glueck, 500 Criminal Careers; 1,000 Juvenile Delinquents; 500 Delinquent Women; Unravelling Juvenile Delinquency.
105. Senate Report (p. 880).
|
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"redpajama_set_name": "RedPajamaCommonCrawl"
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{"url":"https:\/\/tex.stackexchange.com\/questions\/519680\/issue-with-alignment-when-using-multi-column-and-hskip","text":"# Issue with alignment when using multi-column and @{\\hskip}?\n\nThere is a slight offset of alignment of the right column header (which is a multicolumn entry) and the column to which it is supposed to be left-aligned below. If the @{\\hspace{1em}} is removed from the tabular, the columns line up great, but as long as there's an \\hspace or \\hskip there, the second column's multicolumn header is slightly offset.\n\nHow can I add space between columns without ruining the alignment?\n\n\\documentclass{article}\n\n\\begin{document}\n\n\\begin{tabular}{ll@{\\hspace{1em}}ll}\n\\hline\nThing 1 & \u2018I am a quote\u2019 & Thing 4 & \u2018Last column, cell 1\u2019\\\\\nThing 2 & \u2018I am quote 2\u2019 & Thing 5 & \u2018Last column, cell 2\u2019\\\\\nThing 3 & \u2018I am quote 3\u2019 & Thing 6 & \u2018Last column, cell 3\u2019\\\\\n\\end{tabular}\n\n\\end{document}\n\n\nYou have to manually suppress the insertion of the default \\tabcolsep before the second \\multicolumn using @{}.\n\n\\documentclass{article}\n\n\\begin{document}\n\n\\begin{tabular}{ll@{\\hspace{1em}}ll}\n\\hline\nThing 1 & \u2018I am a quote\u2019 & Thing 4 & \u2018Last column, cell 1\u2019\\\\\nThing 2 & \u2018I am quote 2\u2019 & Thing 5 & \u2018Last column, cell 2\u2019\\\\\nThing 3 & \u2018I am quote 3\u2019 & Thing 6 & \u2018Last column, cell 3\u2019\\\\\n\\end{tabular}\n\n\\end{document}\n\n\n\u2022 Interesting. I'm not familiar enough with the internals of tabular (by which I mean not really familiar at all) to know exactly why tabcolsep needs to be suppressed before the second multicolumn, but I'll try to read up on it and, in the meantime, just use this as the fix. Thanks! \u2013\u00a0drewx0r Dec 9 '19 at 0:52\n\u2022 @drewx0r If you type l, LaTeX will insert \\tabcolsep on the left and right of that column, unless there is @{...} on that side. \u2013\u00a0Henri Menke Dec 9 '19 at 2:02\n\nWhy not add some blank columns in between?\n\n\\documentclass{article}\n\n\\begin{document}\n\n\\begin{tabular}{llllll}","date":"2020-12-04 07:59:42","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\": 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\": 0.7515472173690796, \"perplexity\": 4694.622487342524}, \"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-2020-50\/segments\/1606141735395.99\/warc\/CC-MAIN-20201204071014-20201204101014-00177.warc.gz\"}"}
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package com.foc.business.notifier.manipulators;
import com.foc.business.notifier.FocNotificationEvent;
import com.foc.business.notifier.FNotifTrigger;
public class FocNotificationTableDeleteManipulator extends AbstractFocNotificationTableAddManipulator {
@Override
public boolean shouldTreatEvent(FNotifTrigger notifier, FocNotificationEvent event) {
return isSameDBTableAsFocObject(notifier, event);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
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| 5,601
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{"url":"https:\/\/math.stackexchange.com\/questions\/2596284\/cantor-bendixson-rank-of-a-tree-over-omega-possible-values-and-more","text":"# Cantor-Bendixson rank of a tree over $\\omega$ : possible values and more.\n\nThis is exercise 5.12 from Schimmerling's \"A course on set theory\", which states the following:\n\nProve by induction that, for every ordinal $\\delta$ < $\\omega_1$, there is a tree $\\mathcal T$ on $\\omega$ whose Cantor-Bendixson rank is $\\delta$ and $\\mathcal T^\\delta$ = $\\emptyset$.\n\nIt's the \"and...\" part that's been bugging me.\n\nTake $\\delta$ = $\\omega$ and suppose $\\mathcal T$ satisfies those conditions.\n\nThen, for each $\\mathcal n < \\omega$, $$\\mathcal T^n \\neq \\emptyset,$$ otherwise we would have $\\mathcal T^{n+1} = (\\mathcal T^n)' = \\mathcal T^n.$\n\nSo each $\\mathcal T^n \\subseteq \\omega ^{<\\omega}$ is a non-empty tree, and thus $$<> \\space \\in \\space \\mathcal T^n,$$ where $<>$ denotes the empty sequence.\n\nWe get $$<> \\space \\in \\space \\mathcal T^\\omega,$$ since $\\mathcal T^\\omega = \\bigcap\\limits_{\\mathcal n < \\omega} \\mathcal T^n$ by definition, and we have a contradiction.\n\nWe could take any limit ordinal $\\delta < \\omega_1$ and get the same contradiction.\n\nWhat's wrong here?\n\nThanks.\n\nedit: Some clarifications, all are straight from Ernst Schimmerling's book. If more are needed please ask.\n\nFor each $s \\in \\omega ^{<\\omega}$, with $k < \\omega$ being the domain of $s$, define $$N_s = \\{ x \\in \\omega^{\\omega} : x {\\restriction_k} = s \\}.$$\n\nFor a tree $\\mathcal T$ on $\\omega$ define its set of infinite branches, $$[\\mathcal T] = \\{ x \\in \\omega^{\\omega} : x {\\restriction_n} \\in \\mathcal T \\space \\forall n < \\omega \\}.$$\n\nThen the Cantor-Bendixson derivative of $\\mathcal T$ is $$\\mathcal T' := \\{s \\in \\mathcal T : N_s \\cap [\\mathcal T] \\space has \\space at \\space least \\space two \\space elements \\}.$$\n\nFor a tree $\\mathcal T$ on $\\omega$ we can define the following descending sequence: $$\\mathcal T^0 = \\mathcal T,$$ $$\\mathcal T^{\\alpha + 1} = \\mathcal (T^\\alpha)'$$ for any ordinal $\\alpha$ and, for a limit ordinal $\\beta$, $$\\mathcal T^\\beta = \\bigcap\\limits_{\\mathcal \\alpha < \\beta} \\mathcal T^\\alpha.$$\n\nThose are all trees on $\\omega$.\n\nThe Cantor-Bendixson rank of $\\mathcal T$ is defined to be the least ordinal $\\delta$ such that $$\\mathcal T^{\\delta + 1} = \\mathcal T^\\delta.$$ One proves that the above equality is valid for some $\\delta < \\omega_1$, so the CB rank of $\\mathcal T$ is a countable ordinal.\n\nSorry if i'm writing too much but this is the only context where i've studied these things (trees, CB rank etc...).\n\n\u2022 How do you define $\\mathcal T'$? The standard definition is to look not at a tree but at its set of infinite branches and have the derivative be the set of accumulation points (in the usual topology of $\\omega^\\omega$). Ernst is doing something different here. Jan 8, 2018 at 2:11\n\u2022 Ok i'm gonna edit the main post Jan 8, 2018 at 2:24\n\u2022 You are absolutely right. You can prove the result for $\\delta$ successor. For $\\delta$ limit, you can prove the weaker version where $[\\mathcal T^\\delta]=\\emptyset$; can you prove the version where $\\mathcal T^\\delta=\\{\\langle\\rangle\\}$? Jan 8, 2018 at 4:12\n\u2022 Sir thanks for the answer, i'll think about your suggestion tomorrow cuz it is almost 5am here! Cumprimentos. Jan 8, 2018 at 4:46\n\n(This is very late, and the OP hasn\u2019t been around in several years, but I\u2019d like to get the question off the unanswered list.)\n\nYou are correct in thinking that if $$\\alpha<\\omega_1$$ is a limit ordinal, there is no tree $$\\mathcal{T}$$ of Cantor-Bendixson rank $$\\alpha$$ such that $$\\mathcal{T}^\\alpha=\\varnothing$$; the best that we can hope for is a tree $$\\mathcal{T}$$ of Cantor-Bendixson rank $$\\alpha+1$$ such that $$\\mathcal{T}^\\alpha=\\{\\langle\\rangle\\}$$. And in fact this is possible: for each $$\\alpha<\\omega_1$$ there is a tree $$\\mathcal{T}_\\alpha$$ such that $$\\mathcal{T}_\\alpha^\\alpha=\\{\\langle\\rangle\\}$$. Of course this means that for each successor ordinal $$\\alpha+1<\\omega_1$$ the tree $$\\mathcal{T}_\\alpha$$ is of Cantor-Bendixson rank $$\\alpha+1$$ and has empty $$(\\alpha+1)$$-st Cantor-Bendixson derivative.\n\nAs one would expect, the trees $$\\mathcal{T}_\\alpha$$ can be constructed recursively, but some notation helps: for $$\\mathcal{S}\\subseteq\\omega^{<\\omega}\\setminus\\{\\langle\\rangle\\}$$ and $$p\\in\\omega^{<\\omega}$$ let $$p^{\\frown}\\mathcal{S}=\\{p^\\frown s:s\\in\\mathcal{S}\\}$$.\n\nClearly $$\\mathcal{T}_0=\\{\\langle\\rangle\\}$$. Given $$\\mathcal{T}_\\alpha$$ for some $$\\alpha<\\omega_1$$, let\n\n$$\\mathcal{T}_{\\alpha+1}=\\{\\langle\\rangle\\}\\cup\\bigcup_{n\\in\\omega}\\Big(\\left(0^{n+1}\\right)^{\\frown}\\mathcal{T}_\\alpha\\Big)\\cup\\bigcup_{n\\in\\omega}\\Big(\\left(1^{n+1}\\right)^{\\frown}\\mathcal{T}_\\alpha\\Big)\\,;$$\n\nthen\n\n$$\\mathcal{T}_{\\alpha+1}^\\alpha=\\{\\langle\\rangle\\}\\cup\\{0^{n+1}:n\\in\\omega\\}\\cup\\{1^{n+1}:n\\in\\omega\\}=\\mathcal{T}_1\\,,$$\n\nso $$\\mathcal{T}_{\\alpha+1}^{\\alpha+1}=\\{\\langle\\rangle\\}$$, as desired.\n\nNow suppose that $$\\alpha<\\omega_1$$ is a limit ordinal and that we have constructed $$\\mathcal{T}_\\xi$$ for all $$\\xi<\\alpha$$. Let $$\\langle\\alpha_n:n\\in\\omega\\rangle$$ be a strictly increasing sequence of ordinals whose limit is $$\\alpha$$, and let\n\n$$\\mathcal{T}_\\alpha=\\{\\langle\\rangle\\}\\cup\\bigcup_{n\\in\\omega}\\big(\\langle n\\rangle^\\frown\\mathcal{T}_{\\alpha_n}\\big)\\,;$$\n\nclearly $$\\mathcal{T}_\\alpha^\\alpha=\\{\\langle\\rangle\\}$$.\n\nIt\u2019s not too hard to picture what\u2019s going on here. $$\\mathcal{T}_1$$ is just a pair of branches, $$\\{0^n:n\\in\\omega\\}$$ and $$\\{1^n:n\\in\\omega\\}$$, intersecting at the root $$\\langle\\rangle$$. $$\\mathcal{T}_{\\alpha+1}$$ is built by making each non-root node of $$\\mathcal{T}_1$$ the root of a tree isomorphic to $$\\mathcal{T}_\\alpha$$. $$\\mathcal{T}_\\alpha$$ for limit $$\\alpha$$ is built from the bush $$\\{s\\in\\omega^{<\\omega}:|s|\\le 1\\}$$ by making each node $$\\langle n\\rangle$$ the root of a tree isomorphic to $$\\mathcal{T}_{\\alpha_n}$$.\n\n\u2022 Hi professor, could I ask your assistance here, please? Perhaps I found an answer to a question I posed but I am not sure it is correct. Forgive for the bother. Apr 13 at 19:11","date":"2022-08-09 05:07:33","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\": 43, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9642792344093323, \"perplexity\": 184.2760901608619}, \"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-2022-33\/segments\/1659882570901.18\/warc\/CC-MAIN-20220809033952-20220809063952-00400.warc.gz\"}"}
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Trump initially announced his intention to name "THE MOST DISHONEST & CORRUPT MEDIA AWARDS OF THE YEAR" (all caps emphasis his) on Tuesday, January 2 - the same day he fired off multiple confounding tweets, including an alarming threat at North Korea that was apparently prompted by a segment on Fox News. He added that the victor of such an award should get a "fake news trophy". The president gave an exclusive, 30-minute interview, with no aides around, to The New York Times' Mike Schmidt last week while at his club Mar-a-Lago in Palm Beach, Florida.
"TIME "MISTAKENLY" reported that President Trump removed a bust of Martin Luther King, Jr from the Oval Office", the email said as it asked the participants to "let the President know if there is another story you think should be crowned as the 2017 KING of Fake News".
The ad includes a photo of Trevor as well as some quotes calling the show "monotonous ... liberal dogma" and "a fake news show".
Trump had promised to hold the mock awards show to castigate mainstream news organizations for their coverage of his presidency. Maybe-maybe-there was a chance that we would, for at least a couple minutes, stop paying attention to Donald Trump. It's more like that wrestling video in which Trump body-slams a guy with a CNN head-something that will excite his base, which already distrusts the press and loves the president's media-bashing.
It is unclear how Mr Trump's awards will be judged.
|
{
"redpajama_set_name": "RedPajamaC4"
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| 5,763
|
{"url":"https:\/\/www.kingscountyradioclub.com\/roy\/","text":"# Roy\u2019s Place For Web Links:\n\n### Magnetic Logic \u2013 How To Get Hysteresis Loss To Work For You (A Great YouTube Video)\n\n*********************************************************************\n\n### \u2026And Here\u2019s A Very Nice YouTube Veritasium Video On The Kibble (watt) Scale Used\n\n.\n\n(That small white dot in the middle is Oumuamua \u2013 do you see the little alien waving \u201chello\u201d out of the porthole?)\n\n.\n\nThis is Oumuamua\u2019s trajectory into our Solar System and then out again.\n\n## Correlation Does Not Imply Causation!\n\n#### Just because almost every person that has died has lied down on a mattress\u00a0within the previous 24 hours, doesn\u2019t mean that mattresses kill you.\n\n(Or has this entry been paid for by the notorious, secretive mattress manufacturing industry?????)","date":"2021-01-16 21:05:17","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\": 0.4537148177623749, \"perplexity\": 4795.218514125982}, \"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-2021-04\/segments\/1610703507045.10\/warc\/CC-MAIN-20210116195918-20210116225918-00029.warc.gz\"}"}
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Nutrients in 85% Lean Ground Beef
By Rose Haney Updated November 27, 2018
Nutrition Guide for a Bacon & Egg Croissant Sandwich
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Ground beef has received a bad rap over the years because of its high saturated fat content. But ground beef has a number of beneficial nutrients your body requires, so you should not necessarily cut it out of your diet altogether. Leaner cuts of beef are easily found at your local supermarket. Just look for ground beef with less than 15 percent fat content, and eat it in moderation.
According to the Food Guide Pyramid, the recommended single serving size for lean meat is 3 ounces, based on a 2,200 calorie diet. The USDA National Nutrient Database for Standard Reference reports that a 3-ounce serving of lean 85 percent lean ground beef served as a broiled patty contains 197 calories. A 3-ounce serving of the same cut, but crumbled and pan-browned, contains 218 calories.
Fat Content
A main concern is the fat content of foods, especially saturated fat. Saturated fat is mostly found in animal products such as beef. Recommendations from MayoClinic.com suggest limiting your daily saturated fat intake to 10 percent or less of your total calories, because saturated fat consumption is linked to type 2 diabetes and heart disease. A 3-ounce broiled patty of 85 percent lean ground beef contains 5 grams of saturated fat. Additionally, it contains 5.7 grams of unsaturated fat, .4 grams of polyunsaturated fat, and .9 grams of trans fat. This serving size also includes 76 milligrams of cholesterol. MayoClinic.com suggests that a healthy diet limits cholesterol intake to fewer than 300 milligrams per day, because excessive cholesterol is also linked with increased risk of stroke and heart disease.
Lean ground beef is an excellent source of protein. Protein is essential for growth, tissue repair, energy and proper immune function. The USDA Dietary Reference Intake guidelines suggest 10 to 35 percent of your daily calories should come from protein. A 3-ounce broiled patty of 85 percent lean ground beef contains 22 grams of protein. According to the McKinley Health Center, protein that comes from animal sources, such as ground beef, provides all the essential amino acids your body needs.
Minerals and Vitamin Content
Ground beef contains a number of mineral and vitamins that are beneficial for you. A 3-ounce serving of 85 percent lean ground beef is a rich source of selenium, iron and phosphorus. It provides 32.5 percent, 29.5 percent and 27.6 percent, respectively, of your daily requirements for these minerals. Additional minerals in a single serving include calcium, magnesium, potassium, sodium, zinc, copper and manganese. Ninety-two percent of your daily B12 intake and 25 percent of your daily B6 intake is also present in a single serving. Additional B vitamins include thiamin, riboflavin, niacin and pantothenic acid. Minor amounts of vitamins E and K are also present.
The Food Guide Pyramid: Dietary Guidelines
USDA National Nutrient Database for Standard Reference: Ground Beef Calculator
USDA: Dietary Reference Intakes for Energy, Carbohydrate, Fiber, Fat, Fatty Acids, Cholesterol, Protein, and Amino Acids
Rose Haney received her joint doctoral degree in clinical psychology from San Diego State University and University of California, San Diego. Her clinical and research specialties are neuropsychology and neuroimaging. She has been published in several scientific journals and has presented her work at numerous national conferences. She has been freelance writing since 2008.
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1 The 9 best indoor cycling bikes for your home gym
2 What Nutrients Are in Brisket?
3 The Nutrition in a Beef Chuck Boneless Top Blade Steak
4 Is Salami Healthy?
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 7,515
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\section{Introduction: geodesic length separation in negative curvature}
On negatively curved manifolds, the number of closed geodesics of length $\leq T$
grows exponentially in $T.$
(We refer the reader to \cite{MS, PP, PS} for a comprehensive discussion about
the growth and
distribution of closed geodesics).
The abundance of closed geodesics leads to the natural question about the
sizes of gaps in the length spectrum.
In the current note we present a number of results related to this question.
In some situations we are able to control
the gaps from below, while in other we show that such control is not possible
in general.
We note that a presence of exponentially large multiplicities in the length spectrum
of a Riemannian manifold (which can be considered as a limiting case of small
gaps) changes the level spacings distribution of Laplace eigenvalues on that manifold,
see e.g. \cite{Luo:Sarnak}.
For generic Riemannian metrics, the length spectrum is {\em simple}
\cite{Abr,Anosov-generic}, so for
any closed geodesic $\gamma$, only $\gamma^{-1}$ will have the same length.
So, by the Dirichlet
box principle, there exist {\em exponentially small} gaps between the lengths of different
geodesics.
Accordingly, it seems interesting to investigate manifolds where the gaps between the
lengths of different geodesics have exponential {\em lower bound:} there exist
constants $C,\beta>0$, such that for any $l_1\neq l_2\in {\rm
Lsp}(M)$ (length spectrum of the negatively curved manifold $M$), we have
\begin{equation}\label{separation}
|l_2-l_1|>Ce^{-\beta\cdot\max(l_1,l_2)}.
\end{equation}
This assumption
is satisfied for arithmetic hyperbolic groups by the trace separation criterion
(cf. \cite{Takeuchi} and \cite[\S 18]{Hej}).
In Section~\ref{ScDiop2D}
we explain (see Theorem \ref{ThAlgGen}) why the assumption \eqref{separation}
holds for hyperbolic manifolds whose fundamental group has algebraic elements.
In particular, the surfaces satisfying \eqref{separation} form a dense set in the corresponding
Teichmuller space.
On the other hand the existence of arbitrary small gaps is {\em topologically generic}
as is shown in
Theorem \ref{ThTop-HS} for surfaces of constant negative curvature and in
Theorem \ref{thm:smallgapneg} for the space of negatively curved metrics endowed
with $C^r$-topology, for any $r>0.$
While arbitrary small gaps are topologically generic, it is plausible that the gaps
are not too small
for almost every metric. One result in this direction is presented in
Section \ref{ScH2-KR} there we obtain an explicit
lower bound for the gaps valid for almost every hyperbolic surface.
Length separation between closed geodesics is relevant for the study of
wave trace formulas on
negatively-curved manifolds: to accurately study contributions from exponentially
many closed geodesics to the
wave trace formula, it is necessary to separate contributions from geodesics which differ
either on the length axis, or in phase space. We remark that a suitable version
of \eqref{separation} always holds in {\em phase space}: small tubular
neighbourhoods of closed geodesics
in phase space are disjoint, as shown in \cite{JPT}.
Since there exist metrics for which the size of the length gaps
cannot be controlled (Theorem \ref{thm:smallgapneg}), the authors
in \cite{JPT} established microlocal
wave trace formula, and used the separation of closed trajectories in
phase space in the proof.
\section{Diophantine results for hyperbolic manifolds.}
\label{ScDiop2D}
\subsection{Distances between algebraic numbers.}
In this section we consider gaps in the length spectrum for manifolds whose fundamental group
admits algebraic generators. But first we provide a few general results about the algebraic numbers.
\begin{lemma}
\label{LmSmallestRoot}
If $\alpha$ is a root of $P(x)=x^D+a_{D-1} x^{D-1}+\dots+a_0$ then
$$ |\alpha|\geq \frac{|a_0|}{\left(1+\sum_{j=0}^{D-1} |a_j|\right)^{D-1}}. $$
\end{lemma}
\begin{proof}
Let $\alpha_j$ be the roots of $P$ counted with multiplicities.
We claim that $|\alpha_j|\leq R:=1+\sum_j {|a_j|}.$ Indeed if $|x|>R$
then since $R>1$ we get
$$ |P(x)|\geq |x|^D-\sum_{j=0}^{D-1} |a_j| |x|^j\geq
|x|^{D-1}\left(|x|-\sum_{j=0}^{D-1} |a_j|\right)>0. $$
The result follows since $\prod_j |\alpha_j|=|a_0|.$
\end{proof}
Given a field $K$ which is an extension of ${\bf Q}$ of degree $d$ let ${\mathfrak{H}}(L, N, p)$ be the set of all elements of $K$ of the
form $\frac{\beta}{N^p}$ where $\beta\in{\mathbf{O}}_K$ and for each automorphism $\sigma_j$ of $K$ we have
$|\sigma_j(\beta)|\leq L.$
\begin{lemma}
\label{LmAuto}
If $0\neq \alpha\in {\mathfrak{H}}(L, N, p)$ then
$$ |\alpha|\geq \frac{1}{L^{d-1} N^p}. $$
\end{lemma}
\begin{proof}
Indeed $|\beta| L^d\geq 1$ because $\displaystyle \prod_{j=1}^d |\sigma_j(\beta)|\geq 1. $
\end{proof}
Let ${\mathfrak{I}}(L, N, p, D)$ be the set of numbers which satisfy
$$ \alpha^{E}+a_{E-1} \alpha^{E-1}+\dots+a_0=0 $$
where $E\leq D$ and $a_j\in {\mathfrak{H}}(L,N,p).$
\begin{corollary}
\label{CrAlgAlg}
If $0\neq \alpha\in {\mathfrak{I}}(L, N, p, D)$ then
$$ |\alpha|\geq \frac{1}{L^{d-1} N^p (DL+1)^{D-1}}. $$
\end{corollary}
\begin{proof}
Since $\alpha\neq 0$ we can assume after possibly reducing the degree of the polynomial that
$a_0\neq 0.$ Then the result follows by combining Lemmas \ref{LmSmallestRoot}
and \ref{LmAuto}
\end{proof}
\begin{prop}
\label{PrSumAlg}
(see e.g. \cite[Section 5.8]{vW})
There exists constants $C$ and $q$ such that
if $\alpha_1, \alpha_2\in {\mathfrak{I}}(L, N, p, D)$ then
$\alpha_1+\alpha_2$ and $\alpha_1-\alpha_2$ are in
${\mathfrak{I}}(CL^q, N, pq, D^2).$
\end{prop}
Combining Proposition \ref{PrSumAlg} with Corollary \ref{CrAlgAlg} we obtain
\begin{corollary}
\label{CrAlgDif}
If $\alpha_1, \alpha_2\in {\mathfrak{I}}(L, N, p, d)$ then either
$\alpha_1=\alpha_2$ or
$$ |\alpha_1-\alpha_2|\geq \frac{c}{L^{q(d-1)} N^{pq} L^{D^2}}. $$
\end{corollary}
\subsection{Manifolds with algebraic generators of $\pi_1$}
We now formulate the main result of this section.
\begin{theorem}
\label{ThAlgGen}
Let ${X}$ be a hyperbolic manifold such that the generators of $\pi_1({X})$ belong to
$PSO_{n,1}(\bar{{\bf Q}}).$ Then \eqref{separation} holds.
\end{theorem}
We remark that in dimension 2 groups
satisfying the assumptions of
Theorem \ref{ThAlgGen} form a dense set in the corresponding
Teichmuller space $T_g$. This can be established, for example, by the arguments of Section~\ref{ScH2-KR}.
If $n\geq 3$ then \cite[Theorem 0.11]{GR} building on earlier results of of Selberg \cite{Selberg} and Mostow (\cite{Mostow}) shows
the conditions of Theorem \ref{ThAlgGen} are satisfied for all finite volume hyperbolic manifolds.
Hence we obtain
\begin{corollary}
\eqref{separation} holds for finite volume hyperbolic manifolds of dimension $n\geq 3.$
\end{corollary}
The proof of Theorem \ref{ThAlgGen} is similar to the proof of
Proposition 3 in \cite{GJS}, where it is shown that the
rotation matrices in ${\rm SU}(2)\cap M_2(\overline{{\bf Q}})$
satisfy the {\em Diophantine condition} defined in \cite{GJS}.
Related results for other Lie groups were established in \cite{ABRS,Br11,Varju}.
Related questions were also discussed in \cite{Glu}.
\subsection{Proof of Theorem \ref{ThAlgGen}}
\begin{proof}
Let $\gamma_1$ and $\gamma_2$ be two closed geodesics. Let $l_j$ be the length of $\gamma_j,$
$W_j$ be the word fixing $\gamma_j,$ $B_j$ be the matrix corresponding to $W_j,$
$m_j$ be the word length of $W_j$ and $r_j=l_j/2.$ To establish
\eqref{separation} it suffices to show that
\begin{equation}
\label{SeparationExp}
\left| e^{r_1}-e^{r_2}\right| \geq {\bar C} e^{-{\bar c} \max(r_1, r_2)}.
\end{equation}
Without a loss of generality we assume that $m_j\gg 1.$
By (\cite[Lemma 2]{Milnor}) we know that
\begin{equation}
\label{EqWL-GL}
\frac{l_j}{C} \leq m_j \leq C l_j
\end{equation}
so \eqref{separation} if trivial unless $m_j$ and $m_2$ are comparable. Let us assume to fix our ideas
that $m_2\geq m_1.$ By assumption there is a finite extension $K$ of ${\bf Q}$ and numbers
$L$ and $N$ such that all entries of the generators belong to
${\mathfrak{H}}(L, N, 1).$ Accordingly the entries of $B_j$ belong to
$${\mathfrak{H}}((L(n+1))^{m_j}, N, m_j).$$
Closed geodesics on ${X}$ correspond to {\em loxodromic} elements of
$\pi_1({X})\subset PSO_{n,1}$ (also called {\em boosts})
that that fix no points in ${\mathcal{H}}^n$ and fix two points in $\partial{\mathcal{H}}^n$.
It is shown in the proof of \cite[Thm. I.5.1]{Franchi} that $B_j$ has precisely two positive
real eigenvalues $\alpha_{1,j}=e^{r_j}$ and $\alpha_{2,j}=e^{-r_j}$; all other eigenvalues of
$B_j$ have modulus one. Since the coefficients of the characteristic polynomial of $B_j$
are the sums of minors we have
$$ e^{r_j}\in {\mathfrak{I}}((L(n+1))^{(n+1)m_j} (n+1)!, N, m_j (n+1)). $$
Reducing to the common denominator we see that both $e^{r_1}$ and $e^{r_2}$ belong to
$$ {\mathfrak{I}}((L(n+1))^{(n+1)m_2} N^{m_2-m_1} (n+1)!, N, m_2 (n+1)). $$
Now \eqref{SeparationExp} follows by Corollary \ref{CrAlgDif} and \eqref{EqWL-GL}.
\end{proof}
\begin{remark}
In dimension two the proof can be simplified slightly by remarking that
$2\cosh(l_j/2)={\rm tr} B_j\in K$.
An alternative proof of Theorem \ref{ThAlgGen} could proceed by
using explicit formulas for the lengths of closed geodesics on hyperbolic manifolds
(see e.g. \cite[(3), p. 246]{Prasad:Rapinchuk}) and the estimates for linear forms
in logarithms (see e.g. \cite[Chapter 2]{BW}).
The proof we give is more elementary, using only basic facts about
algebraic numbers and matrix eigenvalues; and fairly concrete.
\end{remark}
\section{Small gaps for surface of constant negative curvature.}
Let $$G_g=\{(A_1, \dots A_{2g})\in (SL_2({\bf R}))^{2g}:
[A_1, A_2] [A_3, A_4]\dots [A_{2g-1}, A_{2g}]=I\}. $$
\begin{theorem}
\label{ThTop-HS}
The set of tuples $(A_1, A_2\dots A_{2g})\in G_g$
where \eqref{separation} fails is topologically generic.
\end{theorem}
\begin{proof}
Let $\gamma_A$ denote the closed geodesic whose lift to the fundamental cover joins $q$ and $A q.$
Let $\mathbb{L}$ denote the length spectrum of the geodesics $\gamma_A$ where $A$ belongs to a subgroup
generated by $A_1$ and $A_2.$ Note that for a dense set of tuples it holds that for each $\delta$
there exists $L$ such that for $l>L$ the set $[l, l+\delta]$ intersects $\mathbb{L}.$
One way to see this is to consider the
geodesics $\gamma_{A_1^k A_2^m}.$ Their length have asymptotics
$$ \kappa(A_1, A_2) k \lambda_1+m \lambda_2 $$
where $e^{\lambda_j}$ is the leading eigenvalue of $A_j.$
Note that for typical $A_1, A_2$ we have $\kappa(A_1, A_2)\neq 0$ and $\lambda_1$ and $\lambda_2$
are non commensurable. Consider a geodesic ${\bar\gamma}=\gamma_{A_3 A_1^n}$ where
$n$ is very large. By the foregoing discussion there exists $l\in \mathbb{L}$ such that
$|l-L_{\bar\gamma}|<\delta.$ Now consider the perturbations of $A_3$ of the form
$ A_3(\eta)=\left(\begin{array}{cc} 1 & \eta \\
0 & 1
\end{array}\right) A_3 . $
Assume that
$A_3 A_1^n=\left(\begin{array}{cc} a & b \\
c & d
\end{array}\right) .$
After applying a small perturbation if necessary we can assume that all entries of this matrix have the same
order as its trace. Then
$$ {\rm tr}(A_3(\eta) A_1^n)={\rm tr}(A_3 A_1^n)+\eta c, $$
so by a small perturbation we can make $L_{\gamma_{A_3(\eta) A_1^n}}$ as close to
$l$ as we wish. Now the result follows by a standard Baire category argument (cf. Section \ref{SSVarCurvSG}).
\end{proof}
\section{Constructing metrics with small gaps in the length spectrum}
\label{SSVarCurvSG}
This section is devoted to the proof of the following fact.
\begin{theorem}\label{thm:smallgapneg}
For any $r>3$ for any negatively-curved $C^r$ metric $g$,
for any function $F(t)$ (which we assume is monotone and
fast decreasing), and a number $\delta>0$, there exists a metric
${\bar g}$, such that $||{\bar g}-g||_{C^r}<\delta$
and there exists an infinite sequence of pairs of
closed ${\tilde g}$-geodesics $\gamma_{1,j},\gamma_{2,j}$ with
$L_{\bar g}(\gamma_{i,j})\to\infty$ as $j\to\infty$, and
\begin{equation}\label{smallgap}
|L_{\bar g}(\gamma_{1,j})-L_{\bar g}(\gamma_{2,j})|<\min
\{F(L_{\bar g}(\gamma_{1,j})),F(L_{\bar g}(\gamma_{2,j}))\}.
\end{equation}
\end{theorem}
This shows that, in general, one {\em cannot} obtain good lower
bounds for gaps in the length spectrum for
a $C^r$ open set of negatively curved metrics.
Theorem \ref{thm:smallgapneg} follows from the lemma below by a standard Baire category argument.
\begin{lemma}
\label{LmEqLen}
Given a metric $g$ and numbers $L$ and $\delta$ there is a metric
${\tilde g}$ such that $||g-{\tilde g}||_{C^r}\leq \delta$ and there are two ${\tilde g}$-geodesics $\gamma_1$ and $\gamma_2$ such that
$$L_{\tilde g}(\gamma_1)=L_{\tilde g}(\gamma_2)>L.$$
\end{lemma}
We also need the following fact
\begin{lemma}
\label{LmComp}
Let $g$ and ${\tilde g}$ be two negatively curved metrics such that $||g-{\tilde g}||_\infty\leq \delta$ and
$\gamma$ and ${\tilde\gamma}$ be two closed geodesics for $g$ and ${\tilde g}$ respectively of lengths
$L$ and ${\widetilde{L}}$. If $\gamma$ and ${\tilde\gamma}$ are homotopic then
$$ \frac{L}{1+\delta}\leq{\widetilde{L}}\leq L(1+\delta). $$
\end{lemma}
\begin{proof}
Recall that for negatively curved there exists a unique geodesic in each homotopy class and this geodesic is length
minimizing. The second inequality follows since the length of $\gamma$ with respect
to ${\tilde g}$ is at most $L(1+\delta)$ and ${\tilde\gamma}$ is shorter. The first inequality follows from the second by interchanging
the roles of $g$ and ${\tilde g}.$
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:smallgapneg}]
We claim that given metric $g$ and numbers $k\in \mathbb{N}$ and $\delta>0$ there exists a metric ${\bar g}$ such that
$||{\bar g}-g||_{C^r}<\delta$ and for each $j=1\dots k$ there are geodesics $\gamma_{1,j}, \gamma_{2,j}$ such that
\begin{equation}
\label{EqKGaps}
L_{\bar g}(\gamma_{i,j})>j, \quad |L_{\bar g}(\gamma_{1,j})-L_{\bar g}(\gamma_{2,j})|\leq
F(\max(L_{\bar g}(\gamma_{1,j}), L_{\bar g}(\gamma_{2,j}))).
\end{equation}
It follows that the space of metrics satisfying \eqref{smallgap} is topologically generic and hence dense.
It remains to construct ${\bar g}$ satisfying \eqref{EqKGaps}.
By Lemma \ref{LmEqLen} we can find $g_1$ such that $||g-g_1||_{C^r}<\frac{\delta}{2}$
and there are two geodesics $\gamma_{1,1}$ and $\gamma_{2,1}$ such that
$$ L_{g_1}(\gamma_{i,1})>1 \text{ and } L_{g_1}(\gamma_{1,j})=L_{g_1}(\gamma_{2,j}). $$
For $j\geq 1$ we apply Lemma \ref{LmEqLen} to find $g_j$ such that
$$ ||g_j-g_{j-1}||_{C^r} \leq\min\left(\frac{\delta}{2^j}, \min_{l=1}^{j-1}
\frac{F(L_{g_l}(\gamma_{1,l}))+1)}{L_{g_l}(\gamma_{1,l}) 2^{j-l+1}}\right) $$
and there are two geodesics $\gamma_{1,j}, \gamma_{2,j}$ such that
$$ L_{g_j}(\gamma_{1,j})=L_{g_j}(\gamma_{2,j})>j. $$
Then $g_k$ satisfies the required properties since, by Lemma \ref{LmComp},
the lengths of $g_{i,l}$ have changed by less than
$F(L_{g_l}(\gamma_{1, l})+1)/2$ in the process of making consecutive inductive steps.
\end{proof}
\begin{remark}
In particular if we continue the above procedure for the infinite number of steps then the limiting metric
will satisfy the conditions of Theorem \ref{thm:smallgapneg}.
\end{remark}
The proof of Lemma \ref{LmEqLen} relies on two facts.
If $\gamma$ is a closed geodesic let $\nu_\gamma$ denote the invariant measure for the geodesic flow supported on
$\gamma.$ Let $h$ denote the topological entropy of the geodesic flow. Let $\mu$ denote the Bowen-Margulis
measure. Recall \cite{PP} that $\mu$ the measure of maximal entropy for the geodesic flow. It has a full support
in the unit tangent bundle $SM.$
\begin{lemma}\cite[Theorem 6.9 and Proposition 7.2]{PP}
\label{LmMor}
$ Lh e^{-Lh} \sum_{L(\gamma)\leq L} \nu_\gamma $ converges as $L\to\infty$ to $\mu.$
\end{lemma}
\begin{lemma}
\label{LmAvoid}
For each $q_0\in M$ there exists ${\varepsilon}$ such that for each $L$ there is periodic geodesic $\gamma$ such that
$L(\gamma)>L$ and $\gamma$ does not visit an ${\varepsilon}$ neighborhood of $q_0.$
\end{lemma}
\begin{proof}[Proof of Lemma \ref{LmEqLen}]
Pick a small ${\bar\delta}$ and large $L.$
By Lemma \ref{LmAvoid} there exists a closed geodesic $\gamma_1$
such that $L_g(\gamma_1)>L$ and $d(q(\gamma_1(t)), q_0)>{\varepsilon}).$
Let $\gamma_2$ be a closed geodesic such that $L_g(\gamma_1)<L_g(\gamma_2)<L_g(\gamma_1)+{\bar\delta}$
and $\gamma$ spends at least time $\mu(B(q_0, {\varepsilon}/2)/2 L_g(\gamma_1)$ inside $B(q_0, {\varepsilon}/2)$
(the existence of such a geodesic follows from Lemma \ref{LmMor}). Take
${\tilde g}^\eta=(1-\eta z(q)) g$ where $z(q)=1$ on $B(q_0, {\varepsilon}/2)$ and $z(q)=0$ outside $B(q_0, {\varepsilon}).$
We can choose $z$ so that $||z||_{C^r}=O\left({\varepsilon}^{-r}\right).$ Then $||g-{\tilde g}^\eta||_{C^r}=O(\eta/{\varepsilon}^r).$
Let $\gamma_2^\eta$ be the closed geodesic for ${\tilde g}^\eta$ homotopic to $\gamma_2.$ Note that
$\gamma_1$ is a geodesic of ${\tilde g}^\eta$ for each $\eta$ and
$L_{{\tilde g}^\eta}(\gamma_1)\equiv L_g(\gamma_1).$ Also
$$L_{{\tilde g}^\eta}(\gamma_2^\eta)\leq L_{{\tilde g}^\eta}(\gamma_2)\leq L_g(\gamma_1)+{\bar\delta}-\frac{\mu(B(q_0, {\varepsilon}/2) L_g(\gamma_1) \eta }{2} .$$
Accordingly there exists $\eta<\frac{2{\bar\delta}}{L \mu(B(q_0, {\varepsilon}/2))}$ such that
$L_{{\tilde g}^\eta}(\gamma_1^\eta)=L_{{\tilde g}^\eta}(\gamma_2)$ as claimed.
\end{proof}
In the proof of Lemma \ref{LmAvoid} we need several facts about the dynamics of the geodesic flow which
we call $\phi_t.$ Recall \cite{Anosov} that $\phi_t$ is
uniformly hyperbolic. In particular, there is a cone field ${\mathcal{K}}(x)$ and $\lambda>0$ such that for $u\in {\mathcal{K}},$
$||d\phi_t(u)||\geq e^{\lambda t}||u||.$ Moreover the cone field ${\mathcal{K}}$ can be chosen in such a way that
if $x=(q,v)$ and $u=(\delta q, \delta v)\in {\mathcal{K}}(x)$ then
\begin{equation}
\label{ConeGeom}
||\delta q||\geq c ||\delta v|| \text{ and }\angle(\delta q, v)\geq\frac{\pi}{4}
\end{equation}
We call a curve $\sigma$ {\it unstable} if $\dot{\sigma}\in{\mathcal{K}}.$ By the foregoing discussion if
$\sigma$ is un unstable curve then the length of the projection of $\phi_t(\sigma)$ on $M$ is
longer than $c e^{\lambda t}.$
\begin{proof}[Proof of Lemma \ref{LmAvoid}] We first show how to construct a not necessary closed geodesic
avoiding $B(q_0, {\varepsilon})$ and then upgrade the result to get the existence of a closed geodesics.
The first part of the argument is similar to \cite{BS, D}.
Pick a small $\kappa>0.$ Take an unstable curve $\sigma$ of small length $\kappa.$
We show that if $\kappa$ and ${\varepsilon}$ are sufficiently small then $\sigma$
contains a point such that the corresponding geodesic avoids $B(q_0, {\varepsilon}).$ Let $T_1$ be a number such that
$|\phi_{T_1} (\sigma)|=1$ where $\phi$ denotes the geodesic flow. Note that $T_1=O(|\ln\kappa|).$
Also observe that due to \eqref{ConeGeom} there exists a number $r_0$ such that if ${\tilde\sigma}$ is an unstable curve and
$x\in {\tilde\sigma}$ is such that $d(q(x), q_0)<{\varepsilon}$ then for all $y\in {\tilde\sigma}$ such that $C{\varepsilon}\leq{\bf d}(y,x)\leq r_0$
we have
$${\bf d}(q(\phi_{t} y), q_0)>{\varepsilon}$$
for $|t|<r_0$
where ${\bf d}$ denotes the distance in the phase space (just take $r_0$ much smaller than the injectivity
radius of $q_0$).
Thus the set
$$ \{y\in \phi_{T_1}(\sigma): d(q(\phi_{-t} y), q_0)\leq{\varepsilon} \text{ for some } 0\leq t\leq T_1\} $$
is a union of $O(|\ln\kappa|/r_0)$ components of
length $O({\varepsilon}/\kappa^a)$ for some $a>0.$
Therefore if $\kappa\ll 1$ and ${\varepsilon}\ll \kappa$ then the average distance between the components is
much larger than $\kappa.$ So we can find $\sigma_1\subset \phi_{T_1} \sigma$ such that
$|\sigma_1|=\kappa,$
and if
$y \in \sigma_1$ then $d(q(\phi_{-t} y), q_0)>{\varepsilon} $ for each $ 0\leq t\leq T_1.$
Take $T_2$ such that $|\phi_{T_2}\sigma_1|=1.$
Then we can find $\sigma_2\subset \phi_{T_2} \sigma_1$ such that
$|\sigma_2|=\kappa,$
and if
$y \in \sigma_2$ then $d(q(\phi_{-t} y), q_0)>{\varepsilon} $ for each $ 0\leq t\leq T_2.$
We continue this procedure inductively to construct arcs $\sigma_j$ for all $j\in \mathbb{N}.$
Taking $$x=\bigcap_{j=1}^\infty \phi_{-(T_1+T_2+\dots+T_j)} \;\; \sigma_j$$
we obtain a geodesic avoiding $B(q_0, {\varepsilon}).$
To complete the proof we need
\begin{lemma}
\label{LmClosing}
(Anosov Closing Lemma) (see \cite[Section 18]{HK})
Given $\eta>0$ there exists $\delta>0$ such that if for some $t_1, t_2$
such that $|t_2-t_1|$ is sufficiently large we have
${\bf d}(\gamma(t_1), \gamma(t_2)<\delta$ then there exists a closed geodesic ${\tilde\gamma}$ such that
$|L({\tilde\gamma})-|t_2-t_1||<\eta$ and for each $t\in[t_1, t_2]$ there exists $s$ such that
${\bf d}(\gamma(t), {\tilde\gamma}(s))<\eta.$
\end{lemma}
Take $\delta$ corresponding to $\eta={\varepsilon}/2.$ Consider points $\gamma(j L)$ where $j=1\dots K.$
By pigeonhole principle if $K$ is sufficiently large we can find $j_1, j_2$ such that
${\bf d}(\gamma(j_1 L), \gamma(j_2 L))<\delta$ and so there exists a closed geodesic ${\tilde\gamma}$ avoiding
$B(q_0, {\varepsilon}/2).$ Since ${\varepsilon}$ is arbitrary, Lemma \ref{LmAvoid} follows.
\end{proof}
Suppose now that $\dim(M)=2.$ Let ${\mathcal{H}}_r(M)$ denote the space of $C^r$ metrics with positive topological entropy.
This set is $C^r$ open (\cite{K}) and dense. (If genus$(M)\geq 2$ then every metric has positive topological
entropy \cite{K}. For torus the density of ${\mathcal{H}}_r(M)$ follows from \cite{Ban} and for sphere it follows
from \cite{KW}).
\begin{theorem}
\label{ThSmallGapDim2}
The set of metrics satisfying \eqref{smallgap} is topologically generic in ${\mathcal{H}}(M).$
\end{theorem}
\begin{corollary}
The set of metrics satisfying \eqref{smallgap} is topologically generic in the space of all metrics on
$M.$
\end{corollary}
\begin{proof}[Proof of Theorem \ref{ThSmallGapDim2}] By \cite{K} if $g\in {\mathcal{H}}_r(M)$ then there is a hyperbolic
basic set $\Lambda$ for the geodesic flow. Since Lemmas \ref{LmEqLen}, \ref{LmMor}, \ref{LmAvoid} and
\ref{LmClosing} remain valid in the setting of hyperbolic
sets the proof is similar to the proof of Theorem \ref{smallgap}.
(In the proof of Lemma \ref{LmAvoid} we need to take $\sigma_1$ so that it crosses completely
an element of some Markov partition $\Pi$ such that all elements of $\Pi$ have unstable length between
$\kappa$ and $C\kappa.$ The number of eligible segments now is not $O(1/\kappa)$ but
$O(1/\kappa^a)$ for some $a>0$ but this is still much larger than $|\ln\kappa|.$)
\end{proof}
\section{Small gaps for hyperbolic surfaces, continued}
\label{ScH2-KR}
Here we show that for Lebesgue-typical hyperbolic surface the gaps in the length spectrum cannot be too small.
Our argument in similar to \cite{KR}.
Related results are obtained in \cite{Varju}.
\subsection{Small values of polynomials.}
\begin{prop}
\label{PrChExt}
(see e.g \cite[Section 3.2]{MH})
Consider a degree $D$ polynomial $P(x)=a_D x^D+a_{D-1} x^{D-1}+\dots a_0.$ Then
$$ \sup_{[-1, 1]} |P(x)|\geq \frac{|a_D|}{2^{D-1}}. $$
\end{prop}
\begin{cor}\label{CrMultiCheb}
Let $0\neq P\in{\bf Z}[x_1, x_2\dots x_n],$ $\deg(P)=D$ then
$$ \sup_{[-1, 1]^n} |P(x)|\geq \frac{1}{2^{D-1}}. $$
\end{cor}
\begin{proof}
By induction. For $n=0$ or $1$ the result follows from Proposition \ref{PrChExt}.
Next, suppose the statement is proven for polynomials of $n-1$ variables. If $P$ does not depend
on $x_n$ then we are done. Otherwise
let $k>0$ be the degree of $P$ with respect to $x_n$. Then
$$ P(x)=a_k(x_1, \dots, x_{n-1}) x_n^k+a_{k-1}(x_1,\dots, x_{n-1}) x_n^{k-1}+\dots+a_0(x_1,\dots, x_{n-1}) $$
where $a_k$ is the polynomial with integer coefficients of degree $D-k.$ Let
$$({\bar x}_1, \dots {\bar x}_{n-1})=\arg\max_{[-1, 1]^{n-1}} |a_k(x_1, \dots, x_{n-1})|. $$
Then
$$ \sup_{[-1, 1]^n} |P(x_1,\dots x_{n-1}, x_n)|\geq \max_{x_n\in [-1, 1]}
|P({\bar x}_1,\dots {\bar x}_{n-1}, x_n)|\geq |a({\bar x}_1, \dots,{\bar x}_{n-1})| 2^{1-k}$$
$$\geq
2^{1+k-D} 2^{1-k}=2^{2-D} $$
completing the proof.
\end{proof}
\begin{prop}
\label{PrRemez}
{\sc (Remez inequality)} (see \cite{BG} or \cite[Theorem 1.1]{Y})
Let $B$ be a convex set in ${\bf R}^n,$ $\Omega\subset B,$ and $P$ be a polynomial
of degree $D.$ Then
$$ \sup_{B}|P|\leq C_B {\rm mes}^{-D} (\Omega) \sup_{\Omega} |P|. $$
\end{prop}
\begin{cor} \label{CrRemez}
Under the conditions of Proposition \ref{PrRemez}
$$ {\rm mes}(x\in B: |P(x)|\leq {\varepsilon})\leq \left(\frac{C_B {\varepsilon}}{\sup_B |P|}\right)^{1/D}. $$
\end{cor}
\begin{proof}
Apply Proposition \ref{PrRemez} with $\Omega=\{x\in B: |P(x)|\leq {\varepsilon}\}.$
\end{proof}
\begin{cor}
\label{CrDiop}
If $P_N\in {\bf Z}[x_1, x_2, \dots, x_n]$ are polynomials of degree $D_N$ and ${\varepsilon}_N$ is a sequnces such that
$\displaystyle \sum_N {\varepsilon}^{1/D_N}<\infty $ then $|P(x_1, \dots x_n)|<{\varepsilon}_N$ has
only finitely many solutions for almost every $(x_1\dots x_n)\in {\bf R}^n.$
\end{cor}
\begin{proof}
It suffices to show this for a fixed cube $B$ with side 2. Then Corollaries \ref{CrMultiCheb} and
\ref{CrRemez} give
$$ {\rm mes}(x\in B: |P_N(x)|\leq {\varepsilon}_N)\leq \left(C 2^{D_N} {\varepsilon}_N\right)^{1/D_N}
=\bar{C} {\varepsilon}_N^{1/D_N}$$
so the statement follows from Borel-Cantelli Lemma.
\end{proof}
\subsection{Polynomial maps on $SL_2({\bf R})$}
\begin{cor}
\label{CrDiopSL2}
Let $m$ be a fixed number.
(a) Let
$P_N\in {\bf Z}((a_1, b_1, c_1, d_1),\dots, (a_m, b_m, c_m, d_m))$ be polynomials of degree $D_N.$
For $A_1, \dots A_m\in SL_2({\bf R})$ with
$A_j=\left(\begin{array}{cc} a_j & b_j \\ c_j & d_j\\\end{array}\right) $ let
$$ H_N(A_1, \dots, A_m)=P_N((a_1, b_1, c_1, d_1),\dots, (a_m, b_m, c_m, d_m)). $$
If $\displaystyle \sum_N {\varepsilon}_N^{1/((m+2) D_N)}<\infty$ then
$ |H_N(A_1\dots A_m)|<{\varepsilon}_N $
for only finitely many $N$ for almost every $(A_1,\dots A_m)\in (SL_2({\bf R}))^m. $
(b) Given $g\in\mathbb{N}$ let
$$G_g=\{(A_1, \dots A_{2g})\in (SL_2({\bf R}))^{2g}:
[A_1, A_2] [A_3, A_4]\dots [A_{2g-1}, A_{2g}]=I\}. $$
Let $P_N\in {\bf Z}((a_1, b_1, c_1, d_1),\dots, (a_{2m}, b_{2m}, c_{2m}, d_{2m}))$
be polynomials of degree $D_N.$ Let
$$ H_N(A_1, \dots, A_{2g})=P_N((a_1, b_1, c_1, d_1),\dots, (a_{2g}, b_{2g}, c_{2g}, d_{2g})). $$
Assume that $H_N$ is not identically equal to 0 on $G_g.$
If
$\displaystyle \sum_N {\varepsilon}_N^{\delta_N}<\infty$
where $\delta_N=\frac{1}{(4g-2)(g+2) D_N}$ then
$ |H_N(A_1\dots A_{2g})|<{\varepsilon}_N $
for only finitely many $N$ for almost every $(A_1,\dots A_{2g})\in G_g. $
\end{cor}
\begin{proof}
(a) It suffices to prove the statement under the assumption that $|a_j|>\delta$ for some fixed $\delta>0.$ Then
$d_j=\frac{1+b_j c_j}{a_j}$ and so
$$ H(A_1,\dots A_m)=\frac{{\widetilde{P}}_N((a_1, b_1, c_1),\dots (a_m, b_m, c_m))}{\prod_{j=1}^m a_j^{d_N}} $$
where ${\widetilde{P}}_N$ is a polynomial of degree ${\widetilde{d}}_n\leq (m+2) d_N.$ Thus if $|P_N|\leq {\varepsilon}_N$ then
$|{\widetilde{P}}_N|\leq {\widetilde{{\varepsilon}}}_N:=\frac{{\varepsilon}_N}{\delta^{(m+2) D_N}}.$ Since
$$\sum_N {\widetilde{{\varepsilon}}}_N^{1/{\widetilde{d}}_N}\leq \frac{1}{\delta} \sum_N {\varepsilon}_N^{1/(m+2) D_N}<\infty $$
the result follows from Corollary \ref{CrDiop}.
(b) Rewriting the equations defining $G_g$ in the form
$$ [A_1, A_2]\dots [A_{2g-3}, A_{2g-2}] A_{2g-1} A_{2g} A_{2g-1}^{-1}=A_{2g} $$
we can express the entries of $A_{2g}$ as rational functions of the entries of the other matrices.
Arguing as in part (a) we can reduce the inequality
$|P_N(A_1,\dots A_{2g-1}, A_{2g})|<{\varepsilon}$ to
$|{\widehat{P}}_N(A_1,\dots A_{2g-1})|<{\widehat{{\varepsilon}}}_N$ where ${\widehat{P}}_N$ is the polynomial of degree $(4g-2) D_N.$
Now the result follows from part (a).
\end{proof}
\begin{cor}
For each $\eta>0$ for almost every $A_1, \dots A_m\in SL_2({\bf R})$ the inequality
$$ ||W(A_1, \dots A_m)-I||>(2m-1)^{-|W|^2 (m+2+\eta)} $$
holds for all except for finitely many words $W.$
\end{cor}
\begin{proof}
If $||W(A_1, \dots A_m)-I||\leq {\varepsilon}$ then all entries of $W-I$ are ${\varepsilon}$ close to I. Conisdering for example,
the condition $W_{11}(A_1, \dots A_m)-1$ we
get a polynomial of degree $|W|.$ Therefore, by Corollary \ref{CrDiopSL2} it suffices to check that
$$ \sum_W (2m-1)^{-\frac{|W|^2 (m+2+\eta)}{|W|(m+2)}} <\infty $$
but the above sum equals to
$$ \sum_D (2m-1)^D (2m-1)^{-D-D(\eta/(m+2)}=\sum_D (2m-1)^{-\eta D/(m+2)}<\infty. \qedhere $$
\end{proof}
\begin{cor}
For ${\mathcal{A}}=(A_1\dots A_{2g})\in G_g$ let $S_{\mathcal{A}}$ be the surface defined by ${\mathcal{A}}.$ Given
a word $W$ let $l(W, {\mathcal{A}})$ be the length of the closed geodesic in the homotopy class defined by $W.$
Then for each $\eta>0$ the following holds for almost all ${\mathcal{A}}\in G_g$
There exists a constant $K=K({\mathcal{A}})$ such that for each pair $W_1, W_2$ either
$$l(W_1, {\mathcal{A}})=l(W_2, {\mathcal{A}})\text{ or } $$
\begin{equation}
\label{QuadExp}
|l(W_1, {\mathcal{A}})-l(W_2, {\mathcal{A}})|\geq K (4g-1)^{-[(2g+4)(4g-2)+\eta] \max^2(|W_1|, |W_2|)}.
\end{equation}
\end{cor}
\begin{remark}
Recall that \cite{Randol1} shows that for any hyperbolic surface the length spectrum has unbounded multiplicity
so there are many pairs of non conjugated words there the first alternative of the corollary holds.
\end{remark}
\begin{remark}
Note that \eqref{EqWL-GL} shows that $l(W_1, {\mathcal{A}})$ can be close to $l(W_2, {\mathcal{A}})$ only if the lengths
of $W_1$ and $W_2$ are of the same order. Thus \eqref{QuadExp} implies that for almost every ${\mathcal{A}}$ there are
constants $K,$ $R$ such that
$$ |l(W_1, {\mathcal{A}})-l(W_2, {\mathcal{A}})|\geq K e^{-R({\mathcal{A}}) l^2(W_1, {\mathcal{A}})}. $$
\end{remark}
\begin{proof}
Let $P_{W}({\mathcal{A}})={\rm tr}(W({\mathcal{A}})).$ Since $P_W({\mathcal{A}})=2\cosh(l(W,{\mathcal{A}})/2),$ it follows that if $l(W_1({\mathcal{A}}))$ is
close to $l(W_2({\mathcal{A}}))$ then
$$l(W_1, {\mathcal{A}})-l(W_2, {\mathcal{A}})|\geq C |P_{W_1}({\mathcal{A}})-P_{W_2}({\mathcal{A}})| e^{-|W_1|D} .$$
Therefore it suffices to show that if $l(W_1,{\mathcal{A}})\neq l(W_2, {\mathcal{A}})|$ then
$$ |P_{W_1}({\mathcal{A}})-P_{W_2}({\mathcal{A}})|\geq {\widetilde{K}} e^{-|W_1|D} (4g-1)^{-[(2g+4)(4g-2)+\eta] \max^2(|W_1|, |W_2|)}. $$
Since $\eta$ is arbitrary, we can actually check that
$$ |P_{W_1}({\mathcal{A}})-P_{W_2}({\mathcal{A}})|\geq K (4g-1)^{-[(2g+4)(4g-2)+\eta] \max^2(|W_1|, |W_2|)} .$$
To verify this we will show that for almost all ${\mathcal{A}}\in G_m$ the inequality
$$ |P_{W_1}({\mathcal{A}})-P_{W_2}({\mathcal{A}})|< (4g-1)^{-[(2g+4)(4g-2)+\eta] \max^2(|W_1|, |W_2|)} $$
has only finitely many solutions. Let
$P_{W_1, W_2}({\mathcal{A}})=P_{W_1}({\mathcal{A}})-P_{W_2}({\mathcal{A}}).$ It is a polynomail of degree
$\max(|W_1|, |W_2|).$ So by Corollary \ref{CrDiopSL2}(b) it suffices to check that
$$ \sum_{W_1, W_2} (4g-1)^{-\frac{[(2g+4)(4g-2)+\eta] \max(|W_1|, |W_2|)}{(4g-2)(g+2)}}<\infty $$
There are at most $(4g-1)^{2k}$ pairs $(W_1, W_2)$ with $k=\max(W_1, W_2)$ so the last sum is estimated by
$$ \sum_{k} (4g-1)^{2k} (4g-1)^{-\frac{[(2g+4)(4g-2)+\eta]k}{(4g-2)(g+2)}}=
\sum_k (4g-1)^{-\frac{\eta k}{(4g-2)(g+2)}}<\infty $$
proving the result.
\end{proof}
\section{Open problems.}
(1) A suitable version of Theorem \ref{ThAlgGen} should hold for other symmetric spaces.
In particular, recall that arithmetic manifolds appear as fundamental
domains $G/\Gamma$ where $G$ is a connected semi-simple algebraic
${\bf R}$-group without compact factors of ${\bf R}$-rank $\geq 2$, and $\Gamma$
is a lattice in $G$ (cf. \cite{Margulis74,Margulis75,Margulis77}).
Thus we expect that a version of Theorem
\ref{ThAlgGen} should hold in higher rank setting. Note however, that for higher rank symmetric spaces
closed orbits are not
isolated but appear in families.
(2) The proof of Theorem \ref{thm:smallgapneg} relies on localized perturbations. Therefore it does not work in the
analytic category. We expect that Theorem \ref{thm:smallgapneg} is still valid for analytic metrics but the proof
would require new ideas.
(3) It is likely that an explicit lower bound for the gaps in the length spectrum could also be obtained
for {\em prevalent} set of negatively curved metrics (see \cite{Kal} for related results) but we do not
pursue this question here.
\medskip
\centerline{\bf Acknowledgements.} \smallskip
The authors started discussing questions about gaps in the length spectra in 2005 when both
were participating in the work of the thematic program ``Time at work'' at the Institut Henri
Poincar\'e in 2005.
The authors thank D. Popov for stimulating their interest in this problem, and
Y. Yomdin and V. Kaloshin for discussions related to
Sections \ref{ScH2-KR} of the present paper. The authors would also like to thank
A. Glutsuk, N. Kamran, A. Katok, P. Sarnak and L. Silberman for stimulating discussions.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,036
|
OF HOLY SCRIPTURE***
Transcribed from the 1901 Society for Promoting Christian Knowledge
edition by David Price, email ccx074@pglaf.org
Addresses on the Revised
Version of Holy
Scripture.
BY
C. J. ELLICOTT, D.D.,
BISHOP OF GLOUCESTER,
AND HON. FELLOW OF ST. JOHN'S COLLEGE, CAMBRIDGE.
PUBLISHED UNDER THE DIRECTION OF THE TRACT COMMITTEE.
LONDON:
SOCIETY FOR PROMOTING CHRISTIAN KNOWLEDGE,
NORTHUMBERLAND AVENUE, W.C.; 43 QUEEN VICTORIA STREET, E.C.
BRIGHTON: 129 NORTH STREET.
NEW YORK: E. & J. B. YOUNG & CO.
1901.
PREFATORY NOTE.
The following Addresses form the Charge to the Archdeaconry of
Cirencester at the Visitation held at the close of October in the present
year. The object of the Charge, as the opening words and the tenor of
the whole will abundantly indicate, is seriously to suggest the question,
whether the time has not now arrived for the more general use of the
Revised Version at the lectern in the public service of the Church.
C. J. GLOUCESTER.
_October_, 1901.
CONTENTS.
PAGE
ADDRESS I. EARLY HISTORY OF REVISION 5
,, II. LATER HISTORY OF REVISION 17
,, III. HEBREW AND GREEK TEXT 48
,, IV. NATURE OF THE RENDERINGS 81
,, V. PUBLIC USE OF THE VERSION 117
ADDRESS I.
EARLY HISTORY OF REVISION.
As there now seem to be sufficient grounds for thinking that ere long the
Revised Version of Holy Scripture will obtain a wider circulation and
more general use than has hitherto been accorded to it, it seems
desirable that the whole subject of the Revised Version, and its use in
the public services of the Church, should at last be brought formally
before the clergy and laity, not only of this province, but of the whole
English Church.
Twenty years have passed away since the appearance of the Revised Version
of the New Testament, and the presentation of it by the writer of these
pages to the Convocation of Canterbury on May 17, 1881. Just four more
years afterwards, viz. on April 30, 1885, the Revised Version of the Old
Testament was laid before the same venerable body by the then Bishop of
Winchester (Bp. Harold Browne), and, similarly to the Revised Version of
the New Testament, was published simultaneously in this country and
America. It was followed, after a somewhat long interval, by the Revised
Version of the Apocrypha, which was laid before Convocation by the writer
of these pages on February 12, 1896.
The revision of the Authorised Version has thus been in the hands of the
English-speaking reader sixteen years, in the case of the Canonical
Scriptures, and five years in the case of the Apocrypha--periods of time
that can hardly be considered insufficient for deciding generally,
whether, and to what extent, the Revised Version should be used in the
public services of the Church.
I have thus thought it well, especially after the unanimous resolution of
the Upper House of the Convocation of Canterbury, three years ago {6},
and the very recent resolution of the House of Laymen, to place before
you the question of the use of the Revised Version in the public services
of the Church, as the ultimate subject of this charge. I repeat, as the
ultimate subject, for no sound opinion on the public use of this version
can possibly be formed unless some general knowledge be acquired, not
only of the circumstances which paved the way for the revision of the
time-honoured version of 1611, but also of the manner in which the
revision was finally carried out. We cannot properly deal with a
question so momentous as that of introducing a revised version of God's
Holy Word into the services of the Church, without knowing, at least in
outline, the whole history of the version which we are proposing to
introduce. This history then I must now place before you from its very
commencement, so far as memory and a nearly life-long connexion with the
subject enable me to speak.
The true, though remote fountain-head of revision, and, more
particularly, of the revision of the New Testament, must be regarded as
the grammar written by a young academic teacher, George Benedict Winer,
as far back as 1822, bearing the title of a Grammar of the Language of
the New Testament. It was a vigorous protest against the arbitrary, and
indeed monstrous licence of interpretation which prevailed in
commentaries on Holy Scripture of the eighteenth and nineteenth
centuries. It met with at first the fate of all assaults on prevailing
unscientific procedures, but its value and its truth were soon
recognized. The volume passed through several successively improved
editions, until in 1855 the sixth edition was reached, and issued with a
new and interesting preface by the then distinguished and veteran writer.
This edition formed the basis of the admirable and admirably supplemented
translation of my lamented and highly esteemed friend Dr. Moulton, which
was published in 1870, passed through a second edition six years
afterwards, and has, since that time, continued to be a standard grammar,
in an English dress, of the Greek Testament down to this day.
The claim that I have put forward for this remarkable book as the
fountain-head of revision can easily be justified when we call to memory
how very patently the volume, in one or another of its earlier editions,
formed the grammatical basis of the commentaries of De Wette and Meyer,
and, here in England, of the commentary of Alford, and of critical and
grammatical commentaries on some of St. Paul's Epistles with which my own
name was connected. It was to Winer that we were all indebted for that
greater accuracy of interpretation of the Greek Testament which was
recognized and welcomed by readers of the New Testament at the time I
mention, and produced effects which had a considerable share in the
gradual bringing about of important movements that almost naturally
followed.
What came home to a large and increasing number of earnest and
truth-seeking readers of the New Testament was this--that there were
inaccuracies and errors in the current version of the Holy Scriptures,
and especially of the New Testament, which plainly called for
consideration and correction, and further brought home to very many of us
that this could never be brought about except by an authoritative
revision.
This general impression spread somewhat rapidly; and soon after the
middle of the last century it began to take definite shape. The subject
of the revision of the Authorised Version of the New Testament found a
place in the religious and other periodicals of the day {10a}, and as the
time went on was the subject of numerous pamphlets, and was alluded to
even in Convocation {10b} and Parliament {10c}. As yet however there had
been no indication of the sort of revision that was desired by its
numerous advocates, and fears were not unnaturally entertained as to the
form that a revision might ultimately take. It was feared by many that
any authoritative revision might seriously impair the acceptance and
influence of the existing and deeply reverenced version of Holy
Scripture, and, to use language which expressed apprehensions that were
prevailing at the time, might seriously endanger the cause of sound
religion in our Church and in our nation.
There was thus a real danger, unless some forward step was quickly and
prudently taken, that the excitement might gradually evaporate, and the
movement for revision might die out, as has often been the case in regard
of the Prayer Book, into the old and wonted acquiescence of the past.
It was just at this critical time that an honoured and influential
churchman, who was then the popular and successful secretary of the
Society for the Propagation of the Gospel, Rev. Ernest Hawkins,
afterwards Canon of Westminster, came forward and persuaded a few of us,
who had the happiness of being his friends, to combine and publish a
version of one of the books of the New Testament which might practically
demonstrate to friends and to opponents what sort of a revision seemed
desirable under existing circumstances. After it had been completed we
described it "as a _tentamen_, a careful endeavour, claiming no finality,
inviting, rather than desiring to exclude, other attempts of the same
kind, calling the attention of the Church to the many and anxious
questions involved in rendering the Holy Scriptures into the vernacular
language, and offering some help towards the settlement of those
questions {12}."
The portion of Scripture selected was the Gospel according to St. John.
Those who undertook the revision were five in number:--Dr. Barrow, the
then Principal of St. Edmund's Hall, Oxford; Dr. Moberly, afterwards
Bishop of Salisbury; Rev. Henry Alford, afterwards Dean of Canterbury;
Rev. W. G. Humphry, Vicar of St. Martin's in the Fields; and lastly, the
writer of this charge. Mr. Ernest Hawkins, busy as he was, acted to a
great extent as our secretary, superintended arrangements, and encouraged
and assisted us in every possible manner. Our place of meeting was the
library of our hospitable colleague Mr. Humphry. We worked in the
greatest possible harmony, and happily and hopefully concluded our
Revision of the Authorised Version of the Gospel of St. John in the month
of March, 1857.
Our labours were introduced by a wise and attractive preface, written
mainly by Dr. Moberly, in the lucid, reverent, and dignified language
that marked everything that came from the pen of the late Bishop of
Salisbury.
The effect produced by this _tentamen_ was indisputably great. The work
itself was of course widely criticized, but for the most part favourably
{13}. The principles laid down in the preface were generally considered
reasonable, and the possibilities of an authoritative revision distinctly
increased. The work in fact became a kind of object lesson.
It showed plainly that there _were_ errors in the Authorised Version that
needed correction. It further showed that their removal and the
introduction of improvements in regard of accuracy did not involve,
either in quantity or quality, the changes that were generally
apprehended. And lastly, it showed in its results that _scholars_ of
different habits of thought could combine in the execution of such a work
without friction or difficulty.
In regard of the Greek text but little change was introduced. The basis
of our translation was the third edition of Stephens, from which we only
departed when the amount of external evidence in favour of a different
reading was plainly overwhelming. As we ourselves state in the preface,
"our object was to revise a version, not to frame a text." We should
have obscured this one purpose if we had entered into textual criticism.
Such was the tentative version which prepared the way for authoritative
revision.
More need not be said on this early effort. The version of the Gospel of
St. John passed through three editions. The Epistles to the Romans and
Corinthians appeared in 1858, and the first three of the remaining
Epistles (Galatians, Ephesians, and Philippians) in 1861. The third
edition of the Revision of the Authorised Version of St. John was issued
in 1863, with a preface in which the general estimate of the revision was
discussed, and the probability indicated of some authoritative procedure
in reference to the whole question. As our little band had now been
reduced to four, and its general aim and object had been realized, we did
not deem it necessary to proceed with a work which had certainly helped
to remove most of the serious objections to authoritative revision. Our
efforts were helped by many treatises on the subject which were then
appearing from time to time, and, to a considerable extent, by the
important work of Professor, afterwards Archbishop, Trench, entitled "On
the Authorised Version of the New Testament in connexion with some recent
proposals for its revision." This appeared in 1858. After the close of
our tentative revision in 1863, the active friends (as they may be
termed) of the movement did but little except, from time to time, confer
with one another on the now yearly improving prospects of authoritative
revision. In 1869 Dean Alford published a small handy revised version of
the whole of the Greek Testament, and, a short time afterwards, I
published a small volume on the "Revision of the English Version," in
which I sought to show how large an amount of the fresh and vigorous
translation of Tyndale was present in the Authorised Version, and how
little of this would ever be likely to disappear in any authoritatively
revised version of the future. Some estimate also was made of the amount
of changes likely to be introduced in a sample portion of the Gospels. A
few months later, a very valuable volume ("On a Fresh Revision of the New
Testament") was published by Professor, afterwards Bishop, Lightfoot,
which appeared most seasonably, just as the long-looked-for hope of a
revision of the Authorised Version of God's Holy Word was about to be
realized.
All now was ready for a definite and authoritative commencement. Of
this, and of the later history of Revision, a brief account will be given
in the succeeding Address.
ADDRESS II.
LATER HISTORY OF REVISION.
We are now arrived at the time when what was simple tentative and
preparatory passed into definite and authoritative realization.
The initial step was taken on February 10, 1870, in the Upper House of
the Convocation of Canterbury. The Bishop of Oxford, seconded by the
Bishop of Gloucester, proposed the subjoined resolution, which it may be
desirable to give in the exact words in which it was presented to the
House, as indicating the caution with which it was framed, and also the
indirectly expressed hope (unfortunately not realized) of the concurrence
of the Northern Convocation. The resolution was as follows:
"That a committee of both Houses be appointed, with power to confer
with any committee that may be appointed by the Convocation of the
Northern Province, to report upon the desirableness of a revision of
the Authorised Version of the New Testament, whether by marginal
notes or otherwise, in those passages where plain and clear errors,
whether in the Hebrew or Greek text originally adopted by the
translators, or in the translations made from the same, shall on due
investigation be found to exist."
In the course of the debate that followed the resolution was amended by
the insertion of the words "Old and," so as to include both Testaments,
and, so amended, was unanimously accepted by the Upper House, and at once
sent down to the Lower House. After debate it was accepted by them, and,
having been thus accepted by both Houses, formed the basis of all the
arrangements, rules, and regulations which speedily followed.
Into all of these it is not necessary for me to enter except so far as
plainly to demonstrate that the Convocation of Canterbury, on thus
undertaking one of the greatest works ever attempted by Convocation
during its long and eventful history, followed every course, adopted
every expedient, and carefully took every precaution to bring the great
work it was preparing to undertake to a worthy and a successful issue.
It may be well, then, here briefly to notice, that in accordance with the
primary resolution which I have specified, a committee was appointed of
eight members of the Upper House, and, in accordance with the regular
rule, sixteen members of the Lower House, with power, as specified, to
confer with the Convocation of York. The members of the Upper House were
as follows: the Bishops of Winchester (Wilberforce), St. Davids
(Thirlwall), Llandaff (Ollivant), Salisbury (Moberly), Ely (Harold
Browne, afterwards of Winchester), Lincoln (Wordsworth; who soon after
withdrew), Bath and Wells (Lord Arthur Hervey), and myself.
The members of the Lower House were the Prolocutor (Dr. Bickersteth, Dean
of Lichfield), the Deans of Canterbury (Alford), Westminster (Stanley),
and Lincoln (Jeremie); the Archdeacons of Bedford (Rose), Exeter
(Freeman), and Rochester (Grant); Chancellor Massingberd; Canons
Blakesley, How, Selwyn, Swainson, Woodgate; Dr. Jebb, Dr. Kay, and Mr. De
Winton.
Before, however, this committee reported, at the next meeting of
Convocation in May, and on May 3 and May 5, the following five
resolutions, which have the whole authority of Convocation behind them,
were accepted unanimously by the Upper House, and by large majorities in
the Lower House:
"1. That it is desirable that a revision of the Authorised Version
of the Holy Scriptures be undertaken.
2. That the revision be so conducted as to comprise both marginal
renderings and such emendations as it may be found necessary to
insert in the text of the Authorised Version.
3. That in the above resolutions we do not contemplate any new
translation of the Bible, nor any alteration of the language, except
where, in the judgement of the most competent scholars, such change
is necessary.
4. That in such necessary changes, the style of the language
employed in the existing version be closely followed.
5. That it is desirable that Convocation should nominate a body of
its own members to undertake the work of revision, who shall be at
liberty to invite the co-operation of any eminent for scholarship, to
whatever nation or religious body they may belong."
These are the fundamental rules of Convocation, as formally expressed by
the Upper and Lower Houses of this venerable body. The second and third
rules deserve our especial attention in reference to the amount of the
emendations and alterations which have been introduced during the work of
revision. This amount, it is now constantly said, is not only excessive,
but in distinct contravention of the rules which were laid down by
Convocation. A responsible and deeply respected writer, the late Bishop
of Wakefield, only a few years ago plainly stated in a well-known
periodical {21} that the revisers "largely exceeded their instructions,
and did not adhere to the principles they were commissioned to follow."
This is a very grave charge, but can it be substantiated? The second and
third rules, taken together, refer change to consciously felt necessity
on the part of "the most competent scholars," and these last-mentioned
must surely be understood to be those who were deliberately chosen for
the work. In the subsequently adopted rule of the committee of
Convocation the criterion of this consciously felt necessity was to be
faithfulness to the original. All then that can justly be said in
reference to the Revisers is this,--not that they exceeded their
instructions (a very serious charge), but that their estimate of what
constituted faithfulness, and involved the necessity of change, was, from
time to time, in the judgement of their critic, mistaken or exaggerated.
Such language however as that used in reference to the changes made by
the Revisers as "unnecessary and uninstructive alterations," and
"irritating trivialities," was a somewhat harsh form of expressing the
judgement arrived at.
But to proceed. On the presentation of the Report it was stated that the
committee had not been able to confer with the Northern Convocation, as
no committee had been appointed by them. It was commonly supposed that
the Northern President (Abp. of York) was favourable to revision, but the
two Houses, who at that time sat together, had taken a very different
view {22}, as our President informed us that he had received a
communication from the Convocation of York to the effect that--"The
Authorised Version of the English Bible is accepted, not only by the
Established Church, but also by the Dissenters and by the whole of the
English-speaking people of the world, as their standard of faith; and
that although blemishes existed in its text such as had, from time to
time, been pointed out, yet they would deplore any recasting of its text.
That Convocation accordingly did not think it necessary to appoint a
committee to co-operate with the committee appointed by the Convocation
of Canterbury, though favourable to the errors being rectified."
This obviously closed the question of co-operation with the Northern
Convocation. We sincerely regretted the decision, as there were many
able and learned men in the York Convocation whose co-operation we should
have heartily welcomed. Delay, however, was now out of the question.
The working out of the scheme therefore had now become the duty of the
Convocation that had adopted, and in part formulated, the proposed
revision.
The course of our proceedings was then as follows:
After the Report of the committee had been accepted by the Upper House,
and communicated to the Lower House, the following resolution was
unanimously adopted by the Upper House (May 3, 1870), and in due course
sent down to the Lower House:
"That a committee be now appointed to consider and report to
Convocation a scheme of revision on the principles laid down in the
Report now adopted. That the Bishops of Winchester, St. Davids,
Llandaff, Gloucester and Bristol, Ely, Salisbury, Lincoln, Bath and
Wells, be members of the committee. That the committee be empowered
to invite the co-operation of those whom they may judge fit from
their biblical scholarship to aid them in their work."
This resolution was followed by a request from the Archbishop that as
this was a committee of an exceptional character, being in fact an
executive committee, the Lower House would not appoint, as in ordinary
committees, twice the number of the members appointed by the Upper House,
but simply an equal number. This request, though obviously a very
reasonable request under the particular circumstances, was not acceded to
without some debate and even remonstrance. This, however, was overcome
and quieted by the conciliatory good sense and firmness of the
Prolocutor; and, on the following day, the resolution was accepted by the
Lower House, and the Prolocutor (Bickersteth) with the Deans of
Canterbury (Alford) and Westminster (Stanley), the Archdeacon of Bedford
(Rose), Canons Blakesley and Selwyn, Dr. Jebb and Dr. Kay, were appointed
as members of what now may be called the Permanent Committee.
This Committee had to undertake the responsible duty of choosing experts,
and, out of them and their own members, forming two Companies, the one
for the revision of the Authorised Version of the Old Testament, the
other for the revision of the Authorised Version of the New Testament.
Rules had to be drawn up, and a general scheme formed for the carrying
out in detail of the whole of the proposed work. In this work it may be
supposed that considerable difficulty would have been found in the choice
of biblical scholars in addition to those already appointed by
Convocation. This, however, did not prove to be the case. I was at that
time acting as a kind of informal secretary, and by the friendly help of
Dr. Moulton and Dr. Gotch of Bristol had secured the names of
distinguished biblical scholars from the leading Christian bodies in
England and in Scotland from whom choice would naturally have to be made.
When we met together finally to choose, there was thus no lack of
suitable names.
In regard of the many rules that had to be made for the orderly carrying
out of the work I prepared, after careful conference with the Bishop of
Winchester, a draft scheme which, so far as I remember, was in the sequel
substantially adopted by what I have termed the Permanent Committee of
Convocation. When, then, this Committee formally met on May 25, 1870,
the names of those to whom we were empowered to apply were agreed upon,
and invitations at once sent out. The members of the Committee had
already been assigned to their special companies; viz. to the Old
Testament Company, the Bishops of St. Davids, Llandaff, Ely, Lincoln (who
soon after resigned), and Bath and Wells; and from the Lower House,
Archdeacon Rose, Canon Selwyn, Dr. Jebb, and Dr. Kay: to the New
Testament Company, the Bishops of Winchester, Gloucester and Bristol, and
Salisbury; and from the Lower House, the Prolocutor, the Deans of
Canterbury and Westminster, and Canon Blakesley.
Those invited to join the Old Testament were as follows:--Dr. W. L.
Alexander, Professor Chenery, Canon Cook, Professor A. B. Davidson, Dr.
B. Davies, Professor Fairbairn, Rev. F. Field, Dr. Gensburg, Dr. Gotch,
Archdeacon Harrison, Professor Leathes, Professor McGill, Canon Payne
Smith, Professor J. J. S. Perowne, Professor Plumptre, Canon Pusey, Dr.
Wright (British Museum), Mr. W. A. Wright of Cambridge, the active and
valuable secretary of the Company.
Of these Dr. Pusey and Canon Cook declined the invitation.
Those invited to join the New Testament Company were as follows:--Dr.
Angus, Dr. David Brown, the Archbishop of Dublin (Trench), Dr. Eadie,
Rev. F. J. A. Hort, Rev. W. G. Humphry, Canon Kennedy, Archdeacon Lee,
Dr. Lightfoot, Professor Milligan, Professor Moulton, Dr. J. H. Newman,
Professor Newth, Dr. A. Roberts, Rev. G. Vance Smith, Dr. Scott (Balliol
College), Rev. F. H. Scrivener, the Bishop of St. Andrews (Wordsworth),
Dr. Tregelles, Dr. Vaughan, Canon Westcott.
Of these Dr. J. H. Newman declined, and Dr. Tregelles, from feeble health
and preoccupation on his great work, the critical edition of the New
Testament, was unable to attend. It should be here mentioned that soon
after the formation of the company, Rev. John Troutbeck, Minor Canon of
Westminster, afterwards Doctor of Divinity, was appointed by the Company
as their secretary. A more accurate, punctual, and indefatigable
secretary it would have been impossible for us to have selected for the
great and responsible work.
On the same day (May 25, 1870,) the rules for the carrying out of the
revision, which, as I have mentioned, had been drawn up in draft were all
duly considered by the committee and carried, and the way left clear and
open for the commencement of the work. These rules (copies of which will
be found in nearly all the prefaces to the Revised Version hitherto
issued by the Universities) were only the necessary amplifications of the
fundamental rules passed by the two Houses of Convocation which have been
already specified.
The first of these subsidiary rules was as follows:--"To introduce as few
alterations as possible in the text of the Authorised Version
consistently with faithfulness." This rule must be read in connexion
with the first and third fundamental rules and the comments I have
already made on those rules.
The second of the rules of the committee was as follows:--"To limit, as
far as possible, the expression of such alterations to the language of
the Authorised and earlier English versions." This rule was carefully
attended to in its reference to the Authorised Version. I do not however
remember, in the revision of the version of the New Testament, that we
often fell back on the renderings of the earlier English versions. They
were always before us: but, in reference to other versions where there
were differences of rendering, we frequently considered the renderings of
the ancient versions, especially of the Vulgate, Syriac, and Coptic, and
occasionally of the Gothic and Armenian. To these, however, the rule
makes no allusion.
The third rule speaks for itself:--"Each Company to go twice over the
portion to be revised, once provisionally, the second time finally, and
on principles of voting as hereinafter is provided."
The fourth rule refers to the very important subject of the text, and is
an amplification of the last part of the third fundamental rule. The
rule of the committee is as follows:--"That the text to be adopted be
that for which the evidence is decidedly preponderating; and that when
the text so adopted differs from that from which the Authorised Version
was made, the alteration be indicated in the margin." The subject of the
text is continued in the fifth rule, which is as follows:--"To make or
retain no change in the text on the second final revision by the Company
except _two-thirds_ of those present approve of the same, but on the
first revision to decide by simple majorities."
The sixth rule is of importance, but in the New Testament Company (I do
not know how it may have been in the Old Testament Company) was very
rarely acted upon:--"In every case of proposed alteration that may have
given rise to discussion, to defer the voting thereupon till the next
meeting, whensoever the same shall be required by one-third of those
present at the meeting, such intended vote to be announced in the notice
for the next meeting." The only occasion on which I can remember this
rule being called into action was a comparatively unimportant one. At
the close of a long day's work we found ourselves differing on the
renderings of "tomb" or "sepulchre" in one of the narratives of the
Resurrection. This was easily and speedily settled the following
morning.
The seventh rule was as follows:--"To revise the headings of chapters and
pages, paragraphs, italics, and punctuation." This rule was very
carefully attended to except as regards headings of chapters and pages.
These were soon found to involve so much of indirect, if not even of
direct interpretation, that both Companies agreed to leave this portion
of the work to some committee of the two University Presses that they
might afterwards think fit to appoint. Small as the work might seem to
be if only confined to the simple revision of the existing headings, the
time it would have taken up, if undertaken by the Companies, would
certainly have been considerable. I revised, on my own account, the
headings of the chapters in St. Matthew, and was surprised to find how
much time was required to do accurately and consistently what might have
seemed a very easy and inconsiderable work.
The eighth rule was of some importance, though, I think, very rarely
acted upon: "To refer, on the part of each Company, when considered
desirable, to divines, scholars, and literary men, whether at home or
abroad, for their opinions." How far this was acted on by the Old
Testament Company I do not know. In regard of the New Testament Company
the only instance I can remember, when we availed ourselves of the rule,
was in reference to our renderings of portions of the twenty-seventh
chapter of the Acts of the Apostles. In this particular case we sent our
sheets to the Admiralty, and asked the First Sea Lord (whom some of us
knew) kindly to tell us if the expressions we had adopted were nautically
correct. I believe this friendly and competent authority did not find
anything amiss. It has sometimes been said that it would have been
better, especially in reference to the New Testament, if this rule had
been more frequently acted on, and if matters connected with English and
alterations of rhythm had been brought before a few of our more
distinguished literary men. It may be so; though I much doubt whether in
matters of English the Greek would not always have proved the dominant
arbiter. In matters of rhythm it is equally doubtful whether much could
have been effected by appealing to the ears of others. At any rate we
preferred trusting to our own, and adopted, as I shall afterwards
mention, a mode of testing rhythmical cadence that could hardly have been
improved upon.
The concluding rule was one of convenience and common sense: "That the
work of each Company be communicated to the other, as it is completed, in
order that there may be as little deviation from uniformity in language
as possible."
All preliminaries were now settled. The invitations were issued, and,
with the exceptions of Canon Cook, Dr. Pusey, and Dr. Newman, were
readily accepted. Three or four names (Principal Douglas, Professor
Geden, Dr. Weir, and, I think, Mr. Bensley), were shortly added to those
already mentioned as invited to join the Old Testament Company, and, in
less than a month after the meeting of the committee on May 25, both
Companies had entered upon their responsible work. On June 22, 1870,
both Companies, after a celebration of the Holy Communion, previously
announced by Dean Stanley as intended to be administered by him in
Westminster Abbey, in the Chapel of Henry VII, commenced the
long-looked-for revision of the Authorised Version of God's Holy Word.
The Old Testament Company commenced their work in the Chapter Library;
the New Testament Company in the Jerusalem Chamber.
The number of the members in each Company was very nearly the same, viz.
twenty-seven in the Old Testament Company, and, in nominal attendance,
twenty-six in the New Testament Company. In the former Company, owing to
the longer time found necessary for the work (fourteen years), there were
more changes in the composition of the Company than in the case of the
latter Company, which completed its work three years and a half before
its sister Company. At the close of the work on the New Testament
(1880), the numbers in each Company were twenty-six and twenty-five; but
owing to various reasons, and especially the distance of many of the
members from London, the number in actual and regular attendance was
somewhat reduced as the years went onward. How it fared with the Old
Testament Company I cannot precisely state. Bishop Harold Browne, after
his accession to the See of Winchester, was only able to attend twice or
three times after the year 1875. In that year Bishop Thirlwall died, and
Bishop Ollivant ceased to attend, but remained a corresponding member
till his death in 1882. Vacancies, I am informed, were filled up till
October 1875, after which date no new members were added. The Company,
however, worked to the very end with great devotion and assiduity. The
revision occupied 794 days, and was completed in eighty-five sessions,
the greater part of which were for ten days each, at about six hours a
day.
I can speak a little more exactly in reference to the New Testament
Company. The time was shorter, and the changes in the composition of the
Company were fewer. At the end of the work a record was made out of the
attendances of the individual members {35}, from which it was easy to
arrive at the average attendance, which for the whole time was found to
be as much as sixteen each day. The number of sessions was 101 of four
days each, and one of three days, making a total of 407 days in all.
More than 1,200 days were thus devoted to the work of the revision of the
Authorised Versions of both Testaments. The first revision, in the case
of the New Testament lasted about six years; the second, two years and a
half. The remaining two years were spent in the consideration of various
details and reserved questions, and especially the consideration of the
suggestions, on our second revision, of the American Revisers, of whose
work and connexion with the English Revisers it will now be convenient to
speak.
* * * * *
The idea of a connexion with America in the great work of revision was
nearly as early as the movements in Convocation of which an account has
been given. It appears that, in the session of Convocation in July,
1870, it was moved in the Lower House by Lord Alwyne Compton (afterwards
and now Bishop of Ely) that the committee of Convocation should be
instructed to invite the co-operation of some American divines. This was
at once agreed to by both Houses, and measures were taken to open
communications with America. The correspondence was opened by the acting
Chairman of the New Testament Company (the present writer) in a letter to
Dr. Angus (dated July 20, 1870 {36}) who was about to visit the United
States, empowering him to prepare the way for definite action on the part
of American scholars and divines. This he did in a letter ("Historical
Account," p. 31) sent round to American scholars, and especially by
communication with Dr. Philip Schaff of the Bible House at New York, who,
from the first, had taken the deepest interest in the movement. This
active and enterprising scholar at once took up the matter, and operated
so successfully that, as he himself tells us in his valuable and accurate
"Companion to the Greek Testament and the English Version" (New York,
1883), a committee of about thirty members was formally organized Dec. 7,
1871, and entered upon active work on Oct. 4, 1872, after the first
revision of the Synoptical Gospels had been forwarded by the New
Testament Company.
Our Old Testament Company was no less active and co-operative. As they
tell us in the Preface prefixed to their revision, "the first revision of
the several books of the Old Testament was submitted to the consideration
of the American Revisers, and, except in the case of the Pentateuch
(which had been twice gone through prior to co-operation) the English
Company had the benefit of their criticisms and suggestions before they
proceeded to the second revision. The second revision was in like manner
forwarded to America, and the latest thoughts of the American Revisers
were in the hands of the English Company at their final review." Both
our English Companies bear hearty testimony to the value derived from the
co-operation. In the case of the New Testament Company, the "care,
vigilance, and accuracy" which marked the work of their American brethren
is distinctly specified.
But little more need be said of the American Companies. They were soon
fully organized, and, so far as can be judged by the results of their
work, carefully and judiciously chosen. The Old Testament Company
consisted of fifteen members, Dr. Green, Professor in Princeton, being
Chairman: the New Testament Committee consisted of sixteen members, three
of those who had at first accepted having been obliged, from ill-health
and stress of local duties, to resign. Dr. Woolsey, Ex-President of Yale
College, was Chairman, and Bishop Lee, of the Diocese of Delaware, one of
the most faithful and valuable participators in the work, a member of the
Company. Dr. Philip Schaff, Professor of Sacred Literature in the Union
Theological Seminary, New York, was also a member, and was President of
the whole undertaking, Dr. George Day of Yale College, a member of the
Old Testament Company, being the general secretary. The two Companies
met every month (except July and August) in two rooms in the Bible House,
New York, but without any connexion with the Bible Society, which, as in
England, could only circulate the Authorised Version.
The American Committee, Dr. Schaff tells us, included representatives of
nine different denominations, viz. Episcopalians, Presbyterians,
Congregationalists, Baptists, Methodists and, to the extent of one
member, Lutherans, Unitarians, and Society of Friends. The Episcopal
Church of America was applied to by Bishop Wilberforce with the request
that they would take part in the revision: this was declined. The
American Church however, as we have already shown, was not wholly
unrepresented in the work. The whole Committee was obviously much more
mixed than the English Committee; but it must not be forgotten that
though the English Companies were chosen by Episcopalians, and
Episcopalians, as was natural, greatly preponderated, nearly one-third of
the two Companies were not members of the Church of England. If we
assume that each Company consisted at any given time of twenty-five
members, which, as we have seen, would be approximately correct, the
non-Episcopal members will be found to have been not less than sixteen,
viz. seven Presbyterians, four Independents or Congregationalists, two
Baptists, two Wesleyans, and one Unitarian. Be this however as it may,
it is certain that by the great blessing, we may humbly say, of God the
Holy Ghost, the greatest possible harmony prevailed in the work both here
and in America. Here, as is well known, this was the case; and in
America, to quote one only out of many similar witnesses, one who was
himself a reviser, and the only pastor in the Company (the Old Testament
Company), thus gives his experience, "Never, even once, did the _odium
theologicum_ appear. Nothing was said at any time that required
retraction or apology {41}."
This brief notice of our American brethren may close with one further
comment. Their work began, like ours, with reliance on financial aid
from the many who would be sure to be interested in such an important and
long-desired work. Help in our case was at once readily proffered, but
very soon was found not to be necessary, owing to our disposal of
copyright to the Presses of the two Universities. With the American
Revisers it was otherwise. During the whole twelve years all the
necessary expenses of travelling, printing, room-rent, and other
accessories were, as Dr. Schaff mentions, cheerfully contributed by
liberal donors from among the friends of biblical revision. There
remained, however, a grave difficulty. It was plainly impossible that
such distinguished men as those who formed the two American Companies
could simply act the part of friendly critics of what was sent over to
them without being recognized as fellow revisers in the full sense of the
words. How, however, formally to establish this parity of position was
found to be very difficult, owing to our connexion with the Presses, who
had trade rights which had properly to be guarded. The result was much
friendly negotiation for several months, but without any definite
adjustment {42a}. At last, by the wise and conciliatory action of the
Presses an agreement was arrived at in August, 1877 {42b}, by which we on
this side of the Atlantic were bound not only to send over the various
stages of our work to our American brethren and carefully to consider all
their suggestions, but also to sanction the publication in every copy of
the revision of a list of all the important passages, in regard of text
and renderings, upon which the English and American Revisers could not
finally agree. The American Revisers on their part undertook not to
publish any edition of their own for fourteen years.
The fourteen years have now passed away, but prior to the expiration of
the time the long-needed marginal references were completed, and in
September, 1898, were attached to the pages of all the larger English
copies of the Revised Version of the Holy Scripture, with a short account
of the sources from which they were derived, and of the circumstances of
their delayed publication. As they were somewhat closely connected with
the labours of two of the members of the New Testament Company, and had
received the general approval of that Company, I had real pleasure in
presenting to both Houses of Convocation on Feb. 10, 1899, the completed
body of references, and, in them, the very last portion of every part of
the work of the Company with which I had so long been connected.
The appearance of the references was very seasonable, as it enabled the
Universities to acquire copyright for any of the editions _with these
references_ which they might publish, or cause to be published in
America. The University Press of Oxford has, I know, acted on this
right, but whether in conjunction with the Cambridge University Press or
independently I am not able to say. The right at any rate remains, and
in the sequel may be of greater importance in America than we may now
suppose, as it may tend to discourage the spread of altered editions of
the revision, which from time to time might be brought forward by
irresponsible publishers {44}.
One subject still remains to be noticed in this portion of my address
which cannot be passed over--the revision of the Apocrypha. This the
English revisers were pledged to the University Presses to complete,
before our connexion with them could be rightfully concluded. This
revision, as we know, has been completed, though perhaps not in a manner
that can be considered as completely satisfactory, owing to the want of a
co-ordinating authority. The arrangement, of which a full and clear
account will be found in the preface to the published volume, was briefly
as follows. On March 21, 1879, as the New Testament Company was fast
approaching the completion of its labours, it was agreed that the Company
should be divided into three portions, each consisting of eight members,
to which the names of the London, Westminster, and Cambridge Companies
were to be respectively assigned. The portion of the work that each of
the three Companies was to take was settled by lot. To the London
Company, of which I was a member, the book of Ecclesiasticus was
assigned; to the Westminster Company, the first book of Maccabees, and
subsequently the books Tobit and Judith; and to the Cambridge Company,
the second book of Maccabees and the Wisdom of Solomon.
On the completion of their work, the Old Testament Company assigned to a
special committee chosen out of their number the remaining books of the
Apocrypha, viz. 1 and 2 Esdras, the remainder of Esther, Baruch, Song of
the Three Children, Susanna, Bel and the Dragon, and the Prayer of
Manasses.
It was agreed that each Company and the above-named committee should go
through their work twice, but without the two-thirds condition, and that
each body should send its work when completed round to the rest. The
times, however, at which the portions were completed were by no means,
even approximately, the same. The London Company completed its work in
May, 1883. The Westminster Company finished the first book of Maccabees
in November, 1881, and the books of Tobit and Judith in October, 1882.
The Cambridge Company completed its revision of the second book of
Maccabees in December, 1889, and of the Book of Wisdom, which underwent
three revisions, in November, 1891. The revision of the remaining books,
undertaken by the Old Testament Company, does not seem to have been
completed till even two or three years later. This interval of ten or
twelve years involved in some of the books, especially in reference to
Ecclesiasticus, the clear necessity for further revision. This compelled
me, with the help of my valued friend Dr. Moulton, to go over the work of
my former Company on my own responsibility, my coadjutors in the work
having been either called away by death or too seriously ill to help me.
It was thus with some sense of relief that, on the request of those
connected with the publication of the volume, I presented the Revised
Version of the Apocrypha to the two Houses of Convocation on February 12,
1896.
The rise and progress of the desire for a revision of the Authorised
Version of Holy Scripture has now been set forth as fully as the limits
of these Addresses permit. What now remains to be specified is what may
be called the internal history of this Revision, or, in other words, the
nature and procedure of the work, with such concluding comments as the
circumstances of the present may appear to suggest.
ADDRESS III.
HEBREW AND GREEK TEXT.
We now pass from what may be called the outward history of the Revision
to the inward nature and character of the work of the Revisers, and may
naturally divide that work into two portions--their labours as regards
the original text, and their labours in regard of rendering and
translation.
I. First, then, as regards the original text of the Old Testament.
Here the work of the Old Testament Company was very slight as compared
with that of the New Testament Company. The latter Company had, almost
in every other verse, to settle upon a text--often involving much that
was doubtful and debatable--before they proceeded to the further work of
translating. The Old Testament Company, on the contrary, had ready to
hand a _textus receptus_ which really deserved the title, and on which,
in their preface, they write as follows: "The received, or, as it is
commonly called, the Massoretic text of the Old Testament Scriptures has
come down to us in manuscripts which are of no very great antiquity, and
which all belong to the same family or recension. That other recensions
were at one time in existence is probable from the variations in the
Ancient Versions, the oldest of which, namely, the Greek or Septuagint,
was made, at least in part, some two centuries before the Christian era.
But as the date of knowledge on the subject is not at present such as to
justify any attempt at an entire reconstruction of the text on the
authority of the Versions, the Revisers have thought it most prudent to
adopt the Massoretic text as the basis of their work, and to depart from
it, as the Authorised Translators had done, only in exceptional cases."
That in this decision the Revisers had exercised the sound judgement
which marks every part of their work cannot possibly be doubted by any
competent reader. The Massoretic text has a long and interesting
history. Its name is derived from a word, Massora (tradition), that
reminds us of the accumulated traditions and criticisms relating to
numerous passages of the text, and of the manner in which it was to be
read, all which were finally committed to writing, and the ultimate
result of which is the text of which we have been speaking. That the
formation of the written Massora was a work of time seems a probable and
reasonable supposition. A very competent writer {50} tells us that this
formation may have extended from the sixth or seventh to the tenth or
eleventh century. From the end of this Massoretic period onward the same
writer tells us that the Massora became the great authority by which the
text given in all the Jewish manuscripts was settled. All our
manuscripts, in a word, are Massoretic. Any that were not so were not
used, and allowed to perish, or, as it has been thought, were destroyed
as not being in strict accordance with the recognized standards. Whether
we have sustained any real critical loss by the disappearance of the
rejected manuscripts it is impossible to say. The fact only remains that
we have no manuscript of any portion of the Old Testament certainly known
to be of a date prior to A.D. 916. The Massora, it may be mentioned,
appears in two forms--the _Massora parva_ and the _Massora magna_. The
former contains the really valuable portion of the great work, viz., the
variation technically named K'ri (_read_), and placed in the margin of
the Hebrew Bibles. This was to be substituted for the corresponding
portion in the text technically named C'thib (_written_), and was
regarded by the Massoretes themselves as the true reading. The _Massora
magna_ contained the above, and other matter deemed to be of importance
in reference to the interpretation of the text.
The Revisers inform us that they have generally, though not uniformly,
rendered the C'thib in the text, and left the K'ri in the margin, with
the introductory note, "Or, according to another reading," or, "Another
reading is." When they adopted the K'ri in the text of their rendering,
they placed the C'thib in the margin if it represented a variation of
importance.
These things, and others specified in the preface, should be carefully
attended to by the reader as enabling him to distinguish between the
different characters of the alternative renderings as specified in the
margin. Those due to the Massoretes, or, in other words, the K'ris, will
naturally deserve attention from their antiquity. They are not, however,
when estimated with reference to the whole of the sacred volume, very
numerous. In the earliest printed bible they were 1,171 in number, but
this is generally considered erroneous in excess, 900 being probably much
nearer the true estimate.
We cannot leave the subject of the Hebrew text without some reference to
the emendation of it suggested by the Ancient Versions. But little, I
believe, of a systematic character has, as yet, been accomplished. The
Revisers mention that they have been obliged, in some few cases of
extreme difficulty, to depart from the Massoretic text and adopt a
reading from the Ancient Versions. I regret to observe that it is stated
by one of those connected with the forthcoming American revision of the
Old Testament version that in nearly one hundred cases the marginal
references to the Ancient Versions will be omitted. Reasons are given,
but these could hardly have escaped the knowledge and observation of the
learned men by whom the references were inserted. The Revisers also
mention that where the Versions appeared to supply a very probable,
though not so absolutely necessary, correction as displacement of the
Massoretic text, they have still felt it proper to place the reading in
the margin.
This recognition of the critical importance of the Ancient Versions by
the Revisers, though obviously in only a limited number of cases, seems
to indicate the great good that may be expected from a more complete and
systematic use of these ancient authorities in reference to the current
text of the Old Testament. At present the texts implied in them have, I
believe, never yet been so closely analysed as to enable us to form any
just estimate of their real critical value. They have been used by
editors, as in the case of Houbigant, but only in a limited and partial
manner. Lists, I believe, are accessible of all the more important
readings suggested or implied by the Versions; but what is needed is far
more than this. In the first place we require much more trustworthy
texts of the Versions themselves than are at present at our disposal. In
the case of the Septuagint we may very shortly look forward to a
thoroughly revised text; and a similar remark may probably be made in
reference to the Vulgate, but I am not aware that much has been done in
the case of the Syriac {53}, and of other versions to which reference
would have to be made in any great critical attempt, such as a revision
of the _textus receptus_ of the Old Testament.
If, however, a first need is trustworthy editions of the Versions, a
second need appears to be a fuller knowledge of the Hebrew material, late
in regard of antiquity though it may be, than was, at any rate, available
till very recently. The new edition of the text of the Hebrew Bible by
Dr. Ginsburg, with its learned and voluminous introduction, may, and
probably does, supply this fuller knowledge; but as in regard of these
matters I can speak only as a novice, I can only reproduce the statement
commonly made by those who have a right to speak on such subjects, that
the collation of the Hebrew manuscripts that we already possess has been
far from complete. There appears to have been the feeling that they all
lead up to the Massoretic text, and that any particular variations from
it need not be treated over-seriously; and yet surely we must regard it
as possible that some of these negligible variations might concur with,
and by their concurrence add weight to, readings already rendered
probable by the suggestive testimony of the Ancient Versions. It may be
right for me to add that the whole question was raised in 1886 by Dr.
Green and Dr. Schaff in a circular letter addressed to distinguished
Hebrews in Germany and elsewhere. The answers are returned in German
{55}, and are translated. They are most of them interesting, though not
very encouraging. The best of them seems to be the answer of Professor
Strack, of Berlin.
But here I must pause. The use made by the Revisers of these ancient
documents has called out the foregoing comments, and has awakened the
hope, which I now venture to express, that the critical use of the
Versions may be expanded, and form a part of that systematic revision of
the text of the Old Testament which will not improbably form part of the
critical labours of the present century.
II. We may now turn to the New Testament, and to the revision of the
_textus receptus_ of the New Testament which our rules necessitated, and
which formed a very important and, it may be added, a very anxious part
of our revision.
And here, at the very outset, one general observation is absolutely
necessary.
It is very commonly said, and I fear believed by many to be true, that
the text adopted by the Revisers and afterwards published (in different
forms) by the two University Presses, hardly differs at all from the
afterwards published text of the two distinguished scholars and critics,
one of whom was called from us a few years ago, and the other of whom
has, to our great sorrow, only recently left us. I allude, of course, to
the Greek Testament, now of world-wide reputation, of Westcott and Hort.
What has been often asserted, and is still repeated, is this, that the
text had been in print for some time before it was finally published, and
was in the hands of the Revisers almost, if not quite, from the very
first. It was this, so the statement runs, that they really worked upon,
and this that they assimilated.
Now this I unhesitatingly declare, as I shall subsequently be able to
prove, is contrary to the facts of the case. It is perfectly true that
our two eminent colleagues gave, I believe, to each one of us, from time
to time, little booklets of their text as it then stood in print, but
which we were always warned were not considered by the editors themselves
as final. These portions of their text were given to us, not to win us
over to adopt it, but to enable us to see each proposed reading in its
continuity. How these booklets were used by the members of the Company
generally, I know not. I can only speak for myself; but I cannot
suppress the conviction that I was acting unconsciously in the same
manner as the great majority of the Company. I only used the booklets
for occasional reference. In preparing the portion of the sacred volume
on which we were to be engaged in the next session of the Company, I took
due note of the readings as well as of the renderings, but I formed my
judgement independently on the evidence supplied to me by the notes of
the critical edition, whether that of Tischendorf or Tregelles, which I
then was in the habit of using. This evidence was always fully stated to
the Company, nearly always by Dr. Scrivener, and it was upon the
discussion of this evidence, and not on the reading of any particular
editor, on which the decision of the Company was ultimately formed. We
paid in all cases great attention to the arguments of our two eminent
colleagues and our experienced colleague, Dr. Scrivener; but each
question of reading, as it arose, was settled by the votes of the
Company. The resulting text, as afterwards published by the Oxford
University Press, and edited by Archdeacon Palmer, was thus the direct
work of the Company, and may be rightly designated, as it will be in
these pages, as the Revisers' text.
It is of considerable importance that this should be borne in mind; for,
in the angry vituperation which was directed against the Revisers' text,
it was tacitly assumed that this text was practically identical with that
of Westcott and Hort, and that the difficulties which are to be found in
this latter text (and some there certainly are) are all to be found in
the text of the Revisers. How very far such an assumption is from the
true state of the case can easily be shown by a simple comparison of one
text with the other. Let us take an example. I suppose there are very
few who can entertain the slightest doubt that in Acts xii. 35, St. Luke
tells us that Barnabas and Saul returned _from_ Jerusalem after their
mission was over, and took with them (from Jerusalem) St. Mark. Now what
is the reading of Westcott and Hort?--"to Jerusalem" with the Vatican
Manuscript, and a fair amount of external support. We then turn at once
to the Revisers' text and find that _from_ ([Greek text]) is maintained,
in spite of the clever arguments which, in this case, can be urged for an
intrinsically improbable reading, and, most likely, were urged at the
time, as I observe that the Revisers have allowed the "to" to appear in a
margin.
I regret that I have never gone through the somewhat laborious process of
minutely comparing the Revisers' text with the text of Westcott and Hort,
but I cannot help thinking that the example I have chosen is a typical
one, and does show the sort of relations between the two texts, when what
a recent and competent writer (Dr. Salmon, of Trinity College, Dublin)
considers to be the difficulties and anomalies and apparent perversities
in the text of Westcott and Hort are compared with the decisions of the
Revisers {59}. There are, I believe, only sixty-four passages in the
whole revision, in which the text of the Revisers, when agreeing with the
text of Westcott and Hort, has not also the support of Lachmann, or
Tischendorf, or Tregelles.
I observe that the above-named writer expresses his satisfaction that the
Revised Version has not superseded the Authorised Version in our Churches
{60a}, and that things which were read at Rome in the second century may
still be read in our own Churches in the nineteenth century. This,
perhaps, is a strong way of expressing his aversion to the text of
Westcott and Hort, but it is not perfectly clear that the Revisers' text
has "so closely" followed the authority of these two eminent critics as
to be open, on Dr. Salmon's part, to the same measure of aversion. Until
more accurate evidence is forthcoming that the Revisers have shown in
their text the same sort of studied disregard of Western variations as is
plainly to be recognized in the text of Westcott and Hort, I can only
fall back on my persuasion, as one who has put to the vote these critical
questions very many times, that systematic neglect of Western authority
cannot fairly be brought home to the Revisers. It is much to be
regretted then, that in the very opening chapter of his interesting
volume, Dr. Salmon roundly states that Westcott and Hort exercised a
"predominating influence" on their colleagues in the revision on the
question of various readings {60b}, and that "more than half of their
brother members of the Committee had given no special attention to the
subject." Now, assuming that the word "Committee" has been here
accidentally used for the more usual term Company, I am forced to say
that both statements are really incorrect. I was permitted by God's
mercy to be present at every meeting of the Company except two, and I can
distinctly say that I never observed any indication of this predominating
influence. We knew well that our two eminent colleagues had devoted many
years of their lives to the great work on which they were engaged; and we
paid full deference to what they urged on each reading as it came before
us, but in the end we decided for ourselves. For it must not be
forgotten that we had an eminent colleague (absent only eight times from
our 407 meetings) who took a very different view of the critical evidence
to that of Westcott and Hort, and never failed very fully, and often very
persuasively, to express it. I am of course alluding to my old friend
Dr. Scrivener. It was often a kind of critical duel between Dr. Hort and
Dr. Scrivener, in which everything that could be urged on either side was
placed before the Company, and the Company enabled to decide on a full
knowledge of the critical facts and reasonings in reference to the
reading under consideration.
Now it is also not correct to say of the Company that finally decided the
question, that more than half "had given no special attention to the
subject." If this refers to the matter _subsequently_ put forward by Dr.
Hort in the introductory volume to Westcott and Hort's Greek Testament,
to the clever and instructive genealogical method, and to the numberless
applications of it that have given their Greek Testament the pre-eminence
it deservedly holds--if this be the meaning of the Provost's estimate of
the critical knowledge of the Company, I should not have taken any
exception to the words. But if "the subject" refers to the general
critical knowledge at the time when the Company came together, then I
must gently protest against an estimate of the general critical
capabilities of the Company that is, really and truly, incorrect. All
but three or four are now resting with God, and among these twenty they
were not few who had a good and full knowledge of the New Testament
textual criticism of the generation that had just passed away. Among
them were not only the three experts whom I have mentioned, but editors
of portions of the New Testament such as Bishop Lightfoot and others,
principals of large educational colleges both in England and Scotland,
and scholars like Dean Scott, who were known to take great interest in
questions of textual criticism. A few of these might almost be
considered as definitely experts, but all taken together certainly made a
very competent body to whose independent judgement the settlement of
difficult critical questions could be safely committed.
And, as I venture to think, the text which has been constructed from
their decisions, their resultant text as it might be called, will show
that the Revisers' text is an independent text on which great reliance
can be placed. It is the text which I always use myself in my general
reading of the New Testament, and I deliberately regard it as one of the
two best texts of the New Testament at present extant; the other being
the cheap and convenient edition of Professor Nestle, bearing the title
"Novum Testamentum Graece, cum apparatu critico ex editionibus et libris
manu scriptis collecto. Stuttgart, 1898." This edition is issued by the
Wurtemberg Bible Society, and will, as I hear, not improbably be adopted
by our own Bible Society as their Greek Testament of the future.
The reason why I prefer these two texts for the general reading of the
sacred volume is this, that they both have much in common with the text
of Westcott and Hort, but are free from those peculiarities and, I fear I
must add, perversities, which do here and there mark the text of that
justly celebrated edition. To Doctors Westcott and Hort all faithful
students of the New Testament owe a debt of lasting gratitude which it is
impossible to overestimate. Still, in the introductory volume by Dr.
Hort, assumptions have been made, and principles laid down, which in
several places have plainly affected the text, and led to the maintenance
of readings which, to many minds, it will seem really impossible to
accept. An instance has been given above on page 58, and this is by no
means a solitary instance.
Having now shown fairly, I hope, and clearly the thoroughly independent
character of the text which I have called the Revisers' text, I will pass
onward, and show the careful manner in which it was constructed, and the
circumstances under which we have it in the continuous form in which it
has been published by the Press of the University of Oxford.
To do this, it will be necessary to refer to the rule under which we were
directed to carry out this portion of our responsible work. We had two
things to do--to revise the Authorised Version, and also to revise under
certain specified limitations the Greek text from which the Authorised
Version was made; or, in other words, the fifth edition of Beza's Greek
Testament, published in the year 1698. The rule under which this second
portion of our work was to be performed was as follows: "That the text to
be adopted be that for which the evidence is decidedly preponderating;
and [let this be noted] that when the text so adopted differs from that
from which the Authorised Version was made, the alteration be indicated
in the margin." Such was the rule in regard of the text, and such was
the instruction as to the mode of notifying any alterations that it might
have been found necessary to make.
Let us deal first with the direction as to notifying the alterations.
Now as it was soon found practically impossible to place all the
alterations in a margin which would certainly be needed for alternative
renderings, and for such matters as usually appear in a margin, we left
the University Presses to publish, in such manner as they might think
most convenient, the deviations from the Greek text presumed to underlie
the Authorised Version. The Cambridge University Press entrusted to Dr.
Scrivener the publication of the Received Text with the alterations of
the Revisers placed at the foot of the page. The Oxford University Press
adopted the more convenient method of letting the alterations form part
of the continuous text (the readings they displaced being at the foot of
the page), and entrusted the editing of the volume to Archdeacon Palmer
(one of our Company) who, as we know, performed the duty with great care
and accuracy. Hence the existence of what I term throughout this address
as the Revisers' text.
We can now turn to the first part of the rule and describe in general
terms the mode of our procedure. It differs very slightly from the mode
described in the preface of the Revisers of the Old Testament. The verse
on which we were engaged was read by the Chairman. The first question
asked was, whether there was any difference of reading in the Greek text
which required our consideration. If there was none, we proceeded with
the second part of our work, the consideration of the rendering. If
there was a reading in the Greek text that demanded our consideration it
was at once discussed, and commonly in the following manner. Dr.
Scrivener stated briefly the authorities, whether manuscripts, ancient
versions, or patristic citations, of which details most of us were
already aware. If the alteration was one for which the evidence was
patently and decidedly preponderating, it was at once adopted, and the
work went onward. If, however, it was a case where it was doubtful
whether the evidence for the alteration _was_ thus decidedly
preponderating, then a discussion, often long, interesting, and
instructive, followed. Dr. Hort, if present (and he was seldom absent;
only forty-five times out of the 407 meetings) always took part, and
finally the vote was taken, and the suggested alteration either adopted
or rejected. If adopted, due note was taken by the secretary, and, if it
was thought a case for a margin, the competing reading was therein
specified. If there was a plain difficulty at coming to a decision, and
the passage was one of real importance, the decision was not uncommonly
postponed to a subsequent meeting, and notice duly given to all the
members of the Company. And so the great work went on to the end of the
first revision; the members of the Company acquiring more and more
knowledge and experience, and their decisions becoming more and more
judicial and trustworthy.
Few, I think, on reading this simple and truthful description, could fail
to place some confidence in results thus patiently and laboriously
arrived at. Few, I think, could forbear a smile when they call to mind
the passionate vituperation which at first was lavished on the critical
efforts of the Revisers of the text that bears the scarcely correct name
of the _textus ab omnibus receptus_.
But what I have specified was only the first part of our responsible
work. By the memoranda of agreement between the English Companies and
the American Committee, it had to be communicated to the American Company
of the Revisers of the Authorised Version of the New Testament, among
whom were some whose names were well and honorably known in connexion
with textual criticism. Our work, with the American criticisms and
suggestions, had then to undergo the second revision. The greater part
of the decisions relating to the text that were arrived at in the first
revision were accepted as final; but many were reopened at the second
revision, and the critical experience of the Company, necessarily
improved as it had been by the first revision, finally tested by the
two-thirds majority the reopened decisions which at the first revision
had been carried by simple majorities. The results of this second
revision were then, in accordance with the agreement, communicated to the
American Company; but, in the sequel, as will be seen in the lists of the
final differences between ourselves and the American Company, the
critical differences were but few, and, so far as I can remember, of no
serious importance.
The critical labours of the Revisers did not however terminate with the
second revision. The cases were many where the evidence for the readings
either adopted or retained in the text was only slightly stronger than
that of readings which were in competition with it. Of this it was
obviously necessary that some final intimation should be given to the
reader, as the subsequent discovery of additional evidence might be held
by a competent critic to invalidate the right of the adopted reading to
hold its place in the text. This intimation could only be given by a
final marginal note, for which, as we know, by the arrangement of the
University Presses (see p. 66), our page was now available.
These notes were objected to by one of our critics as quite unprecedented
additions; but it will be remembered that there are such notes in the
margin of the Authorised Version, though of course few in number
(thirty-five, according to Dr. Scrivener), textual criticism in 1611
being only in its infancy.
The necessity for the insertion of such notes was clearly shown in a
pamphlet that appeared shortly after the publication of the Revised
Version, and was written by two members of the Company. The three cases
in which these notes appeared certainly to be required were thus stated
by the two writers: "First, when the text which seemed to underlie the
Authorised Version was condemned by a decided preponderance of evidence,
but yet was ancient in its character, and belonged to an early line of
transmission. Secondly, when there were such clear tokens of corruption
in the reading on which the Authorised Version was based, or such a
consent of authority against it, that no one could seriously argue for
its retention, but it was not equally clear which of the other competing
readings had the best claim to occupy the vacant place. In such a case
there was not, in truth, decidedly preponderant evidence, except against
the text of Beza, and some notice of this fact seemed to be required by
critical equity. The third and last case was when the text which, as
represented in the Authorised Version, was retained because the competing
reading had not decidedly preponderant evidence (though the balance of
evidence was in its favour), and so could not under the rule be admitted.
In such a case again critical equity required a notice of the facts in
the margin."
This quotation, I may remark in passing, is not only useful in explaining
when and where marginal notes were demonstrably needed, but also in
showing how carefully such questions were considered, and how
conscientiously the rules were observed under which our work was to be
carried out.
Such were the textual labours of the Company. They were based on, and
were the results of, the critical knowledge that had been slowly acquired
during the 115 years that separated the early suggestions of Bentley from
the pioneer text of Lachmann in 1831; and, in another generation, had
become expanded and matured in the later texts of Tischendorf, and still
more so in the trustworthy and consistent text of our countryman
Tregelles. The labours of these three editors were well known to the
greater part of the Revisers and generally known to all; and it was on
these labours, and on the critical methods adopted by these great
editors, that our own text was principally formed. We of course owed
much to the long labours of our two eminent colleagues, Dr. Westcott and
Dr. Hort. Some of us know generally the principles on which they had
based their yet unpublished text, and were to some extent aware of the
manner in which they had grouped their critical authorities, and of the
genealogical method, which, under their expansion of it, has secured for
their text the widespread acceptance it has met with both at home and
abroad.
Of these things some of us had a competent knowledge, but the majority
had no special knowledge of the genealogical method. They did know the
facts on which it was based--the ascertained trustworthiness of the
ancient authorities as compared with the later uncial, and the cursive
manuscripts, the general characteristics of these ancient authorities,
the alliances that were to be traced between some of them, and the
countries with which they were particularly connected. This the majority
knew generally as a part of the largely increased knowledge which the
preceding forty or fifty years, and the labours of Lachmann, Tischendorf,
and (so far as he had then published) Tregelles, had placed at the
disposal of students of the Greek Testament. It was on this general
knowledge, and not on any portions of a partly printed text, that the
decisions of the Company were based; these decisions, however, by the
very nature of the case and the use of common authorities, were
constantly in accordance with the texts of Lachmann, Tischendorf, and
Tregelles, and so with the subsequently printed text of Westcott and
Hort.
Such a text, thus independently formed, and yet thus in harmony with the
results of the most tested critical researches of our times, has surely
great claims on our unreserved acceptance, and does justify us in
strongly pleading that a version of such a text, if faithfully executed,
should, for the very truth's sake, be publicly read in our Churches.
That the Revised Version has been faithfully executed, will I hope be
shown fully and clearly in the succeeding chapter. For the present my
care has been to show that the text of which it is a version, and which I
have called the Revisers' Text because it underlies their revision, and,
as such, has been published by the Oxford University Press, is in my
judgement the best balanced text that has appeared in this country. I
have mentioned with it (p. 63) the closely similar text of the well-known
Professor Nestle, but as I have not gone through the laborious task of
comparing the text, verse by verse, with that of the Revisers, I speak
only in reference to our own country. I have compared the two texts in
several crucial and important passages--such for example as St. John i.
18--and have found them identical. Bishop Westcott, I know, a short time
before his lamented death, expressed to the Committee of the Bible
Society his distinct approval of their adopting for future copies of the
Society's Greek Testament Professor Nestle's text, as published by the
Wurtemberg Bible Society.
I have now, I trust, fairly shown the independence of the Revisers' Text,
and have, not without reason, complained of my friend Provost Salmon's
estimate of its dependence on the text and earnestly exerted influence of
Dr. Hort and Dr. Westcott. Of course, as I have shown, there is, and
must be, much that is identical in the two texts; but, to fall back on
statistics, there are, I believe, more than two hundred places in which
the two texts differ, and in nearly all of them--if I may venture to
express my own personal opinion--the reading of the Revisers' Text is
critically to be preferred. Most of these two hundred places seem to be
precisely places in which the principles adopted by Westcott and Hort
need some corrective modifications. Greatly as I reverence the unwearied
patience, the exhaustive research, and the critical sagacity of these two
eminent, and now lamented, members of our former Company, I yet cannot
resist the conviction that Dr. Salmon in his interesting Criticism of the
Text of the New Testament has successfully indicated three or more
particulars which must cause some arrest in our final judgement on the
text of Westcott and Hort.
In the first case it cannot be denied that, in the introductory volume,
Dr. Hort has shown too distinct a tendency to elevate probable hypotheses
into the realm of established facts. Dr. Salmon specifies one, and that
a very far-reaching instance, in which, in the debatable question whether
there really was an authoritative revision of the so-called Syrian text
at about A.D. 350, Dr. Hort speaks of this Syrian revision as a _vera
causa_, as opposed to a hypothetical possibility. This tendency in a
subject so complicated as that of textual criticism must be taken note of
by the student, and must introduce some element of hesitation in the
acceptance of confidently expressed decisions when the subject-matter may
still be very plainly debatable.
In the second place, in the really important matter of the nomenclature
of the ancient types of text which, since the days of Griesbach, and to
some extent before him, have been recognized by all critical scholars, it
does not seem possible to accept the titles of the fourfold division of
these families of manuscripts which have been adopted by Westcott and
Hort. Griesbach, as is well known, adopted the terms Western,
Alexandrian, and Constantinopolitan, for which there is much to be said.
Westcott and Hort recognize four groups. To the first and considerably
the largest they give the title of Syrian, answering to some extent to
the Constantinopolitan of Griesbach; to the second they continue the
title of Western; to the third they give the title of Alexandrian, though
of a numerically more restricted character than the Alexandrian of
Griesbach; to the fourth, an exceedingly small group, apparently
consisting of practically not more than two members, they give the title
of Neutral, as being free alike from Syrian, Western, and Alexandrian
characteristics. On this Neutral family or group Westcott and Hort lay
the greatest critical stress, and in it they place the greatest reliance.
Such is their distribution, and such the names they give to the families
into which manuscripts are to be divided and grouped.
The objections to this arrangement and to this nomenclature are, as Dr.
Salmon very clearly shows, both reasonable and serious. In the first
place, the title Syrian, though Dr. Salmon allows it to pass, is very
misleading, especially to the student. It is liable to be confounded
with the term Syriac, with which it has not and is not intended to have
any special connexion, and it fails to convey the amplitude of the family
it designates. If it is to be retained at all, it must be with the
prefix suggested by Dr. Schaff--the group being styled as the
Graeco-Syrian. But this is of slight moment when compared with the
serious objections to the term Neutral, as this term certainly tends in
practice to give to two manuscripts or even, in some cases, to one of
them (the Codex Vaticanus), a preponderating supremacy which cannot be
properly conceded when authorities of a high character are found to be
ranged on the other side. There are also other grave objections which
are convincingly put forward by Dr. Salmon in the chapter he has devoted
to the subject of the nomenclature of the two editors.
We shall be wise therefore if we cancel the term Neutral and use the term
Older Alexandrian, as distinguished from the later Alexandrian, and so
fall back on the threefold division of Alexandrian (earlier and later),
Graeco-Syrian, and Western, though for this last-mentioned term a more
expressive designation may perhaps hereafter be found.
The third drawback to the unqualified acceptance of the text of Westcott
and Hort is their continuous and studied disregard of Western
authorities; and this, notwithstanding that among these authorities are
included the singular and not unfrequently suggestive Codex Bezae--of
which Dr. Blass has lately made so remarkable a use--the Old Latin
Version, the Graeco-Latin manuscripts, and, to some extent, the Old
Syriac Version, all of them authorities to which the designation of
Western is commonly applied. To this grave drawback Dr. Salmon has
devoted a chapter to which the attention of the student may very
profitably be directed. Here I cannot enter into details, but of this I
am persuaded, that if there should be any fresh discovery of textual
authorities, it is by no means unlikely that they may be of a Western
character, and if so, that many decisions in the text of Westcott and
Hort will have to be modified by some editor of the future. At any rate,
taking the critical evidence as now we find it, we cannot but feel that
Dr. Salmon has made out his case, and that in the edition of which now we
are speaking there has been an undue, and even a contemptuous, disregard
of Western authorities.
Here I must close this address, yet not without expressing the hope that
I may have induced some of you, my Reverend Brethren, to look into these
things for yourselves. Do not be deterred by the thought that to do so
you must read widely and consult many authorities. This is really not
necessary for the acquiring of an intelligent interest in the text of the
Greek Testament. With a good edition (with appended critical
authorities), whether that of Tischendorf or of Tregelles, and with
guidance such as that which you will find in the compendious _Companion
to the Greek Testament_ of Dr. Schaff, you will be able to begin, and
when you have seriously begun, you will not be, I am persuaded, very
likely to leave off.
ADDRESS IV
NATURE OF THE RENDERINGS
From the text we now turn to the renderings, and to the general
principles that were followed, both in the Old and in the New Testament.
The revision of the English text was in each case subject to the same
general rule, viz. "To introduce as few alterations as possible into the
Text of the Authorised Version consistently with faithfulness"; but,
owing to the great difference between the two languages, the Hebrew and
the Greek, the application of the rule was necessarily different, and the
results not easily comparable the one with the other.
It will be best then to consider the renderings in the two Testaments
separately, and to form the best estimate we can of their character and
of their subordination to the general rule, with due regard to the widely
different nature of the structure and grammatical principles of the two
languages through which God has been pleased to reveal His truth to the
children of men.
I. We begin then with the Revised Version of the Old Testament, and
naturally turn for general guidance to the Preface of those who were
engaged in the long, diversified, and responsible work. Their general
principles as to departures from the Authorised Version would appear to
be included in the following clearly-specified particulars. They
departed from the Authorised Version (_a_) where they did not agree with
it as to the meaning or construction of a word or sentence; (_b_) where
it was necessary, for the sake of uniformity, to render such parallel
passages as were identical in Hebrew by the same English words; (_c_)
where the English of the Authorised Version was liable to be
misunderstood by reason of its being archaic or obscure; (_d_) where the
rendering of an earlier English version seemed preferable; and (_e_)
where, by an apparently slight change, it was possible to bring out more
fully the meaning of a passage of which the translation was substantially
accurate.
These principles, which I have been careful to specify in the exact words
of the Revisers, will appear to every impartial reader to be fully in
harmony with the principle of faithfulness; and will be found--if an
outsider may presume to make a passing comment--to have been carried out
with pervasive consistency and uniformity.
The Revisers further notice certain particulars of which the general
reader should take full note, so much of the random criticisms of the
revised text (especially in the New Testament) having been due to a
complete disregard in each case of the Preface, and of the reasons given
for changes which long experience had shown to be both reasonable and
necessary.
The first particular is the important question of the rendering of the
word "JEHOVAH." Here the Revisers have thought it advisable to follow
the usage of the Authorised Version, and not to insert the word uniformly
in place of "LORD" or "GOD," which words when printed in small capitals
represent the words substituted by Jewish custom for the ineffable Name
according to the vowel points by which it is distinguished. To this
usage the Revisers have steadily adhered with the exception of a very few
passages in which the introduction of a proper name seemed to be
required. In this grave matter, as we all probably know, the American
Company has expressed its dissent from the decision of the English
Company, and has adopted the proper name wherever it occurs in the Hebrew
text for "the LORD" and "GOD." Most English readers will agree with our
Revisers. It may indeed be said, now that we can read the American text
continuously, that there certainly are many passages in which the proper
name seems to come upon eye or ear with a serious and appropriate force;
still the reverence with which we are accustomed to treat what the
Revisers speak of as "the ineffable Name" will lead most of us to
sacrifice the passages, where the blessed name may have an impressive
force, to the reverential uniformity of our Authorised Version, and to
the latent fear that frequent iteration might derogate from the solemnity
with which we instinctively clothe the ever-blessed name of Almighty God.
The next particular relates to terms of natural history. Here changes
have only been made where it was certain that the Authorised Version was
incorrect, and highly probable that the word substituted was right.
Where doubt existed, the text was left unchanged, but the alternative
word was placed in the margin. In regard of other terms, of which the
old rendering was certainly wrong, as in the case of the Hebrew term
_Asherah_ (probably the wooden symbol of a goddess), the Revisers have
used the word, whether in the singular or plural, as a proper name. In
the case of the Hebrew term "Sheol" (corresponding to the Greek term
"Hades"), variously rendered in the Authorised Version by the words
"grave," "pit," and "hell," the Revisers have adopted in the historical
books the first or second words with a marginal note, "Heb. _Sheol_," but
in the poetical books they have reversed this arrangement. The American
Revisers, on the contrary, specify that in all cases where the word
occurs in the Hebrew text they place it unchanged in the English text,
and without any margin. The case is a difficult one, but the English
arrangement is to be preferred, as the reader would not so plainly need a
preliminary explanation.
The last case that it here seems necessary to allude to is the change
everywhere of the words "the tabernacle of the congregation" into "the
tent of meeting," as the former words convey an entirely wrong sense.
These and the use of several other terms are carefully noted and
explained by the Revisers, and will, I hope, induce every careful reader
of their revision to make it his duty to study their prefatory words.
The almost unavoidable differences between them and the American
Revisers, as to our own language, are alluded to by them in terms both
friendly and wise, and may be considered fully to express the sentiments
of the New Testament Company, by whom the subject is less precisely
alluded to.
In passing from the Preface to the great work which it introduces, I feel
the greatest difficulty, as a member of a different Company, in making
more than a few very general comments. In fact, I should scarcely have
ventured to do even this, had I not met with a small but very instructive
volume on the revision of the Authorised Version of the Old Testament
written by one of the American Revisers, and published at New York some
fifteen or sixteen years ago. The volume is entitled--perhaps with
excusable brevity--_A Companion to the Revised Old Testament_. The
writer was Rev. Dr. Talbot W. Chambers, of the Collegiate Reformed Dutch
Church of New York, from whose preface I learn that he was the only
pastor in the Company, the others being professors in theological
seminaries, and representing seven different denominations and nine
different institutions. The book is written with great modesty, and as
far as I can judge, with a good working knowledge of Hebrew. The writer
disclaims in it the position of speaking in any degree for the Company of
which he was a member, but mentions that his undertaking was approved of
by his colleagues, and received the assistance, more or less, of all of
them. He was a member of the Company during the last ten years of its
labours.
I can recommend this useful volume to any student of the Old Testament
who is desirous to see a selected list of the changes made by the
Revisers in the Pentateuch, Historical Books, Poetical Books, and
Prophetical Books. These changes are given in four chapters, and in most
cases are accompanied by explanatory comments, which from their tenor
often seem to be reminiscences of corporate discussion. I mention these
particulars as I am not aware of any similar book on the Old Testament
written by any one of the English Company. If there is such a book, I do
sincerely hope the writer will forgive me for not having been so
fortunate as to meet with it.
The remaining comments I shall venture to make on the rendering of the
Old Testament will rest on the general knowledge I have acquired of this
carefully-executed and conservative revision, and on some consideration
of the many illustrations which Dr. Chambers has selected in his
interesting manual. The impression that has long been left on my mind by
the serious reading of the Old Testament in the Revised Version is that
not nearly enough has been said of the value of the changes that have
been made, and of the strong argument they furnish for the reading of the
Revision in the public services of the Church. Let any serious person
read the Book of Job with the two English versions in parallel columns,
and form a sober opinion on the comparison--his judgement I am confident
will be, that if the Revision of this Book be a fair sample of the
Revision generally, our congregations have a just right to claim that the
Revised Version of the Old Testament should be publicly read in their
churches. Ours is a Bible-loving country, and the English Bible in its
most correct form can never be rightly withheld from our public
ministrations.
I shall now close this portion of the present Address with a few comments
on the four parts of the Revision to which I have already alluded--the
Pentateuch, and the Historical, Poetical, and Prophetical Books of the
Old Testament.
What the careful reader of Genesis will not fail to observe is the number
of passages in which comparatively small alterations give a new light to
details of the sacred narrative which, in general reading, are commonly
completely overlooked. A new colouring, so to speak, is given to the
whole, and rectifications of prevailing conceptions not unfrequently
introduced, either in the text or, as often happens, by means of the
margin, where they could hardly have been anticipated. The prophecy of
Jacob as to the future of his children (chap. xlix) will supply an
instance. In the character of Reuben few of us would understand more
than general unsteadiness and changefulness in purpose and in act, but a
glance at the margin will show that impulse and excitability were plainly
elements in his nature which led him into the grievous and hateful sin
for which his father deposed him from the excellency of a first-born.
What has been said of the Book of Genesis is equally applicable to the
remainder of the Pentateuch. The object throughout is elucidation, not
simply correction of errors but removal of obscurity, if not by changes
introduced into the printed text, yet certainly always by the aid of the
margin; as, for example, in the somewhat difficult passage of Exodus
xvii. 16, where really, it would seem, that the margin might rightly have
had its place in the text. Sometimes the correction of what might seem
trivial error, as in Exodus xxxiv. 33, gives an intelligible view of the
whole details of the circumstance specified. Moses put on the veil after
he had ceased speaking with them. While he was speaking to them he was
speaking as God's representative. In Numbers xi. 25 the correction of a
mistranslation removes what might otherwise lead to a very grave
misconception, viz. that the gift of prophecy was continuous in the case
of the whole elderhood. In the chapters relating to Balaam,
independently of the alterations that are made in the language of his
remarkable utterances, the mere fact of their being arranged rhythmically
could not fail to cause the public reader, almost unconsciously, to
change his tone of voice, and to make the reading of the prophecy more
distinct and impressive. Among many useful changes in Deuteronomy one
may certainly be noticed (chap. xx. 19), in which the obscure and
difficult clause in regard of the tree in the neighbourhood of the
besieged city is made at any rate intelligible.
In the historical books attention may be particularly called to the Song
of Deborah and Barak, in which there are several important and
elucidatory corrections, and in which the rhythmic arrangement will be
felt to bear force and impressiveness both to reader and to hearer. In
the remaining Books changes will be found fewer in number and less
striking; but occasionally, as for example in 1 Kings xx. 27, we come
across changes that startle us by their unlooked-for character, but
which, if correct, add a deeper degradation to the outpoured blood of
Ahab in the pool of Samaria.
Of the poetical Books, I have already alluded to the Book of Job and to
the high character of the Revision. The changes in this noble poem are
many, and were especially needed, for the rendering of the Book of Job
has always been felt to be one of the weakest portions of the great work
of the Revisers of 1611. Illustrations I am unable to give, in a cursory
notice like the present, but I may again press the Revisers' version of
this deeply interesting Book on the serious attention of every earnest
student of the Old Testament.
It is difficult to say much on the Revised Version of the Book of Psalms,
as Coverdale's Version, as we have it in our Prayer Book, so completely
occupies the foreground of memory and devotional interest, that I fear
comparatively few study the Bible Version or the careful and conservative
work of the Revisers. This Revision, however, of the version of the Book
of Psalms deserves more attention than it appears to have received. Not
only will the faithful reader find in it the necessary corrections of the
version of 1611, but clear guidance as to the meaning of the sometimes
utterly unintelligible renderings of the version of the Great Bible which
still holds its place in our Prayer Books. To take two examples: let the
reader look at the Authorised Version and Prayer Book Version of Psalm
lxviii. 16, and of lxxxiv. 5, 6, and contrast with both the rendering of
the Revised Version. This last-mentioned rendering will be found, as I
have said, to correct the Authorised Version, and (especially in the
second passage) to remove what is unintelligible in the Prayer Book
version. It may thus be used by the Prayer Book reader of the Psalms as
a ready and easily accessible means of arriving at the real meaning of
the many ambiguities and obscurities which long familiarity with the
Prayer Book Version has led him to pass over without any particular
notice. The revision of the Prayer Book Version has been long felt to be
a very real necessity. To read and to hear read in the daily services of
the Church what, in parts, cannot be understood can never be spiritually
good for reader or hearer. And yet, such is the really devout
conservatism of the bulk of our congregations, that though a careful
revision, sympathetically executed, has been strongly urged by some of
our most earnest scholars and divines, it is more than doubtful whether
such a revision ever will be carried out. If this be so, it only remains
for us so to encourage, in our schools and in our Bible classes, the
efficient explanatory help of the Revised Version. If this is steadily
done, nearly all that is at present obscure or unintelligible in the
Prayer Book Version will no longer remain so to the greater part of our
worshippers.
Of the remaining Poetical Books the revision of the Authorised Version of
the Song of Solomon must be specially noticed. In the common version the
dramatic element is almost entirely lost, the paragraphs are imperfectly
noted, and obscurities not a few the inevitable consequence. In a large
degree these serious imperfections are removed, and the whole tenor of
this exquisite poem made clear to the general reader. The margin will
show the great care bestowed on the poem by the Revisers; and the fewness
and trifling nature of the changes maintained by the American Company
will also show, in a confessedly difficult Book, the somewhat remarkable
amount of the agreement between the two Companies. On the Prophetical
Books I do not feel qualified to speak except in very general terms; and
for illustrations must refer the reader to the large list of the
corrected renderings, especially of the prophecy of Isaiah, in the useful
work of Dr. Chambers, who has devoted at least eleven pages to the
details of the Revisers' work on the Evangelist of the Old Covenant. The
impression which the consideration of these details leaves on the mind of
the reader will be, I am confident, the same as that which is I believe
felt by all professed Hebrew scholars who have examined the version, viz.
that it is not only faithful and thorough, but often rises to a very high
level of poetic utterance. Let any one read aloud in the Revised Version
the well-known passage, chap. xiv. 12-23, already nobly rendered in the
Old Version, and ask himself if the seemingly slight and trivial changes
have not maintained this splendid utterance at a uniform height of
sustained and eloquent vigour.
In the prophecies of Jeremiah and Ezekiel the changes are less striking
and noticeable, not however from any diminished care in the work of
revision, but from the tenor of the prophecies being less familiar to the
general reader. Four pages of instructive illustrations are supplied by
Dr. Chambers in the case of each of the two prophecies. The more
noticeable changes in Daniel and Hosea are also specified by Dr.
Chambers, but the remainder of the minor prophets, with perhaps the
exception of Habakkuk, are passed over with but little illustrative
notice. A very slight inspection however of these difficult prophecies
will certainly show two things--first, that the Revisers of 1611 did
their work in this portion of Holy Scripture less successfully than
elsewhere; secondly, that the English and American Revisers--between whom
the differences are here noticeably very few--laboured unitedly and
successfully in keeping their revision of the preceding version of these
prophecies fully up to the high level of the rest of their work.
II. I now pass onward to the consideration of the renderings in the
Revised Version of the New Testament.
The object and purpose of the consideration will be exactly the same, as
in the foregoing pages, to show the faithful thoroughness of the
Revision, but the manner of showing this will be somewhat different to
the method I have adopted in the foregoing portion of this Address. I
shall not now bring before you examples of the faithful and suggestive
accuracy of the revision, for to do this adequately would far exceed the
limits of these Addresses; and further, if done would far fall short of
the instructive volume of varied and admirably arranged illustrations
written only four years ago by a member of the Company {96}, now, alas,
no longer with us, of which I shall speak fully in my next Address.
What I shall now do will be to show that the principles on which the
version of the New Testament was based have been in no degree affected by
the copious literature connected with the language of the Greek Testament
and its historical position which has appeared since the Revision was
completed. It is only quite lately that the Revisers have been
represented as being insufficiently acquainted, in several particulars,
with the Greek of the New Testament, and in a word, being twenty years
behind what is now known on the subject {97}. Such charges are easily
made, and may at first sight seem very plausible, as the last fifteen or
twenty years have brought with them an amount of research in the language
of the Greek Testament which might be thought to antiquate some results
of the Revision, and to affect to some extent the long labours of those
who took part in it. The whole subject then must be fairly considered,
especially in such an Address as the present, in which the object is to
set forth the desirableness and rightfulness of using the version in the
public services of the Church.
But first a few preliminary comments must be made on the manner and
principles in which the changes of rendering have been introduced into
the venerable Version which was intrusted to us to be revised.
The foremost principle to be alluded to is the one to which we adhered
steadily and persistently during the whole ten years of our labour--the
principle of faithfulness to the original language in which it pleased
Almighty God that His saving truth should be revealed to the children of
men. As the lamented Bishop of Durham says most truly and forcibly in
his instructive "Lessons on the Revised Version of the New Testament
{98a};" "Faithfulness, the most candid and the most scrupulous, was the
central aim of the Revisers {98b}." Faithfulness, but to what?
Certainly not to "the sense and spirit of the original {98b}," as our
critics contended must have been meant by the rule,--but to the original
in its plain grammatical meaning as elicited by accurate interpretation.
This I can confidently state was the intended meaning of the word when it
appeared in the draft rule that was submitted to the Committee of
Convocation. So it was understood by them; and so, I may add, it was
understood by the Company, because I can clearly remember a very full
discussion on the true meaning of the word at one of the early meetings
of the Company. Some alteration had been proposed in the rendering of
the Greek to which objection was made that it did not come under the rule
and principle of faithfulness. This led to a general, and, as it proved,
a final discussion. Bishop Lightfoot, I remember, took an earnest part
in it. He contended that our revision must be a true and thorough one;
that such a meeting as ours could not be assembled for many years to
come, and that if the rendering was plainly more accurate and more true
to the original, it ought not to be put aside as incompatible with some
supposed aspect of the rule of faithfulness. Proposals were often set
aside without the vote being taken, on the ground that it was not "worth
while" to make them, and in a trivial matter to disturb recollection of a
familiar text; but the non-voting resulted from the proposal being
withdrawn owing to the mind of the Company being plainly against it, and
not from any direct appeal to the principle of faithfulness. If the
proposal was pressed, the vote of the Company was always taken, and the
matter authoritatively settled.
The contention, often very recklessly urged, that the Revisers
deliberately violated the principles under which the work was committed
to them is thus, to use the kindest form of expression, entirely
erroneous. I have dwelt upon this matter because when properly
understood it clears away more than half of the objections that have been
urged against our Revision. Of the remainder I cannot but agree with
good Bishop Westcott that no criticism of the Revision--and the
criticisms were of every form and kind "pedantry, spiritless literality,
irritating triviality, destroyed rhythm," and so forth--no criticism ever
came upon us by surprise. The Revisers, as the Bishop truly says, heard
in the Jerusalem Chamber all the arguments against their conclusions they
have heard since; and he goes on to say that no restatement of old
arguments had in the least degree shaken his confidence in the general
results. Such words from one now, alas, no longer with us, but whose
memory we cherish as one of the most wide-minded as well as truth-seeking
of the biblical scholars of our own times, may well serve to reassure the
partially hesitating reader of the Revised Version of its real
trustworthiness and fidelity. But we must not confine our attention
simply to the renderings that hold a place in the text of the Revised
Version. We must take into our consideration a very instructive portion
of the work of the Revisers which is, I fear, utterly neglected by the
general reader--the alternative readings and renderings that hold a place
in the margin, and form an integral portion of the Revision. Though we
are now more particularly considering the renderings, I include here the
marginal readings, as the relation of the margins to the Version could
hardly be fully specified without taking into consideration the margin in
its entirety. As readers of the Preface to the New Testament (very few,
I fear, to judge by current criticisms) will possibly remember,
alternative readings and renderings were prohibited in the case of the
Authorised Version, but, as we know, the prohibition was completely
disregarded, some thirty-five notes referring to readings, and probably
more than five hundred to alternative renderings. In the fundamental
rules of Convocation for the Revision just the opposite course was
prescribed, and, as we know, freely acted on.
These alternative readings and renderings must be carefully considered,
as in the case of renderings much light is often thrown on the true
interpretation of the passage, especially in the more difficult portions
of the New Testament. Their relation however to the actually accepted
Version must not be exaggerated, either in reference to readings or
renderings. I will make plain what I mean by an example. Dr. Westcott
specifies a reading of importance in John i. 18 where he states that the
reading in the margin ("God only begotten") did in point of fact express
the opinion of the majority of the Company, but did not appear in the
text of the Version because it failed to secure the two-thirds majority
of those present at the final revision. This, perhaps, makes a little
too much of an acceptance at a somewhat early period of the labours of
the Company. So far as I remember the case, the somewhat startling
alteration was accepted at the first revision (when the decision was to
be by simple majorities), but a margin was granted, which of course
continued up to the second revision. At that revision the then text and
the then margin changed places. Dr. Hort, I am well aware, published an
important pamphlet on the subject, but I have no remembrance that the
first decision on the reading was alluded to, either at the second
revision or afterwards, in any exceptional manner. It did but share the
fate of numberless alterations at the first revision that were not
finally confirmed.
The American Revisers, it will be observed, agree as to the reading in
question with their English brethren; and the same too is the judgement
of Professor Nestle in his carefully edited Greek Testament to which I
have already referred.
I have dwelt upon this particular case, because though I am especially
desirous to encourage a far greater attention to the margin than it has
hitherto received, I am equally desirous that the margin should not be
elevated above its real position. That position is one of subordination
to the version actually adopted, whether when maintaining the older form
or changing it. It expresses the judgement of a legal, if not also of a
numerical, minority, and, in the case of difficult passages (as in Rom.
ix. 4), the judgement of groups which the Company, as a whole, deemed
worthy of being recorded. But, not only should the margin thus be
considered, but the readings and renderings preferred by the American
Committee, which will often be found suggestive and helpful. These, as
we know, are now incorporated in the American Standard Edition of the
Revised Bible; and the result, I fear, will be that the hitherto familiar
Appendix will disappear from the smaller English editions of the Revised
Version of the Old and New Testament. It is perhaps inevitable, but it
will be a real loss. All I can hope is that in some specified English
editions of the Old and New Testament each Appendix will regularly be
maintained, and that this token of the happy union of England and America
in the blessed work of revising their common version of God's holy Word
will thus be preserved to the end.
But we must now pass onward to considerations very closely affecting the
renderings of the Revised Version of the Greek Testament.
I have already said that very recently a new and unexpected charge has
been brought against the Revisers of the Authorised Version. And the
charge is no less than this, that the Revisers were ignorant in several
important particulars of the language from which the version was
originally made that they were appointed to revise.
Now in meeting a charge of this nature, in which we may certainly notice
that want of considerate intelligence which marks much of the criticism
that has been directed against our revision, it seems always best when
dealing with a competent scholar who does not give in detail examples on
which the criticism rests, to try and understand his point of view and
the general reasons for his unfavourable pronouncement. And in this case
I do not think it difficult to perceive that the imputation of ignorance
on the part of the Revisers has arisen from an exaggerated estimate of
the additions to our knowledge of New Testament Greek which have
accumulated during the twenty years that have passed away since the
Revision was completed. If this be a correct, as it is certainly a
charitable, estimate of the circumstances under which ignorance has been
imputed to us in respect of several matters relating to the Greek on
which we were engaged, let us now leave our critics, and deal with these
reasonable questions. First, what was the general knowledge, on the part
of the Revisers, of the character and peculiarities of New Testament
Greek? Secondly, what is the amount of the knowledge relative to New
Testament Greek that has been acquired since the publication of the
revision? and thirdly, to what extent does this recently acquired
knowledge affect the correctness and fidelity of the renderings that have
been adopted by the Revisers? If these three questions are plainly
answered we shall have dealt fully and fairly with the doubts that have
been expressed or implied as to the correctness of the revision.
First, then, as to the general knowledge which the revisers had of the
character and peculiarities of the Greek of the New Testament.
This question could not perhaps be more fairly and correctly dealt with
than by Bishop Westcott in the opening words of his chapter on Exactness
in Grammatical Detail, in the valuable work to which I have already
referred. What he states probably expresses very exactly the general
view taken by the great majority, if not by all, of the Revisers in
regard of the Greek of the New Testament. What the Bishop says of the
language is this: "that it is marked by unique characteristics. It is
separated very clearly, both in general vocabulary and in construction,
from the language of the LXX, the Greek Version of the Old Testament,
which was its preparation, and from the Greek of the Fathers which was
its development {106}."
If we accept this as a correct statement of the general knowledge of the
Revisers as to the language of the Greek Testament, we naturally ask
further, on what did they rely for the correct interpretation of it. The
answer can readily be given, and it is this: Besides their general
knowledge of Greek which, in the case of the large majority, was very
great, their knowledge of New Testament Greek was distinctly influenced
by the grammatical views of Professor Winer, of whose valuable grammar of
the Greek Testament one of our Company, as I have mentioned in my first
Address, had been a well-known and successful translator. Though his
name was not very frequently brought up in our discussions, the influence
his grammar exerted among us, directly and indirectly, was certainly
great; but it went no further than grammatical details. His obvious
gravitation to the idea of New Testament Greek forming a sort of separate
department of its own probably never was shared, to any perceptible
extent, by any one of us. We did not enter very far into these matters.
We knew by every day's working experience that New Testament Greek
differed to some extent from the Greek to which we had been accustomed,
and from the Septuagint Greek to which from time to time we referred.
But further than this we did not go, nor care to go. We had quite enough
on our hands. We had a very difficult task to perform, we had to revise
under prescribed conditions a version which needed revision almost in
every verse, and we had no time to enter into questions that did not then
appear to bear directly on our engrossing and responsible work.
But now it must be distinctly admitted that recent investigation and, to
a certain extent, recent discoveries have cast so much new light on New
Testament Greek that it becomes a positive duty to take into
consideration what has been disclosed to us by the labours of the last
fifteen years as to New Testament Greek, and then fairly to face the
question whether the particular labours of the Revisers have been
seriously affected by it. Let us bear in mind, however, that it may be
quite possible that a largely increased knowledge of the position which
what used to be called Biblical Greek now occupies may be clearly
recognized, and yet only comparatively few changes necessitated by it in
syntactic details and renderings. But let us not anticipate. What we
have now to do is to ascertain the nature and amount of the disclosures
and new knowledge to which I have alluded.
This may be briefly stated as emanating from a very large amount of
recent literature on post-classical Greek, and from a careful and
scientific investigation of the transition from the earlier
post-classical to the later, and thence to the modern Greek of the
present time. Such an investigation, illustrated as it has been by the
voluminous collection of the Inscriptions, and the already large and
growing collection of the Papyri, has thrown indirectly considerable
light on New Testament Greek, and has also called out three works, each
of a very important character, and posterior to the completion of the
Revision, which deal directly with the Greek of the New Testament. These
three works I will now specify.
The first, which is still in progress, and has not, I think, yet received
a translator, is the singularly accurate, and in parts corrective,
edition of Winer's "Grammar" by Prof. Schmiedel. The portion on the
article is generally recognized as of great value and importance.
The second work is the now well-translated "Bible Studies" of Dr.
Deissmann of Heidelberg {109}. This remarkable work, of which the full
title is "Contributions, chiefly from Papyri and Inscriptions, to the
History of the Language, the Literature, and the Religion of Hellenistic
Judaism, and Primitive Christianity," contains not only a clear estimate
of the nature of New Testament Greek, but also a large and instructive
vocabulary of about 160 words and expressions in the New Testament, most
of which receive in varying degrees illustration from the Papyri, and
other approximately contemporary sources. It must be noted, however,
that the writer himself specifies that his investigations "have been, in
part, arranged on a plan which is polemical {110a}." This avowal must,
to some extent, affect our full acceptance of all the results arrived at
in this striking and laborious work.
The third work is a "Grammar of New Testament Greek" by the well-known
and distinguished scholar, Dr. Blass, and is deserving of the fullest
attention from every earnest student of the Greek Testament. It has been
excellently translated by Mr. St. John Thackeray, of the Education
Department {110b}. It is really hardly possible to speak too highly of
this helpful and valuable work. Its value consists in this--that it has
been written, on the one hand, by an accomplished classical scholar, and,
on the other hand, by one who is thoroughly acquainted with the
investigations of the last fifteen years. As his Introduction clearly
shows, he fully accepts the estimate that is now generally entertained of
the Greek of the New Testament, viz. that it is no isolated production,
as regards language, that had no historic relation to the Greek of the
past or of the future. It was not, to any great extent, derived from the
Greek _translations_ of the Old Testament--often, as Dr. Blass says,
slavishly literal--nor from the literary language of the time, but was
the spoken Greek of the age to which it belonged, modified by the
position and education of the speaker, and also to some extent, though by
no means to any large extent, by the Semitic element which, from time to
time, discloses itself in the language of the inspired writers. This
last-written epithet, which I wittingly introduce, must not be lost sight
of by the Christian student.
Dr. Blass quite admits that the language of the Greek Testament may be
rightly treated in connexion with the discoveries in Egypt furnished by
the Papyri; but he has also properly maintained elsewhere {111} that the
books of the New Testament form a special group _to be primarily
explained by itself_. Greatly as we are indebted to Dr. Deissmann for
his illustrations, especially in regard of vocabulary, we must read with
serious caution, and watch all attempts to make Inscriptions or Papyri do
the work of an interpretation of the inner meaning of God's Holy Word
which belongs to another realm, and to the self-explanations which are
vouchsafed to us in the reverent study of the Book--not of Humanity (as
Deissmann speaks of the New Testament) {112} but of--Life.
I have now probably dealt sufficiently with the second of the three
questions which I have put forward for our consideration. I have stated
the general substance of the knowledge which has been permitted to come
to us since the revision was completed. I now pass onward to the third
and most difficult question equitably to answer, "To what extent does
this newly-acquired knowledge affect the correctness and fidelity of the
revision of the Authorised Version of the New Testament?" It is easy
enough to speak of "ignorance" on the part of the Revisers, especially
after what I have specified in the answer to the question on which we
have just been meditating; but the real and practical question is this,
"If the Revisers had all this knowledge when they were engaged on their
work, would it have materially affected their revision?"
To this more limited form of the question I feel no difficulty in
replying, that I am fully and firmly persuaded that it would _not_ have
materially affected the revision; and my grounds for returning this
answer depend on these two considerations: first, that the full knowledge
which some of us had of Winer's Grammar, and the general knowledge that
was possessed of it by the majority, certainly enabled us to realize that
the Greek on which we were engaged, while retaining very many elements of
what was classical, had in it also not only many signs of post-classical
Greek, but even of usages which we now know belong to later developments.
These later developments, all of which are, to some extent, to be
recognized in the Greek Testament, such as the disappearance of the
optative, the use of [Greek text] with the subjunctive in the place of
the infinitive, the displacement of [Greek text], the interchange of
[Greek text] and [Greek text], of [Greek text] and [Greek text], the use
of compound forms without any corresponding increase of meaning, the
extended usage of the aorist, the wider sphere of the accusative, and
many similar indications of later Greek--all these were so far known to
us as to exercise a cautionary influence on our revision, and to prevent
us overpressing the meaning of words and forms that had lost their
original definiteness.
My second reason for the answer I have given to the question is based on
the accumulating experience we were acquiring in our ten years of labour,
and our instinctive avoidance of renderings which in appearance might be
precise, but did in reality exaggerate the plain meaning intended by the
Greek that we were rendering. Sometimes, but only rarely, we fell into
this excusable form of over-rendering. Perhaps the concluding words of
Mark xiv. 65 will supply an example. At any rate, the view taken by
Blass {114} would seem to suggest a less literal form of translation.
When I leave the limited form of answer, and face the broad and general
question of the extent to which our recently-acquired knowledge affects
the correctness and fidelity of the revision, I can only give an answer
founded on an examination of numerous passages in which I have compared
the comments of Dr. Blass in his Grammar, and of Dr. Deissmann in his
"Bible Studies with the renderings of the Revisers." And the answer is
this, that the number of cases in which any change could reasonably be
required has been so small, so very small, that the charge of any real
ignorance, on the part of the Revisers, of the Greek on which they were
engaged, must be dismissed as utterly and entirely exaggerated. We have
now acquired an increased knowledge of the character of the Greek of the
New Testament, and of the place it holds in the historical transition of
the language from the earlier post-classical to the later developments of
the language, but this knowledge, interesting and instructive as it may
be, leaves the principles of correctly translating it practically intact.
In this latter process we must deal with the language of the Greek
Testament as we would deal with the language of any other Greek book, and
make the book, as far as we have the means of doing so, its own
interpreter.
Having thus shown in broad and general terms, as far as I have been able
to do so, that we may still, notwithstanding the twenty years that have
passed away, regard the Revised Version of the Greek Testament as a
faithfully executed revision, and its renderings such as may be accepted
with full Christian confidence, I now turn to the easier, but not less
necessary, duty of bringing before you some considerations why this
Version and, with it, the Revised Version of the Old Testament, should be
regularly used in the public services of our Mother Church.
ADDRESS V.
PUBLIC USE OF THE VERSION.
We have now traced the external, and to some extent the internal history
of Revision from the time, some fifty years ago, when it began to occupy
the thoughts of scholars and divines, down to the present day.
We have seen the steady advance in Church opinion as to its necessity;
its earliest manifestations, and the silent progress from what was
tentative and provisional to authoritative recognition, and to carefully
formulated procedures under the high and venerable sanction of the two
Houses of the Convocation of Canterbury. We have further seen how the
movement extended to America, and how some of the best scholars and
divines of that Christian country co-operated with those of our own
country in the arduous and responsible work of revising their common
heritage, the Version of God's most Holy Word, as set forth by authority
290 years ago. We have noted too, that in this work not less than one
hundred scholars and divines were engaged--for fourteen years in the case
of the Old Testament, and for ten years in the case of the New
Testament--and that this long period of labour and study was marked by
regularly appointed and faithfully kept times of meeting, and by the
interchange with the Revisers on the other side of the Atlantic of
successive portions of the work, until the whole was completed.
And this Revision, as we have seen, has included a full consideration of
the text of the original languages as well as of the renderings. In the
Old Testament, adherence to the Massorite Text has left only a very
limited number of passages in which consideration of the ancient Version
was deemed to be necessary; but, in the New Testament, as we well know,
questions of textual criticism occupied a large portion of the time and
attention of the Revisers, both here and in America. In regard of the
renderings, we have seen the care and thoroughness with which the
Revision was carried out, the marginal notes in both Testaments showing
convincingly, especially on the more difficult passages, how every
rendering that could be regarded as in any degree probable received its
full share of consideration. Finally, it must not be forgotten that, in
the case of the New Testament, the serious question whether the research
in New Testament Greek since the Revision was completed has, to any
appreciable extent, affected the suggestive light and truth of really
innumerable corrections and changes--this too has been faced, and the
charge fairly met, that just conclusions drawn from the true nature of
the Greek, gravely affecting interpretation, have been ignored by the
Revisers.
So much of the latter part of the last Address has been taken up with
this necessary duty of showing that the changes in renderings cannot be
invalidated by _a priori_ considerations founded on the alleged
insufficient knowledge, on the part of the Revisers, of the nature of the
Greek they were translating, that I have not cited examples of the
light-giving and often serious nature of the changes made in the
Authorised Version. This I regretted at the time; but a little
consideration showed me that it was much better for the cause in which I
am engaged that I should refer you for illustrations of the nature and
value of the renderings in the Revised Version of the New Testament to a
singularly fruitful and helpful volume, published only four years ago,
and so subsequently to the researches in New Testament Greek of which I
have spoken. This volume was written by a member of our Company--now,
alas, no longer with us--whose knowledge of the Greek language, whether
of earlier or of later date, no one could possibly doubt. I allude to
the "Lessons of the Revised Version of the New Testament," by Dr.
Westcott, a volume that has not yet received the full attention which its
remarkable merits abundantly claim, for it.
Of this volume I shall speak more fully later on in this Address, my
object now being to set forth the desirableness, I might even say the
duty, of using the Revised Version in the Public Services of the Church.
After the summary I have just given of the external history of this great
movement, does not the question come home to us, Why has all this been
done? For what have the hundred labourers in the great work freely given
their time and their energies during the four and twenty years (speaking
collectively) that were spent on the work? For what did the venerable
Convocation of our Province give the weight of its sanction and authority
when it drew up the fundamental rules in accordance with which all has
been done? Can there be any other answer than this? All has been done
to bring the truth of God's most Holy Word more faithfully and more
freshly home to the hearts and consciences of our English-speaking
people. And if this be so, how are ministers of this Holy Word to answer
the further question, When we are met together in the House of God to
hear His word and His message of salvation to mankind, how hear we it?
In the traditional form in which it has been heard for wellnigh three
hundred years, or in a form on which, to ensure faithfulness and
accuracy, such labour has been bestowed as that which we are now
considering? It seems impossible to hesitate as to our answer. And yet
numbers do hesitate; and partly from indifference, partly from a vague
fear of disquieting a congregation, partly, and probably chiefly, from a
sense of difficulty as to the rightful mode of introducing the change,
the old Version is still read, albeit with an uneasy feeling on the part
of the public reader; the uneasy feeling being this, that errors in
regard of Holy Scripture ought not to remain uncorrected nor obscurities
left to cloud the meaning of God's Word when there is a current Version
from which errors are removed, and in which obscurities are dissipated.
Why should not such a Version be read in the ears of our people?
This is the question which I am confident many a one of you, my dear
friends, when you have been reading in your church--say the
Epistles--have often felt very distinctly come home to you. Why should
such a Version not be read in the ears of our people? Has it been
forbidden? No, thank God; full liberty, on the contrary, has been left
to us by the living voice of the synod of this Province that it may be
read, subject to one reasonable limitation. Was it not the unanimous
judgement of the Upper House of the Convocation of our Province,
confirmed by the voice of the Lower House {122}--"That the use of the
Revised Version of the Bible at the lectern in the public services of the
Church, where this is desired by clergy and people, is not open to any
well-founded objection, and will tend to promote a more intelligent
knowledge of Holy Scripture"? And further, was not this adopted by the
Lay House of our Province, even when a few doubting voices were heard
{123}, and an interpretation given to the word "use," in the form of a
rider, which, I can confidently say, never entered into the minds or
thoughts of the members of the Upper House? Indeed, though I do not wish
to criticise the decision of the House of Laymen, their appended words of
interpretation fall to the ground. If "use" is to mean "occasional
employment of Lessons from the Revised Version, where, in the interest of
more accurate translation, it is desirable," can any Lessons be found
where the interest of more accurate translation is not patently
concerned? If this be so, what meaning can we assign to "occasional
employment"?
We see then plainly, if we are to be guided by the judgement of the
venerable body to whom the authoritative inception of Revision is alone
to be assigned, that the way to its use in the Public Services of the
Church is open to us all--_where such use is desired by clergy and
people_. Now let us take these words seriously into our consideration.
They clearly mean, however good the Version may be, that there is to be
no sudden and precipitate use of the Revised Version in the appointed
Lessons for the day on the part of the minister of any of our parishes.
If introduced, its introduction must not be simply when it is desired by
the clergyman, but when it is also desired by his people. So great a
change as the displacement of the old and familiar Authorised
Version--for it amounts to this--in the public reading of Holy Scripture
in the Services of the Church, in favour of an altered form of the old
Version (though confessedly so altered that the general hearer would
hardly ever recognize the displacement)--so great a change ought not to
be made without the knowledge, and further, the desire of the
congregation.
But how is the desire for the change to be ascertained? So far as I can
see, there can be only one real and rightful way of bringing about the
desire and the manifestation of it, and that is by first of all showing
simply and plainly how, especially in the New Testament, the alterations
give life, colouring and reality to the narratives of Evangelists, force
and lucidity to the reasonings of Apostles, and, what is of still more
vital importance, deeper insight into our relations to our saving Lord,
clearer knowledge of His blessed life and work here on earth, and
quickened perceptions of our present and our future, and, to a very real
extent, of the holy mysteries of the life of the world to come. When
changes of text and of renderings are shown, and they can be shown, to
bear with them these fuller revelations of God's Holy Word, there will be
no lack of desire, and of the manifestation of it, in any congregation,
for the public use of a Version through which such disclosures as I have
specified can be brought home to the truth-seeking believer.
My fixed opinion therefore is this, that though, after a long and careful
consideration of the subject, I do sincerely desire that the Revised
Version should be introduced into the churches of this diocese, I do also
sincerely desire that it should not be introduced without a due
preparation of the congregation for the change, and some manifestation of
their desire for the change. There will probably be a few churches in
our diocese in which the Revised Version is used already, and in regard
of them nothing more will be necessary than, from time to time, in
occasional addresses, to allude to any important changes that may have
appeared in the Lessons and recent readings of Holy Scripture, and thus
to keep alive the thoughtful study of that which will be more and more
felt to be, in the truest sense of the words, the Book of Life. But, in
the great majority of our churches--though in many cases there may have
been passing desires to read and to hear God's Word in its most truthful
form--no forward steps will have been taken. It is in reference then to
this great majority of cases that I have broken my long silence, and,
before my ministry closes, have resolved to bring before you the whole
history of the greatest spiritual movement that has taken place since the
Reformation; and also to indicate the untold blessings the Revision will
bear to those who avail themselves of it in all reverent earnestness and
devotion.
Thus far I hope I have made it plain that any forward steps that may be
taken can only hopefully be taken when, both in the case of pastor and
people, due preparation shall have been made for what, in the sequel,
will be found to be an enduring spiritual change in the relation of the
soul of the devout hearer or reader to the Book of Life. He will learn
not only faithfully to read the inspired Word, but inwardly to love it.
But what shall we regard as due preparation in the case of pastor and
people? This question, I can well believe, has already risen in the
hearts of many who are now hearing these words, and to the best answer to
it that I am able to give you I will gladly devote the remainder of this
present Address. Let us first consider how any one of you really and
truly desirous to prepare his congregation for the hearing of God's Word
in the form known as the Revised Version--how such a one should prepare
himself for the responsible duty. Prayer for himself and his
congregation in this great spiritual matter should ever be his first
preparation. After this his next care should be to provide himself with
such books as will be indispensable for faithful preparation. First and
foremost, let him provide himself with a copy of what is called the
Parallel Bible, the Authorised Version being on the left-hand side of the
page, and the Revised Version on the right. Next let it be his duty to
read closely and carefully the Preface to the Old Testament and the
Preface to the New Testament. Had this been done years ago, how much of
unfair criticism should we all have been spared? The next step will be
to obtain some competent guide-book to explain the meaning of the
different changes of rendering, the alterations due to readings having
been separately noted. The guide-book, whether in the case of the Old or
of the New Testament, should, in my judgement, be a volume written by a
Reviser, as he would have a knowledge, far beyond what could be obtained
by an outsider, of the reasons for many of the departures from the
Authorised Version.
In regard of the Old Testament I have said in my last Address that I do
not myself know of any guide-book, written by a Reviser, save the
interesting volume by Dr. Talbot Chambers, to which I have been indebted
for much that, being a member of another Company, I could not have
brought forward without his assistance. In regard of the New Testament,
however, it is otherwise. There is a useful volume by my old friend and
former colleague the late Prebendary Humphry; but the volume which I most
earnestly desire to name is the volume already mentioned, and entitled
"Some Lessons of the Revised Version of the New Testament," by the late
Bishop of Durham. This book is simply indispensable for any one desirous
of preparing himself for the duty of introducing the Revised Version of
the New Testament into the Public Services of his parish. It is one of
those rare and remarkable books that not only give the needed
explanation, but also cast a light on the whole spiritual results of the
change, and constantly awaken in the reader some portion of the
enthusiasm with which the Bishop records changes that many an earnest and
devout reader might think belonged only to the details of grammatical
accuracy. I thus cannot forbear quoting a few lines in which the Bishop,
after alluding to the change in Matt. xxviii. 19, _into_ (not _in_) _the
name of the Father and of the Holy Ghost_, and the change in Rom. vi. 23,
_eternal life in_ (not _through_) _Christ Jesus our Lord_, thus speaks
from his inmost soul: "Am I wrong in saying that he who has mastered the
meaning of those two prepositions now truly rendered--'_into_ the name,'
'_in_ Christ'--has found the central truth of Christianity? Certainly I
would gladly have given the ten years of my life spent on the Revision to
bring only these two phrases of the New Testament to the heart of
Englishmen." Is it too much to say that a volume written by a guide such
as this is simply indispensable for any one who prepares himself for
introducing to his people--the government of whose souls has been
committed to him--the Revised Version of the New Testament of our Lord
and Master Jesus Christ.
With the help that I have specified any one of you, my dear friends,
might adequately prepare himself for the duty and responsibility of
taking the next step, the preparation of his congregation for hearing the
Word of God in the form that most nearly approaches in our own language
what prophets, evangelists, and apostles have written for our learning
under the inspiration of God. This preparation may be carried on in many
forms, by pastoral visitations, through our Bible classes, through the
efforts of our mission preachers in the holy seasons, but obviously most
hopefully and persuasively by the living voice of the faithful pastor in
his public ministrations in the pulpit of his church. Parishes differ so
much in spiritual culture that probably no method of preparation could be
specified that would be equally applicable to all. Still in the case of
our country parishes I am persuaded our preparation must come from the
pulpit and in a manner carefully thought out and prearranged. Let me
give some indication of a mode of bringing the subject forward in a
country parish that would call out the desire for the regular use of the
Revised Version in the reading of the Lessons for the day.
Let us suppose a month set apart for the preparation. On the first
Sunday let an account be given of the circumstances, and especially the
authority under which the Revision came into existence. On the second
Sunday let illustrations be given of the nature of the Revision from
those parts in Bishop Westcott's "Lessons of the Revised Version of the
New Testament" which made the deepest impression during the study of that
suggestive and spiritual volume. On the third Sunday let comments be
made on the most striking of the changes in the two appointed Lessons for
the day from the Old Testament. Here the preacher may find some
difficulty, as want of knowledge of Hebrew or of the right interpretation
of the passage in which the alteration is made might prevent his clearly
stating the reasons for it. In such cases a good modern Commentary on
the Old Testament would probably supply the needed assistance. The most
available Commentary I know of for the purpose is the one published by
Messrs. Cassells, and now sold at the low price--for both Testaments--of
thirty-five shillings. On the fourth Sunday, the preacher's subject
should be the most striking of the changes in the two appointed Lessons
from the New Testament. For this there would be abundant help supplied
by the volume of Bishop Westcott, and, if needed, by the Commentary on
the New Testament to which I have alluded.
Now I sincerely believe that if this very simple and feasible plan were
carried out in any parish, two results would certainly follow: first,
that the Revised Version would be desired and welcomed; secondly, that an
interest in God's Holy Word would be called out in the parish and its
Bible classes that would make a lasting impression on the whole spiritual
life of the place. We have many faults, but we are a Bible-loving
nation, and we have shown it in many crises of our history; and thus, I
am persuaded, in a change such as I have suggested, the old love would be
called out afresh, and would display itself in a manner we might never
have expected.
I feel now that I have said all that it may be well for me to have laid
before you. I have used no tone of authority; I have not urged in any
way the introduction of the Revised Version, or that the plan of
introducing it should be adopted by any one among you. I have contented
myself with having shown that it is feasible; and I have definitely
stated my opinion that, if it were to be adopted, it is in a high degree
probable that a fresh interest in the Holy Scriptures would be awakened,
and the love of God's Holy Word again found to be a living reality.
Perhaps the present time may be of greater moment in regard of the study
of Holy Scripture, and especially of the language of the Greek Testament,
than we may now be able distinctly to foresee. I mentioned in my last
Address the large amount of research, during the last fifteen years, in
reference to the Greek of the New Testament and the position which the
sacred volume, considered simply historically and as a collection of
writings in the Greek language of the first century after Christ, really
does hold in the general history of a language which, in its latest form,
is widely spoken to this very day. I mentioned also what seemed to be
the most reasonable opinion, viz. that the Greek of the New Testament was
the spoken Greek of the time, neither literary Greek nor the Greek of the
lower class, but Greek such as men would use at that time when they had
to place in the definiteness of writing the language which passed from
their lips in their converse with their fellow-men. Now, that advantage
will be taken of this, and that it will be used to show that the
spiritual deductions that we draw from the written words cannot be fully
relied on, because old distinctions have been obscured or obliterated, is
what I fear, in days such as these, will often be used against the
faithful reading, marking, and learning of the Written Word. But we
shall hear them, I hope, with the two true conclusive answers ever
present in the soul, the answer of plain human reasoning, and the deeper
answer which revelation brings seriously home to us. In regard of the
first answer, does not plain common sense justify us in maintaining that
the writers meant what they _wrote_, and that when they used certain
Greek words in the mighty message they were delivering to their
fellow-men and to all who should hereafter receive it, they did mean that
those words were to be understood in the plain and simple meaning that
every plain reader would assign to them. They were not speaking; they
were writing; and they were writing what they knew was to be for all
time. Thus to take an example from the passages above referred to of
which Bishop Westcott makes such impressive use, who can doubt, with any
fair show of reason--however frequent may be the interchange of the
particular prepositions in the first century--that, in those passages,
when St. Matthew wrote [Greek text] he did mean _into_; and that when St.
Paul used [Greek text], he did mean _in_, in the simplest sense of the
word?
But to the devout Christian we have a far deeper answer than the answer
we have just considered.
In the first place, does not the manifold wisdom of God reveal itself to
our poor human thoughts in His choice of a widespread spoken language,
just by its very diffusion readily lending itself to the reception of new
words and new thoughts as the medium by which the Gospel message was
communicated to the children of men? Just as the particular period of
Christ's manifestation has ever been reverently regarded as a revelation
of the manifold nature of the eternal wisdom, so may we not see the same
in the choice of a language, at a particular period of its development,
as the bearer of the message of salvation to mankind? Surely this is a
manifestation of the Divine wisdom which must ever be seen and felt
whenever the outward character of the Greek of the New Testament is dwelt
upon by the truth-seeking spirit of the reverent believer.
And is there not a second thought, far too much lost sight of in our
investigation of the written word of the New Testament--that just as the
writers had their human powers quickened and strengthened by the Holy
Ghost for the full setting forth of the Gospel message by their spoken
words, so in regard of their written words would the same blessed
guidance be vouchsafed to them? And if so, is it not right for us, not
only to draw from their words all that by the plain laws of language they
can be understood to convey to us, but also to do what has been done in
the Revised Version, and to find the nearest equivalent our language
supplies for the words in the original?
These thoughts might be carried much further, but enough has been said to
justify the minute care that has been taken in the renderings of the
written word of the New Testament by the Revisers, and further, the
validity of the deductions that may be drawn from their use of one word
rather than another, especially in the case of words that might seem to
be practically synonymous. It may be quite true that, in the current
Greek of the time, many of the distinctions that were valid in an earlier
period of the language were no longer observed; and of this we find many
indications in the Greek Testament. But it must be remembered that we
also find in the Greek Testament a vastly preponderating portion of what
is grammatically correct according to the earlier standard, and often
clear indications that what was so written must have been definitely
meant by the writer. Is it not then our clearest duty, remembering
always that what we are translating is the Gospel message, to do what the
Revisers did, to render each passage in accordance with the recognized
meaning of the words, and in harmony with the plain tenor of the context?
I now close these words and these Addresses with the solemn prayer to
Almighty God that in this great matter, and in the use of that which the
living voice of our synod permits us to use, we may be guided by God the
Holy Ghost, through Jesus Christ, our ever-blessed and redeeming Lord and
God.
* * * * *
[As the use at the lectern of the Revised Version in the Public Service
of the Church may be thought likely to involve expense, I may mention
that the small pica edition of the Bible, at 10_s._ 6_d._ net, and of the
Apocrypha separately, at 7_s._ 6_d._, will be found sufficient in most
churches. The folio edition in buckram of the Bible with Apocrypha will,
I understand, be two guineas, net. Application however should be made to
the University Press of Oxford or of Cambridge, or to the Christian
Knowledge Society.]
OXFORD: HORACE HART
PRINTER TO THE UNIVERSITY
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Footnotes:
{6} The following Resolution was passed unanimously by the Upper House
of the Convocation of Canterbury on Feb. 10, 1899, after the presentation
of the Report of the Committee (well worthy of being read) by the Bishop
of Rochester. The Report is numbered 329, and, with other Reports of
Convocation, is sold by the National Society:--
"That in the opinion of this House the use of the Revised Version at
the lectern in the public service of the Church, where this is
desired by clergy and people, is not open to any well-founded
objection, and will tend to promote a more intelligent knowledge of
Holy Scripture."
{10a} Among others may be named the _Edinburgh Review_ for 1855 on
Paragraph Bibles, in which it was said that it was now high time for
another revision (p. 429); the _Christian Remembrancer_ for 1856 on the
Revision of the Authorised Version (an interesting article); the
_Quarterly Review_ for 1863, intimating that as yet we were not ripe for
any authorised text or translation; the _Edinburgh Review_ for 1865; and
the _Contemporary Review_ for 1868, a careful and elaborate article,
contending that the work must be done by a Commission.
{10b} In February, 1856, when Canon Selwyn gave notice of proposing a
petition on the subject to the Upper House. The proposal in a somewhat
different form a year afterwards was disposed of by a characteristic
amendment of Archdeacon Denison.
{10c} On July 22, 1856, Mr. Heywood, one of the members, I think, for
North Lancashire, in rather an interesting speech, moved for an Address
to the Crown to issue a Royal Commission on the subject. The motion was
rejected, Sir George Grey expressing his conviction that the feeling of
the country was not in accordance with the motion.
{12} Preface to the Revision of the Authorised Version of the Gospel
according to St. John by Five Clergymen, p. xii. As I remark afterwards,
this preface proved to be very attractive, and by its moderation greatly
helped the cause. The book has long since gone out of print, but if any
reader of this note should come across it, this preface will be found
well worth reading, as it will show what was in the minds of many beside
the Five Clergymen five and forty years ago.
{13} See Schaff, _Companion to Greek Testament and English version_, p.
367, note (New York, 1883).
{21} The _Expositor_ for October, 1892, pp. 241-255. The article was
answered by me in the same periodical two months later.
{22} The account of the discussion in the Convocation of York (Feb. 23,
1870) will be found in _The Guardian_ of March 2, 1870. In the comments
of this paper on the action or rather inaction of the Northern
Convocation a very unfavourable opinion was expressed, in reference to
the manner in which the Southern Convocation had been treated. But these
things have long since been forgotten.
{35} It may be interesting to give this list, as it slightly affects
matter that will be alluded to afterwards in reference to the Greek text.
The attendances were as follows: The Chairman, 405; Dr. Scrivener, 399;
Prebendary Humphry, 385; Principal Newth, 373; Prof. Hort, 362; Dean
Bickersteth (Prolocutor), 352; Dean Scott, 337; Prof. Westcott, 304; Dean
Vaughan, 302; Dean Blakesley, 297; Bishop Lightfoot, 290; Archdeacon Lee,
283; Dr. Moulton, 275; Archdeacon Palmer, 255; Dean Stanley, 253; Dr.
Vance Smith, 245; Principal Brown, 209; Principal Angus, 199; Prof.
Milligan, 182; Prof. Kennedy, 165; Dr. Eadie, 135; Bishop Moberly, 121;
Bishop Wordsworth (St. Andrews), 109; Dr. Roberts, 94; Archbishop Trench,
63; Dean Merivale (resigned early), 19; Dean Alford (died soon after
commencement), 16; Bishop Wilberforce, 1.
{36} This letter will be found in a very valuable _Historical Account of
the Work of the American Committee of Revision_ (New York, 1885), p. 30.
This _Historical Account_ was prepared by a special Committee appointed
for the purpose in May, 1884, and was based on documents and papers
arranged with great care by Dr. Philip Schaff, the President of the
American Committee, and printed privately. These two volumes, the
_Historical Account_ and the _Documentary History_, contain the fullest
details of the whole transactions between the American Committee and the
English Companies and also the University Presses.
{41} Talbot W. Chambers, _Companion to the Revised Old Testament_ (Funk
and Wagnalls, New York and London, 1885), Preface, p. ix.
{42a} A full account of the negotiation and copies of the letters which
passed between the American Revisers and our own Revisers will be found
in Part 2, p. 81 sqq. of the _Documentary History_, above referred to in
the note at p. 36.
{42b} A full account of this agreement and copies of the correspondence
with the Universities of Oxford and Cambridge will be found in Part 3, p.
91 sqq. of the _Documentary History_.
{44} Since the above was written, information reaches me that an
_American Standard Revision of the Bible_ either just has been, or
shortly will be, published, which though not simply an incorporation of
the recorded American preferences, as long specified in our copies of the
Revision, is a publication resting on authority, and likely to put a stop
to what is unauthorised. As the reader may like to know a little about
this _American Standard Revision of the Bible_, I will, at the risk of a
long note, mention what I have ascertained up to the present time. The
survivors of the Old Testament Company (Dr. Osgood and others) with the
three surviving members of the New Testament Company (Dr. Dwight, Dr.
Riddle, and Dr. Thayer--very powerful helpers) have co-operated in
bringing out a new edition of the Revision as it has been hitherto
current in America. It will contain about _twice as many_ deviations
from the English Revised Version as appear in the original Appendices;
but, in regard of them, the survivors give this important assurance, that
"the survivors have not felt at liberty to make new changes of moment
which were not favourably passed upon (_sic_) by their associates, at one
stage or another of the original preparation of the work." They specify
that the original Appendix was prepared in haste and did not, in a
satisfactory manner, express the real views of the Committee. They claim
to have drawn up a body of improved marginal references, to have wholly
removed archaisms, to have supplied running headings, to have modified
what they consider unwieldy paragraphs, to have lightened what they
regard as clumsy punctuation, and by typographical arrangements, such as
by leaving a line blank, to have indicated the main transitions of
thought in the Epistles and Apocalypse. These and other characteristics
will be found specified in the American _Sunday School Times_ for August
11, 1901, in an article apparently derived from those interested. Till
we see the book we must suspend our judgement.
{50} See an article by Rev. J. F. Thrupp in Smith's _Dictionary of the
Bible_, vol. ii. art. Old Testament.
{53} Since the above was written a critical edition of the four Peshitto
Gospels has been published by the Oxford University Press, based on the
labours of the late Philip Edward Pusey, and Rev. G. H. Gwilliam, of
Hertford College.
{55} The title of the pamphlet, which contains twelve letters from
distinguished German Professors, with translations, is _The Revision of
the Old Testament_ (New York, Scribner's Sons, 1886).
{59} The title of Dr. Salmon's interesting volume is _Some Thoughts on
the Textual Criticism of the New Testament_ (Murray, London, 1897).
{60a} Salmon, p. 157.
{60b} Ibid., p. 12.
{96} See below, pp. 98, 120.
{97} See the Preface to Dr. Rutherford's _Translation of the Epistle to
the Romans_, p. xi sq. (Lond. 1900).
{98a} Hodder & Stoughton (Lond. 1897).
{98b} Page 18.
{106} See page 32.
{109} _Bible Studies_, by Dr. G. Adolf Deissmann, Authorised Translation
(Clark, Edinburgh, 1901).
{110a} Page 175.
{110b} London, Macmillan, 1898.
{111} _Theologische Literaturzeitung_, xix (vol. for 1894), p. 338.
{112} _Bible Studies_, p. 84 Transl. See, however, the translator's
note, p. 173, where the use of the term is explained.
{114} _Grammar of New Testament Greek_, section 38. 5, p. 118 (Transl.).
{122} See _Chronicle of Convocation_ for February 10, 1899, p. 71 sqq.
{123} At the May Meeting of the present year.
***
|
{
"redpajama_set_name": "RedPajamaBook"
}
| 7,226
|
\section{The algebra $A$}
\noindent Throughout the paper $\mathbb F$ denotes a field.
All unadorned tensor products are meant to be over $\mathbb F$.
An algebra is meant to be associative and have a 1.
\medskip
\noindent
We now introduce our topic.
\begin{definition}
\label{def:t}
\rm
Let $d$ denote a positive integer.
Let $A=A(d,\mathbb F)$ denote the $\mathbb F$-algebra
defined by
generators $\lbrace e_i\rbrace_{i=0}^d$,
$\lbrace e^*_i\rbrace_{i=0}^d$ and the
following relations:
\begin{eqnarray}
&&e_ie_j = \delta_{i,j} e_i,
\qquad \qquad
e^*_ie^*_j = \delta_{i,j} e^*_i,
\qquad \qquad (0 \leq i,j\leq d),
\label{eq:rel1}
\\
&&1 = \sum_{i=0}^d e_i,
\qquad \qquad
1 = \sum_{i=0}^d e^*_i.
\label{eq:rel2}
\end{eqnarray}
Here $\delta_{i,j}$ denotes the Kronecker delta.
\end{definition}
\begin{definition}
\label{def:gens}
\rm
Referring to Definition \ref{def:t},
we call
$\lbrace e_i\rbrace_{i=0}^d$ and
$\lbrace e^*_i\rbrace_{i=0}^d$ the {\it idempotent
generators} for $A$.
We say that the
$\lbrace e^*_i\rbrace_{i=0}^d$ are {\it starred}
and the
$\lbrace e_i\rbrace_{i=0}^d$ are {\it nonstarred}.
\end{definition}
\noindent We now briefly explain how $A$ can be viewed as a
coproduct in the sense of
Bergman\cite{bergman1,bergman2}.
As we will see in Theorem
\ref{thm:tbasis}, the elements
$\lbrace e_i\rbrace_{i=0}^d$ are linearly
independent in $A$ and hence form a basis
for a subalgebra of $A$
denoted
$A_1$. Similarly
the elements
$\lbrace e^*_i\rbrace_{i=0}^d$
form a basis for a subalgebra of
$A$ denoted $A^*_1$.
By construction $A$ is generated by
$A_1, A^*_1$. The $\mathbb F$-algebras
$A_1$ and $A^*_1$
are each isomorphic to a direct sum
of $d+1$ copies of $\mathbb F$.
The elements
$\lbrace e_i\rbrace_{i=0}^d$
(resp.
$\lbrace e^*_i\rbrace_{i=0}^d$)
are the primitive idempotents of
$A_1$ (resp. $A^*_1$).
Since no relation in
(\ref{eq:rel1}),
(\ref{eq:rel2})
involves both $A_1$ and $A^*_1$,
the algebra $A$ is the coproduct of
$A_1$ and $A^*_1$ in the sense of
Bergman\cite[Section~1]{bergman1}.
As part of his comprehensive study of
coproducts, Bergman determined the
units and zero-divisors in $A$
\cite[Corollary 2.16]{bergman1}.
\medskip
\noindent
Our goal in this article is to describe four
bases
for $A$ that we find attractive.
We determine how these bases are related
to each other. We describe how the multiplication
in $A$ looks with respect to these bases.
Using our bases we obtain an infinite
nested sequence of
2-sided ideals for $A$.
Using our bases we obtain an infinite
exact sequence involving a certain
$\mathbb F$-linear map $\partial: A \to A$.
We show that the kernel $F$ of $\partial$
is a subalgebra of $A$ that is free
of rank $d$.
We show that $F$ is generated by the elements
$\lbrace e_i-e^*_i\rbrace_{i=1}^d$.
We show that each of the $\mathbb F$-linear maps
\begin{eqnarray*}
&&F \otimes A_1 \;\to\; A
\qquad \qquad \qquad
F \otimes A^*_1 \; \;\to \;\; A
\\
&&u \otimes v \; \; \mapsto \; \; uv
\qquad \qquad \qquad
u \otimes v \; \;\mapsto \; \; uv
\end{eqnarray*}
is an isomorphism of $\mathbb F$-vector spaces.
We will define our bases after a few comments.
\medskip
\noindent The following three lemmas are about symmetries of $A$;
their proofs are routine and left to the reader.
\begin{lemma}
\label{lem:aut}
There exists a unique $\mathbb F$-algebra automorphism
of $A$ that sends
\begin{eqnarray*}
e_i \mapsto e^*_i,
\qquad \qquad e^*_i \mapsto e_i
\end{eqnarray*}
for $0 \leq i \leq d$.
Denoting this automorphism by $*$
we have $x^{**}=x$ for all $x \in A$.
\end{lemma}
\noindent By an
{\it $\mathbb F$-algebra antiautomorphism} of $A$
we mean an isomorphism of $\mathbb F$-vector spaces
$\rho: A\to A$ such that $(xy)^\rho=y^\rho x^\rho$
for all $x,y \in A$.
\begin{lemma}
\label{lem:antiaut}
There exists a unique $\mathbb F$-algebra antiautomorphism
$\dagger $ of $A$ that fixes each idempotent generator.
We have
$x^{\dagger \dagger}=x$ for all $x \in A$.
\end{lemma}
\begin{lemma}
\label{lem:mapscom}
The maps $*$ and $\dagger$ commute.
\end{lemma}
\noindent Let $X$ denote a subset of $A$.
By the {\it relatives} of $X$ we mean
the subsets
$X$, $X^*$, $X^\dagger$,
$X^{*\dagger}$.
\section{Four bases for the vector space $A$}
In this section we display four bases for the
$\mathbb F$-vector space $A$.
\begin{definition}
\label{def:word}
\rm
A pair of idempotent generators for $A$ is called
{\it alternating}
whenever one of them is starred and the other is
nonstarred.
For an integer $n\geq 1$, by a {\it word of length $n$} in $A$
we mean a product $g_1g_2 \cdots g_n$ such that
$\lbrace g_i\rbrace_{i=1}^n$
are idempotent generators for $A$ and $ g_{i-1}, g_i$
are alternating for $2 \leq i \leq n$.
The word
$g_1g_2 \cdots g_n$
is said to {\it begin} with $g_1$
and {\it end} with $g_n$.
\end{definition}
\begin{example}
\rm For $d=2$ we display the words in $A$ that have length
3 and begin with $e_0$.
\begin{eqnarray*}
&&e_0e^*_0e_0, \qquad e_0e^*_0e_1, \qquad e_0e^*_0e_2,
\\
&&e_0e^*_1e_0, \qquad e_0e^*_1e_1, \qquad e_0e^*_1e_2,
\\
&&e_0e^*_2e_0, \qquad e_0e^*_2e_1, \qquad e_0e^*_2e_2.
\end{eqnarray*}
\end{example}
\begin{definition}
\rm
For an idempotent generator $e_i$ or $e^*_i$ we call
$i$ the {\it index} of the generator.
A word $g_1 g_2 \cdots g_n$ in $A$
is called {\it nonrepeating} (or {\it NR})
whenever $g_{j-1}, g_j$ do not have the same index for
$2 \leq j \leq n$.
\end{definition}
\begin{example} \rm
For $d=2$ we display the NR words in $A$ that have length 3
and begin with $e_0$.
\begin{eqnarray*}
e_0e^*_1e_0, \qquad
e_0e^*_1e_2, \qquad
e_0e^*_2e_0, \qquad
e_0e^*_2e_1.
\end{eqnarray*}
\end{example}
\begin{theorem}
\label{thm:tbasis}
Each of the following is a basis for the $\mathbb F$-vector space
$A$:
\begin{itemize}
\item[\rm (i)]
The set of NR words in $A$ that end with a nonstarred element.
\item[\rm (ii)]
The set of NR words in $A$ that end with a starred element.
\item[\rm (iii)]
The set of NR words in $A$ that begin with a nonstarred element.
\item[\rm (iv)]
The set of NR words in $A$ that begin with a starred element.
\end{itemize}
\end{theorem}
\noindent {\it Proof:}
(i)
Let $S$ denote the set of
NR words in $A$ that end with a nonstarred element.
We first show that $S$ spans $A$.
Let $A'$ denote the subspace of $A$ spanned
by $S$.
To obtain $A'=A$
it suffices to show that $A'$ is a left ideal of $A$
that contains $1$.
To show that $A'$ is a left ideal of $A$,
it suffices to show that
$e_iA' \subseteq A'$ and $e^*_iA' \subseteq A'$
for $0 \leq i \leq d$.
For a word $w=g_1g_2\cdots g_n$
in $S$ and $0 \leq i \leq d$
we show that each of $e_iw$, $e^*_iw$ is contained in $A'$.
Let $j$ denote the index of $g_1$.
Invoking
(\ref{eq:rel2})
we may assume without loss that $i \not=j$.
First assume $n$ is odd, so that
$g_1=e_j$.
Since $e_ie_j=0$ we have
$e_iw=0$, so $ e_iw\in A'$.
Also $e^*_iw=e^*_ig_1g_2\cdots g_n$
is a word in $S$, so $e^*_iw \in A'$.
Next assume $n$ is even, so that $g_1=e^*_j$.
Then $e_iw=e_ig_1g_2\cdots g_n$
is a word in $S$, so $e_iw \in A'$.
Since $e^*_ie^*_j=0$ we have
$e^*_iw=0$, so $e^*_iw\in A'$.
We have shown
$A'$ is a left
ideal of $A$.
The ideal $A'$ contains 1, since
$e_i \in S$ for $0 \leq i \leq d$ and
$1=\sum_{i=0}^d e_i$.
We have shown $A'$ is a left ideal of $A$ that contains $1$, so $A'=A$.
Therefore $S$ spans $A$.
Next we show that
the elements of $S$ are linearly independent.
Let $\cal S$ denote the set of sequences
$(r_1, r_2, \ldots, r_n)$
such that (i) $n$ is a positive integer; (ii)
each of $r_1, r_2, \ldots, r_n$ is contained in
the set $\lbrace 0,1,\ldots, d\rbrace$;
(iii) $r_{i-1}\not=r_i$ for $2 \leq i \leq n$.
Let $V$ denote the vector space over $\mathbb F$ consisting
of those formal linear combinations of $ \cal S$
that have finitely many nonzero coefficients.
The set $\cal S$ is a basis for $V$.
For $0 \leq i \leq d$ we define linear transformations
$E_i: V \to V$ and
$E^*_i: V \to V$. To this end
we give the actions of $E_i$ and $E^*_i$ on
$\cal S$. Pick an element $(r_1, r_2, \ldots, r_n) \in \cal S$.
The actions of $E_i$ and $E^*_i$ on
$(r_1, r_2, \ldots, r_n)$
are given in
the table below.
\begin{center}
\begin{tabular}{c|c|c}
Case & $E_i.(r_1, \ldots, r_n)$ & $E^*_i.(r_1, \ldots, r_n)$ \\
\hline
$r_1=i$, $\;n$ odd &
$(r_1, \ldots, r_n)$ & $(r_1, \ldots, r_n)
-
\sum_{\stackrel{0 \leq j \leq d}{j \neq i}}
(j, r_1, \ldots, r_n)$
\\
$r_1\not=i$, $\;n$ odd & $0$ & $(i,r_1, \ldots, r_n)$
\\
$r_1=i$, $\;n$ even & $(r_1, \ldots, r_n)-
\sum_{\stackrel{0 \leq j \leq d}{j \neq i}}
(j, r_1, \ldots, r_n)$ & $(r_1, \ldots, r_n)$
\\
$r_1\not=i$, $\;n$ even & $(i,r_1, \ldots, r_n)$ & $0$
\end{tabular}
\end{center}
\bigskip
\noindent Using the table,
\begin{eqnarray}
&&E_iE_j = \delta_{i,j} E_i,
\qquad \qquad
E^*_iE^*_j = \delta_{i,j} E^*_i,
\qquad \qquad (0 \leq i,j\leq d),
\label{eq:Erel1}
\\
&&1 = \sum_{i=0}^d E_i,
\qquad \qquad
1 = \sum_{i=0}^d E^*_i.
\label{eq:Erel2}
\end{eqnarray}
Comparing
(\ref{eq:Erel1}),
(\ref{eq:Erel2})
with
(\ref{eq:rel1}),
(\ref{eq:rel2}) we find that $V$ has an $A$-module
structure such that $e_i$ (resp. $e^*_i$)
acts on $V$ as $E_i$ (resp. $E^*_i$) for $0 \leq i \leq d$.
Define the element
$\Delta \in V$ by
$\Delta= \sum_{i=0}^d (i)$, and consider the
linear transformation
$\gamma: A \to V$ that sends
$x \mapsto x.\Delta$ for all $x \in A$.
For each word
$w=g_1g_2\cdots g_n$ in $S$
we find
$\gamma(w)= ({\overline {g_1}} ,
{\overline {g_2}}, \ldots,
{\overline {g_n}} )$ where
$\overline g$ denotes the index of $g$.
Thus the restriction of $\gamma$ to $S$ gives
a bijection $S \to {\cal S}$.
The elements of $\cal S$ are linearly
independent and $\gamma$ is linear,
so the elements of $S$ are
linearly
independent.
We have shown $S$ is a basis for $A$.
\\
\noindent (ii) Apply the automorphism
$*$ to the basis in (i) above.
\\
\noindent (iii), (iv) Apply the antiautomorphim
$\dagger$ to the bases in (i), (ii) above.
\hfill $\Box$ \\
\section{How the four bases for $A$ are related}
\noindent In this section we obtain some identities
that effectively give the transition matrix
between any two bases from Theorem
\ref{thm:tbasis}.
\medskip
\begin{notation}
\label{not:index}
\rm
Let $w=g_1g_2\cdots g_n$ denote a
word in $A$, with
$g_n$ nonstarred.
We represent $w$
by the sequence $(r_1, r_2, \ldots, r_n)$, where
$r_j$ denotes the index of $g_j$ for $1 \leq j \leq n$.
We represent $w^*$ by
$(r_1, r_2, \ldots, r_n)^*$.
\end{notation}
\begin{example}
\rm
We display some words in $A$ along
with their
notation.
\begin{center}
\begin{tabular}{c|c}
word & notation \\
\hline
$e_0e^*_2e_1$ & $(0,2,1)$
\\
$e^*_1e_0e^*_2e_1$ & $(1,0,2,1)$
\\
$e^*_0e_2e^*_1$ & $(0,2,1)^*$
\\
$e_1e^*_0e_2e^*_1$ & $(1,0,2,1)^*$
\end{tabular}
\end{center}
\end{example}
\noindent The next result effectively gives
the transition matrix between any two
bases from Theorem \ref{thm:tbasis}.
\begin{theorem}
\label{thm:trans}
With reference to Notation
\ref{not:index}, and
for each basis vector $(r_1, r_2, \ldots, r_n)$ from
Theorem
\ref{thm:tbasis}(i), the element
\begin{eqnarray*}
(r_1, r_2, \ldots, r_n)
\quad + \quad
(-1)^n(r_1, r_2, \ldots, r_n)^*
\end{eqnarray*}
is equal to
\begin{eqnarray*}
&&
\sum_{\stackrel{0 \leq j \leq d}{j \neq r_1}}
(j,r_1, r_2, \ldots, r_n)
\quad + \quad
\sum_{\ell=1}^{n-1} (-1)^{\ell}
\sum_{\stackrel{0 \leq j \leq d}{j \neq r_{\ell},\; j\neq r_{\ell+1}}}
(r_1, r_2, \ldots, r_{\ell}, j, r_{\ell+1}, \ldots, r_n)
\\
&& \qquad \quad
+\quad
(-1)^n\sum_{\stackrel{0 \leq j \leq d}{j \neq r_n}}
( r_1, r_2, \ldots, r_n,j),
\end{eqnarray*}
\noindent and also equal to
\begin{eqnarray*}
&&
\sum_{\stackrel{0 \leq j \leq d}{j \neq r_n}}
(r_1, r_2, \ldots, r_n,j)^*
\quad + \quad
\sum_{\ell=1}^{n-1} (-1)^{n-\ell}
\sum_{\stackrel{0 \leq j \leq d}{j \neq r_{\ell}, \;j\neq r_{\ell+1}}}
(r_1, r_2, \ldots, r_{\ell}, j, r_{\ell+1}, \ldots, r_n)^*
\\
&& \qquad \quad
+\quad (-1)^n
\sum_{\stackrel{0 \leq j \leq d}{j \neq r_1}}
(j, r_1, r_2, \ldots, r_n)^*.
\end{eqnarray*}
\end{theorem}
\noindent {\it Proof:}
To obtain the first assertion, define
\begin{eqnarray}
\phi_0 &=&
\sum_{\stackrel{0 \leq j \leq d}{j \neq r_1}}
(j,r_1, r_2, \ldots, r_n),
\label{eq:phi0}
\\
\phi_{\ell} &=&
\sum_{\stackrel{0 \leq j \leq d}{j \neq r_{\ell},\; j\neq r_{\ell+1}}}
(r_1, r_2, \ldots, r_{\ell}, j, r_{\ell+1}, \ldots, r_n)
\qquad \qquad
(1 \leq \ell \leq n-1),
\label{eq:phil}
\\
\phi_n
&=&
\sum_{\stackrel{0 \leq j \leq d}{j \neq r_n}}
( r_1, r_2, \ldots, r_n,j).
\label{eq:phin}
\end{eqnarray}
Evaluating
(\ref{eq:phi0})--(\ref{eq:phin})
using
(\ref{eq:rel2}) we find
\begin{eqnarray}
\phi_0 &=& (r_1, r_2, \ldots, r_n)-
(r_1,r_1, r_2, \ldots, r_n),
\label{eq:phi0ev}
\\
\phi_{\ell} &=& -(r_1, r_2, \ldots, r_{\ell}, r_{\ell},r_{\ell+1},\ldots, r_n)-
(r_1, r_2, \ldots, r_{\ell}, r_{\ell+1},r_{\ell+1},\ldots, r_n),
\label{eq:philev}
\\
\phi_n &=& (r_1, r_2, \ldots, r_n)^*-
(r_1, r_2, \ldots, r_n, r_n).
\label{eq:phinev}
\end{eqnarray}
Combining
(\ref{eq:phi0ev})--(\ref{eq:phinev}) we obtain
\begin{eqnarray*}
\phi_0 + \sum_{\ell=1}^{n-1} (-1)^{\ell} \phi_{\ell}
+ (-1)^n \phi_n &=&
(r_1, r_2, \ldots, r_n)+
(-1)^n(r_1, r_2, \ldots, r_n)^*,
\end{eqnarray*}
and the first assertion follows. The second assertion is similarly
obtained.
\hfill $\Box$ \\
\begin{example}
\rm
Assume $d=2$. Then for $n=1$ and $r_1=1$ the assertions
of Theorem
\ref{thm:trans} become
\begin{eqnarray*}
e_1 - e^*_1
&=& e^*_0e_1+e^*_2e_1
-e^*_1e_0 - e^*_1e_2
\\
&=& e_1e^*_0 + e_1e^*_2-e_0e^*_1-e_2e^*_1.
\end{eqnarray*}
For $n=2$ and $(r_1, r_2)=(1,0)$
the assertions
of Theorem
\ref{thm:trans} become
\begin{eqnarray*}
e^*_1e_0+e_1e^*_0 &=&
e_0e^*_1e_0+
e_2e^*_1e_0
-e_1 e^*_2 e_0
+e_1e^*_0e_1+
e_1e^*_0e_2
\\
&=&
e^*_1e_0e^*_1+
e^*_1e_0e^*_2
-e^*_1 e_2 e^*_0
+ e^*_0e_1e^*_0
+ e^*_2e_1e^*_0.
\end{eqnarray*}
\end{example}
\section{The product of basis elements}
Consider the basis for $A$ from
Theorem
\ref{thm:tbasis}(i).
We now take two elements from this basis,
and write the product
as a linear combination of elements from the basis.
\begin{theorem}
\label{thm:prod}
Let $(r_1, r_2, \ldots, r_n)$
and
$(r'_1, r'_2, \ldots, r'_m)$
denote basis vectors from Theorem
\ref{thm:tbasis}(i).
Then the product
\begin{eqnarray}
(r_1, r_2, \ldots, r_n)\cdot
(r'_1, r'_2, \ldots, r'_m)
\label{eq:prod}
\end{eqnarray}
is the following linear combination of basis
vectors from Theorem
\ref{thm:tbasis}(i).
\begin{itemize}
\item[\rm (i)]
Assume $m$ is odd and $r_n \neq r'_1$. Then
(\ref{eq:prod}) is zero.
\item[\rm (ii)]
Assume $m$ is odd and $r_n=r'_1$. Then
(\ref{eq:prod}) is equal to
\begin{eqnarray*}
(r_1, r_2, \ldots, r_n,
r'_2, \ldots, r'_m).
\end{eqnarray*}
\item[\rm (iii)]
Assume $m$ is even and $r_n \neq r'_1$. Then
(\ref{eq:prod}) is equal to
\begin{eqnarray*}
(r_1, r_2, \ldots, r_n,
r'_1,r'_2, \ldots, r'_m).
\end{eqnarray*}
\item[\rm (iv)]
Assume $m$ is even and $r_n = r'_1$. Then
(\ref{eq:prod}) is equal to
\begin{eqnarray*}
&&(-1)^{n+1}(r_1, r_2, \ldots, r_n,r'_2, \ldots, r'_m)\quad + \quad (-1)^n \sum_{\stackrel{0 \leq j \leq d}{j \neq r_1}}
(j, r_1, r_2, \ldots, r_n, r'_2, \ldots, r'_m)
\\
&&\quad +\quad \sum_{\ell=1}^{n-1} (-1)^{n-\ell}
\sum_{\stackrel{0 \leq j \leq d}{j \neq r_{\ell}, \;j\neq r_{\ell+1}}}
(r_1,r_2, \ldots, r_{\ell},j,r_{\ell+1},\ldots, r_n, r'_2, \ldots, r'_m).
\end{eqnarray*}
\end{itemize}
\end{theorem}
\noindent {\it Proof:}
(i)--(iii) Routine.
\\
\noindent (iv) In line
(\ref{eq:prod}), evaluate
$(r_1, r_2, \ldots, r_n)$
using the second identity in
Theorem
\ref{thm:trans}, and simplify the result.
\hfill $\Box$ \\
\noindent
Now consider
the basis for $A$
from Theorem
\ref{thm:tbasis}(ii),
and
the basis for $A$ from
Theorem
\ref{thm:tbasis}(i).
In the next result, we take an element from
the first basis and an element
from the second basis, and
write the product as a
linear
combination of elements from the
second basis.
\begin{theorem}
\label{thm:prod2}
In the notation of Theorem
\ref{thm:prod},
the product
\begin{eqnarray}
(r_1, r_2, \ldots, r_n)^*\cdot
(r'_1, r'_2, \ldots, r'_m)
\label{eq:prod2}
\end{eqnarray}
is the following linear combination of basis
vectors from Theorem
\ref{thm:tbasis}(i).
\begin{itemize}
\item[\rm (i)]
Assume $m$ is even and $r_n \neq r'_1$. Then
(\ref{eq:prod2}) is $0$.
\item[\rm (ii)]
Assume $m$ is even and $r_n=r'_1$. Then
(\ref{eq:prod2}) is equal to
\begin{eqnarray*}
(r_1, r_2, \ldots, r_n,
r'_2, \ldots, r'_m).
\end{eqnarray*}
\item[\rm (iii)]
Assume $m$ is odd and $r_n \neq r'_1$. Then
(\ref{eq:prod2}) is equal to
\begin{eqnarray*}
(r_1, r_2, \ldots, r_n,
r'_1,r'_2, \ldots, r'_m).
\end{eqnarray*}
\item[\rm (iv)]
Assume $m$ is odd and $r_n = r'_1$. Then
(\ref{eq:prod2}) is equal to
\begin{eqnarray*}
&&(-1)^{n+1}(r_1, r_2, \ldots, r_n,r'_2, \ldots, r'_m)\quad + \quad (-1)^n \sum_{\stackrel{0 \leq j \leq d}{j \neq r_1}}
(j, r_1, r_2, \ldots, r_n, r'_2, \ldots, r'_m)
\\
&&\quad +\quad \sum_{\ell=1}^{n-1} (-1)^{n-\ell}
\sum_{\stackrel{0 \leq j \leq d}{j \neq r_{\ell}, \;j\neq r_{\ell+1}}}
(r_1,r_2, \ldots, r_{\ell},j,r_{\ell+1},\ldots, r_n, r'_2, \ldots, r'_m).
\end{eqnarray*}
\end{itemize}
\end{theorem}
\noindent {\it Proof:}
(i)--(iii) Routine.
\\
\noindent (iv) In line
(\ref{eq:prod2}), evaluate
$(r_1, r_2, \ldots, r_n)^*$
using the first identity in
Theorem
\ref{thm:trans}, and simplify the result.
\hfill $\Box$ \\
\section{The subspaces $A_n$}
\noindent
In this section we introduce
some subspaces $A_n$ of $A$, and use them to
interpret
our results so far.
\begin{definition}
\label{def:tn}
\rm
For an integer $n\geq 1$ let $A_n$ denote the subspace of
$A$ spanned by the NR words that have length $n$
and end with a nonstarred element.
\end{definition}
\begin{lemma}
\label{lem:sd}
For $n\geq 1$ we display a basis for each
relative of $A_n$.
\begin{center}
\begin{tabular}{c|c}
space & basis \\
\hline
$A_n$ &
the NR words in $A$ that have length $n$ and end with a nonstarred element
\\
$A^*_n$ &
the NR words in $A$ that have length $n$ and end with a starred element
\\
$A^\dagger_n$ &
the NR words in $A$ that have length $n$ and begin with a nonstarred element
\\
$A^{*\dagger}_n$ &
the NR words in $A$ that have length $n$ and begin with a starred element
\end{tabular}
\end{center}
\end{lemma}
\noindent {\it Proof:}
Immediate from Lemma
\ref{lem:aut},
Lemma
\ref{lem:antiaut},
and Theorem
\ref{thm:tbasis}.
\hfill $\Box$ \\
\begin{lemma}
\label{lem:dimtn}
For $n\geq 1$ each relative of
$A_n$
has dimension $(d+1)d^{n-1}$.
\end{lemma}
\noindent {\it Proof:}
Apply Lemma
\ref{lem:sd} and a routine counting argument.
\hfill $\Box$ \\
\begin{lemma}
\label{lem:eveodd}
The following {\rm (i), (ii)} hold
for all integers $n\geq 1$.
\begin{itemize}
\item[\rm (i)] Suppose $n$ is even. Then $A^{*\dagger}_n=A_n$ and
$A^*_n=A^\dagger_n$.
\item[\rm (ii)] Suppose $n$ is odd. Then $A^{*\dagger}_n=A^*_n$ and
$A^\dagger_n=A_n$.
\end{itemize}
\end{lemma}
\noindent {\it Proof:} Pick a word $w$ in $A$ of length $n$.
If $n$ is even, then $w$
begins with a starred element if and only if $w$ ends
with a nonstarred element. If $n$ is odd,
then $w$ begins with a starred element if and only if
$w$ ends with a starred element. The result follows.
\hfill $\Box$ \\
\begin{theorem}
\label{lem:ds}
Each of the following sums is direct.
\begin{eqnarray*}
&&A = \sum_{n=1}^\infty A_n,
\qquad \qquad
A = \sum_{n=1}^\infty A^*_n,
\\
&&
A = \sum_{n=1}^\infty A^\dagger_n,
\qquad \qquad
A = \sum_{n=1}^\infty A^{*\dagger}_n.
\end{eqnarray*}
\end{theorem}
\noindent {\it Proof:} Combine
Theorem
\ref{thm:tbasis}
and Lemma \ref{lem:sd}.
\hfill $\Box$ \\
\begin{theorem}
\label{thm:interp2}
For $n\geq 1$ and $x \in A_n$,
\begin{eqnarray*}
x +(-1)^n x^* \in A_{n+1}\cap A^*_{n+1}.
\end{eqnarray*}
\end{theorem}
\noindent {\it Proof:}
By Definition
\ref{def:tn} we may assume without loss that
$x$
is an NR word in $A$ that has length $n$ and ends with
a nonstarred element. Now
$x+(-1)^n x^* \in A_{n+1}$
by the first assertion
of Theorem
\ref{thm:trans}, and
$x+(-1)^n x^* \in A^*_{n+1}$
by the second assertion of
Theorem
\ref{thm:trans}.
The result follows.
\hfill $\Box$ \\
\begin{corollary}
\label{cor:interp2}
For $n\geq 1$,
\begin{eqnarray*}
A_n +
A_{n+1}\cap A^*_{n+1}
=
A^*_n +
A_{n+1}\cap A^*_{n+1}.
\end{eqnarray*}
\end{corollary}
\noindent {\it Proof:}
This is a routine consequence of
Theorem
\ref{thm:interp2}.
\hfill $\Box$ \\
\noindent For subsets $X,Y$ of $A$
let $XY$ denote the subspace of $A$
spanned by $\lbrace xy \,|\, x \in X, \;y \in Y\rbrace $.
\begin{theorem}
\label{thm:ptntm}
For positive integers $n,m$ the products $A_n A_m$
and $A^*_n A_m$
are described as follows.
\begin{itemize}
\item[\rm (i)]
Assume $m$ is odd. Then
\begin{eqnarray}
A_n A_m \subseteq A_{n+m-1},
\qquad \qquad
A^*_n A_m \subseteq A_{n+m}+A_{n+m-1}.
\label{eq:tntmodd}
\end{eqnarray}
\item[\rm (ii)]
Assume $m$ is even. Then
\begin{eqnarray}
A_n A_m \subseteq A_{n+m}+A_{n+m-1},
\qquad \qquad
A^*_n A_m \subseteq A_{n+m-1}.
\label{eq:tntmeven}
\end{eqnarray}
\end{itemize}
\end{theorem}
\noindent {\it Proof:}
In (\ref{eq:tntmodd})
and (\ref{eq:tntmeven}) the
inclusions on the left
follow from Theorem
\ref{thm:prod},
and the inclusions on the right
follow from
Theorem \ref{thm:prod2}.
\hfill $\Box$ \\
\noindent In Section 9 we will obtain a more
detailed version of Theorem
\ref{thm:ptntm}.
\section{The ideals $A_{\geq n}$}
\noindent Motivated by
Corollary
\ref{cor:interp2}
and
Theorem
\ref{thm:ptntm} we consider the following subspaces of $A$.
\begin{definition}
\label{def:tgeqn}
\rm
For $n\geq 1$ define
\begin{eqnarray*}
A_{\geq n} = A_n + A_{n+1} + \cdots
\end{eqnarray*}
\end{definition}
\begin{theorem} For $n\geq 1$
the space $A_{\geq n}$
is a 2-sided ideal of $A$.
\end{theorem}
\noindent {\it Proof:}
This is a routine consequence of the
inclusions on the left in
(\ref{eq:tntmodd}),
(\ref{eq:tntmeven}).
\hfill $\Box$ \\
\begin{theorem}
\label{thm:same}
For $n\geq 1$
we have
\begin{eqnarray*}
A^*_{\geq n}=A_{\geq n}, \qquad \qquad
A^\dagger_{\geq n}=A_{\geq n}.
\end{eqnarray*}
\end{theorem}
\noindent {\it Proof:}
For $m\geq 1$ we obtain
$A_m \subseteq A^*_m+A^*_{m+1}$
and
$A^*_m \subseteq A_m+A_{m+1}$
from Corollary
\ref{cor:interp2}.
Therefore
$A^*_{\geq n}=A_{\geq n}$.
For $m\geq 1$ the space $A^\dagger_m$
is one of $A_m$, $A^*_m$ by
Lemma
\ref{lem:eveodd},
and each of $A_m$, $A^*_m$ is contained in
$A_m+A_{m+1}$,
so $A^\dagger_m \subseteq A_m+A_{m+1}$.
In this inclusion we apply $\dagger$ to each side
and find
$A_m \subseteq A^\dagger_m+A^\dagger_{m+1}$.
Therefore
$A^\dagger_{\geq n}=A_{\geq n}$.
\hfill $\Box$ \\
\begin{lemma}
\label{lem:prodideal}
For positive integers $n,m$
the product $A_{\geq n}
A_{\geq m}$ is contained in
$A_{\geq n+m-1}$.
\end{lemma}
\noindent {\it Proof:}
This follows from
Definition \ref{def:tgeqn}
and the products on the left
in (\ref{eq:tntmodd}),
(\ref{eq:tntmeven}).
\hfill $\Box$ \\
\section{The map $\partial:A\to A$}
\noindent Motivated by Theorem
\ref{thm:interp2} we consider the following map.
\begin{lemma}
\label{lem:map}
\rm
There exists a unique $\mathbb F$-linear transformation
$\partial:A\to A$
such that for $n\geq 1$,
\begin{eqnarray}
\partial(x)= x+(-1)^n x^*
\qquad \qquad (\forall x \in A_n).
\label{eq:par}
\end{eqnarray}
\end{lemma}
\noindent {\it Proof:}
By Theorem
\ref{lem:ds} the sum
$A=\sum_{n=1}^\infty A_n$
is direct.
\hfill $\Box$ \\
\begin{lemma}
\label{lem:parset}
With reference to Lemma
\ref{lem:map}
we have $\partial (A_n)\subseteq A_{n+1}$ for $n \geq 1$.
\end{lemma}
\noindent {\it Proof:}
Immediate from Theorem
\ref{thm:interp2}.
\hfill $\Box$ \\
\begin{lemma}
\label{lem:bound}
With reference to Lemma
\ref{lem:map} the following
{\rm (i), (ii)} hold for all $x \in A$.
\begin{itemize}
\item[\rm (i)]
$\partial (\partial(x))=0$.
\item[\rm (ii)]
$\partial(x^*)=-(\partial(x))^*$.
\end{itemize}
\end{lemma}
\noindent {\it Proof:} Without loss we may assume
$x \in A_n$ for some
$n\geq 1$. \\
\noindent (i)
Observe $\partial(x)\in A_{n+1}$ by
Lemma
\ref{lem:parset}, so
$\partial(\partial(x))=\partial(x)-(-1)^n(\partial(x))^*$
by
(\ref{eq:par}).
In line
(\ref{eq:par})
we apply $*$ to both sides and get
$(\partial(x))^*=(-1)^n \partial(x)$.
The result follows.
\\
\noindent (ii)
In line
(\ref{eq:par})
we apply $\partial$ to both sides and use
(i) above to get
$\partial(x^*)=(-1)^{n-1} \partial(x)$.
We observed
$(\partial(x))^*=(-1)^n \partial(x)$ in the proof
of part (i), so
$\partial(x^*)=-(\partial(x))^*$.
\hfill $\Box$ \\
\begin{lemma}
\label{lem:dr}
For $n \geq 1$
the kernel of $\partial$ on $A_n$
is $A_n\cap A^*_n$.
\end{lemma}
\noindent {\it Proof:}
For $x \in A_n$ we show that
$\partial(x)=0$ if and only if $x \in A^*_n$.
First assume $\partial(x)=0$. Then
$x^*=(-1)^{n-1}x$ by
(\ref{eq:par}), so $x\in A^*_n$.
To get the reverse implication, assume
$x \in A^*_n$ and note that $x^*\in A_n$.
Now each of $x, x^*$ is contained in $A_n$, so
$\partial(x) \in A_n$ in view of
(\ref{eq:par}).
But $\partial(x) \in A_{n+1}$ by Lemma
\ref{lem:parset} and $A_n \cap A_{n+1}=0$
by Theorem
\ref{lem:ds} so $\partial(x)=0$.
\hfill $\Box$ \\
\noindent Our next goal is to show that
for $n\geq 1$ the image of $A_n$ under $\partial$
is $A_{n+1}\cap A^*_{n+1}$.
To this end it will be convenient to introduce
some subspaces
${}^+A_n$ and
${}^0A_n$ of $A_n$.
\begin{definition}
\label{def:tnpos}
\rm For $n\geq 1$
let
${}^+A_n$
(resp. ${}^0A_n$)
denote the subspace of $A_n$ with a basis
consisting of the NR words that
have length $n$, and end with one of
$e_1, e_2, \ldots, e_d$
(resp. end with $e_0$).
\end{definition}
\begin{example}
\rm
Assume $d=2$. The basis for
${}^+A_3$ from Definition
\ref{def:tnpos} is
\begin{eqnarray*}
&&e_1e^*_0e_1,
\qquad
e_2e^*_0e_1,
\qquad e_0e^*_2e_1,
\qquad e_1e^*_2e_1,
\\
&&e_1e^*_0e_2,
\qquad
e_2e^*_0e_2,
\qquad e_0e^*_1e_2,
\qquad e_2e^*_1e_2.
\end{eqnarray*}
\noindent The basis for
${}^0A_3$ from Definition
\ref{def:tnpos} is
\begin{eqnarray*}
&&e_0e^*_1e_0,
\qquad
e_2e^*_1e_0,
\qquad e_0e^*_2e_0,
\qquad e_1e^*_2e_0.
\end{eqnarray*}
\end{example}
\begin{lemma}
\label{lem:tposobv}
For $n\geq 1$,
\begin{itemize}
\item[\rm (i)] $A_n = {}^+A_n + {}^0A_n$ {\rm (direct sum)}.
\item[\rm (ii)] The dimension of $\,{}^+A_n$ is $d^n$.
\item[\rm (iii)] The dimension of $\,{}^0A_n$ is $d^{n-1}$.
\end{itemize}
\end{lemma}
\noindent {\it Proof:}
Routine using
Lemma
\ref{lem:sd} and
Definition
\ref{def:tnpos}.
\hfill $\Box$ \\
\begin{definition}
\label{def:sigmap}
\rm
For $n\geq 1$ we define an isomorphism
of vector spaces
$\sigma :
{}^+A_n \to {}^0A_{n+1}$.
To do this we give the action of $\sigma$
on the basis for
${}^+A_n$
from Definition
\ref{def:tnpos}.
Let $(r_1, r_2, \ldots, r_n)$
denote an NR word in $A$ such that $r_n\not=0$.
We define the image of this word under
$\sigma$ to be
$(r_1, r_2, \ldots, r_n,0)$. Note that
$\sigma$ sends the above basis for
${}^+A_n$
to the basis
for
${}^0A_{n+1}$ given in
Definition
\ref{def:tnpos}.
Therefore $\sigma$ is an isomorphism of
vector spaces.
\end{definition}
\begin{lemma}
\label{lem:sigalt}
For $n\geq 1$ and $x \in
{}^+A_n$,
\begin{eqnarray}
\sigma(x) = (-1)^n \partial(x)e_0.
\label{eq:sigalt}
\end{eqnarray}
\end{lemma}
\noindent {\it Proof:}
Without loss we may assume
that $x$ is a vector in the basis for
${}^+A_n$ given in Definition
\ref{def:tnpos}.
Thus $x$ is an NR word $(r_1, r_2, \ldots, r_n)$
such that $r_n\not=0$.
Observe that $xe_0=0$ and
$x^*e_0=(r_1, r_2, \ldots, r_n,0)$.
By this and
(\ref{eq:par}) we find
$(-1)^n\partial(x)e_0$ is equal to
$(r_1, r_2, \ldots, r_n,0)$, which is equal to
$\sigma(x)$ by Definition
\ref{def:sigmap}.
The result follows.
\hfill $\Box$ \\
\begin{lemma}
\label{lem:sigds}
For $n\geq 1$,
\begin{eqnarray}
A_n = {}^+A_n + A_n \cap A^*_n \qquad \hbox{\rm (direct sum)}.
\label{eq:sigds}
\end{eqnarray}
Moreover the dimension of $A_n \cap A^*_n$ is $d^{n-1}$.
\end{lemma}
\noindent {\it Proof:}
We first show that the sum
${}^+A_n + A_n \cap A^*_n$
is direct. By Lemma
\ref{lem:sigalt} and since
$\sigma :{}^+A_n\to {}^0A_{n+1}$ is
a bijection, the restriction of
$\partial$ to
${}^+A_n$
is injective. Therefore the kernel
of $\partial $ on
$A_n$ has
zero intersection with
${}^+A_n$.
This kernel is
$A_n \cap A^*_n$ by
Lemma
\ref{lem:dr}. Therefore
${}^+A_n$ has zero intersection
with
$A_n \cap A^*_n$ so the sum
${}^+A_n + A_n \cap A^*_n$
is direct.
Let $k_n$ denote the dimension of
$A_n \cap A^*_n$.
By our comments so far,
and given the dimensions of
$A_n$ and ${}^+A_n$
from Lemma
\ref{lem:dimtn}
and
Lemma
\ref{lem:tposobv}, respectively,
we obtain $k_n \leq d^{n-1}$, with equality if and only if
$A_n = {}^+A_n + A_n \cap A^*_n$.
To finish the proof it suffices to show
$k_n=d^{n-1}$. We do this by induction on $n$.
First assume $n=1$. We have $k_1 \leq 1$
by our above remarks, and $k_1\geq 1$
since $1 \in A_1 \cap A^*_1$ by
(\ref{eq:rel2}). Therefore $k_1=1$ as desired.
Next assume $n\geq 2$. Let $I_n$ denote the
image of $A_{n-1}$ under $\partial$.
By linear algebra the dimension of
$I_n$ is equal to the dimension of $A_{n-1}$
minus the dimension of the kernel of $\partial$
on $A_{n-1}$. The dimension of
$A_{n-1}$ is $d^{n-1}+d^{n-2}$.
The kernel of $\partial $ on
$A_{n-1}$ is $A_{n-1}\cap A^*_{n-1}$
so its dimension is $k_{n-1}$, which is
$d^{n-2}$ by induction.
Therefore the dimension of $I_n$ is
$d^{n-1}$. By
Theorem
\ref{thm:interp2} and
(\ref{eq:par})
we have
$I_n \subseteq
A_n\cap A^*_n$. In this inclusion we consider
the dimensions and get
$d^{n-1}\leq k_n$. We showed earlier that
$k_n \leq d^{n-1}$ so
$k_n =d^{n-1}$ as desired. The result follows.
\hfill $\Box$ \\
\begin{lemma}
\label{lem:image}
For $n \geq 1$
the image of $A_n$ under $\partial$ is
$A_{n+1}\cap A^*_{n+1}$.
\end{lemma}
\noindent {\it Proof:}
Denote this image by
$I_{n+1}$, and
observe
$I_{n+1} \subseteq
A_{n+1}\cap A^*_{n+1}$ by
Theorem
\ref{thm:interp2}. To finish
the proof we show that
$I_{n+1}$ and
$ A_{n+1}\cap A^*_{n+1}$ have
the same dimension. By
Lemma
\ref{lem:sigds} the dimension of
$ A_{n+1}\cap A^*_{n+1}$ is $d^n$.
By Lemma
\ref{lem:dr} and
(\ref{eq:sigds})
the dimension of
$I_{n+1}$ is equal to
the dimension of
${}^+A_n$, which is $d^n$ by
Lemma
\ref{lem:tposobv}(ii).
The result follows.
\hfill $\Box$ \\
\begin{lemma}
\label{lem:intone}
We have $A_1 \cap A^*_1 = \mathbb F 1$.
\end{lemma}
\noindent {\it Proof:}
Observe
$\mathbb F 1 \subseteq A_1 \cap A^*_1$
by (\ref{eq:rel2}), and
$A_1 \cap A^*_1$ has dimension 1 by
Lemma
\ref{lem:sigds}.
\hfill $\Box$ \\
\begin{definition}
\rm
\label{def:incl}
Let $\iota :\mathbb F \to A$ denote $\mathbb F$-algebra
homomorphism that sends $a \mapsto a 1$ for $a \in \mathbb F$.
Note that
$\iota$ is an injection.
\end{definition}
\begin{theorem}
\label{thm:exact} The sequence
\[
\begin{CD}
\mathbb F
@>>\iota>
A_1
@>>\partial>
A_2
@>>\partial>
A_3
@>>\partial>
\cdots
\\
\end{CD}
\]
is exact in the sense of
{\rm \cite[p.~435]{rotman}}.
\end{theorem}
\noindent {\it Proof:}
This follows from
Lemma
\ref{lem:dr},
Lemma
\ref{lem:image},
and
Lemma
\ref{lem:intone}.
\hfill $\Box$ \\
\noindent We emphasize a few points for later use.
\begin{lemma}
\label{lem:isoemph}
For $n\geq 1$ the restriction of
$\partial $ to
${}^+A_n$
is an isomorphism of vector spaces
${}^+A_n \to A_{n+1}\cap A^*_{n+1}$.
\end{lemma}
\noindent {\it Proof:}
Combine Lemma
\ref{lem:dr},
line (\ref{eq:sigds}),
and Lemma
\ref{lem:image}.
\hfill $\Box$ \\
\begin{lemma}
\label{lem:pol}
For $n\geq 1$ and $x \in A_n$ the following
are equivalent:
\begin{itemize}
\item[\rm (i)]
$x^*= (-1)^{n-1}x$;
\item[\rm (ii)]
$x \in A_n \cap A^*_n$.
\end{itemize}
\end{lemma}
\noindent {\it Proof:}
Combine
(\ref{eq:par})
and
Lemma
\ref{lem:dr}.
\hfill $\Box$ \\
\begin{lemma}
For $n\geq 1$ the map $\partial $ acts on
$A^*_n$ as follows.
\begin{eqnarray*}
\partial(y) = -y-(-1)^n y^* \qquad \qquad (\forall y \in A^*_n).
\end{eqnarray*}
\end{lemma}
\noindent {\it Proof:}
Write $x=y^*$, so that
$x \in A_n$ and
$y=x^*$.
Now compute $\partial(y)$ using
Lemma
\ref{lem:bound}(ii) and
(\ref{eq:par}).
\hfill $\Box$ \\
\section{A subalgebra of $A$}
\noindent In this section we consider the sum
\begin{eqnarray}
\label{eq:tsum}
\sum_{n=0}^\infty
(A_{n+1}\cap A^*_{n+1}).
\end{eqnarray}
We observe by Theorem
\ref{lem:ds}
and
Lemma
\ref{lem:dr}
that
(\ref{eq:tsum})
is the kernel of
the map $\partial : A \to A$.
We will show that
(\ref{eq:tsum})
is
a subalgebra of $A$ that is
free of rank $d$.
\begin{lemma}
\label{lem:grading}
For nonnegative integers $n,m$
the following {\rm (i)--(iii)} hold.
\begin{itemize}
\item[\rm (i)]
$
(A_{n+1}\cap A^*_{n+1})A_{m+1} \subseteq A_{n+m+1}
$;
\item[\rm (ii)]
$
(A_{n+1}\cap A^*_{n+1})A^*_{m+1} \subseteq A^*_{n+m+1}
$;
\item[\rm (iii)]
$
(A_{n+1}\cap A^*_{n+1})
(A_{m+1}\cap A^*_{m+1})
\subseteq
A_{n+m+1}\cap A^*_{n+m+1}
$.
\end{itemize}
\end{lemma}
\noindent {\it Proof:}
(i)
For $x \in
A_{n+1}\cap A^*_{n+1}$
and
$y \in
A_{m+1}$
we show that
$xy \in
A_{n+m+1}$.
First assume $m$ is even.
Using
$x \in
A_{n+1}$
and
$y \in
A_{m+1}$
and the inclusion on the left in
(\ref{eq:tntmodd}), we obtain
$xy \in
A_{n+m+1}
$.
Next assume $m$ is odd.
Using
$x \in
A^*_{n+1}$
and
$y \in
A_{m+1}$
and the inclusion on the right in
(\ref{eq:tntmeven}), we obtain
$xy \in
A_{n+m+1}
$.
\\
\noindent (ii)
For $x \in
A_{n+1}\cap A^*_{n+1}$
and
$y \in
A^*_{m+1}$
we show that
$xy \in
A^*_{n+m+1}$.
Observe that
$x^* \in
A_{n+1}\cap A^*_{n+1}$
and
$y^* \in
A_{m+1}$
so
$x^*y^* \in A_{n+m+1}$
by (i) above.
Applying $*$ we find
$xy \in A^*_{n+m+1}$.
\\
\noindent (iii)
Combine (i) and (ii) above.
\hfill $\Box$ \\
\begin{corollary}
\label{cor:subalg}
The sum
{\rm (\ref{eq:tsum})}
is a subalgebra of $A$.
\end{corollary}
\noindent {\it Proof:}
The sum contains the identity $1$ of $A$ by
Lemma
\ref{lem:intone}. The sum
is closed under multiplication
by Lemma
\ref{lem:grading}(iii).
\hfill $\Box$ \\
\noindent We will return to the subalgebra
(\ref{eq:tsum})
after a few comments.
\begin{lemma}
\label{lem:altform}
For each basis vector $(r_1, r_2, \ldots, r_n)$
from Theorem
\ref{thm:tbasis}(i), the element
\begin{eqnarray*}
(r_1, r_2, \ldots, r_n)
\quad + \quad
(-1)^n(r_1, r_2, \ldots, r_n)^*
\end{eqnarray*}
is equal to
\begin{eqnarray}
(e_{r_1}-e^*_{r_1})
(e_{r_2}-e^*_{r_2})
\cdots
(e_{r_n}-e^*_{r_n})(-1)^{\lfloor n/2 \rfloor}.
\label{eq:redexpand}
\end{eqnarray}
The expression
${\lfloor x \rfloor}$ denotes the greatest integer less than
or equal to $x$.
\end{lemma}
\noindent {\it Proof:}
Expand
(\ref{eq:redexpand}) into a sum of $2^n$ terms.
Simplify these terms using
(\ref{eq:rel1}) and
$r_{i-1}\not=r_i$ for $2 \leq i \leq n$.
\hfill $\Box$ \\
\medskip
\noindent Let $F$ denote the
$\mathbb F$-algebra defined by
generators
$\lbrace s_i\rbrace_{i=1}^d$ and no relations.
Thus $F$ is the free $\mathbb F$-algebra of rank $d$.
We call $\lbrace s_i\rbrace_{i=1}^d$ the {\it
standard generators} for $F$.
We recall a few facts about $F$. For an integer $n\geq 0$,
by a {\it word in $F$ of length $n$} we mean a product
$y_1y_2\cdots y_n$
such that
$\lbrace y_i\rbrace_{i=1}^n$
are
standard generators for $F$.
We interpret the word of length $0$ to be the identity of $F$.
The $\mathbb F$-vector space $F$ has a basis consisting
of its words
\cite[p.~723]{rotman}.
For $n\geq 0$ let $F_n$ denote the subspace of $F$ spanned by
the words of length $n$.
Note that $F_n$ has dimenion $d^n$.
We have a direct sum
$F=\sum_{n=0}^\infty F_n$, and $F_rF_s =F_{r+s}$
for $r,s\geq 0$.
We call $F_n$
the {\it $n$th homogeneous component} of $F$.
\begin{theorem}
\label{thm:freeinj}
With the above notation,
consider the $\mathbb F$-algebra homomorphism $F\to A$
that sends $s_i \mapsto e_i-e^*_i$ for $1 \leq i \leq d$.
This map is an injection
and its image is $\sum_{n=0}^\infty (A_{n+1}\cap A^*_{n+1})$.
Moreover for $n\geq 0$
the image of $F_n$
is $A_{n+1}\cap A^*_{n+1}$.
\end{theorem}
\noindent {\it Proof:}
Let $\varepsilon : F\to A$ denote the homomorphism in question.
We claim that for $n\geq 0$
the restriction of $\varepsilon $ to
$F_n$ is a bijection
$F_n \to A_{n+1}\cap A^*_{n+1}$.
To establish the claim we split the argument
into three cases: $n=0$, $n=1$,
and $n\geq 2$.
The claim holds for $n=0$ by
Lemma
\ref{lem:intone} and since
$F_0=\mathbb F 1$.
To see that the claim holds for $n=1$,
note
that $F_1$ has a basis ${\lbrace s_i\rbrace}_{i=1}^d$.
By
Definition
\ref{def:tnpos}
the elements $\lbrace e_i\rbrace_{i=1}^d$
form a basis for
${}^+A_1$,
so $\lbrace \partial(e_i)\rbrace_{i=1}^d$
is a basis for $A_2 \cap A^*_2$ in
view of
Lemma
\ref{lem:isoemph}.
By Lemma
\ref{lem:map}
we have
$\partial(e_i)=e_i-e^*_i$
for $1 \leq i \leq d$.
Therefore
$\lbrace e_i-e^*_i\rbrace_{i=1}^d$
is a basis for $A_2\cap A^*_2$,
and the claim follows for $n=1$.
We now show that the claim holds for $n\geq 2$.
Using
$F_n=(F_1)^n$,
$\varepsilon(F_1)=A_2\cap A^*_2$,
and
Lemma
\ref{lem:grading}(iii) we obtain
$\varepsilon(F_n) \subseteq A_{n+1}\cap A^*_{n+1}$.
To see the reverse inclusion,
first note by Lemma
\ref{lem:image}
that any element in
$A_{n+1}\cap A^*_{n+1}$ can be written
as $\partial(x)$ for some $x \in A_n$.
We show
$\partial(x) \in \varepsilon (F_n)$.
Without loss we may assume
that
$x$ is a vector
$(r_1, r_2, \ldots,r_n)$ in the
basis for $A_n$
from
Lemma
\ref{lem:sd}.
Combining
Lemma
\ref{lem:map} and
Lemma
\ref{lem:altform}
we find that
$\partial(x)$ is equal to
(\ref{eq:redexpand}). In particular
$\partial(x)=
\varepsilon(z_1)
\varepsilon(z_2)
\cdots
\varepsilon(z_n)
$
where $z_i \in F_1$
for $1 \leq i \leq n$.
Observe
$\partial(x)=\varepsilon(z_1z_2\cdots z_n)$
and $z_1z_2\cdots z_n \in F_n$
so
$\partial(x)\in \varepsilon(F_n)$.
Therefore
$A_{n+1}\cap A^*_{n+1}
\subseteq
\varepsilon(F_n)$.
So far we have $\varepsilon(F_n)=A_{n+1}\cap A^*_{n+1}$.
To show that the map
$F_n \to A_{n+1}\cap A^*_{n+1}$,
$x \mapsto \varepsilon(x)$
is a bijection, it suffices to
show that
$F_n$ and $A_{n+1}\cap A^*_{n+1}$ have
the same dimension. We mentioned
below Lemma
\ref{lem:altform} that
$F_n$ has dimension $d^n$.
By the last line of
Lemma
\ref{lem:sigds} we find
$A_{n+1}\cap A^*_{n+1}$
also has dimension $d^n$.
By these
comments
the map $F_n \to A_{n+1}\cap A^*_{n+1}$,
$x \mapsto \varepsilon(x)$
is a bijection. The claim is now proved for $n\geq 2$.
We have established the claim,
and the result follows in view of
the directness of the
sum
$\sum_{n=0}^\infty A_{n+1} \cap A^*_{n+1}$.
\hfill $\Box$ \\
\noindent For notational convenience let us
identify the free algebra $F$ from above
Theorem
\ref{thm:freeinj}
with
the subalgebra
(\ref{eq:tsum}) of $A$,
via the injection
from
Theorem
\ref{thm:freeinj}.
Our next goal is to show that each of the
$\mathbb F$-linear maps
\begin{eqnarray*}
&&F \otimes A_1 \;\to\; A
\qquad \qquad \qquad
F \otimes A^*_1 \; \;\to \;\; A
\\
&&u \otimes v \; \; \mapsto \; \; uv
\qquad \qquad \qquad
u \otimes v \; \;\mapsto \; \; uv
\end{eqnarray*}
is an isomorphism of $\mathbb F$-vector spaces.
We need a lemma.
\begin{lemma}
\label{lem:part1}
For positive integers $n,m$ the
$\mathbb F$-linear map
\begin{eqnarray*}
(A_n \cap A^*_n) \otimes A_m &\to& A_{n+m-1}
\\
u \otimes v \; &\mapsto & \; uv
\end{eqnarray*}
is an isomorphism of $\mathbb F$-vector spaces.
\end{lemma}
\noindent {\it Proof:}
Let
$\theta$
denote the map in question.
To show that
$\theta$
is bijective, we show
that the dimension of
$(A_n \cap A^*_n) \otimes A_m$
is equal to the dimension of $A_{n+m-1}$,
and that
$\theta$
is surjective.
The dimension of
$A_n \cap A^*_n$ is
$d^{n-1}$
by Lemma
\ref{lem:sigds}, and the dimension of
$A_m$ is
$(d+1)d^{m-1}$ by Lemma
\ref{lem:dimtn}, so the dimension of
$(A_n \cap A^*_n) \otimes A_m$
is $(d+1)d^{n+m-2}$.
The dimension of
$A_{n+m-1}$
is $(d+1)d^{n+m-2}$ by
Lemma \ref{lem:dimtn}. Therefore the dimensions of
$(A_n \cap A^*_n) \otimes A_m$
and $A_{n+m-1}$ are the same.
Next we show that
$\theta$
is surjective.
First assume $n=1$. Then
$\theta$
is surjective
since $1 \in
A_1 \cap A^*_1$ by
Lemma
\ref{lem:intone}.
Next assume $n\geq 2$. By
Lemma
\ref{lem:sd} the space
$A_{n+m-1}$ has a basis consisting
of the NR words in $A$ that have
length $n+m-1$ and end with a nonstarred
element. We show that each of these basis
elements is in the image of $\theta $.
Consider an NR word $w=(r_1, r_2, \ldots, r_{n+m-1})$.
Define $u=(-1)^{(n-1)m}\partial(r_1, r_2, \ldots, r_{n-1})$
and observe that $u \in A_n\cap A^*_n$ by
Lemma
\ref{lem:image}.
Define $v=(r_n, r_{n+1}, \ldots, r_{n+m-1})$
and observe $v \in A_m$.
One verifies $w=uv$
by first using
(\ref{eq:par}),
and then
Theorem
\ref{thm:prod}(i),
Theorem
\ref{thm:prod2}(iii)
if $m$ is odd and
Theorem
\ref{thm:prod}(iii),
Theorem
\ref{thm:prod2}(i)
if $m$ is even.
Therefore
$w$ is the image of $u\otimes v$ under
$\theta$.
We have shown
that
$\theta$
is surjective, and
the result follows.
\hfill $\Box$ \\
\begin{theorem}
Each of the $\mathbb F$-linear maps
\begin{eqnarray*}
&&F \otimes A_1 \;\to\; A
\qquad \qquad \qquad
F \otimes A^*_1 \; \;\to \;\; A
\\
&&u \otimes v \; \; \mapsto \; \; uv
\qquad \qquad \qquad
u \otimes v \; \;\mapsto \; \; uv
\end{eqnarray*}
is an isomorphism of $\mathbb F$-vector spaces.
\end{theorem}
\noindent {\it Proof:}
Let $\psi$ (resp. $\xi$) denote the map on
the left (resp. right).
We first show that $\psi$ is an isomorphism
of $\mathbb F$-vector spaces.
By construction
the sum
$F=\sum_{n=1}^\infty A_n \cap A^*_n$
is direct. Therefore
the sum
\begin{eqnarray*}
F \otimes A_1 = \sum_{n=1}^\infty (A_n \cap A^*_n)\otimes A_1
\end{eqnarray*}
is direct. By Theorem
\ref{lem:ds} the sum
$A = \sum_{n=1}^\infty A_n$
is direct. For $n \geq 1$ we apply Lemma
\ref{lem:part1}
with $m=1$ and find that
the map
\begin{eqnarray*}
&&(A_n \cap A^*_n) \otimes A_1 \;\;\to \;\; A_{n}
\\
&& \qquad \qquad u \otimes v \; \;\mapsto \; \; uv
\end{eqnarray*}
is an isomorphism of $\mathbb F$-vector spaces.
It follows that $\psi$ is
an isomorphism of $\mathbb F$-vector spaces.
The map $\xi$ is an isomorphism of
$\mathbb F$-vector spaces since it is
the
composition
\[
\begin{CD}
F \otimes A^*_1
@>>* \otimes *>
F \otimes A_1
@>>\psi>
A
@>>*>
A
\end{CD}
\]
and each composition factor is an isomorphism
of
$\mathbb F$-vector spaces.
\hfill $\Box$ \\
\section{The subspaces $A_n$ revisited}
\noindent In this section we
present a more detailed version of
Theorem
\ref{thm:ptntm}. Let $n,m$ denote positive integers.
For $m$ odd we consider the $\mathbb F$-linear maps
\begin{eqnarray*}
&&A_n \otimes A_m \;\;\to \;\; A_{n+m-1}
\qquad \qquad \qquad
A_n \otimes A_m \;\;\to\;\; A_{n+m}+ A_{n+m-1}
\\
&& \;\quad u \otimes v \;\; \mapsto \; \; uv
\qquad \qquad \qquad
\qquad \;\;\quad
u \otimes v \; \; \mapsto \; \; u^*v
\end{eqnarray*}
and for $m$ even we consider the $\mathbb F$-linear maps
\begin{eqnarray*}
&&A_n \otimes A_m \;\;\to\;\; A_{n+m}+ A_{n+m-1}
\qquad \qquad \qquad
A_n \otimes A_m \;\;\to\;\; A_{n+m-1}
\\
&&\quad \;u \otimes v \; \;\mapsto \; \; uv
\qquad \qquad \qquad
\qquad \qquad \qquad
\;u \otimes v \; \;\mapsto \; \; u^*v.
\end{eqnarray*}
\begin{definition}
\label{def:eqneq}
\rm
Let $n,m$ denote positive integers.
\begin{itemize}
\item[\rm (i)]
Let
${}^{\not=}
(A_n\otimes A_m)$
denote the subspace of
$A_n \otimes A_m$ that has a basis
consisting
of the elements $u \otimes v$, where
$u=(r_1, r_2, \ldots, r_n)$ is an NR word in $A_n$
and $v=(r'_1, r'_2, \ldots, r'_m)$ is an NR word in $A_m$ such that
$r_n \not=r'_1$.
\item[\rm (ii)]
Let
${}^=
(A_n\otimes A_m)$
denote the subspace of
$A_n \otimes A_m$ that has a basis
consisting
of the elements $u \otimes v$, where
$u=(r_1, r_2, \ldots, r_n)$ is an NR word in $A_n$
and $v=(r'_1, r'_2, \ldots, r'_m)$ is an NR word in $A_m$ such that
$r_n =r'_1$.
\end{itemize}
\end{definition}
\noindent The following result is immediate
from Definition
\ref{def:eqneq}.
\begin{lemma}
\label{lem:basic}
With reference to Definition
\ref{def:eqneq},
\begin{eqnarray}
A_n \otimes A_m &=&
{}^{\not=}
(A_n\otimes A_m)
\quad + \quad
{}^=
(A_n\otimes A_m) \qquad \qquad \mbox{\rm (direct sum)}.
\end{eqnarray}
\end{lemma}
\begin{theorem}
\label{thm:kerok}
For positive integers $n,m$ the following
{\rm (i), (ii)} hold.
\begin{itemize}
\item[\rm (i)]
Assume $m$ is odd.
Then the $\mathbb F$-linear map
\begin{eqnarray*}
A_n \otimes A_m &\to& A_{n+m-1}
\\
u \otimes v \; &\mapsto & \; uv
\end{eqnarray*}
is surjective with kernel
$\;{}^{\not=}
(A_n\otimes A_m)$.
\item[\rm (ii)]
Assume $m$ is even.
Then the $\mathbb F$-linear map
\begin{eqnarray*}
A_n \otimes A_m &\to& A_{n+m-1}
\\
u \otimes v \; &\mapsto & \; u^*v
\end{eqnarray*}
is surjective with kernel
$\;{}^{\not=}
(A_n\otimes A_m)$.
\end{itemize}
\end{theorem}
\noindent {\it Proof:}
(i) A basis for
$\;{}^{\not=}
(A_n\otimes A_m)$ is given in
Definition
\ref{def:eqneq}(i).
By Theorem
\ref{thm:prod}(i)
the map sends each element in this
basis to zero.
A basis for
$\;{}^{=}
(A_n\otimes A_m)$ is given in
Definition
\ref{def:eqneq}(ii).
By Theorem
\ref{thm:prod}(ii)
the map sends
this basis
to the basis for $A_{n+m-1}$ given in
Lemma
\ref{lem:sd}.
The result follows from these comments
and Lemma
\ref{lem:basic}.
\\
\noindent (ii) Similar to the proof of
(i) above.
\hfill $\Box$ \\
\begin{lemma}
\label{prop:concl}
For positive integers $n,m$ we have
\begin{eqnarray}
A_n \otimes A_m &=&
{}^{\not=}
(A_n\otimes A_m)
\quad + \quad
(A_n\cap A^*_n)\otimes A_m \qquad \qquad \mbox{\rm (direct sum).}
\end{eqnarray}
\end{lemma}
\noindent {\it Proof:}
For $m$ odd the result follows from
Lemma
\ref{lem:part1} and
Theorem
\ref{thm:kerok}(i). For $m$ even
the result follows from
Lemma
\ref{lem:pol},
Lemma
\ref{lem:part1},
and
Theorem
\ref{thm:kerok}(ii).
\hfill $\Box$ \\
\noindent The following result will be helpful.
\begin{proposition}
\label{prop:partialmap}
For positive integers $n,m$
the $\mathbb F$-linear map
\begin{eqnarray*}
A_n \otimes A_m &\to& A_{n+m}
\\
u \otimes v \; &\mapsto & \; \partial(u)v
\end{eqnarray*}
is surjective
with kernel
$(A_n \cap A^*_n)\otimes A_m$.
\end{proposition}
\noindent {\it Proof:}
By Lemma
\ref{lem:dr} and Lemma
\ref{lem:image}, the map
$A_n \to A_{n+1}\cap A^*_{n+1}$,
$u \mapsto \partial(u)$
is surjective with kernel
$A_n \cap A^*_n$.
Therefore the map
$A_n \otimes A_m \to (A_{n+1}\cap A^*_{n+1})\otimes A_m$,
$u\otimes v \mapsto \partial(u)\otimes v$
is surjective with kernel
$(A_n \cap A^*_n)\otimes A_m$.
By Lemma
\ref{lem:part1}
the map
$(A_{n+1}\cap A^*_{n+1})\otimes A_m \to A_{n+m}$,
$u \otimes v\mapsto uv$
is a bijection. Composing the two previous maps,
we find that the map
$A_n \otimes A_m \to A_{n+m}$, $u \otimes v \mapsto \partial(u)v$
is surjective with kernel
$(A_n\cap A^*_n)\otimes A_m$.
\hfill $\Box$ \\
\begin{theorem}
For positive integers $n,m$ the following {\rm (i), (ii)} hold.
\begin{itemize}
\item[\rm (i)] Assume $m$ is even. Then
the $\mathbb F$-linear map
\begin{eqnarray*}
A_n \otimes A_m &\to& A_{n+m}+ A_{n+m-1}
\\
u \otimes v \; &\mapsto & \; uv
\end{eqnarray*}
is an isomorphism of $\mathbb F$-vector spaces.
Under this map the preimage of
$A_{n+m}$ is
$\;{}^{\not=}
(A_n\otimes A_m)$ and the preimage of
$A_{n+m-1}$ is
$(A_n \cap A^*_n)\otimes A_m$.
\item[\rm (ii)]
Assume $m$ is odd. Then
the $\mathbb F$-linear map
\begin{eqnarray*}
A_n \otimes A_m &\to& A_{n+m}+ A_{n+m-1}
\\
u \otimes v \; &\mapsto & \; u^*v
\end{eqnarray*}
is an isomorphism of $\mathbb F$-vector spaces.
Under this map the preimage of
$A_{n+m}$ is
$\;{}^{\not=}
(A_n\otimes A_m)$ and the preimage of
$A_{n+m-1}$ is
$(A_n \cap A^*_n)\otimes A_m$.
\end{itemize}
\end{theorem}
\noindent {\it Proof:}
For $u \in A_n$ and $v \in A_m$
we use $\partial(u)=u+(-1)^n u^*$
to obtain
\begin{eqnarray}
\label{eq:breakapart}
\partial(u)v= uv+(-1)^n u^*v.
\end{eqnarray}
\noindent (i) Denote the map by $\eta $.
The restriction of $\eta$ to
$\;{}^{\not=}
(A_n\otimes A_m)$
gives a bijection
$\;{}^{\not=}
(A_n\otimes A_m) \to A_{n+m}$
by Theorem
\ref{thm:kerok}(ii),
Lemma
\ref{prop:concl},
Proposition
\ref{prop:partialmap}, and
(\ref{eq:breakapart}).
The restriction of
$\eta $ to
$(A_n \cap A^*_n)\otimes A_m$
gives a bijection
$(A_n \cap A^*_n)\otimes A_m \to A_{n+m-1}$,
by
Lemma \ref{lem:part1}.
The result follows.
\\
\noindent (ii) Denote the map by $\zeta $.
The restriction of $\zeta$ to
$\;{}^{\not=}
(A_n\otimes A_m)$
gives a bijection
$\;{}^{\not=}
(A_n\otimes A_m) \to A_{n+m}$
by Theorem
\ref{thm:kerok}(i),
Lemma
\ref{prop:concl},
Proposition
\ref{prop:partialmap}, and
(\ref{eq:breakapart}).
The restriction of
$\zeta$ to
$(A_n \cap A^*_n)\otimes A_m$
gives a bijection
$(A_n \cap A^*_n)\otimes A_m \to A_{n+m-1}$,
by
Lemma \ref{lem:pol}
and
Lemma \ref{lem:part1}.
The result follows.
\hfill $\Box$ \\
\section{The subspaces $A_{\leq n}$}
\noindent In this last section we investigate the
following subspaces of $A$.
\begin{definition}
\label{def:leqn}
\rm
For all integers $n\geq 1$ we define
\begin{eqnarray}
A_{\leq n} = A_1 + A_2 + \cdots + A_n.
\label{eq:defleqn}
\end{eqnarray}
\end{definition}
\noindent The following lemma is immediate from the construction.
\begin{lemma}
\label{lem:obv}
For $n\geq 1$ we display a basis for each relative of
$A_{\leq n}$.
\begin{center}
\begin{tabular}{c|c}
space & basis \\
\hline
$A_{\leq n}$ &
the NR words in $A$ that have length at most
$n$ and end with a nonstarred element
\\
$A^*_{\leq n}$ &
the NR words in $A$ that have length at most $n$ and end with a starred element
\\
$A^\dagger_{\leq n}$ &
the NR words in $A$ that have length at most $n$ and begin with a nonstarred element
\\
$A^{*\dagger}_{\leq n}$ &
the NR words in $A$
that have length at most $n$ and begin with a starred element
\end{tabular}
\end{center}
\end{lemma}
\begin{lemma}
\label{lem:dimleqn}
For $n\geq 1$ the relatives of
$A_{\leq n}$ all have dimension
$(d+1)(1+d+d^2+\cdots+d^{n-1})$.
\end{lemma}
\noindent {\it Proof:}
By Theorem
\ref{lem:ds}
and Definition
\ref{def:leqn},
the dimension of
$A_{\leq n}$ is equal to the sum of the dimensions of
$A_1, A_2, \ldots, A_n$.
The result follows from this and
Lemma
\ref{lem:dimtn}.
\hfill $\Box$ \\
\noindent In Lemma
\ref{lem:obv} we gave a basis for each relative
of $A_{\leq n}$.
In a moment we will display another basis.
In order to motivate this new basis we first give
a spanning set.
\begin{lemma}
\label{thm:leqn}
For $n\geq 1$ we display a spanning
set for each relative of
$A_{\leq n}$.
\begin{center}
\begin{tabular}{c|c}
space & spanning set \\
\hline
$A_{\leq n}$ &
the words in $A$ that have length
$n$ and end with a nonstarred element
\\
$A^*_{\leq n}$ &
the words in $A$ that have length $n$ and end with a starred element
\\
$A^\dagger_{\leq n}$ &
the words in $A$ that have length $n$ and begin with a nonstarred element
\\
$A^{*\dagger}_{\leq n}$ &
the words in $A$ that have length $n$ and begin with a starred element
\end{tabular}
\end{center}
\end{lemma}
\noindent {\it Proof:}
Concerning the first row of the table, let $S_n$ denote
the subspace of $A$ spanned by the words in $A$
that have length $n$ and end with a nonstarred element.
We show $S_n=A_{\leq n}$.
By construction $S_n=\cdots A_1 A^*_1 A_1$ ($n$ factors).
By Theorem
\ref{thm:ptntm}
we have
$A_1A_j \subseteq A_j+A_{j+1}$
and
$A^*_1A_j \subseteq A_j+A_{j+1}$
for $1 \leq j \leq n-1$. By this
and
induction on $n$
we find $S_n \subseteq A_{\leq n}$.
To get the reverse inclusion,
note that for $1 \leq j \leq n$ we have $A_j \subseteq S_j$,
and also $S_j \subseteq S_n$
since $ 1 \in A_1$ and $1 \in A^*_1$ by
Lemma
\ref{lem:intone}.
We have verified the first row of the table.
The remaining rows are similarly verified.
\hfill $\Box$ \\
\noindent
For each spanning set in
Lemma
\ref{thm:leqn}, the
set is not a basis for $n\geq 3$,
since the set has cardinality $(d+1)^n$
and this number differs from the dimension
given in
Lemma
\ref{lem:dimleqn}. Our next goal is
to obtain
a subset of the spanning set that is a basis.
\begin{definition}
\label{def:r/nr}
\rm
A word $g_1g_2\cdots g_n$ in $A$ is called
{\it repeating/nonrepeating} (or {\it R/NR})
whenever for $2 \leq j \leq n$,
if $g_{j-1}$, $g_j$ have the same index then
$g_1, g_2, \ldots, g_j$ all have the same index.
\end{definition}
\begin{example}
\rm
For $d=2$ we display the R/NR words in $A$ that have length 3
and end with $e_0$.
\begin{eqnarray*}
&&e_0e^*_0e_0, \qquad
e_1e^*_1e_0, \qquad
e_2e^*_2e_0, \qquad
\\
&&
e_0e^*_1e_0, \qquad
e_2e^*_1e_0, \qquad
e_0e^*_2e_0, \qquad
e_1e^*_2e_0.
\end{eqnarray*}
\end{example}
\begin{definition}
\rm
A word in $A$ is called
{\it nonrepeating/repeating} (or {\it NR/R}) whenever
its image under $\dagger$ is R/NR.
\end{definition}
\begin{example}
\rm
For $d=2$ we display the NR/R words in $A$ that have length 3
and start with $e_0$.
\begin{eqnarray*}
&&e_0e^*_0e_0, \qquad
e_0e^*_1e_1, \qquad
e_0e^*_2e_2, \qquad
\\
&&
e_0e^*_1e_0, \qquad
e_0e^*_1e_2, \qquad
e_0e^*_2e_0, \qquad
e_0e^*_2e_1.
\end{eqnarray*}
\end{example}
\begin{theorem}
For $n\geq 1$ we display a basis for each relative of
$A_{\leq n}$.
\begin{center}
\begin{tabular}{c|c}
space & basis \\
\hline
$A_{\leq n}$ &
the R/NR words in $A$ that have length $n$ and end with a nonstarred element
\\
$A^*_{\leq n}$ &
the R/NR words in $A$ that have length $n$ and end with a starred element
\\
$A^\dagger_{\leq n}$ &
the NR/R words in $A$ that have length $n$ and begin with a nonstarred element
\\
$A^{*\dagger}_{\leq n}$ &
the NR/R words in $A$ that have length $n$ and begin with a starred element
\end{tabular}
\end{center}
\end{theorem}
\noindent {\it Proof:}
Concerning the first row of the table,
let $(R/NR)_n$ denote the set of
R/NR words in $A$ that have length $n$ and end
with a nonstarred element. We show
$(R/NR)_n$ is a basis for $A_{\leq n}$.
Let $(NR)_n$ denote the basis
for $A_n$ given in Lemma
\ref{lem:sd}.
Let
$(NR)_{\leq n}=
\cup_{j=1}^n
(NR)_j$
denote the basis
for
$A_{\leq n}$ given in
Lemma
\ref{lem:obv}.
We now define a linear transformation
$f:
A_{\leq n} \to
A_{\leq n}$.
To this end we give the action
of $f$ on
$(NR)_j$ for $1 \leq j\leq n$.
For a word $(r_1, r_2, \ldots, r_j)$ in
$(NR)_j$
we define its image under $f$ to be
$(r_1, r_1, \ldots,r_1, r_1, r_2, \ldots, r_j)$
($n$ coordinates). This image is contained in
$A_{\leq n}$ by
Lemma
\ref{thm:leqn}.
By the construction $f$ sends the
basis
$(NR)_{\leq n}$ to the set
$(R/NR)_n$.
To show that
$(R/NR)_n$
is a basis for $A_{\leq n}$ it suffices
to show that $f$ is a bijection.
Using the data in Theorem
\ref{thm:prod} and
Theorem
\ref{thm:prod2},
one finds
$(f-I)A_j \subseteq A_{j+1}+ \cdots+ A_n$
for $1 \leq j \leq n$, where $I:A\to A$ is the identity map.
Therefore, with
respect to an appropriate ordering of the
basis
$(NR)_{\leq n}$, the matrix
which represents $f$ is lower triangular, with
all diagonal entries $1$. This matrix is nonsingular
so $f$ is invertible and hence a bijection. Therefore
$(R/NR)_n$ is a basis
for
$A_{\leq n}$.
This yields the first row of the table. The remaining
rows are similarly obtained.
\hfill $\Box$ \\
\small
|
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}
| 1,785
|
David Guy Compton (born August 19, 1930) is a British author who writes science fiction under the name D. G. Compton. He used the name Guy Compton for his earlier crime novels and the pseudonym Frances Lynch for his Gothic novels. He has also written short stories, radio plays, and a non-fiction book on stammering, its causes and cures.
Compton was born in London. Son of Gerald Cross (Actor), and Nuna Davey (Actress). His first published book was the 1962 crime novel Too Many Murderers. His 1970 novel The Steel Crocodile was nominated for the Nebula Award, and his 1974 novel The Continuous Katherine Mortenhoe was filmed as Death Watch by Bertrand Tavernier in 1979.
The 1983 film Brainstorm was very similar in content to Compton's 1968 novel Synthajoy.
In Science Fiction: History, Science, Vision, Robert Scholes and Eric S. Rabkin write:
Compton's work is informed by an acute and subtle moral sense which avoids the extremes of satire and sentiment while compelling us to see the world ethically...he succeeds superbly in preserving certain traditional fictional values and human values in works of genuine science fiction.
Compton was named the 2007 Author Emeritus by the Science Fiction and Fantasy Writers of America.
Bibliography
As Guy Compton:
Too Many Murderers (1962)
Medium for Murder (1963)
Dead on Cue (1964)
High Tide for Hanging (1965)
Disguise for a Dead Gentleman (1966)
And Murder Came Too (1967)
As D. G. Compton:
The Quality of Mercy (1965)
Farewell, Earth's Bliss (1966)
The Silent Multitude (1967)
Synthajoy (1968)
The Palace (1969) (non-SF)
The Steel Crocodile (1970) Alternate title: The Electric Crocodile
Chronocules (1971) Alternate titles: Chronicules and Hot Wireless Sets, Aspirin Tablets, the Sandpaper Slides of Used Matchboxes, and Something that Might have been Castor Oil
The Continuous Katherine Mortenhoe (1974) Alternate titles: The Unsleeping Eye and Death Watch
The Missionaries (1975)
A Usual Lunacy (1978)
Windows (1979)
Ascendancies (1980)
Scudder's Game (1988)
Ragnarok (1991), with John Gribbin
Nomansland (1993)
Stammering: its nature, history, causes and cures (1993) (non-fiction)
Justice City (1994)
Back of Town Blues (1996)
Die Herren von Talojz (1997) German translation only
As Frances Lynch:
Twice Ten Thousand Miles (1974)
The Fine and Handsome Captain (1975)
Stranger at the Wedding (1976)
A Dangerous Magic (1978)
In the House of Dark Music (1979)
References
External links
https://web.archive.org/web/20080117044155/http://www.sfwa.org/awards/2007/index.html#AuthorEmeritus
1930 births
Living people
Writers from London
English science fiction writers
English male novelists
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
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Are you faculty or staff and ready to quit smoking?
Smoking Cessation information for Faculty and Staff can be found here.
Anthem Blue Cross has developed a tobacco cessation program, TLC: The Last Cigarette Program to support employees with smoking cessation efforts. For more information call 866-634-3435.
Kaiser Permanente members can help your efforts to stop smoking through support groups, counseling, classes and a personalized online quit smoking program called Breathe.
After registering for Blue Shield's healthy Lifestyle Rewards members may enroll in an online smoking cessation program including online support groups.
|
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"redpajama_set_name": "RedPajamaC4"
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\section{Introduction}\label{sec:intro}
Given an abelian group $(G,+)$, automorphisms $\varphi$, $\psi$ of $(G,+)$, and an element $c\in G$, define a new operation $*$ on $G$ by
\begin{displaymath}
x*y = \varphi(x)+\psi(y)+c.
\end{displaymath}
The resulting quasigroup $(G,*)$ is said to be \emph{affine over} $(G,+)$, and it will be denoted by $\Q(G,+,\varphi,\psi,c)$.
Quasigroups that are affine over an abelian group are called \emph{central quasigroups} or \emph{T-quasigroups}. We will use the terms ``quasigroup affine over an abelian group'' and ``central quasigroup'' interchangeably. Central quasigroups are precisely the abelian quasigroups in the sense of universal algebra \cite{Sz}.
A quasigroup $(Q,\cdot)$ is called \emph{medial} if it satisfies the medial law
\begin{displaymath}
(x\cdot y)\cdot(u\cdot v)=(x\cdot u)\cdot(y\cdot v).
\end{displaymath}
Medial quasigroups are also known as \emph{entropic} quasigroups.
The fundamental Toyoda-Bruck theorem \cite[Theorem 3.1]{Sta-latin} states that, up to isomorphism, medial quasigroups are precisely central quasigroups $\Q(G,+,\varphi,\psi,c)$ with commuting automorphisms $\varphi$, $\psi$.
\medskip
The classification of central (or medial) quasigroups up to isotopy is trivial in the sense that it coincides with the classification of abelian groups up to isomorphism. Indeed:
\begin{itemize}
\item If $(G,*) = \Q(G,+,\varphi,\psi,c)$ is a central quasigroup then $(G,*)$ is isotopic to $(G,+)$ via the isotopism $(x\mapsto \varphi(x)$, $x\mapsto \psi(x)+c$, $x\mapsto x)$.
\item If two central quasigroups $Q_i = \Q(G_i,+_i,\varphi_i,\psi_i,c_i)$ are isotopic then the underlying groups $(G_i,+_i)$ are isotopic. But isotopic groups are necessarily isomorphic, cf.\;\cite[Proposition 1.4]{Smi-book}.
\end{itemize}
Classifying and enumerating central and medial quasigroups up to isomorphism is nontrivial, however, and that is the topic of the present paper.
\medskip
There are not many results in the literature concerning enumeration and classification of central and medial quasigroups.
Simple idempotent medial quasigroups were classified by Smith in \cite[Theorem 6.1]{Smi}. Sokhatsky and Syvakivskij \cite{SS} classified $n$-ary quasigroups affine over cyclic groups and obtained a formula for the number of those of prime order. Kirnasovsky \cite{Kir} carried out a computer enumeration of central quasigroups up to order $15$, and obtained more classification results in his PhD thesis \cite{Kir1}. Idempotent medial quasigroups of order $p^k$, $k\leq4$, were classified by Hou \cite[Table 1]{Hou}.
At the time of writing this paper, the On-line Encyclopedia of Integer Sequences \cite{OEIS} gives the number of medial quasigroups of order $\le 8$ up to isomorphism as the sequence A226193, and there appears to be no entry for the number of central quasigroups up to isomorphism.
Dr\'apal \cite{Dra} and Sokhatsky \cite{Sok} obtained a general isomorphism theorem for quasigroups isotopic to groups, cf.\;\cite[Theorem 2.10]{Dra} and \cite[Corollary 28]{Sok}, and for central quasigroups in particular, cf.\; \cite[Theorem 3.2]{Dra}, or its restatement, Theorem \ref{Th:Alg}.
Dr\'apal applied the machinery to calculate isomorphism classes of quasigroups of order $4$ (by hand), and Kirnasovsky used Sokhatsky's theory for the calculations mentioned above.
In the present paper, we use a similar approach to obtain stronger enumeration results, taking advantages of the computer system \texttt{GAP} \cite{GAP}.
\medskip
We refer the reader to \cite{Smi-book} for general theory of quasigroups, to \cite{Dra} for a more extensive list of references on central quasigroups, to \cite{Sok} for results on quasigroups isotopic to groups, to \cite{Sta-latin} for results on quasigroups affine over various kinds of loops, and to \cite{SSta,Sz} for a broader context on affine representation of general algebraic structures. The article \cite{HR} gives a gentle introduction into automorphism groups of finite abelian groups and points to original sources on that topic.
\medskip
The paper is organized as follows.
In Section \ref{Sc:Iso}, we formulate an isomorphism theorem for central quasigroups, Theorem \ref{thm:iso}, which is less general than \cite[Theorem 2.10]{Dra} or \cite[Corollary 28]{Sok}, and equivalent to but less technical than \cite[Theorem 3.2]{Dra}. We also present the enumeration algorithm in detail.
In Section \ref{Sc:Cyclic}, we establish our own version of \cite[Theorem 2]{SS} and \cite[Theorem 3.5]{Dra} for cyclic $p$-groups, Theorem \ref{Th:Cp}, providing an explicit formula for the number of isomorphism classes. We were informed that the same result was obtained by Kirnasovsky in his unpublished PhD thesis \cite{Kir1}. Since the automorphism groups of cyclic groups are commutative, Theorem \ref{Th:Cp} also yields the number of medial quasigroups up to isomorphism over finite cyclic groups, and of prime order in particular.
Finally, the results of the enumeration are presented in the Appendix.
\section{Isomorphism theorem and enumeration algorithm}\label{Sc:Iso}
\subsection{Elementary properties of the counting functions $cq$ and $mq$}\label{Ss:Elementary}
For an abelian group $G$, let $cq(G)$ (resp. $mq(G)$) denote the number of all central (resp. medial) quasigroups over $G$ up to isomorphism. For $n\geq1$, let $cq(n)$ (resp. $mq(n)$) denote the number of all central (resp. medial) quasigroups of order $n$ up to isomorphism.
Let us establish two fundamental properties of the counting functions.
First, by the remarks in the introduction,
\begin{equation*
cq(n)=\sum_{|G|=n} cq(G)\quad\text{and}\quad mq(n)=\sum_{|G|=n} mq(G),
\end{equation*}
where the summations run over all abelian groups of order $n$ up to isomorphism.
Second, Proposition \ref{Pr:HK} shows that the classification of central and medial quasigroups can be reduced to prime power orders. As far as enumeration is concerned, Proposition \ref{Pr:HK} implies that the functions $cq$, $mq:\mathbb N^+\to\mathbb N^+$ are multiplicative in the number-theoretic sense.
\begin{proposition}\label{Pr:HK}
Let $G = H\times K$ be an abelian group such that $\gcd(|H|,|K|)=1$. Up to isomorphism, any quasigroup affine over $G$ can be expressed in a unique way as a direct product of a quasigroup affine over $H$ and a quasigroup affine over $K$. In particular,
\begin{equation*
cq(G) = cq(H)\cdot cq(K)\quad\text{and}\quad mq(G) = mq(H)\cdot mq(K).
\end{equation*}
\end{proposition}
\begin{proof}
Any automorphism of $G$ decomposes uniquely as a direct product of an automorphism of $H$ and an automorphism of $K$, cf.\;\cite[Lemma 2.1]{HR}. The rest is easy.
\end{proof}
\subsection{The isomorphism problem for central quasigroups}
Let us now consider the isomorphism problem for quasigroups affine over a fixed abelian group $(G,+)$.
Consider any group $A$. (Later we will take $A=\aut{G,+}$.) Then $A$ acts on itself by conjugation, and $A$ also acts on $A\times A$ by a simultaneous conjugation in both coordinates, i.e., $(\alpha,\beta)^\gamma = (\alpha^\gamma,\beta^\gamma)$.
\begin{lemma}\label{Lm:Orbits}
Let $A$ be a group. Let $X$ be a complete set of orbit representatives of the conjugation action of $A$ on itself. For $\xi\in X$, let $Y_\xi$ be a complete set of orbit representatives of the conjugation action of the centralizer $C_A(\xi)$ on $A$. Then
\begin{displaymath}
\{(\xi,\upsilon):\xi\in X,\,\upsilon\in Y_\xi\}
\end{displaymath}
is a complete set of orbit representatives of the conjugation action of $A$ on $A\times A$.
\end{lemma}
\begin{proof}
For every $(\alpha,\beta)\in A\times A$ there is a unique $\xi\in X$ and some $\gamma\in A$ such that $(\alpha,\beta)$ and $(\xi,\gamma)$ are in the same orbit. For a fixed $\xi\in X$ and some $\beta$, $\gamma\in A$, we have $(\xi,\beta)$ in the same orbit as $(\xi,\gamma)$ if and only if there is $\delta\in C_A(\xi)$ such that $\beta^\delta = \gamma$.
\end{proof}
\begin{lemma}\label{Lm:Action}
Let $(G,+)$ be an abelian group, $A=\aut{G,+}$ and $\alpha$, $\beta\in A$. Then $C_A(\alpha)\cap C_A(\beta)$ acts naturally on $G/\img{1-\alpha-\beta}$.
\end{lemma}
\begin{proof}
Let $U=\img{1-\alpha-\beta}$. It suffices to show that for every $\gamma\in C_A(\alpha)\cap C_A(\beta)$ the mapping $u+U\mapsto \gamma(u)+U$ is well-defined. Now, if $u+U=v+U$ then $u=v+w-\alpha(w)-\beta(w)$ for some $w\in G$ and we have $\gamma(u) = \gamma(v) + \gamma(w) - \gamma\alpha(w) - \gamma\beta(w) = \gamma(v) + \gamma(w) - \alpha\gamma(w) - \beta\gamma(w) = \gamma(v) + (1-\alpha-\beta)(\gamma(w)) \in \gamma(v) + U$.
\end{proof}
We will now state a theorem that solves the isomorphism problem for central and medial quasigroups over $(G,+)$. Instead of showing how it follows from the more general \cite[Theorem 2.10]{Dra}, we show that it is equivalent to \cite[Theorem 3.2]{Dra}, which we restate as Theorem \ref{Th:Alg} here.
\begin{theorem}[Isomorphism problem for central quasigroups]\label{thm:iso}
Let $(G,+)$ be an abelian group, let $\varphi_1$, $\psi_1$, $\varphi_2$, $\psi_2\in\aut{G,+}$, and let $c_1$, $c_2\in G$. Then the following statements are equivalent:
\begin{enumerate}
\item[(i)] the central quasigroups $\Q(G,+,\varphi_1,\psi_1,c_1)$ and $\Q(G,+,\varphi_2,\psi_2,c_2)$ are isomorphic;
\item[(ii)] there is an automorphism $\gamma$ of $(G,+)$ and an element $u\in\mathrm{Im}(1-\varphi_1-\psi_1)$ such that
\begin{displaymath}
\varphi_2 = \gamma \varphi_1 \gamma^{-1},\quad \psi_2 = \gamma \psi_1 \gamma^{-1},\quad c_2=\gamma(c_1+u).
\end{displaymath}
\end{enumerate}
\end{theorem}
\begin{theorem}[{{\cite[Theorem 3.2]{Dra}}}]\label{Th:Alg}
Let $(G,+)$ be an abelian group and denote $A=\aut{G,+}$. The isomorphism classes of central quasigroups (resp. medial quasigroups) over $(G,+)$ are in one-to-one correspondence with the elements of the set
\begin{displaymath}
\{(\varphi,\psi,c):\varphi\in X,\,\psi\in Y_\varphi,\,c\in G_{\varphi,\psi}\},
\end{displaymath}
where
\begin{itemize}
\item $X$ is a complete set of orbit representatives of the conjugation action of $A$ on itself;
\item $Y_\varphi$ is a complete set of orbit representatives of the conjugation action of $C_A(\varphi)$ on $A$ (resp. on $C_A(\varphi)$), for every $\varphi\in X$;
\item $G_{\varphi,\psi}$ is a complete set of orbit representatives of the natural action of $C_A(\varphi)\cap C_A(\psi)$ on $G/\img{1-\varphi-\psi}$.
\end{itemize}
\end{theorem}
Here is a proof of the equivalence of Theorems \ref{thm:iso} and \ref{Th:Alg}: By Lemma \ref{Lm:Orbits}, we can assume that we are investigating the equivalence of two triples $(\varphi,\psi,c_1)$ and $(\varphi,\psi,c_2)$ for some $\varphi\in X$, $\psi\in Y_\varphi$ and $c_1$, $c_2\in G$. Let $U=\img{1-\varphi-\psi}$. The following conditions are then equivalent for any $\gamma\in\aut{G,+}$, using Lemma \ref{Lm:Action}: $c_2 = \gamma(c_1+u)$ for some $u\in U$, $c_2\in \gamma(c_1+U) = \gamma(c_1)+U$, $c_2+U = \gamma(c_1)+U = \gamma(c_1+U)$. This finishes the proof.
\subsection{The algorithm}\label{Sc:Alg}
Theorem \ref{Th:Alg} together with the results of Subsection \ref{Ss:Elementary} gives rise to the following algorithm that enumerates central and medial quasigroups of order $n$. In the algorithm we denote by \texttt{R(H,X)} a complete set of representatives of the (clear from context) action of \texttt{H} on \texttt{X}.
\begin{algorithm}\label{Alg:Main}\ \newline
\noindent Input: positive integer $n$
\noindent Output: $cq(n)$ and $mq(n)$
\begin{verbatim}
cqn := 0; mqn := 0;
for G in the set of abelian groups of order n up to isomorphism do
cqG := 0; mqG := 0;
A := automorphism group of G;
for f in R(A,A) do
for g in R(C_A(f),A) do
for c in R( Intersection(C_A(f),C_A(g)), G/Im(1-f-g) ) do
cqG := cqG + 1;
if f*g=g*f then mqG := mqG + 1; fi;
od;
od;
od;
cqn := cqn + cqG; mqn := mqn + mqG;
od;
return cqn, mqn;
\end{verbatim}
\end{algorithm}
The algorithm was implemented in the \texttt{GAP} system \cite{GAP} in a straightforward fashion, taking advantage of some functionality of the \texttt{LOOPS} \cite{LOOPS} package. The code is available from the second author at {\tt www.math.du.edu/\textasciitilde petr}.
In small situations it is possible to directly calculate the orbits of the conjugation action of $A=\aut{G,+}$ on $A\times A$. For larger groups, it is safer (due to memory constraints) to work with one conjugacy class of $A$ at a time, as in Algorithm \ref{Th:Alg}.
Among the cases we managed to calculate, the elementary abelian group $C_2^5$ took the most effort, about $4$ hours on a standard personal computer.
It might not be difficult to calculate some of the missing entries for $mq(G)$. However, $cq(C_2^6)$, for instance, appears out of reach without further theoretical advances or more substantial computational resources.
The outcome of the calculation can be found in the Appendix.
\section{Quasigroups affine over cyclic groups}\label{Sc:Cyclic}
Let $G$ be a cyclic group. Since $\aut{G}$ is commutative, every quasigroup affine over $G$ is medial.
\begin{theorem}[{\cite[p. 70]{Kir1}}]\label{Th:Cp}
Let $p$ be a prime and $k$ a positive integer. Then
\begin{displaymath}
cq(C_{p^k})=mq(C_{p^k})= p^{2k} + p^{2k-2} - p^{k-1} - \sum_{i=k-1}^{2k-1} p^i.
\end{displaymath}
\end{theorem}
\begin{proof}
Let $G=C_{p^k}$ and $A=\mathrm{Aut}(G)$. We will identify $A$ with the $p^k-p^{k-1}$ elements of $G^* = \{a\in G:\ p\nmid a\}$. We will follow Algorithm \ref{Alg:Main}. Since $A$ is commutative, the conjugation action is trivial and we have to consider every $(\varphi,\psi)\in A\times A$. For a fixed $(\varphi,\psi)\in A\times A$, we must consider a complete set of orbit representatives $G_{\varphi,\psi}$ of the action of $A=C_A(\varphi)\cap C_A(\psi)$ on $G/\img{1-\varphi-\psi}$. Now, $\img{1-\varphi-\psi}$ is equal to $p^iG$ if and only if $p^i\mid 1-\varphi-\psi$ and $p^{i+1}\nmid 1-\varphi-\psi$.
\emph{Case} $i=0$, i.e., \[\varphi+\psi\not\equiv1\pmod p.\]
In this case, we can take $G_{\varphi,\psi}=\{0\}$. How many such pairs $(\varphi,\psi)$ exist? First, let us count those with $\varphi\equiv1\pmod p$. Then $\psi\in G^*$ can be chosen arbitrarily, hence we have $p^{k-1}(p^k-p^{k-1})$ such pairs. Next, let us count those with $\varphi\not\equiv1\pmod p$. Then $\psi\in G^*$ must satisfy $\psi\not\equiv 1-\varphi\pmod p$, hence we have $(p^k-2p^{k-1})(p^k-2p^{k-1})$ such pairs. Since $|G_{\varphi,\psi}|=1$, this case contributes to $cq(G)$ by \[p^{k-1}(p^k-p^{k-1}) + (p^k-2p^{k-1})^2.\]
\emph{Cases} $i=1,\dots,k-1$, i.e.,
\begin{displaymath}
\varphi+\psi\equiv1\pmod{p^i} \quad\text{and}\quad\varphi+\psi\not\equiv1\pmod{p^{i+1}}.
\end{displaymath}
In this case, we can take $G_{\varphi,\psi}=\{0,p^0,\dots,p^{i-1}\}$. How many such pairs $(\varphi,\psi)$ exist? For $\varphi\equiv1\pmod p$, any solution $\psi$ to the congruence above is divisible by $p$, hence there is no such solution $\psi\in G^*$. For $\varphi\not\equiv1\pmod p$, we have precisely $p^{k-i}-p^{k-i-1}$ solutions to the conditions in $G^*$. Since $|G_{\varphi,\psi}|=i+1$, this case contributes to $cq(G)$ by \[(p^k-2p^{k-1})(p^{k-i}-p^{k-i-1})(i+1).\]
\emph{Case} $i=k$, i.e., \[\varphi+\psi=1.\]
In this case, we can take $G_{\varphi,\psi}=\{0,p^0,\dots,p^{k-1}\}$. How many such pairs $(\varphi,\psi)$ exist? Since $\psi$ is uniquely determined by $\varphi$ and neither of $\varphi,\psi$ shall be divisible by $p$, we have precisely $p^k-2p^{k-1}$ such pairs. Since $|G_{\varphi,\psi}|=k+1$, this case contributes to $cq(G)$ by \[(p^k-2p^{k-1})(k+1).\]
Summarized, the cases $i=1,\dots,k$ contribute to $cq(G)$ the total of
\begin{displaymath}
(p^k-2p^{k-1})\left(\left(
\sum_{i=1}^{k-1}(p^{k-i}-p^{k-i-1})\cdot(i+1)
\right)+(k+1)\right),
\end{displaymath}
which, after rearrangement, gives
\[(p^k-2p^{k-1})(2p^{k-1}+p^{k-2}+p^{k-3}+\cdots+p+1).\]
The total sum is then
\begin{align*}
cq(G) &= p^{2k-1}-p^{2k-2} + (p^k-2p^{k-1})((p^k-2p^{k-1}) + (2p^{k-1}+p^{k-2}+p^{k-3}+\cdots+p+1)) \\
&= p^{2k-1} - p^{2k-2} + (p^k-2p^{k-1})( p^k + p^{k-2} + p^{k-3} + \cdots + p+1 )\\
&= p^{2k}-p^{2k-1} - p^{2k-3} - \cdots - p^k - 2p^{k-1},
\end{align*}
which can be expressed as in the statement of the theorem.
\end{proof}
\begin{corollary}
For any $k\ge 1$ we have $cq(C_{2^k}) = mq(C_{2^k}) = 2^{2k-2}$.
\end{corollary}
\begin{corollary}\label{c:prime}
For any prime $p$ we have $cq(p) = mq(p) = p^2-p-1$.
\end{corollary}
Corollary \ref{c:prime} is a special case of \cite[Corollary 2]{SS} for binary quasigroups.
As a counterpart to Theorem \ref{Th:Cp}, we ask:
\begin{problem}
For a prime $p$ and $k>1$, find explicit formulas for $cq(C_p^k)$ and $mq(C_p^k)$.
\end{problem}
|
{
"redpajama_set_name": "RedPajamaArXiv"
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Amblyseius sumatrensis är en spindeldjursart som beskrevs av Ehara 2002. Amblyseius sumatrensis ingår i släktet Amblyseius och familjen Phytoseiidae. Inga underarter finns listade i Catalogue of Life.
Källor
Spindeldjur
sumatrensis
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"redpajama_set_name": "RedPajamaWikipedia"
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This is a work of fiction. Names, characters, places, and incidents either are the product of the author's imagination or are used fictitiously. Any resemblance to actual events, locales, organizations, or persons living or dead, is entirely coincidental and beyond the intent of either the author or the publisher.
The Story Plant
The Aronica-Miller Publishing Project, LLC
P.O. Box 4331
Stamford, CT 06907
Copyright © 2011 by Mark Gilleo
Jacket design by Barbara Aronica Buck
Print ISBN-13: 978-1-61188-034-2
E-book ISBN-13: 978-1-61188-035-9
Visit our website at www.thestoryplant.com
All rights reserved, which includes the right to reproduce this book or portions thereof in any form whatsoever, except as provided by US Copyright Law.
For information, address The Story Plant.
First Story Plant Printing: March 2012
Printed in The United States of America
## Acknowledgments
I would like to take this opportunity to thank more than a few people who supported me in this endeavor. First and foremost, I would like to thank my family and friends. I don't recall a single incident when anyone told me that I was crazy (even if they were thinking it).
In addition to the moral support of family and friends, in particular my wife Ivette, I would like to thank some people who took the time to read the manuscript for this novel, in its various forms, and to provide meaningful feedback.
So for my A-team of readers I would like to thank: Jim Singleton, Fabio Assmann, Michele Gates, Claire Everett, Don Gilleo and Sue Fine. At the risk of missing some others, I would also like to thank the following people for their set of eyes: Ginny Donaldson, Debbie Ingel, Mary Weber, Jim Mockus, Michelle Couret, Ivonne Couret, Ray Rosson and Paula Willson.
Finally, I would like to thank Lou Aronica for taking a chance on me and this book.
## Author's Note
_(This part is true.)_
In late 1999 a woman from Vienna, Virginia, a suburb ten miles from the White House as the crow flies, called the CIA. The woman, a fifty-something mother of three, phoned to report what she referred to as potential terrorists living across the street from her middle-class home. She went on to explain what she had been seeing in her otherwise quiet neighborhood: Strange men of seemingly Middle-Eastern descent using their cell phones in the yard. Meetings in the middle of the night with bumper-to-bumper curbside parking, expensive cars rubbing ends with vans and common Japanese imports. A constant flow of young men, some who seemed to stay for long periods of time without introducing themselves to anyone in the neighborhood. The construction of a six-foot wooden fence to hide the backyard from the street only made the property more suspicious.
Upon hearing a layperson's description of suspicious behavior, the CIA promptly dismissed the woman and her phone call. (Ironically, the woman lived less than a quarter of a mile from a CIA installation, though it was not CIA headquarters as was later reported.)
In the days and weeks following 9/11, the intelligence community in the U.S. began to learn the identities of the nineteen hijackers who had flown the planes into the World Trade Towers and the Pentagon. In the process of their investigation they discovered that two of the hijackers, one on each of the planes that hit the World Trade Towers, had listed a particular house in Vienna, Virginia as a place of residence.
The FBI and various other agencies swooped in on the unassuming neighborhood and began knocking on doors. When they reached the house of a certain mother of three, she stopped them dead in their tracks. She was purported to have said, "I called the CIA two years ago to report that terrorists were living across the street and no one did anything."
The CIA claimed to have no record of a phone call.
The news networks set up cameras and began broadcasting from the residential street. ABC, NBC, FOX. The FBI followed up with further inquiries. The woman's story was later bounced around the various post 9/11 committees and intelligence hearings on Capitol Hill. (Incidentally, after 9/11, the CIA closed its multi-story facility in the neighborhood where the terrorist reportedly lived. In 2006 the empty building was finally torn down and, as of early 2011, was being replaced with another office building).
There has been much speculation about what the government should have or could have known prior to 9/11. The answer is not simple. There have been anecdotal stories of people in Florida and elsewhere who claimed to have reported similar "terrorist" type activities by suspicious people prior to 9/11. None of these stories have been proven.
What we do know is that with the exception of the flight school instructor in Minnesota who questioned the motive of a student who was interested in flying an aircraft without learning how to land, and an unheeded warning from actor James Woods who was on a plane from Boston with several of the purported terrorists while they were doing a trial run, the woman from Vienna, Virginia was the country's best chance to prevent 9/11. To date, there has been no verification of any other pre-9/11 warnings from the general public so far in advance of that fateful day in September.
For me, there is no doubt as to the validity of the claims of the woman in Vienna.
She lived in the house where I grew up.
She is my mother.
_Mark Gilleo. October, 2011.
Washington DC._
## Chapter 1
_Present day_
It's hard to remember the appropriate prayer when you're running from an angry chef waving a meat cleaver. Hadar, sweat streaking down his dark skin, his thick black hair bouncing with every stride, recalled one verse and hoped it was enough. _Allah, you have promised to help us in our time of trouble and need..._ For the fifteen year old, the need was now. As a newly discovered thief, he wished he knew a prayer for forgiveness. Between dodging a slow moving white two-door and stumbling through a small pothole in the roughly paved alley, he realized he had never learned one.
And now was not the time to stop and ask.
Hadar kept his thin arms pumping, his loose-fitting, long-sleeve shirt flapping. He went over the plan in his head and considered where it had gone wrong. He tried to ignore the obvious. Trouble had found him long before his busboy accomplice was spotted heading out the back door of the restaurant with a patron's sports coat rolled up in a dirty tablecloth.
Hadar looked back over his shoulder. The chef was still there, still charging. The obscenities had subsided, the chef's yells replaced by the steady rhythm of feet pounding the ground. Hadar heard the meat cleaver smack the side of something metal. He didn't look back to see if it was intentional. It didn't matter. Either way, the result was the same. He was scared.
As he ran, Hadar's mind flashed back to the beginning. His recruitment to the dark side had been easy. A last minute errand to the market for his mother had been his first step down the wrong road. A dark road. With the daily bread in one hand and change from the purchase in the other, a deep voice had called out to Hadar from a dust covered Mercedes as he made his way home under a setting sun. Initially, Hadar had kept walking.
Then his instincts failed him.
When the same Mercedes pulled into the alley between Hadar and his family's three-room apartment, he froze. He tightened his hand on the brown bag holding half of his family's main course, and made the decision to walk past the car as quickly as he could. These were his alleys. If anyone tried to lay a hand on him, he would vanish like a ghost.
Or so he had thought.
As Hadar passed the car, the window of the Mercedes opened. Fighting to look away, he succumbed to curiosity and glanced into the abyss of the interior. A thousand rupees grabbed his attention and held it. The wad of cash in the driver's hand was more than his father made in a week of bloody-knuckle work as a construction expert in concrete. A thousand rupees. It was a lot of money for a well-worn, middle-aged man. For a fifteen-year-old boy it was a gold ticket on the Hell Express. Hadar took the money, listened to the instructions, and thanked the Devil. There were no rules to the agreement, save one: Don't get caught.
So far, Hadar hadn't. The chef with the meat cleaver was looking to change that. Maybe change a few anatomical features while he was at it. The chef, blood and food stains on the front of his white apron, wasn't giving up. He wasn't fading. His pace was strong, steady. The chef knew the boy had expended his youthful burst. Now it was a matter of endurance, an issue of stamina.
With every step the scale tilted in the chef's favor. The man with the cleaver had been running since his stint with the other CIA. His first pair of running shoes was a twenty dollar knock-off brand from a hole-in-the-wall vendor in Manhattan's Chinatown who offered no refunds and no returns. Three months later he ran through the treads. For a chef from the mountains of Pakistan, the running path along the Hudson was one of the main attractions near the campus of the greatest chef factory in North America: The Culinary Institute of America. Jogging was the only healthy hobby the chef ever had. He could run a 5k in less time than it took to gut and butcher a lamb.
The cigarettes Hadar had started smoking with the money from his deliveries were taking their toll. Who knew the chef would follow his busboy down the alley and see him rifling through the pockets of the liberated jacket? Who knew the mad cook would take up chase? Who knew the chef could run like a Kenyan marathoner?
Thoughts came to Hadar in a flood. He stumbled again as he turned into a slightly uphill alley. _Oh, Allah! Help us to hold fast all together to your path, even in the shaky times_. He took another quick right behind the back of the furniture shop, panting his way past two elderly men who were attaching fabric to a wooden frame. The two old men looked up moments later as six-inches of stainless steel swung by in the hand of the chef.
Another day in Islamabad, another crime.
The city was a perfect dichotomy. Next to the ancient town of Rawalpindi and its crowded hectic streets, downtown Islamabad stood as a modern example of the best and worst of the Middle East and the West. In the early 1960s, realizing it wasn't strategically prudent to have the seat of government in the same proximity as the country's economic center and largest port, the president of Pakistan decided to relocate the capital from Karachi to a swath of undeveloped land in the northern region.
With a design from a Greek urban planner by the name of Doxidas, construction on Pakistan's new capital began. The first government tenants moved in by the end of the decade and the northward surge of bureaucracy continued until the last politician completed his relocation in the early eighties.
But Doxidas, the Greek god of development, was a visionary. He knew that once you moved the government, people and additional jobs would follow. So beyond the main district of Islamabad with its glass buildings and five star hotels, Doxidas had planned for expansion. Thanks to Greek foresight, unlike many burgeoning cities, Pakistan's new capital expanded in controllable pre-planned chunks. Sector by sector downtown Islamabad merged into residential areas, tree-lined streets, and green parkland. Schools and small businesses sprouted up on secondary roads in neighborhoods where neighbors lived middle-class lives in the shadows of Western architectural influence.
These were Hadar's neighborhoods. Islamabad was Hadar's city. His father had helped build it. Hadar had explored it on foot as a youngster, trailing his father to job sites on weekends, watching concrete buildings go up wall by wall. As he got older, he exchanged his sneakers for an old bike and expanded his scope. He knew more than the roads. He knew the alleys and the footpaths, the parks and the trails into the plains where opium and marijuana grew in unmarked fields. It was hard to beat the knowledge gained from the curiosity and natural energy of a teenage boy with an itch to travel to the edges of his world, as far as his body would take him.
Perspiration dripped from Hadar, his shirt clinging to his torso as he closed in on the rendezvous point like a homing pigeon. Block by block he made his way to his personal ATM. For a second he thought about changing routes, making another lap around the neighborhood. His legs wouldn't allow it. He took one final look back and for the first time in over twenty minutes he didn't see the chef. Two more quick turns and he saw his finish line.
The man in the dark shirt standing by the long car looked up as Hadar wheezed towards him. The man quickly shooed away two other boys near his vehicle and focused on Hadar who was staggering like something between a drunk and an out-of-shape fifteen year old who had asked too much from his body.
"Did you get one?" the tall man in a dark button-up shirt and linen pants asked nonchalantly to the gasping teenager.
"Yeah," Hadar choked out.
"Is there a problem?"
Hadar shook his head.
The man condescendingly put his hands on his knees and whispered straight into Hadar's ear. "Do we have a problem?" he repeated, almost hissing. "You're out of breath."
Hadar looked into the darkness of the man's eyes and shivered. Hadar stood, reached in his pocket, and handed-over the phone. "My money," Hadar said, still straining for air.
The man opened the phone and saw that it was working. The signal strength was good. The battery was three-quarters charged. He reached into his own pocket and fished out a thousand rupees.
"Give me the phone," the chef said, slowly coming out of his jog as he rounded the corner.
The man near the car looked at the chef. "Get lost."
The chef was armed. He had just run halfway across the city. He was not going home empty-handed. He knew that reasoning with the boy may not prove fruitful. Reasoning with an adult would be easier. He approached the man in the dark shirt and looked him in the eyes. They were both in their forties. They both had solid, natural builds. They both cut meat for a living.
"He stole the phone from my restaurant, in front of my patrons. It's the third time this year. I can't have my restaurant known as a den of thieves."
"Which restaurant?"
"The Kamran."
"You serve infidels?"
"I serve everyone," the chef answered. "Except thieves," he added.
The man in the dark shirt nodded. He looked at the phone in his hand, the last rays of the day's sun bouncing off the silver casing.
"OK," the man in the dark shirt conceded. He extended his hand with the phone in his palm and the chef swiped it in one smooth motion.
"What are you going to do with the boy?"
The man in the dark shirt looked at Hadar who was regaining his breath.
"He needs to be punished," the chef added.
"He will be."
Hadar's eyes grew wide. He pulled his hands off his hips and stood straight.
The chef looked at his prey and stepped towards Hadar for a final reprimand. He extended his hand, the small antenna from the phone protruding towards the boy. "If I see you near my restaurant again, I will cut off your hand myself." The chef didn't mean it, but hoped the boy believed he did.
"Thank you," the chef said to the man in the dark shirt. They nodded at each other, eyes locked. The chef turned slowly for his long walk back to the restaurant. He would make at least one patron happy. He would walk into the restaurant and return the phone to its rightful owner. And then he would have to throw away burnt food worth ten times the value of the phone. A good reputation doesn't come cheap.
"May Allah be with you," the man in the dark shirt said to the chef.
The chef turned to respond, only to hear the sound of his neck breaking and his favorite heavy blade hitting the ground.
Hadar froze.
The man stooped quickly and picked up the large blade by its wooden handle. He looked down the quiet narrow street as he approached Hadar with the meat cleaver. "I told you, don't get caught." Hadar looked at the knife. Traces of blood ran down the blade next to dried slivers of animal fat.
Hadar stammered. He took one step to the right and the man casually blocked his path. He tried to feign a direction change, but his rubbery legs gave him away. With his last bit of strength the fifteen year old tried to go over the hood of the car.
The man caught Hadar by the hair, turned him around in mid-air, and drove him into the wall. He repeated the motion twice and on the third impact, Hadar's skull cracked.
The man left the boy on the ground, next to a pile of rubbish from an overflowing bin behind a printing shop. He casually walked over and grabbed the phone from the dead chef's hand. "You won't be needing this," he said quietly, slipping the phone into his pocket. He dragged the body of the chef next to Hadar. The young man's lungs were taking shallow, final breaths. As Hadar gasped, the man wiped the meat cleaver on the chef's apron. "Good luck on your voyage, my friends," the man in the dark shirt said in a soothing voice with spooky sincerity.
The man took one last look down the narrow street and opened the door to his car. He started the sedan and checked the rear-view mirror. No one. He put the car into drive, looked up, and watched a metal door swing open. He kept his foot on the brake and eyed the meat cleaver lying amidst a half dozen cell phones on the seat next to him.
An elderly lady with a small shopping bag exited the back of her one-story house a few yards away. She fiddled with the latch on the gate as the car moved slowly forward, small rocks crunching under the weight of its tires. As the car turned the corner out of sight, the driver heard the first scream. He remembered a prayer for forgiveness that Hadar had wished he had known. _Allah, I seek refuge in You from any evil I have committed. I confess to Your blessings upon me, and I confess to You my sins, so forgive me. Verily none forgives sins except You._
## Chapter 2
There is crazy, and then there is crazy. Clark Hayden knew the difference, and the madness in the Immigration and Customs lines at Dulles International Airport barely registered as a blip on his crazy meter.
The middle-aged man in front of him, draped in a dark raincoat, carried on a conversation with himself, fingers in his ears to drown out a crying baby in the distance. A mother with a pitchy, nasal New York accent chased her twin two-year-old boys, grasping for them as they weaved in and out of the people in line before snagging one and reeling him in like a fish with pudgy arms. A young couple plugged into a single iPod was speaking the silent language of love through music. The boy's head bounced slightly to the beat, long strands of pink hair bobbing near the end of his nose, the volume loud enough for Clark to hear every four letter word in the song's lyrics. Behind the rocking couple, a group of Chinese nationals sat on their luggage, rattling back and forth in tonal floods.
Clark took a deep breath and exhaled. He nudged his duffle bag a few inches across the floor with the toe of his gray sneaker. The businesswoman behind him, Blackberry in hand, closed the gap in the line and jammed her rolling suitcase into his Achilles. Another deep breath. Just as the Dalai Lama recommended on the meditation CD he had bought with his last wad of yen from his three-week stint in Tokyo.
Clark pinched a tattered novel between his knees and removed his baggy gray sweatshirt with the Nike swoosh stitched on the front. He was down to a plain white t-shirt with sweaty pits, the last layer of clothes between himself and the heat of the terminal. The last layer between himself and a human with any semblance of pride.
The line in Dulles International's terminal snaked through half a football field of straight-aways and hairpin turns made from temporary barriers with nylon belts stretched between poles. The crowd behind Clark grew steadily, pushing forward from their arrival gates like dough being shoved into a funnel. At Christmas, navigating through Immigration and Customs was like making a trek to the original nativity scene. Time stood still. There was no food or water. Progress was measured one step at a time. All he needed was a camel.
Clark kicked at his bag again with his foot.
In gold letters framed against a blue background, the sign above the door just beyond Immigration read "Welcome to the USA." The customs officer on duty beyond the Immigration checkpoint nodded towards Clark from his stool without the obligatory "next in line." A narcotics dog passed within sniffing distance of his jeans, the handler in a crisp blue uniform looking for any indication of prohibited goodies. The four-legged import-enforcer briefly stuck his nose on the edge of Clark's suitcase and then moved on to its next suspect. The handler followed the dog's lead. _Amazing animal_ , Clark thought. A finely tuned machine, powered by repetitious training, and Milk Bone diligence.
"Where are you coming from?" the customs officer asked with Clark's passport and paperwork in his hand.
"Japan."
"What was the purpose of your trip?"
"I was in a robotics competition."
"Robotics?"
"Yes."
"What kind of robots?"
Clark sighed quietly and opted for his elementary explanation. "We design robots to go through mazes, up flights of stairs, through a few inches of water, over balancing beams. They have to do simple tasks like moving an object from point A to point B." He looked at the customs officer, hoping the simple illustration was sufficient and there was no need to get into gyroscopic balance and infrared vision capabilities.
"Where are these robots?"
"The rest of the team is bringing them back. Well, actually the rest of the team will ship them back. They can be pretty sensitive to travel."
"How long were you gone?"
"Three weeks."
"How did you do in the competition?"
"MIT kicked our butts, but Tokyo University kicked theirs."
The customs officer looked at Clark's worn duffle bag and his single suitcase. "Do you have anything to declare?"
"A bottle of sake."
"Three weeks over the holidays and only one gift?"
"I'm on a student salary. It was either souvenirs or food. I chose the latter."
Clark was rewarded for his wit.
"Please place your bag on the counter," the customs officer responded.
Clark exhaled again, the departing rush of air sounding more like a perturbed sigh than an exercise in relaxation. The customs officer glared over his gray moustache. He dug around in Clark's black duffle bag with one hand and eyed the suitcase, threatening to open it.
Looking towards the next person in the never-ending line, the customs officer gave his automatic response without making eye contact. "You're free to go. Welcome home. Happy holidays."
Clark took his passport and pulled his bag off the long, shiny aluminum table. He walked through the final smoked-glass barrier, happy to be back in the land of forty-plus-inch waistlines.
The automatic exit doors led to a sixty degree drop in temperature from the warmth of the terminal with its teeming bodies. Clark scanned the horizon and focused on the pink sky and the last remnants of the day's sun as it dipped behind the mountains in the distance beyond Leesburg. A nipple-tightening gust of wind rippled Clark's shirt, and a pair of early twenty blondes eyed him as he scrambled to pull his sweatshirt over his dark brown wavy hair. His naturally athletic build concealed, the ladies smiled as they piled into a waiting minivan. Clark adjusted his glasses, an old pair with frames that needed updating, and grinned through the side window of the vehicle as he danced his way across two lanes of traffic. A light layer of post-snow slop concealed the lines in the access road that encircled the airport. He jumped to the curb and yanked his suitcase onto the sidewalk. With his free hand he slapped the trunk of a parked taxi. A minute later he slid into the back seat.
"Where to?"
"Arlington. Between Clarendon and Pentagon City, just off Washington Boulevard."
The cabbie nodded, eyeing the rear-view mirror.
In a half hour Clark would be home, back into the real madness. He was coming home three days early. And it was going to cost forty-five dollars to the taxi driver from Ghana to surprise everyone.
Bing Crosby caroled Clark through the closed door with his dream for a White Christmas, as good as any prediction on Old Man Winter's plan for the D.C. area. The Potomac, Chesapeake, Appalachians, and rolling farmland all converged on metropolitan Washington to make weather gumbo. The precipitation depended on the day's ingredients and how long they were in the atmospheric pot. The Nation's Capital could spit out a minus-five Christmas Eve or a sixty-degree New Years Day. One just never knew. This year the meteorologists were calling for a brutal winter, and so far the forecast was right on target.
Clark pulled the storm door and the weak hinge on the aluminum frame held for a moment before releasing its load, the metal banging into his suitcase. He pushed the interior wooden door open with his hip and the small Christmas festivities in the living room ground to a sudden halt.
Maria Hayden, dressed in green pants and a bright red sweater, heard the front door open as she was returning a ladle to a bowl of fruit punch. She looked across the room, past her holiday guests and overdone decorations, and her knees buckled. She grabbed for the edge of the counter and swiped an unattended glass of eggnog onto the floor. She composed herself as Clark smiled, pulling his suitcase into the room and shutting the door behind him. Unable to speak, tears welling in her eyes, eggnog splattered halfway up her elf pants, Maria stomped her way across the small living room floor and hugged her son until he quietly surrendered.
"Mom, you're crushing me," Clark said like a squeeze-toy running out of air.
For twenty-five years, Maria Hayden had held the record as the oldest woman to give birth in Fairfax Hospital through natural conception and delivery. Sure, there were a dozen older women on a smorgasbord of fertility meds who had given birth since, but Maria Hayden was different. She had avoided the on-ramp to menopause and was in her late-forties when her purported infertile eggs and her husband's dysfunctional sperm decided they didn't appreciate their respective titles. At forty-nine, Maria Hayden gave birth to her first and only child. Her son would be twenty-six in March, but she easily passed for his grandmother. When she was a young mother it had upset her, snide comments made from women less than half her age while they pushed their strollers through the park and wedged their kids into the grocery carts at the supermarket. Now, Maria didn't give a damn about her age. She had bigger concerns.
After a moment of smiles and tears in the middle of the annual Hayden Christmas party, Clark's mother rubbed his cheeks and ruffled his hair. She turned him around, patted his belly, and checked his weight with motherly eyes.
"My son," she finally said, first to Clark, and then to the room as if she was introducing a newborn to the world. Everyone smiled, Bing Crosby moved on to "I'll Be Home for Christmas," and the warmth of the room engulfed Clark like a comfortable blanket.
Clark made the rounds, first with relatives, a number countable on the fingers of one hand. His mother's older sister, Aunt Betty, sporting a new walker, wanted to hear all about "China" and the food. Clark smiled, his teeth exposed, his disappointment hidden, and told her the egg-rolls in Beijing were superb. Someday he would find out.
Clark's slight-of-build second cousin, Eugene, a retiring bald federal employee making a hundred grand a year doing nothing in the truest sense of the word, gave Clark a quick "Welcome back," and then vanished into the oversized cushions on the sofa. Clark shrugged his shoulders slightly, grateful for an abrupt end to the most uncomfortable annual conversation in the mid-Atlantic.
Clark approached his neighbors, a piece of ham hanging off the edge of his thick paper plate. An elderly man and a young woman with a daughter in her lap sat face-to-face in old wooden dining chairs. As Clark approached, the woman, dressed in a traditional Muslim headdress with Christmas colors, looked up through her thick black-framed glasses.
"I like your hijab," Clark said. "Green and red, very appropriate."
"Just thought I would show my festive side for my Christian neighbors. I had to make it myself."
"It's great."
Ariana looked at Clark and the dark circles under his eyes. "It's good to see you. Your mother didn't stop talking about you and your trip," Ariana said, her olive skin radiating in the light from the Christmas candle on the side table.
"Your daughter has gotten bigger."
"As has her vocabulary. She is starting to talk up a storm."
Clark bent his knees and looked the young girl in the eye. "Hi Liana."
Liana, big dark eyes resting over perfectly placed dimples, nodded once and then buried her face in her mother's chest.
Ariana smiled. "She's acting shy, but trust me, once you get her going she never stops."
"Thank you for keeping an eye on my mother, Ariana. I appreciate it."
"Don't mention it. I just stopped by to say hi, to see how she was doing."
Clark looked her in the eyes, through her thick-framed glasses. "I know it's not that easy."
"Nor is it that bad."
Clark smiled to avoid agreeing. "I don't know if you have heard or not, but I have decided to move back home. I should be able to take better care of her in a week or two."
"You're moving home?"
"I don't really have a choice. Aunt Betty is knocking on eighty and she can't drive until she gets hip replacement surgery. Her cousin Eugene is no spring chicken and lives in Annapolis. It's just easier for me to move home. I have one more class to take and a thesis to write, but I can attend the class remotely — _interactive education_ they are calling it at Virginia Tech. I'll have to drive down to Blacksburg occasionally but it won't be that bad. My professors understand."
"As long as you finish school. That's the most important thing you can do right now," Ariana said.
"I'll finish. I've come too far to quit now."
"I hope so."
Clark paused for a moment, watched his mother buzz around the room, and then changed the subject. "Where's your husband?"
"Working, as usual. He said he might stop by if he gets home in time. You know how he is. He's not much for crowds."
Clark looked around at the eight people in the room. "He should be fine then. There isn't much of a crowd here."
Ariana laughed quietly and her two-year-old daughter followed suit with a muffled giggle from the depths of her mother's sweater.
"What was that? Did you say something, Liana?" Clark asked, shaking the girl's foot, her face still hidden.
Liana looked up at Clark and proudly stated. "I like airplanes."
"You do?"
"Yes, we saw them at the airport," Liana said, eyes wide.
"Did you know we have airplanes in the basement?"
The young girl smiled coyly and then turned away again.
"Leave the girl alone and have a drink," Mr. Stanley interrupted, his raspy voice cutting the air. He reached out his arthritic hand and Clark grabbed a seat, the last seed from the Hayden family tree landing in a lopsided wicker rocking chair next to his favorite neighbor.
An octogenarian with a full head of silver hair and World War II stories, Mr. Stanley had been spinning yarns since Clark was in grade school. A million-in-one shot had brought the two together; a misdirected soccer ball fired two houses away that somehow managed its way through a small bathroom window. When Clark knocked on Mr. Stanley's door offering to pay for the window and asking for his ball back, an unlikely friendship had been born.
When Clark became old enough to pull the cord on the mower and push it around the yard with a reasonable amount of predictability, he started earning twenty bucks a week during the warm season. By the time he hit high school, he had added raking leaves, cleaning gutters, and shoveling Mr. Stanley's driveway to his list of money-making duties. His pay increased over the years and when Clark turned sixteen Mr. Stanley began shoving pre-screened Playboy magazines into brown bags as supplemental income.
"How was Japan?" Mr. Stanley asked. "Were the people polite to you?"
"Very polite. Distant, but very polite."
"Don't forget what they did to our boys on Bataan. Treated them worse than rabid dogs."
"I know all about it."
"Don't forget; that's all I'm saying."
"I won't forget, but I'm also trying to look towards the future."
"You gotta keep one eye on the past and one eye on where you're going."
Clark sipped off his eggnog. His brown wavy hair was heading out of control. Two hours waiting on the tarmac at Narita and another thirteen hours in the plane had sucked a few years from his appearance. "I also want to thank you for your help looking after my mother."
"You're welcome," Mr. Stanley said, waving his hand in the air. "We all pitched in."
"You took her to the grocery store."
"I was going anyway. Have to take that Cadillac for a spin just to keep her breathing." Mr. Stanley paused and took a gulp of eggnog from his paper Christmas cup. "Your mother gets along fine, as long as she doesn't forget to take her pills."
Ariana, Mr. Stanley, and Clark fell into a moment of quiet understanding and then Clark spoke. "There is a lot of stress worrying about someone all the time."
"I understand," Mr. Stanley responded, his blue eyes alive.
"We understand," Ariana added.
"I probably shouldn't have even gone to the robotics competition. But we put a year of work into those. And the World University Robotics Competition is a good thing to have on your resume."
"We understand," Mr. Stanley said again. "You don't have to convince us. You're doing the best you can. You play with the cards you are dealt."
Ariana changed the subject. "How did you do?"
"We came in sixth. MIT and Cal Tech were the highest placed U.S. universities. Oslo University came in third. Tokyo University took the title. We made a lot of friends. We were dormed on the same floor as the team from MIT. We got to be pretty close, outside of the competition."
Mr. Stanley spoke. "Always good to make friends. Especially smart ones. You know what they say..."
Clark rolled his eyes as his neighbor and friend prepared to pontificate. "No, what do they say?"
"If you're the smartest person in your group of friends, you need to meet new people."
"There's truth to that," Ariana said.
Clark smiled and then turned his thoughts to his mother. He looked at Liana who was still clutching her mother's bosom.
Mr. Stanley broke the silence. "We're glad you made it home in one piece. And if you need any help while you are making the move back from Blacksburg, you know where to reach me."
Clark knew his mother was more effort than Mr. Stanley and Ariana were admitting, even on a good day. He had been keeping an eye on her since he was in grade school. But there were things to be thankful for. His mother still knew where the two bathrooms in the house were located. She still managed to make it down the small flight of stairs to handle the washer and dryer at the back of the garage. And on any given day she could appear completely normal to an outside observer. It was the sudden emotional swings, the nonsensical outbursts, the moments of blankness that worried Clark and scared the shit out of everyone else.
"Once I get back into the house, things should improve."
Mr. Stanley and Ariana nodded but said nothing. They knew better.
Clark watched his mother mill about the kitchen acting as normal as any seventy-four year old woman with a penchant for baking. As if to prove she had heard the conversation, Maria Hayden stepped towards the dining table with impeccable timing and presented a massive tray of baked goods, a pile of gluttony begging for a glutton. The smell of the already baked Christmas ham was immediately overpowered by cream puffs, hazelnut cookies, and a warm pecan pie.
"Good thing I didn't eat the ham," Mr. Stanley said. "You can always count on your mother and her desserts," he added, drawing air in through his nose.
Clark picked up his glass of eggnog and took a sip. There was a faint hint of booze that wasn't there a minute ago. "It has a little kick to it. Did you make the eggnog?"
"I'm appalled at the suggestion," Mr. Stanley said, smiling.
Clark distracted Mr. Stanley with a pat on the back and a look over his shoulder. With his free hand he lifted the shiny container from his neighbor's hanging jacket pocket.
"Look familiar?" Clark asked.
Mr. Stanley swiped the flask out of Clark's hand and looked around suspiciously. "Of course it does. You don't carry something in your pocket for three decades without knowing what it looks like." He twisted off the top and took a slug.
Clark smelled his eggnog and took another sip. "Probably improved the taste."
"That's the Christmas spirit."
Ariana's husband arrived with a quiet knock and Clark moved to the door. A familiar face smiled under the porch light and Clark pushed open the storm door.
"Nazim. Good to see you."
Nazim flashed a brief smile and gave Clark a disingenuous hug. His waiflike build was barely large enough to reach around Clark's shoulders. "I thought you were coming back next week."
"I was, but I decided to come home early to surprise my mother."
"How was your trip? How was the competition?"
"Great. Really great. We didn't win, but we made a good showing for Virginia Tech."
"Good. Good. You know, there is no substitute for foreign travel to learn more about yourself and your own country."
"You're right. Please come in."
"Only for a minute."
Nazim quickly greeted Maria Hayden as Clark sat back down. Mr. Stanley gave Clark a nod in the direction of Nazim, covertly making a face of disdain.
Nazim moved towards his wife and whispered in her ear. "It's time to leave."
Ariana looked up at her husband. "Just a few more minutes. Let me finish the dessert. I don't want to be rude."
Nazim locked eyes with his wife for a second and then put his hand on his daughter's head. "How is my favorite girl this evening?" Liana looked up at her father and reached for his dark beard. Nazim grabbed his daughter and put her on his forearm.
Maria Hayden came out of the bedroom with her digital camera flashing. She took one of Nazim and Liana, father holding his daughter like a prized possession. She snapped a picture of Clark and Mr. Stanley sitting shoulder-to-shoulder.
"Let's have a group picture," Maria said. "Everyone stand in front of the Christmas tree in the corner."
The people in the room took little steps across the tiny living room, dancing around the coffee table, finding their place in front of the artificial tree with tacky reflective pink tinsel.
"Smile," Maria said.
Maria took the picture. "Now I want to be in one," she said.
Ariana stepped forward and changed places with her. She looked through the viewfinder and took a step back. With everyone in focus, Ariana pushed the button and heard the fake shutter sound of the digital camera impersonating its manual predecessor. Ariana shared the picture with Maria and put the digital camera on the table. "We have to get going. Thank you for inviting us."
"Thank you Ariana. Thank you for everything."
"You're welcome."
Maria Hayden pulled her neighbor toward the corner of the room. "Can I ask a question?"
"Sure."
"Can you see my hair?"
"Sure I can," Ariana answered. "It looks wonderful this evening."
"No, I mean, I feel it growing out of my scalp," she answered in a rising voice. As eyes turned towards her, she scratched at her head and repeated herself several times, her voice reaching a crescendo that overpowered the Christmas CD. And just as suddenly as her voice had risen, it dropped to a whisper before she added, "I hope no one notices."
"No, Maria, you look fine," Ariana answered, smiling to the room and taking Maria by the elbow. "Now, let's see if we can get you your medicine before I leave."
Clark watched as his mother performed her ever-frequent ritual of a mental meltdown. The Christmas party was officially over.
There is crazy, and then there is crazy.
## Chapter 3
Ariana turned on the nightlight and closed the door to her daughter's room. She walked down the carpeted hall towards the light stretching out from the plastic chandelier over the dining room table. Her husband's chair was empty and she quietly called out his name. No response. As Ariana turned the corner to the kitchen and reached for the knob on the cabinet over the counter, eight hundred pages of advertising crashed into her rib cage, sucking the wind from her lungs. As his wife doubled over, Nazim raised the thick Yellow Book with both hands and hit her on her back, driving her body to the floor.
"Don't you ever disobey me in front of others again."
Ariana coughed. There was no blood. This time. She tried to speak but her lips only quivered. Her thick-framed glasses rested on the floor, out of reach. Her brain fought to make sense of what happened, what had set her husband off. It could have been anything. But every curse had its blessing, and for Ariana the blessing was the fact that Nazim didn't hit her in front of Liana. A blessing that the child didn't see her mother being punched. The reason was simple. Nazim was afraid of his daughter. Afraid of what she could say now that she could speak.
The curse was that Ariana never knew when she had crossed the line. She never knew when the next blow was coming. She merely had to wait until they were alone to learn her fate for past indiscretions.
Ariana gasped slowly for air. She didn't cry. The pain she felt in her side wasn't bad enough to give her husband the satisfaction.
"When I say it is time to leave, it is time to leave. There is no room for negotiation in this marriage."
Ariana panted as her mind flashed back to the Christmas party. She immediately realized her faux pas. "I didn't want to be rude to Maria. She spent days making dessert. She is old. Do we not respect our elders anymore?"
Nazim pushed his wife onto the floor with his knee, a reaction Ariana fully expected. "You are my wife. This is about you and me. Our neighbor has nothing to do with it." Nazim looked down at Ariana sprawled on the linoleum and spit on her with more mock than saliva.
"Maria is my friend."
"Well, her son is coming home and she doesn't need you."
Nazim dropped the Yellow Book on the counter with a thud and went to the basement. Ariana gathered herself, pushing her body onto all fours and then pulling herself up by the front of the oven. She looked at the Yellow Book and her blood boiled. It was like getting hit by a cinderblock with soft edges. When it hit flush, it left very little bruising. As her husband intended. For a man of slight build, Nazim could generate power when a beating was needed.
Ariana took inventory of herself, one hand propping herself up on the counter. She had been beaten worse. Far worse. By other men before she met her husband. Her eyes moved beyond the Yellow Pages and settled on the knife set on the counter, the shiny German steel resting in its wooden block holder. She grabbed the fillet knife, caressed the blade with her eyes, and then pushed the thought from her mind.
Her husband called her from the basement and she snapped out of her momentary daze. "Coming," she answered, putting the knife back in its designated slot in the wood. She knew what was coming next. It was always the same. A physical assault followed by a sexual one. She reached up her skirt and removed her panties. There was no sense in having another pair ripped, even if robbing Nazim of the joy would cost her a punch or two.
Christmas, _the season of giving_ , she thought as she made her way down the stairs into the chilly basement.
Maria Hayden took a shower in the main bathroom on the first floor. The aqua-green porcelain of the tub and toilet were leftovers from the fifties that easily gave the bathroom the title of ugliest room in the house. Maybe the ugliest room on the block. The bathroom was wedged between the front door and a small room that had once been Clark's bedroom. His room was now a sewing vestibule. A piece of furniture his mother referred to as a hobby table combined with a chair and a bookcase to occupy most of the floor space. Oil paintings of flowers and mountain scenery now hung on the wall where Redskins posters and a Sports Illustrated Swimsuit Calendar once held prime real estate.
Clark was sitting at the dining table near the entrance to the kitchen when he realized his mother was taking a shower in the old bathroom. He cocked his head to the side and closed his eyes for a moment. He got up from the table and walked into his mother's bedroom, past the large post bed, and clicked the light on the master bathroom switch. It was exactly as he thought. Exactly as he had feared.
The master bathroom had been a labor of love. His father, with the help of two fellow blue-collared acquaintances, had worked on the addition for nearly a year. The bathroom and the new walk-in closet protruding from the back of the house were two exceptions to the original rectangular shape of the house. And Charles Hayden had died a month after laying the last tile.
Clark looked around at the new fixtures and the unused tub with whirlpool jets. A tear welled up in his eye, but didn't fall, the salty fluid refusing to break the edge of his eyelid.
A tube of toothpaste was on the sink and Clark picked it up. The tube was hard, almost petrified. There were remnants of stubble in his father's razor on the sink top under the mirror. His father's eyeglasses were on the small towel shelf, his deodorant lying on its side in the medicine cabinet. Clark wanted to cry, to enjoy a full-fledged wheezing tear-feast, but didn't. It was a promise he had made to his father to be strong. It was a battle he had fought daily for over a year before conquering his emotions.
He thought about putting the glasses, the deodorant, and the razor back under the sink where he had put them a dozen times before. _What the hell?_ he thought, _it's Christmas._ He turned out the lights and returned to the living room.
The cordless phone on the nightstand rang twice before the sleeping hand reached over and knocked it from its perch. Nazim mumbled. Ariana answered with equal gibberish. The phone continued to ring. Beneath the edge of the dust skirt on the queen mattress, the black phone with the digital display rang one last time before the searching hand found its mark.
"Hello," the still sleeping voice said.
"Assalamu alaikum," rang an unforgettable voice from the past.
"Wa alaikum assalam," the now waking voice said on automatic pilot. Arabic was neither participant's mother tongue, but it was the language of Islam, and that was enough.
"It is time."
The mattress shook as the dark form of a body shot into the upright position on the bed.
"How many?"
"Four."
"When?"
"Be ready. Assalamu alaikum wa rahmatullahi wa barakatuh."
The phone went dead.
A minute of silence passed in the darkness. The mattress shook again and Nazim stood and walked to the bathroom to relieve himself.
"Do you need anything?" Ariana asked dutifully. It was the last offer she would ever make to her husband.
"No," he answered as the bathroom door shut. Five minutes later Nazim was asleep.
Ariana went to the kitchen and quietly opened the pantry door. She stood on the small stepping stool and rummaged through the top shelf, gently pushing aside jars of flour, sugar, and wheat germ. Her husband hadn't cooked or cleaned in his life, certainly not in their lives together. She knew the only place to hide something was right under his nose, in a place he would never scratch.
The small glass jar with the airtight seal read "cleaner" on the side, written with magic marker on a strip of masking tape.
Ariana returned to the bedroom and sat on the bed with her back to her husband. She slipped on a pair of green rubber gloves from the kitchen and listened to her husband fall into his rhythmic snore. She watched him sleep and waited for his mouth to droop open, waiting for the snoring that had kept her awake year after year. Then, in minute doses, she sprinkled in her magic potion. She waited, repeated the task, and waited again. It wouldn't be long.
Nazim woke gasping. He reached for his wife and grasped at the empty comforter. He turned the switch on the light next to the bed, but the cord was unplugged. His throat burned. His eyes watered. He strained for oxygen.
"Help," he said, sucking in air as he staggered out of bed.
Ariana turned on a small light on the dresser. She was sitting in a plain wooden chair in a t-shirt and pink cotton panties. Smiling. "There is no help."
Nazim grabbed at his throat.
"What have you done?"
"It was done long ago. Long before we met. You were merely a pawn," she said without emotion.
Ariana watched as Nazim fought for air, for the strength to summon rage. "You bitch!" he gurgled. He grabbed the clock radio and threw it at Ariana. His wife ducked as the black plastic box smashed into the wall.
Ariana stood and stepped towards him. Nazim staggered forward and reached out with both hands, his death rage focused on his wife's neck. He never got close. Ariana hit her husband in the nose with the palm of her hand and felt the bone crush. Nazim's head snapped back and Ariana hit him in the solar plexus with a reverse punch polished by years of training. Nazim's lungs emptied and he began making a deflating noise. He stooped over and Ariana slapped both of her husband's ears with cupped hands, bursting one of his eardrums. Now in total control, control she had fought to exercise for far too long, Ariana snapped her hips and drove a back kick into Nazim's neck. As her husband wheezed, she finished him off with a front kick to the crotch.
By the time Nazim's hands reached his groin, he was dead.
## Chapter 4
The chill from the top of the closed toilet lid gave Ariana goosebumps as she looked down on her husband's slim body in the bathtub. Her ex-husband, she thought to herself with satisfaction. Nazim had been dead ten hours and rigor mortis had already set in around the jaws and in the torso. His eyes were closed and his body seemed to take on the color of the off-white bathroom tiles like a chameleon blending into its surroundings. Gravity played its law-of-physics card and blood pooled on Nazim's backside, the lowest part of his body. His arms were folded over his chest. A faint purple bruise surrounded his Adam's apple. His nose was obviously broken, the bone jagging left, the nostrils laced with traces of dried blood.
But it was a clean death. As clean as she could make it and still keep it up close and personal. She had wiped her husband's face with wet tissues dipped in bleach and removed most of the blood while it was still fresh. The tissues went down the toilet. She wiped two additional drops of blood from the warped hardwood floors in the bedroom and flushed the incriminating evidence. She knew that there would be traces of blood on the micro-level, but she didn't care. A few drops of blood in any home could be explained by a bloody nose, a cut on the finger, shaving with shaky hands. Besides, by the time the authorities started looking for her, it would be too late.
The only real crime scene was the body. As clean as the bedroom was, if the police arrived for reasons she could not imagine, she would have some explaining to do. She checked her watch. The body would be gone soon enough. Long before the temperature in the corpse began to rise again from decomposition. Long before the smell from rotting flesh alerted the neighbors.
She wished she had more time. If she had, she would take care of the body where it was. The flesh, bones, and hair were no match for a bath of hydrochloric acid. But getting her hands on a few gallons of lethal acid on Christmas Day was going to take more imagination than just believing in Santa Claus. Time was a gruesome luxury she didn't have.
She thought about the phone call she had received. She began to wonder long ago if contact with her would ever be made. The first few years of her assignment had passed slowly. Then came 9/11, concealed excitement, and more years of silence. She began to think she had been forgotten forever. But, as she found, it is hard to rest quietly when you are trained to kill and have an I.Q. of 170.
In the hours since her husband's death, Ariana's natural state of heightened awareness had switched into overdrive. There were going to be lies to tell. Lies bigger than the one she had been living. After all, she was a housewife and a mother. These were truths. She was also a killer. Her personal tale up until now had only required her to omit one part of the story. From this moment on, each lie would have the potential power to bring her down, to end her service to a higher power. She reminded herself to keep it all straight. One discovered lie and one suspicious person could start a bloodbath.
"Mom," Liana called out from the other side of the door.
"Just a minute."
"I need to go potty."
"Use the one in the hall. I'll be there in a minute."
Ariana pulled the shower curtain closed. Her daughter's voice reminded her that she had a bigger decision to make than what to do with her husband's body.
Clark stomped his heel into his boot, tied the laces, and stepped outside onto the porch. A thin layer of ice covered the cracked concrete slab and Clark grabbed the iced-over railing as he found his balance. "Christ," he said, settling himself and checking the content of his jacket pocket.
The branches on a pair of dogwoods in the front yard drooped from the weight of their shimmering coat. Clark maneuvered down the small flight of stairs with his Vibram-soled Timberlands and shuffled his feet on the uneven walk that led to the small driveway. He stopped at his mother's car and listened to the silence of the middle-class neighborhood on Christmas morning.
He looked up the street, the sun reflecting off the ice-covered cars stuffed into driveways barely long enough to accommodate two vehicles parked end-to-end. The original middle-class neighborhood had been comprised of classic tract housing built in the fifties. Square box homes with nearly identical floor-plans produced from cookie-cutter molds. At the peak of the post World War II expansion, one-story ramblers with unfinished basements had stretched for a mile in any direction. Clark's parents had lived on Dorchester Lane long enough to remember when the original sod was laid.
Clark looked down the street from his mother's slice of paradise. He could see the changes. The last ten years had seen an explosion in the geometry of the residences. Everyone with money, and half those without, began slapping up additions with such fervor that anyone with a hammer could open a booming home improvement business. The neighborhood began to shift from architectural boredom to eclectic hell. Bathrooms jutted from the front of houses. Garages, most now stuffed with the collected crap of life, were constructed as close to a neighbor's property line as regulations would allow. Bay windows opened next to homebuilt greenhouses, next to two-story decks. Carports and master suites melted together. The neighborhood looked as if a drunken Lego team had won a competition to ruin the original blue-collar, tract-housing charm.
Gone were the kids who used to play in the streets. Classic games like team hide-and-seek and murder-the-man-with-the-ball had been relinquished to the trash heap of neighborhood activities. If Dorchester Lane and its neighboring houses were harboring a secret stash of children, they were hiding in the basements, plugged into their video games.
At the incline at the end of the driveway, Clark pushed forward with his right foot and let gravity carry him to the edge of the street. He slipped and slid his way past his neighbor on the right and turned towards a slightly uphill battle on a sloping icy driveway.
The thermometer in the window faced outward and Clark squinted from beneath the hood on his blue jacket. The red line bumped the bottom of the twenty-six degree mark. With his bare knuckle, Clark knocked on the window of the side door at the top of a small metal staircase. Icicles hung from the gutter above, the points of the inverted spikes threatening downward.
Arthritic feet hobbled to the door after the third set of knocks. The curtain in the small window moved and an eye peered out from the crack. Mr. Stanley fumbled for the small lock on the storm door and yanked the handle.
"Good morning, Clark."
"Good morning."
"Did you come to spread sand and salt on the driveway?"
Clark's eyebrows jumped. "Actually, no. But if you need me to, I will."
"Not necessary. I was just testing you. There's a kid a few blocks over who usually stops by when it snows. Charges forty bucks to shovel."
"Not bad cash."
"Inflation since the last time you worked for me."
"Nothing gets cheaper."
Mr. Stanley shivered and hustled Clark inside. Clark pushed the door closed behind him and the blitz of senior citizen scent assaulted his nose. Clark took a deep breath. It was just like tearing a Band-Aid off a hairy arm. Some things you have to do quickly.
Mr. Stanley's was a combination old folk's home and bachelor pad. Every molecule was frozen in motion, each electron covered in its own layer of dust. The air was warm, the lack of movement stifling. The smell was between musty and old, sweet and sour. It wasn't anything a year with the windows open wouldn't cure.
"Here's something I picked up for you in Tokyo," Clark said, pulling a small bottle from his jacket pocket. "I meant to give it to you last night, but things kind of ended suddenly."
Mr. Stanley looked at the bottle and held it at arm's length to read the label. Particles of sparkling bits moved about in the liquid at the bottom of the bottle. Mr. Stanley turned and held the small glass container to the light.
"What is it?"
"It's Japanese sake," Clark said, waiting to deliver the punch-line. "It has flakes of pure gold in it."
"Gold?"
"Pure gold."
"Why in the hell would I want to drink gold?"
A damn good question, Clark thought. A question he had been wondering since he bought it. Clark appealed to Mr. Stanley's more practical nature. "Just think of it as something extra to go with the alcohol."
"Seems like a perfectly good waste of gold to me."
"I guess. But the Japanese aren't the first to consume it. Different civilizations have been eating it for years. As a metal, gold is somewhat unique. It is biologically benign. Passes right through the digestive system without any adverse effects."
"You know your metals."
"My father's hobby."
"Do you want to try it out?" Mr. Stanley asked, mocking as if he twisted the cap.
"It's eleven o'clock Christmas morning."
"But it's Christmas night somewhere. What else do you have to do?"
"I'll take a rain-check."
"All right, but I can't guarantee there will be any left by the time you make up your mind. Except for the gold flakes in my daily constitution."
"I'll risk it."
"How's preparation for the big move?"
"I haven't even been home for twenty-four hours. There are some things to do. First I need to see what is going on down at 'the hole.' Blacksburg is a four hour drive and all my roommates are either still in Japan or back visiting their families."
"The hole?"
"That's what some of my roommates' girlfriends have been calling the house we live in."
"I see."
"With my roommates there's no guarantee that the house is even still there. But the rent is paid through January, so I will be moving back in over the next couple of weeks. Whatever fits in the back of the Civic. One load at a time."
"Son, you know you probably don't have to move back home. Your mother should be fine, if you wanted stay in Blacksburg for a few more months."
"I probably would if Aunt Betty hadn't broken her hip. My mom is not getting any better. Or younger. It is time. I see her slipping."
"Ahh. We're all slipping a little, and someday, even you will start to slip."
Clark forced a smile and nodded towards the sake bottle. "Merry Christmas, Mr. Stanley."
"Merry Christmas, son," Mr. Stanley responded, feeling as if he had crossed some unspeakable boundary. He changed the subject. "Could you help me with something?"
"Of course."
Clark followed Mr. Stanley to a small bedroom down the hall that had long ago been converted to a study. There were piles of books in the corner. Enough paper was scattered around the room to take out an acre of timber. Crossword puzzles covered the desk. A magnifying glass rested on yesterday's edition of The New York Times. A high-watt sunlamp was clamped to the edge of a bookcase behind a leather chair. A painting rested on an easel in the corner, the portrait of a woman almost complete.
"Who is the woman?" Clark asked.
"It's my wife when she was young."
"I don't think I've ever seen a picture of her at that age."
"Certainly not an oil portrait. There is only one in existence as far as I know, and you are looking at it."
"Nice."
Mr. Stanley pulled a box from the top of the small bookcase under the window.
"Can you help me change the ceiling light in the hall? I broke the cover over a month ago."
"How did you do that?"
"I was cleaning and put the end of the broom right through it. Forty years that thing has been up there. Took me forever to find a replacement. Most of them are made of plastic these days."
Clark pulled the light cover from the small cardboard box. "No one's coming by to help you around the house?"
"I don't need any help."
"What about your nephew?"
"He stops by once in a while, but hell, he is more helpless than I am. Fifty years old and he can barely take care of himself."
"You could have asked Nazim."
"I don't trust that guy. Never did."
"What's not to like? He's quiet, helpful, and keeps his property clean. I mean, we got people parking their cars on cinderblocks in this neighborhood, in case you haven't noticed. This guy actually plants grass in his yard every year."
"The neighborhood is not what it used to be."
"Yeah, but I don't think a neighbor would turn you down if you asked for help."
"I don't want that guy in my house."
Clark shrugged his shoulders then removed a pile of books from the seat of a wooden chair and dragged it into the hallway. He removed the remnants of the previous light shade, carefully catching the few inches of glass near the screw in the center. He put the new cover in place and tightened the screw.
"What are your plans for the day?" he said, stepping off the chair.
"Going to my brother's in McLean."
"That's good. Family is good."
"Family is good. The in-laws that come with that family can be hell."
"I wouldn't know."
"You want something to drink?"
Clark didn't, but tried to be polite. "Water would be great."
"Water? Hell. You will join me in a coffee."
"Then a coffee would be great."
Clark followed Mr. Stanley as he walked backed through the house to the kitchen. Clark turned at the dining nook and looked at the backyard. Old trees and rusted chain-link fences marked the territory of his neighbors' properties. Mr. Stanley's old metal shed leaned to the right, remnants of the last snow following the slant in the roof.
"I see the neighbor finished his construction."
"Coleman's Castle?" Mr. Stanley asked from the kitchen.
"Is that what they're calling it?"
"No, that's what I'm calling it."
"Well, it's one hell of an addition."
"They're calling those McMansions."
"I know," Clark said.
"Everyone on the street has some kind of addition, but Mr. Coleman's house takes the cake. Two and a half stories, including the loft. A nice big chimney with huge glass windows. It looks like an addition from Better Homes and Gardens. Something you would see in a ski chalet. Except his neighbors in the back are the only ones who get to enjoy the view. And the only thing Coleman gets to look at is the ass-side of our property. From the front of his street, the house still looks like a rambler. A one-story rambler with a big penis sticking up out of the back. The damn thing looks ridiculous."
"He has the right to build on his house."
Mr. Stanley continued his rant. "In fifteen years there will be no middle-class neighborhoods left within a hundred miles of D.C. It's going to be either a million dollar mansion or a flop house where you chase the roaches away long enough to fall asleep. You know, two blocks away four houses were just bought and bulldozed for the lot. Some software CEO. The real estate market may have crashed, but not for the rich."
"People don't have to sell."
Mr. Stanley chose not to hear Clark.
"Rich people are funny. Even the rich like to claim they are middle class. Families with three kids, five cars, seven cell phones and ten televisions. Ask any of them and they will tell you they are middle-class." Mr. Stanley paused to see if Clark was listening, then continued. "I'll tell you right now, if you can't see in at least one of your neighbor's windows, or can't hear your neighbor giving the goods to his wife through the walls, you are not middle-class. If you own two homes, and I don't care if one of them is a shack in the mountains, you are not middle-class. Middle-class neighborhoods are defined street by street and block by block. People who live month-to-month and have to save to rent a house for a week at the beach for their summer vacation."
"What about the preacher at your church? Rumor has been for years that he has a second house."
"That man is going to hell. You can't be a man of the cloth, claim the souls of your parishioners to save, preach against the seven deadly sins, and have a beach house. It just doesn't work that way. If you have enough money to help yourself to a beach house, you have enough money to help those in need. Bastard."
"I don't know about calling a man of the cloth 'a bastard.'"
"Excuse me. I meant stingy, greedy, _reverend_ bastard."
Clark tried to end the bitching that old men seem to perfect over the course of their lives. He looked at the small bird that landed on a bird feeder in the middle of the yard. A sea of seeds were scattered on the ground under the feeder.
"Been bird watching?"
"I have a pair of big blue jays visiting the yard that I haven't seen before. They are just remarkable."
"Shouldn't they have migrated?"
Mr. Stanley looked genuinely offended. "Smart ass."
Clark looked up at Mr. Coleman in the distance as the neighbor entered the tower of his castle and sat down. A few seconds later Coleman tipped a bag of junk food upward and the orange foil of the bag covered his face.
Clark stared out the back window as Mr. Stanley delivered the coffee. They reminisced about the neighborhood and the time Clark hit a yellow jacket nest buried in the back yard with the lawn mower. Twenty-two stings and a trip to the emergency room in a Cadillac going ninety.
They heard a car door shut and both men looked across Nazim and Ariana's back yard. Ariana was standing at the end of the driveway. A large box lay on the ground next to the car.
"Aren't you going to go help your neighbor?" Mr. Stanley asked.
"Do I have a choice?"
"You could have some of that gold sake and leave the heavy work to her husband."
"I gotta get going anyway." Clark pulled the zipper on his jacket to just below his chin.
Mr. Stanley didn't want his guest to leave. "How is everything else?" he asked, touching the coffee cup adorned with a Christmas tree to his lips.
"Well, I will be twenty-six soon and I'm spending my winter break moving back into the basement of the house where I grew up. Granted this is a small improvement over last year when I was still reeling over my dad. I have spent the last six months taking a double load of classes so I can get out of school as fast as I can to start working, which according to everyone I know, I will probably hate. And, oh, I found a stack of letters in the kitchen from the IRS, who is threatening to sue my mother unless she can explain discrepancies in my father's tax returns for the last five years." Clark thought about mentioning his mother's mental capacity, but that was like flogging a horse that was never going to find its feet. "Things really couldn't be much worse," he added.
"Things can always get worse."
"Good morning," Clark called out making his way down his third slippery driveway of the morning. "Can I help you with that?"
Ariana's eyes opened wide and she stole a glance over her shoulder. Her heart raced and her blood ran fast through panic-stricken arteries and veins. One end of the box was on the rear-seat and she was pushing the other end with both hands. Two smaller boxes were on the ground near the trunk of the car.
Clark approached smiling. "I was visiting Mr. Stanley and saw you with the boxes from the window," Clark continued as he balanced on the icy ground with his arms.
Ariana pushed the large box across the back seat of the car and shut the door with authority. As she turned to greet Clark she flashed her best housewife grin. Small beads of sweat formed under her eyes, beneath her thick black-framed glasses.
"Good morning, Clark. Merry Christmas."
"Happy holidays," Clark answered.
"How is your mother doing?"
"Fine, fine. Thank you for your help last night."
"Don't mention it."
Clark looked into the car at the large box. Ariana smiled again.
"Can I give you a hand?"
Ariana paused for a moment. "Sure. Thanks. There are only two boxes left."
Clark walked to the back of the car and Ariana opened the trunk. He bent at the knees and grappled with the bottom corner of the box.
"Where's Nazim?" he asked through a half-grunt, putting the bottom of the box on the edge of the trunk.
"He took Liana for a walk. He said the cold was invigorating," Ariana shook her head as if her husband were alive and crazy.
"I hope Liana is wearing ice skates."
Ariana watched as Clark picked up the second box and the bottom sagged.
"Just put it anywhere." She held her breath and hoped the obvious questions would remain in oblivion.
"What's in the boxes?" Clark asked innocently as the last box thudded into place in the trunk.
"Old stuff I have been meaning to get rid of."
Clark nodded. "I have a bit of that to take care of myself," Clark said.
"I thought today would be a quiet day. A good day to get some end-of-the-year cleaning accomplished. And with my husband out of the house, I can throw away a few things. Or at least put them in the car."
"It's a good day to be inside if you ask me."
"Unless you want to be alone outside."
Clark cocked his head to the side and a hint of perplexity showed on his face. Ariana ignored the reaction and noticed Clark wasn't wearing gloves.
"Well, next time you should make Nazim carry the heavy stuff. Those boxes are pretty hefty."
"I'll tell him you said so."
As Clark walked away, Ariana's eyes pierced his backside. Helpful neighbors could be a bitch.
## Chapter 5
The beat up black Volkswagen Golf turned around at the north end of Dorchester Lane and made the trip down the street in the opposite direction. They watched the houses pass by in descending number order and slowly pulled up to the curb in front of number 202.
Karim, Syed, and Abu looked at each other. Their dark black eyes exuded a seriousness that sent shivers down the spines of everyone they encountered, most recently the young girl behind the counter of the gas station mini-mart near Culpepper, Virginia.
Syed, his head touching the fabric on the ceiling of the car, looked at the number scribbled on the torn corner of white paper. "That's it," he said to the two other men. He folded the Northern Virginia street map and shoved it into the small pocket on the lower half of the door.
Karim killed the lights and turned off the engine. All three men looked around at their surroundings and tried to get their bearings. "Who goes?" Karim asked, his hands still on the wheel. His unshaven face hid the gauntness in his cheeks, the physical effects of weeks on the road easier to hide than the psychological ones.
No one moved. Karim spoke again. "Either someone volunteers, or we all go." Thirty hours after meeting in a truck stop outside of Houston, trust was earned slowly. Suspicion, even of one another, was their shared lifeline. "Then we all go," Karim added with a very slight accent. Syed and Abu nodded in silent agreement.
Karim looked around. "Let me pull the car on the other side of the street and away from the front of these two houses. I would prefer to be out of sight and heading in the right direction if we have to leave suddenly."
The neighborhood was quiet. The clock on the dash read 2:13. Give or take an hour, they were on schedule. It was easier said than done when every step was revealed by a distant master on intermittent communication according to an unknown timetable.
Syed opened the passenger door and unfolded his six-foot-three frame from the small German compact. Abu, the smallest of the three, pushed open the back door and a loud screech rang out into the night. All three men ducked as if the sound were gunfire, as if stooping would save their lives. "Quiet," Karim said, the white of his eyes visible in the darkness.
"If we had a better car..." Abu answered
"The car is fine," Karim answered, usurping the role of leader. "Let's go."
Syed led the way across the yard and the three men turned their back to a stiff wind. Flakes of snow whipped by in streaks of white. They knocked lightly on the door, waited, and knocked harder. "Where are they?" Syed asked, towering over Abu and Karim.
"What makes you think it is a 'them?'" Karim asked. "We have no idea who is on the other side of that door."
Abu, his cheek scarred, checked the handle of the ten-inch knife in the small of his back.
Maria Hayden's eyes had opened with the screeching car door. She slept light. Always had. The side-effects of the myriad medications she took daily had stolen a decade of shut-eye from her life. A faint knock on the door forced her to sit up in bed. The second knock led her to slip on her bathrobe and walk in short strides to the front door. She watched through the stretched tunnel-view of the peephole as the hand of a dark-skinned man reached towards her door and knocked again. Her heart skipped a beat. She let go of the edge of her robe and leaned closer to the peephole. She placed her palms on the wood panels of the door and switched to her right eye. The fourth knock was even louder and Maria jumped slightly, her right hand hitting the chain on the unlatched chain lock over the deadbolt.
The subtle but unmistakable sound of metal hitting the door brought the unknown standoff to a pause. Karim glanced up at Syed's face then shifted his stare to Abu. The group froze. After seconds that passed like hours, Karim broke the silence.
"Assalamu alaikum. We have traveled a great distance and pray for your hospitality."
Abu shook his head.
Karim repeated the standard Arabic greeting followed by the specific words he had memorized long ago.
Syed spoke to his two accomplices. "Something is not right. Let's go."
Abu quickly agreed with the taller Syed. "He's right; let's go." Karim took one look at the Hayden's front door and the numbers 202 that ran down vertically. Without speaking he nodded and flicked his head towards the car. Maria Hayden watched as the three men vanished beyond the vision of her peephole. Hands shaking, she took one more look outside the edge of the living room window and saw nothing but darkness. She walked back into her bedroom and opened the phone book to the listings in the blue pages.
The three men waited in the darkness of the black car to see if the neighborhood had noticed them. A porch light three houses away was the only company for the increasing wind and heavier flakes of snow. Karim started the car as Syed checked the scrap piece of paper with the house number on it. Abu cursed from the backseat. "Let's get out of here."
Karim hit the headlights and Syed yelled.
Standing in front of the car in a winter jacket and black hijab, Ariana stared at the three men with an expressionless face.
"What the hell?" Karim asked as Ariana approached the side of the car. She knocked on the driver's side window which Karim rolled down with the manual handle.
"I believe you have the wrong house."
Karim and Abu looked at Syed who sheepishly consulted the number on the paper for the tenth time. "It's the house number I was told."
Ariana stared through him. "I'm going to give you instructions and you are to follow them exactly. There is a Giant supermarket about a mile from here off Curtis Trail Boulevard. It is open twenty-four hours a day. It's not too far from 395 and there are always other cars. Park in the side lot, away from the main road. Go into the store and get some bread, milk, and whatever else you want to eat. I will meet you in the side parking lot when you get out of the store. Be ready to get whatever you need from the car. You will be leaving it there."
"Why milk and bread?" Karim asked.
"Because I need milk and bread," Ariana answered.
"How do we know we can trust you?" Abu piped in from the back seat.
"Because I haven't killed you already," Ariana answered.
The three men in the car paused momentarily. Karim nodded and put the car into first gear. Ariana watched as the black VW drove out of the neighborhood. She went into her house, grabbed a black cylindrical item from under the sink and headed for her car.
Maria Hayden picked the beige phone off the stand on the bedside table. Hands shaking, she unfurled the curly wire that ran between the handset and the base of the phone. She pinched the receiver between her ear and shoulder and carefully punched the numbers as she read them from the phone book. She listened to three rings before a woman's voice answered from the other end.
"Central Intelligence Agency."
"Hi," Maria said nervously. "My name is Maria Hayden and I would like to report some terrorists."
The soothing voice on the other end of the phone showed no emotion. "Yes. What is your location?"
"Arlington, Virginia."
"And where are the terrorists?"
"They were at my front door."
"What were they doing?"
"They were knocking."
"Can you describe them for me?"
"Three Middle Eastern men. One was tall. One was shorter. The other had a beard."
"When did this occur?"
"About five minutes ago."
"Where are they now?"
"They disappeared?"
"Disappeared?"
"Yes."
"Ms. Hayden?"
"Yes."
"How do you know they were terrorists?"
"What else would they be?"
"Of course," the woman replied from the three person call center staffed in an unnamed building in downtown McLean not far from Langley. "Could I get your address and phone number, please?"
"Sure," Maria answered. She gave the information slowly, carefully spelling out her address in full and repeating her phone number twice.
"Ms. Hayden, if you see these men again I suggest that you call 911. They can respond much faster than we can."
"911?"
"Yes, ma'am, the general police emergency number."
"But you are the CIA; it's your job to catch terrorists."
The voice sighed ever so perceptibly. "Yes, Ms. Hayden. I will pass your information along."
"I see," Maria Hayden replied, not sure if the conversation was over.
The woman with the soothing voice let Maria know it was. "Have a good night, Ms. Hayden."
"Good night," she answered into the dead line.
Ariana parked next to the black VW and looked around. The side parking lot of the Giant supermarket ran to the back of the building and sat in a small gully that could be tricky to navigate in the snow. Ariana stepped from her beige Toyota Camry and made one lap around the VW, taking inventory. She walked away from her car and as she turned the corner to the large brick structure she hit the auto-lock button on her keychain. Her car alarm chirped as she headed for the store entrance.
Karim, Syed, and Abu were in the frozen food section, throwing a combination of vegetables, cheese pizzas, and organic complete meals into their cart.
"I can't believe our contact is a woman," Abu said with distain. "They are soft." In the light of the frozen food aisle, the pock-mock scar on the right side of his face was obvious, almost gruesome. The flesh had a texture that was neither human nor reptilian, but somewhere in between. "And I don't like being threatened by a woman."
Karim spoke. "We don't know anything yet. She might just be an intermediary. Her sex doesn't matter. Only her conviction concerns me. Nothing else."
"I assure you that I am not soft," Ariana said.
Syed and Abu jumped and let the door to the freezer slam shut. "Shit," Syed said, adding some intelligence to the conversation. "That is the second time you have snuck up on us."
"I know," Ariana said. "I am unimpressed."
Karim looked at Ariana, their eyes locking. He slowly redirected his eyes downward and she felt his eyes cover her body from her hijab to her toes.
"There are supposed to be four of you," Ariana said.
"As you can see, there are only three," Abu said.
"Did we lose one?" she asked.
"I was told to pick up two," Karim answered.
"Who told you?"
"A voice on the phone."
"How did they reach you?"
"I went to an Internet café and checked a pre-arranged email account. In that account there was a draft message. A message that had never been sent so it couldn't be monitored. I followed the instructions in the draft email. I bought a pay-by-the-minute cell phone at a 7-11, and saved the number in a different draft folder in another pre-arranged email account. Then I waited for a call. They told me where I could pick up a car. I got the car and met these two at a truck-stop outside of Houston."
"How did you two know where to go?"
"We were told before we crossed the border."
"Crossed the border? Together?"
"No," Abu answered. "We met in Houston."
"So there are only three."
"Yes," Abu answered.
"Meet me outside at your car in five minutes," Ariana said over her shoulder as she walked away.
The three men rounded the corner, each with a brown bag in their arms.
"Put the food in the trunk of my car," Ariana said, motioning towards the Toyota. "Open the back of your car and get your stuff."
"What do we do with the car?"
"I said we are going to leave it." She reached into her pocket and shoved the electric screwdriver into the screw on the license plate. The screwdriver buzzed for a few seconds, went silent, and then buzzed again. Karim took three worn backpacks from the back of the VW and threw them in the Toyota. Ariana removed the single license plate and threw it in the trunk of her car with the three backpacks and groceries.
"Get in," she said.
"Do you think it is a good idea to leave the car here?" Abu asked.
"I think it is a better idea than parking what is likely a stolen car in my driveway."
Karim smiled as he got in the passenger seat.
"Now what?" Abu said.
"Now we wait. I was told to receive four people and there are only three of you."
## Chapter 6
Beautiful trouble arrived in a burgundy four-door Ford 500 wearing a business suit, glasses, and carrying a handful of legal accordion folders. Clark heard the car door slam and he picked up his pace. He shoved the old Hoover in the hall closet and pulled the sliding metal doors shut. He walked past the open bathroom door and saw his mother applying the finishing touches to her lipstick in front of the mirror.
"Are you ready, Mom?" Clark asked.
"Finished," she answered, turning off the light.
Clark made one pass around the dining table and took a last glance at the stack of files. He looked at the set of mechanical pencils resting next to the calculator and decided they were too geeky for his own good. Studious was good, geeky was not. He swiped the pencils off the table and threw them in the small drawer in the corner cabinet. He plucked a pen off the kitchen counter and threw it on the table as he headed for the front door.
The word "geek" bounced around in Clark's head as he eyed the studious-looking IRS auditor with silky auburn hair coming up the front stairs. _Well, this may not be so bad after all_ , he thought.
Clark took quick steps to the door in anticipation of a closer look at his guest. His first sight was an eyeful of an official Department of Treasury IRS Auditor badge extended into his face.
"Lisa Prescott. IRS auditor."
Clark smiled. Just like in the movies.
"Hi. I'm Clark Hayden, please come in. I hope you didn't have too much trouble finding the place."
"The neighborhood is a little counter-intuitive. I called you from the gas station on Route 110 and it still took twenty minutes to get here."
"Yes, these old neighborhoods can be tricky."
"The dead ends don't help."
"We like to call them cul de sacs," Clark said. "Dead end is passé. We're shooting for something more yuppie. Change the clientele in the neighborhood."
IRS Auditor Prescott looked over at the dining area table and nodded almost imperceptibly. "Is that where I should go?" she asked.
Clark cleared his mind of lust and snapped to attention. "Yes. May I take your coat?"
Lisa removed her outer layer. Clark tried not to stare. She couldn't have been much older than he was. Maybe she had a year on him, certainly no more than two. Clark took her coat, looked around, and put it on the arm of the sofa.
Lisa found a dining room chair and sat down, her bust concealed behind her oversized brown folders. Clark noticed a cup of coffee on the table that he hadn't seen the IRS auditor bring in. He read the wording on the side of the cup aloud.
"Jammin' Java."
"Yes."
"I like that place. It's hard to find a good coffee shop that plays live music."
"It is the only one that I know of," Lisa answered. "But what I really like is open mic night."
"You sing?"
"No, but I go sometimes to listen. There's more than just singing."
Maria Hayden came from the kitchen and introduced herself. She followed with further niceties. "I put the kettle on. Can I interest you in a cup of tea?"
"I have a little coffee left, but some tea would be nice," Lisa answered with a beautiful smile underneath her button nose.
If Clark had stopped to listen, he would have heard the proverbial gloves hitting the floor as the minx in front of him sharpened her bureaucratic claws.
Clark positioned himself in the chair directly across from Auditor Prescott. "Can I call you Lisa?"
"No you may not."
"How should I address you?"
"Ms. Prescott."
"You can call me Clark."
"Fine, Clark it is. As you may know, I'm investigating your mother and father for tax discrepancies over the last two tax years. As I mentioned over the phone, they filed a joint 1040, so even though your father is deceased, and please accept my apologies, your mother is still responsible for the information presented on those forms. Her signature is there. She benefited from any misrepresented tax information that would have enriched your father's financial standing. I think it is safe to say that these infractions are, what I would consider, severe."
"What does 'severe' mean?"
"Generally I classify severe as any infraction that could result in jail time."
Clark looked over at his mother in the kitchen and then back at Auditor Prescott. In a quiet voice he spoke with conviction. "My mother is seventy-five years old and she has diminished mental capacity. It would never happen. No one in their right mind would put her in jail."
"Never say never, Mr. Hayden."
"Clark."
"Clark," Prescott repeated. "Do you have power of attorney over your mother's affairs? Without it, I am not authorized to speak with you."
"No, I don't. My aunt does. My mother will answer your questions and I will sit next to her."
"Fine. But I will not be directing any questions towards you. If you choose to obtain a Power of Attorney, I will need a copy for my files."
There was something about being reprimanded by someone his own age that made the whole exchange more insulting than it was. Clark didn't know what to make of the IRS auditor. Her eyes showed kindness and warmth. Her tongue was as sharp as a razor. Auditor Prescott ran some hair behind her left ear. "Can we get started?"
"When my mother finds her seat."
Small manila folders poured from the auditor's larger accordion style one. Prescott stacked the folders into piles under a system that only she understood. When she finished arranging her papers in accordance with her thoughts, she began.
"So, Ms. Hayden. Do you understand why I am here?"
"Yes. You think I'm cheating on my taxes."
"Well, not necessarily you, and not necessarily in the present. But there are several discrepancies that we have found in looking at your taxes for the last two years."
"I see," Maria responded sarcastically.
"We have reason to believe that either you or your husband has failed to claim substantial income on your returns for the last two tax years."
"Says who?" Maria Hayden answered.
"Says the IRS."
Auditor Prescott pulled out several pieces of paper. "These are deposits made into two different bank accounts over the last two tax return years. You can see that they were endorsed by your husband."
Maria looked at the signature and nodded. Clark pulled out a previous year of his parents' tax returns to compare the signatures. He too nodded slowly.
"One of the accounts was in the now-defunct Riggs Bank. Most of the deposits were cash."
Clark interrupted. "Lisa, can I ask a question?"
"Ms. Prescott."
"Yes, Ms. Prescott. May I?"
"Please."
"How did you get this information?"
Lisa Prescott looked at Maria and answered the question. "Ms. Hayden, you were identified as a subject for audit at random. The initial audit is run by an IRS auditor. A criminal investigator is assigned if one is needed."
"I see," Maria said again.
"The IRS has many avenues for identifying and pursuing tax evaders. We have the $10,000 limit on wire transfers and certain other financial transactions, and we allow for anonymous sources to provide information on tax evaders. Audits are still the largest revenue generators."
"I'm sorry, but did you just say that you allow anonymous sources to turn in other citizens?" Clark asked.
"Yes."
"Who would do this?"
"An anonymous source means that we don't know whom."
"How the hell does that work? Anyone can just turn in another person as an anonymous source?"
"Absolutely. There is a service center in California that only handles anonymous submissions."
"I would like the address," Clark said.
"Why?"
"I need to report 435 members of Congress and 100 Senators."
Lisa Prescott smiled involuntarily, her cute face framed by wisps of auburn hair.
Gotcha, Clark thought. Not even an IRS nutcracker was immune to well-timed wit and charm.
"The address is public information. I can provide it, or you can look it up," Lisa said, shuffling to a new stack of paper. The dimples on her cheeks strained to remain hidden. She stole a glance over the frame of her glasses as Clark put his hand on his mother's arm. "Of course, there needs to be first-hand familiarity with the situation in order for an anonymous submission to have merit. It is not a witch hunt."
"I understand" Clark said. "So, from what you have said, my father was stashing away money and not paying the taxes on it."
"I cannot answer that question to you, Clark. Your mother needs to ask it."
Clark put his hand on his mother's shoulder and whispered into her ear. Maria Hayden repeated the question closely enough to get the point across.
Auditor Prescott responded. "It appears that way. It is not a difficult formula. Add up the deposits, look at the tax forms and the declared wages, and see the difference. These deposits exceeded your parents' income, as stated on their tax returns."
"I see," Clark said, stealing another glance at his mother to see if she understood.
The interrogation and banter went on until lunch. In the process, Clark learned a valuable lesson that all Americans figure out sooner or later: if the government wants to fuck you, it is only a matter of when and what position they prefer. KY is optional and applied purely at the government's discretion.
The conversation dragged on through two pots of tea, twice as many bathroom breaks, years of tax forms, stock holdings, pension plans, IRAs, major purchases, and the deed on the house. Clark kept up with the numbers and the total was anything but impressive. His parents were blue-collar. Even with the accounts and deposits in question, they spent their whole lives just getting by. A lifetime of labor under scrutiny for some twenty thousand dollars of mystery money.
Clark spoke. "I would like to take a crack at summarizing our meeting here today. My parents have lived, God rest my father's soul, for a combined hundred and forty years, and the IRS is interested in twenty-two thousand dollars that appears, as you put it, to have come out of nowhere?"
"You summed it up for your mother nicely. That is correct."
"So you say."
"So the paperwork says. Numbers don't lie."
"Numbers lie all the time. Don't you read the newspapers? You can have multiple conclusions from the same statistical study."
"I don't believe everything that I read."
"You said it," Clark proclaimed. "Neither do I."
"Well, today is just a preliminary meeting. I will give you a week to get your ducks in a row and to examine the discrepancies we discussed. Perhaps there is a reasonable explanation that neither of us is yet aware of. But these deposits are no mistake. Whether they can be explained is another matter altogether."
Clark ignored the last statement. "So we get to meet again?"
"I think it's safe to say."
Clark walked Ms. Prescott to her car against her protest. He waited until she hit the alarm on the American sedan and then opened the driver's door. Lisa Prescott smiled and walked past Clark to the back door. She placed her folders in the rear seat and turned towards Clark who was still standing at attention like a chauffer driver.
"Thank you," she said, offering the first genuine kindness of the day.
"I knew there was real person in there trying to escape."
"My job doesn't allow me to be a real person."
"Maybe not, but I sized you up before you hit the front steps."
Lisa Prescott stopped and looked Clark in the eyes. "Assuming you're right, and that there is a real person trying to escape, what gave me away?"
"You were singing to yourself as you walked up the sidewalk. A real bitch would have stopped singing in the car." Clark paused for a second. "Sorry, I probably should have chosen another word."
"Consider us even."
"Would it be inappropriate for me to ask you out?"
"Yes. It's against regulations."
"Not mine."
"I'm prohibited from personal relationships with those under my investigational jurisdiction."
"Investigational jurisdiction? Try to say that five times really fast. You must have a team of lawyers over there in the Treasury Department thinking up terms."
"We have a few."
"Well, technically you are investigating my parents, not me. I'm just an intermediary."
"You will likely obtain Power of Attorney."
"A mere technicality."
Lisa Prescott sighed, smiled and bit her lip. "I'll see you again in a week."
Clark walked through the small hall that ran from the kitchen to the back of the garage. The concrete floor in the narrow passage was as cold as the frozen earth outside. His breath billowed out a white mist that stretched forward a foot before dissipating. He wiped his sneakers on the small green plastic mat and stepped into the laundry room his father had built with the attached garage shortly after Clark was born.
Clark flicked the light switch and stepped from the laundry room into a garage that had never housed a car. Light trickled in through two small windows on the outer wall and through a set of glass panes on the roll-up door. A smell hung in the air that Clark associated as a mix of burnt metal and oil. It was the smell of a machine shop, and not all the Lysol at a Costco could change that.
Clark walked around the shop in the presence of his father's image, a ghost still standing at the machines, intermittently breaking to look over blueprints at the small work table. Clark pulled the cover off the lathe and let it fall to the concrete floor. The lathe was a six-foot beauty, a machine designed to spin a piece of metal or wood horizontally at rapid speeds while applying a cutting blade to the side of the object, resulting in the production of spheres and cylinders.
Clark ran his hand over the cutting control, just as he had when he and his father had crafted a handmade Louisville slugger replica in junior high school. Clark turned the knob on the control and the geared wheel moved effortlessly. His father called the grease he applied to the gears "Toyota slick" and he ordered the lubricant directly from the manufacturer. It lasted forever, his father had said, and Clark was starting to believe him. Clark reminded himself to run the machines and check the de-humidifier in the corner that emptied into the yard through a hole in the wall of the garage.
Clark moved to the stand-alone milling machine with the Computer Numeric Control pad. The semi-automated CNC pad could guide a drill bit into a solid object, accurate to within a thousandth of an inch. A fraction of a human hair. More than precise enough to drill the wheel holes on a winning pinewood derby car in Boy Scouts.
The machines were part of his father, as much as the old wool jacket and matching gray hat that he wore six months out of the year. Clark had tried to get his mother to sell the machines and there was no shortage of interested buyers. Many of his father's friends had inquired about the machines in the months after the funeral. His mother refused to discuss it. And Clark knew that if his mother was still keeping an old tube of toothpaste in the bathroom, she sure as hell wasn't ready to let go of something that symbolized her husband for fifty years. Maybe the trouble with the IRS would change her mind. The machines could settle some of the current bill to Uncle Sam. Cover some prescriptions, too.
Clark shuffled slowly through a set of standing toolboxes and shelves. He passed a huge stack of metal stored on the shelves in the corner, scraps of leftovers that included prototypes, car parts, and other bits and pieces whose origins were a secret Clark's father had taken to the grave. An old drill-press divided the unused precious metal. Aluminum was the metal of his father's profession, but all machinists worth their salt had a stash of the good stuff. Titanium. Magnesium. Inconel. Clark ran his hands over the shelves of unused metal — cubby holes of tubes, squares, cylinders, and rings. There wasn't a shape on Earth his old man couldn't reproduce. And there were a thousand pieces no longer on this Earth that his father had made with sweat and his machines.
Clark caressed the handle on the band saw and snapped out of his daze. He stepped past the small desk against the wall and opened the dented metal filing cabinet in the corner. He pulled out stacks of folders from the top drawer and flipped through the tabs. Blueprints, orders, invoices, copies of checks. Clark grabbed the folder labeled "invoices and billing," took another adoring look around the room, and hit the lights on the way out the door.
Clark leafed through the financial papers from his father's filing cabinet as his mother finished the dishes. She wiped her hands on the towel on the counter, put her apron on a hook in the wall, and disappeared into the bedroom. She reappeared with the family photo album and found her seat next to Clark on the sofa.
"Do you remember this?" she asked showing a photo of an elementary school-aged Clark at a petting zoo trying to run from a goat with an appetite for his sweatshirt.
"How could I forget? I still hate goats."
"Your father saved you."
"Yeah, Dad gave the goat a knock on the head it probably still remembers."
"That animal was possessed, I tell you," his mother said, making a sign of the cross.
Tears ran down Maria's cheeks as she flipped the black pages of the thick book, photos from a lifetime of events pinned to paper with triangular corner holders. She had just finished her second Christmas without her husband of fifty years. Her second Christmas as a widow. Clark offered his shoulder to cry on. For himself, he was done crying. He looked at a picture of his father dressed in a suit and then pried the photo album from his mom's fingers and put it on the table.
## Chapter 7
Ariana looked out the kitchen window at the back end of the white seventeen foot moving truck. The words Piedmont Delivery were written in burgundy letters that arched across the side of the vehicle. The reverse lights were on and an intermittent beep blared with force that belied the size of the truck.
Ariana stuck her head around the corner of the kitchen. Her three guests were sitting in the living room. Karim and Syed were examining a map of the D.C. area spread across the coffee table. Abu was sitting on the floor, trying to make his way through a copy of USA Today. They were studying the enemy. Three hundred million potential targets, not including the 1.8 million Muslims in the US population.
"Everyone in the basement," Ariana said.
Karim looked up from his hunched over position. "What is it?"
"No questions. Get in the basement. All of you."
Abu slammed the paper shut with a rustle. Karim looked at the short-fused member of the group. "Let's go. And bring your paper with you. You need the reading practice."
"I don't need to read to know how to kill."
Ariana walked up behind the seated Abu and put a finger in the small crevice between the bottom of his ear lobe and his jaw. She pushed his head into her knee and let the agony of the pressure point rush through his neck. Abu squirmed in pain. Ariana loosened her grip slightly and spoke. "No, you don't need to read to know how to kill, but you need to know how to read to avoid getting us killed before we complete our task."
Abu reached up with his hand and tried to swat away Ariana's grasp. As he did, Ariana stepped back and Abu rolled back from his seated position onto the carpet.
Syed and Karim smiled slightly and stood. Abu rubbed the side of his neck and gathered himself, searching for the dignity he had just lost.
The basement door shut and Ariana reached for her jacket as the footsteps of her guests faded down the stairs. Her black hijab in place, she pulled on her coat and hustled out the door.
In the truck, James Beach flicked off the radio and slammed down the pedal-mounted emergency brake. He grabbed the single key from the ignition, pushed the heavy door open, and lowered himself to the ground in one large step. His tattered Baltimore Orioles jacket looked absolutely new compared to the state of his faded, threadbare jeans. James reached for a small folder of paperwork wedged behind the driver's seat and shut the door.
Ariana, trace remains of adrenaline still in her blood from the confrontation with Abu, rounded the corner of the van with a smile.
"Can I help you?" she asked the man with the disheveled brown hair and piercing blue eyes. "I'm not expecting any deliveries." Ariana measured the delivery man quickly. Six-foot one, one-hundred and ninety pounds, athletic. In need of a change of clothes and a shower.
James stepped forward with the folder in his hand and Ariana noticed the poorly done tattoo on his right wrist protruding just below the end of the sleeve.
"I'm here to be received," James said as if he had uttered something profound.
Ariana stepped back, creating room between herself and the deliveryman. "I'm sorry?"
"I'm here to be received."
Ariana stared into the man's eyes. James Beach stared back.
"I'm going to ask you a very important question. Are you sure that you have the right address and the right message?"
The cold stare from Ariana gave James goosebumps. He opened his folder and read from the script. "Assalamu alaikum. I have traveled a great distance and pray for your hospitality."
"Then I shall receive you," she said.
James stuck out his hand and Ariana looked at it without moving.
"Why the truck?"
"What?"
"Why the truck? This neighborhood has eyes. Small yards. Curious minds. You should have at least disabled the reverse alarm."
"I only had an address. I didn't know what kind of neighborhood it was."
"What's in the truck?"
James Beach smiled. "You're going to love it."
James dug in his jeans for the brass key to the large Master Lock hanging on the back of the truck. He turned the key and pulled down on the square block that housed both ends of the u-shaped steel bar. With another smile, James pushed the rolling door up. Stacks of large brown burlap sacks about the size of garbage bags filled the back of the van halfway to the ceiling. Ariana pushed on the side of a sack to feel the texture. The bag gave way slightly. She ran her hand down the outside of the bag and felt a twinge of excitement.
"Is that what I think it is?"
"Yes."
"How much is in there?"
"Four thousand pounds, give or take."
Ariana quickly did a calculation in her head that only a handful of people on Earth would know how to do, and even fewer who would relish in it. The results of her calculation would make the average person lose their lunch, or at the very least, their appetite for dinner.
"You want to see?" James Beach asked, slowly reaching for the buck knife from his back pocket. Ariana shifted her stance slightly preparing for a possible attack. James Beach noticed the attention gained by his movement and removed his hand from his pocket, empty.
"Lock the door on the truck and follow me inside," Ariana said. James Beach headed for the door as Ariana looked up the street under the midday sun and wondered how many eyes were on her.
James Beach sat on the old cot in the basement with Syed on his left and Karim on his right. Abu sat backwards on an old dining room chair, his legs spread to each side. Ariana pulled the curtains in the small windows that were high on the wall, level with the ground outside.
"He's American; we can't trust him," Abu started.
"I agree," Syed added, stroking his chin.
Karim remained silent and watched as Ariana unfolded a large map and put it on the floor.
James spoke in his defense. "I'm American but I am also a Muslim."
"It's not the same," Abu added.
"Why not?"
"Because you do not share a thousand years of spilled blood at the hands of the infidels. You _are_ an infidel."
"Do I need to have relatives who have died in the name of Islam? I thought our faith was measured by faith. Faith and action. What else is there?"
"Pure blood. You are American, right down to your blue eyes."
James opened his mouth to respond and Ariana intervened. "Enough," Ariana repeated. "We don't have time for this. I will deal with it later. There may be reasons that an American was sent. For one, his name would not raise suspicion if, for example, he were the driver of a truck that was pulled over."
"What's the name on your driver's license?" Abu asked.
"James Beach. But I have also taken the name Moham-med Al-Jabar."
"Enough," Ariana repeated. "We have a more urgent problem. I need to take care of the truck outside."
"I can drive it," James offered.
Ariana thought. "You will wait here with everyone else. I will be gone a few hours. If anything happens to me, I may need to call." She pointed at Karim. "Only you answer the phone."
"Why him?" Abu asked.
"Because his voice is similar to my husband's."
She turned her attention back to James who spoke. "I have a pre-paid phone. It is clean."
Ariana's mind was on the cargo in the back of the truck. "No. From this moment on, every step we take needs to be erased. There are no retreats. No going back. No duplication of activities. If we use a computer, the computer gets destroyed before we move to the next phase. If we use a cell phone, it gets destroyed. I will get us all new phones this afternoon. It will take me a few hours. Once I have the phones, I will give them to you when the time is right. I also need everyone's clothes size. Shirts, pants, shoes. Head, feet, waist, inseam, chest. Everything. But first, I need to get that truck out of the driveway. That is a delivery truck. Every minute it sits in the driveway it draws more suspicion. Every minute it spends in the driveway without someone delivering something is another second that I have to cover with a lie. Delivery trucks generally deliver and leave. I'm a Muslim woman and the delivery man is a good-looking American. People could start talking. Even if they are joking, it may help them remember an important detail later."
The knock on the kitchen door made everyone jump. "Shit," Ariana said. "No one move. I'll be right back." Ariana ran up the stairs and peeked out the window.
The dark blue government-issued hat bounced around the small window in the door. "Is that you, Mel?" Ariana asked.
"Yes," he said stretching the word into two syllables. "I have a package for you," Mel the postman said from under the brim of his USPS winter hat.
"Just leave it on the porch,"
"I need a signature."
Ariana opened the door and offered her arm for the signature. Mel handed her a pen and Ariana scribbled her name in the small blank on the green receipt. "You having something delivered?" Mel asked motioning towards the truck in the driveway.
Ariana looked quickly at the name on the side of the truck and the wheels in her mind spun. Piedmont Delivery. "Yes, my husband ordered a handmade dresser from an arts and crafts place near Front Royal."
"Super, super. New furniture is always nice."
"Yes, it's quite a piece. Thanks for the package, Mel."
"Thank you. Have a great day."
Ariana looked at the package and watched as her mailman walked down the driveway next to the truck. She cringed as Mel Edgewood studied the name on the side of the van before he cut across the corner of the yard to Mr. Stanley's.
## Chapter 8
Allan Coleman's thick belly sat on the edge of his desk, resting just in front of the keyboard. His body strained to breathe, bands of fatty tissue pushing his stretch pants to their limits. Unhealthy bulges of cellulite rode up his chest, a deep fold of flesh drawing the boundary between the top of his stomach and flabby man-boobs. Oversized buttocks oozed from the sides of the chair under the armrests, his fat ankles squeezed into a pair of white sneakers.
The newly built office in Coleman's Castle was a roadmap to obesity. An empty bag of Frito Twisters protruded from the trash can supported by a box from the dozen Krispy Kreme doughnuts he had eaten earlier in the day. A thirty-two ounce plastic cup rested on the corner of a small table, away from his computer and the spaghetti of wires that ran along the floor in every direction. His nails were chewed short and dandruff dripped off his shoulder, rolling down his arm in the direction of his elbow.
By most accounts, Allan Coleman was wasting his life. His cholesterol level was over 300. His blood pressure was a monumental 220 over 160. His doctor told him to lose weight. Allan ate to compensate for the stress of the doctor's orders. Gastric bypass was suggested. Allan refused. He believed he had an exercise routine. Every thirty minutes he walked to the small refrigerator on the far side of his office and retrieved a Coke Classic. With the completion of the two-story addition and the onset of winter, he augmented his usual workout by going down one flight of stairs, opening the side door, taking two steps, and retrieving another log to put on the fire. It was as much exercise as he cared to do.
The oversized ergonomic chair was little compensation for the natural stress his frame endured by carrying an extra average-sized human on his skeleton. Allan Coleman rolled his neck, cracked his knuckles, and quickly pounded out an email to a team of programmers and help-desk gurus halfway around the world. Allan was a work-induced shut-in. He couldn't even recall the year of his last date, unless he counted Little Debbie — the vixen of desserts who held a strategic, immediate gratification advantage over the second most important woman in his life — Betty Crocker.
"Ping," the speaker on his computer screamed with another message. He was living in a high-tech world where email wasn't fast enough. The homegrown instant messaging system forced on its employees by his company was the bane of Allan Coleman's existence. Thirty seconds to reply was the standard. It didn't matter if you were in midstream of urinary relief or performing the Heimlich on a lunch guest. Any longer than thirty seconds and his phone would start ringing. Allan didn't live on Eastern Standard Time. He was on telecommuter time. And his conversation of the night was identifying a software bug with a pack of Indian programmers in Mumbai.
But what Allan didn't qualify for in the superficial world of size two dresses and six-pack abdomens, he made up for in knowledge of ones and zeros. They were the building blocks of digital life, the binary DNA of every computer system and piece of software in the world. As the head of level five technical support for the graveyard shift, working for the last American computer maker, Allan got the tough cases. The unsolvable electronic mysteries that his underlings in India couldn't handle.
Allan's eyebrows jumped at the rumble in his own stomach and he reached to the right and grabbed his black wireless headset. As his intestines churned, he fumbled to hang the upper portion of the device over his ear. He adjusted the mic so that it was a comfortable distance from his mouth, close enough so he could be heard, far enough away so that he could cover it with his free hand and so that it wouldn't impede food consumption. He turned off the speaker function on the expensive phone, stood, and adjusted the angle of his flat-screen monitor. He was now free to move about the cabin. He wasted no time in running straight up the aisle.
The third wave of pain hit as he was turning to face the still open door and pulling on the elastic of his pants. The exodus began before his butt hit the seat. He fumbled for the mute button on his hand-free device and tried to concentrate on the conversation as unearthly sounds resonated around the new bathroom. His computer pinged twice and he tried to look through his now tearing eyes at the flat screen across the room. There were two beeps on the line and Allan let them ring unanswered. He had bigger problems.
Twenty minutes later, still in the seated position on the throne, Allan stammered through a few clarifying points, trying to control his bowels for a moment of silence.
With shaking legs, he stood and flushed the evidence of severe internal bleeding into the sewer system and on its way to the Chesapeake Bay watershed. He exited the bathroom weakly, sweat dripping, trying to catch up with the conversation, giving orders to colleagues who were looking out their window in Mumbai at one of the largest metropolises on the planet. Allan looked out the back of his castle at the single light in the kitchen window of his neighbor's house.
"You don't have time to get the flu," Allan said to himself as a case of chills ran down the skin on the back of his neck. He walked slowly to the corner and reached for another log to throw into the large stone hearth on the far side of the room. He pushed the log into the fire with a poker and returned across the floor to his computer and a new set of "pings."
Allan tried to open the attachment on an incoming email. His vision blurred and he mis-clicked three times before he found his target and the Excel spreadsheet opened. He continued to fumble with the mouse, his first inclination that something was wrong, something beyond the leftover Chinese take-out he had eaten as a late night snack. A man like Allan, someone who had spent ten years with a mouse attached to his finger, eighteen hours a day, didn't make a habit of mis-clicking. The pointer finger on his right hand was the only part of his body that was Olympic-athlete caliber. He noticed his head beginning to throb and his throat was sore. Eighty-hour workweeks were catching up with him.
By three in the morning, Allan had lost his appetite. He made two additional trips to the bathroom and was now feeling pain in his kidneys on both sides of his back. He went to the medicine cabinet in the bathroom for another dose of Advil and stared at what looked like traces of blood on his lips. He opened his mouth and pulled on his cheeks, worried by the bleeding gum lines. He rinsed his mouth with water and swallowed the ibuprofen-based pain killer. He washed it down with twelve fresh ounces of Coke.
Allan vomited bile to begin his last call of the night. By the time he wedged himself back into his seat he had lost most of the feeling in his arms. He struggled for the prescription bottle he kept at the back of his cluttered desk. He ripped the top off the brown plastic bottle and pills spilled over his desk and keyboard. He scooped up two and put them under his tongue. He leaned his head back in his chair and waited for the magic to work. The pain was everywhere, his back, his head, his kidneys. His heart felt as tight as a marathoner's calf. He tried to breathe, tried to imagine relaxing on a beach.
His computer screamed "ping, ping, ping" as the big man's head crashed into his keyboard and his chair rolled back.
Sitting in the dark, Ariana watched her neighbor from between the blinds in her bedroom. She was sitting in the same spot she had sat the first time she saw her neighbor watching her, his eyes focused on the back end of a high-powered pair of binoculars. Unfortunately for her neighbor, he had caught her practicing her martial arts. She had thought about calling the police, but didn't want to be on record with any law enforcement agency. She knew there would be a time for paybacks.
She marked the time in her head and waited to see if Allan Coleman would reappear in the window. Twenty minutes later, with the lights still on in the room and no sign of the King of Coleman's Castle, Ariana let a small smile escape her lips.
## Chapter 9
Ariana placed a piece of duct tape over the drain in the sink, just as she had done to the drain in the shower. Another thick layer of duct tape was crossed in an X over the toilet seat. The handle on the toilet was covered with additional straps of the adhesive-backed, all-purpose material. Finished, she looked around at her handiwork. She caught a glimpse of herself in the mirror and she liked what she saw. Her eyes were calculating, working. She felt alive.
At the foot of the stairs to the basement, Ariana turned on the lights and clapped her hands several times. Her guests were on the floor, sleeping as much as the rest of the neighborhood in the hours before dawn. Karim stirred first, squinting through one eye before forcing the second open.
"Turn off the lights," Abu said, his head under a pillow. James Beach rolled over on his side, still in his sleeping bag in the corner.
Ariana clapped again. "Everyone wake up. Now!"
The room stirred with a sudden sense of urgency. Syed was the first on his feet, standing almost at attention in a loose-fitting pair of boxer shorts. The white t-shirt he had on was too short for his torso and his midsection showed.
"Pack up everything you brought with you. Everything. Leave nothing behind. When we walk out of this basement, we will not return."
Fifteen minutes later, the room was spotless. Each member of the group had their backpacks and a sleeping bag in hand. The rest of the room was sterile. A sofa and two old dining chairs were pushed against the wall. A small table sat in front of the sofa.
Satisfied, Ariana said, "Get everything and bring it upstairs. James you go up the stairs first. Then Abu, then Syed, and then Karim. I want you to walk in single file towards the bathroom in the hall and stop at the door."
"What is this?" Abu asked.
"Shut-up and do it," Karim said. "For once, try not to think that you know everything."
The four men got in line and Ariana stood behind them. "Let's go," she said, and the formation moved up the stairs.
In the hall, James stopped at the bathroom door as instructed. His backpack was on his right shoulder, his sleeping bag under his left arm. Behind him were Abu, Syed, and Karim. The four men and their belongings filled the narrow hall. Ariana walked past the men and stopped on the other side of the bathroom door. She turned around and continued the routine she had devised earlier in the day.
"Place your belongings on the ground in front of you and remain standing."
Four backpacks thudded quietly as they hit the carpeted hall. In turn, Ariana picked the bags off the floor and dragged them down the hall into her bedroom. Next she carried the sleeping bags from in front of her guests. The four men remained, clothed only in their sleepwear, standing in the hallway. James' blue eyes grew wide.
She opened the bathroom door and looked at the only American. "Get in and strip down. Naked. Leave your clothes on the floor. The door will remain open. You will take a thirty second shower, long enough to thoroughly wet your hair. You will then turn off the water, get out of the shower, and I will give you new clothes to wear. All the drains are sealed, as is the toilet."
"Is this necessary?" James asked. "What does this prove?"
"Yes, it is necessary. It may not prove anything, but you are still going to do exactly as I say."
"What if I refuse?"
"Then I'll kill you."
"You can't kill all four of us."
Ariana took one step towards James. She reached up and put her finger at the base of his throat before pushing downward and hooking the inside of his sternum. Immediately both of James' hands shot towards his neck. "I didn't say I would kill all four. I said I would kill you." She pushed her finger deeper into James' body cavity and he gasped. Then she let him go and took two steps back. "Besides, the four of you can't do anything without me."
James stepped into the bathroom and undressed. Ariana waited until he was in the shower with the water running before collecting his clothes and throwing them down the hall towards the bedroom.
When the water stopped, Ariana handed him his new set of clothes through the still-open door. When James was dressed, Ariana nodded and he stepped from the bathroom. "Sit down to my left," Ariana said.
In turn, all four men went through the sanitation process. When they were finished, all were dressed in jeans and other American clothes typical for a winter night.
The four moved to the living room in silence as Ariana continued to execute her plan.
"We are leaving tonight. This is going to be the routine. Everyone will leave with the clothes on their backs. Nothing else. Everything you brought with you is gone."
"What about our passports?" Karim asked.
"They are gone. I may be able to get us driver's licenses for IDs if we need it. That depends on time. Right now you need to understand that everything you brought with you into this home will stay in this home. No exceptions. Our success will be determined by our deaths, and we will not need these items."
"Where are we going?" Abu asked.
"I'll be taking you to our new location. We need to hurry. We need to finish before the sun comes up. Given our time constraints, your travel accommodations may not be first class."
It was before six in the morning when Ariana helped Karim into the trunk of her Toyota Camry, against a protesting Abu.
"Both of us won't fit."
"Both of you have to fit."
"Why?"
"Because it's going to be light in an hour. The neighborhood is going to come to life and I don't have enough time to take you individually." She paused for a moment, pushed on Karim's back and said, "Think thin."
Karim climbed into the trunk, his legs laying over Abu's. He moved the sleeping bag that their heads were resting on and tried to get settled.
"You fit better than Syed did," Ariana said.
"Well, he got to ride by himself."
"We are out of time. I can't take you one-by-one."
"Can't you just blindfold us?"
"And drive the two of you around the nation's capital? Two blindfolded Arabs?"
"I can go without the blindfold," Karim offered.
"The others traveled in the trunk. You two will also travel in the trunk."
Karim continued to protest and Ariana reached into her pocket and pulled out a black rectangular device the size of a cell phone. Karim started to speak but his voice was cut short by 80,000 volts of electricity. As Karim's body arched and twitched, Abu tried to climb over his incapacitated traveling partner. Ariana quickly redirected the stun gun to Abu's exposed neck and he joined Karim in uncontrollable thrashing.
"Watch your head and arms," she said as she shut the trunk quietly. She took one look around the dark neighborhood and wondered if she would need to come back.
Ariana crossed the train tracks on Georgia Avenue and entered into a non-residential area highlighted by a dozen businesses bordering on bankruptcy. She turned left down an alley just inside the D.C. city limits near Takoma and snaked behind two old brick buildings. She rounded the corner near a lot with crushed cars stacked ten feet in the air, the rolls of razor-wire on the fence costing as much as the junk it protected.
At the end of a small road the lot widened into an open patch of asphalt and mud, a mixture of blacktop and exposed ground. She pulled the Camry over near the last warehouse on the well-hidden block. Unpainted cinderblocks with a brick façade supported a two-story metal roof. The building stretched forty yards across. On the left was a single bay door with two lone windows six feet off the ground and completely blacked-out from the inside with plastic trash bags and tape. In the middle of the warehouse stood two huge bay rollup doors, each large enough to park an oversized tow truck. The large windowless doors took up most of the front of the warehouse. Near the right hand side of the warehouse Ariana slipped her key into another solid metal, windowless door and disappeared as the spring-loaded hinge pulled the door closed behind her with a resounding thud. She took one look around the warehouse and checked on the Piedmont Delivery truck in the otherwise empty floor. She hit a green button on a hanging control panel and the door on the left chugged upward. She went back to her car and drove the vehicle into the car bay.
When the large bay door shut, she exited the car and walked to the rear of the vehicle.
"Rise and shine," she said, opening the trunk.
Karim nearly leaped from his confines, pulling himself out by the edge of the trunk.
"What the fuck was with the stun gun?" Karim asked, the veins in his neck and forehead pulsating. Abu joined in the verbal assault and tried to exit the trunk but his legs were asleep, the blood trying to find its way back into his lower extremities.
"The stun gun was a necessary evil. You'll both live."
"You touch me again and that is more than I will be able to say for you," Abu said, still in the trunk.
Ariana pulled out the stun gun and a blue electric current danced between the delivery prongs. "I'm the only one in the position to make threats," Ariana added, cooling Abu's temperament.
Karim helped Abu up by the armpits and both men looked around at the warehouse.
"Where are we?" Abu asked.
"Home," Ariana answered.
A small seam of light escaped from beneath a door on the far side of the concrete expanse. Karim followed Ariana toward the light as Abu limped slowly behind them. They approached the door and Ariana knocked once before opening it.
## Chapter 10
Officer Jim Singleton pulled himself out of his police car and walked up the driveway. He hated house calls. He had, in fact, ignored the first inquiry from a worried out-of-state relative who had tried to reach the resident in question. The man's employer called next, concerned that its star employee was incapacitated. When Officer Singleton got a third call from the man's brother, the fifteen-year veteran decided he would personally stop by the house in question on his way to lunch. Not because it was his duty, but because it was near his favorite gyro restaurant.
Walking up the three steps of the front porch, Officer Singleton, crumbs from breakfast still in his brown beard, knocked on the glass in the upper-half of the white storm door. "Arlington County Police, can you come to the door please?"
He waited for a moment and looked at the neighbors on both sides of Coleman's Castle. He knocked again. "Arlington County Police Department, I would like to have a word with you."
Officer Singleton's memory served up a fresh reminder of the time he kicked in a front door after being called to a house for domestic abuse. As the door smashed open, Officer Singleton had pulled his gun on a local doctor delivering a doggie-style, free medical exam to one of his office assistants. Knocking three times became the standard for all future house calls.
Singleton walked to the side door of the house at the end of the driveway. He knocked once more, and reached for the knob. The door swung open.
Singleton identified himself three more times from the new stainless steel kitchen. He listened intently for any sound as sunlight shined through the bay window on his dark blue uniform. The house echoed with an eerie silence. Five years pushing papers hadn't dulled his senses. He knew there was a body on the premises. The only thing left was to find it.
Out of respect for the dead, Officer Singleton announced himself as he entered every room. He scanned the original three-bedroom, one-bath layout, and stood at the top of the stairs to the basement. He looked up at the skylight over the new foyer in the back of the house, and walked up the staircase which opened into an office with a view of the neighborhood. Officer Singleton enjoyed the view just long enough for his brain to register the large body on the floor in front of the desk.
"Good God," he said, stooping to find a pulse on Allan Coleman's neck. At thirty-six hours after death, the victim was cold. Not in-the-refrigerator cold, but certainly chicken-on-the-counter cool. Officer Singleton wrestled Allan onto his back and looked into the grimace of pain still frozen on his face. His eyes were almost bulging, his mouth open in a painful, stretched grin. The officer looked around the room. He grabbed the tri-fold leather wallet off the desk and checked the driver's license. Allan R. Coleman. Singleton looked at the face of the deceased and compared it to the picture. The DMV-issued photo was far more appealing than the one now on the victim's face. He checked the contents of the wallet and pulled out three hundred and forty dollars. So much for a robbery.
He put the wallet on the desk and saw the white pills. His mind switched into detective gear and he noticed several more pills lodged between the keys of the keyboard. He grabbed one with his fingers and brought it to his nose. He scanned the work area, checked the floor, and found the prescription bottle on the other side of the victim's head. Nitroglycerin.
Officer Singleton called dispatch and reported that he had found the body of the individual reported missing earlier in the day. He took one look at the obese man, set the bottle on the table, and walked downstairs to wait. Case closed.
Clark heard the ring and searched for his cell phone under a pile of papers on the dining area table. He answered on its fifth ring.
"I thought I saw your car in the driveway," the voice said.
"Yeah, just arrived with another load of stuff."
"Got a few minutes?"
"What's up?"
"Grab something warm. I'll meet you out front."
Clark did as he was asked. Two minutes later he was standing at the foot of Mr. Stanley's driveway. The World War II veteran was ambling down the drive in a huge dark parka with a white furry fringe around the hood. Clark looked at his neighbor and realized just how far down the scale of importance fashion was for a man in his mid-eighties.
"Where are we going?"
"Around the corner. It's time to be nosey neighbors."
Clark walked with Mr. Stanley through the small blacktopped path that ran through county land between his house and the Krause residence on the far side of the dead end.
"You know, I remember when this park was nothing but woods. The trees were the only buffer between the W&OD railroad and the housing development."
"How long ago was that?"
"Most of the rail traffic was before even my day, but I think the last train passed through in, oh, must have been 1968 or so."
"Well, at least we don't have train whistles screaming through the neighborhood. I'll take a bike path anytime," Clark said, referring to the recreational area that now extended along the former tracks, stretching from D.C. to Leesburg, forty miles west.
Clark added, "I haven't been in this park in years."
"It hasn't changed much, except for the paths. Once they laid down blacktop, they opened the park to dog walkers and strollers."
"That might be better than the former clientele. When I was a kid, we used to find all kinds of stuff back here. Beer cans, condoms, underwear."
"I'm not sure that's any better than dog shit."
"Maybe, but some of those condoms we found were used."
"You can't prove any of them were mine," Mr. Stanley said, straight-faced.
Clark laughed loud enough for it to echo in the leafless trees. "Well, it was good to be a kid. We used to catch crayfish, play war, splash in the creek."
"Nothing like getting wet and dirty as only boys can do with God-given material."
"For a while, and then one day we found ourselves older, dumber, and more adventurous. I am not sure whose idea it was, but one day we decided it would be more fun to build a swing to _go over_ the creek than it was to trounce through it. I learned an important lesson that day."
Mr. Stanley pulled the left side of his parka back away from his ear so that he could hear Clark as he talked. "Which was?"
"If you are going to build a rope swing, don't use a garden hose. Particularly if your friend is Jimmy Shultz who weighed a hundred and fifty pounds in the sixth grade. Hoses stretch more than you think. Not only was that the day I learned about the elasticity of a stolen garden house, but it was also the day that I learned the expression 'to get racked.'"
"Sounds like Jimmy Shultz learned the real lesson."
"I guess you are right about that."
"You know, you were probably the last generation to play outdoors. Nowadays, between video games and child molesters, kids don't play outside."
They approached an intersection of paths and Mr. Stanley took a right. A minute later they were on the street behind Dorchester Lane. An ambulance with flashing lights was parked in front of Allan Coleman's house. Before Clark asked, Mr. Stanley answered.
"I saw the flashing lights out the back window. Figured it was my duty to at least come and see what was going on."
"Your duty?"
"My duty as a good neighbor."
"I thought it was nosey neighbor."
"Call it what you want."
"I just did."
Clark had yet to win an argument with Mr. Stanley. The reason was simple. The cagey neighbor didn't fight fair. Whenever the tide in the debate turned against him, Mr. Stanley quit talking. Sometimes he blamed it on his aging ears. Most of the time he simply acted as if the conversation was over; and if one-half of a two-person conversation deemed it over, it was.
Coleman's neighbors from the next street over huddled near the corner of the dead man's property line where a chain-link fence met a small wooden post. A middle-aged woman wearing only a sweater hugged her teenage children for warmth, support, or both.
Clark talked with Mr. Stanley as they strolled up the sidewalk, still fifty yards away from the scene. "I think the show is over."
"Why do you say that?"
"No one seems to be in a hurry. The EMT driver is sitting behind the wheel, talking on the radio. The cop on the front porch is more concerned with his nicotine intake than with what's going on in the house."
As if on cue, Officer Jim Singleton cupped his hand over the end of his cigarette trying to light it against the wind.
Clark stepped toward another middle-aged mother, appropriately dressed in a black down jacket. "What happened?"
"Looks like Mr. Coleman had a heart attack," the woman's son said. His eyes were glued to the side door of the house. His mother hushed him and put her fingers over her lips.
"How long has he been dead?"
"The cops won't say," the boy answered through teeth with braces.
The mother rolled her eyes and looked down at her son. "It appears that Mr. Coleman died in the last day or so. Natural causes. He obviously wasn't the most health-conscious individual."
Everyone's suspicion was confirmed with the arrival of the long black car that opened from the rear.
"This is going to be interesting," Clark whispered to Mr. Stanley. "This guy was huge."
Mr. Stanley flicked his head and Clark followed. "Good afternoon, officer. My name is John Stanley. Former Marine Captain. What's the story?"
"Retired Marine?"
"Yes sir. Retired with enough shrapnel in my ass and legs to set off the detectors at the airport."
Clark looked at Mr. Stanley and bit the inside of his cheeks.
"Did you know Mr. Coleman?"
"He was my neighbor. We exchanged greetings from time-to-time."
"Well, Captain, Mr. Coleman passed away sometime in the last two days, most likely from natural causes."
"Who called you?"
"His brother and his employer."
Officer Singleton was summoned from inside the house. "If you would excuse me."
Inside the house, it took four people to get Allan Coleman into the extra-large body bag and six to get him onto the stretcher. With the joints of the stretcher protesting, the six adult men struggled to maneuver Coleman's three hundred and seventy-eight pounds of deadweight down the stairs.
Inside Coleman's office, no one seemed to notice the stack of voyeur-themed porn on the floor in the corner under a recent _Time_ magazine, or the high-powered binoculars that rested on the windowsill overlooking the neighborhood.
## Chapter 11
The natural light from the outside was tracking across the ceiling in the morning sun, waking the cavernous room from its gray darkness. The new rays of indirect sunlight struggled to push through the windows near the top of the wall into the garage turned warehouse.
Abu and Syed were busy cleaning, their arms tired, their fingers wrinkled from the damp towels they had used to wipe most of the surfaces of the warehouse, as far up as they could reach. Karim was on his hands and knees with a scrub brush, working to remove hardened bird droppings that dotted the floor near the base of a main support beam. Ariana walked by and looked up at the ventilation hood above. She put it on the list of things to secure.
James drew the short straw and got the bathrooms, of which there were two. The American Muslim was at work on the small bathroom near what would become the group's sleeping quarters. The single sink and toilet were covered in mildew from months of non-use. The shower stall was a science experiment-in-progress, a green fungus growing out from the grout lines to reach half-way across each tile.
Next to the crude bathroom facility was another heavy steel door. Ariana opened the door and the hinges squeaked. A wave of must greeted her. She checked the floor for water and touched the wall. It was cold and moist. She looked up at the window twelve feet above. The confining walls and the lone trace of light above sent Ariana back in time. She shut her eyes and saw the steel door shut behind her. She felt the club come down on her head and the boot hit her in the ribs. She shivered, and opened her eyes, thankful she had avoided the memory of her final lesson in her training years before. Thankful for unconsciousness as she lost the last remains of her innocence.
Mentally back in the room, she walked off the dimensions between the walls. When she was finished she smiled, the specifics of the warehouse, down to the cubic-foot space, were as she memorized. But for some reason the room she was in now felt smaller, tighter. With her soldiers in mind, she imagined the configuration of the barracks. She stepped out of the room for a moment and tried to pinpoint the direction of Mecca with as much accuracy as she could. The far right corner, she thought to herself. She smiled. With bunk beds, she could have received twice as many men.
Satisfied, Ariana made her way across the warehouse, glancing at her Toyota Camry and the Piedmont Delivery van that occupied the garage turned warehouse floor. She put her key in the lock, entered the small office in the corner and shut the door behind her. She flicked the lone light switch on the wall near the door and the ten-by-ten foot room received its first light in three months. The glass walls of the office looked out onto the floor of the warehouse. Two sets of metal filing cabinets stood in the corner side-by-side. A lamp and a black phone sat on the desk in the middle of the room. The outdated calendar on the wall depicted a large-busted woman with a handheld drill bending over the side of a motorcycle. It was hard to tell if it was an advertisement for the motorcycle, the tool, or for silicone implants.
Ariana noted what she needed to have a business up and running. Paper, pens, a computer, folders. Her eyes looked around the room, envisioning a fully functioning office. Or at least a room that would give the appearance of one. She picked up the landline phone and the dial tone greeted her ear. She put the phone back on its cradle and checked the desk drawers. They were locked, just as she had left them.
She opened the office door, looked around at the warehouse and her human resources. Then she curved her thumb and middle fingers to make a ring, pushed them to her lips, and let out a whistle that froze the room.
"Everyone stop what you are doing."
Karim stood from his battle with the dried bird crap on the floor. Abu and Syed dropped their rags. James emerged from the bathroom on the back left.
"I wanted everyone to gather so we can go over some ground rules."
"I hope it doesn't include a weekly cleaning list," Syed said, rubbing his left triceps.
"It does. But hopefully we won't be here too long," Ariana paused for a moment, waiting for James to get closer to the office door, and then continued. "Welcome to our home away from home. As you can probably tell by the hydraulic lifts in the floor and the other interior design features, this warehouse was once an automotive garage. It has a long-term lease, paid through an untraceable party, and it is as safe a haven as we are going to find. There are no neighbors, but we are in an urban area. You will have to take my word for this. If I catch anyone stepping outside the confines of these walls, I will kill them."
She didn't wait for a reaction.
"I chose this location for a variety of reasons, so if anyone has any complaints, look no further for a target for your ire. I have been here off and on for the last year or so. Checking on the place, making sure the electricity and water still function. I also cleaned from time to time, believe it or not."
"There are six rooms in total. We have the main warehouse floor, where we are standing. There are two bays with twenty-foot doors. Large enough for any operation we may need. Behind me is the office. It has a working landline phone that none of us will use. I will make some acquisitions for the office in an attempt to modernize the atmosphere. But by and large the office is to maintain appearances only. None of you will go into the office unless accompanied by me."
Abu and Syed rolled their eyes.
"To your left, my right, there are three doors on the wall. As you have discovered, the room at the back of the warehouse is a bathroom. There is a shower stall, a toilet and a single sink. I expect everyone to keep this room clean. As well as yourselves. I cannot have my men smelling like they haven't bathed in a month. It would draw unneeded attention when the time comes to execute our plan.
"The door next to the bathroom is your sleeping quarters. There is a window. I have affixed a mihrab on the wall so that you will know which direction to pray. I will also get a space-heater, some beds, and other items so that you will feel comfortable. There should be no personal effects from anyone outside of the sleeping quarters. I will be sleeping in the office.
"The oversized third door on the left side is locked and is to remained locked. Its contents are of no consequence to any of you. Ditto goes for the truck. The door is locked and I have the only key."
Done with the left side of the warehouse and the office, Ariana walked towards a set of double doors on the right side of the former automotive floor.
Ariana explained as she went through the motions. "Beyond these double doors is a smaller room that was owned and operated by a printing company." The doors swung open and the men stepped forward to look down the length of the room. "There is a door at the front of the shop, but it has been sealed shut. It cannot be opened by normal means. Don't waste your time trying. There is a bathroom in the back, also with its own shower. I had these double doors installed when the lease for the property was executed."
"Why do these places have showers?" Syed asked.
"Hygiene and safety precautions. A lot of facilities that work with chemicals have showers for emergencies. Which probably explains the print shop. In any case, we have two bathrooms at our disposal. For now, we only use one."
Syed shrugged as if the answer were sufficient.
Abu interjected the obvious. "But if we have two..."
"We use one," Ariana said without negotiation.
"Any questions so far?"
The men fumed and said nothing.
"Good. Until we are operational, when I am out of the warehouse, I expect all of you to remain in the sleeping quarters. Consider it home. If you must use the bathroom, please do so one at a time, and be brief. Also, feel free to use the bathroom for wadu. We will all need our faith. And keep the talking to a minimum. Discuss the weather. Nothing personal. Nothing revealing."
The men looked at each other suspiciously.
"Understood?" Ariana asked.
The men nodded.
"Ok. Let's not lose sight of why we are here. I will also speak with each of you individually. Later this evening I will be running some errands and picking up a few things. If there is something that you absolutely must have, let me know and I will consider it. Now, get back to work."
The group grumbled as it broke and returned towards their work. Ariana stepped forward and grabbed Karim by the arm. "Not you. We need to have a word." Karim followed Ariana into the office and Abu and Syed paused long enough to peek back over their shoulders.
"Please have a seat," Ariana said as she shut the door to the office.
Karim did as he was told and forced his backside onto an uncomfortable wooden chair.
Ariana sat down and opened her purse. She slowly slid a pre-paid cell phone across the desk in the direction of Karim. "If anything happens while I am out, you will be responsible for calling me."
"Of course," Karim answered. He looked over his shoulder at the warehouse and could feel Abu and Syed's eyes straining to see through the open Venetians as they resumed wiping the walls. Ariana shook her head slightly, and Karim understood her warning to be careful.
"Unless you have to contact me, I want you to keep that phone powered off and out of sight."
Karim looked at the phone and opened the cover. "Untraceable?"
"Completely. Especially if we use them only once."
"Are you sure?"
"Positive. I've tested them from time to time. Not only are they untraceable, but if you call someone, the display shows only 'incoming call.' The laws are changing quickly though. In another year it will be impossible to get a pre-paid phone. If you plan on using a legitimate ID anyway."
"How are the other preparations?"
"Fine. I still have things to do, and will need some time to make some meaningful purchases."
"You look worried."
"There is a small complication."
"How small?"
"I'm concerned that someone will start to notice a family has gone missing. I need to call a neighbor."
"That's not a good idea."
Ariana thought for a moment before concluding. "We have no choice."
Karim noticed the use of the word "we" and he smiled.
Ariana continued. "A family doesn't just get up and leave their house unattended. Neighbors will start to worry. People will start to ask questions and get suspicious. Someone will call the police."
"I see your point."
"And with a child missing, well, you are talking about fodder for national news."
Karim rubbed his beard and stared out the window of the office. "Sounds like you could have been better prepared."
"I had no warning. My family was one part of the equation that was fluid."
"I understand. Make the call to your neighbor, but be brief, Ariana. We have come too far. We have waited too long."
Ariana nodded, though it was not in deference. "If the others ask what we were talking about, tell them we were discussing special skills you have or training you have received. I will ask each of them in turn so that no one is overly suspicious. Once I am done with their debriefing, I will need to run out for supplies."
Karim got up from his chair and slipped the cell phone into his pocket. As he headed for the door, Ariana added, "Tell James to hurry up with the bathroom and tell Abu he is next for his debriefing."
Ariana hung the CVS bag on the hook on the wall. The narrow bathroom with the single shower stall was designed for mechanics to clean up after work. If the walls could speak they would attest to having never seen what was unfolding before their grout-laden eyes. She dug through two hundred dollars worth of self-improvement and then loosely separated them into different piles on the freshly cleaned floor.
She placed a bag of clothes on the top of the closed toilet lid. She removed the outer layer of her black headscarf, revealing most of her cheeks, her ears, and the side of her neck. She pulled the inner tube of her hijab from the back of her head and her dark brown hair fell to her shoulders and radiated in the light from the bulb above the sink.
Her bushy eyebrows were the first to go. It was the one thing she couldn't change with Nazim as her husband. Clothes could be changed. Hair could be curled. Glasses could be removed. But semi-permanent change was something she couldn't allow herself until now.
She plucked her eyebrows, one hair at a time. The first pluck made her eyes water. As she continued, the small pricks of pain became therapeutic. When she was done plucking the meat of the brows, she shaved the edges with an eyebrow razor. She splashed water on her face and admired the high arching brows that had taken the place of her previous untamed, near uni-brow, pair.
Next she picked up a box with a high-priced fashion model on the side. She opened the tab on the box and pulled out a plastic cap and tied it over her hair, the strings on the cap tying under her chin. With the same pair of tweezers she had used to assault her brows, she spent half an hour pulling strands of her hair through the holes in the tight-fitting shower-cap like covering. When she finished, she read the instructions on the back of the box for a second time. She pulled out a small bag, prepared the concoction as prescribed, and pasted a mix with the consistency of honey onto the strands of hair she had pulled to the outside of the rubber cap.
Forty minutes later, Ariana admired herself in the mirror.
Her knee-length skirt was the most revealing thing she had worn outside of the bedroom in decades, before she knew better. Her low-cut black sweater with the swooping neckline showed off a very perky set of C-cup proportions, the strap of a new black bra peeking over the edge of her shoulder. Her make-up had been a labor of love. It had taken a few tries to get the combination just right: eyeliner, eye shadow, blush, and red lipstick. Too much make-up and she would look like a hooker. Too little, and it would defeat her efforts to look like anything but a conservative Muslim woman. She jettisoned her glasses for her contacts. She put a cubic zirconia on her neck and the sparkling diamond look-alike fell nicely between her breasts. Just another attention grabber to get the male population staring at her cleavage. Clairol Auburn dye #7 had flavored her hair just enough. She was no longer a Middle Eastern housewife. She was an Americanized woman of unknown ethnic background.
And she was a knockout.
## Chapter 12
FBI Agent Chris Rosson was sitting in a small room with a glass wall. A new hire to the Bureau fidgeted slightly in his chair across the table.
Agent Rosson's perfectly combed gray hair told the new recruit nothing about his age. The agent's first gray had appeared at the widow's peak on the left side of his head at the age of fifteen. He plucked it out after the novelty had worn off, but another gray, joined by a battalion of identical invaders, soon took its place. In high school, his friends would pull them out in algebra class, during recess, in detention. All to no avail. By the time he was twenty, Agent Rosson had more grays than his grandfather. His Aunt Millie told him that he wasn't the first. There was a great uncle in Atlanta who was gray and dead by his mid-twenties. At fifty-two, Rosson had at least avoided the latter.
The new hire was a recent graduate of the Academy and, in the Bureau's expanding effort to match skills and interest with position, the wet-behind-the-ears bureaucrat was making the rounds and asking questions to personnel from different sections. As the new recruit fired off various inquiries from the FBI-approved hiring manual list, Agent Rosson thought about interrogations, his mind transforming the clear glass interior wall into a two-way mirror.
"What was the most interesting case you ever worked on?"
There were few things Agent Rosson did better than reminisce about the good old days in the Bank Robbery Division. "The Cowboy Bandit."
"The Cowboy Bandit?" the new hire with tightly cropped dark-blonde hair repeated.
"That's right. The Cowboy Bandit. This guy robbed dozens of banks, year after year, in full view of the camera, and we never even got close."
"He was never caught?"
"Never caught, but later identified," Agent Rosson clarified, taking a sip of coffee from the white Styrofoam cup on the table. His jacket was back in his cubicle; his shirt wrinkled, obviously the second day it had been worn since its last trip to the dry cleaners. Single, he wasn't opposed to wearing his shirts for three or four days, if the weather was right and the pits didn't get too sweaty. He had ironed his own shirts once and vowed never to do it again. Not when the Korean dry cleaners down the street was charging eighty-nine cents a pop.
"I'm not sure I follow."
"The Cowboy Bandit left a notarized letter for the good people of the Bureau in his will documents."
The young new hire, a conservative white kid in a suit that had been on a store rack the day before, looked surprised. "How many criminals inform the Bureau of their deeds after they die?"
"It happens. But this was the only time I was personally involved in a case that did."
Agent Rosson sized up the new agent. "Sure would make it easier for us to solve the crimes if everyone took that approach."
"Yes, sir. I guess it would."
"You don't have to call me 'sir.'"
"Yes, sir."
Agent Rosson shut his eyes briefly and shook his head. Then he continued. "The interesting part was that the Cowboy Bandit was actually a woman. She dressed like a man, wore a fake beard, big sunglasses, even looked like she had a chew of tobacco in her cheek, though she never spit. Not that it would have mattered. This was long before DNA testing was used for law enforcement purposes."
"No one ever theorized that she might be a woman?"
"You would have thought. But then again, she would have been the _Cowgirl_ Bandit. All I know is I spent a thousand hours watching bank security films. Looked at hundreds of still shots. Saw her from the front, the side, the back. She was good. She missed her calling as a male impersonator in Vegas. Probably could have made as much money as she stole."
"How much did she get away with?"
"Forty-one banks in eleven years. I think the total take was just under $600,000."
"That comes out to about sixty grand a year."
"Give or take. All the banks were in Texas. Started in San Antonio, moved to Dallas, went back down to Austin, and then hit sporadically all over from El Paso to Houston."
"They mentioned in the Academy that half of all bank robberies go unsolved."
"True. The number was even higher in the eighties. I think the success rate peaked at a seventy four percent in 1989. Hell, there were times back in the day when I thought about robbing them myself."
"Why so high?"
"Lots of reasons. At the top of the list was the similarity between the M.O. of most bank robbers and customer behavior. The average bank robber writes, or has something written on a piece of paper, approaches the teller, and waits for the teller to process the transaction. Every customer in a bank does essentially the same thing. There is no suspicious behavior. Bank employees are told not to resist and to hand over the money. So for all intent and purposes, the transactions look normal. Banks are also typically located in high traffic areas. No one wants a bank in the middle of nowhere because a bank needs to be where the customers are. Banks are in the parking lots of shopping centers, on major roads, in the center of town. They are transient in nature. The average bank customer parks his car in the parking lot, walks in, does a transaction, and is gone in two minutes. Ditto for the average bank robber. The guys who get caught are usually the ones who go in with guns blazing."
"What about the silent alarms and dye packs. The academy says they're standard in all banks."
"The silent alarms are good, if you can get to them unnoticed. They are usually located far under the counter, or in a drawer, or under a cover, so the teller doesn't trip them accidentally. You can't have them right under the lip of the counter or too close to the teller's feet. Banks did for a while, and the cops were forever chasing false alarms from tellers with big feet and fat thighs. If a bank robber lets the teller know that he is watching for any movement towards the silent alarm, the teller usually won't hit the button until the robber is out the door."
"And the dye packs?"
"Even a bad thief can spot one pretty easily. They feel different. If you hold a stack of bills with and without a dye pack, you would be hard pressed to find anyone over the age of eight who couldn't tell the difference."
"So why did you give up on bank robbery? Why the Anti-Terrorism Task Force?"
"The Executive Director for Counterterrorism came into the room during a monthly Bank Robbery Division briefing and asked if anyone was interested in two of the FBI's newer divisions: Anti-Terrorism and Cybercrime. There was a lot of talk about refocusing the FBI towards bigger and badder criminals. Me and a couple of buddies raised our hands."
"That was it?"
"That was it. Rubbing a magic lantern and having a genie pop out wouldn't have been any faster."
"And the training?"
"We had a few classes on the structure of terrorist organizations, sources of funding, myths about religious fanatics. The simple truth is that we are in new territory. There are no rules and sure as hell no rulebook."
"What do you do, day-to-day?"
"Read the news, follow leads, most of which are dead ends, write reports. Enter data into the tracking database."
"Sounds exciting," the young man said taking a sip of water.
"There is nothing glamorous about the day-to-day operations of the Bureau. The busts are great, but there is a lot of legwork needed to get the bust. Unless you get lucky."
"How many times have you been lucky?"
"A dozen, though to be fair I should probably subtract two from my lucky count. One for each of my ex-wives."
The young man looked at the clock on the wall. "Time is up."
Agent Rosson looked at his watch, an old clunker that went through batteries like a moth in a closet of wool sweaters. "If you have any follow-up questions on the Bureau, let me know."
"Thank you, I will. I have two more sessions today and need to put in for my preference by the end of the week."
"Don't get your hopes up. There is still a good chance the Bureau will ignore your preference."
"I'll keep that in mind."
"Good luck to you," Agent Rosson said standing. He threw his empty cup towards the trash can in the corner and it bounced off the rim onto the floor.
"No, good luck to you," the new hire said as he watched the cup find the floor.
Chris Rosson, thirty-year FBI agent and three years from mandatory retirement, banged his computer mouse on his cluttered desk and tried to steer the cursor towards his email inbox.
The first email in his inbox was a notification that his request for a new computer had finally been processed. It had only been nine months since the current dinosaur on his desk had frozen for the first time, a small snafu which forced him to spend two days with a geeky kid from technical support with questionable hygiene. Per the email, Agent Rosson was now on the computer waiting list, a mere two months from new hardware. In the meantime, he practiced hard love with his six-year-old Compaq.
"Piece of shit," he said loud enough for the occupants in neighboring cubes to hear.
"The List" came via email every Monday. One hundred and eleven was the magic number for the week. One hundred and eleven calls and letters made to the CIA and passed forward to the FBI.
Every Monday morning the Director of the Joint Terrorism Task Force, the JTTF, received the email and perused it with his morning coffee. This morning was no different. Between a call to his wife and a visit to the barber in the basement, the Director sent the email to his chief bureaucrat underling who divvied the leads and sent it on. All the leads from the CIA were handled locally by members of the FBI's JTTF. The Director was adamant about the leads not going to field offices. He wanted the responsibility for that little list to remain in the building. After a career in the Federal Bureaucratic Institute, he knew it was hard to tear an agent a new ass long-distance. And if something on the list were overlooked or mishandled, there would be a long list of asses in jeopardy.
Chris Rosson was on the rotation for "The List," the only moniker for the CIA email that wasn't too offensive to utter in public. For the last four months, Rosson had averaged fifteen leads a month. Not a large number, but still enough work to be an inconvenience when added to his usual duties.
But the list was more than just leads. The FBI had thousands of leads of their own. Tens of thousands. And with the current number of Anti-Terrorist agents in flux between a hundred and a hundred and fifty, the ever-growing list of leads would always exceed manpower.
Not that the Bureau wasn't trying. It had hired hundreds of analysts since 9/11. They were packed into office buildings around the Beltway from Tysons to Clarendon to Pentagon City to Bethesda. Young men and women with no knowledge of the world but with college degrees and computer skills hired by defense contractors with innocuous names like Enteon, Pitre, MACI. Companies that produced nothing but paperwork.
These college grads worked nine-to-five prioritizing leads according to a matrix that an old man with a security clearance and no intelligence background had created in a room full of equally under qualified nodding heads. Leads were initially filtered by concrete intelligence parameters such as suspicious name, suspicious location, backgrounds related to the Middle East. For a college grad from Indiana, everyone one except Larry Bird fit the bill. On more than one occasion, usually during happy hour, Agent Rosson had publicly claimed, "If the U.S. avoids another 9/11 in my lifetime, it will be just dumb luck."
So far the luck was holding out.
Rosson clicked on the email and opened the attached Excel spreadsheet. He looked for his name and saw the four new leads he had this week. Two from New York, one from Boston, and one from Detroit. He printed out the three-page document and perused the rest of the list for kicks. All the information would have to be entered into the FBI tracking database, each with a case number and an assigned agent.
On the third page he stopped on the last entry. The street name seemingly popped out from the page. He checked the name of the agent assigned to the lead and then looked over at the stack of files in the corner tall enough for a full-grown, armed-to-the-teeth terrorist to hide in. _What the hell,_ he thought. He bound from his desk and followed the maze of blue and gray fabric walls to the far side of the floor to a row of cubes with a semi-obstructed view of Pennsylvania Ave.
"Good morning, Agent Taylor," Rosson said with a smile. The middle-aged agent with wire-framed glasses looked up.
"Agent Rosson. What can I do for you?"
"I saw you on the list from the Agency."
"The shit list stops for no man."
Rosson nodded. The headache he had woken with was finally subsiding. "I wanted to know if you'd be interested in swapping a lead with me."
Agent Taylor laughed from his seat in an impeccably organized cube. Pens and pencils were lined up next to perfectly stacked legal papers. "Swapping a lead? What, do you have something in Miami?"
Agent Rosson smiled, his gray hair almost shining in the reflection from the light directly overhead. "No. I have the less desirable but still promising areas of Boston, Detroit, and New York."
"Christ, Rosson. It's winter. If one of those leads pans out I might have to head north. We are in the middle of one of the coldest seasons in memory, in case you haven't noticed." Agent Taylor knew he had a fish on the line, but he wasn't sure if he wanted to catch it or not. "Which of my leads are you interested in?
"The one in Virginia."
"Hell, no," Taylor responded. "Hell, no. That lead is almost on my way home. Besides, the director plays by the book. He takes that list very seriously. You do the investigation for the leads assigned to you. I'm responsible for that lead and for entering it into the task force tracking system. That system generates a report that goes to the director."
"Well, actually, there is a provision that would allow me to request the lead from the director."
"The 'special knowledge' provision? Unless this lead has something to do with a terrorist robbing a bank, I doubt you would have a chance."
"I do have special knowledge that could assist with this lead."
"What's that?"
"The house on the list is on Dorchester Lane. I sort of know the neighborhood. I grew up a few blocks away."
Agent Taylor looked up at Rosson from his seat and peered intently through his glasses.
"No swapping, but if you want it that badly, I'll give it to you."
"OK."
"I'm not finished," Agent Taylor said, pushing his glasses up his nose. "If I agree, you have to get permission and fill out the paperwork for both of us. Then you have to do the data entry. And I want to be copied on any related correspondence."
"Fine," Agent Rosson said.
"Besides, it's probably nothing. Most of the leads from the Agency are good American citizens who have watched too many movies and think that the local cab driver they had that morning was planning an attack."
"Just the same, thanks."
"No, thank you."
Agent Rosson walked away dreading the exception report he would have to file just to make one phone call to Maria Hayden.
## Chapter 13
The smell from an open bag of Taco Bell chalupas hung in the air around the small table in the back of the store. Boxes of hiking boots, winter hats, and gloves lined the far side of the room, inventory that had arrived in time for the latest snowfall but which the crunchy granola generation staff had yet to put in order. A set of surveillance monitors rested on a shelf above the water fountain near the hall that led to the emergency door at the back of the massive outdoor adventure store.
Rick Peterson, his long blonde hair pulled into a ponytail, looked up at the monitor that relayed an image of the retail floor near the camping equipment. Without taking his eyes off the monitor he guided his backside onto an empty metal chair. "You owe me three bucks," he said to his coworker, Bruce, a college graduate with a peach fuzz goatee and no plans for the future other than to see the world. At minimum wage it would take him a decade.
"You still owe me five from last week."
"Then I owe you two. Unless you want to pay me the three you owe me for today, and I can pay you back eight next time."
"I will take my chances with you owing me two."
"You don't trust me?"
"Should I?"
"If you want to eat," Rick said, pushing the bag of fast food towards him on the table.
"Then I guess some trust is in order."
Rick pulled out a chalupa and slid it across the worn black table.
"What do we have here?" Rick said, his ponytail draped on his shoulder.
Bruce turned his head towards the surveillance camera. "She's wearing a winter coat. You can't even see her ass."
"Oh, I can see her ass. I have x-ray vision for asses and perky tits."
"What's the wager?"
"Ten bucks says I can get her to take off her jacket. Another ten says she has an ass you would grab, and tits you wouldn't know what to do with."
Bruce took another bite of his chalupa, hot sauce squirting out onto the corner of his mouth. His eyes were fixated on the screen. "Looks like I won't need to bet you. She's taking off her jacket."
On the monitor, Ariana peeled back her winter coat. Her push-up bra settled half of the bet that wasn't made.
"Nice tits."
"Perfect."
Unaware of the surveillance camera, Ariana put her jacket in the crook of her arm and bent over to look at lanterns on the lower shelf.
"And a great ass too."
"It'll do."
"It'll do? Dude, you're not someone who can be too picky. You would do crusty Sarah in woman's shoes, and she hasn't gotten any since you were born."
"I don't like women with dust on them."
"Fifty bucks says I can get this girl's phone number."
"You can't even pay for the dollar lunch menu at Taco Bell."
"Twenty."
"Done. But I want some collateral."
"I'll show you collateral," Rick said grabbing his crotch. He brushed crumbs from his shirt and took a long slug from the straw in his pink lemonade. He paused for a moment, gargled with the drink in his mouth, and left the back room to enter the floor of the store.
He beelined it for the camping gear, zooming past a rack of NorthFace fleeces and Gortex outerwear guaranteed to keep you warm and dry in a blizzard at thirty below zero. Ariana was standing at the end of the aisle, a small stove in her hand.
"Can I help you?" Rick asked, his eyes devouring the woman he saw as prey and a ticket to a quick twenty bucks.
"No, thanks. I'm just looking," Ariana said, glancing slightly down the aisle before turning her attention back to the display of stoves.
"Going camping?" Rick asked, unfazed.
"Maybe," Ariana answered without looking over. "And I don't need any help."
Rick heard the hint and chose to ignore it. "Well, choosing the right stove can be tricky, depending on what kind of camping you want to do. Some of these work better at a higher altitude. And of course weight has a lot to do with your decision. You don't want to carry more weight than you have to."
"I see. Thank you for your time."
"Where are you going camping? I have been hiking all over the U.S. Been to Everest base camp. Climbed Machu Pichu last spring. If you take the original Inca trail it is a real test of endurance."
Ariana moved to the next stove.
"You don't want that one. It's more expensive than the others, but the canisters hold less gas."
"Thank you."
In the back room, Bruce watched as his coworker tried desperately to make a connection with the woman on the screen. The twenty bucks would be a small subsidy for his weekend beer bill.
On the floor, Ariana had reached the point where not answering the salesman's questions could be seen as something to be remembered for.
"Which do you recommend?" Ariana asked, thawing without getting too warm. "There are so many to choose from," she added, still giving the store clerk as much of a profile view as she could.
"Where're you going?"
"I'm going camping in Havasupai, Arizona."
"I love Havasupai. The waterfalls are just awesome. Have you been?"
"No, I've never been. But I've always wanted to go to the Grand Canyon."
"Well Havasu is not really the Grand Canyon. I mean, it is better than the real Grand Canyon, but not the same Grand Canyon you see in travel brochures and in movies. Havasupai is on the west end of the Canyon. You have to hike ten miles and go through an Indian reservation. It is an oasis in the middle of the canyon really."
"I've seen pictures."
"Where are you from?" Rick asked.
Ariana didn't answer the question and smiled instead.
"I noticed a very slight accent. Maybe Pakistani. I have always been good with languages and accents. One of those gifts that I am not sure what to do with."
Ariana turned the attention back to a stand-alone gas burner. "How about this one?"
"That one is pretty good. Easy to use, easy to travel with. Replacement canisters are easy to find. It doesn't have the largest flame base, but if you are looking for something light, that's a solid choice."
Ariana ran through her mental checklist of things to buy while considering whether to break the salesman's neck in the middle of the camping supply section. With a foot sweep and directed downward momentum, she could make it look like an accident. Except for the cameras. And her need for supplies.
Rick plowed forward. "So you didn't tell me. Was I right? Are you from Pakistan?"
_Are you kidding me with this guy?_ Ariana thought. She hadn't had anyone comment on her accent in five years. Her fluency was native and she had studied long and hard not to have an accent. "Something like that," Ariana said.
"I knew it. We should have made a bet."
"What kind of bet?" Ariana said, now facing Rick. Her eyes twinkled with an energy that Rick felt in his Royal Robbin hiking pants.
"Dinner."
"Then I guess you would have owed me dinner."
Rick was too keyed up on testosterone to realize he was facing a thousand potential ways to die, and none of them involved whips, chains, or bed sheets.
Ariana pulled the packed Camry into the parking bay on the right next to the Piedmont delivery truck. The four men emerged from the sleeping quarters when the driver's side door shut.
"Help me unpack the car," Ariana said, holding a paper bag full of spray paint and another bag with painting supplies. She was slightly irritated and the four men noted the change in her usual cool demeanor. Maybe it was the tight skirt and sweater. A change in appearance can alter one's attitude.
"What did we get?" Karim asked.
"I bought cots to sleep on. Sleeping bags. Backpacks. Duffle bags. I bought two camping stoves, some pots to cook in, some pack-and-go plates and silverware. I have enough food in the trunk to feed five people for a month. I got identical watches for everyone. We will have to synchronize the time on the watches down to the second. I also picked up some office supplies and two space heaters."
"And in the bags? Karim asked, motioning towards Ariana's arms.
"Spray paint for my car. Time to go from beige to blue. It doesn't have to be pretty, just a different color. We also need to paint over the side of the delivery truck. White."
Abu went to the car to remove the food. Syed and Karim helped unload the backseat. James pulled the foldable cots off the floor of the backseat.
"Put the food in the corner. That will be our kitchen, Ariana directed as she walked towards the office. "It's near the bathroom and we won't have to drag our dishes across the warehouse to wash them."
As the men took trips to unload the food and equipment into the designated areas, Karim approached the office door. "Did you run into trouble? You look perturbed."
"Nothing I couldn't handle. But I need to limit my visits to the outside world. There's too much potential for trouble."
"What else do we need?"
"There are a few things that I need to have delivered, but I have to purchase them in person."
"Regulated items?"
"You'll know when they arrive."
## Chapter 14
Maria Hayden moved between the dining area and kitchen, her hands never empty. She was busy answering questions and, more importantly to her, playing the role of host for her only guest. The buzz of the alarm clock at five-thirty in the morning had done nothing to quell her enthusiasm. She saw the alarm as the starter's pistol to a day in the kitchen, a race to prove her cooking prowess on an unfamiliar stomach. A mixer with rapidly drying dough remnants on its blades rested on the counter next to an old cookie jar with a cracked lid. A rolling pin covered in flour protruded from the single basin sink. A glass jar of sugar was open, its red cap pushed to the back of the stove. And if a half-dozen desserts weren't enough to keep her occupied, she had started a batch of her top secret, garlic-laden spaghetti sauce.
Maria finished sprinkling a heavy dose of confectioners sugar on a cake fresh from the pan and brushed her hands on her apron. She rounded the corner from the kitchen and filled Agent Rosson's cup with another tank of dark brew.
"Cream?"
"Yes, please," Agent Rosson answered for the third time in as many cups. He was being served, and served well, but he had already noted in his notebook the mental lapse with the repeated question about cream in his coffee.
Maria reached for the milk on the table and poured it into her guest's mug.
"Do you mind if I ask you a few more questions?" Agent Rosson asked.
"No, not at all," Maria said. "Have I mentioned how wonderful your hair is? It is so thick. Thick and fabulously white."
"No, you hadn't mentioned it Ms. Hayden. Thank you for the compliment. A full head of gray hair has few admirers."
"You look like Sean Connery."
Agent Rosson wasn't sure about the actor having either gray hair, or a full head of it, but kept the speculation to himself. He hadn't made much progress in the hour he had been sitting there. Maria Hayden was the master of redirect. Not knowing if it was intentional or not, Agent Rosson plowed forward by going back to the beginning one more time.
"When did you see the three men in question?"
"Oh, there is no question about it, I saw three men."
Agent Rosson grinned. "And when was this?"
"A few weeks ago. A weeknight. Maybe a Tuesday."
"And could you describe them?"
"I thought I did. They were Middle Eastern. Dark skin, dark eyes. One was tall."
"And you saw them through the peephole?"
"Yes, I only saw them through the peephole."
"Then what happened?"
"They knocked on the door and I watched them for a minute. Then they said something about hospitality and disappeared."
"Disappeared?"
"Vanished into the night," she answered with drama.
"I see."
"Would you like another muffin?"
"No, thank you."
"Do you have any other ethnic neighbors?"
"Sure, sure. There is a German couple up the street and a Korean family that bought two houses next to each other on the next block. And then there is Nazim and Ariana, across the street, 203 Dorchester.
"Nazim and Ariana?"
"Yes. They live next door to Mr. Stanley. Nicest couple you have ever met."
"Are you sure it wasn't Nazim and — what did you say — Ariana? Are you sure it wasn't them at your door?"
"Positive. I said it was three people, didn't I?"
"Yes, you did." Agent Rosson paused for a bite of a scone and another sip of coffee.
"Do you mind if I use the bathroom?"
"Please. It is the first door on the right," Maria said, pointing to the opposite side of the living room.
Agent Rosson waited for his bladder to drain, something that took longer with each passing year, and shook his friend before flushing. He washed his hands and moved closer to the mirror to check the nick on his chin from an aggressive, pre-caffeine swipe of his razor earlier in the morning. As he stuck out his chin, his elbow hit the mirror, and the spring-release medicine cabinet door opened. Agent Rosson moved his left hand from the shaving injury and peaked behind the mirror into the medicine cabinet.
The entire middle shelf was lined with dark brown prescription bottles. Agent Rosson turned the first bottle on the left and read the label. Xanax. He looked around the closed bathroom and felt a twinge of guilt that he easily pushed passed. He moved his thick fingers to the next bottle and repeated the motion across the bottles on the bottom shelf, facing each prescription outward so he could read them. Xanax, Valium, Prozac, Asendine. Lithium Carbonate. Flaunxol. All of them made out to Maria Hayden.
"Someone is on serious medication," he whispered.
He slowly shut the door to the medicine cabinet and found himself staring back in the mirror. _What?_ he said to himself. _It was an accident_.
And he almost believed it.
The sound of the front door opening startled him, and he looked around the bathroom as if he were a criminal trying to clean the crime scene of evidence. He heard a male voice through the closed bathroom door and took one last look around.
Clark jumped a little as Agent Rosson stepped from the bathroom. Clark looked at his mother in the kitchen and then back at Agent Rosson, not sure what to make of the situation.
"Clark, we have a lunch guest," a voice wafted from the kitchen.
"I can see that," Clark answered. He turned toward Agent Rosson and introduced himself. "Hi, I'm Clark Hayden."
Agent Rosson stepped forward. "Agent Rosson, Special Agent, FBI."
"FBI?"
"That's correct."
"Well, that's not what I expected."
Agent Rosson smiled. "Understandable."
"You know the IRS was here earlier in the week."
"I guess you're lucky."
"How's that?"
"Most people always want to know when the government is going to do something for them, put all their tax money to good use. I guess you and your mother are getting the velvet glove treatment. Home visits by two government agencies in the same week."
"I think I'd refer to it as the lubricated latex glove treatment."
Agent Rosson shrugged his shoulders almost imperceptibly. "Well, I'm here following up on a phone call from your mother regarding a threat to Homeland Security."
Clark looked at his mother who was placing cookies on a plate, and then motioned towards the dining area table. Agent Rosson nodded in response, following Clark across the rectangular room.
"Did your mother mention that she called the CIA?"
Clark looked over his shoulder and shook his head. "No she didn't. She did mention something about strange men at the door a few weeks ago. I assume that's what we're talking about."
Agent Rosson opened his notebook. "I should preface this conversation by saying that any information divulged here is highly confidential."
"Understood."
Clark could smell the coffee in the air. He wanted his mom to stay in the kitchen and called to her, "Mom, would you mind making some tea?"
Maria Hayden wiped her hands on her apron as she stuck her head around the corner. "Is black tea, ok?"
"Sure."
Agent Rosson watched the interaction between the mother and son. "Did you see the people whom your mother called about?"
"No, I wasn't home, but I'm staying in the basement and you can't hear anything from down there anyway."
Agent Rosson wrote in his notebook. "Well, several weeks ago your mother called the CIA to report suspicious terrorist activity in the neighborhood. This information was passed onto the FBI, and I was assigned to investigate."
Clark dropped his voice and eyed the doorway to the kitchen before dropping his voice. "Sometimes it's hard to see, but my mother has diminished mental capacity. I think you may be wasting your time."
"Just the same, there are questions I need to ask."
Clark unzipped his blue ski jacket. "Fire away."
"Did you see the men your mother claims were knocking on the door?"
"Again, no."
"Have you seen anything suspicious in the neighborhood? Anything out of the ordinary?"
"One of our neighbors passed away earlier in the week."
"How did he die?"
"Heart attack, I think. He was a pretty large individual."
"When did this happen?"
"The other day. Monday maybe. Well, the ambulance and police found him on Monday. I'm not sure when he actually passed."
"Where did the deceased live?"
"On the next street over."
"And what did you say his name was?"
"I didn't. His name was Allan Coleman. At least, that's what he was known as. I never really met him. I said 'hi' over the fence a few times and ran into him at the store once. The man loved Ho-Hos, judging by the contents of his grocery cart."
Agent Rosson scribbled.
"Have you seen any — as your mother put it — strange Middle Eastern men in the neighborhood?"
"No. We have Pakistani neighbors across the street."
"What do you know about them?"
"The husband is kind of distant. Doesn't talk much. The wife is very kind. She helps out my mother quite a bit. They have a daughter named Liana. She is two or so."
"What do you know about the husband?"
"He works in a garage or gas station as a mechanic. Somewhere in P.G. County I think. Like I said, he doesn't talk too much."
"Ever see guests at your neighbors? Relatives?"
"No, now that you mention it. I think I met the husband's brother once a few years ago. I don't remember his name."
"Nothing else that comes to mind?"
"Not really. I mean, they are a pretty typical family. They take care of their property. They don't make much noise. The wife helps out in the neighborhood. Between you and me, they are as likely terrorists as my mother is."
Agent Rosson looked at the table full of goodies and smiled. "Well, I certainly don't think your mother is associated with Al-Qaeda, unless she is their official dessert maker."
"Un-official dessert maker."
Agent Rosson laughed a little, the notch in his belt tighter than he remembered it being, even over the holidays. For twenty five years, he had taken great pleasure in telling people he had the same waist measurement that he did in college. Then he hit fifty and he had to alter his ego-line. Another hour with Maria Hayden and he would be at the tailor, getting seams expanded.
"Let me leave you my card. If you see anything suspicious, please don't hesitate to give me a call. We can never be too careful these days."
Clark reached out and took the card from Agent Rosson's hand. "I will, but trust me, my neighbors aren't terrorists."
Maria Hayden popped her head around the corner. "I can make us lunch if you would like?"
"I already ate," Clark responded.
"I'm stuffed. I couldn't eat another thing," Agent Rosson answered.
"Ok, but I insist you take a walnut muffin for the road. They are one of my specialties."
Clark nodded and Agent Rosson took the hint.
"I'm sure they are."
"I will put them in a bag for you."
Agent Rosson smiled. Just like mom.
"Do you think you will be able to find them?" Maria asked as Agent Rosson packed his notebook into a small leather briefcase and stood to put on his coat.
"Who?"
"The terrorists."
"Oh, I'm sure there is nothing to worry about, Ms. Hayden. Thank you for your time and for the food."
"Come back by anytime."
Clark watched from the window until Agent Rosson reached the government-issue black sedan parked on the street. He unlocked the door with his remote keyless entry keychain and placed the muffins on the passenger seat. He checked the number from his notes and looked at the house across the street. _Number 203_. He called into the office to check his voicemail and clicked the phone shut without returning any calls. _You might as well check it out while you are here_ , he thought.
Rosson knocked on the front door with authority and waited. He looked at the door, with its shiny brass knocker and eyed the small mail slot midway up the door. He flipped the slot subconsciously with his left hand and knocked again. He waited an additional minute before walking in front of the house on the uneven sidewalk until he reached the driveway. He turned left and ascended the short staircase to the side door of the kitchen.
He peered over the curtain on the lower glass pane and peeked into the home. The kitchen was clean. The dining area spotless. He had an obstructed view of the living room and a long look down the dark hallway. He knocked again though his intuition had already answered the obvious: no one was home.
Agent Rosson spent the afternoon going through files accessed through the FBI's central database, the Terrorist Screening Center (TSC). Most of his efforts were focused on Nazim, then Ariana, and when his information didn't yield anything suspicious, he turned his attention to the personal background of Maria Hayden.
After thirty minutes of reading Maria Hayden's run-ins with the law, one of which included walking down the shoulder of the beltway in her bra, Agent Rosson decided he had read enough. He opened the Ziplock bag and pulled out one of Maria Hayden's self-proclaimed specialty walnut muffins. _At least she can cook_ , he thought as he put the muffin on a napkin and went to the breakroom to pull a chilled Diet Coke out of a slowly dying community refrigerator. Back at his desk, he shoved a quarter of the muffin into his mouth and chewed. As his saliva glands worked to moisten the dense bread, his taste buds came to life, emitting a silent alarm for something in his mouth that shouldn't be there. Hacking the remains of the walnut muffin onto his napkin, Rosson recognized the taste. Maria Hayden, walnut muffin expert, had put cloves of garlic into the dough. "Crazy old bird," Rosson said aloud. He slugged half of his Diet Coke without stopping and pulled up the form report for an "unreliable source" on the CIA generated lead list.
## Chapter 15
Clark looked at the incoming call message on the phone and prepared to berate the solicitor with the unknown number for calling a phone on the "do not call" list.
"Hayden residence."
"Hello, Clark?"
"Speaking."
"It's Ariana."
Clark checked the number on the phone again.
"Hi Ariana. How are you?"
"Fine. Fine. I have some news. We had a family emergency and had to return home rather suddenly."
"Home, home?"
"Yes, we flew out the other day in a bit of a hurry and have been on the road for the last seventy-two hours."
"I hope it's nothing serious."
"Well, my father has been in poor health for a while. It wasn't entirely unexpected."
"I hope he pulls through."
Ariana paused, considering how far she wanted to take the lie. "Right now, it doesn't look good."
"Sorry to hear that."
"Thank you. I had a favor to ask. I was wondering if you could keep an eye on the house for a few weeks?"
"Of course. What do you need me to do?"
"Just keep an eye on the place. Water the plants. I have canceled the paper and am canceling the mail tomorrow."
"Does my mother have a key to your house?"
"No, but there is one on the side porch in the storage room. It's on the top of the doorframe on the inside of the door. At least it should be."
"I think I can find it."
"If it's easier, you can take the plants to your house. There are three of them that need water. One in the living area, one in Liana's bedroom, and one in the master bedroom."
"How often do you water them?"
"Once a week during the winter. I left the thermostat at 65, so they should be ok for now. Maybe you could water them in a few days."
"Not a problem. Anything else?"
"No, that should be all."
"Do you have a number where I can reach you?"
Ariana paused. "Not at the moment. I'll have to call you back. We're staying at my aunts. She doesn't have a phone."
"Where are you calling from? Maybe I can take down that number."
Ariana's mind raced for second before she calmed it. _Be careful, she reminded herself. Every lie has repercussions_.
"I'm calling from the hospital. One of the doctors is an old family friend."
"You want to give me that number? Just in case."
"I'll have to call you back, Clark. Someone in a white jacket needs to use the phone."
Clark stopped his premature search for paper and pen. "Ok. I'll take care of the plants and keep an eye on things."
"Thanks. I'll be in touch."
Ariana hung up the phone and threw it to Karim who was sitting in the chair on the other side of the desk.
"Destroy it," she said, expressionless.
"Do you think that was a good idea?"
"I told you I had no choice."
"You could have asked another neighbor."
"I could have, but that would have been suspicious. I've spent a lot of time with his mother." She paused, her mind drifting before coming back. "It was necessary."
"Then why do you look worried?"
"He asked for a number where I could be reached."
Karim's dark eyes flickered with a hint of danger. "What do you know about this kid?"
"He isn't a kid. He is very adult. Grew up with older parents. Spent most of his life taking care of his mother. He is very bright."
"Maybe too bright."
Ariana looked around the small office. "Don't worry about this neighbor. I have him covered."
"And if he turns into a problem?"
"Then I'll take care of it."
"Killing another neighbor is not prudent."
Ariana looked over with a quizzical glance as if to indicate Karim was being preposterous. "I'll kill them all if I have to."
## Chapter 16
Clark set the table for a quiet lunch. A working lunch —working on getting the IRS off his mother's back while working on getting into the IRS auditor's pants. He hoped the dress shirt he was wearing would help on both fronts, but if he could only conquer one of two missions, he knew which one it would be. You can only go so long without getting laid. Eventually you just lower your standards until someone fits the bill, get your shag in, and then raise the bar right back where it was. If the IRS hottie blew him off, he would be dropping the bar more than a few inches.
His mother was in her bedroom, ironing clothes that she hadn't worn since her husband's death. Clark listened to her sing to herself as she ironed, the song out of tune and the words a little jumbled. But it was his mother, and her voice, bad singing and all, had a pleasant psychological effect on him.
The sound of the car door shutting made his heart skip a beat and he checked himself in the mirror on the wall on his way to the door. It was time to shine, to dust off the old Hayden charm that he and his father were famous for when the need arose.
Clark flung the door open with a smile, and a wrinkled skin IRS auditor on the other side of the storm door flashed her tobacco stained teeth. Excitement ran from Clark's face as he opened the door.
"Good afternoon Mr. Hayden. My name is Patricia Moody. I am the IRS auditor taking over the case for Auditor Prescott."
Clark didn't hear another word for the next hour until the replacement agent was safely back in her car, the cloud of tobacco stench that followed her finally removed from the living room.
The comic stepped onto the small stage to a smattering of polite applause. His face turned slightly red, the color accented by the lighting system that focused its rays towards the stage and those bold enough to step on it. A hand-me-down suit from his older brother hung loosely off the comic's shoulders, the sleeves a tad too long, the pants a little baggy in the waist. Something he could grow into. A dab of hair gel was perfectly sculpted in the front of his black wavy hair. There was no pretense in the comic; he resembled what he was — a teenager in a borrowed suit. In his left hand was a small notebook, crib notes for the crib he was in. It was his first crack at stand-up and the sparse crowd was hoping for something better than the one-man-band-without-an-instrument performance they had just suffered through. The fact that the performer was tone-deaf didn't help with the audience who was typically forgiving in nature.
Clark walked in off the sidewalk and swiped at the droplets of rain on his gray wool overcoat. He approached the cashier and ordered a hot black tea, something to take the chill off. The male cashier with the pierced eyebrow and red hair handed him his change and said, "It's open mic night in the back. Free of charge. Take a look if you like. We are booked through the first hour, but after that we still have openings."
"I don't sing or dance."
The cashier leaned forward. "Neither do most of the people coming on stage."
"Maybe next time," Clark said, grabbing his cup and a cardboard insulation sleeve with Jammin' Java printed on the front.
The young man on stage was hitting the microphone with the palm of his hand, expertly exemplifying the surefire way humans have devised to jumpstart sensitive electronic equipment.
Clark worked his way from the well lit service counter in the front of the shop to the darker back area where a few rows of folding chairs were haphazardly lined up facing the stage. The tables on the edge of the room were occupied and Clark found a seat in the back row of chairs as the young comedian in the borrowed suit moved past his introduction into his routine.
"I would like to start with a joke that is short but sweet. A man comes into the bedroom with a sheep under his arm. He wakes his wife and says, 'Honey, this is the pig I have been sleeping with.'"
"His wife opens her eyes and says 'That's not a pig.'"
"The husband quickly replies, 'I wasn't talking to you.'"
The crowd snickered. Clark laughed out loud. He looked around the room, and the light emanating from an open notebook computer illuminated a face he recognized. The comic launched into a series of infidelity jokes, and Clark approached the small table and the woman who had her nose in her notebook computer.
"I missed you today."
The young lady with the auburn hair and button nose looked up and said, "Clark Hayden."
"Auditor Prescott. Or may I call you Lisa?"
"No, you may not," she answered smiling.
"Your replacement told me you had moved on to bigger and better things."
"I got a job as an IRS special agent. Criminal investigator. I put in for the position over a year ago."
"Criminal investigator? Do you get to carry a gun with that title?"
"Government issue."
Clark stumbled. "You're kidding."
"Not at all. But I haven't been trained yet. Range practice starts next week."
"Then I guess it's safe to sit down."
"This week," Lisa said, far more friendly than she had been in his mother's house, grilling Clark and his mother over mystery money.
"Are you working?"
"Not really work. I had a few things to finish up and didn't feel like sitting at home."
"So you came to open mic night?"
"Sure, why not? I told you I love this place. But I'm guessing you remembered this and that's why you're here."
Clark blushed a little. "You left me little choice when you pulled the ol' switcharoo with Ms. Moody."
"Mrs. Moody."
"She's married? That's hard to believe."
"Be nice."
"How can you work in here?"
"It is pretty quiet, really. Despite the fact you have people reading poetry, telling jokes, playing the guitar, usually acoustic."
"There's a piano on stage."
"Who doesn't like a piano? Besides there are usually some characters in the crowd. Someone interesting to talk to."
"Do I qualify as interesting?"
"The evening is young. We'll see."
The comic on stage started on a long joke that kept Clark and Lisa at rapt attention. "A man comes home early from work and busts through the door of his condo expecting to find his wife having an affair. The man goes room-to-room looking for his wife's lover and finds nothing. He stops in the kitchen, looks out on the balcony, and sees a pair of hands holding on to the railing. Infuriated, he goes onto the balcony and smashes the fingers of the hanging man with his fists. After a few seconds, the man hanging on the rail falls ten stories and lands in some bushes. The husband looks down and sees the man has survived. Still in a rage, the husband goes into the kitchen and pushes the oversized fridge across the floor and through the balcony railing. The refrigerator falls ten stories and lands on the man in the bushes, killing him instantly.
"A few seconds later the dead man is standing before the Pearly Gates with St. Peter. There is another man in line and St. Peter is checking the list in his Godly notebook. St. Peter says, 'Well, unfortunately, we only have room in heaven for one more soul today. So what I am going to do is have both of you tell me how you died, and the one who died with the most unusual story gets in. The other guy has to wait in purgatory until tomorrow, and let me tell you, purgatory isn't that nice.'
"The two recently departed looked at St. Peter and then at each other. Each man shrugs his shoulder in agreement and St. Peter nods towards the first man.
"The man who appeared first began. 'Well this afternoon I was out on the balcony of my eleventh floor condo washing the windows of the sliding glass door. I tripped over a potted plant, lost my balance and fell over the railing. Amazingly I caught myself on the rail of the balcony below me. Being a good Christian,' the man said with a wink, 'I thanked God for saving me. The next thing I know the neighbor from the floor below me comes out onto his balcony and starts smashing my hands. I try to hold on, but eventually I fall ten stories and hit the ground. Amazingly, I live, and once again I thank the Lord for saving my life.'
"St. Peter has stopped writing in his notebook and is staring at the man in disbelief.
"'So I am lying in the bushes and I look up and I see a refrigerator coming over the balcony. The next thing I know, here I am.'
"St. Peter looks at the man in amazement. 'That is one hell of a story. I think that's going to be hard to beat.' He turns toward the second man in line and says, 'Let's hear what you got.'
"The other man clears his throat and looks at St. Peter. 'Imagine this... you're naked and hiding in a refrigerator...'"
Clark and Lisa looked at each other and she laughed first. "Interesting material for a high school kid," she said.
The sophomoric comic went on for another five minutes and walked off the stage to a standing ovation. Clark turned towards Lisa. "You interested in grabbing something to eat?"
"I have to get going."
"Where do you live?"
"Pentagon City. Arlington Ridge Road."
"If you don't mind going over the bridge, I know just the place."
Lisa thought about the offer. "No discussing your parents' case."
"Come on, you aren't working on my parents' case any longer. You're out of reasons."
"Maybe I don't like you," she said, revealing her dimples.
Clark smiled as he stood. "We both know that's not true."
## Chapter 17
Clark and Lisa walked into Zed's Ethiopian restaurant in Georgetown twenty-five minutes later. Traffic was at a standstill on the bridge over Rock Creek Parkway and brake lights stretched the five blocks towards the Key Bridge.
Two groups formed a semi-circle around the unmanned podium near the entrance to the restaurant. Clark looked at the photos on the wall while everyone else waited for a waitress to appear. The prominent picture on display near the door was then Governor George W. Bush with his chimpanzee smile standing next the Ambassador of Ethiopia.
"Who is he?" Lisa asked, pointing at another picture.
"That's Sugar Ray Leonard, former boxing champ and local sport product. He grew up in Maryland somewhere." In the photo the boxer had his arms up in a defensive stance with the owner of the restaurant, the picture taken on the sidewalk just outside the glass door from where Clark was standing.
A smattering of pictures of famous politicians and TV personalities ringed the small waiting vestibule. A group picture of an unidentified sports team was on the wall near the first step of the stairway that led to the private dining room on the second floor. Clint Eastwood, Mike Tyson, Rudolph Guiliani. Zed had a following. The food was good, the atmosphere unique, and everything on the menu was under twenty bucks.
The groups in front of Clark were seated, and when the hostess returned Clark spoke, "A table for two, please."
Long fingers wrapped around a gold-colored pen and checked a table off the map of the restaurant. "This way, please," the hostess said, her dark Ethiopian features contrasting with her traditional headdress.
Clark and Lisa followed the hostess through the crowded restaurant to a table next to the window overlooking the action at the convergence of M Street and Pennsylvania Avenue.
"Your waiter will be right with you."
"No hurry. Thank you."
Moments later, the waiter appeared as promised and set two menus on the table along with a set of napkins, sans silverware.
Clark opened the two-page menu and eyed Lisa. She was wearing a button-up white sweater, the mid-region attracting the attention of Clark's eyes, if only for a moment.
"I'm glad I was able to see you again," Clark said.
Lisa looked around the restaurant. "Thank you for the invitation to dinner."
"Thank you for accepting the invitation."
"Thank you for agreeing to my ground rules. There will be no discussing the case."
"I think that is enough thanking each other for one evening." _Unless I get really lucky_ , Clark thought. _Then I might throw in a prayer to boot_.
Zed's chicken and beef sampler with chicken Doro Watt, beef Kaev, and Alich came on one large plate. "Enjoy," the waiter said as he left.
"Where are the forks?"
"It's Ethiopian. We eat it with our fingers."
Lisa's eyes widened slightly, her eyebrows jumping momentarily.
"Never eaten beef with your hands before?" Clark asked.
"Why would you say that?"
"You flinched a little."
"Flinched."
"Yeah, flinched. You flinched a little when I said we eat with our fingers."
"I don't think I'm much of a flincher."
Clark let it go, and ripped a piece off of the small spongy bread that served as an eating utensil. "You have to use the bread to scoop up the food."
Lisa watched as Clark wrestled with a sauce-covered piece of chicken before popping it into his mouth. Lisa pulled up the sleeve on her white sweater and cornered a piece of beef near the edge of the plate.
After the meal, Lisa leaned back in her chair. "Taking someone to an Ethiopian restaurant on the first date is a make or break strategy."
"Why do you say that?"
"Well, you're eating with your hands, which can turn some people off right away. Then you are eating off the same plate, not exactly the most hygienic thought. Then there is the fact that not many people even know what Ethiopian food is. It conjures up images of a desert country... eating rice and whatever.
"I figured, why waste time to see what you are made of. If you can handle this as a first date, you must be pretty adaptable."
"Adaptable? Is this what you are looking for in a woman?"
"It's right up there with cute and smart."
Lisa laughed and Clark knew things were going places. Maybe not tonight, maybe not to her place, but in his mind the launch sequence had begun. All he needed was patience, the right coordinates, and a nod from Mission Control.
The walk along the frozen C&O Canal was brief. Another small rain shower forced them into Teaism for some warmth. They worked their way to the back of the restaurant, Clark gently steering Lisa with one hand on her waist. They sat down in a dark corner on large wooden tree stumps that had been converted into chairs.
After several minutes of perusing the tea list, Lisa spoke. "So tell me, what kind of man was your father?"
"I thought we weren't discussing the case?"
"I wasn't asking for the case."
"I guess we have been working this backwards. First you find out about my parents' finances, then you meet mom, and now we are on a date. Not exactly the normal progression of things."
Lisa listened and her blues eyes sparkled under the low hanging lamp over the hand-crafted table.
"My father was a good man. He didn't always know what to do with me, but he was a good man."
"Were you trouble?"
"My father was in his fifties when I was born. He'd spent most of his adult life believing a physician who had told him that he would never have children. I came along and kind of ruined his retirement plans."
"You weren't close?"
"No, we were close. It was just different. You can't play ball in the backyard with your father when he starts pushing seventy."
"No, I guess not."
"But we learned to have a relationship in other ways. You know, the first time I understood what my father did for a living, I saw him in a different light. I was thirteen and my father proudly sat me down on the sofa in the living room in front of the TV. On the screen was a picture of the Space Shuttle orbiting the Earth with its payload doors wide open. I listened as my father explained the sequence of events and when the satellite separated from the fuel module, my dad jumped to the television and pointed, 'There. That big circular piece right there was made in our very garage.'"
"Not many people could say that."
"That's what he said. Until then I had never thought much about what my father did. I had never really understood all those hours he spent in the garage with blueprints and metal and wires. And those huge machines. For me, the garage was just a room in the house I went to when I needed a screwdriver or a hammer. But after that, I started paying attention. Satellites became a common thread of a lot of conversations."
"Satellites? Not your everyday conversation."
"My father knew everything there was to know about satellites and model airplanes. The former was his job, the latter was his hobby. Quite a combination, really, and not one that many kids are interested in. Still, those conversations about satellites kept us close through the spring and summer when the Redskins weren't playing."
"So, what do you know?" Lisa asked, moving her body forward a little as if she were going to hear a secret.
"About satellites?"
"That's the topic."
"I know more than I should. Satellites fly around on different orbital paths and at different altitudes depending on what the satellite is designed for. Equatorial satellites fly between a hundred and a thousand miles up. Stationary satellites, by comparison, are positioned over twenty thousand miles above Earth. A lot of satellites used by TV stations are in stationary orbit, just sitting up there beaming back info for a growing couch potato population. The first communication satellite in stationary orbit was used to broadcast the 1964 Olympic Games from Japan."
"I did not know that."
"A good Jeopardy question."
"I guess so."
"The Space Shuttle flies at an altitude of about two hundred miles above Earth. At that altitude the shuttle acts like a satellite, from a law-of-physics perspective. It travels around the Earth at over 17,000 mph. At that speed, the astronauts on board experience a sunrise or sunset every hour. The International Space Station also flies at an altitude of two hundred miles, which makes sense given that the Space Shuttle services the space station."
"I've seen that on TV."
Clark smiled. "Spy satellites, often referred to as spysats, and military satellites fly considerably higher, in the range between 600 to 1,200 miles."
"What kind of satellites did your father build?"
"Some were for commercial purposes. TV satellites, weather satellites, communication satellites. But the good stories were the military satellites. Satellites that could read something the size of a license plate at night from 600 miles up. Satellites that could spot underground geological phenomena. Underground nuclear tests, for example... And then there was the Divinity Satellite."
"What did that one do?"
"The Divinity Satellite was so secret that it was built in a hundred different locations and pulled together through five sub-assembly plants. The final assembly took place on a military base, under the watchful eye of B-2 bombers and a fully armed tank unit."
"So what did it do?"
"According to my father, the Divinity Satellite was built for one purpose only — to communicate with God. He said there was no other explanation."
Lisa laughed and Clark's heart melted just a little.
"Even now, I look up at the sky sometimes and think about the pieces of metal my father made which are circling the planet hundreds of miles in the air. It's hard to see satellites here in the D.C. area because there is too much light pollution, but if you get out in the country, away from the lights, you can actually see satellites zipping across the sky at night. They look like tiny, fast-moving stars."
"Fascinating."
"What else did your father make?"
"All kinds of stuff. Armor piercing plates for intelligence vehicles. Made to order suitcases to hold communications equipment."
Lisa looked curiously at Clark. "Don't take this the wrong way, but why was your father making this type of stuff in his garage?"
"Are we talking about the IRS investigation?"
"No. I'm just curious. I know, according to IRS records, your father ran a business out of his home."
"That's correct," Clark answered. "Hayden, Ltd. After he officially retired, he continued to work from his shop in the garage. He usually worked on jobs that had been subcontracted from larger companies. He knew a lot of people and they kept him busy. The defense contractor industry is a tight-knit group. Most of the big defense contractors have offices around the beltway, in Maryland, in Virginia. They reside in large non-descript buildings that cast shadows on the cars stuck in traffic on 495. My dad worked at E-Systems for years, on Route 50 near Fairfax Hospital. His company, and others, would get work for huge contracts, tens of thousands of man-hours. In order to meet the deadline for these projects, the defense contractors sub-contracted out portions of the jobs to smaller companies. In turn, these smaller companies sub-contracted out portions of their work to others. If you are good at what you do, and are a trusted individual in the community, which my father was, you could always find work. And that is how spy satellites, or portions of them, ended up being built in my garage."
"Pretty amazing."
"Don't get me wrong, my father spent his career bouncing around between big-name defense contractors in the area, which, love them or hate them, are legitimate companies. But given his line of work, and the machines he had in the garage, he was in a profession where he could take his work home with him. Though it wasn't until after he retired that he opened his own business, for supplemental income."
"I'm starting to get a picture of a man who worked very hard."
Clark nodded and looked away in reminiscence. "My dad had another good story he used to like to tell. One day, when he was still working at that place over there on Route 50, a big truck pulled up to the loading dock at the end of the day. A guy in dress pants and a nice shirt got out from behind the wheel of a truck."
"A truck driver in dress pants?"
"Exactly. So this guy walks to the back of the vehicle as my father approached from the shop area. As the door opened my father started to scratch his head. In the back of the truck, stacked floor to ceiling, were dozens of school desks."
"School desks?"
"Wooden school desks."
"And... ?"
"The well-dressed driver starts waving around a work order that he has and claims he needs immediate help. My dad starts calling upstairs to see if the order is legitimate and after a few calls, and some ass-chewing, they start unloading the desks. When they finish unloading the contents, the driver goes into the cab of the truck and pulls out a blueprint. Everyone gathers around and the man explains that he wants the wooden desks turned into rungs to be used in rope ladders."
"Why?"
"As my dad explained it, this guy was not into questions and answers. My dad, who was pretty quick to assess things, surmised that the ladders were for some covert CIA mission and that the desks were being used because they were untraceable sources of wood. So if one of these rope ladders got left behind, in whatever mission they were part of, the material couldn't be traced back to its source. Or at least, someone would have a hell of a time trying to connect a wooden rung on a rope ladder with an old school desk bought from public school surplus."
"Makes you wonder what the government is up to..."
"You work for them."
"But I'm one of the good guys."
"So was my father."
"Anything else I need to know about you?"
"I'm the control man for Virginia Tech's Robotics Team."
"The control man?"
"The driver. The guy with the remote control."
"Because your father flew model airplanes?"
"You were paying attention. My father tried to instill his love of model airplanes on me, but it didn't stick. But I had a gift for the controls."
"I wouldn't mind trying to fly a plane."
"Well, when you start flying model airplanes, you have to have a buddy system."
"What's that?"
"It's like driving school. If you go to driving school the instructor has a brake and a gas pedal on his side of the car, just in case. Well, when you are flying a remote control airplane that can cost up to a thousand dollars, generally the guy who bought or built the plane doesn't want some novice pilot crashing it on its maiden voyage. So they set up a buddy system. Two remote controls running on the same frequency. One of the remote controls has a switch that allows the controller of the experienced pilot to override the controller of the novice pilot. When I was young I used to fly occasionally and my father was always standing next to me with his own controller, ready to take over if I lost control of the airplane. My controller was this God-awful bright red color with a bright red flag on the end of the antenna announcing to the world to pay attention. It was the equivalent of driving a car with a 'student driver' sign plastered across the back of the vehicle. Standing out there on the radio control pitch with that bright red controller and matching antenna flag. It was embarrassing for a kid."
"Did you ever crash?"
"I almost crashed a few times, but my old man saved me."
"So you didn't like flying airplanes, but you like driving robots?"
"It's a lot easier to drive a robot. You are only dealing with two dimensions."
"What else do I need to know about you?"
Clark stumbled a little before continuing. "There's something else you may find either extremely interesting or extremely boring."
"How can I refuse a segue like that?"
"In high school I set the national record for memorizing Pi. I memorized Pi to the 1,679th decimal place."
"Good God. Why?"
"No reason really. I have a gift for remembering numbers."
"I would say so."
The new couple took turns sipping their tea and the waitress delivered the bill.
Clark pulled out his wallet as Lisa dug into her purse. She scribbled on the back of a "buy nine, get the tenth drink free" punch card from Jammin' Java. "Here is my home number, though I guess, according to what you say, you might not need me to write it down."
"Are you saying there is another date in our future?"
"There could be," Lisa answered coyly.
Clark looked at the card. "This is great. Either way, I get a free latte with only two more drinks at full price."
## Chapter 18
Diplomatic immunity is great if you can get it. Travis Keyes had it and flaunted it. He parked his BMW in tow-away zones, had turned drinking and driving into a hobby, and had received more than one citation for taking a shortcut over the sidewalk when the traffic was at a standstill. But that was in Dhaka, where the sidewalks and roads melted together during the raining season into puddles that could swallow both a car and its occupants.
Pakistan was a little different. There were more rules that needed to be "officially" followed. The rules never changed, just the adherence to them. For Travis Keyes, Pakistan was far more advanced than Bangladesh. More advanced meant better perks. Better housing, better food, better amenities. But better perks also meant less freedom to break the rules. Sure, diplomatic immunity still gave him carte blanche to break all the rules he wanted, but more people noticed. More people complained. It wasn't good diplomacy.
The latest perk Travis had discovered was a firm-bodied office assistant in the Commercial Foreign Service Department. The dalliance had started in the embassy with a friendly glance, which escalated into an even friendlier office blowjob, and culminated with a downright personal, take-her-from-behind on the thirty-foot conference room table used to sign low-level treaties. Tonight was going to be different. Maybe something as mundane as the missionary on his own bed followed by a good night's sleep. He had two hours to decide.
Travis Keyes clicked the remote control and the BBC reporter came to life. His top-floor apartment with meager views was in the secured compound on the wealthy side of the city. Like everything else in his life, his TV was the best money could buy. Pumped in by satellite, his large screen flat panel was capable of pulling in two hundred and ninety-seven channels. Everything from the Discovery Network to Japanese game-shows where contestant ran through obstacle courses designed to temporarily incapacitate the participants at every wrong step. His TV provided a twenty-four hour a day, non-stop, crap-o-rama fest. The exceptions to the rule were his lifeline to the real world: BBC and CNN. He needed the coverage that both channels provided, live news fixes that never stopped running, never went off the air — an endless supply of real-world morphine, the one drug that most foreign service officers needed after a long day of dealing with the local population.
Travis, his smoothly combed dark hair still in perfect position, loosened his red tie and took off the jacket to his navy blue suit. He went to the kitchen and opened the refrigerator. The lone remaining Bass Ale stared at him from the sparsely populated shelf.
The expatriate life was good if you could get over the little things. The big things were part of the package — a different country, a different language, a different culture. They were the givens that expatriates knew were part of the job. There was no more sense in complaining about them than for a felon to argue about the bars on his cell.
It was the little things that made the difference. A real Sunday paper with the morning coffee. Not having to wake up at four in the morning to watch the Superbowl. The ability to get a pizza that wasn't slathered in mayonnaise, or sprinkled with fish or corn or crickets or whatever the local delicacy was.
The best way to cope was to focus on what you could get rather than what you couldn't. And Uncle Sam with his 300 million tax-paying supporters gave well. While the pizzas may have been crap, the free housing, the tax breaks, and the all expenses paid lifestyle meant that Travis Keyes would return to the U.S. with a quarter million dollars in the bank. Uncle Sugar was sweet, indeed.
Travis' butt found the sofa, and a small squeak escaped from the leather-on-leather friction between the seat and back cushions. He looked up at a news story of a wayward dolphin swimming up the mouth of the Thames and flipped through the day's short stack of mail. Travis paused at the last envelope and the wheel in his mind began to turn. He shoved his finger under the tab and ran it the length of the envelope, pulling out the bill with the subtle enthusiasm of a detective on the hunt.
He quickly scanned the list of calls. Six to his mother, one to his brother, one to his financial planner, and one to his old college roommate to place a bet on the Lions in their first playoff appearance in years. He looked at the last two calls. "I'll be damned."
He took a sip of his only surviving beer and reached for the phone on the table. He glanced at the clock on the wall and calculated the overseas time as quickly as it took him to hit the first numbers on the phone.
The phone on the other end rang and Travis Keyes waited for someone to pick up. When the message machine system answered, he paused, thought about leaving a nasty message full of implications, and hung up instead. Diplomacy was his profession.
Five minutes and half a bottle of beer later he said, "Fuck it." He called the same number back, this time leaving a detailed message with his name, number, and why he was calling. Then he called the other number on the bill.
## Chapter 19
Clark opened the door to the outside storage closet at the top of the short staircase that led to the side door. Every house on the street with a side kitchen entrance had the small storage space, a convenient architectural idea that disappeared from tract housing blueprints by the time Flower Power was in full swing. Clark pushed the handle of a snow shovel to the side and put one foot inside the closet, reaching up with his right hand. He ran his fingers along the top of the inside of the doorframe and found what he was looking for near the left corner.
He fumbled with the key and a moment later he was inside his neighbor's house. The house was warm and Clark unzipped his jacket but kept it on. He announced his presence to the empty home and listened to the complete silence return his call. You can never be too polite.
Clark made his way through the kitchen and checked the gas on the stove. It was something he had learned to do at an early age, the product of growing up with a "forgetful" mother in the house. He found a red plastic watering can on the floor of the pantry in the kitchen and turned on the spigot to let the cold water run for a minute. Watching the water made Clark thirsty and he ducked his head under and took a gulp. It was an old habit, one that he had picked up from his father, a man whose hands were often too dirty, greasy, and grimy to touch the cabinets and get a glass. Clark was the only person he knew who still drank water from the tap, glass or not. People bitching about the price of gas seemed to forget that they were paying twice that amount for bottled water. Four billion people on the planet would kill for the quality of U.S. water. Except for Americans.
Clark swallowed the chilly water and put his finger in the stream from the spigot to check the temperature. He was convinced that room temperature water was better for plants, though the last biology class he had taken was in the 10th grade and he had barely passed.
He strolled through the living area and eyed a potted plant on the floor, the sunlight from the window tickling the upper leaves. Clark poured water into the plant and the dry soil effortlessly soaked up the elixir of life. He headed down the hall with the watering can in his hand, the spout dripping slightly.
Liana's room was painted pink with matching frills on the bedcover. Stuffed animals clung to one another, threatening to tumble and fall from a chair in the corner. The plant in Liana's bedroom was smaller but just as thirsty as its larger sibling in the living room.
Clark moved down the hall towards the master bedroom and cautiously pushed open the door to a room he had never seen, much less entered.
The dark wooden bed was made to perfection, its white comforter almost taut. Clark hit the lights and located the plant in the corner near the window. He walked across the hardwood floors, their beams creaking as they absorbed his weight.
Clark slowly directed the stream of water into the pot and jumped as the phone next to the bed came to life. Water spilled on the hardwood and Clark set the watering can on the floor and went into the small master bath in search of a towel. The phone rang four times before the answering machine turned on, the recorded stern voice of Nazim echoing across the room. "You have reached the Shinwari residence. We are unable to take your call at the moment. If you leave a message we will get back to you as soon as we are able." The English message was followed by a briefer one in Urdu.
Clark stood near the phone for a moment staring at the black digital message machine with its lone blinking red light. When the message machine cut-off, Clark stood in eerie silence. He turned his attention back towards the plastic watering can and finished what he came to do. As he shut the door behind him, he silenced the little devil sitting on his shoulder telling him to "go ahead, listen to the message one more time."
The devil seemed to forget that Clark didn't need to.
Karim was sitting in a leather-backed chair in the corner office of the warehouse watching the local news on a small color TV while reading a stack of newspapers that Ariana had picked up during her morning errands, which was primarily a surveillance drive around the neighborhood. She made one lap around the warehouse on foot, and then drove through the maze-like industrial park she and her four-man team now called home. She drove slowly to Georgia Avenue, the main thoroughfare of traffic, and then came back to the warehouse from a small one-way entrance near a shady neighborhood on the other side of the industrial park. She kept documents in a folder on the front seat of her car, papers that showed the company she worked for and their address in the industrial complex, in the unlikely event that she was questioned by anyone with a badge on their uniform. But the south side of Takoma on the D.C. side of the line wasn't harboring any gold mines or diamond deposits. Rabid dogs and barbed wire were the security systems of choice.
But Ariana also knew that a warehouse with secure doors and high windows, while undeniably effective at keeping the outside world out, was equally effective at keeping the inside world in. It was a basic tenant of defense, extolled in _The Art of War_ , and tried and tested in a thousand skirmishes since. As many military groups learned in the jungles of Vietnam, and the family of the deceased learned in state-side funerals, the rule was clear: defend your bunker, but not so well as to cut-off your own escape route.
The door to the office was cracked when Ariana opened her laptop and punched the keys for her password. Connected to the Internet with a wireless broadband card she had stolen from a temporarily unoccupied laptop at the local coffee shop, she logged on to her Logitech Live 8i Home Security System.
Karim watched over the edge of the newspaper. When the screen opened, Karim asked. "You have a camera set-up?"
Ariana sat in blue jeans, sneakers and a black knit sweater. Her breasts were outlined nicely and Karim's eyes dipped as Ariana answered smugly. "Yes. A motion-activated system. There are three cameras. One in the basement, one in the living room, and one in my bedroom."
"Expensive system?"
"Not at all. Most of the computer peripheral companies these days have some sort of web-camera security system. A lot of them are marketed as baby-sitter cameras. People are suspicious these days. They want to know if their children are being treated right and that they aren't in Molester Day Care. There have also been a few cases of shaken-infant syndrome that have gotten a conviction based purely on evidence from a baby-sitter camera."
"Sad statement for society."
Ariana bit her tongue and chose not to rant. She had seen the best and worst of both societies in the U.S. and at home. "Anyhow, I made a few upgrades to an off-the-shelf system. The camera in the basement is imbedded in a fully functional smoke detector on the ceiling. The one in the living room is incorporated into a picture frame. The one in the bedroom is in the new clock radio on the bedside table."
"How much did the upgrades costs?"
"What do you care? I have access to money."
"As do I."
Ariana stared at Karim, measuring the comment. She answered. "I bought the three hidden camera devices at a little shop called _Spies Are Us_ in Georgetown, just off Wisconsin Avenue, not far from the Soviet Embassy."
"An appropriate location for a spy paraphernalia shop."
"Quite. Makes you wonder."
"Paid cash?"
"Of course. And I looked good. Told them I was giving them as a gift for my husband. Told the store clerk that we were voyeurs, of sorts."
"I hope you're kidding."
"About which? Being a voyeur or telling the clerk that is what I wanted the camera for?"
"Either."
"I just went in, asked a few questions and walked out. I was wearing a hat, muffler, and sunglasses, and I was the most underdressed person there. That shop is not interested in what people are doing with the equipment they sell."
"The shop may not be interested in what you are doing, but there could be people who are interested in who's going into that store. Private detectives, the police."
Ariana chimed in sarcastically. "You're right. The police may even be interested in brothers who want to spy on their sisters. Office workers who want to catch employees stealing the coffee money. The college student who wants to video his roommate cheating with his girlfriend. Real national security stuff."
Karim stopped talking for a moment, sulking. "All I'm saying is that you could have used another shop. I mean, a spy paraphernalia shop down the street from the Russian Embassy?"
"I heard you the first time," Ariana said squinting as she stared at the computer screen.
"How often are you checking this?" Karim asked.
"The cameras have a wireless feed to a laptop I left in the attic which transmits through the wireless router in the house, which I left on. The cameras are all 0.5 Lux, which means they work pretty well at night and indoors. They all shoot monochrome video with autofocus." Ariana paused to see if Karim was still listening. "So, if any of the cameras in the house detect motion, it starts recording for as long as there is sustained movement in the field of sight. The camera turns off after sixty seconds of non-motion. The laptop has over 750 gig of memory, so the camera can record for eight hours, on all three camera feeds, in low res. But if someone is in the house for that long, we will have problems."
"No alarm?"
"Not exactly. If motion is detected in the house, the camera switches to record mode and the computer sends me an email. Of course, I can always log into the system and view the rooms under security at any time. But in this case, the computer is sending me a text message to one of the pre-paid cell phones I bought."
"What did the message say?"
"Possible intrusion... 10:37."
"What set it off?"
Ariana turned the computer screen a little.
Karim pulled the chair to the end of the desk. He inhaled Ariana's scent and she caught him. "Knock it off. This is business."
Karim took his second tongue-lashing of the hour, this one more playful than the last.
Ariana spoke as she watched the taped video feed on the computer. "It's my neighbor."
"The boy from across the street?"
"He's not a boy."
"The young man from across the street?"
Ariana rolled her eyes.
Both watched as Clark entered the living room with a red plastic watering can.
"He's watering the plants. Just like I asked him to."
"Let's hope that is all he's doing."
Ariana looked around the rest of the house from the view in the camera. There was some mail on the floor near the front door and Clark stooped to pick it up and put it on the table in the small foyer area.
"Nice neighbor," Karim said. "Very considerate."
Ariana watched the screen, expressionless. The view on the computer blinked once, and then re-opened with Clark entering the bedroom.
"Nothing unusual."
"No. Not yet."
The video showed Clark turning on the lights. Small mistake, Ariana thought. _I should have left the blinds fully open_. It was a minor detail, but something she committed to memory.
"He's watering the plant," Karim said, taking his turn at giving the dialogue. "Stop right there."
"We'll go back when we finish."
"Something startled him. He's going into the bathroom."
Clark came back into view of the camera.
"He is staring to his left."
"The phone," Ariana said. "The phone must have rung."
They both watched as Clark approached the phone and fumbled around the black edges of the device.
"He must be looking for the volume." _Mistake number two_ , Ariana said to herself. On the screen Clark stood next to the phone with the digital voice recorder. Then he stepped away, finished watering the plant, and left the room.
"How long ago was this?"
"The security system sends out an email once triggered. It took me a couple of minutes to log on." Ariana fumbled with the mouse for a minute. In the lower right hand corner of the screen, the time and date appeared. "It happened six minutes ago."
"We need to hear what's on that message," Karim said, his dark eyes suddenly darker, more brooding.
Ariana had already pulled out a new pre-paid cell phone card and was calling her home number. She punched the code for the answering machine, clicked the speakerphone button on the top of the phone, and held her breath.
"When they finished listening to the call, she played the message again. She grabbed a pencil and scratched a note in shorthand on the notepad on the desk.
"Fuck, fuck, fuck," Karim said before adding a few mother-tongue swear accessories that included a unnamed person's father and a goat.
Ariana sat back in her chair, and pursed her lips.
"What do we do now?"
"Right now we do nothing. He has nothing. He heard a phone call from someone in Pakistan. I'm in Pakistan, as far as he knows. The message is now deleted."
"You should have disabled the home phone."
"I couldn't. I need to be reachable. Eventually, the phone will get cut-off when no one pays the bill. But until then, people need to be able to call the house and leave messages for myself or my husband."
Karim took a deep breath. "I, too, have people who need to reach me," Karim said.
"Who?"
"Contacts."
Ariana looked at Karim with her dark eyes. "What have you done?"
Karim's head dipped slightly. "I made a call."
"When?"
"From your house in the neighborhood. The day you drove the moving truck here. The day James arrived with the truck."
"Why? Did I not make it clear that we are following my program? I am in charge. I told you to answer the phone if it rang. I did not tell you to use the phone."
"It was necessary... I was suspicious. Besides, I knew we would be leaving that location."
Ariana sighed. "I can only keep up with my own lies. I cannot keep up with yours."
"Without me, you would be nowhere."
"Is that what you think?"
"That is what I know."
"You don't know everything. You think I was sitting on my hands playing housewife in suburbia waiting for you?"
"Weren't you?"
Ariana paused. "I have made plans of my own. You may think you know me, but that was a long time ago. A different time and a different place. I'm not the same girl you 'rescued,' as you once put it."
"It was a long time ago, but you have not changed."
"Change cannot be measured from the outside. Real change can only come from within."
"Perhaps," Karim conceded. "I still need to use a phone."
Ariana opened the desk drawer and pulled out a pre-paid Nokia. She threw it overhand across the desk and Karim snatched it out of the air. She slid a pre-paid calling card across the desk like a Vegas dealer and Karim stopped it with his shoe as it reached the edge.
"You use it here and now, in front of me."
"Suspicious?"
"I wasn't until now."
## Chapter 20
The door to the sleeping quarters pushed outward into the dark warehouse floor. The shadow of the office lurked in the distance, lights off, blinds pulled. The outline of a man in sweatpants and t-shirt pulled the door shut and took three slow, but natural, strides towards the bathroom entrance. An arm reached into the small bathroom and quietly flipped the light switch. The light fixture over the sink illuminated the surroundings briefly before the closing door left only a sliver of white in the crack between the door and the cold concrete floor below.
The man waited. He could feel his heartbeat. The silence was thick and heavy, his ears searching for signs of movement. A low rumble from a passing truck teased the limit of his hearing, the vehicle in the distance, well beyond the confines of the warehouse.
He gave himself five minutes to complete the task. It was a reasonable period of time for a middle-of-the-night bowel movement. Any longer and he risked someone waking up and noticing he was gone. He would have preferred a daytime reconnoiter, but with three men keeping an eye on each other, and natural suspicion running high, the opportunity was not making its way to the surface.
Moving faster than he had before, he moved down the far wall of the warehouse. He passed the door to the sleeping quarters and could hear one of his bunkmates still snoring heavily through the door. A few paces later and his hand felt the outline of another door. The oversized steel door had an additional latch attached to the wall and secured with a padlock.
He paused, looked across the warehouse floor. The truck in the parking bay blocked his view to the office, just as he had noticed the previous day. Calming his breath, he pulled a small stiff wire from the pocket of his t-shirt. He ran his fingers across the front of the door and felt for the lock. His fingers danced over the metal. His mind went into work mode. _A shrouded padlock with dual ball locking. Most likely a five-pin cylinder_. He took another deliberate breath and inserted the wire.
Two minutes to go before the alarm clock in his head went off.
He went through the first cylinder of the lock like a Ginsu knife through a beer can on a late-night infomercial. The second and third pin fell with only the slightest adjustment. The final two pins protested briefly but succumbed to the deft touch honed by years of practice, some legal, some not. He glanced over at the warehouse floor and pulled down on the body of the lock with his free hand. Only a small click escaped into the cavernous void of the warehouse.
Moving faster, he removed the lock from the latch that kept the oversized door sealed, looping the u-shaped bar through the hole without metal touching metal.
He reached for the latch, and pulled gently. He picked the lock on the knob in less than ten seconds. He paused and turned the knob on the door on the left with the precision of a criminal dueling with a combination lock on a bank vault. He felt the tension in the knob and reversed direction slowly, releasing the knob when it reached its original position.
Something was not right.
He took the wire he had used to pick the locks and straightened it in his hand to its fullest length. He ran the stiff wire in the crack around the outside of the door, beginning on the lower right and working up and to the left. He ran it across the top of the door and then down. Halfway down the door, adjacent to the knob, his wire caught resistance. He pulled the wire from the crack and reinserted it near the floor and worked upwards. The same resistance met him just below the lock. _A security system_ , he thought. _Very clever and very untrusting_.
With less than thirty seconds remaining on his internal clock, he closed the latch, replaced the lock, and pushed the U bar into its locking mechanism with an audible click. With ten seconds remaining his hand opened the bathroom door and quickly turned off the light. By the time the countdown hit zero, he was back in the sleeping quarters fluffing his pillow.
## Chapter 21
Mr. Stanley had the routine of an old man. Compared to other demographics, the old-man routine varied less and, when it was forced to change, there were complaints, dentures in or not. Mr. Stanley got up every morning at six sharp wearing matching pajama tops and bottoms, usually plaid, though he had nothing against stripes. This morning his black and green plaid outfit almost matched the slippers he kept in perfect line at the end of the bed. Lining up his shoes was a habit from the military. He had learned early that searching in the dark for your boots when the air raid siren screamed was not very martial. On nights when the bombs never ceased, he had slept with his boots on.
Mr. Stanley stood with creaky knees, the shrapnel wounds to the muscles in his legs concealed by his pajama pants, and walked around the wooden bed frame to retrieve his glasses from the lone bed stand on Mrs. Stanley's side. He had given his wife, God rest her soul, that side of the bed on their wedding night. It was the first of many marital concessions. Even in passing, Mrs. Stanley kept her side of the mattress. For fifty-six years of marriage, and the five years since, that side of the bed wasn't his.
Mr. Stanley's old-man routine continued in the bathroom. With slippers on his feet and his eyeglasses on the edge of the washbasin, he splashed water on the face, followed by a two-step shuffle to the john for some standing relief. It didn't matter if he got up twice in the middle of the night or not. Three times wasn't out of the question. The wisdom that came with being in your mid-eighties was priceless. The body that came with being eighty-plus needed improving.
Finished in the bathroom, he exchanged his slippers for a pair of slip-on insulated duck shoes, and read the thermometer on the outside of the window. He grabbed his coat off the white peg in the wall and wrapped it around his body, still in his PJs. He let both sides of his jacket flap, unfastened. With a lack of fashion sense forgiven to old men, Mr. Stanley walked out the door to get the paper at the end of the driveway. The crazy old man in pajamas and army boots out to get his paper at the crack of dawn. Except that he wasn't crazy, and they weren't army boots. But who wants to be picky?
Mr. Stanley poured a morning dash of booze directly into his coffee from a half-liter bottle of the aptly known Brinley Gold Coffee Rum. "So your mother called the FBI?"
"No, she called the CIA."
"You said FBI."
"She called the CIA, but the guy who came to our house was FBI. The CIA focuses on affairs outside the borders of the United States," Clark said with conviction.
"Thanks for the update."
"I guess you already knew that."
"Son, I did intelligence before there was a CIA. I never told you about the German Bridge, did I?"
"I'm not sure."
"Well, that's a 'no.' No one ever forgets the German Bridge."
Clark sat down and ran his fingers through his wavy hair.
"I was stationed in France for eighteen months during World War II."
"This much I know. You've mentioned French women a few thousand times in the past."
"Well, as much as I love French women, there are none in this story."
"Then I definitely haven't heard it."
Mr. Stanley, still in his pajamas, raised one eyebrow. "As I was saying, I was in my early twenties, stationed in France. Paris had already been liberated, for lack of a better word, but we were driving, or following, depending on who you asked, the remains of the happy Hitler clan back to their homeland. Our unit was isolated, as was the enemy, and we were running low on supplies and holed up for a few days awaiting orders. It was the middle of summer and it was hot. We were tired and we were sure as hell the Germans were tired. Anyway, we were in the Marne River valley, a beautiful stretch of land with a fabulous river running right down the middle of it. The banks of the river were steep with erosion; the water was about fifty or sixty yards across, and maybe twenty feet deep in the middle. After a few days we get some supplies and get our orders to move out and find the German unit we were trailing. But by the time we were armed, fed, and ready to go, the Germans had disappeared. A major who I didn't particularly care for pulled me and a buddy of mine out of our cots in the wee hours of the morning and told us to get ready for a counter-intelligence operation."
"Counter-intelligence?"
"That's what he called it. I remember looking at the man and thinking how tired he appeared. Tired from lack of sleep, tired from being bombed, tired from getting shot at."
"All good reasons to be tired." Clark took a sip of his coffee, already drawn into the story which he estimated as only half-true.
"So me and this other private — Mike Fearson from Prescott, Arizona — we go off tramping into the woods at night in search of the Germans who had slipped out of the valley somehow."
"That sounds dangerous."
"You're damn right it was dangerous. We had pistols, flashlights, a couple of outdated maps, and enough food for about a day in the wilderness. But we were also young, and when you are young, well, sometimes the dangerous things don't seem that dangerous. So, anyhow, Mike and I spent a few nights wandering through this valley. We would come back to our unit during the day, but every night the major ordered us back out. We spotted a few bridges, or rather what was left of them, but nothing that would get the unit across the river. We had already sunk one truck and were working on sinking a half-track, in one of the shallowest areas we could find."
"How hard is it to find a unit of Germans in tanks trapped in a valley?"
"Well, the group we were after wasn't much bigger than ours. War attrition. Probably a patchwork unit trying to get the hell out of Dodge. At any rate, we assumed the Krauts were armed and dangerous, which means you kind of have to sneak up on them a bit."
"What happened?"
"After a couple of days, Mike and I were out around eight or nine in the evening, just as the sky was getting dark. We had probably covered five or six miles, one-way. Each day we went out, we ventured a bit farther than the day before. Anyhow, there was a little dirt road next to a narrow part of the river. Mike and I were kneeling down, taking a slug from our canteens, and we look out over the river and there is a man walking on water."
"Walking on the water?"
"You heard me correctly."
"Are you going to tell me you met Jesus?"
Mr. Stanley shook his head. "No, but we watched this guy walk back and forth. He did a little fishing, standing there, on the water, in the middle of this river."
"Mike and I approached as quietly as we could and scared the shit out of this guy who immediately went into a tirade in French."
"Could you understand him?"
"Seeing that he wasn't a sexy little femme, my French vernacular was useless. The French man scooted across the top of the water until he got to the shore, grabbed his tackle and stormed off, probably pissed because we scared the fish, though he was the one doing all the yelling."
"Mike and I walked down to the water. The river was gorgeous. The type of place you'd want to retire. The water was deep, with even deeper pools, near a large bend where the river took a left. Good fishing water. Mike walked down the edge of the steep embankment and I noticed some track marks, and they weren't from vehicles off any of Uncle Sam's production lines. Mike was about thirty yards away and he sees something that I can't. Next thing you know he jumps out over the river and lands, water up to his calves. I was about to yell, 'I'll be damned,' but Mike Fearson looked at his feet and finished my thought. He said with a smile. 'An underwater bridge. I guess we know how those Germans got out of the valley.'"
Clark looked mildly confused. "So the Germans built a bridge under the surface of the water?"
"That's right. Brilliant, actually. The next day we followed the river down, crossed over, and went after our friends. By the time we reached them, another unit from the South had taken care of the dirty work."
"What's the moral of the story?"
Mr. Stanley paused. "Not everything is as it appears."
"That's true. But how do you tell when things are indeed exactly as they appear?"
"Are we talking about your mother?"
"Yeah. What if she did actually see terrorists in the neighborhood?"
"You sound like you wish it were true."
"Maybe a little. It can't be easy having people tell you that you're crazy for thirty some years. It would be nice, in an odd way, if she were right."
"I think I'd rather have her be right about something else." Mr. Stanley paused. "But, I don't know what she saw. It could have been Nazim, but I guess you know how I feel about him. The question is what do you think about what your mother claims to have seen? No one knows her better than you."
"I don't know. She claimed to have seen the Easter Bunny one year when I was eight."
"The Easter Bunny?"
"Yeah, the Easter Bunny. Have I neglected to tell you the Easter Bunny story?"
"I'm not sure."
"Well, that is a 'no.' No one forgets the Easter Bunny story."
"I guess I need to hear the Easter Bunny story then."
Clark started. "I was eight years old and it was early Easter morning. My mom had always taken the time to put together Easter baskets for my father and me, and she hid them somewhere in the house. Well, on this particular night, in the revelry of the evening, I assume my mother had forgotten to take her medication. A recurring holiday theme, I guess."
Mr. Stanley nodded.
"At some point during the wee hours of the morning my mother starts screaming, waking my father and me. We come rushing out of our rooms and my mother is in the living room, pointing into the coat rack, swearing that she just saw the Easter Bunny disappear into its hole behind my dad's favorite coat."
Mr. Stanley looked at Clark, who slowly smiled and then broke into a laugh.
Mr. Stanley spoke first. "So we have a story about the Easter Bunny and a story about a French guy walking on water."
"Jesus could have been French."
"Doubtful. What did the FBI guy say?"
"He said he would check into things."
"Well, see what he says."
"What do you think?"
"Probably nothing," Mr. Stanley responded.
Clark took a sip from his glass. "You think this country will ever forget 9/11?"
Mr. Stanley took another drink from his hi-octane, morning concoction. "I can sum up 9/11 pretty simply. On that morning, nineteen hijackers got on four planes, armed with ninety-eight cent box-cutters. Less than twenty dollars worth of weapons. With those twenty-dollars in box-cutters, they killed almost three thousand people and destroyed two buildings worth billions of dollars, and damaged another. And what was the reaction?" Mr. Stanley waited for a second and continued. "Don't answer. I'll tell you. The reaction was two wars and the creation of the largest department in the history of the U.S. government."
"The DHS," Clark answered.
"That's right. The Department of Homeland Security. 150,000 employees, another 150,000 contractors, and a budget of several hundred billion dollars over the last few years. Unfortunately, DHS has no chance of being effective."
"You aren't instilling me with confidence."
"Well, let me start with the problem that was never fixed. The weak link in that attack, from a pure tactical level, was the security screening process at the airport, and more specifically the personnel at the security screening. Let me ask you, how much faith do you have in a high school graduate to catch a determined, well-planned terrorist?"
"None."
"You're damn right. None."
"Maybe, but overtaking a plane and flying it into a pre-determined building is much harder now. The doors are hardened and secured. There are air marshals. The pilots will go into evasive maneuvers and anything not strapped down in the cabin is in for one hell of a ride."
"I imagine that's true. But a terrorist could still use a private plane packed with explosives or something even more sinister. A melt-your-face-off type of chemical. And if the crazies out there are still interested in taking down a plane, the next thing we are going to see is a terrorist with a bomb up his ass. He is going to make Wile E. Coyote proud."
Clark gave a nervous chuckle and Mr. Stanley continued.
"I have spent most of my life in D.C. I have a lot of friends and a lot of relatives who work for the government, both local and federal. I will tell you one thing for certain. The government is really only good at one thing: creating unnecessary work and not having the right people to do the work that matters."
"Now you're scaring me."
"You don't think it's true? I'll tell you what. Pick a government agency. Any of them. Go down to the front of that building on a work day and watch the people who go in. Look at them and give an honest assessment if you want these people to save your life. Ask yourself if you think these people are capable. I will bet my Cadillac that nine out of the ten people you see will not instill you with enough confidence to bet your lunch money, much less your life. We are talking about people who are incapable of getting out of the building if there were a fire. And then you take these people and add bureaucracy. Any American who thinks we are safer after 9/11 is in a state of delusion and denial."
"You're saying we are sitting ducks."
"Yes I am. We have borders to the north and south that you can walk across, and somehow, the government thinks that it is un-American to secure these borders with fencing. And if that is not their argument, then they say it is too expensive. Hell, we spent more on the military in Iraq in a month than it would cost to build a fence on both of our land borders. That is one month. Now that would be money well-spent. And someone needs to explain to me how Saddam Hussein and the search for invisible weapons of mass destruction had anything to do with 9/11. Sure Iraq has turned into a hornet's nest, but that was after the fact and because of what we did. We have sullied the good name of the U.S., created the most unstable country in modern times, and pissed off just about everyone in the world."
"And you think retribution is coming."
"Oh, it's coming. And there is nothing we can do to stop it."
Clark looked sullen. "That's about all the depressing thoughts I can digest for the day."
"Wait till you get older. They get more depressing."
"Well," Clark said, standing. "With that, I'm off." He made his way to the door with Mr. Stanley in his wake. The newspaper was on the table and Clark stopped and pointed. "Are you done with that?"
"Except for the crossword."
"Would you mind if I took the rest? Someone in the neighborhood keeps stealing ours from the curb on Wednesdays."
"The coupons are in the Wednesday paper."
"Then there's one cheap bastard in our neighborhood. A paper only costs fifty cents for home delivery."
"And it isn't even worth reading."
"Thanks."
"You're welcome. And listen, if you want to know what is going on in the neighborhood, there is one person who knows more than even I."
"Mel," Clark answered as he turned the knob.
Ariana got up before her soldiers. It was the sign of a good general. A superb military leader. Or so she had once read. It was also the result of a naturally small bladder and a personality that had migrated from generally untrusting to mob-informant suspicious.
She pulled the blinds on the office and slipped on her running pants and a sweatshirt. There were some things the Americans had perfected and comfortable clothes were at the top of the list. She folded her cot, and pushed it between the filing cabinet and the wall. She rolled up her sleeping bag and put it under the small table in the corner, out of the way.
The warehouse was cold and her breath lingered. She made her way to the community bathroom and the chill of the toilet seat took her breath away. Finished, she proceeded through her surveillance checklist as she had every morning since establishing her new residence with her adopted kin. She poked her head out the front door of the warehouse and admired the rusted fence ambiance of her neighborhood. She opened the unlocked double doors on the far side of the warehouse, and walked the length of the room that had been a former printing shop.
She walked to the back of the truck in the parking bay and eyed the large padlock on the roll-down door. Needlessly, she reached for the padlock and tugged firmly. From behind the truck she looked at the oversized door to the one room she had given strict orders not to enter. _Her room_. Her personal plan for martyrdom, though it would be a voyage through the tunnel of light she wouldn't take alone.
The rubber on the bottom of her running shoes kissed off the freezing concrete as she approached her room. She pulled on the lock, and it held firm in the latch. She smiled at the prospects of her plan. Her smile faded before she took another step.
At her feet was a piece of wire about eight inches long, its thickness near that of a paperclip. She reached down, bending at the knees as women do, and picked up the wire. She examined both ends in the sparing light of the warehouse and took the wire to the bathroom for closer examination. Under the 60-watt bulb over the old sink, she squinted at the scratches on one end of the wire, the abrasion riding out over an inch from the end. She turned away from the light and placed the wire against the width of the door. The end of the scratches matched the distance perfectly.
Ariana folded the wire with her hands until it was small enough to easily put into her pocket.
Rules were meant to be broken, she understood. _Just not her rules_.
## Chapter 22
A gentle dawn mist hung in the air over the parking lot as the group of a dozen men in dark blue uniforms gathered around the back door of the white van. With the trepidation of a snake charmer's audience, the semi-circle of blue-collar workers stepped back as the rear door of the van flew open. One-hundred-plus pounds of salivating Rhodesian Ridgeback barking at eye level had that effect.
The handler of the dog yelled, not without the intended theatrics, "Easy, Raspberry, easy." The mention of the dog's name pushed the beast and its barking into a higher gear. Rob Crowe, Raspberry's owner, pulled mightily on the thick chain that tightened around his dog's neck. The dog's barking turned to a stifled wheeze as the chain dug into his thin fur and choked off the canine's air.
"Stand back. We're coming out."
The dozen men, with one exception, moved back another generous step. Rob Crowe, wearing a wrinkled uniform and adorned with greasy black hair, let Raspberry out the back of the van and jumped to the pavement. A few of the men, emboldened by the thick chain, taunted the dog from a distance. "Here, doggie, doggie. Cute little Raspberry."
Rob Crowe gave the dog two more feet of leash and Raspberry lunged forward. The crowd silenced.
"OK, are we ready?" Rob asked, his tattooed forearm wrapping around the leash.
Mel Edgewood, the man under the spotlight, spoke to the audience. "Let's move the dog between the two trailers so we don't have a mishap."
No one disagreed.
Mel, short and powerful, walked between the two unused eighteen wheeler trailers and strolled to the back of the make-shift alley. The dilapidated chain-link fence at the far end of the enclosure marked the edge of government property. The blacktop was wet from the morning dew and Mel wiped his boots on the side wheel of the immobilized tractor trailer.
Rob Crowe, Raspberry tugging his arm as if he understood the implications of a good show, followed Mel around the corner and stopped ten feet away. The dozen colleagues formed a semi-circle safely behind Raspberry and his handler.
Ten and twenty dollar bills quickly changed hands. Most of the money was on the canine. Terry Porter, a senior member of the group who had long since walked his last postal route, took all bets. He knew what the others didn't. The smart money was always on the champion. The young guys were rooting for a thrashing. The old guy with gray hair and a beard knew a champ when his saw one. Betting against Mel was like betting against Ali in his prime.
"Here are the rules," Mel Edgewood spoke with his back against the fence. The trailers on each side put him at the end of the alley with nowhere to run. "I go thirty seconds with Raspberry here. If the dog rips my uniform or draws blood, I lose. The only thing I am allowed to use in this challenge is this bag, and any other appendages that I was born with and feel worthy of risking."
Mel bounced the thick leather mail-carrier bag on his shoulder. "I almost forgot," he added, reaching into the leather bag. He pulled out a plastic grocery bag and held it in the air. "A little incentive for my competition." He removed a huge slab of meat and stepped towards Raspberry. The dog went wild and Rob pulled back on the reigns as Raspberry went up on his hind legs. "Hold him, Rob," Mel said. "Hold him."
Mel stepped forward and let Raspberry get a good whiff of the bloody meat. Not wanting to tease the dog beyond showmanship, Mel walked back to his position and tossed the meat on the ground near the fence behind him. "And if the dog makes it to the steak, I lose."
"Let me know when you are ready," Rob said.
"Give me a sec to get set."
Mel's infatuation with his Postal Service leather carrier bag hadn't been love at first site. He still remembered feeling the weight of the empty bag for the first time, nearly thirty-five years ago. "It's heavy," he had responded to the supervisor who was in charge of on-the-job training during the end of the peace generation heyday.
"You're carrying the U.S. mail, son," the trainer had screamed. "That mail is the lifeblood of this country. What do you want to do, carry it in a paper sack?"
Mel's appreciation for the bag didn't come until his first on-the-job dog attack. With leather an eighth of an inch thick, the bag was multi-purpose. When used as a shield, battering ram, or football sledge, the thick bag took the fight out of most dogs. Even the big nasty ones tended to give up after a well-placed smash to the snout, which was usually followed by a backward somersault and a canine whimper. For someone who had never been athletic, Mel was a natural at "smacking the livestock," as they called it in unofficial postal vernacular.
His technique was perfect, though there were few people analyzing postal dog defense in comparison to, say, a baseball swing. But if smacking the livestock did become a sport, Mel would be the Mickey Mantle of the leather bags. He kept his technique simple. He sunk his weight on his powerful short frame, lowered his center of gravity to canine level, and tightened the bag to his arm. From there, it was a matter of physics. Over the years, Mel had tamed every roughneck dog on his beat. And it earned him his position as the guest speaker at impromptu seminars on how to treat man's best friend when they got unruly. The first time he showed his colleagues how to handle a beast, the monster had been a golden retriever who had nipped a neighbor's son when he tried to touch its dish during dinner. But as time passed and the "training seminars" continued, the dogs got bigger and meaner. He was sure a few had been rabid. Not only did the dogs get larger, but so did the ante and the payoffs. For Mel, Raspberry was merely next on the list.
The crowd got restless as Mel did a set of deep knee bends. "Come on, Mel," an overweight coworker yelled over the still-barking dog. "Raspberry needs to eat."
Mel winked at the crowd and then nodded at Rob. "Let him go anytime you're ready, Rob. Somebody start the clock when he does."
"I will count down from three," the man with the wad of cash in his hands said.
As Rob fumbled with the leash, Mel stared Raspberry in the eyes.
"Come on, Raspberry. Come and get it."
The huge dog, free from its leash, lunged forward and leapt at Mel. The career postal employee sprang forward to meet the dog, the leather bag pounding into Raspberry's sharp pearly whites. The dog spun around and landed on its paws. Raspberry snapped over his shoulder as he turned back around and Mel met the beast with a low crouch. The dog tried to bite the bag, failing to sink its teeth into the taut leather wall that Mel held tightly. Raspberry went high again, snapping inches from Mel's face. The veteran postman held his position. The dog spun and Mel countered. Raspberry nipped at Mel's boots, and Mel kicked the dog while smashing the bag into the canine with his thigh. The crowd went wild. Bets were doubled down. Raspberry was flinging saliva in all directions.
With ten seconds left, Mel encouraged his adversary. "Come on, puppy. Come and get some meat." The dog responded with another charge and Mel battered him back. As the crowd counted down, Mel gave Raspberry a lesson in postal dog law, successfully pinning the dog against the wheel of the trailer with the bag and all his strength.
"Time's up. Come get your dog, Rob." Raspberry's owner came over and attached the leash as Mel kept the beast trapped. Mel backed away with the bag between him and the dog, and Rob wrapped the chain around his arm several times.
"Go get your meat," Mel said to Raspberry, stepping to the side and pointing to the steak. The dog moved slowly past, as if he understood the fight was over and he had lost, and set his teeth into the thick piece of red meat.
Mel walked up to the gang of men and collected the hundred bucks he had put on himself. It was just the start of the day.
"Now get to work, everyone," Mel yelled as he put the cash into his pocket. "I haven't been late on deliveries in ten years and I'm not going to start today," he added with pride.
The large leather bag with the thick strap tugged on Mel's shoulder. He slammed the sliding door on the truck closed and bounced the bag once with his whole body to balance the weight. He was thankful the holiday season was over. The cards, gifts, and packages of the "season of giving" added another twenty pounds to his average load. Twenty pounds. Not a lot of weight until you humped it five miles, up stairs, down driveways, around fences. For the soon-to-retire U.S. postman with a half million miles logged on two pairs of boots a year, Christmas was the season of joy and Advil.
Ten years was a long time to do anything, twenty was an eternity, and thirty years at the Postal Service earned you a gold watch made in China and a free medical check-up from the neck up. Mel Edgewood had been dragging his heavy leather bag for forty-two years. At sixty-three, he was an old-timer. By his age most of the carriers had been put out to pasture; stuck pushing a knee-high pile of mail towards the first-stage sorting machine, sitting at a table in a windowless room trying to read addresses off shredded pieces of mail.
Not Mel.
As he told his wife, his boss, and the young waitress at the diner where he got his morning coffee — the day he couldn't walk his route was the day he would quit.
That day was approaching. Either God would take him, or his route would.
Clark met Mel at the door. "Right on time as usual."
"I don't like to be late."
"I don't think I've ever seen you late, though 'late' is relative."
"I tried to make it to the same house within a five or ten minute window. Barring any mechanical difficulties, of course."
"Of course."
Mel dug into his leather bag, put some envelopes in a folded magazine, and handed them to Clark.
"You moved back in already?"
Clark looked at his postman suspiciously.
"Where did you hear that?"
"Oh, I have ears."
Clark considered the statement and shrugged his shoulders involuntarily. "I guess you do."
"There's no guessing involved. I'm a mail carrier for the United States Postal Service. Believe it or not, we have our fingers on the pulse on American society. Do you know any other organization that touches virtually every American, every day of the week?"
"I never thought about it," Clark said, other questions churning in his mind.
"Did you know that the mail carriers for the Postal Service are credited with saving more lives every year than firefighters? We know when people are home, when they are away. We know when an elderly woman who lives alone hasn't picked up her mail. We make calls to the police, save people from burning buildings, perform CPR..."
"The eyes and ears of the neighborhood."
"Among other things. Like delivering the mail. For less than fifty cents you can have a piece of paper picked up from your house and delivered to your grandmother's house three thousand miles away. Three thousand miles for less than fifty cents. If there's a better bargain out there, tell me what it is."
Clark suppressed the urge to tell Mel that you can do the same thing, for free, via email and it takes less than a second. Instead, he took the inquisitive approach. "Can I ask you a question or two?"
"Shoot."
"What do you know about Nazim and Ariana across the street?"
"Nice couple. His last name is Shinwari and hers is Amin. You know it always struck me as kind of strange that they didn't have the same last name."
"Because they are Muslim?"
"Exactly. To the average non-Muslim, the religion seems, shall we say, a bit sexist. With all those veils and all. You would think that the man would want the woman to take his name."
"Maybe, maybe not. In Korea the women keep their maiden names and the children also adopt it. Every culture is different."
"And I'm not claiming to be a religious or cultural expert. Just telling you what I see."
"What else?"
"They get mail from overseas occasionally, usually from Pakistan. They are pretty quiet. Keep to themselves. They have one daughter, and I don't think I ever saw them with a visitor."
"That's true. I think I saw Nazim's brother once, but I am not sure."
"Some people like to keep to themselves. No crime in that. Hell, it would probably make the world a better place if a lot more people minded their own business."
"I know a few people who fall into that category."
"Everyone does."
"Out of curiosity, did they put a stop on their mail?"
Mel Edgewood raised his eyebrow a little and smiled wryly. "I can't give that information out. It could tell a potential thief that no one is home."
"Do I look like a potential thief?"
Both men smiled and Clark continued with his line of questioning. "So you never saw anything that struck you as unusual?"
"Not really. But, hey, I'm not a peeping Tom either. I guess the husband is a mechanic and occasionally they get an odd mail-order catalogue."
"Like what?"
"I'm not sure I should be telling you. Mail is sacred," Mel answered quietly, looking around as he did.
Clark wasn't sure if he was joking or not so he played it safe. "Mel, how long I have known you?"
"Fifteen years, more or less."
"You can trust me."
Mel looked at Clark who smiled back. "Well, a couple of years ago I remember thinking that maybe Nazim was into farming."
"Farming?"
"I remember delivering a few catalogues from John Deere and another one from Eggers. Big equipment catalogues. Not the kind of machines you would push around the back yard. Large-scale farming operations. I think it stood out in my mind for a couple of reasons. First off, they are Muslim and I don't see many Muslim farmers here in the US. Secondly, they live in this neighborhood and sure as hell don't have a back forty that I'm aware of. And, for whatever reason, they don't receive much advertising mail. Maybe they don't use junk-mail triggers like credit cards."
"I think we know they aren't farmers. Hell, I'm not even sure that Nazim knows how to fix cars and he's a mechanic."
"What do _you_ think about him?" Mel asked.
"Real quiet. I get the impression that he's a man who doesn't like to be told what to do."
"Short-tempered?"
"Don't know exactly what it is about him. You ever been near someone who you knew just wasn't right, but couldn't put your finger on it?"
"Half of my coworkers."
"I don't know if I should laugh or not."
"Go ahead. I do." Mel straightened his winter postal hat and bounced the bag on his shoulder. "What are you looking for, Clark?"
Clark explained the visit from the FBI and the call to the CIA. Mel listened staidly.
"Not sure what to tell you, there. You could always dig around a little on your own. Hell, you could be a hero."
"I don't want to be a hero. I just don't want to be the guy standing around after the city has been leveled saying, 'I wish I would have checked those neighbors out more carefully.'"
"No, you wouldn't be very popular. But then again, snooping on your neighbors, particularly helpful ones, well that's not easy either. Leaves you feeling a little dirty. The kind of dirt that water and soap doesn't wash off. You know, after 9/11 there was an idea bouncing around the bureaucracy on the other side of the Potomac about having postal mail carriers serve as spies on the general public."
"You're kidding."
"Not at all. The plan was nixed before it made the approval rounds but the effect was essentially the same. If we see something fishy on our routes we call the police and the postal inspection service."
"Should I feel better?"
"You would if you saw me with a 120-pound Rhodesian Ridgeback."
## Chapter 23
The folding cots were pushed against the outside wall of the small room, snuggled into the four corners, leaving an open clover leaf shape in the middle of the floor. The green canvas on the cots was stretched tautly over their foldable wooden frames. The fabric was faded from its original shade of olive green, the hinges slightly rusted, the wood darkened with age. The stamp on the left leg of the frame was still legible, the black ink designed to endure rain and snow and incoming mortar rounds. Syed thought the U.S. Army surplus cots were appropriately ironic. Abu did not. He had spent the afternoon scraping away the reminder that he was sleeping in an infidel's bed. When he finished carving, he swiped at the floor, pushing the splinters across the room. James Beach watched as his roommate turned over on his back.
"Hey, Abu."
"What?"
"How about keeping this place clean? There are four of us in this room."
Abu began cutting his nails with the same six-inch knife he had used to carve his cot.
"You need to worry about yourself. Abu will worry about Abu," he said without looking up.
James stood from his cot and turned up the small space heater. Syed, feet hanging off the end of his cot, snored quietly along the far wall, farthest away from the door. With his boot, James pushed the wood splinters on the floor towards the bottom of Abu's bed.
Knife in his hand, Abu watched James. "You need to be careful, American."
"You keep calling me American. I keep telling you that I am both an American and I am a Muslim. A Muslim just like you. I pray five times a day, just like you. I fast during Ramadan, just like you. I embrace Allah and his one true messenger, Mohammed, just like you. I obey the pillars: Shahadah, Salah, Zakah, Sawm, Hajj. And I am willing to die for my cause, just like you."
Abu was unmoved. "We shall see if you are willing to die or not."
"Every committed Muslim has to answer that question for himself."
"But you were born a Christian, no?"
"I was born a Christian, raised agnostic, and saved by Islam."
"Christians are weak in faith. It is our duty to show this to the world."
"Enough," Karim said, pulling the pillow off his face. "You two are driving me crazy. Abu, pick up the wood shavings on the floor."
"I don't have to listen to you."
The light from the small window high on the wall shone downward and hit Abu flush in the cheek. Karim looked at the scar on Abu's face. "Just clean it up. This is my room too."
James Beach looked up at the metal roof and the four cinderblock walls. Beyond the small single window and the light bulb over the door, there was nothing. He got a case of the chills as the confining images of the room ran through his mind like a film through a broken projector. He shook his head to clear the haunting cobwebs of his past.
Karim sat up in bed. "You feeling all right?"
"Fine, just a case of the chills. It's damp in here."
"We have a roof over our head, a heater in the corner, and people to talk to. It could be worse."
"Yes, it could be worse," James agreed. Karim looked hard at the American. There was a certain understanding that they had both been to the same place. Not the same location, but through the same trials of the soul. Karim spoke, "I'm going to wash in the bathroom, wadu for the afternoon salah. When I get back, someone else can go. Ariana says only one person out of the room at a time."
"Why do you get to go first?" Abu asked.
"Because I spoke first," Karim answered. He stood, reached into the new duffle bag under his bed, and pulled out a plain white towel. As the door shut, Abu spoke to James.
"So, what's in the truck, infidel?"
"Fuck you, Abu."
"Come on. You say we are in this together. Share with your Muslim brother."
"I was told not to say."
"By whom? Ariana? She is a Muslim woman. She is incapable of running an operation consisting of four Muslim men, even if one of them is American."
"I wouldn't be so sure."
"Ok, don't tell me what is in the truck. Tell me how many it can kill. Or do you not want to think that way, American?"
James' eyes penetrated Abu's, hatred stirring in the American's baby blues. "I'm not a scientist, I can't answer that question." James paused and ran both hands through his brown hair. His head dipped and then he continued, "I was told, if used correctly, the death toll could be over ten thousand."
Abu perked up. Syed magically sat up in his cot, his snoring cut off in mid-stream.
"Ten thousand?"
"What is it?"
"Something perfectly legal and perfectly deadly."
The conversation paused. All three men were now sitting on their cots, their feet on the floor in the communal area in the middle of the room.
"Ten thousand. More than three 9/11s," Syed said to himself in wonderment.
The three men looked at each other and smiled sinister grins, flashes of dimples and teeth pleased with the possibility of mass death. Each relished the moment for a minute, the individualistic bickering replaced by the brotherhood of a common goal.
"So Syed, how long have you been here?" James asked.
"We're not supposed to talk about our lives."
"I told you what was in the truck."
"Technically, you didn't tell us what was in the truck," Abu chimed.
Syed thought about the question and then answered. "I have been here a few weeks. But I first came to the U.S. as a high school exchange student."
"No kidding?"
"Yes. I spent my junior year in high school in Michigan."
"And this time?"
"Different name, different identity, different purpose."
"How did you get here?"
"The long way. I flew from Karachi to Dubai and then took a flight from Dubai to Bangkok."
"Why Bangkok?"
"Because that is where I was told to go. I got picked up at the airport by a guy on a tuk-tuk, one of those motorcycle-taxi things with four seats. He drove me to Khao Sahn road, the main backpacker area of town, and we stopped in the back of a small shop where a guy took my picture. I followed my guide out the back door of the shop, through some back alleys to a small hotel a block off Khao Sahn road. I waited there for three days, staying inside most of the time, just venturing out for food in the evening. Three days later someone knocked on my door and handed me a new American passport with my photo in it and a train ticket to Malaysia."
"How much did the passport cost?" James asked.
"I don't know, I didn't pay. From Bangkok I went south to Surat Thani, changed trains, and then made my way over the border into Kota Baharu, Malaysia. The east side of Malaysia is very beautiful and very Muslim. Beautiful mosques near beautiful beaches."
"We have allies in Malaysia," Abu added.
Syed ignored him.
"From Kota Baharu I took a series of buses to southern Malaysia and caught a boat to Jakarta. Then I got on a cargo freighter."
"What kind of cargo?" the American asked.
"Everything, from what I was told. I mean, the cargo is in cargo holds, in containers, so you can't really see anything. The ship flew under a Chinese flag, and one of the shipmates told me that they were carrying counterfeit cigarettes, among other things."
"Cigarettes?"
"Fakes. The Chinese are making them by the billions and selling them around the world. Some of them right back to the U.S."
"They told you this?" Abu asked.
"It took three weeks to get from Indonesia to Guatemala. That is a long time for people to talk."
"That it is," James said. "Look at us. We have been cooped up for a few days and already we are talking."
Syed nodded and continued. "We landed in Puerto Quetzal, on the Pacific Coast of Guatemala. I put $300 cash in my passport at the immigration office, my new passport was stamped, and I walked right in. Three hundred dollars lighter, of course."
"What passport did you use to get into Malaysia and Indonesia?" Abu asked, showing he was paying attention.
"I used my real passport in Malaysia and Jakarta. They are Muslim countries. There is no suspicion."
"From Guatemala, how did you get here?"
"I took a chicken bus through the middle of Guatemala. They have these old buses that are converted U.S. public school buses. Some of them even have the school system's name still painted on the side. Mine was from Pulaski County. They were the exact same type of buses I used to ride to school when I was an exchange student, except for the chickens and the smell of piss. In Guatemala I stayed in a town called Flores for one night. Went to see the Mayan ruins and the pyramids in the jungle for the day, you know, the ones where they filmed scenes from the second Star Wars movie."
"I don't remember that part of the movie," James replied.
"Well, if you want to see it, go to Tikal. Closest town is Flores. It even has an airport. Nothing in Tikal but jungle, howler monkeys, and pyramids."
"You're an idiot. You shouldn't have stopped to sightsee," Abu grunted.
"I like Star Wars. Besides, this was the jungle. No police around for fifty miles. And I was waiting for someone to contact me."
"Let him finish," James said. "Why Guatemala?"
"It's lawless and the little law they have is completely and undeniably corrupt. And Guatemala is a well-worn path of drug smugglers. The South American drug cartels have built dozens of small landing strips in the jungle Peten region of northern Guatemala. They land their planes, unload the cocaine, and then destroy the planes right there in the jungle. It is cheaper than trying to fly them back. The cocaine is loaded onto trucks and SUVs and driven near the border. The groups break-up and cross the border on foot. Then they meet another set of vehicles on the other side of the border and the drugs travel north through Mexico to the U.S."
"Bullshit," Abu said, with a strong accent.
"I paid five thousand dollars and followed the drug smugglers through the jungle into Mexico. I spent two days in Chetumal, waiting for my next ride, and then we picked up a dozen people in Saltillo, Mexico. From there a mini-bus bus drove us near the U.S. border and let us out. We crossed the Rio Grande twelve miles west of Eagle Pass Texas about a month ago."
"You crossed the Rio Grande? Why not just use your American passport?"
"Probably could have, but why? I don't plan on leaving this country alive, so there is no reason for me to announce my arrival."
"Was it as easy to get in as they say?"
"Where I crossed the Rio Grande, the river was a foot deep. There were planks of boards balanced on cinderblocks, a bridge of sorts, but you could have just as easily driven an SUV through the water. I followed the boards and didn't even have to get my feet wet."
"No fence?" Abu asked.
"No. There were some posts where a barb-wire fence used to stand, and there were some signs on the posts announcing the U.S. border, no unauthorized entry. But we were running so fast at that point you could barely read the signs." Syed paused then continued. "I walked to the border town of Selig and stayed in a hotel for a few days. Showed them my American passport and got a room no problem. Took a Grayhound to Houston and waited."
"How about you, Abu?"
Karim walked in the door with a wet towel draped over his shoulder. Syed, Abu and James looked up with guilty faces.
"What were you discussing?" he asked.
"World travel," James said.
"Ariana warned us not to talk. It could compromise the mission. If one of us gets caught, the less we know, the safer we will all be."
"We are fine," Syed said, standing to stretch his lanky frame.
"You're fine? Let's see how fine you are after the CIA has run a car battery through your testicles. Let me see how fine you are when you have been awake for ninety-six hours, hanging by your arms, tied to a pole. Tell me how fine you are after you have spilled your guts and emptied your head and disclosed every minute detail."
"The Americans don't torture."
"No, the Americans hire other countries to torture. Do you think your balls will know the difference between a U.S. battery and an Egyptian one?"
Syed looked down at his groin. Abu looked away. James smirked. "Good then, we all agree. Everyone keep your mouths shut about yourselves. I don't want to know anything about any of you."
Darkness fell as Ariana entered the warehouse. She walked to the dormitory, as she called the one-room hotel for her team, and stuck her head in.
The red streaks in her hair were not as noticeable as the push-up bra and her black skirt.
She knocked on the door and pushed her way in. "I have guys delivering some things, so sit tight."
"We have been sitting tight for six hours," Syed said. "Just a few more minutes."
The sound of wenches and hydraulics reverberated around the warehouse. A deep male voice with a hillbilly twang interjected with the occasional command. _Left, right, back_. The sound of heavy machinery moving heavy machinery lasted for an hour.
When the machines quieted, Ariana heard the man in the blue overalls call her with his Southern accent.
With her best ass-shake and her push-up bra pushing for all it was worth, she made her way to the open door on the eighteen-foot truck. The man with the blue overalls and a moustache that grew thick past the corners of his mouth was standing on the hydraulic lift.
Ariana smiled at him with her bright red lipstick. She flipped her hair with her hand and ran a few strands behind her ear. She leaned forward a little as the delivery guy looked down her shirt at her cleavage.
"Are you sure that is where you need these machines? They weigh a few thousand pounds a piece, so unless you have a forklift and some hydraulic lifters, you can't just push them across the floor."
"They won't need to be moved," Ariana replied as bubbly as possible.
"There is more room here in the main floor space," the man in the overalls said looking around.
"My boss was pretty clear. He wants them in the side room," Ariana said, smiling like the clueless secretary she was playing in her role for the day.
The man in the overalls smiled back, looking down at the perky set that Ariana showed with pride. "I need you to sign the papers for the machines. Security reasons. Post-9/11 regulations. Everything is falling under federal scrutiny these days. Even heavy machinery."
"I understand."
"If you have any problems, call the office at the bottom of your copy."
"Do you have a card?" Ariana asked.
"I have a generic company card," the man in the overalls answered. "But I can put my cell phone number on it, if you want."
"That would be great. You have been very helpful," Ariana said, licking her lips slightly.
"Your boss is a lucky man," the delivery worker said, his foot now on the hydraulic lift.
_Luckier than you_ , Ariana thought. _You'll be dead by the end of the week_.
## Chapter 24
The metal table in the corner of the warehouse was the only designated movable piece of community property. The sleeping quarters of the four men was communal, but the gray lines of private property were there if you looked for them. Invisible trip wires that each man knew existed and by which each man controlled their explosives. Syed and Karim got along well. Abu had only respect for himself. James, perhaps from growing up in America, had the most thorough understanding of the meaning of private property.
When Ariana called the four men to the table just after evening prayer, the cell members looked concerned. Since their arrival in cinderblock paradise, their daily routine had been the same. Up for prayer, breakfast, prayer, waiting, lunch, prayer, prayer, dinner, more waiting and more prayer. The newspaper that Ariana brought into the warehouse was passed around feverishly every morning. After she read the front page, the local news, and the obituaries, each man grabbed a section of the paper, and then swapped what they had with the first person finished reading another section. There was a single Koran on the knee-high corner table in the sleeping quarters, and when the newspaper had been visually consumed they took turns reciting their favorite passages.
But what the men had really learned to practice was fighting boredom. The truck that had delivered equipment into the warehouse was a much needed break from the monotony. It was almost as if Santa had slid down the chimney and sprinkled anti-boredom dust.
Ariana walked out of the far room where the machines were installed an hour before and shut the door behind her. She stopped in the office and picked up a medium-sized cardboard moving box that she had kept from her shopping excursion to REI.
Still dressed as a Western woman with Western morals, she walked towards the men sitting at the table. She didn't sway, she didn't swing. The professional flirtation she had exhibited with the gentleman who had delivered the machinery was now replaced with the professionalism of running a sleeper cell that was in the midst of an awakening.
"Is everyone comfortable?" she asked, still fifteen feet away. Her voice echoed off the wall. Her face was stern, in thought, but still present in the moment.
"What was delivered?" Abu asked. "Something that will help us do our jobs, I hope."
"Yes, something that will help some of us do our jobs."
"It's about time."
"Patience. Patience. We'll have one attempt to do this correctly, and if we fail, then we'll leave this world with unfinished business."
"Sitting in our rooms all day is unfinished business."
Karim interrupted Abu. "Hear her out. We're gathered for a reason."
All eyes turned back to Ariana as she looked down at the box in her hands. She placed the box on the table and caressed the back of the open chair in front of her but didn't sit down.
"I want to lay down some operational rules. Just so there is no misunderstanding. I have spoken with all of you individually. I have assessed your skills. Skills are the only thing that matter from this point forward. We will use first names, and first names only. We will not divulge any personal information to one another. It's not prudent. And could, in fact, be detrimental to our cause."
Karim nodded to indicate to the others that he knew best. James Beach flashed annoyance. Abu managed to bite his tongue.
"I'll now share the information you need to know, starting with Abu, who has shown he's eager to actually _do_ something."
Heads turned towards Abu who looked as if he were going to growl.
"Abu is a bomb maker. Specializes in improvised explosive devices. Has experience in household weaponry which, for the rest of you, means making bombs out of typical household goods and cleaners."
Ariana opened the top of the box and pulled out several small plastic containers with twist-on lids. Each was about the size of a large shot glass. Different color liquids sloshed around in the containers as she set them on the table.
Abu's eyes lit up with a mix of trepidation and excitement.
Ariana paused when there were five containers on the table. "Here are five liquids. They are slightly different in color as you can see. Slightly different viscosity. All of the containers are unmarked."
Abu smiled. The three other men looked at the containers on the table as if they had front row seats at the start of a magic show.
Ariana then reached back into the box and began placing five identical containers, each filled with powder, on the table.
Abu nodded as if he knew what was coming.
"Five different powders, five different containers," Ariana added for the audience. The atmosphere was becoming giddy. Then Ariana removed a small glass jar with a twist lid. She placed the glass in the middle of the table between the line of liquid containers and the line of powders.
"Abu here has one minute to create a bomb from the ingredients on the table."
"Do I get to open the tops first? It may take a minute just to get the lids off."
"The tops are not screwed tightly."
"And if I don't make it?"
"You know the answer," Ariana said, her eyes once again in that dark, soulless realm. "Time starts now."
Abu's hands moved with speed and precision. One-by-one, he grabbed each plastic container with one hand, and twisted the top with the other. All the lids unscrewed as easily as promised.
"Forty-five seconds," Ariana said, still standing.
Abu passed each powder container under his nose, pausing slightly between the second and third. He dipped his finger into number four, and put the remnants on his finger to his tongue. He selected two from the five containers and put them to the side.
"Thirty seconds."
Abu stood from his seat and grabbed two of the liquid-filled containers, which were harder to handle than the powder. The liquid in the one in his left hand flirted with the top edge of the container. He smelled the two in his hands, and then dipped his nose to the table to smell the others without having to pick them up. He eyed the liquid with the bluish hue, and lifted the glass to his lips. "Food coloring in water," he said shaking his finger as if to condemn the offending cheat. He drank the liquid in one gulp, just for show, and reached for the glass jar on the middle of the table.
"Fifteen seconds," Ariana said. Karim slid his chair back from the table ever so slightly.
Abu took one last smell of the two powder containers he had set to the side and emptied them into the jar. He took the container with a liquid that moved with the consistency of olive oil and added it to the jar with the two powders. As the liquid hit the powders, the mixture started to fizz. Abu quickly sealed the lid. From his seat, he rolled the jar forcefully across the floor. When the jar met the edge of the wall forty feet away, Abu raised his hand and began a countdown with his fingers. When he reached one, he put his head in his hands. Everyone at the table followed suit as the jar exploded, shards of glass tinkling off the side of the parked truck some ten feet away.
"Very nice," Ariana said.
"What was that?" James asked.
"An old chemistry trick," Ariana said without elaborating. "With a few twists thrown in."
The three men looked at Abu. A short fuse with a deadly talent. A dangerous combination.
"Abu, clean up and sweep the floor when we are done."
As Ariana continued, Abu put the lids back on the remaining benign liquids and powders. "Syed here is trained in conventional weaponry. Handguns, rifles. Highly accurate to eight hundred yards with the proper equipment."
"A thousand yards, with no wind," Syed added.
"Make that a thousand yards with no wind," Ariana clarified. Ariana pulled a knife from the box and put it on the table. She nodded at Syed. Syed picked up the knife and weighed it in his hands. "What do you want me to hit?"
"The door frame to the office."
"It's twenty yards. At that distance I would use a gun."
"It is twenty three yards. Can you do it? That's the question."
"How many times do you want the knife to rotate?" Syed asked, perturbed at the test.
"Your choice," Ariana answered.
Without standing Syed pulled his lanky arm back over his shoulder and flung the knife forward. The razor sharp blade rotated four times in the air and stuck into the door frame of the office. "That should be about five feet off the ground. A neck shot to a man of average height."
"Why the neck?" James asked.
"A knife could bounce off a human skull. I would prefer to kill, not wound."
Ariana nodded and moved on. She reached back into the box and put a large padlock onto the table. "Next is James. A man of multiple talents. Son of a locksmith. An electrician by trade. A pickpocket by habit. A pugilist by nature."
James smiled at his description.
Ariana walked behind James and he stared straight ahead with his blue eyes.
She put a hand on each of his shoulders and looked down at his hair. "You have ten seconds to pick the lock."
James pulled his hands from beneath the table. Pinched between his forefinger and his thumbs was a paperclip. He unbent the clip and slipped the end of the wire into the lock. By the time Ariana reached three, the locking mechanism released and the U-bar clicked open.
"Where did you get the paperclip?" Abu asked skeptically.
"He pulled it off the lid of the box," Ariana answered. "Very deft. And quite insightful. Nice anticipation of a problem to come."
"You gave it away when you tested Syed and Abu. I knew something was coming."
"What about him?" Abu asked, nodding towards Karim. All four men had perked up. The impromptu talent show had them rapt at attention. Severe isolation and boredom are easily tamed by the slightest stimulation, as any child without siblings and with a backyard and a magnifying glass can attest to.
Ariana pulled a piece of paper from her purse and put it on the desk in front of Karim. "Karim here is a little more cerebral. A little more creative. Consider him the Michelangelo of our group."
Ariana dropped a ball-point pen on the table next to the plain sheet of white paper. "A hundred dollar bill, please."
Karim's left hand plotted the outline of a hundred dollar bill with a near perfect rectangle. In under a minute the bill was complete with a Ben Franklin, slightly off center to the left, just as in the real bill. The room was silent as Karim sped through the five places on the bill where the denomination was written on the front. _The United States of America_ came next, and then the serial number in the upper left-hand corner. The Secretary of Treasury's signature came next, a perfect forgery that even the Secretary would have had to look at twice. The Federal Reserve seal followed and then Karim went to work on the intricate outer frame of the bill. At the five minute mark, Ariana spoke. "That's enough. I think we all get the picture."
James looked at Karim and the drawing of the bill. "That's fucking incredible. All from memory."
"I've had practice."
"No shit," James replied.
Ariana stood proudly over the table and then walked to her seat.
"Are there any questions?"
James answered. "If we aren't supposed to discuss ourselves, why the show-and-tell?"
"As I said, it was necessary to know that we all bring different skills to the table. Our mission will depend on trusting one another. Trusting that each person has both the skills and the fortitude to perform under pressure. I think I have answered, or rather, I think you have answered the question as to whether the person next to you has the skills necessary for this team."
"What about the fortitude?" Abu asked.
"A very good question. Indeed, that is why we are here."
Ariana reached into the box again and pulled out a bag of small, slightly oval, asymmetrical objects with a chocolate-swirl appearance.
"We're going to play a game now. The objects you see here contain one of the deadliest poisons know to man."
James swallowed hard once. Syed started to perspire. Abu shifted in his chair. Karim quickly counted what he could see in the bag. Fifteen.
Ariana's eyes made their way around the table. "Everyone here is going to eat three. In turn, we are going to consume this whole bag. Decide among you who will choose mine and I will go first."
"I'll choose it," James said eagerly.
"Why you?" Abu retorted.
"Because I have no allegiance to her, as you have said many times."
"Let her close her eyes and choose it herself," Syed said diplomatically.
All men nodded. Syed reached for the bag and Ariana shut her eyes. Syed rotated the bag once in the air and the three other men nodded again in silent approval. Syed guided Ariana's hand to the top of the open bag. "The bag is under your hand. Take one when you are ready."
Ariana reached into the bag deliberately and grabbed the first object she touched. She put the object, with its detailed, chocolate swirl-like design in her mouth and opened her eyes. Looking around the table she swallowed without changing her facial expression.
Ariana opened her mouth to show it was empty, swishing her tongue around as the men inspected from their respective positions around the table. "The secret," she said, "is not to chew. If you don't chew, I can almost guarantee that you will live." She paused for affect. "If not, well, there are no guarantees." Then she added, "Who's next?"
Abu shrugged his shoulders. "I'll go next." He put his hand in the bag without taking his eyes off Ariana. He pulled an object from the top of the small pile that had gathered in the corner of the plastic bag, put it in his mouth, and followed with an immediate swallow. "No problem."
The atmosphere went from fear to aplomb. James was not to be outdone. Without asking, he reached for the bag, plucked one, and threw it in the air. He opened his mouth, moved his head to the left, and caught the object in the back of his throat, where it disappeared.
"Almost as smooth as opening the lock," Syed said.
"Almost," James answered with a smile.
Syed and Karim followed suit and then the rotation began again with Ariana. Two minutes later the bag was empty.
"Does that answer the faith question once and for all?" Ariana asked, her eyes focusing on Abu.
"As long as we all wake up in the morning," James replied.
## Chapter 25
The street light peeking through the crack in the curtains hit Clark in the eyes. Outside the window, young Asian males raced their souped-up, high-pitched Hondas with blue neon lights past the Pentagon on Route 395. Fidgeting for the second night in a row, Clark's mind begged for sleep. Lisa was snoring lightly, her head facing away from him. Her auburn hair crept over the pillow, tickling his face, pushing sleep further away. With insomnia winning again, Clark stood from the bed. Lisa rolled over once and staked her claim to the real estate down the middle of the queen mattress. The red digital clock on the night stand displayed 2:33.
Clark looked out the window over the far end of Pentagon City Mall towards the Capitol in the distance beyond the Pentagon. The roads around the Pentagon curved and swerved wildly, the result of post-9/11 paranoia and the government's attempt to keep traffic farther from U.S. military headquarters. The fact that it was a plane that hit the Pentagon, and not a truck, was irrelevant. Clark watched as a high-octane rice-rocket zoomed by, followed by flashing lights seconds later. As the mix of lights disappeared down the onramp for the GW Parkway, Clark walked down the carpeted hall to the kitchen for water.
Glass in hand, he strolled into the small den on the far side of the living room. Folders were neatly stacked on the floor. The shadow of a floor lamp seemed to grow from the corner behind a leather club chair. Clark opened the laptop on the desk, pressed his thumb on the power button, and sat down in the proper posture-inducing Aero Chair. With the melodic Microsoft boot-up music fading into the background of another motor racing outside the apartment, Clark searched the computer's desktop for the Internet Explorer icon, his eyes running over names of unfamiliar applications that he had seen Lisa use. He fought the urge to poke around, and checked his personal Yahoo email account before mindlessly surfing the web. Ten minutes later, with his little devil sitting on his shoulder egging him on, Clark succumbed to the Satan of Snoop.
He opened several applications, clicked a few icons with his mouse, and closed them without making headway. When he hit the icon for the IRS' Master File Database Access, the screen opened with a soft red hue, and a pop up window demanded a password. Beneath the password, in bold font, was a warning: Unauthorized access to this application can result in prosecution by the Department of Treasury.
Clark opened the small drawer in the middle of the desk, as he had seen Lisa do countless times, and looked at the Post-It notes that were stuck to the cover of a green plastic folder. IRS criminal investigator Lisa Prescott, like the rest of the world, needed a lesson in security. She may have carried a gun but she left her passwords written down without a lock.
Clark read the Post-It notes and tried the first password on the list. On his third attempt, the screen turned from soft red to light green. He looked across the living room of the condo towards the darkness of the hall and considered what he had done. _Screw prosecution_ , he thought. He was more worried about a serious breach of trust between him and the sleeping beauty in the bedroom he was falling in love with. He also knew there was something he needed to check. Just so he could sleep at night, if he could put the light coming through the curtains and the street racers outside out of his mind.
The Master File Database Access (MFDA) for the IRS is the largest single non-commercial database in the world and the second largest overall after Wal-Mart's inventory database. Clark was sitting on the front porch to the house of personal tax data and the door was wide open. He poked around the menus on the first page and read through the dozen search criteria from which to choose. Last Name, Social Security Number, Address, Date of Birth. He punched in his mother's address, and a list of supporting data filled the screen. He was sure Lisa had spent a few hours perusing the same screens he was now viewing for the first time. Somehow, the thought of his parents being investigated made his breach of trust feel righteous. If one looks hard enough for justification, one will find it.
He paused and listened for noise from the bedroom, but the sound of another police siren outside echoed off the walls of the den turned office. He hit the button for a new search and the computer returned to the previous screen. With trepidation and guilt overridden by the still-present devil on his shoulder, he changed the house address on the previous screen by one number.
Nazim Shinwari and Ariana Amin. Married in 1999. Property records showed that the house was also bought in 1999 with a current outstanding mortgage just north of $235,000. All tax forms were complete and filed on time. The couple had never been audited. _Of course not_ , Clark thought. _The IRS is too busy auditing dead Americans_. Liana was listed as the couple's only dependent.
Clark went back to the opening page and searched by name. He ran a query for Nazim Shinwari as an individual and found a glut of information. He was a U.S. citizen, naturalized in 1992 in Detroit. His occupation as a car mechanic seemed genuine, as did the salary Clark imagined would go with the position. Born in Pakistan in 1969. Started paying taxes in 1990 in Dearborn, Michigan. No relatives listed.
Clark plugged the name Ariana Amin into the name field and left the address and other fields blank. An hour glass appeared on the screen and a few seconds later the computer listed nine different individuals with the same name throughout the U.S. All those were listed under their current addresses. His neighbor was the only one in the Washington area. Her records indicated that she hadn't paid taxes prior to her joint return with Nazim after they were married. Her social security number was issued in 1996. As Clark read the next field his eyes bugged. No known relatives. Clark's mind raced back to his conversation with Ariana.
"What are you doing up?" Lisa asked from down the hall.
Startled, Clark almost fell from the chair.
Lisa, wearing only a long t-shirt, was rubbing her eyes as she walked past the entrance to the kitchen.
Clark's face lit up in fear and he fumbled to log out of the application. "Just checking email. I couldn't sleep," he mumbled as his index finger speed-tapped the button on the mouse.
Lisa got closer and Clark realized the computer would not shut down in time. He jammed his finger on the power-off switch, and as Lisa's hand hit Clark's shoulder the computer went dark. "I'm finished."
"Good. Come back to bed. Keep me warm."
Clark's racing heart from his breach of trust episode combined forces with the sudden increase in bloodflow to his manhood, and he sprang from his chair. "Now there's an offer I can't refuse."
As Clark followed the well-shaped legs of his new girlfriend, the functioning portion of his brain was still at the computer. _Maybe my mother isn't so crazy after all_.
## Chapter 26
The steady pulse of an AK-47 emptying its magazine pierced the air over the bustling swarm of people negotiating prices. In any other market a high-powered semi-automatic, even a perfect replica, would have sent patrons to the floor scurrying on their hands and knees. Children would have their ears covered, their bodies draped by protective mothers. But in the Sakhakot market, where twenty people a year are killed by raining bullets, no one even flinched as the AK-47 spent its load. The buyers and sellers knew that the risk of lethal projectiles was part of the trade.
Al-Zahim, hands tough as leather with steel-like calluses, wiped down a .357 magnum as he waited for his potential client to empty the full magazine into the air. When the gunfire stopped, Al-Zahim spoke. "I will throw in a case of ammunition for free."
"Ammunition is hardly worth carrying back home."
"What good is a gun without ammunition?"
"I can get ammunition anywhere."
"Not real ammunition with an official Russian Army stamp."
The patron, dressed in a traditional salwar kameez, ignored the statement. "How much for the gun?"
"Thirty."
"That's too much. The guy on the corner is selling two for forty," the customer said, gesturing towards a storefront in the distance.
"Maybe he would. But do you want a gun, or do you want a near-perfect piece of machinery? I have been in this shop for over thirty years. I make weapons that are precise replicas. Some say they are better than the real thing. Built by hand with identical material and then blessed by Allah." Al-Zahim, the art of the sale perfected over the decades, paused for effect. "But I think you already know this."
The customer stared down the sight of the weapon while pointing it across the crowd at a man selling grenades for a shoulder-launching RPG.
"You have a reputation," the patron responded, pointing the gun into Al-Zahim's store.
Al-Zahim gently pushed the barrel of the gun towards the ground. "No, I have a good reputation."
"How much for two?"
"Sixty."
"One for thirty and two for sixty?"
"That's correct."
"Forget it."
Al-Zahim leaned forward and pulled the assault rifle from the grip of the customer. "If you change your mind, you know where to find me. If you don't see me, I'll be in the back working."
The customer huffed once and walked to the next storefront.
"Did you lose a customer?" a voice called from the workshop in the back of the store.
"He'll be back," Al-Zahim answered. He grabbed a towel off the back of his chair and wiped the ever-present dust off his wares. More gunfire exploded from a .50 caliber machine gun and the trees on a hill on the far side of the market danced, leaves and branches falling to the Earth.
Al-Zahim pushed aside the small canvas curtain and entered the back of the store. His brother was working on another AK-47. A piece of metal that would become the gun's barrel was pinched firmly in a vise as Al-Zahim's younger brother lovingly filed the metal to perfection. The back of the store was hot. The air was stale. Al-Zahim's brother, in his fifties but no longer counting the years, sweated profusely. His shirt stuck to his back, his hair dripping with perspiration.
A young boy appeared at the back of the shop near the heated foundry where metal was forged, hammered, and re-forged. His sandaled feet shifted on the dirt path that ran behind the strip of shops. His clothes were sweaty but otherwise clean, his face serious.
"Are you Al-Zahim?" the boy asked.
"I am."
"I have a letter for you."
"Who are you?
"I was told not to say."
"How old are you?"
"Nine."
"Would you like something to drink?"
"No. I was told to give you this letter and return as quickly as I can."
The boy held out his hand. Pinched between his grubby fingers was a folded piece of paper. Al-Zahim exchanged glances with his brother and stepped forward to retrieve the note. The boy extended his hand, let go of the letter, and vanished.
Al-Zahim picked his glasses off the workbench and slipped them over his ears. He opened the letter and caressed his graying beard as his eyes darted from letter to letter, word to word. When he reached the end, he started over at the beginning. Halfway through his second pass, a tear ran down his cheek and his hands started to shake.
His brother stood from his short stool and put his hand on Al-Zahim's shoulder. "What is it, brother?"
"My daughter is going to be a martyr," he said. "Allah Akbar."
His brother read the note and smiled.
"That is wonderful news."
"Now everyone will see that my decision was the right one."
"I never doubted you."
"My wife did."
"It's too bad she did not live long enough to see your success."
"It is time to celebrate," Al-Zahim said, tears of joy still on his face. He reached into a wooden box similar to a foot locker and pulled out two grenades. He threw one to his brother and pulled the pin on the one in his left hand. They both exited the rear of the shop and lobbed the grenades into the bed of a small stream that ran twenty yards from the back of the shops. The Earth shook and mud rained down in large thick drips as the brothers embraced. Celebration, Sakhakot style.
## Chapter 27
The small communal metal table with an unknown history in the corner of the warehouse was well-worn. Two of its folding legs were dented, bent in some past mishap and then straightened for further use. The matching chairs were equally abused, the thin padding torn from its moorings, the caps on the end of the legs long since missing in action.
Abu was stirring the pot for dinner. The night's menu included three ten-ounce servings of freeze-dried chicken and rice, heated on a camping stove. Each man was consuming the daily recommended allowance of calories, per Ariana's direction, supplemented with multi-vitamins. Syed, the lankiest of the group, had even managed to gain a few pounds. Like the others, his caloric intake was not offset by any exercise. Sleeping, eating, praying, and waiting were not on the back of any diet shake as prescribed ways to lose weight.
The freeze-dried food, rehydrated with water from the small sink in the group bathroom, was nutritional. With the daily vitamins, the men were as well-fed as the climbers at Everest Base Camp who were downing matching diets in temperatures near zero.
The three other men sat down at the table as Abu filled the bowls and passed them around. Ariana wedged herself into the corner between James and Karim, her back to the wall, a view of the floor.
Abu spoke for the group. "When are we going to get started with the machines that were delivered?"
"You will know soon enough," Ariana answered stoically.
"Blowing things up is easy. I only need a few items."
"You are naïve, Abu. You think I have you here to blow something up? I wouldn't need you for that."
Karim tried to keep the peace. "Abu, just shut up and eat."
"She keeps telling us about a plan, but I haven't seen anything yet."
"Look around you, Abu. Search for what you can't see on the surface. Why do you think we are living like this?"
Abu's eyes darted around the warehouse and he shrugged his shoulders.
Ariana spoke. "Have you heard the expression to 'leave a footprint'?"
"Yes," Abu answered. Syed and Karim looked around the table but said nothing.
Ariana spoke forcefully but calmly. "We are eating freeze-dried food to prevent generating trash. We are minimizing our baths to conserve water. We are using camping lanterns when possible to conserve electricity, though we will have to use quite a bit of electricity in the coming days. But by then, it should be too late."
"That's not a plan."
"It is survival," Ariana retorted, almost hissing. "We are not leaving a footprint, we are conserving our resources. We are minimizing our existence. I don't want anyone to get suspicious about a warehouse in the middle of a run-down neighborhood. This is our bastion of safety. We don't need someone becoming curious about spiking electricity, a blown transistor, lights, the garbage men wondering where the trash is coming from."
James shoved a spoonful of chicken and rice into his mouth and spoke before he swallowed. "I don't think the CIA has the interest or ability to monitor some warehouse somewhere for spikes in electricity. And I certainly don't think that the garbage men would give a few bags of trash another thought."
"Who are you to say for sure? After 9/11 there was a plan by the CIA to work with the electric companies to flag any commercial address for spikes in power usage beyond what was historically used."
"Urban myth," James said again, chewing with his mouth open and smacking his lips slightly. "I was an electrician. There are too many addresses, too many possibilities."
Abu's mind went back to the truck. "If the truck is not full of explosives, what is it? VX Gas? Chlorine? Fertilizer?" As Abu chewed his food, the scar on his cheek danced with animation.
"If I show you what's in the truck, you'll tell me what happened to your face," Ariana said.
Everyone at the table stopped eating except for Ariana who scooped a mouthful of food with her spoon.
"What does my face have to do with it?"
"I'm curious. I know you like to talk about your past," she added.
Syed, James and Abu all looked at Karim. "I didn't tell her anything about you guys having conversations," he said.
"Deal," Abu said. "I tell you what happened to my face if you tell me what's in the truck."
Ariana grabbed a packet of salt, and tore it open. She poured it into her hand like a magician, the salt disappearing into a closed fist. "Imagine something that could fit into the palm of my hand and kill a thousand people. Something with no smell and no taste. More importantly, something with no antidote. No remedy. The victim is faced not only with death, but gets to live with their impending doom without hope. That is real terror."
Karim and Syed put their spoons on the table and were riveted to Ariana.
"That's definitely worse than getting killed in an explosion," James said, joining the conversation.
Karim glanced at James who wiped his lips with a napkin. James pulled the white napkin away from his face and the paper smeared with a streak of crimson. He swiped the napkin across his lips again and Karim interrupted Ariana's speech. He pointed at James. "Your nose is bleeding."
James tipped his head back and placed the napkin against his nostrils.
Ariana continued. "So the afflicted will wait for their organs to shut down, one by one."
James Beach's shoulders started to shake and his head lurched forward. A raspy attempt at a word filled his throat. He raised his head and Abu jumped from his chair. Blood was dripping from the American's nose and mouth, oozing down his jaw line.
"Do something," Syed said, rising from his chair as well.
Ariana stood and got behind James as his body began to tremble. She wrapped on arm around his neck and lowered him to the ground.
"We need to get a doctor," Abu said. "Something's wrong with him."
"No one can help him," Ariana said, dragging James away from the table and dropping him on the cold cement floor.
Syed turned pale and he looked down at his half-eaten bowl of chicken and rice. "What have you done?"
"He could not be trusted," Ariana answered as James legs kicked haphazardly. His body went into a bone-wrenching spasm and then relaxed. Blood trickled from his nose onto the floor. Ariana asked for another napkin and then wiped the blood from the limp man's face.
"Is he dead?" Syed asked.
"He will be," Ariana said calmly. "But not just yet," she added with a smile. She placed James' head on the concrete floor, and found her seat at the table. Karim looked over at Ariana expressionless. Ariana looked up at Abu and Syed who were still out of their seats, frozen in place, staring down at their fallen brother-in-arms.
She picked up her spoon and motioned towards the two standing men.
"Sit down. Eat. You're going to need your strength."
## Chapter 28
With few exceptions — spies, informants, and undercover agents topping the short list — nothing is more suspicious than someone guilty trying to act inconspicuously. Clark's knowledge of covert operations was limited to information gleaned from a trip to the spy museum, a few Ian Fleming novels, and watching the Jason Bourne movies. His choice of a bright orange Virginia Tech sweatshirt with a hood that dangled out the back of his blue winter coat was not going to go unnoticed by anyone with even marginal vision.
His fingers found his neighbor's key more easily on his second visit to the storage closet near the side door, and the key slipped into the lock like an old patron slips into his neighborhood bar.
The air inside the house was cooler than on his last visit and he immediately noticed the lack of heat. Compared to the balmy temperatures his mother kept at Chez Hayden across the street, Ariana's was frigid.
He started his mission in the kitchen with the small cabinets near the pantry. The top drawer was the junk drawer and it burgeoned with the usual collection of crap: scissors, unused postage stamps, a stash of colored rubber bands. Clark shut the junk drawer and looked at the front of the refrigerator. A monthly calendar for January, printed on a plain piece of office paper, was attached to the refrigerator door with a translucent suction cup. The calendar was blank. Not a notation on any day. January was void of a single doctor's appointment, birthday, or anniversary. Quite an achievement, Clark thought. The blue and white magnet on the left side of the door caught Clark's attention. He removed the magnet, read its message, and slipped it into the pocket of his jeans.
The small filing cabinet in the corner of the dining area was locked and Clark faced his first real decision. _How far are you willing to go?_ He bent over at the waist and examined the lock eye-to-eye, the hood on his sweatshirt flopping down as his head dipped past horizontal. He rattled the handle on the top drawer and then ran his hands along the back of the filing cabinet. His fingers located two nails where the pressboard back of the cabinet met the hardwood edge of the frame. _I could just pop the back off_ , Clark thought. _Pop the back off, take a peak at some papers, and put it right back on. No one would ever know. The lock would remain locked and maybe, just maybe, I could get some sleep_.
Clark picked up the stack of mail on the floor near the foyer. The mail slot in the front door was still slightly open, the metal hinge pinching the corner of a Capitol One credit card offer. Clark felt a small stream of air coming through the mail slot and wondered about his botanical friends in the living room and down the hall. He would check on them, and the thermostat, just as soon as he was finished with a more important task.
Clark put the inner dialogue he was having about the filing cabinet on the back burner and headed down the hall. He passed the open door to Liana's room and stepped into Ariana and Nazim's bedroom. He paused briefly to look at the two photos on the dresser, one a leather framed photograph of Nazim in a tux. The second photo was Nazim holding his toddler daughter. The husband looked relaxed, happy. Clark's eyes darted around the room and his mind registered the lack of Ariana photographs.
Whatever guilt Clark had about snooping around his neighbor's house ballooned when he opened the top dresser drawer and a silk pair of pink panties stared up at him. When you cross the line, you cross the line, and sometimes once the line is crossed it's just easier to plow forward than it is to step back.
Clark put his hand on the panties and pushed them to the side. He dug around through the underwear and assortment of socks like a squirrel looking for a nut he buried in October but couldn't remember where when the hunger struck in February.
The second drawer was t-shirts and jeans and the bottom drawer was a collection of dark dress socks and boxer shorts that seemed too small for a full-grown male to wear. _God, Nazim was skinny_ , Clark thought.
The search through undergarments turned up nothing useful other than insight into Ariana's color and fabric preferences. Clark wasn't sure what he was looking for, but the locked filing cabinet in the living room was beckoning.
He had been in his neighbor's basement once, when Nazim had taken him downstairs to show him his recently purchased thirteen-drawer standing tool chest, replete with a 540-piece mechanics toolset.
Clark clicked on the light in the small shop near the bottom of the stairs. He pulled the top drawer to the tool chest and shuffled through a set of metric wrenches before moving to the second drawer. The tools seemed untouched, the stainless steel gleaming in the light from above. For a mechanic who had dropped three thousand dollars on a righteous toolset with a matching cabinet, everything looked suspiciously unused.
Clark rifled through the drawers, moving from one to another in search of the most basic tool in the collection. As Clark grabbed the flat head screwdriver with the red and white handle from the third drawer, the force of the explosion from the far end of the basement sent him chest first into the standing tool chest, his head hitting a shelf above. He staggered back from the initial impact, falling to one knee. He tried to take a breath and gasped. Blood trickled from a cut above his eyebrow, blocking the vision in his left eye. Clark shook his head as smoke filled the basement. In the clearing fog, his mind issued its first all-points bulletin. _Explosion. Smoke. Fire. Shit_.
Clark found his balance as he stood and felt his way around the corner to the edge of the staircase. The Smokey the Bear training he had received in elementary school had concluded with rule number five: _Stop, Drop, and Roll_. The rule for an explosion in the basement was obvious. _Get the hell out_.
As the first stage of panic set in, Clark felt his way around the corner to the stairs. Blood in one eye, smoke in the other, he pulled his cell phone and his thumb punched 911. He reached the kitchen, opened the door to the pantry and took a frantic glance around for a fire extinguisher. The smoke billowing from the basement intensified and Clark felt the heat from the growing flames on the floor below.
Through the vision in one eye, Clark moved to the side door on the kitchen he had entered through. He turned the knob and pulled. It didn't budge. He turned the lock, twisted the dead bolt and tried again. Nothing. He sent his elbow through the small window, reached down for the knob from the outside and cursed as his twisting fingers again met resistance.
His heart pounding, Clark peeled off his outer coat and held it to his face. The smoke turned a thicker black and Clark took three large strides for the dining area. He knocked over Liana's high chair, pushed Ariana and Nazim's dining room table towards the chest high window on the back of the house, and climbed onto the artificial wood veneer of the table. His first kick broke the window. The second kick sent the window frame and the screen to the ground eight feet below. Clark crouched on the table, his head touching the plastic chandelier, and moved to edge of the window frame, one foot on the table, one foot towards safety. A second explosion sent Clark through the window opening in the direction of Coleman's Castle.
## Chapter 29
The first fire truck arrived as the plastic from the siding burned black, adding to the thicker, whiter smoke that poured from the seams of the house. A hole in the middle of the roof danced with larger flames, the massive gap giving the impression that a volcano in the living room had blown its top. Flames licked the vinyl siding near the blown out windows of the main floor.
Clark stood on the curb, his aching body wrapped in his coat, stained with blood from the cut above his eye. He nodded as the last team of firemen dismounted from their vehicle in full fire-brigade gear.
Still dazed, ears ringing, Clark crunched across the frozen yard, past the ambulance and towards the nearest firefighter. He tapped the man firmly on the shoulder with an open hand. The fire-resistant jacket felt heavy and rough to the touch. The firefighter turned, identified Clark as a neighbor, and yelled "Get back."
The voice was firm. Professional. Female.
Clark nodded to show that he had heard the command, and then leaned forward and yelled. "I just want you to know that no one is in the house. They are out of the country."
The firewoman looked Clark up and down, nodded and pointed to a patch of grass in Clark's neighbor's yard. "Over there," she said before going in search of the onsite commander.
Hoses were unfurled from the truck, and the clapper valve on the hydrant on the far side of the street was removed. Within a minute, water was surging through the hose, disappearing into steam as it went through the hole where the intact roof had been. Clark watched with guilt as a three-man unit axed through the side door of Nazim's and Ariana's. The shingles over the side porch curled from the heat. Looking at the flames leaping from the hole in the roof, Clark's eyes then moved next door. One adventurous flame to a single tree and the fire would be within spitting distant of his favorite neighbor's house.
Moving behind the impromptu fire-line established by the female firefighter, Clark made his way the thirty yards to Mr. Stanley's house and then jumped up the stairs to his neighbor's front porch.
Mr. Stanley answered on the first knock, completely dressed.
"There's a fire next door."
Mr. Stanley looked at Clark admiring the slightly singed side of his coat, the blood, and the dull shocked expression on his face. "Were you helping put it out?"
When the fire was extinguished there were two additional holes in the roof and 20,000 gallons of water in the basement. The crowd around the house, behind the fire scene tape, had grown to over 50. Over the din of activity, Clark spent the next ten minutes talking to the fire chief in his official red-colored Suburban.
"The neighbors are out of town?" the fire chief asked for the third time.
"Yes. They are in Pakistan. The wife asked me to water her plants and keep my eye on things."
"And you were in the house when you heard an explosion."
"I was in the basement when the first explosion hit." Clark gestured to the cut over his eye. "I hit my head on a shelf. By the time I made it upstairs to call 911, the fire was out of control. I tried to get out the door, couldn't get it unlocked and was on my way out the back window when the second, larger explosion hit and helped me on my way."
"Were you smoking?"
"No, sir."
"Where were the plants you were watering?"
"There are plants in the living room and bedrooms."
"And the basement."
"I'm sorry?"
"You said you were watering plants, and that you were in the basement."
Clark lied. "I was checking to see if there were any plants in the basement."
The fire chief eyed Clark and pondered the scenario. "You sure you weren't in there smoking dope and things got out of control?"
"No, sir. No dope, no lighters, no matches."
The fire chief nodded. "You want to go to the hospital and get that cut checked out?"
"The paramedics gave me the once over. A butterfly bandage was enough. I'm good. A little shaken up. For a minute I didn't think I would get out of the house. I'm not sure how the doors got locked. I didn't touch the deadbolt."
"Panic does strange things to people. Did you know that ninety percent of the people who drown in car accidents involving water drown because they forget to unbuckle their seatbelts? Panic."
"Well I panicked."
There was another pause. "Until I determine the cause of the fire I'm putting you in my report as someone who was watching the house, who was present during the explosion, and who called 911."
"That's fine." Clark paused. "I was watching their house."
The Fire Chief looked over at the house through the window of the vehicle and tried to break the tension. "And technically the plants have been watered."
"I guess," Clark smirked.
"Do you know how to reach your neighbors?"
"I don't, but I'll see if I can find someone who does."
"Give me a call at the station if you find anything."
"I will."
"And don't be too hard on yourself. We get a lot of fires this time of year. It was probably Christmas decorations that caught fire and hit a gas can or something else people shouldn't store in their house."
"Probably not." Clark answered. "They didn't celebrate Christmas."
"In that case, I can scratch it off the list of probable causes."
Clark got out of the Fire Chief's car and made his way across the wet street, weaving between the spectators and fire equipment. No one noticed Ariana's re-painted Camry parked four houses up with its lights off. Dressed in jeans and a sweatshirt, Ariana frowned. _Lucky bastard_.
## Chapter 30
Scott Caldwell was tired. In the seat of his pickup sat his brownbag lunch and his crumpled work overalls. He yawned as he took the ramp for exit seven to Central Avenue. For seventeen years he had worked in the most inhospitable urban sprawl the D.C. area had to offer. His mechanic shop was located on a stretch of Central Avenue that was anything but central when it came to a business location. Next door was an empty lot, just beyond that a Chinese restaurant with bars on the windows that did a brisk business selling forty ounce beers with an orange chicken special for $4.99. Next door, on the other side, a used car lot offered its wares guaranteeing credit to anyone with a job. Across the street was the financial district, a run-down stretch of roadside shops with signs in the windows offering check-cashing services.
Scott pulled his truck into the manager's spot, an unmarked parking space near the side door of his shop. He stepped from his red pickup and grabbed his lunch and work clothes from the seat next to him. It was 8:57 in the morning when his key went into the first lock. He turned on the lights and looked around the office to confirm that none of his neighbors had managed their way through the three deadbolts or barred windows. He opened the front office door and made sure the Help Wanted sign was still visible to the outside world from the other side of the Venetian blinds.
In the double bay of his shop sat two cars. The first was a 91 Toyota Corolla with over 300,000 miles on the odometer. He was working on installing a new starter, new o-rings, and a new gasket to a car that otherwise had one tire in the junkyard. The owner of the car was a man in his seventies, and Scott Caldwell didn't have the heart to tell him it would be cheaper to buy another car. So he was doing the work at cost.
In the other bay was Scott's personal obsession, a 76 Camaro he had bought from a man outside of Gaithersburg. It was a work-in-progress and at his current rate of reconditioning it would be road-ready by the following spring. Fall at the latest.
Standing in the first bay, next to the Camry, Scott slid out of his jeans and into his work overalls. He turned on the radio on the workbench near the back of the shop and switched the coffeemaker on with enough water for six cups. Just enough caffeine to get him through lunch.
The phone in the office rang and he picked up the extra handset, which had long since stopped ringing, off the charger on the workbench.
"Caldwell's Garage and Body Shop. Scott speaking."
"Good morning," the voice said before stalling momentarily.
"Good morning," Scott answered, waiting to hear the symptoms, and often the associated saga, of a car in need of service.
"Good morning," Clark said again. "My name is Clark Hayden and I wanted to ask you a few questions."
"Shoot," Scott said, expecting to sharpen his pencil and give an estimate on repairs for a car problem as seen through a non-mechanical mind.
"I live in Arlington, Virginia, and I am trying to locate a neighbor of mine. His name is Nazim Shinwari. I got a call from his wife last week and she said they had returned to Pakistan to attend to family matters. She asked me to look after her house, which I did. Unfortunately, there was a fire at their residence last night and I don't have a contact number for them."
"Are you some kind of bill collector?" Scott asked.
Clark ducked the unexpected accusation. "No, not at all. You can look me up in the phone book. The Hayden residence is right across the street from Nazim's. Or you could call the fire department. Check the news this morning."
"When did you say his wife called you?"
"Week before last."
"Did she mention Nazim?"
"She said they had returned to Pakistan. Why?"
"Well, Clark. It was Clark, correct?"
"Clark Hayden."
"Well, Clark. I also got a call from Ariana about the same time. She also told me that they had returned to Pakistan to attend family business."
Clark became hopeful. Finally, someone with the same story. Someone who knew how to contact Ariana and get the simple explanation that had thus far been elusive. "Did you get a number for Nazim?"
"Nazim is dead. His wife informed me he died in a car accident on a road an hour outside of Karachi. Apparently it was a bad accident. Hit head-on by a passing truck. There wasn't much left."
Clark felt a nauseating pain in his gut. "And when did you say she called?"
Scott looked up at the calendar over the workbench. He checked the schedule for Nazim, and noted where he had crossed out Nazim's name and replaced it with his only other employee. "Exactly twelve days ago. Monday morning."
Clark thought for a minute and put the days into order. "Son of a bitch," he said aloud.
"Pretty much how I felt."
## Chapter 31
The cars poured from the mosque parking lot, blocking traffic on the thirty-five mph main thoroughfare. The police officer on traffic duty stood on the yellow lines, his reflective vest shining in the afternoon sun. His face was covered with a partial ski mask, his nose exposed and frozen, clouds of warm breath engulfing his face with each exhalation. His left hand was extended as he stopped traffic with his palm and a stern look he had practiced to perfection at the academy. The officer's hand waved the religious faithful onto the main road, giving right of way to the spiritually righteous over the average rush-for-a-latte driver.
Five cars back from the traffic cop, Clark sat in his Honda, the rear-end of a large SUV blocking both his view and the sun. It took another ten minutes before Clark pulled into a spot near the front of the mosque. He removed the blue and white magnet from his pocket and matched the name from the magnet to that on the front of the mosque. He was in the right place.
Clark approached the front stairs nervously. He gingerly stepped into the foyer of the mosque and silently took off his shoes. A young man, dressed in a traditional salwar kameez, a Pakistani pants and shirt combination, appeared from behind a wooden door to the left of the large foyer. The pajama-like pants with tapered legs ruffled slightly as the young man approached.
"May I help you?"
Clark put the age of the young man between late teens and early twenties. The overgrown peach fuzz near his wispy sideburns was only one indicator. "Yes. I'm here to visit Imam Alamoudi."
"May I have your name?"
"Clark Hayden. I called yesterday."
"Let me see if the imam is available."
Before Clark could extend his hand in greeting, the young man turned and walked across the open floor of the mosque. Clark stood in his gray wool socks with orange toes and watched as the young man disappeared through another door on the far end of the prayer hall.
The foyer was decorated with mosaic tiles. There were several plaques with Arabic carved into the rich wood display. The building was quiet, an amazing accomplishment given that fifteen minutes ago there were two hundred religious faithful praying in unison.
Clark could feel Imam Alamoudi's presence when he entered the prayer hall. It was not a mystic spiritual force or an overdeveloped Middle Eastern chi that he felt. It was the vibration in the floor from three hundred pounds of flesh wrapped tightly in another salwar kameez.
Imam Alamoudi had an unkempt beard and hair that dropped midway between his ear and shoulder. There was a hint of a receding hairline, though given the Imam's height it was hard to determine the degree of follicle retreat.
The imam approached, gave Clark a once-over from head-to-toe, and extended his hand.
"Welcome to Al-Noor Masjid."
"Thank you for seeing me."
"What can I do for you?"
"I'm a neighbor of Nazim Shinwari and his wife Ariana."
"Of course, of course. They are wonderful people."
"Yes they are. My mother has been ill and Ariana has been helping out a great deal."
"That is what neighbors do. More importantly, that is what Muslims do. It is one of our tenets."
"In the interest of bettering neighborly relations, I wanted to learn a little more about Islam and, if possible, take a tour of your mosque."
The imam gave Clark a judgmental pat-down with his eyes. The imam started to speak, paused briefly, and then nodded. "Please come in."
Imam Alamoudi began his tour near the entrance to the prayer hall. "The English word 'mosque' derives from its Arabic equivalent, masjid, which literally means 'place of prostration.' Hence the name Al-Noor Masjid meaning the 'Al-Noor Mosque.'"
"What does Al-Noor mean?"
"The light," the imam answered, his gaze heavy. "Before we start, are there any rules governing what I should or shouldn't do?" Clark asked. "I don't want to offend anyone."
"The rules for the mosque are simple. In general, men and women should dress conservatively, usually covering both the arms and the legs."
"Easy when the weather is like this," Clark said.
The imam looked at Clark humorlessly, indicating that the only visitor in the building was not being as serious as he should have been.
"Sorry," Clark said. "Anything else?"
"Shoes are left at the entrance to the prayer area. This is done so that the rugs are not soiled. Women may be asked to cover their hair when visiting a mosque and some mosques will loan scarves to visitors, or those who otherwise need them."
The imam motioned towards an open doorway to the right of the foyer. "There is also a washroom for the followers to perform wadu, also known as ablution. Washrooms for wadu can range from sinks to elaborate rooms with built-in floor drains and faucets. In some form or another, they are found in every mosque. Cleanliness is vital to prayer."
"In Christianity the expression is 'Cleanliness is next to Godliness.'"
"Yes. It's the same principle, but Muslims actually practice it. Muslims wash their hands, faces, and feet before prayer as a way to purify and prepare themselves to stand before God. Five times a day."
Clark took the insult for his religion and moved forward. "It's quite beautiful inside."
"Thank you. The main function of the mosque is to provide a place where Muslims can perform their obligatory five daily prayers. It also serves as the vestibule to hold prayers on Fridays, the Muslim day of communal prayer. There are also two main Muslim holidays called Eids, which loosely translate as 'festivals.' The first festival is Eid ul-Fitr, which is the festival to celebrate the end of the holy month Ramadan. The second one is Eid ul-Adha, which celebrates Ibrahim's willingness to sacrifice his son."
"We have Abraham in Christianity too."
"I am aware," the imam said, irritated. He stroked his beard as if to consider whether to continue the tour.
"Is a mosque considered holy?" Clark asked.
"It is a dedicated place of prayer, but there is nothing sacred about the building. There is no altar, or its equivalent, in a mosque. A Muslim may pray on any clean surface. I'm quite sure you have seen Muslims on TV praying outside, in public."
"Yes, I have. On television," Clark confirmed.
The imam continued. "There are several distinct features of a mosque. The first is the musalla, or prayer hall. Every mosque is designed with the orientation of Mecca in mind. As you know, Muslims pray facing the direction of Mecca. In North America, this means that most Muslims face northeast. Prayer halls are open spaces which help to accommodate as many worshippers as possible. As you can see, there are no seats and no pews. During prayer, worshippers sit on the floor in lines. They may also stand and bow in unison."
"Do the men and women pray together?"
"A good question. Men and women form separate lines when they stand in prayers. Some mosques will have a separate area entirely for women. Our mosque has a single musalla, so women and men pray together, but separately."
A door closed in the distance and the imam looked across the prayer hall in the direction of the noise. "All mosques also have some sort of mihrab, or niche marker, on the wall that indicates the precise direction of Mecca. The mihrab is usually decorative and is often adorned with Arabic calligraphy. It is usually curved to echo the voice of the prayer leader back to the worshippers, as even the imam must face Mecca when praying, thus leaving his backside facing the congregation. Many mosques also have a minbar, or pulpit, to the right of the mihrab. During the Friday prayer service, the imam delivers a sermon from the minbar."
"Perhaps I could attend a Friday prayer service."
"You would be welcome. If you do, there are some expressions you may hear, traditional Islamic greetings. Perhaps you have heard 'assalamu alaikum?'"
"Yes. It means 'peace be with you.' And the response is 'wa alaikum assalam...' 'and with you be peace.'"
Clark heard the imam's stomach growl, and the large man in the robe-like attire rubbed his belly gently. "I haven't had lunch."
"Me either. Once in a while it's good to remind ourselves what it is like to be hungry," Clark added, feeling more religious.
"Fasting is an important part of Islam. One of the five pillars of our religion," Imam Alamoudi said proudly, though there was no indication of a recent adherence to reduced food consumption.
"Most mosques also have a minaret, a tower used to issue the call to prayer, or adhan. In North America, the minaret is largely decorative. But in Muslim countries it is a vital part of the mosque. In Muslim countries, the mosque is usually centrally located in the middle of town, and most people walk to their daily prayers. Here in America, most people drive from a farther distance. As such, the minaret and adhan are largely symbolic here in the U.S."
Clark looked at the floor-to-ceiling bookcase on the wall and the imam answered the quizzical look on his guest's face. "Most mosques have a collection of books; sometimes they have a library. These books usually include works on theology, Islamic philosophy, law, and the followings of the Prophet Muhammad. There are always copies of the Koran available as well. Would you like a complimentary copy?"
"Yes, that would be great," Clark answered honestly. So far, he was impressed with what he saw. He had to admit that Islam and the Al-Noor Masjid were running circles around the modern Catholic Church when it came to religious depth. _Pray fives times a day, everyday?_ Not unless you were a nun, or a priest, or just knocked up your girlfriend, Clark thought.
"Other common features found in the mosque are schedules displaying the times of the five daily prayers and large rugs or carpets covering the musalla floor."
"I see that there's a school here, as well."
"Yes. While a mosque's primary function is that of a place of worship, it can also serve as school, a day care center, and a community center. Some mosques offer Koranic instruction, as well as Arabic classes. It's not that different from a Catholic Church with a school."
"Or a Jewish community center associated with a synagogue," Clark added.
The imam grunted his response, "Perhaps."
The tour continued for another five minutes, concluding back near the foyer where it started. The same young man who had met Clark at the door had returned and was now standing next to the imam.
"This is Farooq, he is one of my understudies. His name means 'he who distinguishes truth from falsehood.'"
"Great name," Clark said.
Clark and Farooq shook hands, something that the understudy wasn't interested in at their first meeting in the foyer.
Farooq then turned his back towards Clark and whispered upwards into the imam's ear.
"Is there anything else I can answer for you today?" Imam Alamoudi asked as Clark removed his shoes from their place on a large shelf used as a shoe rack.
It was time for Clark to get to the real question. "I was wondering if you would know how to reach Nazim or Ariana? I understand that they have gone home to attend family business, and they asked me to watch over their house. I don't have a contact name or number to reach them and, unfortunately, there was a fire at their residence the day before yesterday."
Imam Alamoudi felt the ambush but remained stoic. Without another word to Clark he looked at Farooq. "See if you can help this young man in his request."
"Yes, Imam," Farooq answered as Clark slipped on his shoes and zipped up his jacket.
## Chapter 32
FBI Agent Chris Rosson was hot on the trail of his latest suspect. The evidence was straightforward. The perp had started with small assaults and minor infractions. Invasion of private property, trespassing, and loitering — judging by the turds left on the floor underneath his desk.
When Agent Rosson opened his bag of Cheetos and found not only had the air already escaped through a hole in the bottom, but that the contents of the bag had been pilfered, he called in the experts. Thanks to Rosson, there were now a dozen rat traps scattered around the fifth floor of FBI headquarters. The maintenance crew referred to them as "mice enticers," as if the distinction between rat and mouse meant anything when the vermin are scurrying over your shoes, comfortably planted from nine to five at the most advanced investigative entity on Earth.
The Director of the Joint Terrorism Task Force, Eric Nerf, had also spent the majority of his career in the pursuit of rats, usually those on the next rung of the bureaucratic ladder. It had been fifteen years since he last donned a bulletproof vest and a rain jacket with his employer's name emblazoned on the back. His shirts were pressed. His tie collection managed to keep up with the latest fashion, following the trends from thin to thick, from bland to colorful. His slicked-back black hair was gelled and dyed to perfection. As a bureaucrat he had exchanged his gun for a pen, to use his sense of investigation for general suspicion of those who worked for him as much as real criminals.
His forays onto the floor where his staff toiled were limited to the far aisle of cubes, those that stood between his corner office and the elevators. So when the Director appeared at Agent Rosson's cube and cast a shadow on the agent's workstation, Rosson's heart took a couple of un-syncopated beats.
"Rosson. I need a word with you."
"Yes, sir," Rosson answered to the back of his superior's suit as Eric Nerf turned the corner and walked towards the meeting rooms on the other side of the floor. The Director may have made a rare trip into his employee's lair but he wasn't going to demean himself by having a conversation in a cube. That just wasn't done by someone of his stature, of his rank, of his mindset. He was an office guy. He met in offices or conference rooms and he discussed matters with others who had offices. There was nothing for him in the world of cubes. He had briefly been there and had no plans on going back. As a government employee, his rise through the organization was totally independent of his ability to manage people or relate to them on their level. He was the man in charge and, as such, his subordinates were to follow orders. And orders came just as easily through email. Unless the Director was passing along a good ass-chewing, as happened to be the case.
"Shut the door behind you," the Director said as Rosson entered the room. The room was small and windowless. It had no electrical outlets, no carpet, and gray paint on the walls. Agent Rosson took a seat at the end of the table and the Director put his butt on the corner of the metal table.
"What's going on, sir?"
"Rosson, have you been working the list of leads we get from the CIA? That less than popular list of calls and letters that the general public sends to the CIA contact address and call center."
"Yes, sir. I have been working my portion of the list for the last five months or so. My rotation is supposed to end next month."
"Any progress?"
"Progress?"
"Anything that has struck a chord of plausibility with you? Any leads that made you think, 'maybe I got something here?'"
"No sir. Had a jewelry store in the Bronx that we checked out for being a potential Halawa broker. But that was in October. I have been keeping the system up to date with those leads. You can check on my comments and progress."
"I don't need to. I have always looked at that list as a dog-and-pony show. It's a feel-good list, though I am well aware of the less-than-feel-good names the agents have for it. The list is a pacifier for the general public and a get out of jail free card for our friends at the CIA."
"I'm sorry, sir?"
"The CIA, by virtue of this list, is able to send us every crackpot inquiry that comes to them from unsolicited domestic sources."
"I get the feeling you are about to tell me that I have missed something."
Eric Nerf puckered his lips and made a sucking sound with his mouth like a fish gasping for breath. "I just got off the phone with the Deputy Director of Clandestine Services at the CIA. It seems there was information on the CIA list that has since been deemed 'of interest to national security.'"
Rosson ran his hand through his prematurely gray hair. A man with great perception in most situations, he was nervous because he didn't know if he was going to get patted on the back or lose part of his derrière.
"Do you remember a house here in the D.C. area on the list?"
"Sure, it was a few weeks ago. In Arlington. I switched with Agent Taylor because I knew the neighborhood. The address sort of jumped out at me. I lived nearby when I was in junior high school. Moved before my freshman year."
"Did you visit the house?"
"Affirmative. I stopped by and spoke with the woman who made the call."
"And..."
"It was a dead-end. The woman was on at least half a dozen different medications for psychosis. Meds for schizophrenia, depression, bi-polar disorder."
"How did you obtain this information?"
Agent Rosson didn't blink. "Her medicine cabinet was open when I went to use the john. Though I didn't mention the medication in my notes for that report."
"Why not?"
"Just because she has the medicine, doesn't necessarily mean that she is using it. Besides, someone could misinterpret that information as being obtained illegally."
The Director nodded and picked at his teeth with his fingernail. "Was there anything unusual about her claim?"
"Nothing more unusual than the average call that comes through that channel. The woman claims she saw three Middle Eastern-looking men at her door in the middle of the night. She only saw them through the peephole. She said they muttered something about hospitality through the door before they disappeared."
"Disappeared?"
"I believe the phrase she used was 'vanished into the night.'"
"So you didn't buy her story?"
"Well, her son showed up as I was about to leave and I spoke with him. He seemed to have his act together. A grad student at Virginia Tech. I asked him if he had seen anything unusual and he said no, but that they did have Pakistani neighbors — a family of three — husband, wife, and daughter. He said they were a typical family. I asked a few more questions and took a peek at the house across the street where the Pakistani neighbors lived. No one was home, but there was nothing out of the ordinary. The son admitted that his mother had some obvious issues. I closed the report as an unreliable source."
"I see."
Agent Rosson considered sharing his culinary interlude with the garlic muffin but decided against it. "Where is this conversation going, sir?"
The Director of the FBI Joint Terrorism Task Force cleared his throat. "The information I am about to share is highly confidential and is not to go outside this room."
"I will leave it here."
"The CIA hit on a match for the address across the street from one Maria Hayden. It appears that a clandestine agent for the CIA has infiltrated a potential sleeper cell and has taken up residence at the same address from the list we received from the CIA. This intelligence asset represents six years of work including, as I was told, imprisonment of the agent as part of his cover."
"You're kidding."
"I don't kid. The Deputy Director made it very clear that there is to be no action taken toward this address or anyone coming or going from the residence. It was, as he put it, a matter of national security. Any attempt to investigate this address or apprehend anyone at this address will be dealt with harshly."
"Meaning?"
"Meaning you have twenty eight months until retirement at eighty percent of your salary. So don't fuck it up."
"Not a problem sir. I closed the FBI's end of the investigation the day after visiting the location."
"That's good. That's good. I'm relieved," Director Nerf said with concern. "But I am going to need you to amend your report."
"Amend?"
"Re-post."
"Exactly what am I re-posting?"
"Go back in to the system and change the reason for closing the account. Amend the 'previous unreliable source' notation with 'closed at the direction of the Central Intelligence Agency.'"
Rosson was processing the information in his head. "You want it backdated?"
"Can you do that?"
"I can change the date that I closed the report. The date that I altered the report — the date I accessed the report — that I cannot change. So if someone is looking at the report, they will not be able to see what was written before but they will be able to see the date that something was modified."
"Do it."
Rosson was quiet for a second and then asked. "Why is the CIA running the domestic list through their databases? I mean, correct me if I am wrong, but the FBI has jurisdiction over domestic terrorism. If the CIA is using the list, why bother sending it to us?"
"Deniability. If they miss something, they blame it on us, claim that the information was the FBI's jurisdiction. If they catch something and we don't, they can point their finger at us for not catching it. And if something big, something important were to slip past our filters, we will be the ones to blame. The heads will roll right here in this office. Right here on this floor."
"So much for working with one another."
"You never believed that interagency cooperation bullshit, did you? Shame, shame. You think decades of institutionalized hatred and mistrust for one another is going to disappear because the President and Congress tells it to? You can see how that worked with the civil rights movement."
"A little different, but I see your point."
"You know, there is a saying in Washington that there is only one thing worse than another 9/11," the Director said.
"What's that?"
"Being blamed for it."
## Chapter 33
The wind howled along the waterfront in Georgetown. The curvy two-section five-story building that ran parallel to the Potomac housed a conglomeration of restaurants, law firms, and condos that cost well into seven figures. A fountain dissected the open air between the building sections and during the heydays of summer the seats around the pools of water were filled with young people holding hands and slobbering over their favorite ice cream. When the weather was nice, the restaurants facing the river opened their patios to throngs of young professionals who flocked to the deck chairs and tables like seagulls to a spilled order of French fries. The captains of expensive boats pulled up to the docks, luring the beautiful people from their tables for a trip to Roosevelt Island and whatever else they could talk or impress their subjects into doing. Downhill from the main drag on M Street, with its bars and boutiques, the waterfront offered a view of the Key Bridge to the right and the Kennedy Center to the left. It was an address for the kings, and those who wanted to imagine they were royalty for the afternoon.
With the wind whipping down the Potomac in mid-winter form, the action on the Waterfront was muted. The occasional brave jogger could be seen running down the large concrete path, their breath visible to the restaurant patrons who were warm and cozy behind the glass windows.
When a female Georgetown medical resident with a sensitive nose jogged downwind past the docks, she paused in motion. Legs still pumping in her high-tech thermal pants, her torso covered in a vest with her school's name, she removed her earphone jacks and inhaled. With that, the woman with dark hair and dark features stopped running in place.
Following her nose, she walked over to the retaining wall that kept the water of the Potomac, when it flooded, away from the expensive real estate. The jogger looked down at the empty boat dock. She first saw what looked like a leg protruding from the dock at water level. She moved several steps upstream for a better look. Peering over the edge, she calmly reached into her pocket, pulled out a thin cell phone, and called 911.
The dead body was stuck near the corner post of the dock, the back of the victim wedged under the planked walkway. With spring still months away, and the Potomac alternating consistency between ice water and a Slurpee, the body could have gone unnoticed for another month. Except for the smell. Nature's way of pointing the living to the dead.
The ambulance and police converged on the scene, lights flashing, sirens off. News traveled quickly. A couple of lawyers from a law firm on the fourth floor grabbed their coats and headed for the elevator. Where there was death, there was fault, and where there was fault, there was money to be made.
A small crowd braved the cold and stood on the retaining wall from a distance, watching the police pull the fully clothed body from the frigid water. When the coroner flipped the corpse on its back, a gagging sound mixed with the sound of a woman crying. The body's face was white, the mouth open.
Detective Earl Wallace, career officer with the D.C. Metropolitan Police Department, cornered the jogger who was perturbed that her run had been interrupted. He sucked in his gut a little as he made his way to the witness and introduced himself. Early fifties, black, and graying, Detective Wallace still maintained a boyish face. His knees and diet, however, protested the mere thought of an exercise routine.
"How did you find the body?"
"I smelled it," the jogger answered. The wires to her headphones now resting over each shoulder. Her red running cap was in her left hand.
"How do you know what a dead body smells like?"
"I'm a resident at Georgetown hospital. We have dead bodies from time to time. But I also did a pathology rotation in the morgue to see if I could stomach it."
"How did that go?"
"I survived. Made it through med school."
"Most of the bodies in med school are preserved."
"Yes, I know," the jogger answered, stretching her legs.
The coroner came over and whispered into the detective's ear. The detective nodded in response. The coroner zipped the black bag closed, its contents out of sight.
"Let me get your name and contact information and then you are free to go."
The jogger did deep knee bends as she gave the detective her personal information, and then jogged off in the direction of Georgetown University.
It took the medical examiner less than an hour to pinpoint drowning as the cause of death.
Detective Wallace, resident expert on dead bodies from the D.C. Robbery and Homicide unit, walked into the small office area of the morgue and hit the buzzer near the desk to let the medical examiner know that someone was waiting.
The medical examiner hit his foot on a lever that opened the door to the examination room and popped his head through the opening. He was wearing an apron smeared with human muck, streaks of undistinguishable, unmentionable bits, combined with the more discernable blood and dirt. Two dabs of powerful mentholated ointment stretched from the underside of the medical examiner's long curved nose. The medical examiner immediately recognized the officer.
"Detective Wallace. Please come in. Good to see you again."
"It's never good to see me. I only come when someone is dead. But thanks just the same."
"Like the Grim Reaper," the doctor said, returning to the table with the drowned body. The metal table was wet and water ran into the built-in drainage channels in the corners. The body was on its back, naked and fully exposed.
"What can you tell me?"
"White, male."
"Obviously," Detective Wallace said, looking down at the man's groin.
"I always start with the obvious," the doctor said with a smile. "White, male, in his mid-thirties. Cause of death, drowning, as indicated by foam in the airways and nostrils. Though that could change."
"What? The foam or the cause of death?"
The medical examiner looked over the frame of his glasses at the detective.
Wallace smiled and walked around towards the head of the deceased. "When we fished him out, he had a wallet, but there was no I.D. No driver's license. The wallet still had sixty bucks in it."
The doctor nodded. "Makes sense. Probably wasn't a robbery. I can't find any sign of foul play. No GSW, no stab wounds. No cuts. In a majority of drowning cases we find ancillary injuries. A lot of drowning victims have contusions on their head. They hit their noggin on the side of the boat, on a rock, whatever. Of course, many drowning victims also have post-mortem injuries from bumping into things in the water. This guy on the table has a few, but nothing of real consequence. There is always a danger of misdiagnosing ante-mortem and post-mortem injuries with drowning victims. Drowning victims always float face down. Given the weight of the head and the face-down position, some post-mortem injuries can bleed. But I don't think that is the case with this guy."
"He was found wedged under a dock."
"That shouldn't make a difference. The current probably carried him there and his clothes got stuck on something protruding from the dock structure. There was nothing to indicate otherwise."
"So no post-mortem means..."
"He probably didn't go too far."
"So it's not a body that came through Great Falls."
"Absolutely not. It fell somewhere east of Chain Bridge. There are too many rocks farther upstream. The body may have not traveled far at all."
"Anything else?"
"There was some residual blood in his mouth and sinus cavity that I haven't found a cause for yet. He has some scars on his knuckles. Battered hands. Looks like he was a man who wasn't afraid to mix it up. Maybe a boxer, maybe a barroom brawler. Also has a tattoo on his arm," the doctor said, pointing with the sharp end of the scalpel.
"Stating the obvious, again?" Wallace asked.
"I guess I am," the doctor answered, standing up from his hunched position.
Wallace looked at the clothes and shoes resting in the metal pan on the tile floor near the table. He picked up a pair of jeans and water dripped from them. "His dress seems normal for winter in this area. Jeans, worn hiking boots, an old winter coat."
"Yes, nothing unusual there. But it doesn't tell me how he got into the river. Hard to explain with the cold weather we have been having. Not many people are out there taking walks by the banks. Not even too many people up near the canal."
"And the fish aren't biting."
"That's true. Which is good for me. Drowning victims look better in the winter when the fish haven't been picking at them."
"What about toxicology?"
"We did a prelim, but didn't find anything. The full results should be done in a few days. I will check for the usual suspects."
"What do you think?"
"I know he drowned. His lungs were soggy; his stomach was full of Potomac water. Could have been unconscious when he drowned, but I don't know that for sure yet. But then, I guess figuring out the circumstances under which he drowned is your job."
"I guess it is."
"Can you give me an approximate time of death?"
"I'm having one of the assistants research the water temperature for the last couple of weeks. I should have a guess for you in a few hours."
"Off the top of your head?"
"Don't hold me to it, but I would say two, three days max. I think we saw fifty degrees over the weekend for a minute or two, so somewhere in that timeframe. I mean, it has been cold and there has been a fair amount of ice on the river already."
"Maybe that was the intention. Maybe someone dumped this guy in the river hoping it would freeze completely and that it would be another month before he popped to the surface."
"Possible. Sinister, but possible."
"Can I get some fingerprints and dentals? I still need to I.D. this guy."
"Already did. The prints and dental x-rays are on the counter," the doctor said, gesturing to the small work area in the corner. "The prints are a little messy, but thanks to freezing water, you might be able to get something from them."
"Thank you, doctor."
"Oh, I have one more thing..."
"What's that?"
"His natural hair color is blonde. The brown is just a dye."
Wallace walked into the station and handed the prints to his occasional partner, Detective Nguyen, a second generation Vietnamese-American. He was impeccably dressed in a light brown suit, shoes shined. His hair was short and he had a goatee, both attributes an intentional effort to look serious and older. "Could you run these through the database and see if anything comes up."
"Still in a tiff with forensics?"
"Not that it's any of your business."
Detective Nguyen, still a baby by police-tenure standards, looked at the set of prints. "Not exactly clean."
"You wouldn't be either if you spent a few days in the Potomac."
"Are these from the guy who ruined lunch down at the Waterfront?"
"Same guy. White, male, in his thirties. Tattoo. Died from drowning. Could be accidental or homicidal. We are waiting for toxicology."
"Good thing that body didn't come floating up there during the summer. All those tourists and visitors. Someone would have lost their lunch."
"A small victory for us, a poor consolation for the victim," Wallace said. He nodded towards the envelope he had just handed to Nguyen. "How long to run those through the database?"
"A day. But if it's a priority, I can get it pushed through."
"A day is fine. Someone will probably call in to I.D. the guy anyway before the prints are run. The M.E. thought the guy had been dead for a few days at the most. He doesn't look homeless. Someone will be looking for him."
Two hours later Nguyen appeared at Detective Wallace's desk. Nguyen's part-time partner was on what he called REM patrol, something he was known to do after lunch. "Sergeant?"
Wallace opened his eyes. "Yes."
"I got a hit on the prints."
Nguyen led Wallace to the far side of the precinct, through a maze of wooden chairs and ringing phones, and down a flight of stairs to the basement. Detective Nguyen pushed open the door to a glass room and a wall of new sounds rushed forward to greet them. Wallace followed Nguyen to the far corner. "I ran the prints myself and got a hit from the National Prison System database."
"Who is he?"
"His name is James Beach. Convicted felon. Did four years in Petersburg, Virginia. Long rap sheet. Most recently did time for drug charges and kidnapping. They caught him outside Bristol, Virginia with a trunk full of meth and a fifteen-year-old girl."
"A boy scout."
"Has served time for assault with a deadly weapon and auto theft. Had a couple of DUIs."
"Make that a choir boy."
"For the last eighteen months, he has been quiet. No arrest. No infractions. Meets with his parole officer in Richmond every month. I have a call into his P.O., and am waiting to hear back."
"Let me know what you find out."
"What are you going to do?"
"Call Petersburg. See if there is anything else we need to know about him."
## Chapter 34
Ariana peeked her head inside the warehouse before fully committing herself to the doorway. It was a habit she forgot she still possessed. She remembered the first time she had caught a rifle butt to the head for inattentiveness to her surroundings. Or rather, she remembered the headache and welt that persisted for a week after impact. It had been a non-lethal blow, from a teacher, but it made its point. When you are in operational mode, don't let your guard down. Even at home.
She stepped inside, and moved towards the large green button on the hanging controller pad near the door. Her eyes roamed the limited horizon as she raised the roll-up door on the far parking bay. She pulled her car inside and shut the door behind her.
As she walked away from her Toyota towards the office, Karim exited the community bathroom.
"Bathroom break?"
"Don't worry, we are following protocol. As you can see, everyone is still in the room."
Ariana walked over to the sleeping quarters' door and opened it. "I need to speak with Karim for a moment. We can start on lunch after prayer."
Syed and Abu looked up at Ariana from their seat on their cots. On the floor was a pile of playing cards. "Fine," Abu answered. "In another fifteen minutes, everything Syed owns will be mine."
"Lucky you," Ariana answered. "From here I can see all his worldly possessions."
Syed tried to smirk but it came out as a scowl.
"Just a few minutes."
In the office, Ariana shut the door and removed her coat. She had left before dawn, without make-up, wearing jeans and a button down sweater. Her nipples stiffened to the cold in the room and Karim noticed immediately.
"The package has been delivered," she said, drawing the blinds on the windows in the office.
"Your neighbor?"
"With any luck both the young man and his mother."
"Did anyone see you?"
"No. I was careful."
"How long?"
"A day or two. If it works."
"You don't sound confident."
"It was not a perfect product. That will require a little more time."
Karim sat in the chair and Ariana moved behind him, drawing a finger across his neck.
Karim felt her touch and his body responded below the waist.
"Killing the boy and his mother is a risk. Maybe an unneeded one. It will draw attention to us. To you. We are in no immediate danger from your neighbor. He knows nothing. A little nosey perhaps."
"I assure you that he'll be less trouble dead than he is alive."
"Perhaps."
"Besides it was necessary," Ariana said. She moved slowly behind him and he tried to turn to track her. She gently pushed his head so it was staring forward. "And exhilarating," she added slowly.
"The thrill of the hunt? Isn't the thrill better when the prey knows he is being hunted?"
"Not always. Killing can be enough of a thrill regardless of the prey." Ariana moved to the front of Karim and her sweater was open, her firm breasts straining from beneath a black bra.
"How long has it been?" Ariana asked.
Karim tried to count the years in his head, but his attention was elsewhere. "Too long. A lifetime ago."
Karim stood from his chair and Ariana met him in the distance between them. He leaned to kiss her and she turned her head. His lips redirected towards her neck and she let out a quiet moan.
A minute later Ariana's jeans were on the floor, her legs supported over Karim's elbows. She scratched at his exposed chest as her hips raised to meet his thrusts. She pushed him back as she felt him nearing the point of no return, and guided him to the chair. Straddling him, his face in her bosom, she pulled his hair as she took control.
_The Wood Artisans_ cabinet and furniture workshop was down the street from historic downtown Manassas, thirty miles from D.C. The train station that had once been a stopover on the way westward, now served commuters heading east into Washington on the Virginia Rail Express. The VRE ran trains during rush hour, taking some of the burden off of eastbound I-66, a road at such overcapacity that an inline skater with a missing wheel could keep up with the pace of traffic.
The Wood Artisans sat two blocks from the station, past the brick storefronts and the Starbucks, on the other side of the tracks. The warehouse stretched for a New York City block, which was the rough equivalent of three Manassas blocks. The front half of the warehouse was a showroom, offering custom made furniture ranging from bedroom ensembles to dining room sets that cost more than a compact car. Behind the decorated wall of the showroom was a cavernous workshop where a dozen carpenters practiced their trade in a dying industry.
Ivan Kozlowski was a fourth generation carpenter.
Seated at a long workbench on the back wall of the warehouse, Ivan grabbed the newspaper which was still wrapped in the plastic bag that protected it from the elements until the neighborhood residents found the energy to fetch them from the end of their driveways. The era of the paperboy was dead. And with his death was the end of doorstep delivery. Papers were now thrown from moving vehicles driven by middle-aged immigrants or retirees. But curbside delivery did have its advantages.
Ivan pulled the newspaper from the plastic bag and an imperceptible puff of white escaped from the paper as it hit the workbench, mixing with a thin layer of wood dust that covered everything in the warehouse. Ivan ripped open a small bag of sugar and poured it into his coffee. He pulled his daily dose of donuts from the nondescript white bag and placed them on a napkin. He swiped at the front page of the paper and the dust on the photo of the President on the front page. He wiped his hand across the workbench one more time before grabbing the paper and shaking it, just to the left of his gourmet breakfast.
As he did every Wednesday, Ivan spent the fifteen minutes before work going through the mid-week advertisements in The Post. The fifty cents he saved by stealing his neighbor's paper would cost him his life.
## Chapter 35
Clark's blue Honda Civic turned left at the pair of oak trees off the two-lane country road. He passed through the open gate and admired the crest of the mountains in the distance as the car crept up the long driveway. He pushed the open map next to him into the back seat and read the number scribbled on a piece of paper lying in the passenger seat. He checked the address on the house one last time. After a tank of gas, two pit-stops, two maps, and three different sets of directions from the locals, Clark had reached his destination.
He zipped his jacket and stepped from the driver's side. He stretched his back as he walked towards the white country house and up the stairs to the large front porch. The screen door was hanging cockeyed, the upper hinge holding onto the wooden frame with more authority than the lower one. A set of weathered rocking chairs sat empty, angled towards one another, a small table pinched between them.
Clark knocked on the frame of the precariously hanging screen door and waited before knocking again. A stiff wind blew through his denim jeans and Clark cursed, thankful that he wasn't living in the mountains of southwest Virginia, at least for the winter. The Blue Ridge Mountains in the summer were great for hiking and fishing. Spring brought wildflowers into full bloom. Autumn was simply breathtaking. But winter, with the trees barren and the ground frozen, was not his favorite time of year.
After knocking twice on the door, Clark peered into the living room window from the front porch. He took a few paces down the porch, the planks of wood creaking, and peeked into the dining room. He knocked on the window and announced his presence one more time. Another cutting breeze was the only response.
He walked back towards the front door, flipped the lid on the mailbox attached to the house, and peaked into the empty metal container. He looked at the leaves and twigs strewn about the porch and on the green welcome mat at the foot of the door. In his unprofessional detective opinion, no one was home, nor had they been for a while.
He began to wonder if the call he had made to a Foreign Service Officer in Pakistan, a man who felt passionately about his recently stolen cell phone, was just an additional scene in the _Candid Camera_ episode that was rapidly becoming Clark's winter vacation.
He had called Pakistan over the weekend, out of morbid curiosity. Curiosity stirred up from a mentally deficient woman in her seventies. A woman whom he happened to love more than anyone in the world.
The man in Pakistan had given him a second number made from the same missing cell phone. Clark took the number to Switchboard.com, a website that matches addresses and phone numbers, and vice versa. When the second number from the stolen phone matched with an address, and that address was in Virginia, Clark knew he would be in his car first thing in the morning.
In hindsight, Clark would have preferred to sleep in and read the Post with a cup of Joe.
Clark took one lap around the yard and the eerie stillness of the old frame farmhouse made Clark nervous. The yard was deserted, the summer grass long since dead. Brown patches of weeds dotted the edge of the yard where it blended into the first layer of pasture. He announced himself as he made his way around the house, not sure of what he would say if someone actually answered. Clark thought about his explanation and then considered that it's not every day someone drives to a farmhouse to ask about a cell phone call made from a stolen phone in Pakistan. Clark mumbled something to himself that sounded like a self introduction and then caught himself. "What the hell are you doing?" he asked aloud.
Five minutes later Clark walked back towards his car, the empty house yielding no warm bodies, no greetings, no coffee, no clues. Clark's hands were chilled by the air. As he unzipped the pocket to his blue winter jacket his keys fell to the ground. A breeze blew over his back as he reached for the keys in the gravel driveway.
A pile of odd-looking stones caught Clark's attention and he grabbed one along with his keys. He brought the stone to eye-level. It felt light. The shape was asymmetrical. Clark looked down at the pile near the left front fender of his car. Then he raised his glance back at the empty yard and beyond the house.
He returned his focus to the ground, his eyes dancing across the small objects that littered the end of the driveway and continued into the yard. He shuffled his feet in measured steps, following the trail for a few yards in front of his car before bending over again and retrieving two more objects. As he ran his hand across the ground, he noticed a dozen more of the curious objects resting in the indented strips of matted dead grass where a vehicle had been driven. The sound of a slamming door in the distance broke his concentration and Clark stood from his crouch. He was getting spooked. He waited for a moment, took another lap around the house, and decided he had seen enough.
## Chapter 36
The Nelson County Sheriff's office was wedged between the Lovingston General Store and the Lovingston Post Office in a row of old townhouses turned storefronts. Combined with a single-pump gas station across the street and the sawdust-floor watering hole half a block down, the handful of buildings hovering around a sidewalk statue of General Lee made up bustling downtown Lovingston and served the ten thousand habitants of the rural Virginia county. Charlottesville was down the road, twenty minutes away by Nelson County time, the standard answer for how long it took to get anywhere in the sprawling jurisdiction. If it was more than twenty minutes away, it was a safe bet the person giving directions had never been there.
Sheriff Laskey leaned back in his old wooden desk chair and the spring supports squeaked as they absorbed his weight. The sound of the chair was part of the office, as much as the old black phone with the authentic bell ringer, the dark wood bookcases on the walls, and the Mayberry-like jail cell in the back room that held the occasional drunk for a few hours until they sobered up. Real legal perpetrators were sent down the road to a larger facility in Charlottesville. Incarceration at the sheriff's jail didn't include a meal plan and the law just wouldn't permit long-term guests with empty stomachs.
Sheriff Mike Laskey worked ten minutes from the house where he had been born on a worn kitchen table his grandfather had built from trees on the Laskey family property. His father held his mother's hand as the local veterinarian tended to the business-end of the delivery and, on a sweltering summer day in 1947, Michael J. Laskey joined the world of the living with a wail. The umbilical cord was cut with a butcher's knife sterilized in a pot of boiling water, and young Laskey was three months old before he saw his first doctor. It was an aversion to medicine he never got over. Not even when gangrene set in on the middle toe of his left foot after a failed dancing lesson with a cow.
The main valley of Nelson County started south of Charlottesville and ran like retreating Confederate soldiers until the mountains pinched in on three sides. Sheriff Laskey had seen his county change over the years. Charlottesville, in Albemarle County, was wealthy. The city and its surrounding areas were pollinated with old money that spread generation to generation. Families with prestigious sounding names and the genealogical pedigree to prove them, bought up prime rolling acreage for horse farms and wineries. In the last two decades the good ol' boys who ruled the roost in Charlottesville had spread their golden wings and started swooping down on Nelson County.
It started with Wintergreen, a resort for weekend getaways, as if families living in a mansion with a pool and horses needed a place to relax and unwind. With a spa, golf courses, and ski runs, Wintergreen, smack dab in the middle of Nelson County, was a haven for the wealthy and a nightmare for a two-man sheriff office. Rich kids with too much money raced their convertible graduation presents along the winding roads, occasionally wrapping the finest German engineering around hundred-year-old tree trunks. Teenagers high on the designer drug-of-the-year ran across the golf course at night screaming, some of them believing they could fly and flapping their arms as if they were ready to prove their point. Swinger parties were known to occur, the participants some of Charlottesville's finest citizens who frolicked naked in scenes from Caligula, all in view of their neighbors through soaring glass windows.
The rest of Nelson County was a different story. Where the encroaching neighbors to the north rode their horses around their vineyards, most Nelson County residents took great pride in cultivating vintage cars in their yards. Dilapidated vehicles without engines, tires, and doors grew easily in some of the most fertile land in the mid-Atlantic.
The landscape was different in old Nelson County and so was the sheriff's clientele. The wife who cold-cocked her husband with a frying pan for cheating. The five-man hunting party that managed to get separated and shoot each other in the woods despite perfect weather conditions and broad daylight. The father who beat his son within inches of his life for changing the channel on the satellite television, the one real amenity in their double-wide trailers.
Sheriff Laskey dealt with them all. In his mind, they were all the same. He knew the law and he loved the Bible, and somewhere between the two he found peace in his job.
And the peace was about to be ruined.
For the last half-century, Nelson County had been infamous for one thing. A small meteorological record that is unlikely to be broken unless God reneges on His word and sends Noah back down to work on a second rendition of the Ark.
In 1969, Hurricane Camille smashed into the Mississippi coast as a full-fledged Category 5 monster. The storm surge was estimated at twenty-five feet with wind gusts topping 200 mph. Entire blocks were wiped clean. Brick apartments were removed from their foundations and swallowed by the sea or blown inland bit by bit. One man purportedly escaped from the attic of his two-story home, barely squeezing out a small window before the house was engulfed by rising water. He lived over a mile from the coast. Still three days away, no one could have guessed the wraith Camille was going to let loose on a small county in central Virginia.
When the first raindrops started to fall in Nelson County, Sheriff Laskey, who had not yet been voted into his current title, was busy with the tractor in the old red barn behind the family house. As darkness fell, so did larger drops of rain. By the time the radio was crackling out warnings and the TV had issued an emergency broadcast, it was too late to run. Over the next twelve hours, thirty-six inches of rain fell onto Nelson County's mountains and rolling hills. There were stories of mothers who carried their babies face down so the children wouldn't drown. A man trapped outside claimed to have survived by burying his mouth in the hollow of a tree to breathe as the storm reached its crescendo. Rain came down in sheets, no longer drops. Laskey gathered his wife and daughter in the hall on the first floor of the farmhouse to pray by candlelight. The shingles on the roof were no match for the relentless downpour and between Psalms and Proverbs, Laskey and his wife hopped from room to room on the second floor, catching drips from the ceiling in pots, pans, and buckets. The rain made noises Laskey had never heard, noises he had never knew were possible. It was as if the house was sitting directly under a waterfall, the long unbroken flow of water relentless.
Sometime before dawn, as Laskey emptied another bucket into the bathtub on the second floor, he heard the rumble. His wife sat up in bed and called his name as the rumble turned into a crash. Laskey ran downstairs, flung open the door, and looked out over the old porch into the night. He saw nothing beyond the flooded yard, a river where the driveway had once been.
The crash had lasted thirty seconds before it slowly disappeared into thunder and rain. There was nothing to do but wait.
Dawn brought little reprieve from the deluge, but by afternoon the word of tragedy had spread to every home with a working phone. The mountains in western Nelson County, drowning in flood waters the country had never seen, had crumbled like mashed potatoes being washed off a dirty plate after supper. Twenty feet of top soil, and everything living on it, had crashed down on the county, swallowing two hundred residents as they slept. Their houses were never found and the official death toll was merely a best guess by emergency workers a week later.
Now, forty some years later, Nelson County was about to gain notoriety for something non-weather related.
The phone on Sheriff Laskey's desk rang, and the sheriff arranged the hat on his head before he leaned forward in his squeaky chair to answer it.
"Laskey, Nelson County Sheriff's office,"
"Sheriff. This is detective Earl Wallace of the D.C. Metropolitan Police."
Laskey rocked back in his chair to another squeak. "D.C. police you say?"
"Yes, sir."
"We don't get many calls from D.C. down here."
Detective Wallace thought about the statement for a second and continued. "I guess not."
"What can I help y'all with?"
Detective Wallace noticed the contraction "y'all," and wondered just where the geographical line was when the population dropped "you" in favor of the southern extension. "Well, I have a dead body here in D.C. that we pulled out of the river. We ran the prints and found that they belonged to James Beach, a former convict in Petersburg, Virginia."
"Petersburg is on the other side of the state," Sheriff Laskey stated plainly.
"Yes, sheriff, I know. I already spoke with the assistant warden at Petersburg. I'm interested in locating one of his cellmates at Petersburg. Are you familiar with a Nelson County resident named Jackson Price?"
Sheriff Laskey nodded to himself as he spoke. "Yeah. I know Jackson Price, all right. Also known as J.P. around here, among other things."
"Class citizen?"
"He's been on the Nelson County catch and release program for years."
"What do you know about him?"
"What don't I know about him might be a shorter conversation."
Detective Wallace laughed. "I'll take the short version."
"Jackson Price came from a fairly wealthy family in Nelson County. His father ran a couple of car dealerships. One in Staunton. One in Charlottesville. One in Lynchburg. He was a good man. Raised two sons by himself after his wife ran off with a doctor from Richmond."
"Was a good man?"
"God rest his soul. Mr. Price passed away, oh, must be eight, ten years ago. Cancer."
"Sorry to hear that."
"Yeah, well, his sons were nothing but trouble before he passed and their behavior didn't get any better with their father in the cemetery. The older brother disappeared a few years ago with a couple of warrants on his head, but the Price family still has a home here. Over a hundred acres in southern Nelson County. A nice piece of land. Rolling hills and pastures. They grew apples for a while, until Mr. Price passed and Jackson decided to grow another more profitable crop."
"Marijuana."
"You got it. Got sent to the big house for cultivation and possession with the intent to distribute."
"Have you seen him around?"
"I've seen him a few times since he got out last year. Seems to be staying out of trouble, I guess. No one has called me about anything, if that's any indication."
"Would you mind seeing if you can track him down for me?"
"You expect he had something to do with the floater?"
"He may, may not. Just trying to figure out how an ex-con ended up in the Potomac."
"J.P. may be a stoner and a rebel, but I doubt he killed anyone. He just doesn't have it in him. His older brother, well, now, he's a different story."
Detective Wallace jotted in his ever-present notebook. "Just the same, I would be grateful if you paid Jackson a visit, or see if you can locate his whereabouts. You know how ex-cons are."
"Thick as thieves."
"It's like a brotherhood."
"Let me get your number, and I'll take a spin by the Price farm later this afternoon."
"Much obliged, Sheriff," Detective Wallace said with his best cowboy western impersonation.
Sheriff Laskey shut the door on the brown cruiser with his title emblazoned down the side in bold silver letters. He pulled out onto Route 29 south, and drove past the barbecue pit restaurant on the edge of Lovingston proper before turning on Route 808 for the winding two lane road that ran past the Price farm. True to the words of every county resident, he pulled up to the open gate exactly twenty minutes later.
His cruiser kicked up dust as it rolled down the gravel and dirt drive that meandered through a short grove of mature trees. At the end of the trees the road cut right and traversed an open field that led to a picturesque two-story farmhouse.
The sheriff's car came to rest on the left side of the house. The sheriff pulled his lanky frame from the vehicle and took a deep breath of the afternoon air before walking towards the front door. The sun was at its winter apex in the sky. He stepped on the front porch and premonition tied a knot in his stomach. There were no outwards signs that anything was amiss. There were no newspapers piled on the porch, not that the Price brothers would have bothered to read them if they had been delivered. There was nothing in the outward appearance to tell Sheriff Laskey that something was wrong. Nothing except the dull warning in his stomach, a feeling that had proven time and time again to be more accurate than real evidence.
He knocked on the door, waited, and knocked again. He found himself checking his weapon with his right hand and unsnapping the small leather strap that kept the gun in his holster. _It's just a hunch_ , he said to himself. _Nothing but a hunch_.
He tried the door knob with no luck and stepped off the wooden porch with his hand still on his holstered weapon. He peered into a sitting room through a side window on the house and carefully walked to the back of the property, his eyes darting with alertness. He moved around a small plastic table and chairs resting on a patch of stone slabs that had been arranged to form a make-shift patio. He pressed his nose to the window on the back door and twisted the locked knob.
His nose against the glass, his warm breath fogging up the window, Laskey jumped as the sound of crashing wood rushed him from behind. His gun was in hand with the safety off before he turned around. Senses jumped to high alert. Laskey stepped from the porch, his weapon by his side. A few seconds passed and another collision echoed across the backyard. The sheriff stared out towards the large gray barn behind the house. He gingerly moved across the yard, less anxious than he had been a moment earlier. To him, old barns were something he had grown up with. Even angry old barns that were barking out warnings. Shadows from the sun through the naked tree branches covered the ground as he approached the edge of the barn. Laskey peeked his head around the corner, momentarily looking down the sights of his pistol before dropping it into a two handed waist high position.
"Nelson County Sheriff's Office," he announced to the empty stalls, the hayloft, and the work area in the back left corner.
Silence.
He walked through the barn, his boots landing on remnants of old hay and dirt. At the far end of the barn he stopped at the workbench and took inventory of the tools that hung on the wall. A large wood saw hung by its handle. An oversized wrench dangled from a hook. An assortment of hammers, files, and vises filled the area in no particular order. A basket of old horse bridles rested on the floor near the rear entrance of the barn.
The large door in the opposite corner of the barn slammed shut with surprising authority, the metal plate near the lock smacking hard against the door frame. Laskey whirled towards the noise. The sheriff, sweat beading beneath his gray hairline, pushed open the rear door and stuck a nearby pitchfork into the ground to stop the door from slamming in the wind. Silence restored, he stared out at the rolling acres that made up the Price family farm. He looked towards a large pile of weeds near a neglected fence line and scanned the horizon for anything suspicious. In the distance he could see the ski runs on Wintergreen and the snow covered tree-line that ran along the Blue Ridge Parkway at four thousand feet.
He brought his focus from the mountains in the distance and zoomed in on the Price farm. His eyes narrowed, the wrinkles in his forehead gathering near those between his eyebrows and the top of his nose.
Sheriff Laskey walked to the edge of the field and stomped over the dead grass to a row of withered plants, some half-heartedly standing, leaning in the winter wind, waiting for the arrival of a spring they wouldn't live to see. He reached down and picked a plant off the ground. The core stem was firm, hardy. He grabbed the stalk and held it next to his body to estimate the height. The plant towered his six-two frame. "Just what in the hell have we been growing here?" he asked.
The sheriff reached into his pocket and pulled out a bag of Red Man chewing tobacco. Brain food. He pushed his thumb and the first two fingers of his hand into the corner of the foil-lined bag and pulled out a wad of dark brown, intertwined tobacco leaves. He shoved them into the right side of his cheek and bit down several times to release the juice. He spit once, and moved the wad into the pocket of his gum-line.
"Now let's see what we have going on," he said to himself in almost a whisper.
With a hundred acres of land, the sheriff had no intention of walking the entire farm. When he finished his chew, he would be finished with his walk. For ten minutes he walked across the Price farm among the never-ending sea of dead plants, stopping occasionally to examine their remains, to pick at their seeds, the hopeful offspring for a future generation. When he reached the start of the old apple orchard, a faint smell in the air brought him back to reality.
He saw the boots first, the dangling toes of the shoes protruding from the side of an apple tree in the distance. As he approached, the large brass belt buckle in the shape of the initials J.P. stared back at him at eye-level. Jackson Price, or what was left of him, was three feet off the ground, his neck snapped, an old rope holding his weight from a branch above. His hands tied behind his back. "Godammit," the Sheriff said. He dislodged the chew from his cheek and spit it on the ground.
He picked up his radio and called his deputy sheriff. He checked for dirt on the bottom of Jackson's boots and then took a look at the man's face and tried to remember what he looked like before death had come calling. A stiff breeze kicked up and Sheriff Laskey made his way back to the house. He stood on the front porch and waited for the Cavalry. A cooler wind and the sirens in the distance were the only reply.
_The Pig and Whistle_ was crowded, by Nelson County standards. Two men in camouflage, obviously on the return leg from stalking one of God's creatures that may or may not have been in season, stood in line at the small counter that served made-to-order sandwiches. The larger and dirtier of the two men asked old Mrs. Dalton what the sandwich-of-the-day was. Mrs. Dalton, hair tied back in a white hairnet with a matching white apron, gave her standard response. "Country ham on country wheat." It was an answer she had given everyday for thirty years and one that most of the population of southern Nelson County knew before it was given.
A teenage girl with an oversized jacket that made her look like a red Michelin man stood at the single soda cooler, trying to decide between three varieties of Coke. _The Pig and Whistle_ had long since excluded itself from the Pepsi versus Coke battle played out in commercials and advertisements. The eight hundred square foot store didn't have room for variety. And there was no competition within ten miles to force them to do anything they weren't good and ready to do. At seventy-five, Mr. and Mrs. Dalton were past the time in their lives when they saw change as a good thing.
When Sheriff Laskey walked into the store, Mr. Dalton was hunched over the counter reading an old issue of the _Nelson County Times_. His white hair was combed over and the sheriff got a bird's eye view of Mr. Dalton's effort to conceal the effects of the recessive maternal gene. "Good afternoon, Sheriff."
"Good afternoon."
"What can we do ya' for?"
The two men in camouflage grabbed their sandwiches from Mrs. Dalton and squeezed past the sheriff to the register. Mrs. Dalton tracked the men from behind the counter and rang them up. Sheriff Laskey watched them as they left the store and headed for the mud-covered pick-up on the far side of the parking lot.
"You seen anything strange around here?"
"This gotta do with the body up at Price's farm?"
Sheriff Laskey's eyebrows jumped slightly. "News travels fast. Particularly to someone who is reading last week's paper."
"Small county. Don't need a paper for most of the happenings round here. Which one of them boys did you find up there?"
"Jackson."
"Figured as much. I was betting it was J.P."
"Betting with whom?" Mrs. Dalton chimed in.
"You seen him around?" the sheriff asked.
"Time to time. Haven't seen him for a few weeks. Haven't seen his brother in over a year."
"You ever see him with another guy? Someone who wasn't local?"
"Sure." Mr. Dalton paused to look at his wife who was listening intently. He looked out across the store at the Michelin girl who was still contemplating the soda cooler. Then he continued. "There was another guy who came in here with him a few times. Blond guy. Looked kind of like a surfer, though I can't say that I've ever seen one in real life."
"Anything else?"
Mr. Dalton looked over his shoulder at his wife for confirmation. Mrs. Dalton nodded her head slightly. "No. The guy was blond and had a few tattoos. But heck, everyone has a tattoo these days, so I don't reckon that narrows down your options much."
"Not much."
The girl in the thick red jacket at the back of the store walked forward and put her Coke Classic on the counter.
"You Ted Sherman's kid?"
"Yes, sir," she said from the depths of the coat with its huge rolls of contained goose down.
"You seen anything strange around here recently? People you don't know?"
"No, sir. No stranger than the usual."
Sheriff Laskey smiled.
Mrs. Dalton spoke. "There was a young man in here about an hour ago. Grabbed a sandwich, got gas, and asked for directions to Charlottesville."
"Charlottesville?"
"Seemed lost."
"You get many people in here asking for directions?"
"Sure, we get plenty. But most of them are looking for a fishing hole or a hiking trail or the ski runs. City folk and people from Charlottesville mostly. But we don't get many down here looking _for_ Charlottesville. Not many people are traveling from the South. I got the impression he was from the city."
"What was he driving?"
"A little blue import. Toyota or Honda. I don't know much about cars. I think it had a Virginia Tech sticker in the window, but my eyesight is failing me. But it was a sticker with Virginia Tech colors anyway."
Mr. Dalton looked over his shoulder at his wife, surprised at the sudden sharing of observation skills he didn't know she possessed. "Probably nothing. A rich kid down at Wintergreen for the weekend who took a wrong turn."
"Probably."
"So, Sheriff, how did he die?"
"It looks like he hung himself. Tied off to one of the old trees in the apple orchard." Laskey looked down at the counter and away from Mr. Dalton. "The critters got at him a bit. Wasn't real nice."
"Jesus. Good thing his parents weren't around to see that."
"God rest all their souls."
Officer Wallace fumbled for the cell phone in the passenger seat of his unmarked D.C. police cruiser. He was in the middle of DuPont Circle, heading towards Georgetown, and he answered on the second ring.
"Detective Wallace."
"Good evening, Detective. This is Sheriff Laskey of the Nelson County Sheriff's office."
"Good evening, Sheriff."
"I have some good news and some bad news on the person you asked me to find."
Detective Wallace knew what those words meant. But if the good sheriff was willing to pitch a slow ball right across the plate, it would have been unprofessional not to at least take a swing at it. "Let me guess. The good news is that you found him. The bad news is that he's dead."
"Hung himself in an old apple orchard."
"Well, I certainly couldn't have guessed that ending. How long has he been dead?"
"A while."
"You sure it's him?"
"I don't have DNA proof, if that is what you mean. Won't have that for a while. But I know it was him. Had on a big J.P. belt buckle. Known that kid his whole life. It was him."
"Let me know when you have confirmation, if you don't mind. Not that I don't believe you."
"Sure thing, Detective. There was something else. Two things actually. First, it seems as if the deceased had been growing something on his farm."
"Sounds like his hobby. That's what sent him to Petersburg."
"It wasn't marijuana this time. I'm not sure what it is exactly. Big tall plants that look like weeds. But they were grown in rows, real orderly. Most of them are dead of course, being that it's winter. There were some seeds on the ground that I'm going to have someone take a look at. See if they can tell me what they are."
"What was the second thing?"
"There was an open Koran on the dining room table of the house."
"A Koran?"
"Yep. Don't think I have ever seen one before. Fancy book. Gold writing on the cover. Arabic on one side of the page and an English translation on the other. At least I think it was Arabic."
"You have a Muslim community there in Nelson?"
"Detective, this is King James Bible country. But I don't think Jackson Price had been going to church."
"Sounds like he was praying for something."
Laskey was leaning over his desk looking at a stack of notes when Debbie Ingle entered and added another to the pile. Debbie, the sheriff office's lone secretary and unarmed employee, exited the room quietly. The sheriff was thinking and she knew that meant to leave him alone. Debbie was in her forties but looked younger with her blonde hair in a short earlobe-length cut. She was married to a local math teacher who was a sort of a renaissance man among Nelson County circles. He taught math, but was also known for his poetry and plumbing prowess.
Debbie sat behind her large wood desk, dressed in her usual work attire — black slacks and a white button up blouse. She occasionally mixed in a skirt when she was feeling sexy, but the color scheme never changed. It was like working with a waitress. But what she lacked in fashion sense, she made up for in organization. She took phone calls, kept records, and made copies of everything that passed through her hands. Her husband had taught her how to use a computer and half of the office's files were now scanned and saved electronically.
"Did they get J.P.'s body to the medical examiner's yet?" Laskey asked from the next room in a loud but professional voice.
"I haven't heard anything, but I will make a call and check," Debbie answered as the sheriff came from his office in the back room.
"I need to check on something. I'll be back in an hour or so. Going over to the Seed and Feed."
"You walking?" Debbie asked as the sheriff headed towards the front door, the opposite direction from his car parked in the lot behind the building.
"Yep. Let's call it a foot patrol," the sheriff said pulling open the door.
"Ok, Sheriff. I'll radio you if I hear anything."
"And see if anyone knows where we can reach Jackson's two-bit brother. I need to notify next of kin, if I can find him."
_Lovington Seed and Feed_ was a half-mile from the sheriff's office, past the gas station and just beyond a large field that had once been a horse stable but was now overrun with waist-high weeds. The field offered a nice backdrop for the Seed and Feed, the large red barn and slightly leaning silo providing a picturesque scene for any artist who was able to envision the field of weeds as a swath of winter wheat.
The sheriff threw in another chew of Red Man and stood on the sidewalk for a minute, staring up and down the "main drag," a street that was less of the first and more of the latter.
Kenny Buckner, dressed in jeans, work boots, and a green flannel shirt, grabbed the last bale of hay and threw it in the back of the red Ford 150 pickup. He tossed the rope from one side of the truck bed to the other and proceeded to hogtie the half-dozen blocks of hay.
"Good morning, Kenny."
"Good morning, Sheriff."
"How's business?"
"Still afloat. Been a cold winter, which hurts us a bit, but we'll make it."
"Sometimes making it is enough."
Kenny finished tying the load in the truck and tugged at the taut ropes one last time.
"Can I get you a cup of coffee, Sheriff?"
"As long as it's inside."
Kenny Buckner put two Styrofoam cups of coffee on the table in the small Seed and Feed office. Sheriff Laskey reached into the pocket of his jacket and removed an evidence bag. "Ever seen these before?" he asked, putting the bag on the table.
"Is that evidence? Isn't it against protocol to walk around with evidence?"
"What have you been watching, CSI?"
"Yeah. The wife loves it. Thought she was a forensic specialist after the first season."
Sheriff nodded towards the bag. "Got those off the Price's farm. Seems they were growing them. But I'm not sure what they are."
"Sorry to hear about J.P. He and my son were friends until J.P got the devil in him. They say he killed himself."
"That's what it looks like. The body is on the way over to the medical examiner's for confirmation. Been a long day."
Kenny Buckner took a sip of coffee and opened the bag. He reached in and held one of the seeds to the light. "Looks a bit like a coffee bean."
"Thought the same thing myself."
Kenny rolled the bean in his finger and then pulled out his buck knife from the sheath on his waist. "Do you mind?"
"Go ahead."
Kenny pushed down hard on the seed and it cracked under the pressure of the sharp blade. Kenny wiped the small residue of oil off the blade onto his dirty jeans. Each man picked up half of the seed.
"Looks like a seed to me. Nothing strange."
"I agree. Except for the fact that Jackson Price was growing them on the family farm. And there ain't been nothing legal growing on that farm since the death of J.P.'s father."
## Chapter 37
Clark had stopped to look at his discovery twenty minutes north of Charlottesville. The multi-colored gravel he had found all over the yard he had now re-classified as a seed or a bean. His precious cargo in need of an explanation was currently riding shotgun in the cup holder of his Honda Civic. The drive back to D.C. took just over three hours, half as long as it took him to find the Price family farm on his journey down the Blue Ridge Mountains.
His jaunt to Nelson County generated more questions. Who the hell was calling both his neighbor and a farm in Virginia? Why? Right now the only clues he had were the three large beans vibrating in the cup holder near the parking brake. The objects made him nervous, curious. As the lines in the road zoomed passed, Clark imagined himself getting in an accident, losing his beans in a sea of broken glass and twisted metal, mumbling to ambulance personal to locate his magic beans because he had questions he wanted answers to.
As his mind raced, he did his best to keep his foot light. He felt relief at the first sign for the Beltway. There was something about coming home, even when home was measured at five miles per hour, bumper-to-bumper.
The small print on the sign outside the Merrifield Garden Center on Lee Highway read, "If you can't find it here, we'll get it for you. If we can't get it for you, it can't be found."
Clark pulled in at the far end of the parking lot near the edge of the white one-story building. Three large greenhouses in the back of the lot stared down at the shorter white building. Beyond the greenhouses, acres of bushes, plants, trees, and sod stretched in three directions.
The Merrifield Garden Center was holding onto some of the most prime real estate in the region. A quarter mile from the Dunn Loring Metro station, development surrounded the nursery on three sides. But Merrifield Garden Center wasn't in the selling-out business. They were in the growing business. And with most of the other garden centers and nurseries pushed into the far limits of the suburbs, Merrifield Garden Center had, through attrition, become a monopoly of sorts.
The automatic doors in the front of the store opened and Clark smiled at the middle-aged woman working as the greeter and director of traffic. "Can I help you find something today?"
"Yes, I'm looking for your seed section."
"Seeds are in the far back right of the store, just before the door to the fertilizers."
"So just follow my nose?"
"I guess you could," the woman answered with a smile.
Clark meandered through the store, checking out the selection of potted flowers with his mother in mind. Then his mind moved on to Lisa and he knew he would end up with two different arrangements. He passed a small display of garden pools, complete with fish, and turned left at the large palm tree which was as out of place in the D.C. winter as he was with a handful of beans.
The wall of seeds stretched forty feet and Clark whistled quietly as he took in the view. There were hundreds of bags hanging on shiny silver metal hooks. Clark walked down the wall, looking over the heads and shoulders of other patrons who were more certain about their horticultural needs. The seeds were listed in alphabetical order which did Clark no good as he had no idea what he had. A handwritten sign hung on the hook where marigolds were previously sold. The sign read "Due to a recent surge in the purposeful ingestion of marigold seeds for recreational use, we will no longer be carrying them."
Clark spent twenty minutes pulling bags from the display on the wall and comparing them to the bean in his hand. After reaching the midway point in the wall, he gave up and looked for help. In the hanging plant section he found the same middle-aged woman who had greeted him at the front door and cornered her as she finished doling out advice to a yuppie with a cartful of his wife's marching orders.
"Did you find the seeds you were looking for?" she asked.
"No, I sure didn't. Maybe you would know what kind of seed this is," Clark said as he pulled one from his pocket.
The woman picked up the almost dime-sized seed and held it towards the light. She ran her fingers across the unusually textured skin and she admired the chocolate swirl appearance. "I don't know what this is. But it sure looks more like a bean than a seed."
"Maybe it is a bean. I'm not even sure what the difference is, really. I don't know much about gardening."
"I know someone who does. Follow me," she said. She marched through a set of double doors and past a long stretch of cacti that lined a single shelf on the wall. Head up, apron flapping, she continued walking, her eyes focused forward on the bean firmly pinched between her forefinger and thumb.
Without stopping she turned the corner through an "Employees Only" doorway and Clark followed closely behind. She passed through a sparsely decorated break-room with a round white table and a vending machine, and headed straight towards an office on the other side of the room. She stopped at the doorway and rapped slightly on the frame.
"Jerry, there is someone here who has a question maybe you can answer for him."
"Send him in."
"You're in good hands now. Jerry will take care of you." The woman pressed the bean into Clark's hand. Her job finished, she disappeared.
Clark stepped into the office and found himself staring at the top of Jerry's head as the man looked through a magnifying glass. Jerry paused and looked up at Clark, whose head snapped back a little in reaction to the thickest set of eyeglasses he had ever seen. Jerry had a chiseled face and dirty muscular hands. The kind of hands you would expect from a man who spent a lifetime in a garden center. His well-kept black curly hair wrapped around the top of his ears. The man needed a new pair of glasses.
"Hi, Jerry. Nice to meet you. I'm Clark."
"Clark, grab a seat," Jerry said, now looking back through the magnifying glass. "I will be with you in just one second."
"What are you looking at?" Clark asked after a minute of silence.
"Not exactly sure yet. I'm trying to figure out what kind of herbivorous insect has been nibbling on my ferns like a salad. Never seen this character before."
"Are you a bug specialist?"
Jerry smiled. "Nope, not a bug specialist. I own this place. And as its owner I am trying to determine just what's eating my inventory."
Clark smiled back and Jerry pushed the magnifying glass to the corner of his cluttered desk. Jerry looked up again and Clark fought the urge to tell him his glasses made him look like the bug he was trying to identify. Clark took in the room. A bookcase on the wall sagged under the weight of a hodgepodge of topics from lumberjacking to flowering perennials to insecticides.
"What can I help you with Clark?"
"I was wondering if you knew what kind of bean or seed this is?"
Clark put the seed on the table and Jerry picked it up with his thick fingers. Dirt was encrusted under the edge of his nails, the skin on his hands was rough and cracked. "Where did you get this?"
"Found it on a farm in Virginia."
"And you have no idea what it is?"
"None."
Jerry got up from his seat and headed for the bookcase. He ran his fingers along the spine of his books as he read, whispering to himself as he went along. _Not this one. Not that one. There might be something in here_.
He grabbed a black covered book from the middle of the shelf and placed it on a pile of papers on his desk. He turned the pages deliberately, scanning every picture. "If that is what I think it is, you may want to keep it quiet."
"Why's that?" Clark asked.
Before he could answer, Jerry poked his finger down on one of the pages. "Right here. I think this is what we're talking about."
Clark read the description and plowed roughshod over the Latin pronunciation of the word. Jerry watched as Clark continued to read and when Clark's eyes opened a little, Jerry knew he had reached the good part.
"Holy shit."
"That pretty much sums it up."
"How much of this did you find?"
Clark reached into his pocket and pulled out two more beans. "There were a few, hundreds maybe, scattered on the ground. I took three of them."
"Where in Virginia?"
"Nelson County. South of Charlottesville."
"That's odd."
"Why do you say that?"
"Well, Ricinus Communis likes slightly warmer climates as a rule, but it could grow quite well there."
"Grapes like warmer climates and there are a lot of vineyards in central Virginia," Clark added.
"Yes there are a few vineyards. And you're right, the growing conditions for grapes and our friend here are not that dissimilar. But I don't think these were grown in the winter. It's just too cold."
"Ricin," Clark said aloud, as if his mind was on delay.
"The name Ricinus, and hence ricin, comes from the Latin word for 'tick,' which obviously someone thought these seeds resembled. Our friend here is more commonly known as the castor bean, though it is technically a seed."
"If they were grown during the summer, could they be stored?"
"Sure, sure. Hell, these plants grow wild in some states here in the U.S. They are classified as a weed in Florida, if I remember correctly."
"Classified as a weed? You mean it isn't illegal to grow?"
"Not at all. Illegal to process, not illegal to grow."
Clark read the passage in the book again. "Well, that doesn't sound like a good idea, just allowing anyone to grow them."
"The castor plant has been around for hundred of years. Maybe thousands. They use it in paints and varnishes. It's water resistant and is used as a coating for insulation and guns. And it's used as a medicine. Also as a motor oil."
"Medicine?
"Castor oil. What did you think that was?"
"I never thought about it."
"I remember reading there are over 200 metric tons of semi-processed ricin produced every year. If someone with ill intentions really wanted to get their hands on a lot of ricin, all they would have to do is park outside a castor production plant."
"Where is the nearest castor production facility?" Clark asked.
"India, probably."
"Maybe it would just be easier to grow it."
"Maybe. Either way, stolen or grown, gram for gram, it is a hundred times more deadly than most conventional poisons, man-made or natural."
"Pretty nasty stuff."
"The meanest bean on the planet. You say you only found a few hundred?"
"It was hard to say. They were on the ground, in the grass, on the driveway. I didn't find a mountain of them if that is what you are getting at."
"Well that's good news. In order to get ricin from the bean it needs to be processed, and you would need more than a hundred to make it worth your while."
"Worth your while?"
"Yeah, economically it would be cheaper to get a gun than to get the machinery necessary to process a handful of castor beans."
Clark laughed nervously. "You know what they say — it's not guns that kill people. It's the people pulling the trigger on the guns who kill people."
"Have you ever been shot?" Jerry asked.
"Not yet," Clark answered.
"Well, I haven't either, but I don't think a person with a bullet in their ass really cares if it was the gun or the person behind the gun."
"Probably not."
Jerry looked at one of the beans under the magnifying glass for another minute. "You know, you should probably call the authorities."
"It was on my mind."
"After 9/11 the florist industry received communication from the FBI warning us to keep an eye out for large orders of potentially hazardous plants."
"Like what?"
"There are more than you can imagine. Hemlock, a Shakespeare assassin, looks like a big carrot and is quite deadly. Mistletoe berries will drop you like a ton of bricks. Yew, jasmine berries, the leaves from rhubarb. God didn't skimp when it came to giving plants and animals the ability to evolve with a little kick-ass of their own."
"Don't worry about contacting the authorities. As it happens, I'm on a first name basis with an FBI agent. We had lunch last week."
"Give him a call, just to be safe."
"I will. Thanks for letting me pick your brain."
"No problem," Jerry answered. Before Clark reached the door, Jerry offered parting advice. "Do us all a favor and throw those away, just to be on the safe side. We don't need someone trying to grow them around here. You would feel awful if someone ended up digesting the by-product from those."
## Chapter 38
The windowless side-room stretched the width of the garage, fifteen feet wide and sixty feet deep. The lone working entrance, a set of double doors, was near the back of the warehouse, two steps from the unused one stall bathroom with a sink and shower, and ten paces from the door to the small office in the main warehouse floor.
Each of the three machines resided in a third of the room. The roller was at the far end, the farthest from the double-doors that led to the warehouse. The crusher was next, taking up the middle position on the side-room floor. The pulverizer was nearest the exit.
Each machine weighed over a ton, heavy enough to crack most floors but not nearly heavy enough to cause the thick concrete beneath the warehouse to protest. The warehouse had originally been designed to withstand a couple of cars, the odd truck, and the four-ton wrecker that brought the injured autos to the shop.
Ariana opened the double doors and her three-man team eagerly stepped forward. When you are stuck in a warehouse, and a portion of it is off-limits, natural curiosity reaches unnatural proportions. Just ask James Beach.
Ariana spoke. "It's time to get to work."
"What are they?" Syed asked, stepping past Ariana from the rear. Karim followed Syed and moved for a better view.
"This room is our processing facility. We'll have to do some modification to the environment, but it should work."
"What are we processing?"
"Ricin," Ariana said. "From the seeds of the castor plant."
"Is that what's in the truck?" Syed asked.
"Yes."
"How deadly is it?" Abu asked. "James said he had enough material in his truck to kill ten thousand men."
"It's possible. Ricin is five thousand times more poisonous than cyanide and ten thousand times more deadly than cobra venom. It only takes a very small amount to kill an average size man. But, I won't know the final potential until the seeds are processed. Ricin is only part of the plant. We are not extracting it from the seeds. Meaning once we process the seeds, a percentage of the final product will be ricin. Maybe five percent by weight."
"How do we process it?" Abu asked, looking at the control panel on the crusher.
"I'm going to show you. And, if you follow the rules I set up, only one of us will be in real danger."
"Only _one_ of us?" Abu asked. "We are down to three."
Ariana ignored the second half of Abu's statement. "Yes. Only one."
"Let me guess. It won't be you," Syed said.
Ariana smiled and then pursed her lips. "No, it won't be me."
"Was there ever a moment of doubt?" Abu added.
Ariana moved on. "We have three machines. Production will begin at the far end of the room and will work its way towards this door. The first machine is a high-impact roller made by John Deere for the farming industry."
Ariana walked the men towards the red machine at the end. It was squat, four feet high and six feet square. A wide conveyor belt fed into a narrow opening on the machine. "It looks like a pizza oven," Syed said. "One of those that run a conveyor belt through it."
"The roller is pretty basic. It crushes the seeds and removes the oil. The oil is harmless and will pool in a reservoir on the bottom that will need to be changed as it becomes full. We will store the oil in the metal barrels along the far wall. At this point in the process the crushed seeds will be relatively benign. As long as no one decides to eat one."
"I don't think you have to worry about that," Syed said.
"We may need to let the crushed seeds dry for a day or two. I will decide that once I see the output from the roller. I have manually processed a small number of seeds and I don't think the oil will present us with any problems. It dries quickly with the proper treatment, and according to my calculations, based on the roller's specs, ninety-eight percent of the oil will be removed."
"You have not tried these machines?" Karim asked.
"I tested them to make sure they work." Ariana paused. "And that will have to be enough experience."
Karim shrugged his shoulders in unenthusiastic support.
"The next machine is a little more complicated." Ariana stepped towards the oversized light blue piece of machinery. "This one is the JEX Paw Crusher. It can process ten pounds of material per minute. It weighs a little over three tons. The crushed seeds from the roller will go into the chute on the far side of the machine and will come out the vertical tube on the side. The output tube connects directly into the large bin on the floor next to the machine. This is where the hazard begins.
"Once the seeds have entered the crusher, there is the potential for lethal dust in the air. But this crusher is top-of-the-line. It is designed so that dust is self-contained. It is gasket-sealed on all seams, and has a multiple chamber design with inward facing air vents. There should be very little collateral exposure. Unless there is a malfunction, or the machines need to be cleaned. I think we can avoid the latter. We only need to use these machines once."
The three men looked at the large semi-translucent bin at the end of the crusher. "How do we transfer the seeds from the crusher to this last machine?"
"Well, the last machine, the pulverizer, will extract what is left of the seeds directly from the bin. The machines are designed to be used in tandem. But there still may be some collateral dust."
"How large will the seeds be when they exit the crusher?" Syed asked.
"They won't resemble seeds at all. At its maximum setting, the output should be approximately 150 mesh."
"How big is a mesh?" Syed asked.
"It's not measured that way. There is no such thing as 'a mesh.' A mesh is a unit of measure based upon material passing through a one-by-one inch square. 150 mesh means that there are 150 holes per square inch. The output should be small enough to fit through one square inch with 150 holes."
"That's pretty small," Karim said, now looking at the conveyor belt that fed the first machine.
"By most human standards of measurement 150 mesh is very small. About 100 microns."
"Which is how big?" Syed asked.
"Think along the lines of fine sand." Ariana thought for a moment. "Or on the order of flour."
"I assume at 150 mesh the seeds would be dangerous," Abu asked.
Ariana looked at each man, each face waiting for a response to the question.
"Yes. At 150 mesh, the seeds will be a very fine powder. They will be at least five percent ricin by weight. They will be very dangerous."
"How do we protect ourselves?"
"I'll get to that in a minute," Ariana answered.
The men exchanged glances and the cool dampness of the room felt as if it dropped several degrees.
Ariana continued. "The output bin to the crusher will hold about five hundred pounds of powder. The bin is built on a steel platform frame and it has wheels. As you can see, it can be rolled to the pulverizer."
"The pulverizer? I like the name," Abu added unprovoked.
Ariana rolled her eyes slightly and stepped towards the odd looking machine. She highlighted the bells and whistles on the pulverizer like a model at a car show. "On the far end is the intake tube and on this end we have an elephant trunk-like output tube."
"This is an ultra-fine pulverizer and is designed to take anything from a small grain to a powder and pulverize it. The intake tube on the front will go into the bin of ricin powder, the output from the crusher. There will likely be some leakage, though it should be minimal. The pulverizer has an output between 500 and 700 mesh."
"Which is... ?" Karim asked.
"Something a little smaller than an individual spore of plant pollen. A single red blood cell is about 1200 mesh. For those thinking in terms other than mesh, we are talking about thirty microns. The lower limit of visibility for the naked eye is in the neighborhood of forty microns."
"So it will be invisible?" Syed asked.
"Yes, theoretically, a singular piece of processed seed will be invisible to the human eye. But, obviously, as a powder you would be looking at large quantities and it would be identifiable as a very fine powder. Probably more fine than anything any of us have ever touched or seen. Needless to say, when the final product comes out of the ultra-fine pulverizer, the powder will be the most deadly substance for a thousand miles in any direction."
"Fort Detrick is only fifty miles and that houses ebola, among others," Karim added referring to the military installation that studies deadly virus and biological agents.
"I know," Ariana said. "And my statement stands."
A moment of silence, mixed with excitement, filled the air.
"How long will it take to process the ricin in the truck?" Abu asked.
"Twenty four-hours. Maybe forty-eight."
Abu went back to the key point of the conversation. "If this powder is so fine, how do we prevent from inhaling it? I assume there will be powder everywhere."
"If these machines operate as they should, and believe me they should for the price tag that came with them, then there most certainly will not be powder everywhere. But, in the interest of being thorough, I have a half-dozen gas masks that are military grade. Tough to come by."
"How did you get your hands on those?" Abu asked.
"I don't want to know," Karim answered.
Ariana was in the mood to talk. "All of us will wear gas masks for the duration of the process, but only one of us is in real danger. Timing will be important. If there is an accident, well, then, we will spend our last days taking out as many people as we can."
"Only one person will be in here?" Abu asked, looking around the room.
"Yes. And that person will be fitted with a chemical suit."
"Where did you get a chemical suit?"
"Fire department surplus."
"Why only one?"
"They only had one," Ariana said. "Under the chemical suit, the machine operator will wear a CamelBak."
"What's a CamelBak?" Syed asked.
"It's a backpack filled with water. It has a tube that delivers the liquid to the wearer's mouth. It's popular among bikers and mountain climbers and for people who need their hands during physical exertion."
"I know what they are; I didn't know they were called CamelBaks."
"Now you do. The person in charge of manufacturing will not be able to eat and will not be able to remove the chemical suit for the duration of the process. I have adult undergarments as well."
"That's one way to make them work fast," Karim said a matter-of-factly.
Abu and Syed nodded in agreement.
"The three of us not doing the processing will be in the sleeping quarters on the far side of the warehouse. We will have gas masks on. The door will be sealed from the outside with towels and tape, by the machine operator, before he begins work.
"The machine operator will be in the facility for the duration of the task. It will be him, the seeds, the machines, his suit, his CamelBak, and a garden hose attached to the bathroom sink. Primitive, but that is what we have to work with.
"When the processor is finished, and the ricin is in the delivery mechanisms that I provide, the processor will hose down the room. There is a drain on the floor near the crusher. Just as there is on the floor of the main warehouse. When the person is finished, he will hose the room down again. Top to bottom. And then he will hose it down a third time. He will repeat this until he feels comfortable enough to take his chemical suit off."
"A good way to make sure the job is done right," Syed noted.
"Yes it is," Ariana said. "But keep in mind, this room should not be dusty. There may be some collateral dust between the crusher and the pulverizer, and there may be some between pulverizer and the containers I will provide, but these machines have been chosen for their ability to process without contaminating their environment. The water is merely a way to prevent unintended contamination.
"Once the ricin has been placed in delivery mechanisms, and the room has been hosed to satisfaction, the processor will cover the machines with plastic drop clothes which will be in the shower stall. Then he will shower with the chemical suit on. Using the body wash that is also in the stall.
"The truck will be backed up to the double doors. The containers will be transferred into the truck. After the chemicals have been transferred, he will then shower again, take the chemical suit off, and put on a gas mask. Then the doors to the production room will be shut. At that point we are on the clock."
"How long before we start?" Syed asked.
"I will know in a couple of days. I have to prepare the dispersion containers. And pray for good weather."
Karim spoke. "Then there is only one question left. Who is the lucky person?"
Everyone looked at each other.
"I'll do it," Abu said. "I do not fear death."
## Chapter 39
The writing down the side of the car and Laskey's official uniform, both financed by the fine taxpayers of Nelson County, allowed him to pass through the first barricade near D.C. Police Headquarters on Indiana Ave. Tire-shredders guarded the inner sanctum of the parking lot and without a pass code Laskey made the easy decision to follow the small access road around to the back of the building. He pulled his car in behind a set of Jersey walls and grabbed the plastic evidence bag off the front seat. He got out of the car and stared at the sea of security that engulfed the nation's capital from Capitol Hill to Union Station. He was suddenly thankful for the relative peace of Nelson County.
The guard behind the bulletproof glass spoke as Laskey approached the one-man booth. "How can I help you today?"
"My name is Mike Laskey and I'm the sheriff of Nelson County, Virginia. I'm here to see Detective Earl Wallace."
"What's in the bag?"
"Evidence."
"Anything potentially hazardous?"
Laskey knew the truth was only going to be the beginning of trouble. "No, they are seeds."
"Sign in on the bottom line, Sheriff."
Laskey pulled the clipboard through the security crack at the bottom of the window and signed in with the pen that dangled on a small chain.
"Up the stairs to the second floor. Detective Wallace is on the north side of the building."
Laskey looked up at the sun to determine which direction was north.
"There are signs in the building that will tell you where to go," the officer added. "Turn left at the top of the stairs. Or take the elevator and go right."
"Do I look that I old?"
"Just trying to be helpful. I give the same response to everyone."
Laskey climbed the concrete stairs to the second floor and turned left. An officer in a crisp suit shoved a hand-cuffed perp in gang-banger clothes head first through a pair of swinging doors. Laskey followed in the wake of vulgarities that the suspect unleashed on the officer. The gang-banger continued to yell about human rights abuse as he was thrown into an empty wooden chair.
The female officer behind a desk on the second floor looked at Sheriff Laskey in his best outfit. His brown uniform was ironed. His shield gleamed. His gray hair was combed to perfection.
"I'm looking for Detective Wallace."
"Does he know you're coming?"
"I called him this morning."
The woman behind the desk stood and looked out across the room. The floor buzzed with activity. Desks filled the center of the room and a dozen officers in street clothes typed on their computers and talked on their phones.
"He's in the corner on the far left," the female officer said pointing in the direction of a large man in rumpled clothes.
"The guy with the red tie?"
"That's him."
"Thanks."
Detective Wallace, a twenty-four year veteran with the rank of Sergeant, sensed Sheriff Laskey approaching and ended the phone conversation he was having.
"You must be Sheriff Laskey."
"And you must be Sergeant Wallace."
The two law enforcement officers with fifty-plus years of experience exchanged handshakes.
"Have a seat."
"I would prefer to stand. Been sitting for three hours."
"Fair enough. You want something? Coffee, tea?"
"No thanks. Got somewhere we can talk?"
Detective Wallace looked around at the offices that lined the north end of the floor. "Follow me."
Wallace shut the door as Laskey walked into the office with a glass interior wall. Laskey set the evidence bag on the table and Wallace looked down.
"So these are the little things that have been keeping you up all night?"
"Yes they are."
"What are they?"
"That's just it; I have no idea."
"But they kept you up all night?" Detective Wallace asked in his on-the-job tone usually reserved for guilty faces on the street.
"I have good instincts. Hunches really. Not to mention there was a dead body on the farm where I got these. And there were a lot of them."
"Ah, yes. The dead body. Got one of them myself here to deal with. Though I may be making progress. You said you found a Koran at the house of your deceased."
"That's right. Fancy looking book."
"Well, I got in touch with my dead guy's p.o. and he told me, off the record, that my dead guy had converted to Islam in prison. They've got an imam behind bars in Petersburg and he has quite the following, evidently."
"Was your dead guy a blond, California surfer type?"
"He has brown hair now. Dyed. And he certainly doesn't look like a healthy golden boy anymore. But with blond hair, I guess he could be mistaken for a surfer. Why do you ask?"
"I poked around a little and some people in town saw my dead guy with someone who looked like a surfer."
"Probably the same."
Both men nodded and their eyes dropped to the bag of seeds on the table.
Wallace grabbed the bag and held it to the light in his thick black hands. "Let me make a few calls."
Jerry was chasing an odd-colored ladybug around his desk when Detective Wallace and Sheriff Laskey entered the office to the Merrifield Gardening Center.
Handshakes and credential flashing ensued before Detective Wallace pulled the bag of beans from his pocket.
Jerry glanced at the beans in the bag through the magnification of his thick glasses. "You are the second person to bring those in this week."
"So you know what they are?"
"You ever heard of ricin?
Wallace dropped the bag on the table.
"Well, not exactly ricin, but the raw material for it. Those are castor beans. Which is really a misnomer. They are actually seeds."
"Castor beans?"
"Same bean that makes castor oil. It's used in a million different applications. Body lotion. Lubrication."
"My mother use to give my brother and me castor oil for upset stomachs when we were kids," Sheriff Laskey added.
"I thought ricin would kill you?" Detective Wallace asked.
"Parts of the beans will kill you deader than a door nail," Jerry answered. "The oil won't."
"I don't follow."
"Well, if you process castor beans properly and extract the oil, the leftover mash can be processed into a poison. Ricin. Deadly stuff."
"How much does it take to kill a person?"
Jerry recalled what he read in the book days earlier. "Less than five microns."
The plant and insect expert could see by his guests' reactions that they had no idea what a micron was.
"Five microns would be a fraction of the size of a grain of salt. Something along the lines of a small piece of dust."
"You've never seen the dust in my house," Detective Wallace said.
"No, and I think I'm thankful for that."
Sheriff Laskey tried to put things into perspective. "How many people do you reckon we could take out with this small bag here?"
"I don't know. Dozens I guess, once it gets processed. But you would need a whole lot more to make a weapon of mass destruction, or at least a weapon of mass hysteria."
Wallace felt the hair on his neck stand-up, like an animal's reaction to an impending earthquake. He knew what was coming.
"How much more?"
"Thousands of plants."
Sheriff Laskey added his own hysteria to the equation. "How about a field of plants. Twenty or thirty acres. A plant every few feet."
Jerry's head drooped a little in fear. His mind tried to calculate a reasonable number. "That could be millions of beans. But the scary part is this. Those beans remain deadly for years. If someone were growing them, say on a farm as you suggest, they could reap more than one harvest."
Wallace turned white, for a black guy, and Laskey nodded his head slowly in a moment of shared understanding.
"Tell me you got the name of the person who came in earlier in the week?"
"His name was Clark. He had three beans with him. Said he got them on a farm in central Virginia. Also said he was friends with an FBI agent and that he would take it up with him or her."
"You didn't bother getting a last name?"
"Not really. This kid was white suburbia."
"Terrorists are recruiting all nationalities and ethnicities these days."
Jerry looked at Detective Wallace. "If you were a terrorist and you had deadly material on your hands, would you take it to someone and ask them to identify it? Wouldn't you think he would know what he had? Why raise suspicion?"
"You got me there."
Sheriff Laskey got the investigation back on track. "Do you have security cameras in the parking lot? Maybe we can go through the tapes and you could identify the person who paid you a visit earlier. This Clark character."
"We have surveillance cameras, but I'm not sure they will help. We keep them on a twenty-four hour loop. Had a problem with some vandals a few years back, and a couple of Japanese Red Maples went missing around the same time. Those Red Maples can be expensive."
"I'm sure they are," Detective Wallace said.
"If the tapes are on a twenty-four hour feed, our kid is probably not going to be making an appearance," Laskey said. "So all we have to go on is that this kid's name is Clark."
"Sorry. That's all I can tell you, other than that he is harmless."
"Just the same."
"And if he were up to no good, he probably wouldn't be using his real name."
"Thanks for the deductive reasoning."
"Let me know if I can help."
"Will do."
In the parking lot Detective Wallace looked over the top of his car at Sheriff Laskey before entering the vehicle.
"How many seeds did you find out there on the farm?"
"The ground was littered with them. There were hundreds. Thousands. I don't know. But most of the plants I saw were stripped bare. No telling how many were there before the plants were stripped."
"I think we need some answers and we need them yesterday."
"I was hoping between the dead guy you pulled out of the river, and the dead guy I pulled out of the tree, we would be able to figure it out."
"Maybe, but before the end of the day I'm going to need to run this one up the flag pole. The FBI is going to get involved, Homeland Security, you name it."
"Sounds like a recipe for disaster. Get a bunch of people pissing on each other, overstepping boundaries, drawing lines in the sand. A thousand chefs with a thousand hands staring down at a single pot."
"How do you know so much about bureaucracy?"
"We got politics in small towns. And besides, this Administration has been parading their incompetence for the world to see for a few years now. I may live in the country, but I'm not stupid or isolated... unless I want to be."
"You have any ideas on how to find this kid?"
"Just one. There's a chance he may have stopped at a gas station in Nelson, filled up, and asked for directions. If he paid with a debit or credit card, I can get the name with a single call."
"Then make the call. If we can find this kid, maybe we can drag him to a meeting with the Feds."
Debbie Ingel answered the phone at her desk in the Nelson County sheriff's office.
Sheriff Laskey didn't let her get through her standard greeting. "Deb, this is Mike."
"Yes, Sheriff."
"I need you to call down to _The Pig and Whistle_. Talk to the Daltons. Tell them I need the credit card and debit card receipts from everyone who bought gas or a donut for the last two weeks. Start yesterday and work backwards. The first name on the receipt should be Clark. That should help narrow it down."
"It may take a while."
"I don't have a while. I need it within the hour. Tell Mrs. Dalton to skip making sandwiches today. The world will survive without her ham on wheat."
"Will do."
Sheriff Nelson looked over at Detective Wallace who had pulled the bag from his pocket again and placed them on the roof of the car.
"What ya' thinkin' about?"
"How something so small and so innocent looking could kill a man."
"One bean doesn't worry me."
Debbie Ingel called back twenty minutes later.
"You might want to apologize to the Daltons. Apparently our request screwed up their lunchtime business."
"They'll get over it. I'll look the other way the next time they are caught sellin' beer to underage kids."
"You still need the information?"
"Sure do."
"Well, you hit the nail on the head. There was a Clark Hayden who purchased gas with a Bank of America debit card at _The Pig and Whistle_. Purchased it yesterday morning, right before you went out to the Price farm. You couldn't have missed him by much more than an hour or two."
"I need you to run him through DMV."
"Already did. His current address is listed as 202 Dorchester Lane in Arlington, Virginia."
"Let me get a pen."
Laskey searched his pockets with his free hand. Wallace threw his detective notebook, pencil stuffed into the wire binding, onto Laskey's lap. The sheriff scribbled on the first open page he could find and showed it to Detective Wallace.
The detective nodded as he hit the gas pedal.
## Chapter 40
Are you Clark Hayden?" Detective Wallace asked through the closed storm door.
"Who's asking?" Clark replied. He was wearing jeans that were tattered on the fringe of the legs, wool socks, and a striped sweater that made him look like Charlie Brown. His mother was at his aunt's and he wasn't due to pick her up until the next morning. He had the house to himself for the day, and in the evening was planning on Lisa coming over for sex, under the auspice of watching a DVD.
"D.C. Police," Wallace answered, flashing his badge. The heat from the house created condensation on the glass in the storm window and Clark squinted to see the details of the badge. He sighed heavily and opened the door. "Come in."
Wallace extended his hand, and Sheriff Laskey removed his hat.
"Detective Earl Wallace. D.C. Metropolitan Police. This is Sheriff Laskey, Nelson County, Virginia."
Clark shook hands and looked each officer in the eye as his pucker meter inched up. "Have a seat," Clark offered, gesturing towards the sofa with large cushions. He pulled the old wicker rocking chair to face the sofa, the well-worn coffee table between him and his guests.
"You know why we are here?"
"I could guess, but I'd just rather you tell me. I have been doing a lot of speculating recently. I'm all guessed out."
Detective Wallace looked at the young man in front of him. Twenty-five years old. Six foot. One hundred and eighty pounds. Solid build. Looks athletic, except for the overly geeky glasses with the black frames. "Rumor has it that you have some beans in your possession that could kill a few people."
Clark swallowed hard. "That wasn't the question I was expecting. You working with the FBI?"
"No," Detective Wallace said. He looked at Sheriff Laskey and the men nodded to each other. The sheriff pulled the bag of ricin seeds from the pocket of his winter jacket. "Jerry at Merrifield Garden Center told us you came in with a few of these."
"I did. But I didn't tell him my last name, so how did you find me? It was only yesterday."
"You trying to hide something?"
"Not at all. But people usually don't go around dropping their last name. Unless it's Trump or Hilton and you're going to get a reservation at a restaurant. Unfortunately 'Hayden' doesn't come with any perks."
Sheriff Laskey smiled. "No, I guess not."
"So... how did you find me? Not that I'm entirely sad to see you."
"Investigative techniques," Wallace added. "Believe it or not the badge is real and I'm an actual detective."
"Fair enough." Clark was silent for a moment and his armpits started to perspire. The black detective was sitting directly across from Clark, piercing eyes measuring his every movement. Every twitch. The white sheriff was doing the country version of the same. Clark stared back for a few seconds, his eyes bouncing from one guest to the other. "I found the beans on a farm in Nelson County. But given that the sheriff here is the law in Nelson County, I figure you already know that."
"You pay attention," Sheriff Laskey said. "But we don't call the sheriff 'the law' anymore. Except on _Western Movie Day_."
Wallace laughed out loud and Clark relaxed a little. If these guys were ball-busters they were waiting to bring out the testicle hammer. Clark got the feeling that the men were like him, just looking for answers. Besides, from what he knew, most cops handcuffed first and asked questions later.
"Did you grow them?" the sheriff asked.
"The seeds?"
"Are we talking about anything else that grows?" Wallace added.
"Hell, no. I didn't even know what they were until Jerry the-plant-guy told me."
Laskey looked at Wallace. Chalk one up to old Jerry's intuition.
"What brought you to the farm?" Wallace asked.
Clark squirmed and the proverbial cat reached out and grabbed his tongue. And it was a full-grown tiger that hadn't eaten in a month.
"That's not so easy to explain."
"Give it your best shot."
"Ok. But I'm warning you up front it may sound a little kooky."
"We have been warned," Wallace answered.
Clark delved into the story of his mother calling the CIA, the visit from the FBI about terrorists across the street, his Pakistani neighbors disappearing, the explosion at his neighbor's house, his dead neighbor, the calls from a stolen phone in Pakistan owned by a low-level diplomat.
"So I called the guy in Pakistan and asked him if any other calls were made from his phone. He told me yes and gave me the number. From there it was easy."
"And you had never been to the farm before?"
"No."
"And you wouldn't know anything about a dead guy hanging from a tree in the back field?"
Clark's heart kicked up a notch. "Uh, no. I didn't see any dead guys. Hell, I've only seen two dead bodies in my life: my father's and my grandfather's. I went to the farm and walked around the house once. It was kind of spooky. Desolate. You could just tell no one had been there. I headed back to my car and dropped my keys. I saw these seeds on the ground and picked up a few. Then I heard a door or something banging in the distance and decided to leave."
"That's it?"
"That's it. Like I said, it was spooky. I mean, I grew up in Northern Virginia. I haven't spent much time on farms. But one of my roommates in college used to love to tell the story of how he was trouncing through a farm near some family land in Bath County and got shot by a neighbor with rock salt. For some reason, standing out there on that farm, that story came back to me."
Laskey smiled again. "Hardly anyone uses rock salt anymore. Nowadays they shoot you with real bullets. Though I did get a little rock salt in the ass when I was younger. In hindsight, I probably deserved it."
Detective Wallace chimed in. "OK. So we've established that neither of you deal with urban crime much. I've been a cop for thirty years and have never even seen rock salt... So what happened next?"
"Came back here."
"Here being D.C.?"
"Here being Northern Virginia."
"When did you go to the Merrifield Garden Center?"
"On my way home."
"And after you found out you were carrying ricin, you didn't feel compelled to call the authorities?"
"I did. I called the FBI agent who came to the house a few weeks ago and left him a message. It was at 8:07 last night. He didn't call me back and I called him again at 9:21 this morning and left another message. You can check my cell phone's outgoing calls."
"I believe you. Did you mention ricin in your phone call?"
"No. But I told him that I had some very interesting information on my neighbors."
Wallace's face wrinkled as he digested deep thoughts that came to mind.
Laskey asked a question. "Exactly, how many seeds did you bring back?"
"Three."
"Why three?"
"No reason. Like I said, the whole situation was a little nerve-wracking. Poking around, uninvited, on someone's farm, looking for people who can answer a question about a phone call from Pakistan, made on a stolen cell phone. If my hands weren't cold and I hadn't dropped my keys, we wouldn't be having this conversation. Besides, all the seeds looked the same. There was no reason to bring back a ton of them."
All three men stared down at the bag on the table. "Yes, they do look the same," the sheriff concurred.
"Where are the ones you brought back with you?"
"Downstairs. In my dresser. In my underwear drawer."
"Would you mind getting them?"
"Not at all. Give me a sec."
When Clark returned he put the beans, which where now safely stored in an empty prescription medicine bottle, on the table.
"You never told me how you found me."
"Gas station receipt."
" _The Pig and Whistle_."
Sheriff Laskey nodded. "Nice deduction."
"Good ham and wheat sandwiches."
"The best in Virginia."
Clark took a page from his mother's book of hosting guests and asked if anyone wanted coffee. Laskey and Wallace shook their heads.
"If I may ask, what brought _you_ to the farm?"
"I think that conversation needs to be answered downtown. Let's find your FBI buddy."
Agent Rosson chose the same windowless room where his boss, the Director of the Joint Terrorism Task Force, laid down the law for all investigations leading to Dorchester Lane.
Detective Wallace had his notebook on the table and a plain manila folder in front of him. "This feels like an interrogation room."
"It was. In a previous life. Most of the interrogation rooms were relocated to the first level of the basement. Or to another offsite location," Agent Rosson answered. "Now we use this room primarily for private conversations. The type of conversations that shouldn't leave the premises."
"Understood. At police headquarters we use the john," Wallace said.
"We are a little more civilized here at the Bureau." There was a pecking order in law enforcement and Agent Rosson was making sure that everyone in the room knew their position.
"I think we both got our suits at _Men's Warehouse_ ," Wallace responded. "Two for $149 during the spring sale last year."
Clark watched the banter and tried not to laugh. Grown men in the professional sandbox.
Agent Rosson deflected the attempt to be lumped into the same group. "What are we here to discuss?"
"Ricin," Detective Wallace answered. Everyone in the room, even Clark, could tell the answer caught the agent off guard.
"Ricin?"
"Didn't Clark here tell you? I understand he left several urgent messages for you to call him back."
"There was no mention of ricin in his calls," Rosson answered.
"Well, if you returned his call, maybe you wouldn't look like you were just taken from behind. At any rate, we're here now."
"Please go ahead."
Detective Wallace nodded towards Clark. Clark cleared his throat. "There have been several developments with regard to my neighbors across the street."
"203 Dorchester Lane, Arlington, Virginia."
"That's correct."
Agent Rosson motioned with his hand for Clark to continue.
Clark started with his neighbors supposedly leaving the country and finished with Detective Wallace and Sheriff Laskey at his front door.
"Are you sure it's ricin?" Agent Rosson asked.
Sheriff Laskey pulled out his evidence bag and Clark's prescription bottle of seeds. "We're sure. You want to try one?"
"Tell me, just how did you get involved?" Rosson asked the sheriff.
"I got a call from D.C. asking me to check on a residence in my jurisdiction."
"And your jurisdiction is..." Agent Rosson asked as he squinted at the badge on the sleeve of the sheriff's jacket.
"Nelson County."
"Home of Wintergreen."
"Yes, indeed. Home of Wintergreen, and as of yesterday, home of a ricin plantation."
In his best condescending tone Rosson kept his hand on the conversation throttle. "We'll get there in a minute. You were talking about a call from D.C."
"I got a call yesterday morning from Detective Wallace, asking me to check on the whereabouts of a Nelson County resident."
Rosson turned towards Wallace. "And you were interested in this person, why?"
"He was the cellmate of a body I pulled from the Potomac earlier this week. A dead guy named James Beach. Former guest who spent a couple of years choking down three squares a day on taxpayer money at the Petersburg Correctional Facility in Virginia."
"Anything unusual about the body?"
"Nothing really. Still waiting on the toxicology report, but initial indications are that he drowned."
"And the call to Nelson County?"
"Trying to notify next of kin. Wanted to ask a few questions to someone who might have known him."
"Procedural duties."
"If that's what you call them here at the Bureau," Wallace answered.
There was a brief pause and the door to the room swung open. Eric Nerf, the Director of the Joint Terrorism Task Force walked in the room.
"That _is_ what we call them at the Bureau. In fact, consider your entire visit here a procedural duty."
"Thin doors for an interrogation room," Wallace said, standing from his seat. "We haven't met." Wallace's large frame shadowed the table and his mitt-like hand engulfed the well-dressed director's.
"Eric Nerf, Director of the Joint-Terrorism Task Force for the Federal Bureau of Investigation," he announced to the room.
"Is that your full title?"
"Yes it is."
After shaking hands, Wallace took his seat again, this time further from the table, one ankle resting on his other knee.
"And you are Detective Earl Wallace of the Washington D.C. Metropolitan Police Force. Rank of Sergeant. Robbery and Homicide division. Lead detective on the Senator Day case, but I don't think we need to go into the details of that here."
"Well, if you're chasing the terrorists with the same energy that you're doing your homework on other law enforcement officers, I think we can all sleep easy tonight."
Agent Rosson got his boss up to speed on the conversation with only a minor air of superiority. Clark sat back in his chair, taking in the room and the conversation. He told himself to be stoic but his stomach was bubbling beneath his Charlie Brown sweater.
The director took his favorite position, sitting on the corner of the table. There was something about putting your ass at table-level that created resentment among those seated at the table. It was an insult he had perfected over the years. Wallace, Laskey and Clark all took silent offense.
"Well, I'm fascinated by this conversation. I truly am. But what I'm about to tell you is highly confidential. Consider it top secret. And the only reason I'm telling you is out of professional courtesy and, of course, because I don't want anyone in this room stumbling over something they shouldn't."
"The CIA has informed the Joint Terrorism Task Force, in other words, me, that they have an operative on the case. This operative has worked for years in pursuit of a high-profile target. This agent's contact with the CIA clandestine office is on an "as-capable basis," meaning that there are long periods of time when the operative is dark. Completely off the grid. There are thousands of man-hours at stake in this intelligence effort, not to mention national security. With all due respect to everyone in the room, there is to be no further investigation of anyone involved. Right now 203 Dorchester Lane doesn't exist. Leave it alone."
"Is that an order?" Wallace asked.
"That's a professional request. But if you call your captain, he will convey it as an order, if that would make you feel better."
The room fell into an uneasy silence. Wallace brooded. Laskey fumed; his first foray into the politics of big-time law enforcement was even uglier than he imagined. Clark sat at the table, his mind running through the details, the analysis of the situation in full-steam-ahead mode. His mouth opened before his brain could backtrack.
"What about the ricin?"
"We'll take care of it," the Director said.
Wallace looked at Clark and Laskey. Clark started to speak and Wallace cut him off. "They said they'll take care of it."
Clark missed the hint. "You know, there's no one at 203 Dorchester Lane. The house is empty and has been for several weeks. Not to mention it suffered a severe fire."
"Then it should be easy to let go," Rosson added.
Wallace clapped his hands together. "Well, then, I guess our job here is done."
Clark and Sheriff Laskey followed Detective Wallace's lead and got out of their seats.
"Agent Rosson has some NSAs for you to sign before you leave," the director said, turning his back on Detective Wallace's extended hand.
"One more thing," Wallace said, moving the hand into the manila folder he brought with him. "Pass these prints along to the CIA. They are from the body I fished out of the Potomac. If they match their internal CIA files, tell them their agent is dead."
Agent Rosson was in the director's office, stealing glances out the window at the cabs and black "for-hire" sedans on Pennsylvania Avenue two blocks down from the White House. "What do you think?"
"The case is closed."
"And the ricin? I don't remember that being mentioned."
"I'll take care of the seeds. As far as you are concerned, you never saw them. I have only one concern. That the report in the database for that address remains closed. Just as it was the day you amended it. The CIA wants jurisdiction, the CIA can have jurisdiction."
"Are you going to give them the prints?"
"Of course. But I don't expect them to get back to us one way or the other."
"So, we just wash our hands of it?"
"The way I see it, that report was officially closed before the CIA contacted us. Technically, our hands were never dirty."
## Chapter 41
If that is Washington D.C. hospitality, you can keep it." Sheriff Laskey said, getting into the passenger side of Wallace's car.
"I hate to say it, but that's Washington politics."
"Kind of like grabbing your ankles to me. Y'all have a funny way of catching criminals. I mean, we have the occasional turf war with the state police, but we try not to lose sight of the goal. Putting bad guys behind bars. The way I see it, the three of us in this car just went to the FBI with a bag full of material that could be turned into a weapon of mass destruction and they told us to go home."
"So now what?" Clark asked from the back of the unmarked D.C. police cruiser.
"I thought the director made it pretty clear," Wallace said, testing the air.
Clark spoke. "He told us not to investigate 203 Dorchester Lane. I'm talking about locating a missing neighbor. There is no harm in that."
"You trying to convince us, or yourself?"
"Both..." Clark said.
"What was the reference to Senator Day all about?" Laskey asked.
Wallace let out a grunt-like hum. "You all know what happened to the senator?"
"Sure. Who doesn't."
"Well, I was involved in the case, only I didn't know exactly what the case was until it was too late. I caught a lot of heat for the way things turned out, through no fault of my own. Senators and congressmen have the ability to squeeze just about anyone they want to here in D.C."
"You mean you're on thin ice."
"Let me just say I am still on a lot of people's radar screens. But I try not to let that stop me."
"Here is my thought. What if the director is wrong?" Clark asked.
"You're a young man. The question you should be asking is what if you are wrong?"
"What's the worst that can happen?"
"Non-enemy combatant status. Labeled a traitor. Maybe a vacation in Gitmo. End up in a brig on some military base without ever being officially charged with any crime. I can think of a list of things, and they all start at bad and head downhill to worse."
"I think you two have more to lose than I do. I have the responsibility to locate my neighbor, if for no other reason than to let her know her house is gone."
"You gotta love the youth of today," Wallace said, ignoring Clark in the backseat.
"What about you, Sheriff?" Clark asked.
"To tell you the truth, I got a bad feeling about all of this."
"I think that's a healthy response," Clark said.
"The first thing I'm gonna do is go back to Nelson County and burn what's left of that field where we found the seeds. It might take me a few weeks to torch that many acres without causing a forest fire, but I know just the boys to call in. No fuss, no muss. No one is going to question the sheriff burning a field on a convicted drug dealer's farm. As you said, 'I am the law.' I'm also going to dig around a little and see who knew what about those plants. Check out what kind of equipment they have on the farm. I'm sure they didn't plant or harvest those plants by hand. Probably needed some farming equipment."
"Farming equipment?" Clark asked.
"That's right."
"Large scale farming equipment?"
"Something bigger than a tractor and a back hoe. And they probably needed some fertilizer. I have a few things I can check out without causing too many waves in the pool. Put it down to investigating a murder, which I'm still legally obligated to do."
The collective wheels of the three men were spinning silently as Wallace ran through a red light.
Sheriff Laskey continued. "But I imagine the dead guy from the river killed the dead guy in the tree because someone was going to open their mouth."
"You think the dead guy from the river is a CIA agent?"
"The likelihood occurred to me about mid-way through the conversation with the director, if you can call that ass-chewing a conversation," Wallace said.
"It is if you can consider a proctologist just a doctor," Clark answered.
No one laughed and Clark continued. "Follow my logic for a moment. Let's assume this dead guy is the CIA agent."
"Ok," Wallace answered.
"Then the agent is dead, the CIA doesn't know, and the FBI is off the case. The only thing left is us."
Laskey spoke. "We also have to assume that this CIA guy was in so deep he was willing to kill another American citizen. Jackson Price, the dead guy from the tree, was a troubled soul. No question about it. But he was still an American."
"I have no doubt that a CIA agent would kill another American. For ego, national security, career advancement, money. All that jazz," Clark said plainly.
"Well, if you believe it, then it's got to be true," Wallace jibed.
"Have you watched the news at all in the last few years?" Clark asked.
"The boy's right about that."
Wallace took his turn at verbally working the case. "So, if we follow Clark's logic, there is a sleeper cell running around out there who just jettisoned their CIA agent into the Potomac."
Clark added the obvious. "And they have a shit-load of ricin."
"And we were just told by the FBI, and the CIA indirectly, to butt out," Wallace added.
"I'm not saying that we _do_ anything. Just poke around for some information."
Wallace felt the pinch of his career and the pain of doing the right thing closing in on his heart from both sides. He tried to take a deep breath and failed. "Get my cigarettes and lighter out of the glove box, would you?"
"You a smoker?" the sheriff asked, fumbling with the latch.
"Only when absolutely necessary."
Wallace lit his cigarette and cracked the window on his side. The nicotine rushed his system and he felt relief as he exhaled. He looked in the rear-view mirror at Clark, who was perched forward in the back seat. "What do you know about this neighbor?"
"Nothing that has proven to be true. She was married to a man named Nazim Shinwari. They have a daughter named Liana. That is what I know for certain. The husband is purported to be dead. Everything else is either a lie, or so close to bullshit that I can't tell the difference. But hell, I'm not sure what to think. A month ago I was a student trying to take care of my mother. In the last few weeks I have met the IRS, the FBI, the D.C. Police and a sheriff from Nelson County. I have had a neighbor die, though I am not sure about the connection there, my missing neighbor's house exploded and almost got me in the process, and I have stumbled upon ricin. That's too many strange occurrences for me for one month."
"All right. All right," Wallace said. "Take it easy. Take it one step at a time. Breathe."
Clark tried to collect himself by looking out the window at the snow which had just started to fall. His eyes moved from the snow outside to the inside of the unmarked police cruiser. The computer attached to the dash caught his eye.
"What does that computer on the dash access?" Clark asked.
Wallace's eyes met Clarks in the rear-view mirror. "Does your neighbor drive?"
"Sure."
Detective Wallace pulled over into the International House of Pancakes parking lot in Ballston, next to the Metro Station. "Let's run her through the system. Just an ordinary traffic stop violation." The detective punched in the address and name into the Virginia DMV system. A few seconds later the image of a driver's license flashed onto the screen. The information on the license was replicated in fields on the screen below the image of the photograph.
"That's the information, but that's not her," Clark said leaning forward.
"Right address?"
"That's the right address. Right name. The picture is different."
"She's wearing one of those veils," Laskey added.
"It's a hijab. And my neighbor always wore one. Never seen her without it. But the face is different. Similar, but different."
"With that hibachi it's no wonder."
" _Hijab_. _Hibachi_ is a Japanese grill."
Wallace interrupted. "I'm not surprised. After 9/11 and the investigations that followed, the Feds busted three DMVs for gross violations of procedures in issuing licenses to undocumented individuals. Two of those three DMVs are within four miles of where we sit. One of them was in Georgetown, the other in the Tysons Corner, Virginia. If she wanted a license, she came to the right place to get an illegal one. Heck, the DMV will still allow people to get a driver's license based on the name and address on a phone bill."
"No offense to you two guys, but the government is fucked up. How hard is it to hire people who speak English and have common sense?"
Wallace shook his head. "You have no idea. You want to talk scary, listen to this. For years the INS issued citizenship papers on naturalization certificates with photographs that were attached with glue. No security features. Which means that Joe Alvarez could go through the process of getting citizenship, get sworn in by the government, and walk out onto the street with a certificate granting citizenship to anyone who wanted to put their photo on it. That person could take that certificate and get a passport, a driver's license, whatever. Keys to the kingdom."
"I'm sure they've fixed that process."
"A little too late. It's not like these certificates were recalled... And in case you didn't know, you can get your passport through the Postal Service. A Postal Service employee swears the customer to honesty, and then ships the paperwork to the Department of State. Now, how secure do you think that process is?"
"What are you saying?"
"What I'm saying is that if your neighbor wanted to live her life under an assumed name, she only needed a reasonable I.Q. and some perseverance."
"Well, I'm damn sure she wasn't an average housewife."
Wallace took another drag from his cigarette. "I'll tell you what. You dig around a little and see what you can find out. See if you can locate your neighbor. But if you see anything out of the ordinary, see anyone following you, you mind your business. The sheriff and I will try to solve our murders, which we are obligated to do. Maybe something will come from connecting those dots. In the meantime, if you need help, or hit on anything certain, you let me know."
The interior of the car grew quiet, before Wallace added, "But do me a favor, call me from a pay phone to the main work number at the precinct, and then get transferred to me. Don't be leaving any long messages about terrorists or Dorchester Lane. If anyone asks, I want to be able to say I don't know what the hell you are doing."
Clark smirked. "Of course."
## Chapter 42
After memories of the first dates have faded, after the flowers have drooped and the chocolate has settled into cellulite, the first fight in a relationship is when fluid-exchanging partners really get to know each other. The good façade can only be maintained for so long.
When it comes to the female half of the equation, no man really knows what he has until that façade is temporarily tossed aside. Unfortunately, peeking at the undercurrent of true emotions is much like testing a bulletproof vest with a real person and live ammunition. Some women cry, some women hit, some women sulk, some women brood, some women scream. Clark was about to find out which camp Lisa fell into. He was hoping that Lisa didn't come from the Lorena Bobbit school of anger management.
"How the fuck could you do that? I could lose my job."
"I know, I know. It was stupid. It was irresponsible."
Lisa interrupted his self-reprimand. "It was dishonest."
"I know. But..."
"There is no 'but,' Clark. I should turn you in. You deceiving shit."
_Not bad for a geeky looking C.P.A.,_ Clark thought. "You don't want to turn me in. You could get in trouble. Doesn't the Treasury Department teach you to keep your computer secure? Not keep your password written down where just anyone could find it?"
"Fuck you, Clark. It was in my apartment. The doors were locked. It wasn't stolen by some unknown entity. How about some respect for other's property?"
Clark got down from his verbal podium. "It was a mistake. I'm sorry."
Lisa flopped onto the sofa and stared at the laptop on the small coffee table. "What exactly did you do?"
"I poked around a little."
"Exactly what did you poke around and see?"
"I went into the IRS database."
"The MFDA?"
"It's pretty impressive really."
"Clark. You're an asshole."
"I put in my mother's address just to see what would come up."
"I thought you were checking out your neighbors..."
"I was, but not at first... or I guess I was, but I wanted to see some information that I thought I knew was correct before I started poking around."
Lisa shook her head. "And..."
"Well I saw the information for my parents, their address, names, tax history. Place and dates of birth. Real estate transactions. Pensions. Their account was flagged with a message indicating that they were being audited."
"Did you share this information with your mother?"
"No, not at all."
"Did you print anything out?"
"No, I didn't have time."
Clark winced at his poor choice of words and then shoved his foot deeper into his mouth.
"And you came out of the bedroom and dragged me back to bed."
"You betrayed me and then minutes later had sex with me?"
Clark knew better than to answer that question. He dropped his head in mock shame.
"Go on," Lisa prodded.
"Well, I changed the address to Ariana and Nazim's across the street. It was the same type of information. Did you know that Muslim women typically keep their maiden names?"
"Probably because the men are allowed to keep four wives and it gets confusing if all of them are called Mrs. So-and-so," Lisa retorted, still angry with Clark, and by virtue of that, still angry with all men.
"Yeah, maybe. Hadn't thought of that. Anyway, as I said, Ariana called me from Pakistan and asked me to keep an eye on her house."
"You mentioned that."
"Well, she said that her father was ill and that she was in the hospital. But her files listed her parents as deceased."
"Maybe it was Nazim's father she was referring to. A lot of people refer to their fathers-in-law as 'fathers.'"
"Could have been, but Nazim's father, said father-in-law, was also listed in the no-longer-living section. And then I called Nazim's employer."
"And..."
"The guy said he also received a call from Ariana in Pakistan. He said that Ariana said Nazim had been killed in a car accident after they arrived."
Lisa looked uncomfortable. "Did you tell the FBI guy all of this."
"What do you think? I told everybody but the trashman what is going on."
"And the FBI told you, and the two cops, that the CIA was on the case?"
"Yes. But there are still a lot of answered questions."
"Clark, have you thought for a moment that the FBI and CIA can handle it? That you are getting in over your head?"
"I _hope_ they're right. I hope they do have it covered."
"Then why are we having this conversation?"
Clark stumbled a little. "I just need to check on something. Something the detective said to me. And I could use your help."
"With what? Your idea that your neighbor is suspicious and has been lying to you? No, that's not in my jurisdiction."
"Don't have to be so testy."
"I'm still working through the feeling of betrayal. I can be testy if I want."
"I just thought maybe an IRS criminal investigator might have some criminal investigative powers."
"I do. And we do have some jurisdiction with terrorism. For one, we were trained to identify potential parties in the Hawala system. And of course, we have the ability to search bank records and look for potential money-laundering and terrorist support channels. But for the most part, the execution of that arm is done by FinCen, the Financial Crimes Enforcement Network. It's a group based out of the Tysons area, under the Department of Treasury."
"What is the Hawala system?
"It's an informal system of money transfers used in the Middle East, and other places."
"How does it work?"
"Let's say that you are in New York and you want to transfer money to Saudi Arabia without going through normal channels. The Hawala system allows you to make the transfer through informal channels. All you need is to know a Hawala broker. You bring your money to a Hawala broker in, let's say, Brooklyn, and they will arrange to have the money transferred to a Hawala broker in Saudi Arabia. But the key is that the money is never actually transferred. The whole system works on honesty. One broker agrees to pay the other at a later date, usually through a transfer of funds going in the opposite direction. Both sides keep a running total of what is owed to whom and eventually, with the participation of other brokers, the debts get settled. Imagine a big system of I.O.U.s. The Hawala brokers make money off of small commissions and generally avoid foreign exchanges, thus offering their services at a lower price than above-the-board remittance alternatives. Obviously, the system also lends itself to tax evasion. One could work in the U.S. for cash, send that money overseas without a transaction trail, and the taxman would be cut out of the equation."
"A system like that would never work in the U.S. There just isn't that much honesty."
"I wouldn't be so sure. The mafia has had similar systems. At any rate, the Department of Treasury received training on the financial systems of various regions in the world, all under the auspices of identifying unfriendly foreign elements and catching tax avoiders. The Hawala system was just one of these."
"You ever caught a Hawala broker?"
"I was an auditor until last month and have only been a criminal investigator for a few weeks, so the answer is 'no.'"
"Thank you for the explanation of terrorist financing."
"Why are you such a jerk? You were the one asking."
"You're right. I am just a little wound up, I guess. But there is something else that has been bothering me."
"Do tell."
"We had another neighbor die over the weekend. A new guy in the neighborhood. He died of reported organ failure."
"Where did he live?"
"A few houses up. Away from Ariana and Nazim's. The guy was only forty-three. A carpenter. No history of illnesses."
"Who told you about this?"
"Where do I get all of my information?"
"Mr. Stanley."
"Bingo. But with two neighbors passing away recently, well, it got me thinking about my father. He died from a never completely diagnosed illness that set-in rapidly."
"What a minute. Now, you think Ariana killed your father?"
Clark saw a chink in Lisa's armor. "All I'm saying is that my father has some longevity in his genes. He doesn't have siblings, but most of his relatives make it to their nineties."
"I just don't know, Clark. I just don't know."
"I'll tell you what I told the cops. We don't have to actually _do_ anything. I just want to poke around a little."
Lisa looked straight ahead in a near trance.
"Let me ask you one question, Lisa. Wouldn't you hate to be wrong on this one?"
After a minute of silence, Lisa rolled her eyes and sighed. "Ok, Clark. Let's assume that everything you have found out, and a few things that you are speculating on, are true. Then what do you do?"
"Find out who she is."
"That simple? Find her and then what?"
"I'll know when I find her."
"And how do you propose to do this?"
"I have an idea. But you're not going to like it."
"If it involves me, you're probably right."
## Chapter 43
Paul Cannon was on his sixth cup of coffee, three doses over his usual daily allowance. The stubble on his cleft chin was thick, his breath terrible, his hemorrhoids screaming. And he had never been happier, with the exception of the Red Sox ending their eighty-six year World Series drought. He had certainly never been more satisfied. Only a handful of people in the world had the experience he now had. He had just placed a working man-made craft on another planet in the solar system. The moon was for underachievers, he thought, as he walked out of work and drove past the huge gates with armed guards.
NASA didn't hire losers, and Paul Cannon was the upper echelon of an elite core. He had aced every standardized test he had taken since the sixth grade and his I.Q. was at a level where the best estimate was a mere guess. Only poor vision had kept him from his childhood dream of going into space. When he learned his degenerative eye disorder wasn't repairable, he did the next best thing to becoming an astronaut. He set his sights on becoming a NASA scientist. Even though thick glasses meant he would be collecting accolades for deeds done on Earth, his mind was rarely on the planet he called home.
Without sleep for three days, he hadn't seen his wife and son in a week. None of which prevented him from spending most of the past weekend with the true love of his life. The focus of his adoration stood two feet high, five feet long, nose to tail, and was currently scurrying around the surface of the fourth rock from the sun. The largest Mars exploration robot ever built by man was powered by an engine designed by Paul Cannon. He nurtured it to existence and made it work in reality. Nothing a genius I.Q. and a lifetime obsession bordering on a compulsive disorder couldn't do.
Long since trading in his tinker toy set and science project kits, his new BMW 5-series was Paul Cannon's lone adult toy. And the toy was pissing him off. The stuttering had started a week ago and had gradually progressed into a form of automotive break-dancing. As Paul turned onto the highway near Greenbelt, Maryland, the car beeped yet again and he glanced down at the dashboard. Next to the red check engine light, the yellow oil pressure light illuminated the back of the steering wheel.
"Fabulous," Paul said aloud over the NPR commentator speaking from the radio. He looked at the red digital clock in the dash and wondered if he would make it to his son's eighth birthday party, scheduled to begin in ninety minutes with the grand entrance of a two-hundred-dollar-an-hour clown. A quick stop to grab a birthday present, and hopefully outdo the clown, was the only thing on his mind. Other than why he hadn't listened to his wife and bought the Lexus.
_Big Al's Hobby Shop,_ on the border of D.C. and the Maryland state line near Takoma Park, was three blocks west of the metro station, nuzzled into a strip of otherwise sketchy establishments that included a martial arts dojo that taught a Brazilian form of self-defense known as capoeira. The shady neighborhood did little to deter customers from Big Al's, which had the largest selection and cheapest prices of anyone on the East Coast. The store had once been the corner drug store, replete with a real soda fountain, before it became the prime location of failed drug store chains from _Dart Drug_ to _Drug Fair_ to _Peoples_.
Most of Big Al's customers were self-admitted geeks. The patrons were mostly men who spent more time on their hobbies than they did on their wives, kids, or full-time jobs, if they were lucky enough to have any of them. But if you needed a new rotor on a remote control helicopter or an artist easel or the latest philatelic collection book, Big Al's was the place, and Big Al himself was the man.
The king of pricing and selection, Big Al was the pauper of organization. Products lined his store in haphazard fashion, the shelves burgeoning, towering to a full height of ten feet. Big Al, a name that was anything but a misnomer, had long taken to hiring local teenagers who were both spry and coordinated enough to scurry up the ladders he kept in the store to pull items off the acrophobia-inducing shelves. And while the owner may have been too large and too old to climb the ladders himself, he knew where virtually every item in the store was located. The secret was easy. Everyday before he left for home, Big Al pushed his considerable waistline through the aisles and read the contents of every shelf to himself out loud. It was inventory by rote, and that suited Big Al just fine. It was as precise as his accounting method which had generated two decades of favorable tax returns.
Paul Cannon parked his car at a meter, half a block from Big Al's. He glanced around the street and back at his car as he fed a quarter into the meter, and then walked briskly towards the direction of the hobby shop. He had ten minutes to decide what to get, and his mind was already racing through the hodgepodge layout of his favorite store. Paul Cannon, the brilliant NASA mind, knew the contents of Big Al's hobby shop almost as well as its owner. And he knew that the metric conversion calculator in the glass counter in the back of the store was a must have. A gift any eight-year-old, son of a NASA scientist, would appreciate.
Big Al was half-perched on a stool behind the register counter when Paul Cannon threw open the door.
"Good afternoon, Paul," Al said. His closely cropped white beard was trimmed to perfection. His face was red, the capillaries bringing blood to the surface of the skin as it tended to do for those who favored their after-work sauce. "Can I help you find something?"
"No thanks, Al. I only have a few minutes. Got the kid's birthday in less than an hour and don't even have a card."
"Saw your robot on the news today. Pretty cool stuff."
Paul paused long enough to smile. "Thanks. We're all pretty excited."
"What's on the boy's wish list?"
"According to his mother, he wanted a clown."
"I'm all out of clowns this week."
"Then I guess I'll have to look around," Paul retorted, turning the corner of the far aisle that led to the do-it-yourself chemistry kits aimed at high school students but suitable for the son of a genius. Paul, his khaki pants and blue oxford wrinkled from three consecutive days of wear, pulled several kits from the shelf. He barely noticed the other customer slide past him in the aisle. It was the wake of her perfume that caught his attention. Box in hand, Paul Cannon turned as the woman stopped to gaze at various batteries locked in a standing glass case.
Paul Cannon stole a glance at the woman's recently curled and highlighted hair and admired her well-figured profile. As his eyes went from her hair to her waist to her stockings and back up, Ariana's face turned towards her stalker.
Paul Cannon's eye's jumped before gradually turning into an inquisitive squint. "Don't I know you?"
Ariana didn't flinch. She slowly turned back towards the glass case of expensive batteries, her attention still focused on the man next to her, his reflection in the glass case more than clear enough for her to identify him.
"Are you sure I don't know you?"
"No," Ariana said tersely, almost in a whisper.
Paul Cannon would not relent. Being right was hardwired into his ego, particularly his ego with regard to memory. And his ego wouldn't let it go.
"Wait... I got it! I may not remember your name, but I definitely remember you from..."
"Not another word," Ariana hissed. Her dark eyes were soulless, her voice on the verge of evil. "I don't know you. You don't know me." Ariana, turned her back and continued down the aisle in the direction of the door.
By the time Paul Cannon regained his composure, Ariana had exited the store and was on the way to her car.
Ten minutes later, Paul Cannon left the store with his metric calculator and chemistry kit encased in generic floral wrapping paper. He made his way to his BMW, threw his presents in the trunk, and got behind the wheel of the vehicle that had been nothing but trouble.
Across the street and down the block, Ariana watched from behind her oversized sunglasses as Paul Cannon drove down the street in the direction of home. With twenty minutes until clown-time he never once looked in the rear-view mirror.
## Chapter 44
The three hearses were parked in the back of the Money & Ceasar Funeral Home as Clark pulled into the otherwise empty lot. He straightened his tie in the rear-view mirror, a fashion offering to the deceased in case he had the misfortune of running into one on their home turf.
An old black man washing the cars dipped a brush into a bucket of sudsy water and nodded to Clark as he passed.
"How often do you wash those?" Clark stopped and asked out of curiosity.
"Every other day if the weather is nice and they've been in service. If they haven't been out from under the carport, I don't clean em.'"
"You wash them by hand in the winter?"
"I take 'em to the car wash when it gets _really_ cold, but I try to wash 'em by hand most of the time."
"This isn't _really_ cold?"
"Really cold is anything below freezing. Can't wash a car with ice. We are near forty-five today. Gives me something to do."
"I guess everyone's last ride should be in a clean car," Clark joked.
"You got that right. Can't be driving the dead around in dirty cars. Bad for business. Bad for karma."
"Washing them every other day is bad for the finish."
"No worrying there. The owner buys a new one every few years."
"Which owner, Money or Ceasar?"
The man looked up at the sign on the back of the building. "Hell, old man Ceasar died forty years ago. The only one left is Money." The man in the black overalls lured Clark closer with a flick of his head.
"You here for a service? Did someone pass?"
"No, I'm here for some information."
The old man looked around as if he were about to spill a state secret on embalming. "You know what the family calls these vehicles when they think nobody can hear them?"
"No idea."
"Money wagons."
"Jesus," Clark said, catching himself in mid-blasphemy.
The old man laughed in a deep baritone. "Money likes his money. Hell, I don't even think Money was his birth name. Probably just liked the sound of it. But I tell you this, you'll hear all kinds of things whispered at Money family gatherings. Things that will make your skin crawl, make you lose your dinner, and get you thinkin' about cremation." He finished with another laugh and Clark wasn't sure whether to believe the man or not.
Clark knew the funeral parlor business had a bad reputation. Over the years, news reports on the expose beat had included the funeral parlor business with increasing regularity. At the top of the sin list was price fixing on coffins and screwing over the elderly with exorbitant service add-ons that were free to those with tighter reigns on their faculties. Death was good business, unless you had a funeral parlor in a "regentrified" area of town. Yuppies just weren't prone to kicking the bucket fast enough to break-even.
"How many funerals do you have a week?" Clark asked.
"Depends on how many people have been dying. You get a bad year with the flu and we have a lot of elderly pass. Two services a day is not unusual in the winter."
"You drive the cars?"
"Wash them and drive them."
"You ever get used to it?"
"What's that?"
Clark paused as an image of his father flashed through his mind. "Death. Dealing with dead bodies."
"I guess it takes some gettin' used to. It's not for everybody, that's for sure. Had a young man a couple years back who only made it through one day of work. He had a problem locking the legs on the casket carrier and when he went to pull the casket from the car, bam! The casket popped open and out fell a body about the same age as the employee. The deceased had been in a car accident and gone through the windshield. Shook the boy up pretty good."
"Poor guy."
The old man nodded in silent agreement and then added, "But he should have locked the wheels, like I told him to."
"That thought will haunt me all day."
"You know, the problem people have with death is that they don't think it is going to happen to them. Funeral services, funeral homes, cemeteries... they are places people try to avoid. Trying to avoid thinkin' about the unavoidable. I try to remind my kids that there isn't one person left on Earth who was here one hundred and twenty-five years ago."
"I see your point."
"Just food for thought."
The old man dipped the brush back into the water and gave his full attention to the back wheel. Clark took the change in focus as a hint and headed up the sidewalk towards the back door.
The door opened silently on well-greased hinges and Clark stepped into the main hall and proceeded slowly across the burgundy carpet. A large chandelier hung overhead, the sparkling crystals compensating for a room with a permanently somber mood.
There were four viewing rooms off the main hall. All of them held caskets. Clark wondered if they were full or if the funeral parlor used them as showrooms to drum up business in an industry where demand is determined by God and not the consumer opening their wallet.
There was a grand staircase to the far end of the main hall, and Clark walked forward slowly as if trying not to disturb the air.
"Good afternoon. May I help you?" a voice echoed through the hall.
Clark visibly jumped.
"My apologies. I didn't mean to startle you." Phil Money, a week past his fiftieth birthday bash, stepped from the small office near the front door and walked towards Clark, hand extended. He was wearing a dark suit, not quite black, but in the neighborhood of the darkest charcoal. The suit was accentuated with a subdued blue tie, knotted in a double Windsor. Clark saw a thick gold watch dangling from the cuff of his wrist and he thought about the old man washing the cars outside in the middle of winter. Money likes his money.
"My name is Clark Hayden. I wanted to have a word with you, if I could?"
"You are the young man who called this morning."
"Yes, I am."
"Please. If you would step into my office," Phil Money said, gesturing with an open hand.
Clark walked into the office and found a seat on one of the two sofas near the bay window looking out the front of the building.
"May I interest you in some tea or coffee?"
"Tea would be fine."
Money opened a cabinet on the far wall and removed two cups and saucers. He opened the cabinet next to the cups and the shelves burgeoned with tea alternatives. "We have Earl Gray, Darjeeling, green tea, caffeinated and decaffeinated, ginseng, chamomile, and oolong..."
"I'll have Earl Gray," Clark answered, unsure he would recognize any further choices.
"Fine selection. I will have one myself." Money flipped the switch on a modern white Braun water heater that fit the décor of the room. "It will only be a moment."
"Thank you," Clark answered, trying to remain as polite as his host. It was a tough act to follow. Money's delivery was polished, his manners impeccable, his grooming exemplary, his selection of tea unmatched. What looked like an office could have passed as a Far East tea den.
"What may I help you with this morning, Mr. Hayden?"
"I'm looking for a contact name and number for a neighbor who recently passed away. I was told that you don't give out information over the phone on former customers, and I didn't have a fax machine to put the request in writing."
"Yes, I do apologize for our policy here at Money & Ceasar. We try to avoid getting involved in the circumstances of our customers, as you put it. Society is so litigious these days. Money & Ceasar tries to avoid being used as a pawn in the affairs of the deceased. Gold diggers and money grubbers come out of the woodwork when there has been a death in a family, particularly when there is an inheritance to squabble over. As a rule, we avoid giving out personal information."
Clark tried to diffuse the suspicion. "The deceased was a neighbor. His name was Allan Coleman. I was unfortunately out of the country when he passed, and have been unable to locate his family." Clark had a momentary feeling of guilt for lying about the timing of the death and his trip overseas, but forged ahead. "I wanted to pass along my condolences to his family."
"I see," Money said without moving. "You were close to this neighbor?"
"Not super close, but he was a neighbor. He lived on the next street over."
"And none of the other neighbors have forwarding information?"
"I asked, but it seems that Mr. Coleman was a bit of a recluse."
Money gave Clark the once-over and decided that the young man was trustworthy. "You are looking for next-of-kin?"
"I'm looking for someone to whom I can send a condolence card."
The hot water heater beeped once and Money stopped what he was doing to pour the water and steep the tea. He placed a cup on the table in front of Clark and excused himself to the inconspicuous laptop that was on the corner of his desk. He typed in Allan Coleman and a one page file filled the screen. He took a sip of his tea and said, "His brother's name is Greg Coleman, and he lives in Warrenton, Virginia. Shall I write down the address?"
"Please," Clark said. "That would be super."
Money wrote on a memo pad with the company letterhead, the gold pen tracing out the address in perfect cursive letters.
"There you are," Money said, handing the paper to Clark as he rejoined his guest on the opposite sofa.
Clark folded the paper and put it in his front pocket.
Money took another sip of his tea and cleared his throat slightly. "So Mr. Hayden, have you considered purchasing a plot or making other post-life decisions? You're never too young to consider your post-life future."
Clark swallowed hard to refrain from spitting his tea on the table.
Greg Coleman answered the phone groggily even though it was three in the afternoon.
"Mr. Coleman?"
"Speaking," the gruff voice said on the other end of the phone.
"Mr. Coleman, my name is Clark Hayden and I was a neighbor of your brother, Allan."
Silence filled the line for a moment before Greg Coleman spoke. "Yes."
"I wanted to extend my condolences to you and your family for your loss. Your brother was a good neighbor."
"Clark was it?"
"Yes."
"I appreciate your call, Clark, but I know the neighborhood didn't think highly of my brother. It took him two years and three trips to the Town Council before he could get a permit to build that monstrous addition on his house."
"I don't know anything about it. Your brother lived on the next street over from my parents', well, my mother's house. I grew up there, but I have been living at school until recently."
"Thank you, then. I appreciate the phone call."
"Before you hang up, I wanted to ask another question," he blurted.
Greg Coleman was thinking about his brother, the wounds still fresh. He didn't want to have a conversation on the topic. He certainly wasn't ready for what Clark was about to say.
"What's your question?"
"Did your brother have an autopsy done before he was buried?"
Greg Coleman's hand gripped the phone and the blood coursed through the tightening veins in his neck. "What the hell is this, a prank call?"
"No, sir. Not at all. I wish it were. I lost my father last year and I wouldn't jerk you around on the death of your brother."
"Then why the questions? Nothing is going to bring my brother back. Let him and our family rest in peace."
"Please, Mr. Coleman. It's important. I wouldn't ask if it weren't."
Greg Coleman grunted a little. "No, we didn't have an autopsy performed. My mother was against it. As you probably know, my brother wasn't in the best of health. The police found him dead on the floor with his heart medication spilled on the desk. It didn't take Einstein to figure out what happened. He was two-hundred pounds overweight, didn't exercise, smoked, and had a plethora of medical ailments that were trying to kill him. He died because he didn't take care of himself."
"I understand," Clark said, feeling the sorrow over the phone. "But I think there is a chance your brother didn't die of natural causes."
## Chapter 45
The two-woman flight attendant team marched down the aisle in their light blue uniforms giving orders like a female Gestapo unit. Tray backs were pushed forward and seats snapped into their upright position. A high-school student watching a DVD on his laptop grudgingly shut it off after his third warning, the last of which implied the use of an in-flight emergency exit for non-compliance.
Clark stared out the window as the plane banked towards his side of the aircraft. He looked down at the city below, the high-rise buildings in the business district stretching upward, the tallest structures for hundreds of miles. Beyond the center of town, the city expanded in nearly symmetrical blocks — a river, train tracks, and the occasional highway dissecting the geometric perfection. The flight attendants found their seats as the captain made an announcement for final approach, "We will be touching down in Kansas City in approximately ten minutes and should have you at the gate five minutes ahead of schedule." Clark looked at Lisa who was asleep, and had been since before the plane left the Reagan Airport runway.
The National Record Center for Immigration and Customs Enforcement resided in Lee's Summit, a quiet suburb just outside Kansas City. The interior of the building shined with state-of-the-art architectural features designed to wow the politicians who continued to fund the project. Carpeted floors, glass walls, and exposed ducts splashed in funky colors made the N.R.C. at Lee's Summit one of the most progressive buildings in the government. The exterior was harder to appreciate. The N.R.C. was 200,000 square feet of office space constructed into the opening of an existing cave. Thirty million files wedged into the earth in the middle of the country's heartland. The government knew no boundaries when it came to spending money.
"I can't believe I let you talk me into this," Lisa said, pulling her jacket closed.
"It's information that we could have obtained through a Freedom of Information Act request, if we could wait three months. Besides, I told you we may well come out of this looking like heroes," Clark responded.
"Or felons," Lisa answered.
Lisa walked through the automatic doors as Clark stared at the entrance to the building from the outside. "Let's go, Clark," Lisa said, not agitated, but sounding like she was. Criminal Investigator Lisa Prescott pulled her badge and showed it through the thick glass to the Immigration and Customs Enforcement employee. The elderly pale white woman with a terrific set of dentures leaned far enough forward to see the picture on the badge and then moved back into her seat.
"I'm here to see Don Christie."
"Who are you with?" the woman asked. Clark looked at Lisa, wondering exactly what the lady's position was. Obviously, it wasn't checking identities.
"The FBI," Lisa responded in jest.
"And him?" she asked with a flick of the thumb in Clark's direction.
"He is my witness. I need him to identify the person in question."
The old woman pushed a clipboard through the counter level hole in the glass. "Please sign in and have a seat. Someone will be with you in a moment."
Lisa and Clark signed in and the door opened with a buzz long before they made a move for the chairs.
"You the IRS agent?"
"That's me," Lisa answered. "Call me Lisa."
Clark's mind flashed back to their first meeting and Lisa's call-me-Agent-Prescott mandate.
"And him?" Don Christie asked.
Clark took a crack at introducing himself. "Clark Hayden. Here to provide insight into the investigation as a material witness."
Don Christie shook Clark's hand with a knuckle grinding grip and nodded to the woman behind the glass who pushed through two visitor's badges.
"These must be worn at all times."
Don Christie waddled as he walked. It wasn't a strut or a show of force. It was the way the bones of his skeleton interacted with one another. Clark walked next to Lisa and considered that their escort, from the backside, could almost pass as a silverback in a suit.
Don spoke over his shoulder as he walked, his voice bellowing. "Welcome to the National Records Center, or what we call the N.R.C. We used to call it INS NRC, but technically the INS doesn't exist anymore. It's all rolled under the Department of Homeland Security and Immigration and Customs Enforcement. In reality, we haven't quite gotten over the hump with the new name. INS still rolls off the tongue."
"Yes it does," Clark said.
"Besides, most of the documents we use still have INS designated on them. We have over thirty million files under our supervision, with millions more arriving every year. We have been consolidating records from regional INS offices for years, and the boxes keep pouring in. We have storage on site and in two locations around the city. We are also in the process of formatting all the files for electronic storage and applying barcode technology for faster retrieval."
"In the process, you said?" Clark asked.
"Well, as you can imagine, it takes a while to scan thirty million folders, some of them an inch thick."
"How many have been converted electronically?"
"I'm afraid it's not that simple. Each region had its own filing system. Some filed according the INS file number. Some filed them alphabetically. Some filed them by year and month."
"I don't think I like where this is going," Lisa said, unbuttoning the top button on her blouse.
"Probably not," Don answered. He pushed a glass door open into a large work area. As he walked around the edge of workspace, Don continued the tour.
"Most of the people here are working on the classification of documents and electronic formatting. Unfortunately it is a manual process. We have to take every folder out, undo the center clips and then run them through scanners and put the folders back together. Then there is data entry for each folder so that we can retrieve the information from the database."
Clark rolled his eyes imperceptibly. "So how many folders have been entered into the database?"
"About two million. What name are you looking for again?"
"Ariana Amin."
"The beginning of the alphabet."
"Is that good or bad?" Clark asked.
"Bad, I guess. We started our electronic conversion at Z."
"But, of course," Lisa added.
Don raised an eyebrow and continued. "Well, the folders with the last name of Z were the smallest in number. We started there for a sense of accomplishment. That, and we were mandated to have certain percentages of the alphabet finished by certain intervals. So Z is completely in the database. Easy to search."
"And the 'A' section?" Clark asked.
"We've started, but haven't finished. So you may get lucky, and you may not." Don led the pair around the corner to a set of stainless steel elevator doors. "We're going down two stories to the main storage facility."
"Any chance this place was originally built to withstand a nuclear war?" Clark asked.
"Not that I know of. If there were a nuclear war, I hope the country would have higher priorities than keeping these folders safe." The elevator opened with a small ding. "After you," Don said, ushering his guests into the elevator. The silver doors opened on sub-floor two into a hall that stood behind a glass wall. The glass overlooked a huge floor of metal bookcases full of boxes and folders.
Clark spoke first. "Holy shit. That is a lot of files."
"That's generally the reaction we get down here in Sub-Two from our visitors. Most people have never seen anything like it."
Clark and Agent Prescott pressed up against the glass and gawked at the hangar-sized room large enough to park a pair of 747's end-to-end and wing-to-wing.
Lisa muttered in disbelief. "This could take all year."
"I hope not," Don answered. "I get off at six and I'm the only one here today with authorization to bring visitors to Sub-Two."
"Then take us to the 'A' section."
Don Christie turned down the third aisle from the end and ran his finger along several boxes reading the names and numbers on the outside.
"I think this is the area to start. This row runs from 'AG' to 'AS.' All of what you see is not in the database yet. These are the original hard copies.
Clark looked down the aisle and up at the top of the seven-foot shelves. "Are they in order?"
"In places they might be, in some places, probably not. Remember, these files came from different regions, so if the folders are still in the box, there is no telling what system they used. It all depends."
Don looked at Lisa. "Let me know if you need anything, I will be around. Just ask for me."
"Can we get any help?" Clark asked.
"I'll see if I can't have someone come down after the staff meeting."
"Thanks."
"Good luck."
"So, what do you think?" Clark asked.
Lisa answered. "There were nine Ariana Amins in the computer system." She pulled out the list of names and addresses. "All we have to do is locate the correct box and file for each of the nine people on the list. Then we will know where they got citizenship."
Clark ran his fingers along the outside of the box and squinted at the scribbling on some of the labels.
"It may not be that easy. Let's say that Ariana Amin, from Virginia in the IRS system, got citizenship in Miami before she had tax information and then moved to Washington D.C. In the IRS system, it would show her current address, but not necessarily her address when she applied to the INS."
"So, if she hasn't moved, the location would be the same," Lisa paused. "Which percentage of the population moves in a year?"
"That's not the question. The question is, which percentage of terrorists would move after gaining citizenship or legal residency?"
Lisa looked at Clark and answered. "All of them."
"Yeah."
"What do you suggest?"
"Intuition," Clark answered. He shut his eyes and took five steps down the aisle towards the center. Eyes still shut, he stuck out his right hand and tapped a box. "I'll start here."
Lisa looked at Clark as if he were crazy before shutting her eyes, spinning around once, and grabbing the first box in front of her as she completed the turn.
It took two hours to go through a dozen boxes. File by file. Page by page. Photo by photo. At their current rate, Criminal Investigator Prescott would reach mandatory retirement before they made it half-way down the aisle. Empty Styrofoam cups that once held coffee were stacked behind Agent Prescott in a poor attempt at a pyramid. Pouring through the boxes, each person checked the photo on the file and then the name. Clark stood from his crossed-legs position on the floor. "This could take a lifetime. And my butt is already numb from this floor."
"We don't have a lifetime and don't blame the floor. It's kind of nice to be in another city and feel that D.C. isn't the only place with a chill."
"I don't know if I would call it nice."
"Yeah, that's true. Snow just isn't as much fun as it is when you are younger. The only thing I know enjoying this winter is my neighbor's new puppy. He loves the snow. I watch him play from the window sometimes. Watching him brings out the pure joy of snow."
Clark paused for a moment at the thought of Lisa's neighbor's dog frolicking in the snow. Something clicked. He picked up the files on the floor and put them back in the box that was in front of him. He paused again, cocking his head to the side as if a voice was calling out to him from the edge of his hearing. Clark looked down the aisle at the insurmountable task and walked past Lisa who was flipping through folders at a feverish pace.
Clark scanned the handwriting on the outside of several boxes and moved farther down the aisle. He read, paused, and then read again before stopping in front of six boxes, the same words written on each in red permanent marker. INS. Boston District Office, JFK Federal Building. 1996-2001. There were six more boxes dated 2002-2007. Beneath the title of each box was a designation for a part of the alphabet. Clark's eyes moved to the second box in the first set of dates. "AG" through "AP."
In the back of his mind, Lisa's words tumbled. _A new puppy playing in the snow_.
"Lisa, when was the first time you saw snow?"
"I don't remember. My parents used to take me to Maine in the winter to visit my uncle. I guess I was playing in the stuff since I was little. Why?"
Clark pulled a box off the shelf. The white cardboard bottom made a thud as it hit the cool tile floor. "You just said something that reminded me of a story Ariana told my mother and me," Clark started. He sat down again on the floor, the nerves in his butt giving notice that they were sore in addition to cold. "It was a random story, but something I guess I never forgot. She said that a friend of hers, who had never seen snow, ran outside one morning to play in a foot of fresh powder. Ariana took the time to put on boots and gloves and headed for the door when her friend came running back into the house. Ariana asked her friend where she was going and her friend said 'back inside.' When Ariana asked why, her friend answered plainly. 'I didn't realize snow would be so cold.'"
"What's your point?"
Clark dug through the file with verve and a certain amount of anticipation. He passed a folder with the name "Amen, Jesus," written on the outside, and stopped to look at the picture, out of curiosity. Who wouldn't?
"Well, I guess the reason the story stuck with me is simple. If you have never seen snow, never been exposed to snow, didn't come from a climate where they had snow, I guess you wouldn't necessarily realize that snow is cold."
"I guess it's possible. Now, what's your point?"
Clark raised his pointer finger and then dove back into the files. Ten seconds later he found his first "Amin." His heart rate picked up. Abdul Amin, born in Egypt. Al-Mohamed Amin, from Saudi Arabia. Assad Amin, Jordan. Ariana Amin, Pakistan.
"My point is that when Ariana told the story about the first time her friend saw snow, she said she was in Boston," Clark said tapping the side of the folder with a smile.
"Is it her?"
Clark opened the folder and read frantically. "The name is the same. Same nationality."
Agent Prescott pulled her stiff frame off the floor with the help of the nearest bookcase.
Clark fished for the photos which had become detached from the top page but were still in the crack of the folder which he held between his hands. He turned the picture upright and looked at it, moving it directly under a light that was fifteen feet above. "It's not her."
"Are you sure?"
"Pretty sure. She is not wearing a hijab, which Ariana always wore, but the nose is a little different. They are similar, but it isn't her."
"Let me see that file."
"Don't trust me?"
"I don't trust anyone."
"You need therapy."
"I've been. It didn't help."
"What did you go for?"
"Shooting my partner."
"I'm not your partner."
"Lucky you."
"What do we do?"
"You think it is her? There are eight others on the list."
"Well, I remember Ariana telling me that she had been in the States about ten years. But that was a few years ago. The IRS file shows the first date for her as a taxpayer as 1997. So her age is pretty close."
Lisa spoke. "Let me make a call and see if I can't get more details on Boston Ariana here." Lisa opened her cell phone and the screen told her what she already knew. Caves were not a good place to get a signal.
"What should I do in the meantime?"
"Keep reading."
When Lisa returned, Don Christie was by her side. Lisa reached for the folder and handed it to Don Christie. "We would like a copy of this folder. Every page."
"Did you find something?"
"Just an address."
"Are we done here?"
"We can either check out this lead, or spend the rest of the week in this aisle."
"Just say the word."
"We're booked on a plane for Boston that leaves in two hours. It gets in after midnight."
## Chapter 46
Imam Alamoudi entered the inner sanctum of the mosque and headed for the living quarters. A Persian rug worth twenty grand spread from the doorway to the window, wall-to-wall. A large cushioned sofa was placed under a set of windows, seats reserved for the elderly. Pillows balanced precariously on the end of the sofa. Generous portions of larger multi-colored pillows littered the floor in neat piles on the hand-woven rug. The pillows were for the imam and his guests. At over 300 pounds, the imam's favorite activity was lying on the floor, his belly hugging the tightly knitted Persian rug, his eyes fixated on the satellite feed of Al Jazeera that was pumped in from the six foot satellite dish on the roof.
Imam Alamoudi pressed the intercom on the wall and ordered lunch in his native Urdu. He shut the door behind him and loosened the belt on his salwar, his pants. He didn't see Ariana until her hands were on his shoulder. Imam Alamoudi's naturally-stressed heartbeat quickened.
"Long time no see?"
"You should know better than to bring trouble here."
"I didn't bring trouble here."
"Your neighbor was here."
"When?"
"Last week."
"He came on his own. I never even mentioned where I pray. And if not for my husband, I would have never come to this mosque. Prayer is between woman and Allah. The mosque is only a building."
"You're incorrect. Prayer is between _man_ and Allah."
Ariana repeated herself and then added. "What did the boy want?"
"He wanted to know how to reach you. He said that you asked him to watch your house. Now he wants to know how to contact you. He said there was a fire in your home."
"Nosey fucking American."
The imam looked at Ariana, judging her vernacular and delivering a stern warning with his eyes.
"I know, I know. I have been living with a group of men."
The imam headed for the sofa and motioned for Ariana to join him.
"What brings you here?"
"Money."
The imam walked across the room and his weight sunk into the cushions on the sofa.
"And men."
"Why not blood too?"
"Don't tempt me."
"How much do you need?"
"Fifty thousand."
"That is not a trivial amount."
"But can you get it?"
"Of course."
The imam slung his weight back into a standing position and walked across the room to a small table in the corner. He opened the drawer and removed a cash box the size of a large dictionary. With his back to Ariana he punched in the combination and opened the top.
Ariana's hand reached the cash before the imam's. The imam grabbed for Ariana's wrist and found himself with his arm pinned behind his back, hand empty.
"I have never seen your trained side. Very impressive."
"You have no idea," Ariana replied.
"Take what you need."
Ariana smiled as she took the box back to her seat on the sofa. She counted out fifty thousand in neatly stacked hundred dollar bills wrapped in thick rubber bands. Then she threw two additional stacks on the pile. "Just in case," she added as she put the box on the cushion next to her.
"You know, when I first heard about you, I doubted whether it was true. You play a housewife and mother very well."
"Not everything is as it seems."
"This is very true. In your case, anyway. I assume by the presence of your neighbor that a plan is underway."
"Yes. We are past the point of no return."
"Have you covered your tracks?"
"I have been preparing for years. The most recent developments are just icing on the cake."
"How many men do you need?"
"Six if I can have them. Less if there are less available. I need young men with unshakable faith."
"I have several in training. A few even speak your native tongue."
"I need native faith."
"I will give you the best I can find."
"How much time do you need?"
"I need them by next Tuesday."
"Then you shall have them by next Tuesday."
"Where?"
The imam thought for a moment. "There should be a pencil and paper in the cash box."
Ariana handed them to him and the imam scribbled an address. He handed the note to Ariana, who read it. "Are you serious?"
"The owner is one of the faithful."
Ariana stood expressionless until the imam spoke again. "I assume this is the last time we shall meet."
"This will be the last time. If we see each other again, the outcome will not be good for you."
"Then I wish you well."
## Chapter 47
Wellesley, twenty minutes outside Boston when the traffic moves, was still ten minutes away after an hour of bumper-to-bumper. Between Natick and Newton, the quaint town of Wellesley was home to the eponymous all-women's college catering to elite intellectuals and average intellectuals with elite family pocketbooks. At over forty grand a year, Wellesley was one of the most expensive schools in the country.
And while downtown Wellesley had remained a bastion of wealth, an idyllic slice of New England Americana, the surrounding area had spent the last two decades being consumed by the ugly dragon of urban sprawl. Wellesley, a sanctuary of knowledge and wealth, now rested firmly in the beast's stomach. The town square was still picturesque, but two miles down the road, across from the rival cross-town high school, the new Super Wal-Mart had tarnished the remnants of the small town atmosphere. Only the college's three billion dollar endowment was keeping the city planners at bay, the threat of more strip malls at a distance.
Lisa read the map on her lap as Clark weaved among some of the most hostile drivers on the planet. By all accounts, New York drivers were tame in comparison to their northeast corridor neighbors who took delight in tailgating, cut-offs, and middle-finger salutes. New York drivers were tough, but Boston drivers relished in rudeness. Rush hour in Bean Town was not for the faint of heart. Clark swerved away from a monster pothole and into the next lane of traffic. Eyes wide, Lisa glanced out the driver's window for a close-up view of a delivery truck covered in road salt and winter grime, the metal side of the vehicle close enough to wipe clean with her sleeve.
She scooted over in the direction of the driver's seat as Clark found his lane again. Ten minutes later they pulled in front of a two-story, white frame house with no garage. A chain-link fence enclosed the treeless front yard. The white siding was in need of a fresh coat of paint and the gutters sagged from the edge of the roof.
"Number 812. That's us," Clark said to his girlfriend, zipping his jacket before opening the door and exiting into a balmy single-digit wind-chill.
A student in a Boston College sweatshirt opened the door after three knocks. "Yo," the student said, his voice trailing off as he realized his visitors were strangers.
"Good morning," Clark said. The young man in the BC sweatshirt with screaming red hair looked through bloodshot eyes at Clark with suspicion. His eyes then turned to Lisa and his face perked up.
"Who are you?"
"My name is Clark Hayden and I'm looking for someone who used to live at this address. I wanted to ask you a few questions."
The student looked at Clark suspiciously. "Who's your partner?"
Lisa stepped forward and pulled out her IRS badge. "IRS criminal investigator, Lisa Prescott."
Clark longed for a badge to flash.
Lisa's badge had its intended affect and the sleepy young man on the other side of the doorway took a huge step towards coherence. Police at the door tended to scare most people. The IRS at the door made people tighten their sphincters.
With the resident's full attention, Clark plowed forward. "I'm a graduate student at Virginia Tech. You can Google me if you want. I'm on the robotics team; there should be a picture of me out there."
The student looked behind himself into the house and around the room.
"Can we come in for a minute? It's freezing out here," Clark added.
The redheaded BC student just stood there, his toes peaking out from the shredded fringe of his jeans.
Lisa added her soft touch. "Relax, we are not interested in you, your taxes, or whatever weed you may have stuck between the couch cushions."
The BC student stepped back from the door and Clark and Lisa walked into the room, hurriedly shutting the door behind them. Beer cans were scattered on the small dining room table in the corner. Pieces of newspapers littered the sofa. A small lamp with a crooked shade stood precariously next to the sofa. There wasn't a text book in sight.
"You live here, I take it..."
"I hope so. Otherwise I would have some explaining to do."
"You rent this place?"
"Lease with an option to buy."
Clark ignored the smartass statement. "Who's the owner?"
"The old woman next door. Mrs. Crowley."
"Is she around?"
"She is, like, ninety. She's usually home. She takes a walk in the afternoon. But I don't see her outside much when it's this cold."
"How are Mrs. Crowley's faculties? Is she lucid? Forgetful?"
"Well, she doesn't forget when the rent check is due, if that tells you anything."
Lisa laughed a little.
"Do you get along with her?"
"She's a little hard of hearing, which makes her a pretty good neighbor. She doesn't complain about noise at night. Why, did something happen next door?"
"No. Well, not that I am aware of anyway. I'm looking for information on a neighbor of mine in D.C. She disappeared and I'm trying to dig through her past a little to see if I can locate her."
The young man's eyes darted between Lisa's and Clark's.
Lisa asked another soft question. It was something she had learned in her IRS interrogation course. When you run into resistance, ask a simple question. Soften them up a little.
"How long have you been living here?"
"This is my second year, so about three semesters."
Clark pulled out a sketch of Ariana that Mr. Stanley had drawn with the same easel he had drawn a portrait of his wife. "I know this is a long shot, but have you ever seen this woman around?"
The redheaded student grabbed the picture between his freckle-laden hands and quickly handed it back to Clark. "Never seen her before."
"You sure?"
"Yeah, I would remember if I had seen a woman in a hijab around here. There might be a few at Wellesley, but I don't spend much time there, obviously. You have to have a female escort just to get in the gate."
"I'd think it would be a good place to meet women," Clark said.
"Wellesley girls don't date BC guys. They go for Harvard all the way. Everyone knows it. Until a few semesters ago, both Harvard and MIT used to send buses to Wellesley to pick up girls for the weekend. These buses would come right to the main gate on Friday night and drive the women across town. On Saturday and Sunday another bus would come and drop the girls back off. I imagine some of them got shagged all weekend."
Clark had a lot to say but bit his tongue. Girlfriends tended to suck the testosterone out of most male conversations. On the surface, at least.
"You see any other Muslim women around?"
"There are a few Muslims at BC, but they are nice people. Quiet. Only one or two wear a hijab. And the woman in your picture there isn't one of them."
Clark pressed. "She would have been younger than she is in the picture."
"Dude, I haven't seen her."
Lisa knew it was time to cut their losses with BC's finest.
"Which side is Mrs. Crowley on?"
"The ugly green house to the right. You can't miss it."
"Thanks for your time."
A minute later, Mrs. Crowley opened the door and looked up at her visitors from her hunched position. She had a small cane in her left hand, the decorative end of the walking stick wrapped in a ball of old fingers and wrinkled skin.
"Good afternoon, Mrs. Crowley. My name is Clark Hayden, and this is my girlfriend, Lisa. We are looking for a tenant of yours who used to live next door."
"You with the police?"
"No, we live in Washington D.C. and we're trying to locate a neighbor of mine who has gone missing."
"You should call the FBI. They handle missing people."
"I have, but there are national security issues at stake which prohibit the authorities from assisting with this parti-cular investigation."
"The FBI won't help?"
"Pretty much."
"Then just say so. No reason to get all fancy pants on me with your national security issues and authorities crap," Mrs. Crowley said, barking the last part of the sentence out in a reasonable impersonation of a TV detective.
Lisa laughed.
"Well, come on in, I'm not trying to heat the rest of the neighborhood."
Clark and Lisa stepped up the single stair and wiped their feet before entering the warm house.
Mrs. Crowley moved across her small living room with ease. Clark wondered if the cane was for show, or for smacking people upside the head.
"Please have a seat."
Mrs. Crowley liked flowers. More specifically, she liked roses. On the wall over the rose-patterned sofa was an oil painting of a large bouquet of the same. The throw pillows on the rocking chair were knitted with another bouquet of a slightly different color. The upright piano on the far side of the room was covered with rose-patterned doilies, on which a small vase of glass roses gathered dust.
Clark took a seat on the roses on the sofa. Mrs. Crowley put her weight into motion in the rocking chair. The room was hot but Mrs. Crowley had on a sweater and a shawl, the latter of which was a light blue, nearly the same hue as her hair.
"Can I get you some coffee?" she said with a pronounced Boston accent. "Some tea?"
"Whichever is easier for you," Clark answered.
"What kind of answer is that? Boiling water is boiling water."
"Coffee would be great. Thank you."
"See? How hard was that?"
Clark now suppressed a laugh. The charming coffin-dodger was definitely using her cane to crack skulls.
"And for you?" Mrs. Crowley asked Lisa.
"Coffee, black. Please."
Mrs. Crowley walked to the kitchen and returned a moment later. "So, how can I help you this morning?"
"I'm looking for information on someone who was a tenant of yours next door."
"I've had a lot of tenants. Some of them good, some of them bad. Which one are you looking for?"
"Ariana Amin."
"Ariana?"
"You recognize the name?"
"Of course," Mrs. Crowley said, wringing her hands together. Clark pulled a photograph from the manila folder and handed it to Mrs. Crowley. "Is this picture Ariana Amin?"
"Yes, that's her."
"She listed your house next door as her address in her INS documentation."
"Did she?"
"Yes."
"What can you tell me about her?"
"Well, for starters, she's dead."
Clark tried to not look startled. Lisa took over the questioning after a brief pause. "What do you remember about her?"
"She was quiet. Kept to herself, mostly. She had two roommates, but I am not sure how friendly they were. She hung out with other young women from time to time. Used to see them stop by and pick her up."
"Wellesley students?"
"Hard to say. There are so many colleges and universities in the Boston area. Harvard, UMass., MIT, Boston University."
Lisa nodded and Clark pulled out the hand drawn sketch of his former neighbor. "Was this girl one of her roommates, or maybe a friend?"
Mrs. Crowley looked at the picture through her bifocals. "No. I don't recall having ever seen her before. Who is she?"
"We aren't exactly sure. Her name, as far as we can tell, is also Ariana Amin."
"What did she do?"
"We don't know exactly."
"Then why are you looking for her?"
Clark looked at Lisa, who nodded. "I think she killed my neighbors and then vanished."
Lisa spoke with officialdom. "We call her an 'unsub,' or Unidentified Subject."
"Thank you for the bullshit explanation," Mrs. Crowley said.
"What can you tell me about Ariana's death?"
"Well, she disappeared. They found her abandoned car burned out in a remote area near Marlborough. They never did find her body."
"Did anyone come to claim her belongings?"
"Someone did, yes. I can't remember if it was a brother or a cousin, but it was someone around the same age, maybe a little older. It wasn't her father, I don't think. That, I would have remembered."
Ms. Crowley paused.
"The police figured she was kidnapped and murdered. Buried in the woods. There are a lot of trees in western Massachusetts, a lot of places to dispose of a body."
"Is that what the police said?"
"That is what I said. The story was all over the news for weeks. The car was burned to a skeleton and there wasn't much left to investigate. The police came to interview me and they interviewed some friends of hers, but it was routine."
"When was this?"
"Oh, let me see. It was in April 1998. I don't remember the date. Sometime around Easter."
The boiling teapot brought Ms. Crowley to her feet. She went to the kitchen and Lisa and Clark could hear the rattling of china, the refrigerator closing, the soft clanking of silverware.
"April 1998."
"I heard," Clark said.
Lisa looked at the file from the INS. "The last document in the folder is her naturalization certificate. It was awarded April 17th, 1998."
"So she stole her identity?"
"That would be my guess."
"So who is she?"
"I have no idea."
## Chapter 48
Clark sprawled on the queen bed in the Holiday Inn near Newton, Massachusetts waiting for Lisa to finish blow-drying her hair in the bathroom. He turned off the TV and checked the weather on the back page of the local section of the Boston Globe. Old man winter was still feeling ornery, and the forecast for the rest of the week was calling for more arctic air.
He glanced at the weather map of the U.S. and allowed himself to wonder what it would be like on a beach in Hawaii. After a month of ice and snow, Waikiki would be perfect. Clark flipped back to the second page and perused the small hodgepodge of one paragraph articles. The first story was about a sixth grader who came to school with a loaded .44 magnum. _Dirty Harry goes to Show-and-Tell_ was the title of the news snippet. The second article caught Clark's attention and his stomach sank as he sped through the opening sentences. The burned out remains of missing NASA scientist Paul Cannon's BMW had been found in a remote suburb of Northern Virginia. The scientist, a Boston native and MIT graduate, had been employed by NASA in Greenbelt for seven years and was considered a pioneer in new energy propulsion systems. The missing man had also once held the record for reciting Pi to the 10,000th decimal place. _That's where I recognize the name_ , Clark thought.
Clark reminisced about his attempt to recite Pi and then his mind jumped back to the opening sentences. _A missing man with a burned out car found in a remote area_.
Clark felt cold sweat bead up on his neck. "No fucking way."
## Chapter 49
Sealed in the sleeping quarters, Ariana, Syed, and Karim waited. Per protocol, each person wore a military-grade Hycar gas mask with a polyurethane shield lens. Each mask had a nosecup to prevent fogging and a standard mechanical speaking diaphragm which allowed the wearer to speak clearly without undo effort. The head harness was adjustable without the need to deal with unwieldy rubber straps. When it came to warfare, Uncle Sam had the best gadgets.
Syed spoke through the mask, his words intermittent with the muffled wheezing of air through the mask. "Those machines. Do you think Abu can operate them properly?"
Ariana nodded. They had been in the room for over nine hours and Ariana struggled with boredom. Syed and Karim had grown accustomed to the room, the silence, the downtime. Ariana was still on the steep section of the learning curve. "We did a trial run with coffee beans. The output was good. The machines themselves aren't difficult to operate. I set the machines to the required specs, which should remove further variables in the equation. All Abu has to do is run the machines. Whether or not he follows the procedures, well, that's a different story."
"What are his chances?"
Ariana tilted her head slightly to the side.
"Fifty-fifty."
"You said the risk was low."
"If he follows instructions, he may make it."
Silence, and breathing, followed.
"You should have told him," Syed said.
"He knows what he signed up for. We all do," Karim answered, joining the conversation.
"Let's say you inhaled a lethal dose of the processed powder. How long would you have to live?" Syed asked.
Ariana answered, speaking slower than usual though her words were completely audible. "An educated guess is forty-eight hours. There really isn't much research on the subject, for obvious reasons. I think most of the knowledge we have comes from testing animals."
"How does it kill?"
"At the cellular level. Ricin acts as a ribosome-inactivating protein."
"Which means?"
"It breaks down the ribosomes in proteins, until protein synthesis ceases."
"Is there a less technical explanation?"
"Ricin poisoning is very complicated. It prevents the cells in the body from making proteins, and that will kill the cell and eventually the person. It causes clumping and breakdown of the red blood cells, hemorrhaging in the digestive tract, damage to vital organs. Within eight hours of inhalation, symptoms would start to appear. The symptoms would be different, depending on the method of ingestion, but for inhalation it would start with respiratory distress, coughing, tightness in the chest. Nausea, sweating, and fever would follow. Seizures and hallucinations might occur. Within two or three days the kidneys, liver, and spleen would stop working."
"And there's no treatment?"
"There is treatment, but no cure. Medicine can treat the symptoms, and ricin is not one hundred percent fatal. A person can recover on their own. Ricin in various forms is actually used medicinally. In particular, it is used in treating some forms of cancers and in bone marrow transplants."
"So it may not be lethal?"
"A few people may get lucky."
Syed considered the thought. "I guess that's the difference between a mass killing and an assassination. I'm trained to kill, and given the right weapon no one is getting lucky."
Ariana looked at the soldier sitting on an army cot. "Don't worry; there will be enough for you to kill as many as you want."
No one spoke for a minute and then Syed began anew. "I don't understand why we are using ricin. There will be no violent images, no destruction. We need a spectacle of violence to attract the next generation of fighters. To show the world that we are serious."
"Ricin is what we have."
"But a nuclear bomb is what we need."
"It is planned," Karim said. "But highly enriched uranium is hard to come by. We are relying on a small supply stream from southern Russia, passed through north Ossetia and Georgia. The Russian nuclear stockpile is for sale... if you have enough money and ask for small quantities. Large requests make everyone a little nervous. Too nervous. You may remember last year a Georgian man walked into a restaurant with three ounces of HEU in a plastic bag. Unfortunately, he sold it to an agent of the CIA."
"How can you carry it in a plastic bag?" Syed asked.
Ariana answered. "You can touch highly enriched uranium without protection. Unlike plutonium which is very deadly."
"How much uranium is needed for a bomb?"
"It takes over a hundred pounds of HEU to produce a bomb equivalent to the one dropped on Hiroshima. Ounce by ounce we will make it. We almost had six pounds of HEU in one transaction, but it was intercepted by Czech security," Karim said.
"It would be nice to have it now."
"Yes, it would," Karim answered.
Another moment of silence passed before the loquaciousness reappeared.
Karim started. "We have other efforts underway. We have men scouring the caves on the border between Kenya and Uganda. The Kitum cave on Mount Elgon is particularly hopeful. Wild elephants have been known to frequent the Kitum cave to lick its walls, which are laden with salt. It is also one of the known locations for the Ebola and Marburg virus. On two separate occasions the virus has been traced to this single cave."
"How do you capture the virus? You would need specialized equipment. Specialized medicine. Doctors."
"We are not trying to _capture_ the virus. We are trying to _contract_ the virus. We have a network of men who are prepared to contract the virus, travel to another location, and then spread the virus to the next person in the chain. We could have infected men in London, New York, Washington, and Paris in less than a week."
"That still lacks a certain mass appeal. The faithful are not as moved by sick bodies in bed as they are by body pieces in the street."
"Consider yourself a pioneer," Karim said.
A long silence followed and Ariana spoke. "There is something else to consider about our plan."
"What is that?" Karim asked.
"We need more men."
## Chapter 50
The quad in the middle of campus was quiet. The benches along the crisscrossing sidewalks were empty, the graffiti of white Greek letters standing out against the dark wood of the seatbacks. The guerilla advertising on unofficially designated benches was overlooked by university police. Limited "sanctioned vandalism" it was called. To give the campus some character.
Washington D.C. may have been suffering through a cold spell, but Boston was in the freezer next to the gamey meat. The local news was having fun with rumors of polar bear sightings in Boston Commons, America's oldest public park. The frozen Charles River acted as a wind tunnel to the campus of MIT, pushing gusts from the hills in the distance into downtown Cambridge. Unassailable, the winds tore through the largest conglomeration of scholastic intelligence in the free world before continuing on to pelt urban Bean Town.
Old universities ooze knowledge. They perspire with culture. They drip history. They breathe civility. Walk the halls of Princeton or Yale or Harvard or Oxford, and the average person feels smarter. Knowledge hangs in the air, hundreds of years of education forming some unidentifiable aura of cerebral worthiness.
The 430,000 square foot Stata Center on Vassar Street on the MIT campus was different. Stainless steel walls, interspersed with modern brown brick siding, jutted from the building at odd angles giving the feeling that the building was growing from the Earth haphazardly, like unkempt grass. Ninety degree angles were rare in the behemoth building whose construction was partially funded by Bill Gates. The only central recurring theme in the architecture was the lack of one. The madness of the first several floors of the building housed everything from artificial intelligence to linguistics. The towers on either side stretched upwards in swooping curves. Classrooms with inward sloping walls gave some students vertigo. George Jetson would have been right at home.
Clark and Lisa walked the hallway and stopped near the elevators to Tower D, the bulletin boards still covered with colorful fall semester notices and advertisements pinned to the corkboard. Cars for sale. Houses for rent. Roommates wanted. Guitar lessons available. Happy Hour with a band called Randy Dick, in honor of the singer and drummer.
Clark looked around at the glass interior walls and the team of intellects that worked like bees in a translucent hive. "It's not what I expected," Clark said to Lisa as they waited for the elevator.
"What did you expect?"
"History. Oil paintings on the wall. Wooden staircases. This place lacks a certain je ne sais quoi, which I believe is French for 'really old stuff.'"
"For someone so smart, you say some pretty stupid things."
"Just going for the laugh."
"I'm not sure we have time to be laughing."
The elevator opened in the basement of Tower D and the architectural feeling Clark had been searching for became an even more distant hope. He looked at the signage on the wall and followed the arrow towards _Robotics Research_.
"No wonder we lost the robotics competition. Virginia Tech is practicing robotics in stone buildings with walls that perspire and these guys are developing robots in Buck Rogers' apartment."
Clark knocked on the open doorframe under the Robotics Research sign. A girl in jeans and an MIT sweatshirt turned towards the door.
"Hi. I'm Clark Hayden and I'm looking for Mayank Malhotra."
"Mayank?"
"Yes. I called ahead; he should be expecting me."
The girl got up from her seat and shook hands with Clark and Lisa. "I'm Tara Patel. Mayank's this way."
The girl went through a pair of swinging doors, the type hospital staffers open with the end of a gurney, and led Clark and Lisa down a hall lined with shelves of electronic equipment and boxes. She turned left at the end of the hall, went through another set of doors, and the eight-foot ceiling of the passageway opened to thirty feet. A miniature car with a blinking light on top zoomed by on the floor in front of Tara's feet.
"Welcome to the most expensive playroom on the East Coast. Wait here and I'll find Mayank."
A remote control helicopter came into view and hovered in the air a few yards in front of Clark. A small flash of light emitted from the helicopter before it disappeared into the far corner of the room.
Tara returned and passed Clark and Lisa on her way to the door. "He's on his way," she said, smiling as she walked through the double swinging doors.
As promised, Mayank arrived a moment later with a remote control device in his hand, the antenna extended.
"Clark, good to see you again."
"Thanks for seeing me."
"I'm wondering if I should pat you down or blindfold you. For all I know you are here on a university robotics intelligence mission."
"I wish I were."
Clark introduced Mayank to Lisa.
"Did Clark tell you we met in Tokyo? We competed with each other in the World Robotics Championship."
"Yes, he mentioned it. It's some place you have here."
A yellow flag at the end of the antenna on the remote control had a number written on it. Lisa asked "What's the number for?"
"Frequency," Clark answered. "You can't have different toys running on the same frequency. It makes control a little difficult."
"Bad things happen," Mayank said, elaborating with his free hand slamming into the side of the controller to indicate a crash. He put the remote control on a chest-high table near the door. "Let's go somewhere we can talk."
Clark and Lisa followed Mayank back to an office with three glass walls and no windows.
"It is quite a building," Lisa added.
"We are part of CSAIL, the Computer Science and Artificial Intelligence Laboratory. We have a bit to work with."
"That would be an understatement."
"So, Clark, what's going on? You said on the phone you were in Boston and it was important."
Clark gave the Reader's Digest condensed version of the story. He started with the missing family who claimed to have returned home suddenly, touched on two dead neighbors in the last several weeks, a house fire, a car mechanic in search of a new employee to replace his dead one, a visit from the FBI after his mom called the CIA, and the beans of death he pulled off a farm in Virginia.
Mayank Malhotra looked at Clark with a little concern. "And the IRS is involved, how?" he asked, looking at Lisa. Lisa smiled and Mayank felt a little blood flow south.
"She's investigating my parents for tax evasion."
" _Was_ investigating," Lisa clarified.
"Right, _was_ investigating."
"Now we are seeing each other."
"Given those two alternatives, I would say you have chosen wisely," Mayank answered.
"She's helping me poke around a little," Clark added, trying to ignore the joke he left hanging out there, over the plate like a slow pitch softball.
"And you think your neighbor is an MIT graduate, why?"
"Paul Cannon," Clark answered.
"The NASA scientist?"
"The same."
"I don't see the connection."
"Neither would have I," Clark started. "But we think this woman, my neighbor, stole the identity of a Wellesley student in 1998. The student went missing out near the University of Massachusetts. Her car was found burned out. Her body was never found."
"Just like Paul Cannon."
"Exactly."
"Sounds like a hunch."
"It's more than a hunch, it's the culmination of a series of hunches. A cacophony of hunches, bells, whistles, and sirens that have been going off in my head for the last month. All I need is to confirm something."
"The girl's identity," Lisa chimed in.
"And then what?"
"Then I'll find her."
"How are you going to do that?"
"I don't know just yet."
"Maybe I've seen too many movies, but if you have two dead neighbors, maybe you won't need to find her. Maybe she will find you."
"Let's see what we've got," Mayank said, turning towards his computer. "We have any guesses on the girl's name?"
"No."
"Then how do we know we are looking for the right person?"
"We don't know her name, but we believe she is Pakistani. We think she graduated sometime between 1996 and 2000."
"That's not much to go on."
"How about Paul Cannon? Can you tell us when he graduated?"
"You really think they're connected?"
"It's my theory."
"I hope you're wrong. MIT is a small community. And one that is very well-connected through research, alumni, internships, email, websites, blogs, etc. I hope it's nothing insidious between our MIT brethren."
"I read that Paul Cannon graduated from MIT in 1998. Maybe we can start with that year." Clark pulled out the hand-drawn picture of Ariana Amin done by Mr. Stanley. "And this is a rough picture of what she looks like."
"Did you do that?"
"No, why?"
"Looks more like a portrait than a police sketch."
"I'll let the artist know."
Mayank turned his attention back to the computer. "This database has every graduate from the university since its inception," Mayank explained as he typed, an action he augmented with the occasional click of the mouse.
"Of course the data from the old days is just a list of vitals: name, permanent address, grades. It wasn't until 1996 that we really started doing cool things with the student data."
"The advent of the Internet," Clark said.
"That's right. 1996 was the advent of the Web browser anyway. Of course, MIT had the internet from before Al Gore. But with the internet, came a formal university intranet, or rather an institutional wide interest in its intranet. And with that, the underground programmers came to the surface and everyone's job got a little easier around here."
Mayank continued to explain. "We can search by name, social security number, student ID number, year of graduation, field of study."
"Can you search by sex?"
"You know... I don't know." Mayank pecked around for a minute. "Not a searchable criterion. But women make up less than ten percent of the student body. In 1998, the number was even smaller. It shouldn't take long."
"It has taken forever to get this far," Clark said under his breath.
Mayank Malhotra typed into the keyboard and then clicked with his mouse. "Here we are. Paul Cannon, graduated in 1998 with a Ph.D. in alternate energy propulsion."
"Can we search for students in that program and related programs?"
"Let's start with related programs. That sounds like a specialty Ph.D., even by MIT standards." Mayank clicked a few more boxes and a list of names appeared on the screen in blue.
"Do any of them look familiar?"
All three looked down the list. Names from every corner of the globe, and a few from off the radar, populated the screen. Taiwan, Bangladesh, Moldova, Australia, Uruguay, Germany. It was the cream of the crop, the topping on the best intellectual sundae the world had to offer. A global alliance of brainpower.
"Try the third one from the bottom," Clark said.
"Not sure if that is a man or a woman," Mayank said. "Could be either." He clicked on the link and a picture popped into the upper left hand corner of the screen. The man with the turban and moustache was neither the right sex nor the right religion.
Two more clicks, two more misses. Mayank scrolled down and read the next name on the list.
"Safia Hafeez. Definitely a woman's name."
Mayank clicked the link and a picture appeared on the screen. No one spoke. Clark and Lisa leaned over Mayank's shoulder, getting closer to the screen.
"Is that her?" Lisa asked.
All three looked at the picture on the screen and the hand drawing on the desk.
Clark leaned toward the monitor. "Hello, Safia."
"Are you sure?" Lisa asked.
"Yeah, that's her. I've never seen her without a Hijab. She looks good."
Mayank read the bio on the bottom of the screen. "Fulbright Scholar. Studied applied liquid propulsion. A second Ph.D. in Chemical Engineering. Master's in Electrical Engineering."
"Fuck. Nice combination," Clark said to himself out loud, his voice trailing off. Mayank looked up as if that word had not been spoken in the hallow halls of MIT since the sixties and the peace generation.
"Let me translate for my partner here," Lisa said calmly. "Any chance we can talk to one of her professors?"
"I can call around and ask. Class is still out of session, but there are a few professors around. Particularly in the science department. They have experiments and studies that go pretty much year-round. We have a dedicated staff and dedicated students who support that staff."
Clark suddenly felt like he was being recruited. "Can we try to reach one of the dedicated? We need to find this Safia Hafeez."
"Let me get my coat and we can walk over to the main chemistry building."
Professor Mike Ching was in his office with his soft-sided leather briefcase on his shoulder and his office keys in his hand when Mayank knocked on the door.
"Professor Ching?"
"Yes."
"Do you have a minute?"
"I was on my way out the door."
Lisa Prescott pulled out her badge and showed it to the professor. "It's important."
"That's an IRS badge."
"And I said it was important."
"I paid my taxes."
It was apparent that Professor Ching abided to the common misconception that the person asking the question was somehow inferior to the person being asked the question. Lisa was about to change that.
"Well, if you don't want to be audited for the next twenty years, give us a minute of your time."
Mayank fidgeted. Grilling a professor wasn't on his list of semester projects.
The professor put his briefcase on his desk. "Please sit down."
The office was decorated à la Fred Sanford meets Marie Curie. Papers and books filled every corner of the room. A mobile hung from the ceiling in the corner, protons, neutrons and electrons in some configuration that would baffle the average Joe, but was covered in MIT freshman chemistry. A 3-D periodical chart with all the elements was attached to the wall in the corner.
"What can I help you with?" Dr. Michael Ching asked in perfect English. Clark plopped his butt in an old wooden chair. He looked up at Mayank who was eavesdropping, still standing at the door.
Lisa took control of the conversation. She turned towards Mayank.
"Mayank, thanks for your help. If we could excuse ourselves for the moment, we would like to keep this conversation close to our vest, if you don't mind the gambler's parlance."
"Oh, sure. I understand. Let me know if there is anything else you need."
Clark tried to soften the blow. "Yes, there's something you could do. Could we get any addresses she listed, as well as a list of classes, the professors, and whoever you can identify as classmates?"
"I'll have it by the time you're done. But I want half of any reward."
"Deal," Clark answered, before turning towards Professor Ching.
"What can you tell us about Safia Hafeez?"
"Safia Hafeez. That's a name I haven't heard in a while. Why are you interested in her?"
"We're interested in her as a material witness. We think she's been living under an assumed identity for several years. We'd like to know whatever you can tell us."
Dr. Ching sighed and ran both hands through his hair, one on each side of his head. His eyes settled on Clark.
Clark preempted the question he saw coming. "I'm just here for identification purposes. As a material witness, if you will."
"A material witness for a material witness," Dr. Ching said curiously. His black hair had streaks of gray, but his face was youthful. His eyes were penetrating. He paused briefly as if considering speaking, and then moved his bag to the side of his desk and leaned back slightly in his chair.
"Safia Hafeez was a gifted student."
"Aren't most of the MIT students gifted?"
"Sure. But I'm speaking in relative terms. From a natural perspective, Safia Hafeez had a gift. There are all kinds of protégés in this world. Mathematical, musical, artistic. Things that cannot be explained by anything other than Godly intervention. Or, if you are atheist, we can call it dumb luck. There is nothing that can explain a seven-year-old boy who walks by a piano for the first time, sits down, and plays a perfect stanza to Beethoven's fifth symphony. It just doesn't happen. And when it does, everyone tries to capture the magic in a bottle, as if it is something we can keep, analyze, harness, replicate."
"And Safia Hafeez was like this? She was a protégé?"
"She was unlike anyone I have taught at this university. She had an electrical engineering background. I got the impression she had been around engineers growing up. She had an understanding of electricity, mechanics, and chemicals that was unmatched. And it was seemingly untaught, or maybe self-taught. When I lecture, most students try to keep up. They take what I tell them, digest it, and try to apply things I teach in the classroom to experiments in the lab. Not Safia. When I lectured, she took random notes, and the rest she committed to memory. She drew her own conclusions about compounds and what would work best under circumstances of pressure, friction, heat. On the surface, it may sound very easy. When you talk about combining this knowledge to send a man to the moon, it becomes a little more complicated. Safia could talk about chemistry, biochemistry, physics, electrical engineering, as easily as you and I would talk about the weather. And that was _before_ she got into the program."
"What exactly did she work on?"
"Chemical engineering and propulsion systems."
Clark shook his head slowly, unconscious he was doing it.
"What's the problem?"
"We think Safia may have nefarious intentions against the general public," Lisa said.
Professor Ching's face turned pasty. "That's not good news."
"Why do you say that?" Clark asked.
"Because if this woman wanted to start trouble, she has the expertise to take out a thousand people with the household chemicals in the average broom closet."
Clark and Lisa left the professor at his desk with a business card in his fingers. Dr. Ching looked out the window, staring at the heavier flakes of snow that had started according to the weatherman's forecast. _Safia Hafeez_ , he whispered to himself. _God I hope they are wrong_.
Clark and Lisa walked briskly down the hallway, as if distancing themselves from where they were would bring them closer to where they needed to be. Neither knew where that was.
Clark spoke as they hit the stairs. "Looks like we have one potentially dangerous bitch loose on the streets."
"You know what scares me?"
"What?"
"She has a twelve-year head start," Lisa said. "We may never catch up. We may never see her again."
"I hope you're right. I hope we never see her."
## Chapter 51
Most major cities have a street named International Drive or Road or Court, but there are few that live up to their billing like the one in D.C. Wedged into a horseshoe-shaped strip of pavement in northwest Washington, beyond the glamour of the major embassies, International Drive housed embassies from Israel to Bahrain to Ghana to Ethiopia. They weren't the most distinguished embassies in the capital city's repertoire, that title long since claimed by the G-8 member nations and their prime real estate on what was affectionately known as Embassy Row. But where Embassy Row was sprinkled with old-money residences, International Drive was an embassy monopoly. It was void of local riff-raff, even those in Bentleys.
With over a dozen embassies packed into a quarter-mile of looped blacktop, the embassy protection detail on International Drive was a walk in the park for the myriad diplomatic security personnel. With the exception of the Israel Embassy, which tended to focus on its Mossad contingency on anything that moved with intelligence value, the other embassies on International Drive had a certain level of shared security. Given the close proximity of so many embassies, a bomb for one was a bomb for all.
"It's quiet in here," Clark said.
Mr. Khan, Counselor of Community Affairs for the Pakistan Embassy, let the silence shower the room for a moment, as if Clark had laid a new thought out for inspection. "Yes, it is rather quiet. But then again, embassies are not what you see in the movies. Most of them are quiet, serious places. We try to represent the best of our respective countries. Yelling, screaming, fighting... well, these things are not useful in situations requiring diplomatic solutions."
"True. True. I guess the movies do give embassies a bit more flare than they really have."
"Hollywood is not in the honesty business."
Clark feigned a look of concern. "I'm not sure who is in the honesty business these days. In fact, I'm not sure there's an honesty business at all."
The counselor looked at Clark suspiciously. "So, Mr. Hayden. I understand that you have a few questions about the Fulbright scholarship."
"Yes, sort of. I had several questions I was hoping you could answer for me. The Fulbright scholarship is just one aspect."
"I think perhaps your own government may be in a better position to answer your question."
"My question was more geared to Fulbright scholars from Pakistan."
The counselor sighed. "I see, well, as you may or may not know, Fulbright scholarships are awarded to the brightest and the best of the academic world. Fulbright scholars are students who we believe will lead a generation of leaders in the area of law, medicine, academia."
Clark noticed that Counselor Khan's words were crisp, clean, exact... and with a hint of a British accent.
"Your English is impeccable. If I may say so without sounding condescending."
"I was educated in Britain."
"That would explain it," Clark added. He was running out of things to compliment the counselor on. He had started with the picture on the wall when they entered the room, had moved to office furniture, and was now on the counselor's language skills. The local curry shop in Shirlington was next on the list. He dug for something better. "And the Fulbright scholarship is sponsored through the U.S. Department of State..."
"Correct. In the case of Pakistan there is additional funding through the U.S. Agency for International Development and Pakistan's Higher Education Commission."
"How many scholarships are awarded to Pakistanis every year?"
"The number is fluid, dependent upon funding approved by the presidential approval board."
"The U.S. President."
"Correct. The President of the United States."
"So Fulbright scholars are funded by the U.S. Government and the selection of the scholars is done by a President-appointed board?"
"Correct. And to answer your previous question, Pakistan had over 150 Fulbright scholars last year. One-third of those were Ph.D. students."
_Wow, only 150 potential U.S. government-funded terrorists in training,_ Clark thought. _Beautiful_.
"And what are the requirements of the Fulbright scholar? What's the overall purpose of the program? What does the U.S. get out of it?"
"It can be argued that the main goal of the Fulbright program is to influence the future leaders of foreign nations. Once a student has lived in the U.S., and tasted the democracy of the U.S., they are far more likely to be U.S. sympathizers. They will never look at their home country in the same way as before. At the very least, they will have a more global view of the world."
"Perhaps," Clark said. He thought his neighbor might not be on board with that assessment.
"Another stipulation of the Fulbright program is that each scholar must promise to return to their home country for a given period of two to three years, depending on the scholarship they receive. They are not permitted to work in the U.S. until this home country requirement has been fulfilled. This is necessary to ensure that the home country receives some benefit from the program as well. Otherwise, in many instances, the brain drain would be instantaneous. Under the current guidelines, the home country gets at least two years of service from the scholar."
"And what if they don't go home?"
"They have to re-pay their education expenses."
"How many choose this route?"
Counselor Khan laughed. "Not many. The average salary in Pakistan is under three thousand dollars per year. Very few could afford to pay the cost of tuition, particularly at a private U.S. school. Pakistan is still a very poor country. We have a less than a fifty percent literacy rate."
"Interesting," Clark responded before dropping his bomb. "I'm here on a strange request, or quest, as it has turned out. One of my neighbors is an American citizen, or possesses dual citizenship. I'm interested in a 'Welfare and Whereabouts' request."
"A 'Welfare and Whereabouts request?'"
"Yes. I have an acquaintance in the consular services world and he told me that would be the appropriate wording. My neighbor, and her family, recently returned to Pakistan to tend to family affairs. I received a phone call from them asking if I would look after their home, water their plants. The usual. Their house was recently heavily damaged by fire, and I have no way to reach them. I was wondering if the Pakistan Embassy could help in my search."
The counselor looked thoughtfully at Clark. "If they are American citizens in Pakistan, it may be better to go through the U.S. Embassy in Islamabad."
"I'm afraid I don't have much to go on, in this case. There is no record of the neighbor having any residence or relatives in Pakistan. I only have a name."
"Which is?"
Clark watched carefully for any recognition in the counselor's face. "Her name is Ariana Amin." If the counselor recognized the name he had missed his calling on the World Poker Tour. There was not a flinch. Not a twitch. Not a momentary flash of recognition. Clark dealt the river card. "But I believe her real name is Safia Hafeez. She was a Fulbright scholar, sponsored by the Pakistani government, and she studied at MIT."
The counselor's expression did not change. "I'm afraid the name doesn't ring a bell."
Clark sat there for moment. "Would it be possible to search your records and see if you can locate a last known address or contact for my neighbor? I would think that she would want to know that everything she owns in this country has been either charred beyond recognition or soaked in 20,000 gallons of water."
The counselor pulled out a piece of paper and scribbled some notes. "I will see what I can do," he said, pausing in between sentences as he wrote. "But there are no guarantees that I will be able to find anything."
"All I can ask is that you take a look."
"That is a fair request."
As Clark exited the Embassy, the large black metal gate to the adjacent parking lot was slowly shutting behind an equally black Lexus sedan. The car gently pulled into the spot closest to the building's employee entrance, just beyond a sidewalk and behind an eight-foot security fence. A guard decked out with the latest military hardware eyed Clark as he made his way from the visitor's entrance to the main gate booth armed with two identical soldiers. Clark took in his surroundings as he mulled over the conversation he just had inside the Embassy. _Maybe I was wrong,_ he thought. _Maybe, I am just jumping to conclusions._
Clark nodded as he walked the sidewalk towards the main guard booth and the property's exit. As he turned towards the direction of the main street, the back door on the black Lexus opened. Clark was less than ten yards away when the precocious two-year-old exited from the back of the car, waved her lollipop, and smiled.
Clark stopped.
"Hi," Liana said, clearly, her hand gripping her red candy-on-a-stick.
The woman exiting the back of the car panicked, exchanging concerned glances between Clark and Liana.
"Wait!" Clark yelled as the woman quickly scooped up Liana and disappeared into a pair of plain white double doors. Clark, heart racing, took one step off the designated sidewalk in the direction of the security fence and was immediately reprimanded by the guard who was now on full alert. The AK-47 was still on his shoulder, but the eyes and body language told Clark all he needed to know.
"Not one more step," the guard said clearly, pointing down at the sidewalk, the official line of demarcation.
"But, I know that girl," Clark blurted.
"Step back," the guard repeated, one hand now on the grip of his holstered firearm.
Clark took a large step back. The guard put his finger to his ear, listened for directions via the wireless security device, and then looked back at Clark. "It is time for you to leave," the guard said, unequivocally.
"But, I need to speak with Counselor Khan."
"No sir, you do not. You are to leave the premises immediately."
Clark drove home, hands shaking. He was scared. The kind of scared that kept therapists in business. There was only one possible explanation for seeing Liana at the Pakistan Embassy. His instincts had been correct. Counselor Khan, with his perfected poker face, was lying through his teeth. There was an undeniable connection between Ariana and the Embassy. Clark wondered how deep it went.
His stomach knotted and a wave of nausea washed over him. There was only one thing he could think of. Finding a safe location for his mother... if such as thing as a safe location existed anymore. When you consider state-sponsored terrorism, hiding places become scarce.
It took a majority of the afternoon to convince his mother it was in her best interest to go to her brother's house in Annapolis. The conversation was like rationalizing with a toddler who was in the "why?" phase of childhood. And unfortunately for Clark, Maria Hayden was past the point where she could simply be lifted off her feet and strapped into a car seat.
Clark pressed down on the suitcase until the latches were aligned. With a final grunt and additional ass weight, the old American Tourister finally complied.
"Did you get my medicine?" Maria asked for the fifth time.
"I have all your medication, Mom. It is in the backpack. I have your shampoo, your make-up and your hair dryer. I packed five pairs of shoes and enough clothes for a month-long trip around the world. We are good to go."
"And why am I going to Annapolis again?"
Clark had given the truthful answer multiple times, but his mother, on this day at least, was not absorbing the facts. Clark finally succumbed to a brief explanation. "Because there are some things I need to do, Mom."
"But I've been living alone for the last year. I don't need you to take care of me."
"You're going to have to trust me on this."
"I trust you. I trust you," Maria Hayden said as Clark dragged the third suitcase towards the door.
"I have another question," Maria said, sitting down on the sofa as if she were unaware of the impending departure.
"What, Mom?"
"Why am I going to Annapolis and did you pack my medicine?"
## Chapter 52
Saturday night brought in half of the Kabob Keeper's weekly business, but Tuesday night brought in the weekly quota of weirdoes from the local design school. Nestled between the backside of the _Crystal City Restaurant_ , a strip joint with a family-friendly name, and a 7-11 that hadn't shut its doors since 1969, the _Kabob Keeper_ took up residence among a set of shops that had once been under the single roof of a carpet store. The parking lot was tight, the spaces full, and the smell of puke and urine from the back door of take-it-off central tended to waft over during the summer.
Ariana parked her blue Toyota at a space with a broken meter on 23rd street, a few hundred yards from the Kabob Keeper, and walked once around the block. The pulsating rhythm of pole-dancing tunes could be heard in the parking lot, an old chair with a sordid history left to prop open the back fire door. The 7-11 parking lot had surveillance cameras, and although she could never avoid all the cameras in a city with eyes on every corner, she wasn't above avoiding the ones she knew about. But time was short. And, if things went well it wouldn't matter how many cameras had captured her image. Dressed as an Americanized woman, complete with Levis and a sweater, Ariana felt like a foreigner as she entered the restaurant. She knew every food by sight, could taste the dishes with her eyes, knew the ingredients by the smell. Her mind drifted back to her youth: carrying bowls of sevian, nehari, mango, and milk tea to her grandparents across the street in her native homeland.
A dropped dish brought her back to the restaurant. She felt a tension that heightened her awareness. The Kebob Keeper may have been named to appeal to the masses — the office workers, young professionals, and college kids who wolfed down the curries and kebobs, but the eyes of the workers told Ariana something else.
Ariana nodded as she approached the counter and the elderly man in a sweaty white t-shirt nodded in return before turning his back and retrieving a loaf of nan from the inside of the cylindrical clay oven.
"Excuse me," Ariana said over the din of orders being relayed. "I'm looking for Khalid." She pulled out the handwritten note she had received from the imam and moved it gently between her thumb and forefinger.
The man put the nan bread on a tray with the long thin metal rod that he wielded effortlessly. He looked at the group of design students, all with different color hair, waiting at the far end of the counter for their food. His focus on the waiting customers drew Ariana's attention to the other end of the counter, and the man grabbed the note from Ariana's hand. It was a professional move, Ariana noted, and one she shouldn't have fallen for.
The old man unfolded the note as he turned to face the soda dispensers. He finished reading and looked over his shoulder at the clock on the wall. "You are right on time," he said in Urdu.
With the nod of his head he motioned her behind the counter.
"The imam only sent three," the old man in the t-shirt said in a whisper as they approached the office in the back of the store. The heat from the kitchen, initially a welcome from the cold, became more oppressive near the rear of the restaurant. Ariana understood why the old man's shirt was clinging to his body. The four-man kitchen staff moved swiftly through the heat, perspiration on their faces. The clank of metal pans was interrupted by a ringing bell indicating that an order was ready.
Through the stainless steel shelves, Ariana felt eyes watching her every move.
The old man pushed the office door open and the three volunteers sat up in their chairs. They were seated at a round table. Used Styrofoam plates balanced in a stack in the small wastecan near the desk. The smell of lamb filled the room, hanging thick in the air. Ariana was suddenly hungry. Freeze-dried meals had made great improvements over the years but they couldn't compete with handcooked meals. Even those made in quantity.
Ariana nodded to her host and the old man shut the door as he left. Three young faces looked up at her. They all had similar features. Pointed noses, narrow chins.
"Brothers?" Ariana asked after a moment of silence.
"Yes," the oldest of the three answered. "My name is Farooq. We have met."
Ariana recognized the young man's wispy sideburns and peach fuzz, intermingled with the random long whiskers. "You are the imam's understudy."
"Yes," Farooq answered.
"How old are you?"
"Twenty-one. My brothers are eighteen. They are twins."
"Fraternal..."
"Yes, fraternal."
The pudgier of the two brothers dipped his head and spoke. "My name is Jameel." He was wearing a tattered New York Giants jacket and a pair of sneakers that he tapped on the floor in a smooth steady beat.
The slender twin looked up with naturally wide brown eyes. "My name is Omar."
Ariana nodded. "You know why you are here?"
"Yes," Farooq answered.
"I need an answer from all of you."
The twins both raised their voices and replied, "Yes."
"Farooq, Jameel, and Omar," Ariana said rhetorically, as if measuring their names and the suitability to the task. The brothers looked at each other and smiled as if they had been recognized for greatness.
"So everyone knows why they are here. If anyone has cold feet, they need to leave now."
Ariana waited for a response. No one moved.
"I'm glad you have chosen to stay the course. I would not have enjoyed killing all of you in this room. Three brothers would have been messy."
Fear joined the smell of lamb in the air.
"Here's the deal. It's very simple. I will pay each of you $10,000 dollars to help me for the next forty-eight hours. If you agree, however, you will do exactly as I say, exactly when I say it. There will be no negotiations. No complaining. No questions. Unless the questions are operational in nature."
Jameel looked at his older brothers before speaking. "How do we know you have the money? Or that we would live to see it?"
From her purse Ariana pulled a ten thousand dollar stack she had received from the imam. She put the money on the table and the eyes of the younger brothers stared, their eyelids refusing to blink. "I have the money. Living to see it, well, I have no guarantees. That is in Allah's hands. Tell me who should receive it in the case of your death, and I will see to it that they do."
Farooq looked at his twin brothers and shook his head towards them. "We are here for our father and Allah, not for the money."
Jameel and Omar nodded in agreement.
"Very well then," Ariana said. "You have passed the second test. I am looking for men of conviction, not greed."
The young men stared ahead stoically until Ariana asked, "Where is your father?"
"Dead," Omar answered from beneath his wide eyes.
Ariana paced back and forth slowly, her eyes moving deliberately from one recruit to the other. "This is your last chance to run," she added, moving her body to mockingly create an opening towards the door.
She was the only one smiling.
Forty-eight hours. The countdown was on.
## Chapter 53
Clark went through the daily delivery from Mel, stacking and sorting bills and junk mail into separate piles. The thought of sitting down and writing checks, stuffing them into envelopes, getting stamps, and having the postman pick them up was almost comical. He hadn't written a check in over a year. The younger generation had found the Internet and electronic bill payment. And they were not coming back to the "write it, stuff it, lick it, and send it" world.
Clark got to the last letter and his heart sank a little. Neatly printed on the business envelope was the name of his father's company, Hayden Ltd. He ran his finger down the side of the envelope and began reading at the top of the sheet near the word "invoice." There were half a dozen line-items, three with prices in the BMW, low-end Mercedes neighborhood. Clark scanned the dates, looked up at the name on the invoice again, and reached for the phone.
He spent five minutes navigating the hide-the-real-person voice commands of the company's main customer service number. His seventh correct selection came with a reward. "Hunter Scientific," the female voice on the other end of the phone answered with uncommon morning pleasantness.
"Good morning. My name is Clark Hayden, and I need to speak with someone in the billing department."
"This is the billing department."
"Thank God."
"I get that response a lot," the voice said. "How can I help you?"
"I'm looking at an invoice for my father's company and I think there's a mistake."
"There could be. Let me pull it up for you. What's the account number?"
Clark found the account number at the top of the invoice and read it slowly to the woman. As they waited for the information to pull it, the woman gave Clark her speech for the week.
"We have had an unusual number of billing errors this month. One of the ice storms last month took down some trees nearby which hit a transformer. We got a surge of electricity to our computer system here. Caused us to lose more than a few records. It has been a big mess. We have been going through files manually. A lot of end of the year documents are going out. The early bird taxpayers are calling us non-stop for documents."
A one syllable chuckle escaped Clark's mouth. "You think you've had a bad month, let me tell you about mine. I have had run-ins with the IRS, the FBI, and the police. Two of my neighbors have died and another's house burned down." Clark stopped himself before he labeled his missing neighbor a terrorist with connections to the Pakistani government.
"You win," the woman said. There was a pause and the woman segued into a work conversation. "My computer is back up... Now what exactly were you looking for?"
"I'm looking at what I assume is the latest invoice for my father's company. My question is how could my father's company receive an invoice for machines that were delivered a few weeks ago if my father has been dead for over a year and his company only had one employee — my father?"
There was a long silence on the phone. Clark tried to resuscitate the conversation. "Helloooo..."
"That _is_ a good question, indeed," the woman answered. Through the phone Clark could hear the keystrokes being pounded frantically.
"And when was this account closed?"
"I don't know exactly. The summer before last. Eighteen months ago maybe."
"That is interesting," the woman answered, sounding concerned. "My system shows that your father's account was never closed. In fact, it has remained active for the past year and a half with ongoing activity."
The bile in Clark's stomach churned and almost made an unexpected appearance. "Exactly what was ordered?"
The woman read from the list on the screen. "Multiple orders of aluminum, magnesium, titanium. Cylinders. Sheets. Blocks. Total weight over two hundred pounds. There was a separate order for bolts, servos, connectors, sixty feet of wire. Twenty feet of yellow, red, blue. I have a record of seven shipments over the last eighteen months."
"Shipped to this address?"
"Which address is that?"
"The one where I received the invoice. 203 Dorchester Lane, Arlington, Virginia."
"No. That's listed as the previous address of Hayden Ltd. That invoice was probably processed manually and they used the old address. These seven orders were shipped to 9345 Georgia Avenue, Northwest. Warehouse C, Washington, D.C."
"Wait, wait," Clark said, frazzled. "Let me get a pencil."
"I'll hold," the woman in billing said as if she were doing Clark a favor.
Clark ran, his wool socks slipping on the floor as he turned the corner in the kitchen. He stuck his hand into the junk drawer and pulled out a pen. He swiped the Post-it notes off the counter, his mother's guide to her daily routine, and sat down at the table. His hand shook as he fumbled with the top of the pen.
"Go ahead."
Clark scribbled the address on the paper. "Could you fax me the list of shipments? I don't have a fax, but I can give you a number where I can pick it up."
"Not a problem. But you know, the invoices have all been paid in full. There's nothing to worry about."
"I wouldn't count on it."
Lisa knocked on the door once, then entered the house panting. Her face was flush from the cold and the sudden burst of exercise.
Clark got up from the round dining room table which was littered with papers and files. His hair was unkempt, but the bounce in his step had returned. Lisa was glad to see that the energetic guy she had first met had reappeared, evicting the slow moving replica that had taken over her boyfriend with increasing frequency.
"You look alive."
"Amazing, because I haven't slept in a week," Clark said.
Lisa placed some folders on the coffee table. She unzipped her coat and put it over the arm of the sofa. She reached out and gave him a hug. Clark buried his nose in her neck for a moment and then put a kiss where his nose had been. He inhaled her perfume, a vast improvement over his non-showered body.
"I brought my video camera and the fax. Though I have no idea what you are up to. I also found some information that may be helpful," she said, patting the seat next to her on the couch.
"Helpful is something that has been lacking until this morning. Or maybe luck would be a better word."
"What's with the fax?"
"It looks like my helpful neighbor has been helping herself to my father's company, using it as a means to order whatever it is that she was after. I've been contacting different companies for the past hour and she has been busy."
"Why would she do that? If she had connections at the embassy, she could get anything she wanted."
"Maybe she was working on something unsanctioned."
"Maybe. But I'm not so sure terrorism is that discerning. I did a little poking around after you mentioned seeing Liana at the embassy. I ran a check on Ariana's and Nazim's credit cards."
"Smart girl. Still within the domain of the IRS."
"Except that I'm not investigating your neighbors."
"So you fat-fingered a key and got the wrong address. The same thing happened to me."
"And then I continued my way through the wrong file?"
"It happens," Clark said sarcastically.
Lisa shook her head and tried not to smile. "Anyhow, there's no record of Ariana purchasing a plane ticket on any of her credit cards. In fact, there are no charges at all on any of her cards since the first week of January. No withdrawals from the bank either."
"Meaning she had cash, which comes as no surprise." Clark thought about his next statement. "You know, there is a possibility that my parents' problems with the IRS may have been orchestrated by Ariana. The timeline fits."
Lisa put on her auditor's cap. "I'm not sure what the motive would be, but nothing would be a surprise at this point."
"That's good, because I'm all out of my befuddled look." Clark examined the papers on the table. "Anyway, I have been going through my father's files. All the blueprints he has. All the material he ever ordered. I took inventory of the material in the garage. Whatever Ariana was after, she hid her tracks. But the address I was given may have the answers."
"I think the combination of materials and machines being ordered, and the fact that there is ricin involved, maybe you should just eat some crow and call the FBI back. Just to be sure."
"I will, but first I need to borrow your video camera."
## Chapter 54
Abu sat slumped in the desk chair in the office. Ariana watched him from her seat across the room. A small metal green wastebasket was between his legs on the floor. His breathing was labored. He hunched over, coughed, spit blood into the trashcan, and then tried to straighten himself, pushing on the arms of the chair.
Karim motioned Ariana to the door of the office. As she reached the doorway, Karim whispered, "You think he will make it?"
"No. I would guess he has less than twenty-four hours."
"Maybe we should keep him in here. The new recruits don't need to see what is waiting for them. They are young; they may not have the needed conviction once they see the outcome."
"I can solve that."
"I trust that you can," Karim said.
"Get the troops ready. I'll be a moment."
"What are you doing?"
Ariana raised her voice. "You want to sit in?" she asked Abu.
"I did all the work," Abu said in a surprisingly strong voice.
Ariana nodded.
Across the warehouse floor, Karim poked his head in the crowded sleeping quarters. "Make room. Feet off the floor."
The three new recruits and Syed positioned themselves on the four cots in the sleeping quarters. Ariana pushed Abu, seated in the wheeled office chair, to the doorway of the sleeping quarters. She slid past the chair in the doorway, sat on the cot next to Karim, and unfurled the map on the floor.
The brothers glanced at Abu, who smiled weakly.
"I assume everyone here, with the exception of Syed, is familiar with the D.C. Metro?"
The younger brothers, Jameel and Omar nodded. Farooq grunted as if the question were preposterous.
Ariana paused and singled out Syed. "You are the only one who has not ridden the Metro before, so try not to get on the wrong train. You will all be given Metro cards with sufficient funds on them. You will not need to use the vending machines. Simply run the cards through the slot in the turnstile, arrow end first, face up."
Ariana pointed towards the floor. "This is the map of the Metro system."
"There are five lines. Green, Yellow, Blue, Red, and Orange." She paused. "Is anyone here color blind?"
"Color blind?" Farooq asked.
"Yes, color blind. It affects nearly one percent of the population. Problems distinguishing between red and green run at a higher rate. Males are more commonly affected than females."
Heads moved back and forth.
"Good. Then I won't have to worry about anyone getting lost for that reason. There are around eighty stations in the entire system. Over half a million riders per day."
"We will be focused on the Red, Blue, and Orange lines only. The Yellow line is a non-central line and the Green line runs through some of the rougher areas of town, which presents several potential problems."
"Such as?" Syed asked.
"One, a well-dressed Middle Eastern man with a suitcase is likely to either stand out, or get mugged. Neither of which would be good."
"I think I can handle a mugger," Syed responded, running a finger across his throat.
"Yes, perhaps you could." She looked at the younger brothers, two eighteen year olds with more faith and conviction in their eyes than muscle on their bones. "But not all of us are trained."
"Let her speak," Karim said.
"As I said, we are focusing on three lines. The Red, Blue, and Orange. There is only one spot where these three lines intersect: Metro Center. The name should be easy to remember. We will have three teams: Team Red, Team Blue, and Team Orange. I assume everyone can figure out the naming scheme."
Omar, the brother with oversized eyes, chuckled.
"The Blue and Orange lines share the same track and the same platform at Metro Center. The platform between the two lower tracks is where a majority of the targets will be hit. Team Orange team will be arriving from one end of the station and Team Blue will be arriving on the same tracks from the opposite direction.
"The Red line intersects the Orange and Blue lines on tracks above the lower platform. This should make for good fall-out and maximize our kill potential. Syed, the D.C. subway is not like New York. The stations are cavernous, the platforms intersect on bridges. There are minimal tunnels. All things that work in our favor.
"There will be five of you. Each of you will be carrying two bags apiece. Three of you will be carrying suitcases, dressed like businessmen, with tickets to the airport or train station. The other two, Jameel and Omar, will be dressed like students. Duffle bags and backpacks. For those keeping up, we have two profiles: the students and the businessmen. We also have three teams: Red, Blue, and Orange. Does everyone understand?"
All heads nodded.
"I want the twin brothers on the Orange line. Team Orange. Students. Both of you will be going west. One of you will be dressed as a University of Maryland student. The other will be a student at American University. I want one of you on the second car from the front. The other will board the train on the second car from the rear. The appropriate clothes will be provided. You will look like students. Well-groomed, clean-cut students. Karim has all the pocket litter for you as well."
"Pocket litter?" the thinner twin, Omar, asked. In the light from the lone bulb in the sleeping quarters his face seemed oddly elongated, like a stretched balloon, almost caricaturized.
"Pocket litter is papers and I.D. that makes you look legitimate. For this operation they are probably unnecessary, but we will consider it a precaution."
Ariana pointed at Syed. "You are on the Blue line. Businessman. Team Blue. You will be going in the direction of Reagan Airport. I want you to ride in the middle car. It will likely be crowded, so force your way in if you have to. Between you and the twins, you will cover a majority of the lower platform.
"Karim, you and Farooq are on the Red line. Also businessmen. Team Red. Karim, you will be arriving from the south or east, depending on how you look at the map. Farooq, you will be arriving from the north. Both of you aim for the center car. When you exit the train, you will be on opposite sides of the tracks, with the Orange and Blue line platforms beneath you." Karim looked at the eldest brother and their eyes locked in anticipation. Then they both smiled.
Ariana pointed at Farooq. "When was the last time you wore a tie?"
"It's been a while."
"Well, you will have a suit and tie provided to you shortly. Try not to fuss with the tie too much. It gives you away as someone who usually doesn't wear one."
Ariana looked around the room at her soldiers, making deliberate eye contact with Karim and Farooq. "I want you clean-shaven. I have put hair clippers and scissors in the bathroom. Either you do it yourselves, or I will help you.
"This is how it's going to work. The routine will be the same for everyone. Everyone will be dropped off at a pre-selected location, in this case a designated Starbucks that is either across, next to, or otherwise in close proximity to the station I have selected."
"Why Starbucks?" Syed asked.
"Because I don't want anyone having to remember their location... Everyone will buy something to drink and a _Washington Post_. Don't look around, it makes you suspicious. If you need to look at something, look at your watches. This is D.C.; people like to act as if they are busy. I want everyone to arrive at Metro Center on their designated train at 8:30 am. Do not panic if you are late by a minute or two. I will be tracking each of you. Oh, and don't bring your coffee into the train stations. It's one thing that will most definitely attract police attention.
"I will drop each person off from a minivan I have at my disposal. It will take between sixty and ninety minutes to get to all the locations and drop everyone off, which means your potential wait time is ninety minutes, so plan accordingly. Fill in the crossword puzzle. Do sudoku.
"The twins will be first. They will be dropped off in Rosslyn at the Starbucks across from the station. For the twins, it is a seven-minute ride from Rosslyn to Metro Center. At 8:15, Team Orange needs to be heading into the station. The Rosslyn escalator is long and takes an extra minute or two just to reach the trains."
"Why 8:30?" Syed asked.
"Most of D.C. is government employees. We are fortunate that everyone arrives and leaves work at relatively the same time. 8:30 is near peak. 8:40 is probably the busiest time, but for simplicity sake, we are shooting for 8:30. Any questions, Team Orange?"
The fraternal twins shook their heads and answered, "No," in near unison.
"Syed, you are Team Blue. You will be dropped off at Capitol South. There is a Starbucks next to the entrance to the station. You will be eight minutes out from Metro Center. You need to be in the station by 8:20."
"Team Red," Ariana said, pointing towards Farooq. "You will get on at Cleveland Park. There is a Starbucks just past the CVS. You are also eight minutes out. Be on the train by 8:20 as well."
Ariana looked at Karim. "You are also Team Red. You will be dropped off last, in Chinatown. There is a Starbucks with the name written in both English and Chinese. You will be one stop away from Metro Center. One minute of transit time. Be on the platform by 8:25.
"Our main concern is the timing of the trains. Metro does not have a schedule. There is a published timetable but I have yet to see any train run in accordance with it. Trains are habitually late. They stop in the tunnels as a matter of course, a phenomenon that is unique to Metro. In virtually all trains systems in the world the only time trains stop in the tunnels is when there is an emergency. But here, well, it is a variable we will have to contend with.
"What I have tried to do is have everyone six to eight minutes away, and assume a three-minute wait time. Teams Orange and Red — if you get to the station before 8:30, I want you to get off the train, look a little lost, and hold your position at the end of the platform, as if you got off at the wrong station.
"For Team Blue, Syed, mill about as if you are waiting for someone, but stay in the middle portion of the platform. You will be key to the operation and the most centrally located. Regardless of anyone's timing, do not panic. Keep in mind that Metro Center is crowded in the morning. No one should notice you whether you're standing or walking. Metro employees are of no concern whatsoever. They have scraped the bottom of the employment barrel for these men and women."
"Unless you are eating or drinking," Karim added.
"Right."
"I've seen dog patrols before," Jameel added, shedding his New York Giants jacket. The room was getting warm, a combination of the six bodies shoulder-to-shoulder and the topic at hand.
"There's nothing I can do about the dogs. I don't know what substances they have been trained to detect and react to. If you see a K-9 unit coming your way, try to mix with the crowd. Don't run. If the dog reacts to your bag, comply with the officer and give them your cover story. We are running on a pretty tight timeline, so the chance of being questioned in the small window of operation time should be very slim. If you are questioned, be polite, answer the officer, and be prepared."
"What kind of explosives?" Syed asked.
"TATP. Made by yours truly. Also known as the 'Mother of Satan.' Sensitive to heat, friction, and shock." Ariana looked around at her team and then added. "So be careful handling your bags."
"How do we set them off?" Farooq asked.
"I will control the charge remotely. All of the bags are being tracked via a GPS system and my laptop. Once you all arrive at Metro Center, I will detonate the devices. The bags are loaded with directional explosive charges large enough to send their contents half the length of a six-car Metro train. The charges in the backpacks are set to blow out the sides, in both directions, not backwards. The charges in the suitcases will blow outward, towards the front of the suitcase and away from the suitcase wheels and the carrier if he is following usual luggage carrying convention."
"How large will the explosion be?" Syed asked.
"The explosions will be large, but perhaps not lethal, unless you are standing in the way of the suitcase when it blows. The force will turn the top of the suitcase into a dangerous projectile. My goal is to spread the contents of the bags. This is best done through a directional discharge."
"What about the contents?" Farooq asked.
"A powdery substance. That is all you need to know."
"What is the estimated death toll?" Syed asked.
"Four trains, eight cars apiece, and three crowded platforms. Maybe an exposure of two thousand with the initial blast. But the beauty of this is the collateral damage. Once these bags detonate and people realize they haven't been killed by the explosion or shrapnel, there will be mass hysteria. The rush for the exits will be fierce. Every person who has powder on their jacket, their bags, in their hair — everyone will become a human transfer agent. A delivery mechanism. So if two thousand people are impacted at Metro Center, it will be four thousand by the time they hit the street and come into contact with other people. Multiplying the effect of transfer agents, the number could be as high as sixteen thousand exposures by the time the original targets clean themselves off. By the time the bio-hazard teams arrive and figure out what they are dealing with, thirty-two thousand people could be exposed. Health workers. Colleagues. Good Samaritans. All on the way to their deathbeds with nothing but twenty-four to seventy-two hours of despair in front of them."
"Brilliant," Syed said.
Looks of fear crept across the twin brothers' faces. Ariana tried to appease their concerns. "Don't worry. I have an oral vaccine for you. A simple pill. Some of us have already been inoculated." Ariana motioned her hand towards Abu. "This is what happens to someone who refuses. The choice is yours."
Karim looked at Ariana. Her words from an earlier conversation rang in his ears. Indeed, she was no longer the girl he once knew.
"If there are no more questions, I will leave you all to pray and prepare. I will bring everyone their clothes, some money, their Metro cards, and pocket litter."
"When do we go?" Syed asked.
Ariana checked her watch. "We have thirteen hours before detonation."
Karim followed Ariana out of the room, past the slumping Abu who was still breathing, his eyes open and fixated on the three brothers in the sleeping quarters. Syed whispered to them about the importance of being calm under pressure. Not panicking.
Karim pulled Ariana to the side, out of earshot. "That was unnecessary."
"What?"
"Giving those boys hope. Martyrdom is without fear."
"Perhaps. But I cannot have them second-guessing things at the last minute."
Karim thought and rubbed his beard. "Shave it off?"
"Yes," Ariana answered. "I don't want the others to think you are not following protocol. But you won't be going into the Metro tomorrow. You will be with me. We have something else to do."
She reached into the pocket of her oversized sweater and handed him a cell phone.
"This is what I need..." she started, leaning in close.
## Chapter 55
The address is 9345, right?" Clark asked.
"I thought you were a number Jedi," Lisa answered from the passenger seat.
"Just tell me if I am right."
"9345 Georgia Avenue. Warehouse C," she replied.
"We just went straight from the eight hundred block to the eleven hundred block."
"And the neighborhood didn't improve at all."
"Well, we're not stopping to ask for directions."
"Usually I would call you a typical male for that comment." Lisa looked out the window at the brick corner market with its riot doors pulled down to the sidewalk. The bars on the windows protected the store's thick Plexiglass from projectiles and prying neighbors. A homeless man slept under a pile of fabric at the foot of a newspaper machine, the door to the machine long since ripped off. Garbage overflowed from the large trashcan at the corner of the store's property. "But in this case, I'm with you. There's no need to stop, much less ask for directions."
"Let's turn around and give it another look. It has to be here somewhere."
Clark slowed down as the traffic light turned from yellow to red.
"I think you should come back in the daytime. I don't even like stopping at the traffic lights."
"That's why I brought you. You have a gun, Criminal Investigator Prescott."
"I have a gun, but it's at home. Besides, I don't think there is much tax evasion in this zip code."
"Unless you count drug dealers. Like that guy there," Clark said, pointing with his nose.
Lisa glanced in the same direction Clark indicated. A large figure dressed in a black leather coat lurked in the shadows of a doorway to an old hardware store. "How do you know he's a drug dealer?"
"Because it's twenty degrees outside and he's just standing there." A moment later a car pulled up to the stoplight, flashed its high beams, and rolled down its passenger window. The man stepped from the safety of the shadows, exchanged his goods for money, and returned to his covert position.
"Did he just do what I think he did? Right here in front of us?"
"I don't think he was out shaking hands. Doesn't look like a politician running for office to me."
"Jesus."
"Has nothing to do with it."
The light turned green and Clark drove down the block before doing a three-point turn in the middle of the street. "One more pass and I'm calling it quits for tonight."
"You don't have to try to convince me."
Clark slowed down at the end of the old hardware store. A small alley ran between the hardware store and the boarded up building to its left. On the brick wall was a small sign that read 9345, with an arrow pointing ominously into the darkness.
Clark made the turn and Lisa squirmed in her seat. "This does not look good."
"The sign said 9345."
"The sign should have said 'call 911.'"
"Let's just see where it goes."
The Honda crept forward and Clark put on his bright lights. At the end of the building he was forced to turn left by the intimidating fence directly in front and to his right. The car dipped through a rut and the headlights bounced, beaming into the fenced lot ahead.
"Junkyards," Lisa said.
"I see them."
"Look for a place to turn around."
"I'm looking."
Clark followed the small road to the left and the narrow strip of pavement and dirt opened into a larger lot of the same making. Clark's high beams cut across the darkness.
"God, it's dark."
"Like every source of light in the vicinity stopped working with Marion Barry."
Clark turned the wheel to the right and then navigated a large swooping, looping left-hand turn. As he completed the turn his lights flashed against the front of several buildings.
Lisa squinted through the windshield. "I think we found the address."
"Which one?"
"The far right."
"It looks like a garage."
"And it doesn't look like anyone is home."
"I can't imagine why. The neighborhood is fabulous."
On cue, a dog went berserk somewhere in the darkness, followed by a slamming door and human screams. The incident ended with several canine yelps and another slamming door.
"I'll come back in the morning when it's light."
"And you'll come back without me."
"Oh, I don't plan on coming back alone. I have two friends who are coming with me."
## Chapter 56
Clark pulled into the parking lot of the 7-11, past the group of Latino day-workers who swarmed a white van as it approached the edge of the convenience store's property. Clark got out of his car, walked across the gum-spattered sidewalk, and reached for the phone. The dirty, blue and silver communications dinosaur was posted on a pole beneath a 'no loitering' sign written in both English and Spanish.
"I need to speak with Detective Wallace," Clark said to the police operator.
"May I ask who's calling?"
"The same guy who called three times yesterday. I'm providing an anonymous tip on a case he is working on."
"Just a minute."
A full minute passed as Clark blew hot air into his cupped hands, the phone wedged between his ear and his shoulder.
"Detective Wallace."
"Detective. This is Clark Hayden. I have been trying to reach you since yesterday afternoon."
Detective Wallace was standing next to his desk and he looked around the room as he spoke. "I was in Baltimore yesterday. Out of pocket. You didn't leave a message."
"You told me not to."
"You calling from a pay phone?"
"Do you have any idea how hard it is to find a pay phone these days? Much less one that is working?"
"Is that a yes?"
"Yes." Clark watched as six of the day-laborers climbed in the white van. "I have some information I would like to share with you."
"I was wondering if you were going to call. Half of me was hoping you would. The other half was really hoping you wouldn't."
"Any progress on the guy pulled from the Potomac?"
"We are still trying to locate next of kin. Cause of death was officially reported as drowning."
"Did anyone check this guy for ricin poisoning?"
"Yeah. As I said, the _official_ cause of death was drowning. I heard that a couple of suits paid a visit to the medical examiner. We may never get a straight answer on that one. How about you? How did your search go?"
"Well, I think I found the woman I was looking for."
There was a long silence on the phone. "Then I guess that presents me with a dilemma."
"I figured it might. So here's what I was thinking. What if you happen to stumble upon her while in the midst of solving another crime?"
There was a long silence followed by, "Then that would be just dumb luck."
Clark laughed. "I'm not sure if you meant that intentionally or not, but it could be a little of both. A little dumb and a little luck."
"I guess it could."
"I'll be in touch."
Clark called Detective Wallace back an hour later, this time from his cell phone.
"As dumb luck would have it, I made a trip into D.C. and someone broke into my car. Smashed the window."
Detective Wallace sighed and smiled to himself. "What's your location?"
"9345 Georgia Ave, near Tenth and Aspen. Warehouse C, around the back. A left turn behind an old hardware store that looks like it is out of hammers and very soon out of business. It is a good place to get crack at night, though."
"I know the area. I'll be there in twenty minutes. Don't go walking around the neighborhood."
Clark looked around his surroundings. "I wouldn't dream of it." Clark checked his watch, turned up the heat in the car, and made sure the doors were locked.
Eighteen minutes later Clark got out of the car and scoured the lot between the warehouse and the junk car piles next door. Large swathes of blacktop were missing, leaving exposed ground, puddles, rocks. Clark walked to the edge of the lot, expecting a rabid dog to appear on the other side of the fence, ready to mark his territory with whatever piece of Clark's body he could get through the chain links. Clark bent over and picked up a stone the size of a softball. He tossed it in his hand to check the weight and smiled. He took two steps towards his car, wound up, and threw the stone through his passenger side window. _Just covering everyone's ass_ , he said to himself.
Five minutes later, Clark was sitting on the hood of his car when Detective Wallace pulled around the corner.
The detective parked his car near Clark's and pulled himself from the passenger seat by the doorframe. He was dressed in black slacks and a black sweater. His badge was hanging from his belt, a spot of gleaming gold in a black sea of an outfit. There was an obvious bulge on his right hip.
Detective Wallace looked at the broken window on Clark's car. "A smash and grab?"
"I guess," Clark said, sliding off the hood. His butt was slightly warm from the heat dissipating off the engine from the morning's drive.
"You expect me to believe that?"
"That's my story."
"Did they take anything?"
"An old _Poison_ CD is missing. It's a classic."
" _Poison_? You trying to be funny?"
"Maybe."
Detective Wallace looked around. The junkyard was behind him, crushed cars stacked five high. A vacant lot was to the left, the fenced area littered with miscellaneous garbage ranging from tires to old refrigerators to hypodermic needles.
Detective Wallace finished his 360-degree surveillance scan and his eyes stopped where they began, on the closest pile of crushed cars. "I think that's a 72 Cadillac there on the bottom."
"You know your cars. Looks like a 72 accordion to me."
"I had one before I was married."
"A bachelor boat?"
"The women liked it."
"Back in the day," Clark added.
Detective Wallace flashed the look he gave his grandchildren when they did something wrong, but hilarious. "So, what are we doing here?"
Clark pointed to the far warehouse, over fifty yards away. "Warehouse C."
"What's in warehouse C?"
"My missing neighbor."
"How did you find her?"
"I didn't. An ice storm did. It's a long story. My neighbor's real name is Safia Hafeez. But I imagine she stopped calling herself that so long ago she wouldn't even answer to it now. She stole the identity from another girl when they were both students in Boston. The other girl went missing and her body was never found."
Detective Wallace nodded. "Smart girl. You use a missing person, not a dead one. A lot of jurisdictions have started matching death certificates with other systems, like Social Security. But if the person were only missing, and the family never filed a death certificate..."
"It didn't look like the girl had much family."
"Probably chosen for that reason. What else did you find out?"
"She's an MIT graduate. A doctorate in bad news stuff like chemical engineering and propulsion systems. Fulbright scholar. Pakistani national. Wife and mother of one. Has diplomatic contacts at the Pakistan Embassy. She has been living in the U.S. for over the last ten years under the name Ariana Amin. She has also been using my father's company to buy material and equipment."
"For what?"
"I have an idea but no evidence. I spent all yesterday afternoon and last night going through blueprints, invoices, orders." Clark paused. "You ever heard of a hail cannon?"
"No, what is it?"
"These farmers, and now some car dealers, take this big, long vertically standing tube, called a hail cannon, and shoot pressure waves into the air."
"Pressure waves?"
"Yeah, the cannon and its waves are supposed to disrupt the formation of hail in the atmosphere. Great for farmers with sensitive crops and car dealers who lose money on hail damage. Nissan opened a car plant in Mississippi and they even bought one. I guess if you think about it, all those cars have to sit outside until they get shipped to wherever they are going."
"Guess so. And..."
"Anyhow, when I started poking around for info on my neighbor a few weeks ago, I talked to my mailman trying to get an inside scoop. He told me that my neighbor had received some large-scale farming equipment catalogs. It didn't mean anything to me until I got the invoice from the company in Maryland for the machines shipped to this warehouse."
"And..."
"Then I started asking around at different places. Made calls to a few farming equipment companies. Ariana, through Hayden Ltd., had requested info on the hail cannon."
"They told you this?"
"I gave my name as Clark Hayden, and I had all the company information. I played dumb and said we were still interested in purchasing one."
"That would work."
"Except they cost over a million dollars. Anyway, how Ariana used this information, I don't know. Maybe she didn't use it at all. But there are a million companies out there selling a million different things that someone with her background could use. There was no way for me to contact all of them. One thing I do know for certain is that she isn't a farmer."
Detective Wallace nodded his head and looked over at the warehouse. "How do you want to do this?"
"You're the police..."
"Seeing that I'm responding to a smash and grab robbery of one classic _Poison_ CD, I think we go with that."
"Meaning?"
"Didn't you say that you think you saw the person who broke into your car enter the far warehouse?"
"I believe I did."
"Then let's go have a look. See if there were any witnesses... But first, I need to send out a safety line." Detective Wallace pulled his cell phone from his pocket. He called his part-time partner and left a cryptic voicemail on Detective Nguyen's phone giving him the address and circumstances. Wallace finished with a not-so-cryptic, "So if you don't hear from me in an hour, send in a rescue team, bio-hazard, the works." When he hung up, he looked at Clark. "At least one person will know where to look for us."
"Make that one million. My girlfriend has a similar video message from me ready to be posted on YouTube. It will also be sent to the local media and the Institute for Justice, a non-profit law firm in D.C. that specializes in constitutional rights. If something happens to me today there are going to be a shitload of people who know my name, your name, Agent Rosson. Everything."
"Pretty ballsy."
"Yeah, well. I kind of figured 'fuck it.'"
Detective Wallace lifted his sweater on the side and exposed his police issue Glock. "I hope I don't have to shoot anyone today."
"Me, too."
"If things get dicey, you get the hell away from me."
"Don't worry, if you start shooting, you'll need to send out a search team to find me."
Detective Wallace pointed to the building on the far left. "We start on the left and work our way down from there. Walking across that open lot is not a safe approach."
Clark followed behind Detective Wallace, who walked nonchalantly as if he were making a business call. As they passed the middle warehouse and reached the edge of warehouse C, Clark tapped the detective on the arm. "Look at the ground. Fresh tracks."
"I see those, Kimo Sabe. More than one vehicle."
"And one was a truck."
"You sure about this address?"
Clark nodded. "One hundred percent."
Wallace approached the solid metal door with his hand on his weapon.
He pounded hard with his free hand and identified himself as a D.C. police officer. Clark was stooped over at the waist, prepared to run, though he had no idea in what direction or what would trigger his mad dash. Detective Wallace slammed the edge of his closed fist on the door again, waited, and repeated the procedure. After thirty seconds of banging, he turned to Clark. "Looks like Plan B."
## Chapter 57
Ariana pulled the white minivan up to the curb in a no-parking lane in front of the Starbucks, across from the Capital South Metro. Karim was sitting on the floor, the seats long since removed. He sat with his back to the side wall of the minivan, behind the driver's seat, avoiding the exposed metal in the middle of the minivan floor. It was Syed's turn in the front passenger seat, the last stop before martyrdom. As she had done for each departing passenger, Ariana said a prayer.
Then she thrust her hand behind her and Karim placed a gun in it. Smiling, she pulled the handgun forward and passed it to Syed, handle first.
Karim watched from the back of the vehicle as Syed's face lit up with joy.
"The safety is on and it's loaded," Karim said.
Syed gave an instant assessment. "A Beretta Px4 Storm. Nine millimeter. Magazine capacity of seventeen. Nice gun."
"Now put it away," Ariana said, checking the van's mirrors.
"What's it for?"
"It's for the twins, Jameel and Omar."
"Why?"
"If you get a sense they're going to run, or back out, kill them. They are on the same platform as you. If they follow instructions and ride the second cars from the front and the back, then they will be less than thirty yards from your location. Can you hit them from that distance?"
"With that piece of hardware, I can hit them from eighty. But I could just as easily use my hands."
"Whichever. The only reason they were brought in was to carry the bags. I couldn't have one person carrying six bags and have them dispersed over the necessary area. So if you have to kill them, keep the location of their bags in mind. Do not let them out of your sight."
"They are boys, but they will not run."
"Just in case," Ariana said, dipping her head in the direction of the gun.
"Thank you."
"Have a safe journey, my brother," Karim said from the floor of the backseat.
"You have twenty-five minutes. Grab a coffee. Be on time," Ariana added.
Syed opened the passenger door and moved to the sliding side door. He pushed the door open just enough to grab his luggage as Karim pushed the heavy cases from his position inside the minivan. With the luggage on the street, the two men locked eyes through the closing side door.
"Allah Akbar."
"Allah Akbar."
## Chapter 58
Detective Wallace opened the trunk and looked down at the black, three-foot battering ram riddled with scars and scrapes. The paint was chipped, the blunt end slightly rusted.
"Christ, you carry that with you wherever you go?" Clark asked, looking at the thick metal cylinder. He was speaking quickly, almost in a whisper.
"I don't want to hear anything from a guy who carries a robot in his trunk," Detective Wallace said, motioning towards the electronic contraption on the ground next to the car.
"The robot is homework."
"Well, consider this _my_ homework."
"What else do you have in there?"
"Shotgun, vest. Extendable baton, though it's not entirely police issue. Tear gas mask," Wallace said. He pointed at this favorite toy in the trunk. "But this baby right here is known as Betty."
"My aunt's name is Betty."
Detective Wallace gave Clark his police-issue inter-rogation face.
"Ol' Betty, here," Detective Wallace said, joining Clark in half-whisper mode, gesturing with his head towards the battering ram, "has opened more than a thousand doors in her life. She weighs only forty pounds but when swung properly generates forty _thousand_ pounds of force. Unless we are dealing with a blast door or a bank vault, she always gets invited in."
"You need help with that?" Clark asked.
Detective Wallace shook his head. "Stand back, young man. Let me show you what Betty is all about."
Detective Wallace's large frame hunched over the open trunk and he stretched for the battering ram's handles. One hand grip was near the rear end of the device and was used for generating most of the power. The second handle was on the top of the device, near the midway point, its strategic location providing both power and the mild ability to steer the force of the blow towards the intended target.
"Looks heavier than forty pounds," Clark added, ribbing Wallace.
"Are you and your toy ready?"
Clark turned on the remote control in his hand and pushed the paddle forward with his right thumb. The two-foot high robot lurched from its parked position. Clark put the headset to the radio remote control over his ears and nodded. "Let's do this thing."
Detective Wallace approached the front door and tested the weight of the battering ram. He took one practice swing several feet in front of the door, and let out a grunt like a gladiator about to make an entrance into the Coliseum. With the conclusion of his warm-up, Detective Wallace set the battering ram in motion, the arc of the cylinder swinging back and upward.
"Wait," Clark yelled, his ears still covered with his headset.
Detective Wallace tried to reign in the weight of the ram as it surged forward. He stepped back as the battering ram continued forward, scraping the metal door but leaving it intact. Wallace grimaced, his lower back unappreciative of the sudden change in inertia.
"What?" Wallace asked, bouncing the battering ram in his powerful hands, the weight of the tool comfortable on his massive shoulders.
"What if it's booby-trapped?" Clark asked.
Detective Wallace looked at his partner for the day, speechless.
Clark continued. "All I'm saying is that I think this woman has quite a few dead bodies under her belt, and a booby-trap would fit her hobbies perfectly. Who know what she has on the other side of this door. Not to mention the missing ricin."
After a long thoughtful pause, Detective Wallace responded. "Good call." The detective eyed the warehouse exterior. He sized up the building and looked at the large steel roll-up doors on the front of the warehouse. He peeked his head around the corner. "Follow me."
"Where are we going?"
"Let's check out the back."
Detective Wallace took a swing at a small padlock and the door on the chain-link fence near the back of the warehouse swung open. He looked down the back wall.
"No back door," Clark said, stepping to the detective's side.
"Then we go through the wall. It's made of cinderblock."
"How long will it take?"
"Longer than the hinged door, that's for damn sure. But we only need a few feet of space. The rest is up to you and your robot."
"You make it sound like it's a pet."
"I call them like I see them."
"I'll have you know this robot here is a technical marvel. And this is an old model."
"A gamer's wet dream."
"Start pounding, John Henry."
"Is that a racist joke?"
"Heck, no. Was John Henry black?"
Detective Wallace shook his head in disbelief. "Youth."
"What do you know about gaming, anyway?"
"Had a double homicide once over a gaming tournament. Punk kids who couldn't remember the last time they went to school. Sitting around playing each other in video games. Violent games, too. It was a thousand bucks to enter the tournament, financed by drugs, and it went on for three days. The first man out of the tournament shot the host and then strangled the friend he lost to. Strangled him with the cord to his control pad. Learned more about gaming than I ever wanted to know."
Clark examined the wall. The cinderblock was in need of paint and mortar. It wasn't likely to get either.
"I don't need much room for the robot. Aim for the crack that's already there. It should be a weak point."
"Are you a mason?"
"No."
"Good. Just so we got that straight."
Detective Wallace swung the battering ram backwards and as the heavy cylinder reached the apex, Detective Wallace added his own 240 pounds to the physics equation. 40,000 pounds of force sent the head of the battering ram halfway through the wall. Three strikes later the hole was large enough for a basketball.
"Just a little more," Clark said. I can almost put the robot through if I lie him down horizontally."
Detective Wallace grunted as the battering ram lunged forward. "What makes a robot a 'him?'"
"I don't know."
"Sounds queer to me."
"Can we make the hole bigger?"
Detective Wallace took two more John Henry swings and looked at his handiwork. "Big enough?"
"It'll do," Clark answered.
Clark took the robot — an eight-wheeled, two-foot high, self-correcting vehicle with a video panel — and pushed it through the hole.
"What if it lands upside down?" Detective Wallace asked.
"It doesn't matter. It has wheels on all sides. It also has a roll-over feature, so if it gets stuck on something, an arm will extend and push the robot over in the other direction. We designed it so it won't get stuck. Well, at least not easily."
"And you can see what the robot sees?"
"Once I connect the monitor to the remote control and raise the camera on the robot." Clark took a second and plugged a blue cord into the side of his remote controller. He handed the video monitor to Detective Wallace. "Now let's see what we have."
The monitor gave Wallace and Clark a dog-level view of the room on the other side of the wall. Clark adjusted the sight on the robot with a thumb stick on the controller and the view on the monitor focused upwards.
"Looks like an office."
"Yes it does. It's a little dark."
"I can fix that," Clark said. A second later a light illuminated from the robot.
"Neat toy."
"A multi-thousand dollar toy. Years of research. And like I said, this is an old model. The new ones can operate in total darkness, go through water, up stairs."
Clark maneuvered the robot through the office, skillfully avoiding the legs of the table and desk. At the wall, the robot took a right and followed the light into the main room of the warehouse.
"It is a little lighter in here," Wallace said, flashing the video screen towards Clark.
Clark focused the robot's eyes upwards.
Wallace gave his play-by-play, standing next to Clark, sharing the monitor. "Windows." The detective looked up at the back of the building and pointed. "Twenty feet in the air."
Clark nodded and gave the throttle a little juice. The robot zigged and zagged around the warehouse floor as it made a cursory lap through its new environment.
"It's a big open floor," Detective Wallace said looking down into the monitor.
Before Detective Wallace could get the words out, Clark's thumbs jumped from the dual pad controller.
"Looks like we got a body," Wallace said, squinting at the monitor as the shoeless feet came into focus. Clark moved the robot slowly forward and its light shone on the body.
Clark moved until a face entered the screen. Blood trickled from the body's mouth. Clark became ashen. Detective Wallace steadied him by the shoulder. "You all right?"
"Yeah. I'm all right. Didn't expect to see that."
"You get used to it."
"I hope not," Clark said, taking a deep breath of cold air.
"Let's go in for a closer view."
Clark maneuvered the robot past the feet and toggled back and forth so that they could get a view of the whole corpse.
"I would say our body looks like a male, in his thirties. Middle-Eastern descent. Bad scarring on his cheek. Can you pan up?"
Clark did as he was asked.
"Looks like he is handcuffed to one of the main support beams. Not very nice."
Clark kept the robot moving, slowly casing the body in an investigative circle. He steered the robot around the pole, ventured in for a close-up of the cuffed hands, and stopped near the head. Eyes glued to the monitor, his ears focused on the silence coming through his earphones, Clark jumped when the face on the screen turned towards the robot. Clark's response startled the detective, who also jumped, in turn causing Clark to let out a "whaaaaaaa" like Shaggy from Scooby Doo.
"Looks like our dead guy isn't dead," Detective Wallace updated.
Clark pushed the robot to full power and its wheels screeched, running into Abu's side and then backing away.
"Easy. Easy. Don't kill the guy."
Clark settled down and a voice came through on his headphones. "Yeah, don't kill the guy."
Clark looked at the monitor. "We have audio." He pressed a button on the remote control and a small screen flipped up, rising from the main body of the robot in front of Abu's eyes. He pressed another small button and the camera feed on the remote controller indicated it was on with the illumination of a small red light.
"Now he can see us, too" Clark said. "In addition to audio."
"You have an extra earpiece?"
"No," Clark answered. "If you want to hear, lean close. He should be able to hear us speak normally."
Abu cleared his throat and attempted to say something before his voice choked out like an engine on a small plane in a freefall.
Abu swallowed hard. "Neat toy," he said in a raspy voice, looking directly into the screen in the middle section of the robot. He could see both Clark and the detective's faces wedged into the field of view.
Clark introduced himself and then tilted the remote control entirely at Detective Wallace. "And this is Detective Wallace of the Washington Metropolitan Police Department."
"Glad to see you," Abu said, groggily. "Listen carefully. My name is Adahi Uhad, but for the last seven years I have been known as Abu Safi. I'm a clandestine operative working for the Central Intelligence Agency. US-born American citizen."
Gone was the brutish, hot-headed veneer that he had maintained for the last seven years. Back was a well-educated, calculating CIA operative who was trying to give his assessment of the situation as the only witness.
"Is the door booby-trapped? Can we come in?" Wallace asked, looking into the small camera lens on the remote control for the robot.
"The door isn't booby-trapped, but this whole warehouse needs to be treated as a high level bio-hazard site. I wouldn't suggest entering the premises without protective equipment. Level Four protective equipment."
Clark and Wallace looked at the hole in the wall, and then took several large steps back into the open lot.
"Ricin?" Clark asked.
Abu's head nodded slowly.
"You were poisoned?"
"Accidental ingestion. It was a calculated risk. I volunteered to process the ricin with the machines in the side room. I changed the settings and proceeded as slowly as I could. I was hoping to buy time. I processed the ricin, and the output was a fine powder, though not as fine as they think. I put the powder into cylindrical plastic containers that were sealed and then washed. After that, Ariana further configured the containers with explosives and put them in suitcases and backpacks."
"Do you have I.D.?" Wallace asked.
"No, and if I did, it wouldn't be real." The rest of Abu's story came between labored breaths and bloody coughs. "I'm a graduate of the University of California, Berkeley. Born a Muslim, raised an American by Pakistani parents. I have a wife and two kids. I've been undercover for over seven years. I've been out of contact with the agency for over a month. I left for the U.S. from Pakistan in December. All this can be verified. Prior to December, I spent four years in training with Al-Qaeda. Learning explosives. Or enhancing my CIA training, as it turns out. I was injured in a blast two years ago by a fellow bomb-making student who was careless. I was originally inserted into Guantanamo Bay in 2004, completely undercover. Few people at the CIA knew of the mission. I was working a target in the Pakistani government named Karim Al-Housad. He has high-level connections. Diplomatic connections."
Clark looked at Detective Wallace and then back down at the monitor.
"A man?"
Abu nodded. "Yes, a man"
"I think you may have been after the wrong person."
"The thought occurred to me."
"Should we contact someone?" Clark asked into the mouthpiece on his headset.
"I need you to call a number and enter a passcode. Leave a message and give them this address, wherever I am. Nothing can be done for me, except maybe morphine for the pain. Also, tell the CIA that the tracking chip in my left forearm has been disabled."
"Tracking chip?" Clark asked.
"Yes," Abu answered weakly. The pace of his speech increased and decreased with each wave of pain. "The chip was disabled by 80,000 volts of electricity delivered through a stun gun. Very clever, really. Totally unforeseen on my part."
Clark spoke. "You aren't the only one the woman we're looking for has fooled. Her husband, two governments, a university, INS, DMV, a neighborhood."
"You need medical attention," Wallace said.
"If you want to call, you call after I convey what I know. You will need a bio-hazard team, not just an ambulance."
Abu swallowed again then continued. His lucidity was fading and he repeated parts of the story. "My name is Adahi Uhad. The woman you are looking for has approximately six hundred pounds of processed ricin she plans on using. I spent two days processing it. I tried to lessen the lethality by processing it to a larger size than specified, but it is still very deadly and very fine. Deadly enough to have killed me, and I was dressed for the occasion."
Abu coughed. His head rose off the floor and then dropped back down onto the concrete.
"She has lost two members of her team, but she may have more people at her disposal." Abu grit his teeth and inhaled deeply. He continued slowly. "She is going to use the Metro and aim for Metro Center. Explosives will deliver the ricin and will be detonated via cell phone signal to homemade triggers. Black suitcases. Large, blue backpacks. Three men dressed as businessmen. Two younger men dressed as students. At least five terrorists in total. Spread out along the length of both platforms. One of the terrorists is taller than average. A tall Middle-Eastern man, about 6 foot 3 inches, approximately thirty years old, pulling a black suitcase with wheels, in Metro Center. There can't be many targets meeting that description."
"When?"
"Today, 8:30. There was a truck and a car parked here inside the warehouse. There was also something in the far room that I never saw. Maybe a van. I think I heard a sliding door shut."
"What does the truck look like?" Wallace asked.
"A fifteen, seventeen footer. It had Piedmont Delivery on the side, but that was painted over with white spray paint. An amateur job."
Abu's eyes rolled into his head as he kept talking. "Call this number." As Abu recited, Clark nodded into the remote control screen. "Do you need me to repeat it?"
"No, I'm good with numbers."
"Call the number. Tell them my name and location. Tell them about the chip. Someone will come. Tell them it is a bio-hazard site."
Abu's breathing started to labor anew. "Tell my family that I miss them. Tell them that I love them. Tell them I made the sacrifice for them and for Islam. To show them there are people who will die for their Islamic beliefs... when they are righteous ones."
Abu mumbled the word "another," and then coughed. Blood rolled from the corner of his mouth.
"There is something else," Abu rasped. He slowly curled himself upward into a ball, his body wrapped around the support beam he was handcuffed around. His breathing disintegrated into a choke. Ten seconds passed as Detective Wallace and Clark watched the man struggle. Slowly, Abu uncurled. "Check my hand... I found this on the floor of the office."
Clark moved the robot to Abu's hands and focused the camera as close as its magnification would allow. The bright red ribbon with the number on it gave Clark the chills followed by a loosening of his bowels. Clark looked at detective Wallace. "That's not good."
"What is it?"
"I'll explain in the car."
## Chapter 59
Detective Wallace checked his watch and yelled into the dash mounted radio in his car. "No, no, no. I need the wireless communication link to Metro disengaged, and I need all traffic coming into Metro Center held at their location."
"You are not authorized to give that command," the voice said on the other end of the radio. "You will need to have your captain notify the chief of police with that request or supply the appropriate emergency code."
"I'll give you the appropriate code. Disable the fucking link to underground Metro wireless communications and get that station evacuated or you will kill a thousand people."
"Hold just a moment," the radio dispatcher responded.
Detective Wallace kept the radio in his hand as he hit the breaks to avoid a taxi.
Clark opened his flip phone and his thumbs danced through the Central Intelligence Agency number that Abu gave him. The other end of the phone rang once. A recorded, computerized voice greeted Clark with a three-word sentence. "Enter your passcode."
Clark punched the numbers into his phone and a series of high-pitched computerized beeps followed. When the beeps stopped Clark left a rambling message, repeating the name and location of the ill agent. He ended with a designation of the location as a bio-hazardous site and hung up.
"You forgot to mention the tracking chip," Wallace said between orders, barking police codes into the dash-mounted radio. The police cruiser was up to sixty miles an hour and traffic was forcing the detective into NASCAR maneuvers, one hand on the radio handset, one hand on the wheel.
Clark tugged on his seatbelt to make sure there was tension. "If a call to a CIA number with a secret passcode and a CIA agent's name and location doesn't get a reaction, nothing will."
"I would expect a call back."
Clark didn't answer. "You think we should have left him?"
"I don't think we had a choice."
The car bounced and Clark's head ricocheted off his headrest. He glanced at the speedometer and the needle was rising through the seventies as they headed south on Georgia Avenue. A group of homeless wandering across the street scattered as the police cruiser roared past the District's largest shelter.
"You should have said ten thousand deaths. Maybe that would have gotten the dispatcher's attention," Clark added.
"Maybe. You better hang on. I have an idea."
Syed, the soldier-businessman, stepped off the Blue line train. He tugged the large wheeled suitcase over the crack between the train and the platform, holding the second bag, a sizeable carry-on, in his left hand. His new present, the Beretta Px4 Storm, was safely shoved into the back waist of his pants and covered by his suit jacket.
The platform teamed with bodies, six deep. Lobbyists hurriedly elbowed consultants who cut off tourists. Government employees pressed slowly in every direction: on and off the train, up the escalators, down the platform. Syed pulled his bags through a throng of business suits and a retirement group from Oklahoma with matching t-shirts. He found his way to the large concrete bench in the middle of the platform and stood his bags on the tile ground. He recalled what Ariana had said about the direction of the blast, and he adjusted his suitcases for maximum damage, placing them front-to-front.
An Orange line train pulled into the station and Syed pretended to read his paper, looking past the headlines of the business section. He peered over the heads of most of the commuters, thankful for his advantage in height. Passengers on the Orange line train were shoulder-to-shoulder, standing room only. Faces pressed against the glass as the train left the platform and Syed casually looked in each direction for Jameel and Omar. Then he waited for the next train.
Above him, a Red line train pulled into the station and another rush of people pushed down the escalator to the lower platform. Farooq, now cleanly shaven, stepped from the middle car and walked to the wall of the upper platform. He positioned his bags according to Ariana's direction, as Syed had done twenty yards below.
A young couple stood from their seats and Syed sat down on the end of the large slab concrete bench next to his bags. He turned the page in his newspaper and checked his watch. It was 8:27 as the next Orange line train pulled into the station. Syed looked over his shoulder as the train arrived on the platform, brakes squealing slightly. With another surge of bodies, Jameel stepped from the train, the straps from the backpack over his shoulders, a duffle bag in his hands. Syed couldn't see Omar, but knew where there was one brother, there was another. He stood from his seat and peered up at the upper platform. He could see Farooq from the shoulders up.
There was nothing to do but smile at the upcoming carnage, and wait. Three minutes.
Karim drove and Ariana opened her laptop with her Verizon wireless network card and checked the progress of her soldiers. Four blips appeared on the screen, all of them transposed over a Google Map image of the Metro Center area. She could see the office buildings and the roads, as well as an outline of the underground passages and tracks of Metro Center.
"Syed and Farooq are positioned and the Orange Team is just arriving." She checked her watch. "Right on schedule." She reached for the cell phone and turned it on. "Keep driving. South on the GW Parkway."
## Chapter 60
Detective Wallace took a hard right onto Constitution Avenue and gunned it. The big eight cylinder growled as the pistons were fed a heavy dose of gas. Clark was pushed back in his seat and grabbed for the handle over the door as Detective Wallace swerved into oncoming traffic. The car zoomed past the Department of Commerce and the White House, the morning traffic blasting their horns as Wallace weaved his car through the automobile slalom.
At the E street ramp the car rolled hard and the tires screeched. Near the stoplight at the mouth of the Rock Creek Parkway Detective Wallace pushed on the brakes.
"Last chance to get out."
Clark tightened his seatbelt. Wallace nodded.
"This is the unofficial way to get things done."
Detective Wallace hit the gas and the black unmarked patrol car jumped the curb and headed across the grass between Rock Creek Parkway and the Four Seasons hotel. Wallace fought to steer the car as it bounced in the uneven field, the engine roaring, the Thompson Boat Center passing by in a blur on the left. Clark braced himself as the car raced towards a chain-link fence in the distance.
"Slow down, slow down, slow down," Clark said frantically, followed by "too fast, too fast, too fast."
Detective Wallace took his foot off the accelerator and yelled "hang on," as Clark chimed in with "Ohhhh shit."
The car cut through the chain link fence, dug through a patch of gravel, and smashed into the large green electronic box with a shower of sparks. The airbags in the car exploded from their designated locations and Detective Wallace and Clark slammed into the impact-reducing devices. The four foot high electronic box tipped to forty-five degrees but didn't fall.
Detective Wallace took inventory of himself and muttered through the aftermath of the collision. "You all right?"
"Jesus," Clark said, the deflating air bag gathering in his lap. He rubbed his nose and shook his head to clear the cobwebs.
"Are you ok?" Detective Wallace repeated.
"I think so."
"Good." Detective Wallace threw the car into reverse. Gravel and dirt flying beneath the tires, he backed the car up fifty yards, flattened a piece of fence that had been dislodged by the original impact, and hit the gas again. With the second impact the green box fell, wires torn from their moorings.
Wallace backed the car up again and cut the ignition. Clark stumbled out of the cruiser first.
Detective Wallace exited the car and stepped towards his handiwork to admire the destruction. "Mobile phone exchange," Detective Wallace said aloud. "Originally only Verizon had service to the Metro underground. Now, all the major carriers are routed through this transfer terminal."
"Well, I think that took care of it," Clark said. "How did you know where this was?"
"Had a homicide near this spot two summers ago. Rich woman from the Watergate across the way. Never solved that one. The location stuck in my memory, as most things do when you are a detective."
"Does the car still drive?"
Detective Wallace turned, looked at the car, and kicked at the grill. The front bumper dropped to the ground. He peered under the engine block. Multiple streams of liquid sizzled as they dripped down through the heated engine.
"I need to get back to my house," Clark said.
"Why your house?"
"I need to get something."
"Taxi," Wallace said as he started jogging towards the intersection of K Street and Rock Creek Parkway, over the tire tracks he just left.
Clark followed Wallace up the hill, the police cruiser now smoking behind them. The detective, oversized waist bouncing and arthritic knees creaking, pulled out his phone and called his part-time partner with an update. Nguyen answered the phone on the first ring.
"I got your message. You're still alive, I take it."
"Barely. You may hear about this on the news."
"Sounds familiar. Try to keep my name out of it this time."
"I need you to call Metro and have the system emptied. All trains going through Metro Center. We have five terrorists in or approaching Metro Center with explosives and ricin. Middle-Eastern men. Three are dressed as business men with suitcases. Two are students with large backpacks. One of the suspects is six foot three. Thirty years old. Dressed in a suit. Pulling a suitcase on wheels. Shoot first, ask questions later. Hunt down the captain or we are going to have a fucking mess on our hands. Tell the captain if I am wrong, he can have my badge. Call Metro police, call the station directly. Whatever you have to do. I tried to take out cell service to the Metro tunnels, but there may be a back-up trigger."
"I also need a bio-hazard unit sent to the address I left on your phone earlier. 9345 Georgia Ave. Warehouse C. Tell them there is also ricin at that location and an injured intelligence agent. Use extreme caution."
Detective Wallace raised his free hand and a cab driver pulled over. He flashed his badge and the cabbie raised his hands as if he were being held-up.
"Put your hands down," Wallace said through the closed window.
The cabbie let out a sigh of relief and Wallace walked around to the driver's side of the car and opened the door. "Out. Your car is being commandeered."
The cabbie started to protest and Wallace gave him the official don't-fuck-with-the-police expression. The cabbie got out of the car. Detective Wallace looked at the cab number on the roof and then back at the driver.
"You can pick it up later at police headquarters."
Clark apologized and jumped in the passenger seat.
Nguyen was listening to the proceedings over the still-connected phone and repeated Wallace's name several times before the detective answered.
"Where are you going?" Nguyen asked.
"Arlington."
"You don't have jurisdiction in Arlington."
"I do today."
"Get on the phone and tell D.C. and Arlington County that there is a cab being driven by an officer on a police emergency. Don't approach. Make it a BOLO with a request for assistance."
"Cab number?" Nguyen asked.
"4631."
Clark corrected him. "4361."
"Correction, cab number 4361."
"Got it."
"Reach me on my cell. My cruiser is parked near Rock Creek and K Street. No radio."
"Roger that."
"Now Nguyen. Now."
Ariana smiled as she closed her cell phone. "It is done."
Karim muttered a prayer under his breath.
"Get into the right lane," Ariana said. "The exit is up ahead."
## Chapter 61
Farooq tried not to stare at the slow moving pair of Metro police officers entering Metro Center near the main security booth. He had ridden the Metro enough to know the routines; to see what was routine. His professional assessment was exactly that — a routine patrol. The officers, well-carved individuals with former military written all over their demeanors, took in the environment as they walked. Their heads swiveled slowly from left to right, eyeing the tracks, the waiting passengers, the group of youths who should have been in school. As the officers passed Farooq on the upper platform, Farooq smiled and nodded. The officer closest to him flashed a brief professional grin and returned his focus forward. It wasn't until one of the police officer's radio crackled that Farooq's eyes dropped back down to his newspaper. He checked his watch and started to sweat.
On the intersecting platform below, Syed pulled the sleeve on his suit and also checked the time. They had been waiting for five minutes, far too long in his military opinion. Operations were precise. Or at least they started precisely and then quickly disintegrated into mayhem. The platform was crowded. Jameel was thirty yards away, his back facing Syed, his backpack on, his duffle bag in his hand. Syed strained to spot Omar whose backpack poked in and out of sight through the moving commuters.
The sudden wail of an alarm sent people scrambling to cover their ears. Syed's eyes darted up, then left to right as blue and orange trains arrived from opposite directions. People rushed for the exits as panic echoed off the walls. Syed looked up at the platform above and saw a half dozen officers reporting to the scene with their guns pulled. A moment later, the two officers who had just passed Farooq had returned, their guns drawn on the young man with the backpack. Farooq's hands went up a second before his legs were kicked out beneath him and the bottom of a boot landed on his neck, pressing him to the floor.
The alarm wailed, a strong blast followed by a lull and then another wave. "Shit," Syed said aloud. In a well-trained motion, Syed pulled his Beretta from the small of his back. He stepped behind the concrete platform bench, aimed at the larger of his two suitcases, and pulled the trigger.
The explosion ripped through the suitcase and the six people rushing past it. Dust thrust out in all directions, a wave of death canvassing the middle portion of the cavernous tube-like structure of the Metro station, engulfing the upper platform as it made its way outwards.
Syed coughed and tried to catch his breath. He squinted through the aftermath and the temporary silence. Syed's ears trickled blood. His head rang. His hair, face, and suit were covered in a fine white powder. The bench he had taken cover behind was painted with bits of clothes and flesh. Syed shook his head and checked his weapon, balancing himself as he stood amidst the hysteria. He pushed his way through the stunned crowd, the alarm a memory those near the explosion were no longer able to hear. Police officers from the platform above fought their way down the escalators, guns drawn, their commands mixing with the alarm and the unintelligible Metro emergency message. From twenty yards, his aim above the crowd, Syed dropped the first two officers as their torsos appeared, feet first, on the downward escalator. Syed staggered down the platform and spotted the outline of Jameel with his backpack and duffel bag. The young man was frozen, his body rigid, his eyes shut as he waited for his bag to explode. Syed raised his gun, exhaled, and pulled the trigger. Jameel fell to the platform, his duffle bag rolling over the edge onto the tracks. His backpack was still intact, facing away from Syed. The former soldier scanned the madness and continued down the platform, his mind blocking out the rush of bodies that surrounded him. He stepped to the other side of Jameel's twitching body and started to unload his magazine into the large backpack from point blank range.
He didn't hear the warning before the shot from the police issue handgun tore through his skull.
## Chapter 62
So what was the red ribbon the CIA guy had?"
"It's the flag off a radio remote control. One that my father used to fly when I was a kid. It had the same frequency number written on it. I would recognize it anywhere. The flag's color alone actually tells you the frequency range for that device, but my father was very particular. Overly cautious in some regards."
"Why does this woman need a radio control? What about Metro Center and using a cell phone signal as a detonator?"
"Maybe that was plan A. Or maybe that was plan B. But the more I think about it, she may have finally slipped up."
"What do you mean?"
"Well, she may know a lot about explosives and poison and rockets and propulsion systems. But my guess is she doesn't know shit about model airplanes. Or at least that's what I'm hoping for."
Wallace's phone rang. "Wallace here."
"This is Nguyen. An explosion has been reported in Metro Center."
"Fuck," Wallace yelled.
"What?" Clark asked.
"An explosion."
"Shit."
"Keep me updated," Wallace yelled into the phone. His request was met with silence. He looked at the phone and the _Call Failed_ message appeared on the screen.
A prolonged moment of silence engulfed the car.
"Get me home," Clark said.
Clark dug through two standing metal cabinets and ran his eyes along the shelf on the far wall of the hobby room in the basement. He breathed heavily as his hands quickly felt up the nooks and crannies of the room. In a space beneath the stairs Clark's fingers danced over a dust-covered shoebox. He pulled the box through a cobweb and opened the top. "Fucking A," he said to himself in a whisper.
Organized neatly in the box were ten stacks of cash, each wrapped with a thin rubber band. Clark plucked the stack off the top and ran his thumb across the edge of the money, listening to each bill snap as it flipped by. He repeated the action, this time looking at the corner of the bills in motion. Fifties and hundreds. _Enough for a year at school_ , he thought. Maybe even two with room and board. He put the money back, shut the lid, and replaced the shoebox on the shelf. _One problem at a time_.
Clark yanked his head from beneath the stairs and continued his frantic search. He tore open three cardboard boxes stored near the hot water heater, then spotted the large clear plastic container in the corner. _Oh Dad, let this be the one_ , he said to himself as he grabbed the black controller from the plastic box. He snagged a 9.6-volt battery pack off a stack on the shelf in the corner and ran upstairs.
Clark came out of the house and jogged across the front yard to the cab. He flopped into the passenger seat.
"You got what you need?"
"Right here."
"Where to?" Wallace asked.
Clark paused to think. "Reagan National."
"Why?"
"Common sense... and a few other things."
"You want to share?"
"Start driving."
The taxi laid a thick trail of rubber on the street as Detective Wallace pushed the accelerator to the floor.
"Here's what I'm thinking. One, Ariana's daughter told me she liked to watch airplanes. Strange for the daughter of a housewife who never traveled. Two, if Ariana already attacked Metro and is looking for another target, a plane is the easiest. Three, it's what I would do."
Wallace opened his cell phone and tried to dial the precinct with one hand. The call froze in silent limbo. He tried again, this time with 9-1-1. Another call failed message.
The detective turned towards Clark. "Can you get through to 9-1-1 on your phone? I got nothing here."
Clark pulled his phone from his pocket and tried 9-1-1. "Nothing. The call was dropped."
"You know what that means."
"The news is out on the explosion in the Metro and now everyone in DC is trying to use the phone."
"Exactly."
"You don't have a radio?" Clark asked.
"Yeah, it's attached to the dash in the cruiser."
"A lot of good that does us."
"You know what else this means?" Wallace asked.
"It's up to you and me"
"Until further notice."
"Well then I hope I am right." Clark took the small rectangular box with thumb controls and raised the antenna.
"So what do you do with that remote control?"
"This is an old Proline 1100."
"I'll take your word for it."
"The name and model number is written on it. Anyhow, this particular controller in my hands had a long life in the world of model airplanes. But there is something special about this one that Ariana probably doesn't know."
"It tracks terrorists?"
"No. It's part of a buddy system. This is the master control. The one Ariana has, the one with the matching red ribbon in the warehouse, is a slave control."
"Meaning?"
"If we get close enough, I can switch off her transmitter and she will never know. I would be in control of her device, whatever that device is."
"Why would she need a remote control?"
"The better question is why would she have that ribbon if she wasn't going to use a remote control?"
"D.C. is a big city. Lots of high profile targets. Your guess is a shot in the dark."
"Maybe, but so far the CIA, the FBI, and the cops have all been playing catch-up. If I were a betting man, and had money, I would bet on myself this hand."
Clark tried 9-1-1 again and then gave directions. "Take the next right."
"We're still north of the airport."
"She's not going to the terminals."
## Chapter 63
The long black car was parked, rear-end first, in the row of spaces nearest the park entrance. Across the parking lot, the Potomac River, in mid-winter murkiness, chugged downstream towards the Chesapeake. The large brown sign near the riverside park entrance posted the name, Gravelly Point, and the hours of operation.
The white minivan Ariana had driven into the gravel lot minutes earlier was parked over a hundred yards away, in a direct line between the river and her location in the black car. The alignment of the vehicles was perfect, the result of myriad trial runs and hours of trigonometry and calculus.
The gravel lot with the paved entrance held over two hundred cars when it was full, but on a chilly February morning two dozen vehicles sprinkled the lot. Every car had a row of spaces to themselves, each vehicle isolated by choice, like men in a public bathroom with a wall of urinals to choose from. Body heat steamed up the windows on a small import parked at the far end of the lot, the car rocking intermittently.
"What's the target?" Karim asked from the driver's seat of the long black car.
"Flight 1956. Delta Shuttle from LaGuardia. Arrives at 8:53. Passengers include thirty members of the United Nations Terrorism Security Council and a team from Interpol. Some of the best and brightest anti-terrorist minds on the planet."
"Where did you get your intelligence?" Karim asked.
"The price of intelligence isn't what it once was. I obtained the information through more traditional channels — CNN. The summer before last there was a disturbance on a flight from New York to Washington. Some young men who had been up all night partying in New York continued right through to the morning, with Bloody Marys, on the plane. They got a little raucous, grabbed a flight attendant inappropriately, and verbally assaulted the crew. The news reported the incident and also divulged that the passenger list included the UN Security Council on Terrorism."
"You based this plan on one incident on one aircraft?"
"The original idea. But the United Nations website lists ongoing Security Council initiatives. One of those initiatives is a monthly meeting here in Washington. The second Wednesday of the month at 10:00 am."
"They don't fly on a UN-sponsored private plane?"
"Not since the UN adopted an environmentally-friendly protocol requiring members to fly on commercial flights when possible. They are trying to set an example by reducing pollution from private aircraft. I have followed their routine a dozen times over the last year. Right now there are at least eight UN limousines waiting at the terminal's ground transportation exit."
"And that is why today was chosen?"
"I've been planning for a while. Your arrival, and the ricin, was a last minute complication to my plan. Not the other way around. This is my original plan. This is what I have been working on."
Ariana picked up her binoculars and looked through the thick bullet-proof glass of the vehicle. A plane was approaching down the designated flight path over the Potomac between the high rises of Rosslyn and the flight-restricted skies over D.C.
"I was concerned that the Metro incident might cancel flights into the airport. So far, so good. According to plan."
"There has been nothing on the news yet," Karim said, gesturing towards the radio.
"Not yet, but any moment. Security has probably already been notified and will increase patrols. I expect the police to check this area as well. Protocol is predictable. First they will secure the terminal and increase security at check-in. We are 16 minutes after detonation at Metro Center. For now, we can count on bureaucracy to give us another 30 minutes. And that assumes they make a connection between Metro and the airport. There is no one on Earth who can draw that conclusion. That will provide additional time."
"And after this??"
Ariana spoke without taking her eyes off the sky. "Then I want to go home. I want to see my father again. I want to bring my daughter home. I want my daughter to meet her grandfather. I want my father to know that his daughter did not let him down. That his faith from those years ago paid off."
Karim looked out at the minivan in the parking lot. Joggers with winter gear appeared sporadically, popping out from the running paths near the river's edge and disappearing again near the large fence that ran between the lot and the airport terminal in the distance.
"Where did you get the van?"
"Bought it used in Baltimore. Paid cash."
"It seems rather obvious. A white minivan."
"It seems obvious now, sitting there in the parking lot. But you may recall there was a sniper in the D.C. area a few years ago. The news media latched on to the initial report that the first victim was killed from a shot fired out the back of a white truck or van. For weeks the entire region was on high alert... on the lookout for a white vehicle. As it turns out, white is a very popular color for trucks and vans. Drive through south Arlington alone and you will see hundreds of white vans. They line the residential streets between Shirlington, Bailey's Crossroads, and Seven Corners."
Ariana watched as a Southwest plane touched down safely on the runway. Karim picked up the red remote control off the seat. "How does it work?"
"The van is parked directly beneath the flight path. The Delta Shuttle uses only MD-88s."
"McDonnell Douglas?"
"Yes. The MD-88 is forty-five meters from nose to tail with a thirty-two meter wingspan. Its approach speed is 130 knots or 150 mph. The vehicle is parked four hundred feet from the end of the main runway, runway 19. At that distance, the plane's altitude will be less than one hundred and fifty feet when it passes over."
"Amazing."
"What?"
"That they allow the public here."
"It's Washington. Allowing the public to move freely is a show of strength. In any other city this airport would have been closed after 9/11. But not in Washington. Reagan National is only ten minutes from the Capitol, and Congress is not about to drive forty minutes out to Dulles to catch a plane."
Ariana once again looked through her binoculars at the next plane on approach. The aircraft was still a dot in the distance, even through the 8x32 magnification of the Pentax binoculars.
"Tell me about the projectile system," Karim said.
"It is a borrowed design, of sorts, with some homemade modifications. The firing mechanism is compressed acetylene gas. The same component used in welding torches. Highly volatile. Very forceful. Not the optimal substance... but it is easy to obtain, and purchasing it doesn't arouse suspicion. Especially when it's purchased by a legitimate business with a legitimate machine shop."
"The acetylene is compressed in a combustion chamber which is attached through a steel plate to the van by eight bolts. The gas in the combustion chamber is ignited via a remote control trigger — a modified servo — compliments of a deceased neighbor."
"You have a few of them."
"This neighbor was the first," Ariana answered before continuing. "Five vertical barrels are attached to the combustion chamber. Each barrel is four inches in diameter and holds a homemade titanium shell consisting of four smaller individual rounds. Each round is composed of a quarter pound of Tannerite."
"Tannerite?"
"A binary explosive that detonates on impact. Tannerite consists of two components. Individually, each component is non-lethal and perfectly legal to purchase. Together, the components will rip through most metals on impact. The aluminum skin of the airplane will offer little resistance."
"So, you have five tubes, four rounds per tube..."
Ariana finished the math with her eyes behind the binoculars. "At less than one hundred fifty feet, with twenty shots traveling over a thousand feet per second, I estimate more than a fifty percent hit rate. As high as perhaps eighty percent. The time between detonation and impact will be less than a third of a second. I only need one round to hit any number of vital targets. The wings, the ailerons, the elevators, and, obviously, the fuel tanks. Planes on approach are vulnerable."
"You seem sure."
"At this distance and altitude, with this explosive, it's like shooting a paper airplane with a shotgun. Ideally, I'm going for an explosion, but in reality I only need to create a roll of a few degrees. Gravity and the hard surface of the runway will handle the rest."
Ariana peered through her binoculars. She read the words _Delta Connection_ written along the side of the plane and announced. "The plane is approaching."
"Do we know it's the right one?"
"Yes. It took off from New York fifty-two minutes ago. The next one is not due for an hour."
She grabbed the remote control from her lap and moved the control stick on the right-hand side. The sunroof on the van rose by an inch, and then slid back.
"Nice," Karim said.
"Original equipment modified to open remotely." Ariana positioned her thumbs on the controls. "Less than a minute to impact."
From the opposite direction, Detective Wallace steered the yellow cab across two lanes of traffic, narrowly missing an oncoming bus. The cab accelerated down the service road and then slid hard onto the gravel lot as it screamed past the entrance into the park.
"What's the maximum distance on that thing?" Detective Wallace asked.
"Two miles. Maybe farther. The limit, as a practical matter, is your vision. You can technically control a model airplane long after you can no longer see it."
"And you're not going to accidentally bring down a real plane with that controller? You know, they don't allow cell phones on airplanes for a reason."
"Impossible," Clark answered as the car came to a halt in the middle of the parking lot. "Different frequency."
Clark fully extended the antenna on his remote control. The approaching plane banked one final time as the pilot leveled the wings towards a direct descent to the runway. Clark flipped the bright red button on the master control as his father had done a hundred times.
And then he froze.
"What's the problem?" Detective Wallace asked.
"I can't move."
"What?"
"Well, the usual convention of the buddy system remote control is that the master controller takes over _control_ for the slave controller. Usually to avoid a crash."
"And... ?"
"In this case, I don't want to _do_ anything. I want control so Ariana can't do anything. Which means I can't move. Any movement of these controls could be the wrong one."
"Then don't move," Detective Wallace said matter-of-factly. "I'm going to check on these vehicles."
Clark nodded as he stared at the black controller in his hands. "According to the CIA guy, we should start with those that have sliding doors."
"Heads up," Karim said. "We have a police officer getting out of a cab at three o'clock."
Ariana reaffixed the binoculars to her eyes. She focused on the badge attached to the detective's waist and then back to his face.
"And there's a second person in the car."
Ariana moved her eyes again and a close-up profile view of Clark filled her vision. "Impossible," she hissed.
"Who is it?"
"My neighbor."
"How... ?"
"It doesn't matter. He's too late."
Karim nodded towards a short line of police cars heading towards the park entrance. "So much for your response time estimate. I think it is time to go."
"A few more seconds," Ariana said, her thumbs on the control pad as she slipped into her operational zone. The approaching MD-88's landing gear was fully down, its wing flaps extended to increase the drag on the aircraft, slowing its speed.
Ariana's lips moved as she watched the plane continue to descend. She started her countdown from ten as the plane passed the far end of the rugby field. Her lips still silently counting, she smiled as the plane passed over the line of Don's Johns at the near-end of the park. With one swift motion, she pushed both thumbs up on the control pads as the plane passed directly over the van. She repeated the movement, jamming her thumbs forward with more force each time, as the plane roared past on approach,
"Now," Karim yelled. "Now!"
She replied with a tirade in Urdu.
Ariana and Karim, curses flowing, watched helplessly as the target landed safely, the tires touching down on the runway in a small cloud of burned rubber. Ariana let out a guttural animal scream and threw the remote control into the dashboard of the car, breaking off the antenna. She grabbed the binoculars from the seat and looked back at Clark, still in the taxi cab. Then she noticed the metal of the antenna pointing upwards toward the windshield.
The two police cars at the end of the police convoy blocked the entrance to Gravelly Point as the leading cars ripped into the lot. Karim took a deep breath as he drove the long black car slowly forward, approaching the impromptu barricade. "Relax," he said. "And put that controller out of site."
Ariana pushed the remote controller under the seat with the heel of her shoe and threw the binoculars on the backseat. The black car approached and an airport police officer stepped from his car, the lights on the vehicle still flashing in intermittent red, white, and blue.
Karim held the button on the armrest of the driver's side and the thick glass retracted halfway down.
"Identification, please," the officer said, his reddish hair appearing at the doorframe of the black car.
Karim reached into the breast pocket of his suit jacket and removed two diplomatic passports. Smiling broadly, he handed the official documents to the officer. "As you can see, we're diplomats with the Pakistani Embassy. My name is Nazim Shinwari and this is my wife Safia Hafeez."
The officer opened the pages of each passport and examined the photographs on the inside cover. "Enjoying your morning?"
"We were just taking in the sights. Unfortunately, we were called back to the embassy as part of a security alert. Embassy protocol demands that we return immediately. Our progress cannot be impeded."
"Just a moment," the officer said, keeping the passports in his hand as he walked to the front of the car and noted the diplomat license plates. He walked back to the open window and handed the passports back to Karim. "Have a good day."
"Thank you," Karim said.
Ariana leaned across the front seat towards the open window. "Officer, it may be none of my business, but there is a young man in the passenger seat of that parked cab. He is acting suspiciously. I think he has a gun."
The officer looked over at the car as Ariana pointed for guidance. "We'll take care of it. For safety's sake you may want to get moving." As Karim raised the window on the car, the officer reached for his radio.
Clark watched as a second plane passed over, the corners of the remote controller firmly in his hands. Across the parking lot Detective Wallace placed his nose to the window of an empty SUV before moving in the direction of the white minivan.
Concentrating on the remote control in his hands, Clark shook as the yell reached his ears. "Let me see your hands. Hands up. Hands up! Do not fucking move!" Clark followed the last command and looked into the side-view mirror of the cab to locate the voice. Above the warning that _Objects in the Mirror are Closer than they Appear_ , Clark saw the business end of a police-issued nine millimeter pointed at the rear flank of his head.
Karim drove slowly down the service road to the entrance of the GW Parkway. Ariana stared out the window as the police surrounded the yellow cab and Detective Wallace raced across the parking lot to intervene. The detective, badge in his raised hand, arrived on the scene and slowly the officers surrounding the cab lowered their weapons. " _Nine lives_ ," she muttered under her breath.
Karim turned on the radio and the report of an explosion in Metro Center was followed by several beeps and an emergency broadcast message. "The day was not in vain," he said proudly, looking over at Ariana.
Ariana sneered. "Easy to say when you did not waste years of planning. Living as a housewife to a man you despised. Being submissive. Hiding your intelligence."
"It is over now. And your mission was a success."
The long black car merged into traffic heading towards the Memorial Bridge. Ariana coughed and looked up at the next plane on approach to Reagan. She gazed skyward and then followed the plane as it flew over the field in the distance, over the swarm of police now in the parking lot of Gravelly Point. She started to speak, coughed again, and cleared her throat. She noticed the taste of iron thick in her mouth and swallowed hard. She pulled down the sun visor on the passenger side of the front seat and looked into the mirror as the heavy flow of blood from her nose ran over her lips.
## Chapter 64
Clark sat down at the small kitchen table and Mr. Stanley delivered a cup of coffee, no cream, no sugar.
"You read the paper today?" Clark asked.
"No, I just moved it around on the table."
Clark smiled. "And... ?"
"Not watching the news?"
"Haven't turned on the TV in days."
"I'll save you some time. The paper is saying the same thing it's been saying every day since the incident. Finger pointing over the incident at Metro Center. Twenty-six people dead, another hundred in local hospitals in various conditions, ranging from critical to simply "quarantined." Mr. Stanley paused. "You want to fill in the blanks?"
"May get you in trouble. Classified material."
"I'll take my chances."
Clark shrugged his shoulders. "Ariana was dropped off on the doorstep of Georgetown University Hospital in the throes of death. She lived just long enough to tell the authorities absolutely nothing. They believe she accidentally ingested ricin. Her chief accomplice is a Pakistani national with diplomatic immunity. But you aren't going to hear that in the news."
"I suppose not. There's no way the American government is going to blame the Pakistani Embassy in Washington, D.C. of coordinating or supporting a terrorist attack on our home soil."
"More than likely our government will use it as a stick to persuade Pakistan in other matters."
"Now you're thinking like a politician."
"That's an insult."
"I'm glad you think so." Mr. Stanley took a sip of his coffee and put the cup back on its saucer.
"Anyhow, looks like they are hanging the whole incident on this dead Syed character. The CIA is portraying him as the mastermind behind the attack and that faulty homemade triggers on the bombs prevented further disaster."
"No shock there. The CIA is in the business of _misinformation_ as much as they are in the business of collecting information."
"Even so, I'm not sure how they managed to keep a lid on the minivan loaded with explosives at Reagan National. By the time we were removed from Gravelly Point, there were a lot of eyes on the scene: Airport police, D.C. police, tourists."
"I guess I shouldn't expect to see an article on a young man and police officer who prevented a thousand casualties..."
"Saving a thousand lives but losing a hundred is not much of a consolation prize."
"No, I guess it's not. But so far, only 26 are dead."
"Many of those exposed won't make it."
The appropriate level of silence fell on the conversation with the mention of mass causalities. "Are the CIA and FBI done with you yet?" Mr. Stanley asked.
"They didn't demand that I come downtown today."
Mr. Stanley steered the conversation away from Clark. "Did you see they came back to clean out Coleman's Castle?"
"I thought the house was already empty."
"Empty is a relative term. _Now, it's empty_. After they discovered the firewood laced with cyanide and realized that Allan didn't die from lack of vegetables or exercise in his diet, I guess things changed. Last night they sterilized the place. Big trucks with no markings. Movers with masks. They took everything. I saw them wheeling out the refrigerator, the stove, the water heater. Even insulation. They took the whole shed from the backyard, uprooted from its foundation."
"Probably the same team that went through the charred remains at Ariana's."
"Probably."
"You know, at this point, I really just don't want to know any more. I'm sure there are a few bodies out there we'll never know about. Nazim was never found, for one, but you know he didn't make it far. Detective Wallace told me they are investigating the disappearance of a delivery guy from the company that sold Ariana the machines used to process the ricin. He also mentioned that they're looking into the suspicious death of a sales guy at an outdoor adventure store in Bailey's Crossroads. They traced some of the material found in the warehouse back to the store and have the dead employee on surveillance tapes selling camping equipment to Ariana."
Mr. Stanley paused. He wiped a crumb from his morning toast off his burgundy pajama top and took a sip of coffee.
"You still interested in finding out whether Ariana had anything to do with your father?"
Clark looked out the window at a fresh blanket of snow on the tree limbs. "My father lived a long life. Nothing will bring him back. Besides, my mother would never allow for his body to be exhumed. She still has the last tube of toothpaste he used, for crying out loud."
"How about your police officer friend? How did he fair with all this?"
"We all got the same deal. The detective, Lisa, the sheriff, myself. _Keep our mouths shut and all will be forgiven_."
"Not much of a deal for heroes."
"Not much of a choice either."
"And the IRS?"
"I am still working through that one."
"Well, just remember what Ben Franklin said."
"What's that?"
"Death and taxes, son. Death and taxes. The two are inevitable."
## Chapter 65
Clark approached Lisa at a secluded table at the back of Jammin' Java. A band was setting up for the evening's early performance, a trolley full of equipment parked next to the stage. The long haired guitarist was perched on a stool on the far side of the stage, strumming an acoustic guitar and staring out the window at the alley behind the building. A heavy-set man in need of a belt stooped and re-stooped to pluck equipment from the trolley. The rest of the band entourage moved around the stage at a slow steady pace. Mics, speakers, wires, and instruments all carefully found their way to the appropriate locations.
Clark feigned left and then went right, kissing Lisa on the ear as he sat. He placed his backpack conspicuously in the middle of the table.
"How are you holding up?" Lisa asked.
"Good. Today was quiet. Hopefully it is the beginning of a trend. And you?"
"I filed an official report on my involvement in the case directly with the Commissioner of the IRS. I will have an official warning placed in my permanent file, but the warning will be without explanation. If I keep my nose clean, the warning will be removed in two years. A slap on the wrist, essentially."
"But without saying why."
"Exactly."
"Our government... I swear."
"Well, I think both you and I would be in more serious trouble if not for the fact that you saved a few lives and the fact that the CIA and FBI don't want any of this to go public. A grad student, an IRS agent, a police officer, and a sheriff were all one step ahead of the FBI. Technically, the CIA was one step ahead of all of us. I mean, their agent was implanted in a terrorist cell on U.S. soil. That is some modicum of success."
"True enough."
Clark started to speak again, mumbled something unintelligible and then paused.
Lisa rescued him. "What's in the bag?"
"Why do you ask?"
"You put it on the table. You must want to show me something; otherwise you would have put it on the floor."
Clark looked at the bag sheepishly. "Well, you know how I was adamant about my father being on the up and up with regard to paying taxes?"
"Yes, I remember very clearly."
"Well, there is a teeny tiny possibility that I may have been wrong." Clark unzipped the backpack and removed the old shoebox he had found in the basement. He pulled the lid and put it on the table next to the backpack and the shoebox.
"Holy crap," Lisa replied, looking at the contents.
"Yeah."
"How much is in there?"
"Well, I didn't take it out of the rubber bands, but I flipped through the upper corner of the bills. Somewhere just over $67,000."
"Where did you find it?"
"In the basement. But I have been carrying it around in this bag for the last couple of days. I was kind of hoping I would be mugged so I wouldn't have to make a decision on it."
"Why didn't you tell me sooner?"
"I was going to."
"But you were thinking about keeping it? Please tell me that is not the case."
"I would be lying if I said 'no.'"
Lisa sighed and rubbed her temples with both hands. "Well, I will handle it."
"Sorry," Clark said. "There was a lot of temptation there."
"I am going to chalk it up to stress. Who knows, maybe this will count in my favor. I mean, technically, if you did just find this in your house, then I don't see how either of us could end up in any more trouble. But you should have called me sooner."
"I thought about bringing it over to your apartment the other night, but I didn't want to send the wrong message. Some girls would take issue with a man sleeping over and leaving a pile of cash on the bedside table on their way out."
"Some. But this is not a few hundred dollars. Some girls may actually be flattered by this much cash. It could do wonders for a girl's ego."
"How about just a cup of coffee from a broke grad student who happens to also be a super hero in disguise?"
"I guess it will have to do," Lisa replied in feigned disappointment. "For starters."
## About the Author
Mark Gilleo holds a graduate degree in international business from the University of South Carolina and an undergraduate degree in business from George Mason University. He enjoys traveling, has lived and worked in Asia, and speaks fluent Japanese. A fourth-generation Washingtonian, he currently resides in the D.C. area. His two most recent novels were recognized as finalist and semifinalist, respectively, in the William Faulkner-Wisdom Creative writing competition. The Story Plant will publish his next novel, _Sweat_ , in 2012.
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{
"redpajama_set_name": "RedPajamaBook"
}
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Spherillo setaceus är en kräftdjursart som beskrevs av Gustav Budde-Lund 1904. Spherillo setaceus ingår i släktet Spherillo och familjen Armadillidae. Inga underarter finns listade i Catalogue of Life.
Källor
Gråsuggor och tånglöss
setaceus
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,428
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The Five Foundation is an organisation working towards the elimination of the practice of female genital mutilation (FGM). It was founded by Nimco Ali and Brendan Wynne. It was launched in September 2019 in New York and currently has signed-up partners which include Plan International, Action Aid, The ONE Campaign, Save The Children (UK), UN Women (UK) and Women for Women International.
In 2019, The Five Foundation advocated to have FGM included in the UK Children's Act. The Foundation has also been active in raising awareness of a case of a Kenyan doctor's attempts to legalise FGM, persuading Sudan to ban FGM in May 2020, working on a fatwa banning child marriage in June 2019, communicating how the Covid-19 pandemic has increased the risk of FGM, and being a central player in getting the US Stop FGM Act passed in January 2021.
The Five Foundation also advocates for increased funding for grassroots African women's organisations and on the need to end the medicalisation of FGM.
References
External links
Female genital mutilation
Women's rights organizations
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,406
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Governor Wolf Announces New Approvals for Four Small Business Projects
Economy, Jobs That Pay, Press Release
Harrisburg, PA – Governor Tom Wolf announced this month's new low-interest loan approvals through the Pennsylvania Industrial Development Authority (PIDA) program for four projects throughout the commonwealth. The investments will create and retain more than 80 full-time jobs.
"This administration's dedication to supporting Pennsylvania businesses is evidenced by the loans approved this month," said Governor Wolf. "From restaurants to farms to medical device companies, we are committed to assisting all types of business in the state, and our efforts to work with regional partners to support these projects will pay dividends as we continue to create strong communities and a positive business climate."
Through June 2017, PIDA has approved $27 million in low-interest loans for 33 projects that have resulted in $140 million in private investment and supported 1,316 new and retained full-time jobs.
Over the past month, PIDA approved a total of over $2.7 million in low-interest loans to four projects in Adams, Lancaster, Snyder, and Westmoreland counties that will generate nearly $3.5 million in additional private investment.
The following loans were approved during the past month:
The Adams County Economic Development Corporation was approved for a $400,000, 15-year loan at 4 percent, on behalf of John E. Boyer to assist with the purchase of a 232-acre farm located in Gettysburg, Adams County. The farm will produce crops and operate a cattle and calf operation.
EDC Finance Corporation was approved for a $1,300,000, 15-year loan at 3 percent for the first seven years, on behalf of Creek Hill, LLC for the acquisition of a 48,750-square-foot building located on three acres in Leola, Lancaster County. Creek Hill will operate the building as a multi-tenant facility for manufacturing and industrial uses.
Snyder County
SEDA-Council of Governments was approved for a $76,500 loan, at 4 percent for twelve years, on behalf of Robert A. Kirkpatrick. Funding will be used to assist with the building renovations and equipment purchases for BJ's Steak & Rib House, which has been operating in Selinsgrove, Snyder County, for the last 35 years. Through the project, BJ's Steak and Rib House has committed to retain its current twelve full-time employees.
Westmoreland County
Economic Growth Connection of Westmoreland was approved for a $933,343; 15-year loan at 3 percent for the first seven years, on behalf of Xodus Medical, Inc., a manufacturer and distributer of medical equipment and supplies. Funding will be used for the construction of a 30,000-square-foot warehouse building adjacent to the company's current production facility in New Kensington, Westmoreland County. This will allow the company to add more production lines to meet increased demand, marking the company's third expansion utilizing the PIDA program in the past 13 years. PIDA assistance has aided the company in growth from 12 employees in 2004 to the 50 existing positions that will be retained through this project, as well as the creation of 20 new jobs.
PIDA is an independent authority staffed and regulated through the Department of Community and Economic Development. The authority provides capital for building acquisition, construction and renovation work, machinery and equipment loans along with working capital lines of credit, primarily for manufacturers, industrial developers, research and development firms, agricultural processors, and employers looking to establish national or regional headquarters in Pennsylvania.
At today's meeting, PIDA approved to offer eligible qualified applicants a fixed interest rate of 2 percent for real estate financing, equipment loans, and 12-month periods for working capital and accounts receivable lines of credit for a limited time.
For more information about the Pennsylvania Industrial Development Authority and other DCED initiatives, visit dced.pa.gov.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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There's No Peace in the Middle East
The Departure of Michael Scott
"Hundreds of Saudis have rallied in the eastern city of Qatif to blast the regime for its military intervention in #Bahrain. #SaudiArabia." Tweets like these have become the norm in the Middle East, as social networking sites have evolved into the online headquarters for rebellion. Since the initial uprising in January in Egypt, there has been a string of similar incidents in Tunisia, Libya, Yemen, Bahrain and Syria. Protestors in Egypt began their fight on Jan. 25 against President Hosni Mubarak, who was in power for almost 30 years. After violent clashes, mostly based in Tahrir Square in Cairo and resulting in more than 300 deaths, Mubarak resigned on Feb. 11. Soon after, a referendum was held for constitutional changes to ensure competitive elections. Days later, Libyans began protesting Moammar Gadhafi, their leader for 32 years. Gadhafi responded with military force—after which many Libyan officials resigned, condemning his attacks. In March, the U.N. Security Council voted to permit "all means necessary" to establish a no-fly zone and impose a cease-fire. Gadhafi continued his attacks, causing the U.S. and other allies to enforce the no-fly zone. Meanwhile, fighting broke out in Bahrain between the minority ruling Sunnis and the majority Shiites, with the latter citing discrimination. And in Syria, under pressure from protestors, President Bashar al-Assad pledged to lift the state of emergency that had been in place since 1963, and which gave security forces the power to arrest people without a warrant or trial.
Through updates on Twitter and Facebook, the world continues to watch as those in the Middle East fight for better lives.
RELEVANT May 2, 2011
Does Having Faith Make Young Adults Fat?
Flavor of the Bi-Month
Life Life
Life Life, Wellness
Life Career & Money, Life
Life Growth, Life
4 Decisions That Will Change Your Life
Four Questions that Will Point You to Your Purpose
Life Career & Money, Faith + Work, Life
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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On the occasion of the forthcoming 25th anniversary of Casa Olearia Taggiasca (1995-2020) I decided to write a book to celebrate, ironically and unconventionally, this tasty occasion.
Who knows me well is aware that I have always been an "out-of-the-box" soul, distant form the clichés, so, for this book too, written four-hands with Alessandro Maria Ferreri, I took a cue from an original idea I'd been meditating on for a while.
It is an open secret that, nowadays, the chef career has become an incredibly sought-after profession.
Those who, like us, have been behind the scenes, hand in hand with the greatest chefs in the world, from their very beginnings to the achievement of their third Michelin star, know that the fine cuisine requires considerable sacrifices, commitment and, above all, deep passion.
Not only supporters, but also friends with these brilliant and talented professionals, we have frequently had the honor to share moments of their private life and, it was precisely during one of these cheerful occasions that, between a laugh and another, we asked them: "What would you make for breakfast if your girlfriend slept over?"
We expected the most elaborate answer, yet, we were surprised to hear how simple the meals imagined by these chefs for their personal life were: closely tied to childhood memories and prepared just like every mum does every day.
Hence the idea to place, ironically, these starred-stars in an out-of-place scenario, far from their comfort zone. We imagined an odd and unusual tailor-made situation to challenge their abilities in light of a potential unfitness or as a sort of poetic justice.
A mind-blowing experience came out of it: we gathered 15 of our friend-chefs and they immediately fell in love with our project, everyone tied by an underlying thread: the deep devotion for the extra virgin olive oil, the common passion which brought us together in the first place.
From Tokyo to New York, from Paris to Dubai, from Modena to London and Bangkok, a team of photographers, stylists and chefs assisted us in a real tour de force of which this book is the journal.
Everything, completely "OUT OF PLACE", which has become the title of the book, edited by MONDADORI, which will be available both in Italian and English from October 21st2019.
Hope you'll enjoy the reading! It has been an unforgettable experience for us and I express all my gratitude to my friend-chefs, without which, this project would have just remained an idea
Shop Objet La Table About us Food
GALATEO & FRIENDS Via Madonna della Neve 11 18013 Diano Castello (IM) | tel. e fax 0183.497691 | info@galateofriends.it
Concept by Marco Bonaldo | Design Ellelle studio Sanremo | Powered by Esedigital
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,000
|
\section{Introduction}
In ref.~\cite{Crivellin:2021ubm} several ATLAS and CMS resonance searches are combined into six channels. In each channel, the background model predicts a smooth non-resonant spectrum parameterised by four nuisance parameters, collectively denoted by $\boldsymbol\theta$. The signal model, on the other hand, predicts a Crystal-ball shaped resonance in each channel parameterised by seven parameters: a single mass parameter, $m$, common to each channel, and six positive signal strength parameters $\mu_i \ge 0$ for $i = 1 \text{ -- } 6$, one for each channel.
Ref.~\cite{Crivellin:2021ubm} construct a test-statistic based on a profiled likelihood ratio for the background model and background plus signal model,
\begin{equation}\label{eq:lambda}
\lambda \equiv -2 \log{\frac{
\max_{m,\boldsymbol{\mu}, \boldsymbol\theta} \mathcal{L}(m, \boldsymbol{\mu}, \boldsymbol\theta)
}{
\max_{\boldsymbol\theta} \mathcal{L}(\boldsymbol{\mu} = 0, \boldsymbol\theta)
}},
\end{equation}
where $\mathcal{L}(m, \boldsymbol{\mu}, \boldsymbol\theta)$ is the likelihood of the observed data for the model with parameters $m$, $\boldsymbol{\mu}$ and $\boldsymbol\theta$, $\boldsymbol{\mu} = 0$ corresponds to the background model, and the maximisation over the signal strengths is constrained by $\mu_i \ge 0$. Ref.~\cite{Crivellin:2021ubm} find $\lambda = 26.01$~\cite{email}; we haven't verified this result, but assume that it is correct. This test-statistic is used to compute local and global significances. We discuss the interpretation of significances as evidence in ref.~\cite{Fowlie:2019ydo,Fowlie2021Comment}; we focus here only on their correct computation. Although the computation is complicated by the look-elsewhere effect (LEE) and the fact that the signal strength parameters must be positive, the issue with ref.~\cite{Crivellin:2021ubm} is simple: they computed the significances as though they fitted a single signal when in fact they fitted six independent signal strengths.
\section{Local significance}
To compute significances, it is common to resort to asymptotic formulae (see e.g., ref.~\cite{Cowan:2010js}) based on Wilks' theorem~\cite{Wilks:1938dza}. Roughly speaking, this theorem states that test-statistics of the form \cref{eq:lambda} are asymptotically $\chi^2_k$ distributed, where $k$ is the number of parameters describing the signal. The resonance search, however, violates a required assumption (see e.g., ref.~\cite{Algeri:2020pql}), as the mass parameter isn't identifiable under the background model, i.e., when $\boldsymbol{\mu} = 0$, the model doesn't depend on $m$.
By simply fixing the mass parameter to its best-fit value $\hat m$, however, we may compute a local \textit{p-}value\xspace. In this case, the test-statistic splits into independent contributions for each channel,
\begin{equation}\label{eq:lambda_sum}
\lambda = \sum_{i=1}^6
-2 \log{\frac{
\max_{\mu_i,\theta_i} \mathcal{L}_i(\hat m, \mu_i, \theta_i)
}{
\max_{\theta_i} \mathcal{L}_i(\mu_i = 0, \theta_i)
}}.
\end{equation}
There is a further complication, though, as the signal strength parameters must be positive, such that the background model lies at the boundary rather than in the interior of the background plus signal model. This is addressed by Chernoff's modification to Wilks' theorem~\cite{10.1214/aoms/1177728725} and further explored in ref.~\cite{10.2307/2289471}. For each channel, if the best-fit signal at a fixed mass fluctuates downwards, that channel's contribution to the test-statistic in \cref{eq:lambda_sum} vanishes. If it fluctuates upwards, that channel's contribution to the test-statistic is described by a chi-squared distribution with one degree of freedom as usual. This is a mixture distribution called a half chi-square distribution, $\ensuremath{\mbox{\footnotesize\ensuremath{\frac12}}\chi^2}$. The probability density function is (see case 5 in ref.~\cite{10.2307/2289471} or discussion around eq.~52 in ref.~\cite{Cowan:2010js})
\begin{equation}\label{eq:half}
p(\lambda) = \frac{1}{2} \delta(\lambda) + \frac{1}{2} p_{\chi^2_1}(\lambda).
\end{equation}
where $p_{\chi^2_1}(\lambda)$ is the probability density function for a $\chi^2$ distribution with one degree of freedom. The terms in this mixture correspond to downwards and upwards fluctuations and the coefficients are half because downwards and upwards fluctuations are equally likely.
The test-statistic for $n$ channels as in \cref{eq:lambda_sum} for $n = 6$ is thus the sum of $n$ half-chi-squared variates. We denote this as a $\ensuremath{\mbox{\footnotesize\ensuremath{\frac12}}\chi^2}_n$ distribution. The density is (see case 9 with $s = 0$ in ref.~\cite{10.2307/2289471})
\begin{equation}\label{eq:n}
p(\lambda) = \left(\frac12\right)^n \sum_{i=0}^n {n \choose i} p_{\chi^2_i}(\lambda),
\end{equation}
where $p_{\chi^2_0}(\lambda) = \delta(\lambda)$. The coefficients are just the binomial coefficients describing the chances of $i$ upwards fluctuations at the fixed mass in $n$ channels. When $i$ of them fluctuate upwards, the test-statistic is a sum of $i$ chi-squared variates which follows a $\chi^2_i$ distribution. When $n = 1$, this reduces to \cref{eq:half}.
In the $n = 1$ case, the density in \cref{eq:half} leads to the simple result for the local significance (eq.~52 in ref.~\cite{Cowan:2010js}),
\begin{equation}\label{eq:Z_local}
Z_\text{Local} = \sqrt{\lambda}.
\end{equation}
This does not apply when $n \neq 1$ but was used~\cite{email} in ref.~\cite{Crivellin:2021ubm} to calculate $Z_\text{Local} = 5.1\sigma$.
In contrast, using \cref{eq:n}, we obtain $Z_\text{Local} = 4.1\sigma$ when combining $n = 6$ channels as in ref.~\cite{Crivellin:2021ubm}.
We validate this result through Monte Carlo (MC) simulations described in \cref{sec:global}. Thus to avoid overstating the significance and reaching potentially faulty conclusions, we must take into account that we searched six channels.
\section{Global significance}\label{sec:global}
\begin{figure}[t]
\centering
\includegraphics[clip, trim=0cm 0.7cm 0cm 0.6cm, width=0.99\linewidth]{dist.pdf}
\caption{Distribution of test-statistics from MC simulations. The $\ensuremath{\mbox{\footnotesize\ensuremath{\frac12}}\chi^2}$ and $\ensuremath{\mbox{\footnotesize\ensuremath{\frac12}}\chi^2}_6$ distributions are shown by smooth lines for comparison.}\label{fig:dist}
\end{figure}
We previously considered a local significance based on a fixed mass. We should compute a global significance by taking into account every test that was performed and every test that would have been performed were the data different (see problems 1 and 2 in ref.~\cite{Wagenmakers2007} for a pedagogical if partisan discussion). Unfortunately, we don't know the analysis plan of the authors of ref.~\cite{Crivellin:2021ubm} and thus there are imponderable look-elsewhere effects associated with their choices of statistical tests and datasets. Even if the authors of ref.~\cite{Crivellin:2021ubm} began with a particular theory of a new scalar boson in mind, many analysis choices could have been influenced by the observed data. We are not, however, suggesting conscious hacking of any sort and direct readers to the nuanced discussion in ref.~\cite{forking_paths,gelman2014statistical}.
We don't further discuss these issues and instead compute the global significance following the scope of the LEE considered in ref.~\cite{Crivellin:2021ubm} by taking into account the LEE for the mass range that was searched. In this case, we consider the test-statistic as a random field over the mass range and consider its most extreme fluctuation. Ref.~\cite{Crivellin:2021ubm} accounted for this LEE by multiplying the local \textit{p-}value\xspace by a trial factor of 5~\cite{email}. This was an estimate motivated by the $140 \,\text{GeV}$ -- $155\,\text{GeV}$ range and the $1.5\,\text{GeV}$ and $14\,\text{GeV}$ resolutions in the six channels. We instead compute it.
There are rules of thumb for the trial factor based on the range, $\Delta m$, and resolution, $\sigma_m$, e.g.,
\begin{equation}\label{eq:N_thumb}
N \approx \frac13 \frac{\Delta m}{\sigma_m} Z_\text{Local}.
\end{equation}
This rule yields a trial factor of about 14, but it was found and tested only for the case of $n=1$ channels~\cite{thumb}. Thus rather than using a rule of thumb, we compute it through MC simulations and the Gross-Vitells method~(see e.g.,~ref.~\cite{adler:2007,Gross:2010qma,doi:10.1080/10618600.2019.1677474}).
The Gross-Vitells method is a sophisticated treatment based on the expected numbers of up-crossings of random fields above specified thresholds.
To apply it, we required the so-called Euler characteristic densities for the $\frac12 \chi^2_6$ random field and an estimate of the expected number of up-crossings at a specified level. From theorem 1 and remark 2 in ref.~\cite{taylor2013detecting}, the Euler characteristic densities for the mixture $\frac12 \chi^2_6$ are just the same mixtures of the known densities for $\chi^2_n$ random fields (see theorem 15.10.1 in ref.~\cite{adler:2007}). To estimate the expected number of up-crossings, we perform simulations using a toy treatment of a resonance problem with five channels with resolution $1.5\,\text{GeV}$ and one channel with resolution $14\,\text{GeV}$ in a window $140 \,\text{GeV}$ -- $155\,\text{GeV}$. In this toy treatment, we fit the mass and signal strengths of a Crystal ball function on top of a fixed background. The resulting test-statistic should obey the same asymptotic distribution as that in the full treatment. We ultimately find a global significance of about $3.5\sigma$, which corresponds to a trial factor of about 12.
To check the asymptotic results for the local and global significances, we perform MC simulations of the test-statistic. We perform $100\,000$ pseudo-experiments with the toy treatment of the problem described above, computing the test-statistics \cref{eq:lambda,eq:lambda_sum} in each case. By simply counting the number of simulations in which the test-statistics exceeded that observed, we again find $4.1\sigma$ local and $3.5\sigma$ global significances, validating our previous results.\footnote{Note though that this does not validate that the asymptotic limit was valid in the real problem, as our toy treatment used large numbers of expected events in all channels.} The distribution of the test-statistic and the test-statistic at a fixed mass of $151\,\text{GeV}$ are shown in \cref{fig:dist}. We see that the simulations in the latter case closely matches a $\ensuremath{\mbox{\footnotesize\ensuremath{\frac12}}\chi^2}_6$ distribution, as expected.
\section{Summary}
Taking into account the fact that in ref.~\cite{Crivellin:2021ubm} six signals were fitted rather than one and properly computing the LEE, we find the significances for a new neutral scalar are only $4.1\sigma$ local and $3.5\sigma$ global. Thus while this anomaly may remain intriguing, it isn't as significant as first thought.
\section*{Acknowledgements}
I was supported by an NSFC Research Fund for International Young Scientists grant 11950410509. I thank the authors of ref.~\cite{Crivellin:2021ubm} for time spent clarifying details of their paper.
\newcommand{\JournalTitle}[1]{#1}
\bibliographystyle{references}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,485
|
var Movie = require('./Movie.js');
describe('Movie Model', function() {
var testMovieData = {
"poster_path": "/tvSlBzAdRE29bZe5yYWrJ2ds137.jpg",
"adult": true,
"overview": "overview description",
"release_date": "1977-05-25",
"genre_ids": [
12,
28,
878
],
"id": 11,
"original_title": "Star Wars",
"original_language": "en",
"title": "Star Wars",
"backdrop_path": "/4iJfYYoQzZcONB9hNzg0J0wWyPH.jpg",
"popularity": 16.619988,
"vote_count": 4958,
"video": false,
"vote_average": 8
};
var movieModel = new Movie(testMovieData);
it('should contain a string property for poster_path', function(){
expect(typeof movieModel['poster_path'] === 'string').toBe(true);
});
it('should contain a string value for poster_path', function(){
expect(movieModel['poster_path']).toEqual(testMovieData['poster_path']);
});
it('should contain a boolean property for adult', function() {
expect(typeof movieModel['adult'] === 'boolean').toBe(true);
});
it('should contain a boolean value for adult', function() {
expect(movieModel['adult']).toBe(true);
});
it('should contain a string property for overview', function() {
expect(typeof movieModel['overview'] === 'string').toBe(true);
});
it('should contain a string value for overview', function() {
expect(movieModel['overview']).toEqual(testMovieData['overview']);
});
/* Could create a Date object */
it('should contain a string property for release_date', function() {
expect(typeof movieModel['release_date'] === 'string').toBe(true);
});
it('should contain a string value for release_date', function() {
expect(movieModel['release_date']).toEqual(testMovieData['release_date']);
});
it('should contain a Array property for genre_ids', function() {
expect(Array.isArray(movieModel['genre_ids'])).toBe(true);
});
it('should contain a array value for genre_ids', function() {
expect(movieModel['genre_ids']).toEqual(testMovieData['genre_ids']);
});
it('should contain a number property for id', function() {
expect(Number.isInteger(movieModel['id'])).toBe(true);
});
it('should contain a number value for id', function() {
expect(movieModel['id']).toEqual(testMovieData['id']);
});
it('should contain a string property for original_title', function() {
expect(typeof movieModel['original_title'] === 'string').toBe(true);
});
it('should contain a string value for original_title', function() {
expect(movieModel['original_title']).toEqual(testMovieData['original_title']);
});
it('should contain a string property for original_language', function() {
expect(typeof movieModel['original_language'] === 'string').toBe(true);
});
it('should contain a string value for original_language', function() {
expect(movieModel['original_language']).toEqual(testMovieData['original_language']);
});
it('should contain a string property for title', function() {
expect(typeof movieModel['title'] === 'string').toBe(true);
});
it('should contain a string value for title', function() {
expect(movieModel['title']).toEqual(testMovieData['title']);
});
it('should contain a string property for backdrop_path', function() {
expect(typeof movieModel['backdrop_path'] === 'string').toBe(true);
});
it('should contain a string value for backdrop_path', function() {
expect(movieModel['backdrop_path']).toEqual(testMovieData['backdrop_path']);
});
it('should contain a number property for popularity', function() {
expect(typeof movieModel['popularity'] === 'number').toBe(true);
});
it('should contain a number value for popularity', function() {
expect(movieModel['popularity']).toEqual(testMovieData['popularity']);
});
it('should contain a number property for vote_count', function() {
expect(typeof movieModel['vote_count'] === 'number').toBe(true);
});
it('should contain a number value for vote_count', function() {
expect(movieModel['vote_count']).toEqual(testMovieData['vote_count']);
});
it('should contain a boolean property for video', function() {
expect(typeof movieModel['video'] === 'boolean').toBe(true);
});
it('should contain a boolean value for video', function() {
expect(movieModel['video']).toEqual(testMovieData['video']);
});
it('should contain a number property for vote_average', function() {
expect(typeof movieModel['vote_average'] === 'number').toBe(true);
});
it('should contain a number value for vote_average', function() {
expect(movieModel['vote_average']).toEqual(testMovieData['vote_average']);
});
});
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,807
|
Produced by Marius Masi, Don Kretz and the Online
Distributed Proofreading Team at http://www.pgdp.net
THE
ENCYCLOPAEDIA BRITANNICA
ELEVENTH EDITION
FIRST edition, published in three volumes, 1768-1771.
SECOND " " ten " 1777-1784.
THIRD " " eighteen " 1788-1797.
FOURTH " " twenty " 1801-1810.
FIFTH " " twenty " 1815-1817.
SIXTH " " twenty " 1823-1824.
SEVENTH " " twenty-one " 1830-1842.
EIGHTH " " twenty-two " 1853-1860.
NINTH " " twenty-five " 1875-1889.
TENTH " ninth edition and eleven
supplementary volumes, 1902-1903.
ELEVENTH " published in twenty-nine volumes, 1910-1911.
COPYRIGHT
in all countries subscribing to the
Bern Convention
by
THE CHANCELLOR, MASTERS AND SCHOLARS
of the
UNIVERSITY OF CAMBRIDGE
_All rights reserved_
THE
ENCYCLOPAEDIA BRITANNICA
A
DICTIONARY
OF
ARTS, SCIENCES, LITERATURE AND GENERAL
INFORMATION
ELEVENTH EDITION
VOLUME V
CALHOUN to CHATELAINE
New York
Encyclopaedia Britannica, Inc.
342 Madison Avenue
Copyright, in the United States of America, 1910,
by
The Encyclopaedia Britannica Company.
INITIALS USED IN VOLUME V. TO IDENTIFY INDIVIDUAL CONTRIBUTORS,[1]
WITH THE HEADINGS OF THE ARTICLES IN THIS VOLUME SO SIGNED.
A. Bo.*
AUGUSTE BOUDINHON, D.D., D.C.L.
Professor of Canon Law at the Catholic University of Paris.
Honorary Canon of Paris. Editor of the _Canoniste Contemporain_.
Canon Law: _General_;
Cardinal.
A. C. S.
ALGERNON CHARLES SWINBURNE.
See the biographical article, SWINBURNE, ALGERNON CHARLES.
Chapman George (_in part_).
A. E. H.
A. E. HOUGHTON.
Formerly Correspondent of The _Standard_ in Spain. Author of
_Restoration of the Bourbons in Spain_.
Camacho;
Canovas del Castillo;
Castelar y Ripoll.
A. E. S.
ARTHUR EVERETT SHIPLEY, F.R.S., F.Z.S., F.L.S.
Fellow, Tutor and Lecturer of Christ's College, Cambridge.
University Reader in Zoology. Formerly University Lecturer on the
Advanced Morphology of the Invertebrata. Author of _Zoology of
the Invertebrata_. Editor of the Pitt Press _Natural Science
Manuals_, &c.
Chaetognatha;
Chaetosomatida.
A. Go.*
REV. ALEXANDER GORDON, M.A.
Lecturer in Church History at the University of Manchester.
Carranza.
A. H. J. G.
ABEL HENDY JONES GREENIDGE, D.LITT. (Oxon.), (d. 1905).
Formerly Fellow and Lecturer of Hertford College, Oxford, and of
St John's College, Oxford. Author of _Infamia in Roman Law_;
_Handbook of Greek Constitutional History_; _Roman Public Life,
History of Rome_. Joint-author of _Sources of Roman History_,
133-70 B.C.
Censor: _Ancient_.
A. H. S.
REV. ARCHIBALD HENRY SAYCE, D.LITT., LL.D.
See the biographical article, SAYCE, A. H.
Caria.
A. J. G.
REV. ALEXANDER JAMES GRIEVE, M.A., B.D.
Professor of New Testament and Church History at the United
Independent College, Bradford. Sometime Registrar of Madras
University and Member of Mysore Educational Service.
Catechism;
Calvin (_in part_).
A. L.
ANDREW LANG.
See the biographical article, LANG, ANDREW.
Casket Letters.
A. Lo.
AUGUSTE LONGNON.
Professor at the College de France. Director of the Ecole des
hautes etudes. Chevalier of the Legion of Honour. Member of the
Institute. Author of _Livre des vassaux du Comte de Champagne et
de Brie_; _Geographie de la Gaule au VI siecle_; _Atlas historique
de la France depuis Cesar jusqu'a nos jours_; &c.
Champagne.
A. M. C.
AGNES MARY CLERKE.
See the biographical article, CLERKE, A. M.
Cassini.
A. M. Cl.
AGNES MURIEL CLAY (MRS WILDE).
Late Resident Tutor of Lady Margaret Hall, Oxford. Joint-author of
_Sources of Roman History_, 133-70 B.C.
Centumviri.
A. N.
ALFRED NEWTON, F.R.S.
See the biographical article, NEWTON, ALFRED.
Canary;
Capercally.
A. P. C.
ARTHUR PHILEMON COLEMAN. F.R.S.
Professor of Geology, University of Toronto.
Canada: _Geography_.
A. P. H.
ALFRED PETER HILLIER, M.D., M.P.
Author of _South African Studies_; _The Commonweal_; &c. Served in
Kaffir War, 1878-1879. Partner with Dr L. S. Jameson in medical
practice in South Africa till 1096. Member of Reform Committee,
Johannesburg, and Political Prisoner at Pretoria, 1895-1896. M.P.
for Hitchin division of Herts, 1910.
Cape Colony: S _History_ (_in part_).
A. SI.
ARTHUR SHADWELL, M.A., M.D., LL.D., F.R.C.P.
Member of Council of Epidemiological Society. Author of _The
London Water-Supply_; _Industrial Efficiency_; _Drink, Temperante
and Legislation_.
Cancer.
A. S. C.
ALAN SUMMERLY COLE, C.B.
Assistant Secretary for Art, Board of Education, 1900-1908. Took
part in organization of the Textile Manufacturers' Section, St
Louis Exhibition, 1904. Author of _Ancient Needle Point and Pillow
Lace_; _Embroidery and Lace_; _Ornament in European Silks_;&c.
Carpet.
A. V. De P.
A. VAN DE PUT.
Assistant, Victoria and Albert Museum, South Kensington. Author of
_Hispano-Moresque Ware of the XV. Century_; _The Aragonese
Double-Crown and the Borja or Borgia Device_.
Ceramics: S _Hispano-Moresque._
A. Wa.
ARTHUR WAUGH, M.A.
Managing Director of Chapman & Hall, Ltd., Publishers. Formerly
literary adviser to Kegan Paul & Co. Author of _Alfred Lord
Tennyson_; _Legends of the Wheel_; _Robert Browning_ in
"Westminster Biographies." Editor of Johnson's _Lives of the
Poets_.
Calverley, C. S.
A. W. H.*
ARTHUR WILLIAM HOLLAND.
Formerly Scholar of St John's College, Oxford. Bacon Scholar of
Gray's Inn, 1900.
Charlemagne.
A. Z.
ALICE ZIMMERN.
Author of _Methods of Education in the United States_; _The
Renaissance of Girls' Education in England_; _Women's Suffrage in
Many Lands_; &c.
Carpenter, Mary.
B. Bl.
BERTRAM BLOUNT, F.C.S., F.I.C.
Consulting Chemist to the Crown Agents for the Colonies. Hon.
President, Cement Section of International Association for Testing
Materials, Buda-Pesth. Author of _Practical Electro-Chemistry_.
Cement.
B. Ra.
BERNARD RACKHAM, M.A.
Assistant, Victoria and Albert Museum, South Kensington.
Ceramics: S _German, Dutch and Scandinavian._
C.F.A.
CHARLES FRANCIS ATKINSON.
Formerly Scholar of Queen's College, Oxford. Captain, 1st City of
London (Royal Fusiliers). Author of _The Wilderness and Cold
Harbour_.
Castle (_in part_).
C. F. C.
C. F. CROSS., B.SC. (Lond.), F.C.S., F.I.C.
Analytical and Consulting Chemist.
Cellulose.
C. J. J.
CHARLES JASPER JOLY, F.R.S., F.R.A.S. (1864-1906).
Royal Astronomer of Ireland and Andrews Professor of Astronomy in
the University of Dublin, 1897-1906. Fellow of Trinity College,
Dublin. Secretary of the Royal Irish Academy.
Camera Lucida;
Camera Obscura (_in Part_).
C. L.
H. CALDWELL LIPSETT.
Formerly Editor of the _Civil and Military Gazette_, Lahore,
India. Author of _Lord Curzon in India; &c._
Ceylon (_in part_).
C. Pf.
CHRISTIAN PFISTER, D-ES-L.
Professor at the Sorbonne, Paris. Chevalier of the Legion of
Honour. Author of _Etudes sur le regne de Robert le Pieux_.
Capitulary; Carolingians; Charibert; Charles Martel.
C. R. B.
CHARLES RAYMOND BEAZLEY, M.A., D.LITT.
Professor of Modern History in the University of Birmingham.
Formerly Fellow of Merton College, Oxford. University Lecturer in
the History of Geography. Author of _Henry the Navigator_; _The
Dawn of Modern Geography_; &c.
Cam, Diogo;
Carpini (_in part_);
Chang Chun.
C. S. L.
CHARLES STEWART LOCH, D.C.L. (Oxford), LL.D. (ST ANDREWS).
Secretary to the Council of the London Charity Organization
Society since 1875. Member of the Royal Commission on the Poor
Laws. Dunkin Trust Lecturer, Manchester College, Oxford, 1896 and
1902. Vice-President, Royal Statistical Society,
1894-1895-1897-1901. Author of _Charity Organization_; _Old Age
Pensions and Pauperism_; _Methods of Social Advance_; &c.
Charity and Charities.
D. E. J.
REV. D. E. JENKINS.
Calvinistic Methodist Minister, Denbigh. Author of _Life of Lewis
Charles Edwards of Bala_.
Calvinistic Methodists;
Charles, Thomas.
D. F. T.
DONALD FRANCIS TOVEY.
Author of _Essays in Musical Analysis_: comprising _The Classical
Concerto_, _The Goldberg Variations_, and analyses of many other
classical works.
Cantata.
D. G. H.
DAVID GEORGE HOGARTH, M.A.
Fellow of Magdalen College, Oxford. Fellow of the British Academy.
Keeper of the Ashmolean Museum, Oxford. Excavated at Paphos, 1888;
Naucratis, 1899 and 1903; Ephesus, 1904-1905; Assiut, 1906-1907;
Director, British School at Athens, 1897-1900; Director, Cretan
Exploration Fund, 1899.
Cappadocia (_in part_).
D. H.
DAVID HANNAY.
Formerly British Vice-Consul at Barcelona. Author of _Short
History of the Royal Navy_; _Life of Emilio Castelar_; &c.
Carvajal, Luisa de;
Chateau-Renault.
D. Ll. T.
DANIEL LLEUFER THOMAS.
Barrister at law, Lincoln's Inn. Stipendiary Magistrate at
Pontypridd and Rhondda.
Cardiff.
D. Mn.
REV. DUGALD MACFADYEN, M.A.
Minister of South Grove Congregational Church, Highgate. Author of
_Constructive Congregational Ideals_; &c.
Campbell, John McLeod;
Chalmers, Thomas (_in part_).
E. Ar.
EDWARD ARMSTRONG, M.A.
Fellow of the British Academy. Fellow, Bursar and Lecturer in
Modern History, Queen's College, Oxford. Warden of Bradfield
College. Lecturer to the University in Foreign History, 1902-1904.
Author of _The Emperor Charles V._; _Elisabeth Farnese_; _Lorenzo
de Medici_; _The French Wars of Religion_; &c.
Charles V., Emperor.
E. A. J.
E. ALFRED JONES.
Author of _Old English Gold Plate; Old Church Plate of the Isle of
Man_; _Old Silver Sacramental Vessels of Foreign Protestant
Churches in England_; _Illustrated Catalogue of Leopold de
Rothschild's Collection of Old Plate_; _A Private Catalogue of The
Royal Plate at Windsor Castle_; &c.
Cellini, Benvenuto (_in part_).
E. B.*
ERNEST CHARLES FRANCOIS BABELON.
Professor at the College de France. Keeper of the department of
Medals and Antiquities at the Bibliotheque Nationale. Member of
the Academie des Inscriptions de Belles Lettres, Paris. Chevalier
of the Legion of Honour. Author of _Descriptions Historiques des
Monnaies de la Republique Romaine_; _Traites des Monnaies Grecques
et Romaines_; _Catalogue des Camees de la Bibliotheque Nationale_.
Carthage: _Ancient._
E. C.
EDWARD CAIRD, D.C.L., D.LITT.
See the biographical article, CAIRD, EDWARD.
Cartesianism.
E. C. B.
RT. REV. EDWARD CUTHBERT BUTLER, O.S.B., M.A., D.LITT. (Dublin).
Abbot of Downside Abbey, Bath. Author of "The Lausiac History of
Palladius," in _Cambridge Texts and Studies_, vol. vi.
Camaldulians;
Canon: _Church Dignitary_;
Capuchins;
Carmelites;
Carthusians;
Celestines.
E. C. Q.
EDMUND CROSBY QUIGGIN, M.A.
Fellow of, and Lecturer in Modern Languages and Monro Lecturer in
Celtic at Gonville and Caius College, Cambridge.
Celt: _Languages and Literature_.
E. G.
EDMUND GOSSE, LL.D.
See the biographical article, GOSSE, EDMUND
Canzone;
Carew, Thomas;
Cavendish, George;
Chansons de Geste;
Chant Royal.
E. Gr.
ERNEST ARTHUR GARDNER.
See the biographical article, GARDNER, PERCY.
Calydon;
Ceos;
Cephalonia.
E. H. B.
SIR EDWARD HERBERT BUNBURY, BART., M.A., F.R.G.S. (d. 1895).
M.P. for Bury-St-Edmunds, 1847-1852. Author of a _History of
Ancient Geography_, &c.
Cappadocia (_in part_).
E. H. G.
E. H. GODFREY.
Editor, Census and Statistics Office, Department of Agriculture,
Ottawa.
Canada: S _Agriculture_.
E. H. M.
ELLIS HOVELL MINNS, M.A.
University Lecturer in Palaeography, Cambridge. Lecturer and
Assistant Librarian at Pembroke College, Cambridge. Formerly
Fellow of Pembroke College.
Carpi: _Ancient Tribes_.
E. L. W.
SIR EDWARD LEADER WILLIAMS (d. 1910).
Vice-President, Institute of Civil Engineers. Consulting Engineer,
Manchester Ship Canal. Chief Engineer of the Manchester Ship Canal
during its construction. Author of papers printed in _Proceedings
of Institute of Civil Engineers_.
Canal.
Ed. M.
EDUARD MEYER, PH.D., D.LITT. (Oxon.), LL.D. (Chicago).
Professor of Ancient History in the University of Berlin. Author
of _Geschichte des Alterthums_; _Geschichte des alten Aegyptens_;
_Die Israeliten und ihre Nachbarstamme_.
Cambyses.
E. O.*
EDMUND OWEN, M.B., F.R.C.S., LL.D., D.SC.
Consulting Surgeon to St Mary's Hospital, London, and to the
Children's Hospital, Great Ormond Street, London. Chevalier of the
Legion of Honour. Late Examiner in Surgery at the Universities of
Cambridge, London and Durham. Author of _A Manual of Anatomy for
Senior Students_.
Carbuncle.
E. Pr.
EDGAR PRESTAGE.
Special Lecturer in Portuguese Literature in the University of
Manchester. Examiner in Portuguese in the Universities of London,
Manchester, &c. Commendador Portuguese Order of S. Thiago.
Corresponding Member of Lisbon Royal Academy of Sciences, Lisbon
Geographical Society, &c. Author of _Letters of a Portuguese Nun_;
_Azurara's Chronicle of Guinea_; &c.
Camoens;
Castello Branco;
Castilho.
E. Tn.
REV. ETHELRED LEONARD TAUNTON (d. 1907).
Author of _The English Black Monks of St Benedict_; _History of
the Jesuits in England_; &c.
Campeggio;
Campion, Edmund;
Cano, Melchior;
Cassander, George;
Castellesi.
E. V.
REV. EDMUND VENABLES, M.A., D.D. (1819-1895).
Canon and Precentor of Lincoln. Author of _Episcopal Palaces of
England_.
Catacomb (_in part_).
F. C. C.
FREDERICK CORNWALLIS CONYBEARE, M.A., D.TH. (Giessen).
Fellow of the British Academy. Formerly Fellow of University
College, Oxford. Author of _The Ancient Armenian Texts of
Aristotle_; _Myth, Magic and Morals_ (1909); &c.
Cathars.
F. J. H.
FRANCIS JOHN HAVERFIELD, M.A., LL.D. (Aberdeen), F.S.A.
Camden Professor of Ancient History at Oxford University. Fellow
of Brasenose College, Oxford. Fellow of the British Academy.
Member of the German Imperial Archaeological Institute. Formerly
Senior Censor, Student, Tutor and Librarian of Christ Church,
Oxford. Ford's Lecturer, 1906. Author of Monographs on Roman
History, &c.
Celtiberia; Cassiterides.
F. Ll. G.
FRANCIS LLEWELYN GRIFFITH, M.A., PH.D. (Leipzig), F.S.A.
Reader in Egyptology, Oxford University. Editor of the
Archaeological Survey and Archaeological Reports of the Egypt
Exploration Fund. Fellow of Imperial German Archaeological
Institute.
Canopus.
F. H. M.
COL. FREDERIC NATUSCH MAUDE, C.B.
Lecturer in Military History at Manchester University. Author of
_War and the World Policy_; _The Leipzig Campaign_; _The Jena
Campaign_.
Cavalry.
F. Px.
FRANK PUAUX.
President of the Societe de l'Histoire du Protestantisme francais.
Author of _Les precurseurs francais de la Tolerance_; _Histoire de
l'etablissement des protestants francais en Suede_; _L'Eglise
reformee de France_; &c.
Camisards;
Cavalier, Jean.
F. R. C.
FRANK R. CANA.
Author of _South Africa from the Great Trek to the Union_.
Cameroon;
Cape Colony.
F. W. R.*
FREDERICK WILLIAM RUDLER, I.S.O., F.G.S.
Curator and Librarian of the Museum of Practical Geology, London,
1879-1902. President of the Geologists' Association, 1887-1889.
Carbonado;
Cassiterite;
Cat's Eye;
Celestine;
Chalcedony.
G. A. B.
GEORGE A. BOULENGER, F.R.S., D.SC., PH.D. (Giessen).
Assistant in the Department of Zoology, Natural History Museum,
South Kensington. Vice-President of the Zoological Society.
Carp;
Cat-Fish.
G. G. Co.
GEORGE GORDON COULTON, M.A.
Birkbeck Lecturer in Ecclesiastical History, Trinity College,
Cambridge. Author of _Medieval Studies_; _Chaucer and his
England_; _From St Francis to Dante_; &c.
Celibacy.
G. H. C.
G. H. CARPENTER, B.SC.
Professor of Zoology in the Royal College of Science, Dublin.
Author of _Insects: their Structure and Life_.
Chafer.
G. M. W.
GEORGE MCKINNON WRONG, M.A., F.R.S. (Canada).
Professor of History at Toronto University. Author of _A Canadian
Manor and its Seigneurs_; _The British Nation: a History_; &c.
Canada: _History to Federation_.
G. R. P.
GEORGE ROBERT PARKIN, LL.D., C.M.G.
See the biographical article, PARKIN, G. R.
Canada: _History from Federation._
G. W. T.
REV. GRIFFITHES WHEELER THATCHER, M.A., B.D.
Warden of Camden College, Sydney, N.S.W. Formerly Tutor in Hebrew
and Old Testament History at Mansfield College, Oxford.
Carmathians.
H. A. M. S.
HENRY A. M. SMITH.
Calhoun, John C.
H. B. Wa.
HENRY BEAUCHAMP WALTERS, M.A., F.S.A.
Assistant to Keeper of Greek and Roman Antiquities, British
Museum. Author of _The Art of the Greeks_; _History of Ancient
Pottery_; _Catalogue of the Greek and Etruscan Vases in the
British Museum_, vol. ii.; _Catalogue of Bronzes, Greek, Roman and
Etruscan_; &c.
Ceramics: _Greek, Etruscan and Roman_.
H. Ch.
HUGH CHISHOLM, M.A.
Formerly Scholar of Corpus Christi College, Oxford. Editor of the
11th edition of the _Encyclopaedia Britannica_; co-editor of the
10th edition.
Campbell Bannerman, Sir H.;
Canon: _Music_;
Chamberlain, J.
H. De.
HIPPOLYTE DELEHAYE, S.J.
Assistant in the compilation of the Bollandist publications:
_Analecta Bollandiana_ and _Acta Sanctorum_.
Canonization.
H. F. G.
HANS FRIEDRICH GADOW, F.R.S., PH.D.
Strickland Curator and Lecturer on Zoology in the University of
Cambridge. Author of _Amphibia and Reptiles_.
Chameleon.
H. L. C.
HUGH LONGBOURNE CALLENDAR, F.R.S., LL.D. (McGill Univ.).
Professor of Physics, Royal College of Science, London. Formerly
Professor of Physics in McGill College, Montreal, and in
University College, London.
Calibration; Calorimetry.
H. M. V.
HERBERT M. VAUGHAN, F.S.A.
Keble College, Oxford. Author of _The Last of the Royal Stuarts_;
_The Medici Popes_; _The Last Stuart Queen_.
Charles Edward.
H. P. B.
H. P. BIGGAR.
Author of _The Voyages of the Cabots to Greenland_.
Cartier, Jacques.
H. R. H.
HENRY R. H. HALL, M.A.
Assistant in the Department of Egyptian and Assyrian Antiquities,
British Museum.
Ceramics: _Egypt and Western Asia_.
H. Sy.
HENRY SYMONS.
Assistant in the British Museum. Formerly Lecturer in Greek and
Roman History at Bedford College, London.
Chambord, Comte de.
H. T. A.
REV. HERBERT THOMAS ANDREWS.
Professor of New Testament Exegesis, New College, London. Author
of _The Commentary on Acts_ in the _Westminster New Testament_;
_Handbook on the Apocryphal Books_ in the _Century Bible_.
Catechumen.
H. W. R.*
REV. HENRY WHEELER ROBINSON, M.A.
Professor of Church History in Rawdon College, Leeds. Senior
Kennicott Scholar, Oxford, 1901. Author of _Hebrew Psychology in
Relation to Pauline Anthropology_ (in Mansfield College Essays);
&c.
Canticles (_in part_).
H. W. S.
H. WICKHAM STEED.
Correspondent of _The Times_ at Vienna. Correspondent of _The
Times_ at Rome, 1897-1902.
Cavallotti.
H. Y.
COLONEL SIR HENRY YULE, K.C.S.I.
See the biographical article, YULE, SIR H.
Carpini (_in part_).
J. A. B.
SIR JERVOISE ATHELSTANE BAINES, C.S.J.
President, Royal Statistical Society, 1909-1910. Census
Commissioner under the Government of India, 1889-1893. Employed at
India Office as Secretary to Royal Commission on Opium, 1894-1895.
Author of _Official Reports on Provincial Administration on Indian
Census Operations_; &c.
Census.
J. A. H.
JOHN ALLEN HOWE, B.SC.
Curator and Librarian of the Museum of Practical Geology, London.
Callovian;
Cambrian System;
Caradoc Series;
Carboniferous System;
Chalk.
J. A. M'N.
J.A. M'NAUGHT.
Member of the Jury for Carriage Building, Paris Exposition, 1900.
Carriage
J. Bt.
J. BARTLETT.
Lecturer on Construction, Architecture, Sanitation, Quantities,
&c., at King's College, London. Member of Society of Architects.
Member of Institute of Junior Engineers.
Carpentry
J. C. M.
JAMES CLERK MAXWELL, F.R.S.
See biographical article: MAXWELL, JAMES CLERK.
Capillary Action (_in part_).
J. D. Pr.
JOHN DYNELEY PRINCE, PH.D.
Professor of Semitic Languages, Columbia University, New York.
Took part in the Expedition to Southern Babylonia, 1888-1889.
Author of _A Critical Commentary on the Book of Daniel_.
Chaldea.
J. P. D.
SIR J. FREDERICK DICKSON, K.C.M.G.
Reorganized the North-West Province of Ceylon. Editor and
translator of the _Upasampada-Kammavaca_ and the _Patimokha_.
Ceylon (_in part_).
J. F.-K.
JAMES FITZMAURICE-KELLY, LITT.D., F.R.Hist.S.
Gilmour Professor of Spanish Language and Literature, Liverpool
University. Norman MacColl Lecturer, Cambridge University. Fellow
of the British Academy. Corresponding Member of the Royal Spanish
Academy. Author of _A History of Spanish Literature_; &c.
Campoamar y Campooserio;
Castillo Solorzano;
Celestina, La;
Cervantes.
J. H. F.
JOHN HENRY FREESE, M.A.
Formerly Fellow of St John's College, Cambridge.
Calpurnius, Titus.
J. H. R.
JOHN HORACE ROUND, M.A., LL.D. (Edin.).
Author of _Feudal England_; _Studies in Peerage and Family
History_; _Peerage and Pedigree_.
Castle (_in part_);
Castle Guard.
J. Hl. R.
JOHN HOLLAND ROSE, M.A., LITT.D.
Lecturer on Modern History to the Cambridge University Local
Lectures Syndicate. Author of _Life of Napoleon I._; _Napoleonic
Studies_; _The Development of the European Nations_; _The Life of
Pitt_; chapters in the _Cambridge Modern History_.
Cambaceres.
J. M'D.
JAMES MACDONALD, M.A., LL.D.
Vice-President of the Society of Antiquaries of Scotland,
1895-1897. Rhind Lecturer on Archaeology, 1897. Author of _Tituli
Hunteriani: an Account of the Roman Stones in the Hunterian
Museum_.
Chalmers, George.
J. P.-B.
JAMES GEORGE JOSEPH PENDEREL-BRODHURST.
Editor of the _Guardian_ (London).
Chair.
J. P. E.
JEAN PAUL HIPPOLYTE EMMANUEL ADHEMAR ESMEIN.
Professor of Law in the University of Paris. Officer of the Legion
of Honour. Member of the Institute of France. Author of _Cours
elementaire d'histoire du droit francais_; &c.
Chatelet.
J. R. C.
JOSEPH ROGERSON COTTER, M.A.
Assistant to the Professor of Natural and Experimental Philosophy,
Trinity College, Dublin. Editor of 2nd edition of Preston's
_Theory of Heat_.
Calorescence.
J. S. F.
JOHN SMITH FLETT, D.SC., F.G.S.
Petrographer to the Geological Survey. Formerly Lecturer on
Petrology in Edinburgh University. Neill Medallist of the Royal
Society of Edinburgh. Bigsby Medallist of the Geological Society
of London.
Charnockite.
J. T. Be.
JOHN T. BEALBY.
Joint author of Stanford's _Europe_. Formerly editor of the
_Scottish Geographical Magazine_. Translator of Sven Hedin's
_Through Asia, Central Asia and Tibet_; &c.
Caspian Sea (_in part_);
Caucasia;
Caucasus (_in part_).
J. T. C.
JOSEPH THOMAS CUNNINGHAM, M.A., F.Z.S.
Lecturer on Zoology at the South-Western Polytechnic, London.
Formerly Assistant Professor of Natural History in the University
of Edinburgh. Naturalist to the Marine Biological Association, and
Fellow of University College, Oxford. Author of numerous papers in
scientific journals.
Cephalopoda.
J. Wa.
MAJOR-GENERAL JAMES WATERHOUSE.
Indian Staff Corps. Vice-President of the Royal Photographic
Society. Assistant Surveyor-General in charge of Photographic
Operations in the Surveyor-General's Office, Calcutta, 1866-1897.
Took part in the observation of total eclipses, 1871 and 1875, and
of transit of Venus, 1874. President of the Asiatic Society of
Bengal, 1888-1890. Author of _The Preparation of Drawings for
Photographic Reproduction_; &c.
Camera Obscura: _History_.
J. W. D.
CAPTAIN J. WHITLY DIXON, R.N.
Nautical Assessor to the Court of Appeal.
Capstan.
J. W. He.
JAMES WYCLIFFE HEADLAM, M.A.
Staff Inspector of Secondary Schools under the Board of Education.
Formerly Fellow of King's College, Cambridge, and Professor of
Greek and Ancient History at Queen's College, London. Author of
_Bismarck and the Foundation of the German Empire_; &c.
Caprivi.
L. D.*
MONSEIGNEUR LOUIS MARIE OLIVIER DUCHESNE.
See the biographical article: DUCHESNE, L. M. O.
Calixtus I.;
Celestine I.
L. J. B.
LAWRENCE J. BURPEE.
Author of _The Search for the Western Sea_. Joint author (with
Henry J. Morgan) of _Canadian Life in Town and Country_.
Canada: _Literature, English-Canadian_.
L. J. S.
LEONARD JAMES SPENCER, M.A.
Assistant in the Department of Mineralogy, British Museum.
Formerly Scholar of Sidney Sussex College, Cambridge, and Harkness
Scholar. Editor of the _Mineralogical Magazine_.
Cerargyrite;
Cerussite;
Chabazite;
Chalybite.
L. S.
SIR LESLIE STEPHEN, K.C.B., LITT.D.
See the biographical article: STEPHEN, SIR LESLIE.
Carlyle.
L. V.*
LUIGI VILLARI.
Italian Foreign Office (Emigration Dept.). Formerly Newspaper
Correspondent in east of Europe. Italian Vice-Consul in New
Orleans, 1906, Philadelphia, 1907, and Boston, U.S.A., 1907-1910.
Author of _Italian Life in Town and Country_; &c.
Cantu;
Cappello;
Capponi, G. and P.;
Caracciolo;
Carbonari;
Carmagnola;
Carrara;
Cavour.
M. Br.
MARGARET BRYANT.
Chapman, George (_part_);
Charlemagne: _Legends_.
M. G.
MOSES GASTER, PH.D. (Leipzig).
Chief Rabbi of the Sephardic Communities of England.
Vice-President, Zionist Congress, 1898, 1899, 1900. Ilchester
Lecturer at Oxford on Slavonic and Byzantine Literature, 1886 and
1891. President, Folklore Society of England. Vice-President
Anglo-Jewish Association. Author of _History of Rumanian Popular
Literature_; &c.
Cantacuzmo;
Cantemir.
M. H. S.
MARION H. SPIELMANN, F.S.A.
Formerly Editor of the _Magazine of Art_. Member of Fine Art
Committee of International Exhibitions of Brussels, Paris, Buenos
Aires, Rome, and the Franco-British Exhibition, London. Author of
_History of "Punch"_; _British Portrait Painting to the Opening of
the 19th Century_; _Works of G. F. Watts, R.A._; _British
Sculpture and Sculptors of To-day_; _Henriette Ronner_; &c.
Caricature;
Cartoon.
M. J. de G.
MICHAEL JAN DE GOEJE.
See the biographical article: GOEJE, MICHAEL JAN DE.
Caliphate.
M. P.
REV. MARK PATTISON.
See the biographical article: PATTISON, MARK.
Casaubon, Isaac.
N. E. D.
NARCISSE EUTROPE DIONNE, M.D., LL.D., F.R.S. (Canada).
Librarian of the Legislature of the Province of Quebec. Chief
Editor of _Le Courrier du Canada_, 1880-1884. Chief Inspector of
Federal Licences, 1884-1886. Chief Editor of _Le Journal de
Quebec_, 1886. Author of _Life of Samuel Champlain, Founder of
Quebec_; _Life of Jacques Cartier, discoverer of Canada_; _La
Nouvelle France, 1540-1603_; _Quebec et Nouvelle France_; &c.
Champlain, Samuel de.
N. W. T.
NORTHCOTE WHITBRIDGE THOMAS, M.A.
Government Anthropologist to Southern Nigeria. Corresponding
Member of the Societe d'Anthropologie de Paris. Author of _Thought
Transference_; _Kinship and Marriage in Australia_; &c.
Cannibalism.
O. Ba.
OSWALD BARRON, F.S.A.
Editor of _The Ancestor_, 1902-1905.
Cecil.
O. Br.
OSCAR BRILIANT.
Carpathian Mountains (_in part_).
O. M. D.
ORMONDE MADDOCK DALTON, M.A., F.S.A.
Assistant Keeper, Department of British and Medieval Antiquities,
British Museum. Corresponding Member of the Imperial Russian
Archaeological Society. Author of _Guide to the Early Christian
and Byzantine Antiquities_; &c.
Catacomb (_in part_).
P. A.
PAUL DANIEL ALPHANDERY.
Professor of the History of Dogma, Ecole Pratique des Hautes
Etudes, Sorbonne, Paris. Author of _Les Idees morales chez les
heterodoxes latines au debut du XIII^e siecle_.
Capistrano.
P. A. K.
PRINCE PETER ALEXEIVITCH KROPOTKIN.
See the biographical article: KROPOTKIN, P. A.
Caspian Sea (_in part_);
Caucasus (_in part_).
P. C. Y.
PHILIP CHESNEY YORKE, M.A.
Magdalen College, Oxford.
Catherine of Aragon;
Charles I.;
Charles II.
P. La.
PHILIP LAKE, M.A., F.G.S.
Lecturer on Physical and Regional Geography in Cambridge
University. Formerly of the Geological Survey of India. Author of
_Monograph of British Cambrian Trilobites_. Translator and Editor
of Kayser's _Comparative Geology_.
Carpathian Mountains (_in part_);
Caucasus: _Geology_.
P. Vn.
PERCIVAL SYLVANUS VIVIAN.
Author of _Poems of Marriage_. Editor of the _Poetical Works of
Thomas Campion_.
Campion, Thomas.
P. A. M.
PERCY ALEXANDER MACMAHON, F.R.S., D.SC.
Late Major R.A. Deputy Warden of the Standards. Board of Trade.
Joint-General Secretary of the British Association. Formerly
Professor of Physics, Ordnance College, and President of London
Mathematical Society.
Cayley.
R.
THE RT. HON. LORD RAYLEIGH.
See the biographical article: RAYLEIGH, 3rd Baron.
Capillary Action (_in part_).
R. A.*
ROBERT ANCHEL.
Archivist to the Department de l'Eure.
Cambon, Pierre Joseph;
Cathelineau.
R. Ad.
ROBERT ADAMSON.
See the biographical article: ADAMSON, R.
Category (_in part_).
R. A. S. M.
ROBERT ALEXANDER STEWART MACALISTER, M.A., F.S.A.
Director of Excavations for the Palestine Exploration Fund.
Capernaum;
Carmel.
R. G.
RICHARD GARNETT.
See the biographical article: GARNETT, RICHARD.
Cardan.
R. I. P.
R. I. POCOCK, F.Z.S.
Superintendent of the Zoological Gardens, London.
Centipede.
R. K. D.
SIR ROBERT KENNAWAY DOUGLAS.
Formerly Keeper of Oriental Printed Books and MSS. at the British
Museum, and Professor of Chinese, King's College, London. Author
of _The Language and Literature of China_; &c.
Canton.
R. L.*
RICHARD LYDEKKER, F.R.S., F.G.S., F.Z.S.
Member of the Staff of the Geological Survey of India, 1874-1882.
Author of _Catalogues of Fossil Mammals, Reptiles and Birds in
British Museum_; _The Deer of all Lands_; _The Game Animals of
Africa_; &c.
Camel;
Capuchin Monkey;
Carnivora;
Cat;
Cavy;
Cetacea;
Chamois.
R. L. H.
ROBERT LOCKHART HOBSON.
Assistant in the Department of British and Medieval Antiquities,
British Museum. Author of _Porcelain: Oriental, Continental and
British_; _Marks on Pottery and Porcelain_ (with W. Burton); and
_Catalogue and Guide of English Pottery and Porcelain in British
Museum_.
Ceramics: _Medieval and Later Italian_; Persian, Syrian, Egyptian
and Turkish.
R. N. B.
ROBERT NISBET BAIN (d. 1909).
Assistant Librarian, British Museum, 1883-1909. Author of
_Scandinavia, the Political History of Denmark, Norway and Sweden,
1313-1900_; _The First Romanovs, 1613-1725_; _Slavonic Europe, the
Political History of Poland and Russia from 1469 to 1796_; &c.
Canute;
Canute VI;
Casimir III.;
Casimir IV.;
Catherine I.;
Charles I. (_Hungary_);
Charles IX., X., XI., XII. (_Sweden_);
Charles XIII., XIV., XV. (_Sweden and Norway_).
R. Po.
RENE POUPARDIN, D. es L.
Secretary of the Ecole des Chartes. Honorary Librarian at the
Bibliotheque Nationale, Paris. Author of _Le Royaume de Provence
sous les Carolingiens_; _Recueil des chartes de Saint-Germain_;
&c.
Charles the Bold.
R. P. S.
R. PHENE SPIERS, F.S.A., F.R.I.B.A.
Master of the Architectural School and Surveyor, Royal Academy,
London. Past President of Architectural Association. Associate and
Fellow of King's College, London. Corresponding Member of the
Institute of France. Edited Fergusson's _History of Architecture_.
Author of _Architecture East and West_; &c.
Campanile;
Capital: _Arch._;
Cathedral: _Arch._;
Ceiling.
R. S. C.
ROBERT SEYMOUR CONWAY, M.A., D.LITT. (Cantab.).
Professor of Latin in the University of Manchester. Formerly
Professor of Latin in University College, Cardiff; and Fellow of
Gonville and Caius College, Cambridge. Author of _The Italic
Dialects_.
Campania (_in part_).
R. W.
ROBERT WALLACE, F.R.S. (Edin.), F.L.S.
Professor of Agriculture and Rural Economy at Edinburgh
University, and Garton Lecturer on Colonial and Indian
Agriculture. Professor of Agriculture, R.A.C., Cirencester,
1882-1885. Author of _Farm Live Stock of Great Britain_; _Indian
Agriculture_; _The Agriculture and Rural Economy of Australia and
New Zealand_; _Farming Industries of Cape Colony_; &c.
Cattle (_in part_).
R. We.
RICHARD WEBSTER, A.M.
Editor of _Elegies of Maximianus_.
Channing, William E.
ST C.
VISCOUNT ST CYRES.
See the biographical article: IDDESLEIGH, 1ST EARL OF.
Casuistry.
S. D.
SAMUEL DAVIDSON, D.D.
See the biographical article: DAVIDSON, SAMUEL.
Canon: _Scriptures_.
T. As.
THOMAS ASHBY, M.A., D.LITT., F.S.A.
Director of the British School of Archaeology at Rome.
Corresponding Member of the Imperial German Archaeological
Institute. Formerly Scholar of Christ Church, Oxford. Craven
Fellow, Oxford, 1897. Author of _The Classical Topography of the
Roman Campagna_; &c.
Camerino;
Campania (_in part_);
Canosa;
Canusium;
Capena;
Capri;
Capua;
Carales;
Carsioli;
Casilinum;
Casinum;
Cassia, Via;
Catania;
Caudine Forks;
Cefalu;
Centuripe;
Cesena.
T. A. H.
CAPTAIN THOMAS A. HULL, R.N.
Formerly Superintendent of Admiralty Charts.
Chart.
T. Ba.
SIR THOMAS BARCLAY, M.P.
Member of the Institute of International Law. Member of the
Supreme Council of the Congo Free State. Officer of the Legion of
Honour. Author of _Problems of International Practice and
Diplomacy_; &c. M.P. for Blackburn, 1910.
Capture.
T. F. C.
THEODORE FREYLINGHUYSEN COLLIER, PH.D.
Assistant Professor of History, Williams College, Williamstown,
Mass., U.S.A.
Carthage, Synods of;
Chalcedon, Council of.
T. K. C.
REV. THOMAS KELLY CHEYNE. D.LITT., D.D.
See the biographical article: CHEYNE, T.K.
Canaan, Canaanites.
T. M. F.
THOMAS MACALL FALLOW, M.A., F.S.A.
Formerly editor of _The Antiquary_, 1895-1899. Author of
_Memorials of Old Yorkshire_; _The Cathedral Churches of Ireland_.
Cathedral.
T. W. F.
THOMAS WILLIAM FOX.
Professor of Textiles in the University of Manchester. Author of
_Mechanics of Weaving_.
Carding.
W. A. B. C.
REV. WILLIAM AUGUSTUS BREVOORT COOLIDGE, M.A., F.R.G.S.
Fellow of Magdalen College, Oxford. Professor of English History,
St David's College, Lampeter, 1880-1881. Author of _Guide to
Switzerland_; _The Alps in Nature and in History_; &c. Editor of
_The Alpine Journal_, 1880-1889.
Cannes;
Chamonix;
Chartreuse, La Grande.
W. A. P.
WALTER ALISON PHILLIPS, M.A.
Formerly Exhibitioner of Merton College and Senior Scholar of St
John's College, Oxford. Author of _Modern Europe_; &c.
Canon: _Church Dignitary_;
Capo d'Istria;
Carlsbad Decrees;
Chasuble.
W. B.*
WILLIAM BURTON, HON. M.A. (Vict.), F.C.S.
Chairman, Joint-Committee of Pottery Manufacturers of Great
Britain. Examiner for Board of Education in Pottery Design and for
Technological Examinations in Pottery Manufacture. Author of
_English Stoneware and Earthenware_; _Porcelain_; &c.
Ceramics (_in part_).
W. B. D.
WILLIAM BOYD DAWKINS, F.R.S., D.SC.
See the biographical article: DAWKINS, WILLIAM BOYD.
Cave.
W. B. Du.
WILLIAM BARTLETT DUFFIELD, M.A.
Barrister at Law, Inner Temple. Secretary to the Royal Commission
on Canals, 1906-1910.
Chartered Companies.
W. F. C.
WILLIAM FEILDEN CRAIES, M.A.
Barrister-at-Law, Inner Temple. Lecturer on Criminal Law at King's
College, London. Editor of _Archbold's Criminal Pleading_ (23rd
edition).
Capital punishment.
W. F. W.
WALTER FRANCIS WILLCOX, LL.B., PH.D.
Dean of, and Professor of Political Economy and Statistics at,
Cornell University. Formerly Chief Statistician and now Special
Agent of the U.S. Census Bureau. Author of _The Divorce Problem--a
Study in Statistics_; _Social Statistics of the United States_;
&c.
Census: _U.S.A._
W. Fr.
WILLIAM FREAM (d. 1907), LL.D., F.G.S., F.L.S., F.S.S.
Author of _Handbook of Agriculture_. Formerly Agricultural
Correspondent of _The Times_.
Cattle (_in part_).
W. G.*
WALCOT GIBSON, D.SC., F.G.S.
Geologist on H.M. Geological Survey. Author of _The Gold-bearing
Rocks of the S. Transvaal_; _Mineral Wealth of Africa_; _The
Geology of Coal and Coal Mining_; &c.
Cape Colony: _Geology._
W. G. F. P.
SIR WALTER GEORGE FRANK PHILLIMORE, BART., D.C.L., LL.D.
Judge of the King's Bench Division. President of International Law
Association, 1905. Author of _Book of Church Law_. Editor of 2nd
ed. of _Phillimore's Ecclesiastical Law_; 3rd ed. of vol. iv. of
_Phillimore's International Law_; &c.
Canon Law: _Anglican._
W. G. M.
WALTER G. M'MILLAN, F.C.S., M.I.M.E. (d. 1904).
Formerly Secretary of the Institute of Electrical Engineers.
Lecturer on Metallurgy, Mason College, Birmingham. Author of _A
Treatise on Electro-Metallurgy_.
Carborundum.
W. Ha.
REV. WILLIAM HANNA, LL.D., D.D. (1802-1882).
Minister of St John's Free Church, Edinburgh, 1850-1866. Author of
_Life of Dr Chalmers_; _Wycliffe and the Huguenots_; _Martyrs of
the Scottish Reformation_.
Chalmers, Thomas (_in part_).
W. J. G.
WILLIAM JOHN GRUFFYDD, M.A.
Lecturer in Celtic, University College, Cardiff. Examiner in Welsh
to the Central Welsh Board for Intermediate Education. Author of
_Caneuon a Cherddi: An Anthology of Medieval Welsh Poetry_.
Celt: _Literature, Welsh._
W. L.*
WALTER LEHMANN, M.D.
Directorial Assistant of the Royal Ethnographical Museum, Munich.
Conducted Exploring Expedition in Mexico and Central America,
1907-1909. Author of many publications on Mexican and Central
American Archaeology.
Central America: _Archaeology._
W. L. A.
REV. WILLIAM LINDSAY ALEXANDER, D.D., LL.D., F.R.S. (Edin.)
(1808-1884).
Classical Tutor, Lancashire Independent College. Pastor of
Independent Chapel, N. College Street, Edinburgh. One of the Old
Testament Revisers. Author of _A Moral Philosophy_.
Calvin (_in part_).
W. L. G.
WILLIAM LAWSON GRANT, M.A.
Professor at Queen's University, Kingston, Canada. Formerly Beit
Lecturer in Colonial History at Oxford University. Editor of _Acts
of the Privy Council_, Colonial series; _Canadian Constitutional
Development_ (in collaboration).
Canada: _Statistics_;
Cartier, Sir Georges Etienne.
W. M. R.
WILLIAM MICHAEL ROSSETTI.
See the biographical article: ROSSETTI, DANTE GABRIEL.
Canova;
Caracci;
Cartoon;
Cellini, Benvenuto (_in part_);
Charlet.
W. Ri.
WILLIAM RIDGEWAY, M.A., D.SC., LL.D. (Aberdeen), D.LITT.
Fellow of the British Academy. Disney Professor of Archaeology at
Cambridge University. Professor of Greek, Queen's College, Cork,
1883. Ex-President of Cambridge Philological, Antiquarian and
Classical Societies. Author of _The Oldest Irish Epic_; _Origin of
Metallic Currency and Weight Standards_; _The Early Age of
Greece_; &c.
Celt.
W. R. B.
RT. REV. WILLIAM ROBERT BROWNLOW, M.A., D.D. (d. 1901).
Roman Catholic Bishop of Clifton. Provost and Domestic Prelate to
Pope Leo XIII. Co-editor of _English Roma Sotterranea_. Author of
_Early Christian Symbolism_; _Lectures on Sacerdotalism, on
Catacombs and other Archaeological Subjects_. Translator of _Cur
Deus Homo_ and _Vitis mystica_.
Catacomb (_in part_).
W. R. S.
WILLIAM ROBERTSON SMITH
See the biographical article: SMITH, WILLIAM ROBERTSON.
Canticles (_in part_).
W. Wo.
WILLIAM WOOD, D.C.L., F.R.S. (Canada).
Lieut.-Col., Canadian Militia. Formerly President of the English
Section of the Royal Society of Canada and of the Historic
Landmarks Association. Author of _The Fight for Canada_; _The Logs
of the Conquest of Canada_, &c.
Canada: _Literature, French-Canadian_.
W. W. R.*
WILLIAM WALKER ROCKWELL, LIC. THEOL.
Assistant Professor of Church History, Union Theological Seminary,
New York.
Celestine III. and V.
W. Y. S.
WILLIAM YOUNG SELLAR.
See the biographical article: SELLAR, WILLIAM YOUNG.
Catullus (_in part_).
FOOTNOTE:
[1] A complete list, showing all individual contributors, appears in
the final volume.
PRINCIPAL UNSIGNED ARTICLES
California. Carrier.
Cambodia. Cartagena.
Cambridge, Earls and Dukes of. Cassel.
Cambridge, England. Cassiodorus.
Cambridgeshire. Caste.
Campbell, Thomas. Catherine, Saint.
Canary Islands. Catherine II.
Canning, George. Catherine de' Medici
Canterbury. Catiline.
Cape Town. Cato.
Cape Verde Islands. Causation.
Capital (_Economics_). Cavaignac, Louis Eugene.
Capitulations. Cavan.
Carbolic Acid. Cavendish, Henry.
Carbon. Caxton, William.
Cardiganshire. Cedar.
Cards, Playing. Celebes.
Carducci, Giosue. Celsus.
Carinthia. Cemetery.
Carlisle, Earls of. Chambers, Robert.
Carlisle. Chancellor.
Carlos. Chancery.
Carlsbad. Channel Islands.
Carlstadt. Chantrey, Sir Francis.
Carmarthenshire. Charles V., VI., VII. of France.
Carnarvonshire. Charles, Archduke of Austria.
Carnegie, Andrew. Charles Albert, king of Sardinia.
Carnot. Charles Augustus.
Carol. Chartism.
Caroline Islands. Chateaubriand.
End of the Project Gutenberg EBook of Encyclopaedia Britannica, 11th
Edition, Volume 5, Appendix, by Various
***
|
{
"redpajama_set_name": "RedPajamaBook"
}
| 1,746
|
This is really cute. It would make some good presents, especially for a kid to give to class mates.
Very cute. I might try one this afternoon. Thanks Cindy.
That looks so simple and oh so cute! I'm sure I will find something to use it for.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,246
|
Q: Asymptotics of unrooted labeled forests It is well known that the number of unrooted labeled trees on vertex set
$[n]={1,2,...,n}$ is $n^{n-2}$. Let $U(z)$ be the exponential generating function of the sequence of these numbers. Then $F(z)=\text{exp}(U(z))$ is the exponential generating function for the number of unrooted forests of $[n]$.
My question is, how do you prove that $n![z^n]F(z) \sim e^{1/2} n^{n-2}$.
In other words, that is the asymptotic number of unrooted forests on $[n]$. I found this result on page 406 of the Flajolet-Sedgewick book, without proof.
If that helps, $U(z)= T(z) - T(z)^2 /2$ , where $T(z)$ is the exponential generating function for the numbers of rooted trees. But while $T(z)$ satisfies the relation $T=z \ \text{exp}(T)$, I could not find such a relation for F.
A: The idea in the book by Flajolet-Sedgewick is to use singularity analysis. The generating function $F(z)=\exp(U(z))$ with $U=T-T^2/2$ inherits the dominant singularity $e^{-1}$ of $T(z)=-W(-z)$, where $W$ is the Lambert $W$ function. From the local expansion of $T(z)$
$$T(z)=1-\sqrt{2}\sqrt{1-ze}+\frac{2}{3}(1-ze)-\frac{11\sqrt{2}}{36}(1-ze)^{3/2}+O((1-ze)^2),\quad z\rightarrow e^{-1}$$
follows that of $F$:
$$F(z)=e^{1/2}\left(1-(1-ze)+\frac{2\sqrt{2}}{3}(1-ze)^{3/2}+O((1-ze)^2)\right).$$
From there, singularity analysis is straightforward. The asymptotic behaviour of the coefficients comes from the term in $(1-ze)^{3/2}$, leading to
$$[z^n]F(z)\sim \frac{e^{1/2}}{\sqrt{2\pi}}n^{-5/2}e^{n}.$$
That coefficient is the number of unrooted labeled trees divided by $n!$, so that multiplying by Stirling's formula gives the desired $e^{1/2}n^{n-2}$.
A: Here's a proof, following section 4.3 of J.W. Moon's book "Counting Labelled Trees".
First, we have the following formula of Rényi (see this post for a proof and reference) for $f_k(n)$, the number of forests on $n$ vertices consisting of $k$ unrooted trees.
$$ f_k(n)= \binom nk \sum_{i=0}^k \left(-\frac12\right)^i (k+i)\,i!\, \binom{k}{i}\binom{n-k}{i} n^{n-k-i-1}.$$
Note that $U^k/k!$ is the generating function for forests of $k$ trees, so we have:
\begin{align}
\sum_{n=k}^\infty f_k(n)\frac{x^n}{n!} &= \frac{U^k}{k!}\\
&=\frac{1}{k!}(T-T^2/2)^k\\
&=\frac{1}{k!}\sum_{i=0}^k\binom{k}{i}(-1/2)^iT^{k+i}.
\end{align}
Thus $f_k(n)$ is the coefficient of $x^n$ on the right hand side. This can be determined using the following formula:
$$\frac{T^k}{k!} = \sum_{n=k}^\infty\binom{n}{k}kn^{n-k-1}\frac{x^n}{n!}.$$
To prove this, apply the Lagrange–Bürmann formula to $f(x)=T(x)$, $\phi(x)=e^T$, and $H(T)=T^k$ (see also section 4.2 of Moon's book).
To determine asymptotics, we only need to compute the coefficient of $n^{n-h}$ in the formula for $f_k(n)$. Denote this by $c(k,h)$. Then "after some simplification":
$$c(k,1)=\frac{1}{k!}\sum_{i=0}^k(-1/2)^i\binom{k}{i}(k+i)=0,$$
and
$$c(k,2)=-\frac{1}{k!}\sum_{i=0}^k(-1/2)^i(k+i)\binom{k}{i}\binom{k+i}{2}=\frac{(1/2)^{k-1}}{(k-1)!}.$$
Thus $\lim_{n\rightarrow\infty}\frac{f_k(n)}{n^{n-2}}=\frac{(1/2)^{k-1}}{(k-1)!}.$
The result follows after applying Tannery's theorem to sum these limits over all $k$:
$$\lim_{n\rightarrow\infty}\frac{N(n)}{n^{n-2}}=\sum_{k=1}^\infty\frac{(1/2)^{k-1}}{(k-1)!}=e^{1/2},$$
where $N(n)$ is the total number of rooted forests on $n$ nodes, as desired.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,595
|
using System.Collections.Generic;
using System.Net;
using System.Net.Http;
using System.Web.Http;
using System.Web.Http.Description;
using Cards.Extensions.Tfs.Core.Models;
namespace Cards.Extensions.Tfs.Api.Controllers
{
[Authorize]
public class LabelController : ApiController
{
[HttpGet]
[ResponseType(typeof(List<Label>))]
[Route("api/Labels")]
public HttpResponseMessage GetAll(HttpRequestMessage request)
{
var label = new Label();
var result = label.GetAll();
return request.CreateResponse(HttpStatusCode.OK, result);
}
[HttpGet]
[ResponseType(typeof(Label))]
[Route("api/Labels/{id}")]
public HttpResponseMessage GetById(HttpRequestMessage request, int id)
{
var label = new Label();
var result = label.Get(id);
return request.CreateResponse(HttpStatusCode.OK, result);
}
[HttpPost]
[ResponseType(typeof(Label))]
[Route("api/Labels")]
public HttpResponseMessage Add(HttpRequestMessage request, Label label)
{
var result = label.Add(label.Name, label.ColorCode);
if (result != null)
{
return request.CreateResponse(HttpStatusCode.OK, result);
}
else
{
return request.CreateResponse(HttpStatusCode.InternalServerError);
}
}
[HttpPut]
[ResponseType(typeof(Label))]
[Route("api/Labels/{id}")]
public HttpResponseMessage Edit(HttpRequestMessage request, int id, Label label)
{
label.ID = id;
var result = label.Update(label);
if (result != null)
{
return request.CreateResponse(HttpStatusCode.OK, result);
}
else
{
return request.CreateResponse(HttpStatusCode.InternalServerError);
}
}
[HttpHead]
[ResponseType(typeof(void))]
[Route("api/Labels/{id}")]
public HttpResponseMessage Delete(HttpRequestMessage request, int id)
{
var label = new Label();
label.Remove(id);
return request.CreateResponse(HttpStatusCode.NoContent);
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,321
|
Q: Find row-wise minima in sparse matrix I would like to get the minimum nonzero values per row in a sparse matrix. Solutions I found for dense matrices suggested masking out the zero values by setting them to NaN or Inf. However, this obviously doesn't work for sparse matrices.
Ideally, I should get a column vector of all the row-wise minima, as I would get with
minValues = min( A, [], 2);
Except, obviously, using min leaves me with an all-zeros column vector due to the sparsity. Is there a solution using find?
A: This is perfect for accumarray. Consider the following sparse matrix,
vals = [3 1 1 9 7 4 10 1]; % got this from randi(10,1,8)
S = sparse([1 3 4 4 5 5 7 9],[2 2 3 6 7 8 8 11],vals);
Get the minimum value for each row, assuming 0 for empty elements:
[ii,jj] = find(S);
rowMinVals = accumarray(ii,nonzeros(S),[],@min)
Note that rows 4 and 5 of rowMinVals, which are the only two rows of S with multiple nonzero values are equal to the min of the row:
rowMinVals =
3
0
1
1 % min([1 9]
4 % min([7 4]
0
10
0
1
If the last row(s) of your sparse matrix do not contain any non-zeros, but you want your min row value output to reflect that you have numRows, for example, change theaccumarray command as follows,
rowMinVals = accumarray(ii,nonzeros(S),[numRows 1],@min).
Also, perhaps you also want to avoid including the default 0 in the output. One way to handle that is to set the fillval input argument to NaN:
rowMinVals = accumarray(ii,nonzeros(S),[numRows 1],@min,NaN)
rowMinVals =
3
NaN
1
1
4
NaN
10
NaN
1
NaN
NaN
NaN
Or you can keep using a sparse matrix with the fifth input argument, issparse:
>> rowMinVals = accumarray(ii,nonzeros(S),[],@min,[],true)
rowMinVals =
(1,1) 3
(3,1) 1
(4,1) 1
(5,1) 4
(7,1) 10
(9,1) 1
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Q: Regex engine optimization: Is `(?:.)*` really more costly than `.*`? Some context
I'm usually using the website http://regex101.com to test my regex, which provides a "debugger" feature in PCRE that lets you see what the regex engine is doing step by step.
When matching a random string with .*, this debugger tells me the engine follows the constant number of 3 steps.
When matching with (?:.)*, it announces a number depending on the length: 66 steps for something like 0123456789012345678901234567899.
Is (?:.)* really more costly than .*?
It seems that on the latter case, entering the group is considered each time to be a new step, whereas on the former the .* is applied at once.
Is that some sort of "improvement" the website is doing (trying to avoid showing useless cases), or does it match a real internal regex mechanism ? And if so, what's the idea behind the scene?
A: You didn't specify which engine. You mentioned PCRE, but you also tagged with Perl.
regex101 site shows that PCRE handles .* using one operation, but that doesn't mean that one operation is faster than the operations generated by the equivalent (?:.)*. Only benchmarking will tell, but .* will likely be marginally faster due to less overhead.
In Perl, they compile to exactly the same regex program (as you can see below), so they will perform identically.
>perl -Mre=debug -e"'0123456789012345678901234567899' =~ /.*/"
Compiling REx ".*"
Final program:
1: STAR (3)
2: REG_ANY (0)
3: END (0)
anchored(MBOL) implicit minlen 0
Matching REx ".*" against "0123456789012345678901234567899"
0 <> <0123456789> | 1:STAR(3)
REG_ANY can match 31 times out of 2147483647...
31 <901234567899> <> | 3: END(0)
Match successful!
Freeing REx: ".*"
>perl -Mre=debug -e"'0123456789012345678901234567899' =~ /(?:.)*/"
Compiling REx "(?:.)*"
Final program:
1: STAR (3)
2: REG_ANY (0)
3: END (0)
anchored(MBOL) implicit minlen 0
Matching REx "(?:.)*" against "0123456789012345678901234567899"
0 <> <0123456789> | 1:STAR(3)
REG_ANY can match 31 times out of 2147483647...
31 <901234567899> <> | 3: END(0)
Match successful!
Freeing REx: "(?:.)*"
In both cases, the string is scanned for characters than aren't newlines, and that's it.
Note that no matter how many "steps" are taken, this cannot be done in constant time. . doesn't match newlines (without /s), so the regex engine must check each character it's about to match to see whether it's a newline or not.
A: You can use pcretest for seeing the differences. Here is a nice tutorial.
Your first example obviously need few steps, then the second. On the left side with the + you see the position in the pattern, on the right side the matching position in the input.
1.) /.*/CD on str 0123456789012345678901234567899 with debugging modifiers CD
2.) /(?:.)*/CD same str
And that is, what really happens.
A: I have written a small benchmark code to test this scenario. And your both regex returns almost the same time on performance. So not sure about which one is better.
However, I have changed your changed your regex (?:.)* into (.)* and it drastically reduce the performance. I believe it is because of the group capturing. Here is the code:
use Benchmark qw( cmpthese );
cmpthese(-3, {
'.*' => '"kasdaskdhas dhaskdh askdhqwioeuweakjsdhasjdk asjdk ask" =~ /.*/',
'(?:.)*' => '"kasdaskdhas dhaskdh askdhqwioeuweakjsdhasjdk asjdk ask" =~ /(?:.)*/',
'(.)*' => '"kasdaskdhas dhaskdh askdhqwioeuweakjsdhasjdk asjdk ask" =~ /(.)*/',
});
Outputs:
Rate (.)* (?:.)* .*
(.)* 2305921/s -- -34% -35%
(?:.)* 3499870/s 52% -- -1%
.* 3524871/s 53% 1% --
That 1% difference between .* and (?:.)* is noise and meaningless.
A: The website doesn't seem to explain what those steps are!
This has to be an optimisation issue. The question "Is (?:.)* really more costly than .*?" depends on the regex engine in use, and the site is very unlikely to use Perl's regex engine which is built into the perl compiler/interpreter. The optimisation in whatever it does use has chosen to ignore trivial cases like (?:.)* that are unlikely in the real world.
If you need your regex to run faster then you should use Benchmark to compare different patterns, or perhaps Regexp::Optimizer, which will attempt to rewrite your pattern into an equivalent faster one, or Regexp::Debugger which will allow you to see what is going on behind the scenes.
But please don't take these measures until you have written a functional and clear program that doesn't perform fast enough, and have proven that the bottleneck is the regex matching. The regex engine is wholly written in C and you are unlikely to make a big difference to the overall speed of your code by changing the regex patterns that it uses.
A: I'm not an expert on the subject, but from what I can tell, yes, /(?:.)*/ and /(.)*/ are more costly than /.*/.
According to the Perl documentation on backtracking,
A fundamental feature of regular expression matching involves the
notion called backtracking, which is currently used (when needed) by
all regular non-possessive expression quantifiers, namely * , *? , + ,
+?, {n,m}, and {n,m}?. Backtracking is often optimized internally, but the general principle outlined here is valid.
So basically, .* is optimized internally, but I can't find a source that says how.
I also found another source, a blog post by the author of Mastering Regular Expressions, Jeffrey Friedl.
By the way, I guess I should make one mention about how Perl
sometimes optimizes how it deals with regular expressions. Sometimes
it might actually perform fewer tests than what I've described. Or it
perhaps does some tests more efficiently than others (for example,
/x*/ is internally optimized such that it is more efficient than
/(x)/ or /(?:x)/). Sometimes Perl can even decide that a regex can
never match the particular string in question, so will bypass the test
altogether.
If anyone else can explain the optimizations Perl makes in more detail, that would be useful!
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{"url":"https:\/\/tex.stackexchange.com\/questions\/210156\/placing-of-floats-referenced-with-varioref","text":"# placing of floats referenced with varioref\n\nIn my text I use \\vref{<label>} from the varioref package for referencing figures and tables. In most circumstances this works satisfactory. Exceptions are cases, when in a text line \\vref appears near the end of the line such that for example Fig. 1 is placed in one line and on the next page goes to the next line, where a float with attribute [h] is placed. In that case, that below line with Fig. 1 leaves enough room for the float, if the reference is only Fig. 1 and therefore accommodates in one line. If the float is not placed here (below line with reference) but always pushed to the next page and reference is expanded into two lines.\n\nIt seem that this happen because the algorithm of varioref starts always with the assumption that the float is placed on a different page as is \\vref. I wonder if there is a way to convince the varioref to start with the assumption that the float will be placed in the same page as \\vref and if this is not true to expand \\vref with terms on the next page or on the page <num> in the second (or third) run of latex? Or exists some equivalent package with similar features as varioref, which works as I wish?\n\nTemporary I help myself that after compilation in the described cases I manually change \\vref to \\ref. But this is based on guessing and it is contrary to the (La)TeX \"mission\" (take care on text layout).\n\nEdit MWE (based on answer of David Carlisle), which show behavior of varioref is:\n\n% usual I use memoir package\n\\documentclass{memoir}\n\\renewcommand{\\thefigure}{\\thesection-\\arabic{figure}}\n\n\\usepackage{varioref}\n% my shortcuts for referencing\n\\renewcommand{\\fref}[1]{(\\figurename~\\ref{#1})}\n\\newcommand{\\vfref}[1]{(\\figurename~\\vref{#1})}\n\\begin{document}\n\\setcounter{chapter}{11}\n\\setcounter{section}{12}\n\\setcounter{figure}{122}\n\nabc\\\\abc\\\\abc\\\\\n\naaa aaa aaa aaa aaa aaa aaa aaa aaa aa\n% ref produces: \\fref{figure}\nvref produces: \\vfref{figure}\n\\begin{figure}[h]\\centering\n\\fbox{\\parbox[b][5ex][c]{0.5\\hsize}{a figure}}\n\\caption{My figure}\n\\label{figure}\n\\end{figure}\n\\end{document}\n\n\nif I un-comment line with ''ref'' and comment line with ''vref'' than the figure appear on the same page as it is reference on it, contrary the figure is moved on the next page\n\n\u2022 varioref is supposed to detect same-page references, if you have a case where that is not working it would be helpful to make a small complete example document that demonstrates the problem and add it to the question. \u2013\u00a0David Carlisle Nov 2 '14 at 14:02\n\u2022 There are a couple of possibilities on how to interpret your comments above leading to either a bug or a limitation or even a misunderstanding on your part. Please extend your question with a selfcontained complete example to better understand the situation here. I can think of a couple of case where things might get complicated (or even fail) for example if your reference surround the H float but it is pointless to guess. \u2013\u00a0Frank Mittelbach Nov 2 '14 at 20:26\n\nThe only way I can reproduce this is to edit the file between runs.\n\n\\documentclass{article}\n\n\\usepackage{varioref}\n\n\\begin{document}\n\nabc\\\\abc\\\\abc\\\\abc%\\\\ABC\n\naaa aaaa aaaa aaa aaa\naaa aaaa aaaa aaa aaa\nvref produces: Figure \\vref{zz}.\n\naaaa\\\\zzz\n\\begin{figure}[ht]\n\\fbox{XXXX}\n\n\\caption{fff\\label{zz}}\n\\end{figure}\n\nbbb\n\n\\end{document}\n\n\nIf you uncomment ABC the figure goes to page 2 as shown in the first image, varioref picks that up and adds \"on the next page\" if you then comment out the \\\\ABC varioref doesn't notice the figure is now only on the next page because of its added text, so it keeps adding it as shown in the second image.\n\nHowever if you delete the .aux file and run latex twice more, varioref computes everything for the new figure position with a same-page reference.\n\nSo to be sure, after edits delete any old .aux file before running latex until the references are stable.\n\n\u2022 David Carlisle, in my text for referencing i use the following command: \\newcommand\\vfref[1]{(\\figurename~\\vref{#1}), if this is important. Regarding to necessary manual manipulating, your solution is similar to mine :_), of course on different way. This is not big deal (manual manipulation), if you have only few such a cases and doesn't make any revision of written text. Otherwise, the \"automatized\" procedure (as part of package algorithm) is highly desired. In this sense, my question is actually feature request. \u2013\u00a0Zarko Nov 2 '14 at 17:05\n\u2022 @Zarko no mine is not really manual manipulation, you should clean the aux files and rebuild after an edit, otherwise if you come back to the document later when you only have the tex source you will get a different result. if your automated procedure is remove aux file, latex latex latex, then the problem does not arise. I'm not sure that it would be possible to detect this within the same latex run latex cross referencing always uses the values stored on the previous run, so they may always be wrong if the file has been edited \u2013\u00a0David Carlisle Nov 2 '14 at 17:16\n\u2022 Now I see a differences between your approach and mine. In further I will use your suggestion. Thank you. \u2013\u00a0Zarko Nov 3 '14 at 17:06\n\nIt is indeed possible that varioref generates sub-optimal results in special circumstances but your MWE is actually not an example of this. If you run this MWE starting without any .aux file then everything comes out fine, i.e., you first get\n\nBasically your assumtion that varioref starts out with the assumption that the float is placed on a \"different\" page is wrong. Instead it starts out with no knowledge about the float and as a result it just produces the famous ?? as a reference. And this is \"nearly\" as short as it can get. Thus the real reference with normally be longer.\n\nSo in short your MWE can't produce the problem you describe unless you do edit things and start from an existing .aux that is representing a situation with 2 pages.\n\nNow there is the danger that the real reference is even shorter that ?? and it might have been better to generate an unknown reference showing only a single ?, but that has other issues as it doesn't really stand out in proof reading. So there are scenarios (with small probability) that can result in a sub-optimal solution.\n\nThere are also others that involve several references that result in moving text to different places. The one that I came up with is this one:\n\n\\documentclass{article}\n\n\\usepackage{varioref}\n\n\\setlength\\textheight{9.8\\baselineskip}\n\\setlength\\textwidth{2.9cm}\n\n\\begin{document}\n\n\\section{Foo}\n\nIn theory this document could come out with 2 pages only. But the\ndelayed processing makes it 3.\n\nRef to fig1: \\vref{fig1}. % this pushes things out\n\nSo test again to see \\vref{fig1} and and we also reference\nfigure two: \\vref{fig2}\n\n\\begin{figure}[t]\n\\caption{fig1}\\label{fig1}\n\\end{figure}\n\n\\begin{figure}[h]\nThis one is bigger\n\\caption{fig2}\\label{fig2}\n\\end{figure}\n\n\\end{document}\n\n\nIf you run this and look at page 2 you see\n\nthat is the first 2 references are fully on page 1 (because they are short, i.e., ??).\n\nNext time around, however we get\n\nbecause first references pushes things downwards. Worse the ref to fig1 was on page 1 last time so now the string \"on the next page\" is added (which was right but is wrong now). This extra material pushes figure 2 (temporarily) to page 3.\n\nNext time around the reference to fig 1 gets corrected (and the string \"on the next page\" vanishes) but now the ref to fig2 knows this figures is on page 3 and so this time it adds \"on the following page\" which takes up the space just freed:\n\nThis is now both correct and stable (but unfortunately not optimal). Basically it is a local optimum not the global one, but I fear that is something that is simply in the nature of the algorithm and not something you can easily work around (if at all).\n\nAfter all:\n\n\u2022 the generated text length varies in a non-linear fashion with the position of the target\n\u2022 that in turn may change the generated text (as the target position is changed by inserted generated texts\n\u2022 so you have the situation that both call-outs can move (any later ones are affected by generated text earlier on) but also the callback relationship that this in turn affects what needs to be generated as text.\n\nSo a local maximum seems to be the best you can hope for, in fact it is not too difficult to generated an \"impossible\" document, i.e., one that is changing in a way that it is always wrong.","date":"2019-06-25 03:27:03","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\": 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\": 0.8439509272575378, \"perplexity\": 1280.2050060328763}, \"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\/1560627999787.0\/warc\/CC-MAIN-20190625031825-20190625053825-00518.warc.gz\"}"}
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Romano Prodi (wym. ; ur. 9 sierpnia 1939 w Scandiano) – włoski polityk, ekonomista i nauczyciel akademicki, minister i parlamentarzysta, w latach 1996–1998 i 2006–2008 premier Włoch, od 1999 do 2004 przewodniczący Komisji Europejskiej.
Życiorys
Ukończył szkołę średnią Liceo Classico "Ludovico Ariosto" w Reggio nell'Emilia, a w 1961 studia prawnicze na Katolickim Uniwersytecie Najświętszego Serca w Mediolanie. Kształcił się następnie na studiach podyplomowych w Mediolanie i Bolonii, a także w London School of Economics. Od 1963 był asystentem na wydziale nauk politycznych Uniwersytetu Bolońskiego. W 1971 objął na tej uczelni stanowisko profesora, które zajmował do 1999. W pracy naukowej zajmował się zagadnieniami z zakresu rozwoju małych i średnich przedsiębiorstw, a także zależnościami między państwem i rynkiem, kwestiami związanymi z polityką przemysłową i prywatyzacją. Wykładał także m.in. na Uniwersytecie Harvarda i w Stanford Research Institute. W latach 1974–1978 zarządzał domem wydawniczym il Mulino, w 1981 założył przedsiębiorstwo konsultingowe Nomisma. Jako publicysta współpracował m.in. z "Corriere della Sera" i "Il Sole 24 Ore", w 1992 w Rai 1 prowadził program ekonomiczny Il tempo delle scelte.
W działalność polityczną zaangażował się w 1963, wstępując wówczas do Chrześcijańskiej Demokracji. Został w tymże roku radnym miejscowości Reggio nell'Emilia, zrezygnował z tej funkcji z uwagi na działalność naukową. W 1978 wraz z kilkoma innymi wykładowcami z jego uczelni przekazał wskazówki, które miały przyczynić się ujawnienia kryjówki, gdzie przetrzymywano porwanego i później zamordowanego przez terrorystów z Czerwonych Brygad byłego premiera Alda Moro. Romano Prodi powoływał się wówczas na "seans spirytystyczny", co wzbudziło wiele wątpliwości dotyczących faktycznego źródła informacji. Wątpliwości te podnoszono ponownie, wiązały się z ponawianymi od lat 90. oskarżeniami o domniemaną współpracę z KGB. Informacje na ten temat miałyby się znajdować w tzw. Archiwum Mitrochina, zarzuty te według Maria Scaramelli miał także potwierdzić przed swoją śmiercią Aleksandr Litwinienko.
W listopadzie 1978 Romano Prodi powrócił do działalności politycznej. Objął wówczas urząd ministra przemysłu, handlu i rzemiosła w czwartym rządzie Giulia Andreottiego. Sprawował go do końca funkcjonowania tego gabinetu w marcu 1979. W latach 1982–1989 kierował Istituto per la Ricostruzione Industriale (IRI), najważniejszym włoskim holdingiem państwowym, nadzorującym mienie państwowe i procesy prywatyzacji. Na początku lat 90. był konsultantem w koncernie Unilever, w tym samym czasie głównym klientem jego przedsiębiorstwa doradczego był bank inwestycyjny Goldman Sachs. W latach 1993–1994 ponownie był prezesem IRI, przeprowadzając kolejne prywatyzacje. Kontrowersje wzbudziła sprzedaż jednego z przedsiębiorstw spożywczych, które zostało szybko odsprzedane Unileverowi przy współudziale Goldman Sachs. Wobec Romana Prodiego pojawiły się oskarżenia o konflikt interesów. Rzymska prokurator Giuseppa Geremia w 1996 stwierdziła, że materiał dowodowy daje podstawy do przedstawienia ówczesnemu premierowi zarzutów. Postępowanie zostało jednak wkrótce zamknięte, a sama prokurator przeniesiona na Sardynię.
W lutym 1995 zainicjował powołanie centrolewicowej koalicji pod nazwą Drzewo Oliwne. Sojusz ten zwyciężył w wyborach parlamentarnych w 1996, a jego lider uzyskał wówczas mandat posła do Izby Deputowanych XIII kadencji. 17 maja 1996 Romano Prodi objął urząd premiera, który sprawował do 21 października 1998, kiedy to zastąpił go Massimo D'Alema. Ustąpił w związku z opuszczeniem koalicji rządowej przez Odrodzenie Komunistyczne. Zaangażował się następnie w tworzenie nowego liberalnego ugrupowania pod nazwą Demokraci.
W marcu 1999 Romano Prodi został nominowany na przewodniczącego Komisji Europejskiej. Stanowisko to objął 16 września 1999 i zajmował do listopada 2004. W okresie pełnienia przez niego tej funkcji do Unii Europejskiej przystąpiło dziesięć nowych państw, obradował Konwent Europejski i podpisano traktat ustanawiający Konstytucję dla Europy.
W październiku 2005 w wyniku prawyborów, w których otrzymał ponad 74% głosów, został liderem Unii, szerokiej koalicji skupiającej ugrupowania centrowe, centrolewicowe i komunistyczne, pozostające w opozycji do centroprawicy Silvia Berlusconiego. Blok ten zwyciężył w wyborach parlamentarnych w 2006, uzyskując dzięki nowej ordynacji stabilną większość w Izbie Deputowanych i jednocześnie minimalną większość w Senacie. Romano Prodi uzyskał wówczas mandat posła XV kadencji. 17 maja 2006 po raz drugi stanął na czele rządu. Brak spójności między licznymi koalicjantami powodował kolejne kryzysy polityczne. W lutym 2007, po przegranym głosowaniu dotyczącym polityki zagranicznej, Romano Prodi podał się do dymisji. Po kilku dniach kryzys rządowy został zażegnany, a rząd uzyskał wotum zaufania w obu izbach.
W październiku 2007 objął honorową funkcję przewodniczącego nowo powołanej Partii Demokratycznej, z której zrezygnował w kwietniu 2008. Od stycznia do lutego 2008 czasowo wykonywał obowiązki ministra sprawiedliwości. W styczniu 2008 doszło do kolejnego kryzysu rządowego, gdy Unię opuściła partia UDEUR Popolari, co spowodowało utratę większości parlamentarnej w Senacie. W izbie tej 24 stycznia 2008 odbyło się głosowanie za wotum zaufania dla rządu, przegrane w stosunku 156 do 161 głosów. Tego samego dnia Romano Prodi złożył na ręce prezydenta Włoch Giorgia Napolitano dymisję gabinetu. W marcu 2008 zadeklarował wycofanie się z działalności politycznej. 8 maja 2008, po przedterminowych wyborach parlamentarnych, na stanowisku premiera zastąpił go Silvio Berlusconi.
W latach 2008–2014 jako specjalny wysłannik sekretarza generalnego ONZ brał udział w różnych misjach pokojowych w Afryce. W 2009 powrócił jednocześnie do działalności akademickiej m.in. na Brown University.
Życie prywatne
W 1969 ożenił się z Flavią Franzoni. Ma dwóch synów – Giorgia i Antonia. Jego brat, Vittorio Prodi, zasiadał w Parlamencie Europejskim. W młodości brał udział w wyścigach kolarskich, w późniejszych latach jeździł wspólnie z takimi kolarzami jak Eddy Merckx, Mario Cipollini, Gianni Bugno czy Paolo Bettini.
Odznaczenia i wyróżnienia
Kawaler Krzyża Wielkiego Orderu Zasługi Republiki Włoskiej – Włochy, 1993
Krzyż Wielki Orderu Zasługi Rzeczypospolitej Polskiej – Polska, 1997
Krzyż Wielki Orderu Izabeli Katolickiej – Hiszpania, 1998
Krzyż Wielki Orderu Gwiazdy Rumunii – Rumunia, 2000
Komandor Krzyża Wielkiego Orderu Trzech Gwiazd – Łotwa, 2007
Krzyż Wielki Orderu Narodowego Legii Honorowej – Francja, 2014
Order Podwójnego Białego Krzyża I klasy – Słowacja, 2022
Tytuły doctora honoris causa nadane przez kilkadziesiąt uczelni włoskich i zagranicznych
Przypisy
Przewodniczący Komisji Europejskiej
Premierzy Włoch
Włoscy ministrowie (od 1946)
Włoscy parlamentarzyści (od 1946)
Politycy Chrześcijańskiej Demokracji (Włochy)
Politycy Partii Demokratycznej (Włochy)
Włoscy ekonomiści
Absolwenci London School of Economics
Doktorzy honoris causa Katolickiego Uniwersytetu Lubelskiego Jana Pawła II
Doktorzy honoris causa Politechniki Koszalińskiej
Odznaczeni Orderem Zasługi Republiki Włoskiej
Odznaczeni Krzyżem Wielkim Orderu Zasługi Rzeczypospolitej Polskiej
Odznaczeni Orderem Izabeli Katolickiej
Odznaczeni Krzyżem Wielkim Legii Honorowej
Odznaczeni Orderem Trzech Gwiazd
Odznaczeni Orderem Gwiazdy Rumunii
Odznaczeni Orderem Podwójnego Białego Krzyża
Urodzeni w 1939
Wykładowcy Brown University
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<a href="../packages/JAMA%0D%0ACholesky%20decomposition%20class%0D%0AFor%20a%20symmetric,%20positive%20definite%20matrix%20A,%20the%20Cholesky%20decomposition%0D%0Ais%20an%20lower%20triangular%20matrix%20L%20so%20that%20A%20=%20L*L'.html"><i
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Cholesky decomposition class
For a symmetric, positive definite matrix A, the Cholesky decomposition
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Class to obtain eigenvalues and eigenvectors of a real matrix</a></li>
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For an m-by-n matrix A with m >= n, the LU decomposition is an m-by-n
unit lower triangular matrix L, an n-by-n upper triangular matrix U,
and a permutation vector piv of length m so that A(piv,:) = L*U</a></li>
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For an m-by-n matrix A with m >= n, the QR decomposition is an m-by-n
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For an m-by-n matrix A with m >= n, the singular value decomposition is
an m-by-n orthogonal matrix U, an n-by-n diagonal matrix S, and
an n-by-n orthogonal matrix V so that A = U*S*V'</a></li>
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<a href="../packages/JAMA%0D%0APythagorean%20Theorem:%0D%0Aa%20=%203%0D%0Ab%20=%204%0D%0Ar%20=%20sqrt(square(a)%20+%20square(b))%0D%0Ar%20=%205%0D%0Ar%20=%20sqrt(a%5E2%20+%20b%5E2)%20without%20under.overflow.html"><i
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Pythagorean Theorem:
a = 3
b = 4
r = sqrt(square(a) + square(b))
r = 5
r = sqrt(a^2 + b^2) without under/overflow</a></li>
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<ul class="side-nav nav nav-list">
<li class="nav-header">
<i class="icon-custom icon-method"></i> Methods
<ul>
<li class="method public "><a href="#method___construct"
title="__construct :: Create a new PHPExcel_Chart_Layout"><span
class="description">Create a new PHPExcel_Chart_Layout</span>
<pre>__construct()</pre>
</a></li>
<li class="method public "><a href="#method_getHeight" title="getHeight :: Get Height"><span
class="description">Get Height</span>
<pre>getHeight()</pre>
</a></li>
<li class="method public "><a href="#method_getLayoutTarget"
title="getLayoutTarget :: Get Layout Target"><span class="description">Get Layout Target</span>
<pre>getLayoutTarget()</pre>
</a></li>
<li class="method public "><a href="#method_getShowBubbleSize"
title="getShowBubbleSize :: Get show bubble size"><span
class="description">Get show bubble size</span>
<pre>getShowBubbleSize()</pre>
</a></li>
<li class="method public "><a href="#method_getShowCatName"
title="getShowCatName :: Get show category name"><span
class="description">Get show category name</span>
<pre>getShowCatName()</pre>
</a></li>
<li class="method public "><a href="#method_getShowLeaderLines"
title="getShowLeaderLines :: Get show leader lines"><span
class="description">Get show leader lines</span>
<pre>getShowLeaderLines()</pre>
</a></li>
<li class="method public "><a href="#method_getShowLegendKey"
title="getShowLegendKey :: Get show legend key"><span class="description">Get show legend key</span>
<pre>getShowLegendKey()</pre>
</a></li>
<li class="method public "><a href="#method_getShowPercent"
title="getShowPercent :: Get show percentage"><span class="description">Get show percentage</span>
<pre>getShowPercent()</pre>
</a></li>
<li class="method public "><a href="#method_getShowSerName"
title="getShowSerName :: Get show data series name"><span
class="description">Get show data series name</span>
<pre>getShowSerName()</pre>
</a></li>
<li class="method public "><a href="#method_getShowVal" title="getShowVal :: Get show value"><span
class="description">Get show value</span>
<pre>getShowVal()</pre>
</a></li>
<li class="method public "><a href="#method_getWidth" title="getWidth :: Get Width"><span
class="description">Get Width</span>
<pre>getWidth()</pre>
</a></li>
<li class="method public "><a href="#method_getXMode" title="getXMode :: Get X-Mode"><span
class="description">Get X-Mode</span>
<pre>getXMode()</pre>
</a></li>
<li class="method public "><a href="#method_getXPosition" title="getXPosition :: Get X-Position"><span
class="description">Get X-Position</span>
<pre>getXPosition()</pre>
</a></li>
<li class="method public "><a href="#method_getYMode" title="getYMode :: Get Y-Mode"><span
class="description">Get Y-Mode</span>
<pre>getYMode()</pre>
</a></li>
<li class="method public "><a href="#method_getYPosition" title="getYPosition :: Get Y-Position"><span
class="description">Get Y-Position</span>
<pre>getYPosition()</pre>
</a></li>
<li class="method public "><a href="#method_setHeight" title="setHeight :: Set Height"><span
class="description">Set Height</span>
<pre>setHeight()</pre>
</a></li>
<li class="method public "><a href="#method_setLayoutTarget"
title="setLayoutTarget :: Set Layout Target"><span class="description">Set Layout Target</span>
<pre>setLayoutTarget()</pre>
</a></li>
<li class="method public "><a href="#method_setShowBubbleSize" title="setShowBubbleSize :: Set show bubble size
Specifies that the bubble size should be shown in data labels."><span class="description">Set show bubble size
Specifies that the bubble size should be shown in data labels.</span>
<pre>setShowBubbleSize()</pre>
</a></li>
<li class="method public "><a href="#method_setShowCatName" title="setShowCatName :: Set show cat name
Specifies that the category name should be shown in data labels."><span class="description">Set show cat name
Specifies that the category name should be shown in data labels.</span>
<pre>setShowCatName()</pre>
</a></li>
<li class="method public "><a href="#method_setShowLeaderLines" title="setShowLeaderLines :: Set show leader lines
Specifies that leader lines should be shown in data labels."><span class="description">Set show leader lines
Specifies that leader lines should be shown in data labels.</span>
<pre>setShowLeaderLines()</pre>
</a></li>
<li class="method public "><a href="#method_setShowLegendKey" title="setShowLegendKey :: Set show legend key
Specifies that legend keys should be shown in data labels."><span class="description">Set show legend key
Specifies that legend keys should be shown in data labels.</span>
<pre>setShowLegendKey()</pre>
</a></li>
<li class="method public "><a href="#method_setShowPercent" title="setShowPercent :: Set show percentage
Specifies that the percentage should be shown in data labels."><span class="description">Set show percentage
Specifies that the percentage should be shown in data labels.</span>
<pre>setShowPercent()</pre>
</a></li>
<li class="method public "><a href="#method_setShowSerName" title="setShowSerName :: Set show ser name
Specifies that the series name should be shown in data labels."><span class="description">Set show ser name
Specifies that the series name should be shown in data labels.</span>
<pre>setShowSerName()</pre>
</a></li>
<li class="method public "><a href="#method_setShowVal" title="setShowVal :: Set show val
Specifies that the value should be shown in data labels."><span class="description">Set show val
Specifies that the value should be shown in data labels.</span>
<pre>setShowVal()</pre>
</a></li>
<li class="method public "><a href="#method_setWidth" title="setWidth :: Set Width"><span
class="description">Set Width</span>
<pre>setWidth()</pre>
</a></li>
<li class="method public "><a href="#method_setXMode" title="setXMode :: Set X-Mode"><span
class="description">Set X-Mode</span>
<pre>setXMode()</pre>
</a></li>
<li class="method public "><a href="#method_setXPosition" title="setXPosition :: Set X-Position"><span
class="description">Set X-Position</span>
<pre>setXPosition()</pre>
</a></li>
<li class="method public "><a href="#method_setYMode" title="setYMode :: Set Y-Mode"><span
class="description">Set Y-Mode</span>
<pre>setYMode()</pre>
</a></li>
<li class="method public "><a href="#method_setYPosition" title="setYPosition :: Set Y-Position"><span
class="description">Set Y-Position</span>
<pre>setYPosition()</pre>
</a></li>
</ul>
</li>
<li class="nav-header">
<i class="icon-custom icon-property"></i> Properties
<ul></ul>
</li>
<li class="nav-header private">» Private
<ul>
<li class="property private "><a href="#property__height" title="$_height :: height"><span
class="description"></span>
<pre>$_height</pre>
</a></li>
<li class="property private "><a href="#property__layoutTarget"
title="$_layoutTarget :: layoutTarget"><span
class="description"></span>
<pre>$_layoutTarget</pre>
</a></li>
<li class="property private "><a href="#property__showBubbleSize"
title="$_showBubbleSize :: show bubble size"><span
class="description"></span>
<pre>$_showBubbleSize</pre>
</a></li>
<li class="property private "><a href="#property__showCatName" title="$_showCatName :: show category name
Specifies that the category name should be shown in the data label."><span class="description"></span>
<pre>$_showCatName</pre>
</a></li>
<li class="property private "><a href="#property__showLeaderLines" title="$_showLeaderLines :: show leader lines
Specifies that leader lines should be shown for the data label."><span class="description"></span>
<pre>$_showLeaderLines</pre>
</a></li>
<li class="property private "><a href="#property__showLegendKey" title="$_showLegendKey :: show legend key
Specifies that legend keys should be shown in data labels"><span class="description"></span>
<pre>$_showLegendKey</pre>
</a></li>
<li class="property private "><a href="#property__showPercent" title="$_showPercent :: show percentage
Specifies that the percentage should be shown in the data label."><span class="description"></span>
<pre>$_showPercent</pre>
</a></li>
<li class="property private "><a href="#property__showSerName" title="$_showSerName :: show data series name
Specifies that the series name should be shown in the data label."><span class="description"></span>
<pre>$_showSerName</pre>
</a></li>
<li class="property private "><a href="#property__showVal" title="$_showVal :: show value
Specifies that the value should be shown in a data label."><span class="description"></span>
<pre>$_showVal</pre>
</a></li>
<li class="property private "><a href="#property__width" title="$_width :: width"><span
class="description"></span>
<pre>$_width</pre>
</a></li>
<li class="property private "><a href="#property__xMode" title="$_xMode :: X Mode"><span
class="description"></span>
<pre>$_xMode</pre>
</a></li>
<li class="property private "><a href="#property__xPos" title="$_xPos :: X-Position"><span
class="description"></span>
<pre>$_xPos</pre>
</a></li>
<li class="property private "><a href="#property__yMode" title="$_yMode :: Y Mode"><span
class="description"></span>
<pre>$_yMode</pre>
</a></li>
<li class="property private "><a href="#property__yPos" title="$_yPos :: Y-Position"><span
class="description"></span>
<pre>$_yPos</pre>
</a></li>
</ul>
</li>
</ul>
</div>
<div class="span8">
<a id="\PHPExcel_Chart_Layout"></a>
<ul class="breadcrumb">
<li>
<a href="../index.html"><i class="icon-custom icon-class"></i></a><span class="divider">\</span>
</li>
<li><a href="../namespaces/global.html">global</a></li>
<li class="active">
<span class="divider">\</span><a href="../classes/PHPExcel_Chart_Layout.html">PHPExcel_Chart_Layout</a>
</li>
</ul>
<div class="element class">
<p class="short_description">PHPExcel_Chart_Layout</p>
<div class="details">
<div class="long_description"></div>
<table class="table table-bordered">
<tr>
<th>category</th>
<td>PHPExcel</td>
</tr>
<tr>
<th>package</th>
<td><a href="../packages/PHPExcel.Chart.html">PHPExcel_Chart</a></td>
</tr>
<tr>
<th>copyright</th>
<td>Copyright (c) 2006 - 2013 PHPExcel (http://www.codeplex.com/PHPExcel)</td>
</tr>
</table>
<h3>
<i class="icon-custom icon-method"></i> Methods</h3>
<a id="method___construct"></a>
<div class="element clickable method public method___construct" data-toggle="collapse"
data-target=".method___construct .collapse">
<h2>Create a new PHPExcel_Chart_Layout</h2>
<pre>__construct($layout) </pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
<h3>Parameters</h3>
<div class="subelement argument"><h4>$layout</h4></div>
</div>
</div>
</div>
<a id="method_getHeight"></a>
<div class="element clickable method public method_getHeight" data-toggle="collapse"
data-target=".method_getHeight .collapse">
<h2>Get Height</h2>
<pre>getHeight() : \number</pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
<h3>Returns</h3>
<div class="subelement response"><code>\number</code></div>
</div>
</div>
</div>
<a id="method_getLayoutTarget"></a>
<div class="element clickable method public method_getLayoutTarget" data-toggle="collapse"
data-target=".method_getLayoutTarget .collapse">
<h2>Get Layout Target</h2>
<pre>getLayoutTarget() : string</pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
<h3>Returns</h3>
<div class="subelement response"><code>string</code></div>
</div>
</div>
</div>
<a id="method_getShowBubbleSize"></a>
<div class="element clickable method public method_getShowBubbleSize" data-toggle="collapse"
data-target=".method_getShowBubbleSize .collapse">
<h2>Get show bubble size</h2>
<pre>getShowBubbleSize() : boolean</pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
<h3>Returns</h3>
<div class="subelement response"><code>boolean</code></div>
</div>
</div>
</div>
<a id="method_getShowCatName"></a>
<div class="element clickable method public method_getShowCatName" data-toggle="collapse"
data-target=".method_getShowCatName .collapse">
<h2>Get show category name</h2>
<pre>getShowCatName() : boolean</pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
<h3>Returns</h3>
<div class="subelement response"><code>boolean</code></div>
</div>
</div>
</div>
<a id="method_getShowLeaderLines"></a>
<div class="element clickable method public method_getShowLeaderLines" data-toggle="collapse"
data-target=".method_getShowLeaderLines .collapse">
<h2>Get show leader lines</h2>
<pre>getShowLeaderLines() : boolean</pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
<h3>Returns</h3>
<div class="subelement response"><code>boolean</code></div>
</div>
</div>
</div>
<a id="method_getShowLegendKey"></a>
<div class="element clickable method public method_getShowLegendKey" data-toggle="collapse"
data-target=".method_getShowLegendKey .collapse">
<h2>Get show legend key</h2>
<pre>getShowLegendKey() : boolean</pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
<h3>Returns</h3>
<div class="subelement response"><code>boolean</code></div>
</div>
</div>
</div>
<a id="method_getShowPercent"></a>
<div class="element clickable method public method_getShowPercent" data-toggle="collapse"
data-target=".method_getShowPercent .collapse">
<h2>Get show percentage</h2>
<pre>getShowPercent() : boolean</pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
<h3>Returns</h3>
<div class="subelement response"><code>boolean</code></div>
</div>
</div>
</div>
<a id="method_getShowSerName"></a>
<div class="element clickable method public method_getShowSerName" data-toggle="collapse"
data-target=".method_getShowSerName .collapse">
<h2>Get show data series name</h2>
<pre>getShowSerName() : boolean</pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
<h3>Returns</h3>
<div class="subelement response"><code>boolean</code></div>
</div>
</div>
</div>
<a id="method_getShowVal"></a>
<div class="element clickable method public method_getShowVal" data-toggle="collapse"
data-target=".method_getShowVal .collapse">
<h2>Get show value</h2>
<pre>getShowVal() : boolean</pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
<h3>Returns</h3>
<div class="subelement response"><code>boolean</code></div>
</div>
</div>
</div>
<a id="method_getWidth"></a>
<div class="element clickable method public method_getWidth" data-toggle="collapse"
data-target=".method_getWidth .collapse">
<h2>Get Width</h2>
<pre>getWidth() : \number</pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
<h3>Returns</h3>
<div class="subelement response"><code>\number</code></div>
</div>
</div>
</div>
<a id="method_getXMode"></a>
<div class="element clickable method public method_getXMode" data-toggle="collapse"
data-target=".method_getXMode .collapse">
<h2>Get X-Mode</h2>
<pre>getXMode() : string</pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
<h3>Returns</h3>
<div class="subelement response"><code>string</code></div>
</div>
</div>
</div>
<a id="method_getXPosition"></a>
<div class="element clickable method public method_getXPosition" data-toggle="collapse"
data-target=".method_getXPosition .collapse">
<h2>Get X-Position</h2>
<pre>getXPosition() : \number</pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
<h3>Returns</h3>
<div class="subelement response"><code>\number</code></div>
</div>
</div>
</div>
<a id="method_getYMode"></a>
<div class="element clickable method public method_getYMode" data-toggle="collapse"
data-target=".method_getYMode .collapse">
<h2>Get Y-Mode</h2>
<pre>getYMode() : string</pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
<h3>Returns</h3>
<div class="subelement response"><code>string</code></div>
</div>
</div>
</div>
<a id="method_getYPosition"></a>
<div class="element clickable method public method_getYPosition" data-toggle="collapse"
data-target=".method_getYPosition .collapse">
<h2>Get Y-Position</h2>
<pre>getYPosition() : \number</pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
<h3>Returns</h3>
<div class="subelement response"><code>\number</code></div>
</div>
</div>
</div>
<a id="method_setHeight"></a>
<div class="element clickable method public method_setHeight" data-toggle="collapse"
data-target=".method_setHeight .collapse">
<h2>Set Height</h2>
<pre>setHeight(\Height $value) </pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
<h3>Parameters</h3>
<div class="subelement argument">
<h4>$value</h4>
<code>\Height</code>
</div>
</div>
</div>
</div>
<a id="method_setLayoutTarget"></a>
<div class="element clickable method public method_setLayoutTarget" data-toggle="collapse"
data-target=".method_setLayoutTarget .collapse">
<h2>Set Layout Target</h2>
<pre>setLayoutTarget(\Layout $value) </pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
<h3>Parameters</h3>
<div class="subelement argument">
<h4>$value</h4>
<code>\Layout</code>
<p>Target $value</p>
</div>
</div>
</div>
</div>
<a id="method_setShowBubbleSize"></a>
<div class="element clickable method public method_setShowBubbleSize" data-toggle="collapse"
data-target=".method_setShowBubbleSize .collapse">
<h2>Set show bubble size
Specifies that the bubble size should be shown in data labels.</h2>
<pre>setShowBubbleSize(boolean $value) </pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
<h3>Parameters</h3>
<div class="subelement argument">
<h4>$value</h4>
<code>boolean</code>
<p>Show bubble size</p></div>
</div>
</div>
</div>
<a id="method_setShowCatName"></a>
<div class="element clickable method public method_setShowCatName" data-toggle="collapse"
data-target=".method_setShowCatName .collapse">
<h2>Set show cat name
Specifies that the category name should be shown in data labels.</h2>
<pre>setShowCatName(boolean $value) </pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
<h3>Parameters</h3>
<div class="subelement argument">
<h4>$value</h4>
<code>boolean</code>
<p>Show cat name</p></div>
</div>
</div>
</div>
<a id="method_setShowLeaderLines"></a>
<div class="element clickable method public method_setShowLeaderLines" data-toggle="collapse"
data-target=".method_setShowLeaderLines .collapse">
<h2>Set show leader lines
Specifies that leader lines should be shown in data labels.</h2>
<pre>setShowLeaderLines(boolean $value) </pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
<h3>Parameters</h3>
<div class="subelement argument">
<h4>$value</h4>
<code>boolean</code>
<p>Show leader lines</p></div>
</div>
</div>
</div>
<a id="method_setShowLegendKey"></a>
<div class="element clickable method public method_setShowLegendKey" data-toggle="collapse"
data-target=".method_setShowLegendKey .collapse">
<h2>Set show legend key
Specifies that legend keys should be shown in data labels.</h2>
<pre>setShowLegendKey(boolean $value) </pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
<h3>Parameters</h3>
<div class="subelement argument">
<h4>$value</h4>
<code>boolean</code>
<p>Show legend key</p></div>
</div>
</div>
</div>
<a id="method_setShowPercent"></a>
<div class="element clickable method public method_setShowPercent" data-toggle="collapse"
data-target=".method_setShowPercent .collapse">
<h2>Set show percentage
Specifies that the percentage should be shown in data labels.</h2>
<pre>setShowPercent(boolean $value) </pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
<h3>Parameters</h3>
<div class="subelement argument">
<h4>$value</h4>
<code>boolean</code>
<p>Show percentage</p></div>
</div>
</div>
</div>
<a id="method_setShowSerName"></a>
<div class="element clickable method public method_setShowSerName" data-toggle="collapse"
data-target=".method_setShowSerName .collapse">
<h2>Set show ser name
Specifies that the series name should be shown in data labels.</h2>
<pre>setShowSerName(boolean $value) </pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
<h3>Parameters</h3>
<div class="subelement argument">
<h4>$value</h4>
<code>boolean</code>
<p>Show ser name</p></div>
</div>
</div>
</div>
<a id="method_setShowVal"></a>
<div class="element clickable method public method_setShowVal" data-toggle="collapse"
data-target=".method_setShowVal .collapse">
<h2>Set show val
Specifies that the value should be shown in data labels.</h2>
<pre>setShowVal(boolean $value) </pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
<h3>Parameters</h3>
<div class="subelement argument">
<h4>$value</h4>
<code>boolean</code>
<p>Show val</p></div>
</div>
</div>
</div>
<a id="method_setWidth"></a>
<div class="element clickable method public method_setWidth" data-toggle="collapse"
data-target=".method_setWidth .collapse">
<h2>Set Width</h2>
<pre>setWidth(\Width $value) </pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
<h3>Parameters</h3>
<div class="subelement argument">
<h4>$value</h4>
<code>\Width</code>
</div>
</div>
</div>
</div>
<a id="method_setXMode"></a>
<div class="element clickable method public method_setXMode" data-toggle="collapse"
data-target=".method_setXMode .collapse">
<h2>Set X-Mode</h2>
<pre>setXMode(\X-Mode $value) </pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
<h3>Parameters</h3>
<div class="subelement argument">
<h4>$value</h4>
<code>\X-Mode</code>
</div>
</div>
</div>
</div>
<a id="method_setXPosition"></a>
<div class="element clickable method public method_setXPosition" data-toggle="collapse"
data-target=".method_setXPosition .collapse">
<h2>Set X-Position</h2>
<pre>setXPosition(\X-Position $value) </pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
<h3>Parameters</h3>
<div class="subelement argument">
<h4>$value</h4>
<code>\X-Position</code>
</div>
</div>
</div>
</div>
<a id="method_setYMode"></a>
<div class="element clickable method public method_setYMode" data-toggle="collapse"
data-target=".method_setYMode .collapse">
<h2>Set Y-Mode</h2>
<pre>setYMode(\Y-Mode $value) </pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
<h3>Parameters</h3>
<div class="subelement argument">
<h4>$value</h4>
<code>\Y-Mode</code>
</div>
</div>
</div>
</div>
<a id="method_setYPosition"></a>
<div class="element clickable method public method_setYPosition" data-toggle="collapse"
data-target=".method_setYPosition .collapse">
<h2>Set Y-Position</h2>
<pre>setYPosition(\Y-Position $value) </pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
<h3>Parameters</h3>
<div class="subelement argument">
<h4>$value</h4>
<code>\Y-Position</code>
</div>
</div>
</div>
</div>
<h3>
<i class="icon-custom icon-property"></i> Properties</h3>
<a id="property__height"> </a>
<div class="element clickable property private property__height" data-toggle="collapse"
data-target=".property__height .collapse">
<h2></h2>
<pre>$_height : float</pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
</div>
</div>
</div>
<a id="property__layoutTarget"> </a>
<div class="element clickable property private property__layoutTarget" data-toggle="collapse"
data-target=".property__layoutTarget .collapse">
<h2></h2>
<pre>$_layoutTarget : string</pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
</div>
</div>
</div>
<a id="property__showBubbleSize"> </a>
<div class="element clickable property private property__showBubbleSize" data-toggle="collapse"
data-target=".property__showBubbleSize .collapse">
<h2></h2>
<pre>$_showBubbleSize : boolean</pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
</div>
</div>
</div>
<a id="property__showCatName"> </a>
<div class="element clickable property private property__showCatName" data-toggle="collapse"
data-target=".property__showCatName .collapse">
<h2></h2>
<pre>$_showCatName : boolean</pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
</div>
</div>
</div>
<a id="property__showLeaderLines"> </a>
<div class="element clickable property private property__showLeaderLines" data-toggle="collapse"
data-target=".property__showLeaderLines .collapse">
<h2></h2>
<pre>$_showLeaderLines : boolean</pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
</div>
</div>
</div>
<a id="property__showLegendKey"> </a>
<div class="element clickable property private property__showLegendKey" data-toggle="collapse"
data-target=".property__showLegendKey .collapse">
<h2></h2>
<pre>$_showLegendKey : boolean</pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
</div>
</div>
</div>
<a id="property__showPercent"> </a>
<div class="element clickable property private property__showPercent" data-toggle="collapse"
data-target=".property__showPercent .collapse">
<h2></h2>
<pre>$_showPercent : boolean</pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
</div>
</div>
</div>
<a id="property__showSerName"> </a>
<div class="element clickable property private property__showSerName" data-toggle="collapse"
data-target=".property__showSerName .collapse">
<h2></h2>
<pre>$_showSerName : boolean</pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
</div>
</div>
</div>
<a id="property__showVal"> </a>
<div class="element clickable property private property__showVal" data-toggle="collapse"
data-target=".property__showVal .collapse">
<h2></h2>
<pre>$_showVal : boolean</pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
</div>
</div>
</div>
<a id="property__width"> </a>
<div class="element clickable property private property__width" data-toggle="collapse"
data-target=".property__width .collapse">
<h2></h2>
<pre>$_width : float</pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
</div>
</div>
</div>
<a id="property__xMode"> </a>
<div class="element clickable property private property__xMode" data-toggle="collapse"
data-target=".property__xMode .collapse">
<h2></h2>
<pre>$_xMode : string</pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
</div>
</div>
</div>
<a id="property__xPos"> </a>
<div class="element clickable property private property__xPos" data-toggle="collapse"
data-target=".property__xPos .collapse">
<h2></h2>
<pre>$_xPos : float</pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
</div>
</div>
</div>
<a id="property__yMode"> </a>
<div class="element clickable property private property__yMode" data-toggle="collapse"
data-target=".property__yMode .collapse">
<h2></h2>
<pre>$_yMode : string</pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
</div>
</div>
</div>
<a id="property__yPos"> </a>
<div class="element clickable property private property__yPos" data-toggle="collapse"
data-target=".property__yPos .collapse">
<h2></h2>
<pre>$_yPos : float</pre>
<div class="labels"></div>
<div class="row collapse">
<div class="detail-description">
<div class="long_description"></div>
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</div>
</div>
</div>
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<div class="row">
<footer class="span12">
Template is built using <a href="http://twitter.github.com/bootstrap/">Twitter Bootstrap 2</a> and icons
provided by <a href="http://glyphicons.com/">Glyphicons</a>.<br>
Documentation is powered by <a href="http://www.phpdoc.org/">phpDocumentor 2.0.0a12</a> and<br>
generated on 2013-06-02T15:42:48+01:00.<br></footer>
</div>
</div>
</body>
</html>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,180
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<component name="InspectionProjectProfileManager">
<profile version="1.0" is_locked="false">
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<item index="1" class="java.lang.String" itemvalue="oslo.utils" />
<item index="2" class="java.lang.String" itemvalue="Babel" />
<item index="3" class="java.lang.String" itemvalue="docker-py" />
<item index="4" class="java.lang.String" itemvalue="oslo.serialization" />
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<item index="21" class="java.lang.String" itemvalue="Paste" />
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<item index="28" class="java.lang.String" itemvalue="python-neutronclient" />
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| 7,263
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Food Network Magazine created fun new recipes by the dozen.
Deviled eggs are a springtime favorite. They're perfect for an Easter spread or for any special gathering when the weather gets warmer. The Kitchen has an adorable tiny garnish idea that will dress up your eggs--and your table too!
Ideal for easy spring entertaining, Food Network's best five deviled egg recipes are crowd-pleasing appetizers that can be prepared quickly.
Rethink traditional deviled eggs with these nine fresh updates.
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{"url":"https:\/\/www.gamedev.net\/forums\/topic\/701602-cscriptarray-not-checking-for-malloc-fail-during-precache-malloc-fail-handling-in-general\/?tab=comments","text":"# cscriptarray not checking for malloc fail during precache() \/ malloc fail handling in general\n\n## Recommended Posts\n\nHello!\u00a0 This is the first of a few posts I have concerning AngelCode, which has truly been an angel and made it possible to pursue my open source project (more info about that in another post).\u00a0 While looking at cscriptarray (version 2.33.0), I noticed it didn't check for a new pointer after a malloc in Precache(), around line 1694.\u00a0 The fix I added (though I don't think I've tested it specifically yet) is:\n\n\/\/ Create the cache\ncache = reinterpret_cast<SArrayCache*>(userAlloc(sizeof(SArrayCache)));\n\n\/\/\/ BEGIN malloc check\nif (! cache)\n{\nasIScriptContext *ctx = asGetActiveContext();\nif( ctx )\nctx->SetException(\"Out of memory\");\nreturn;\n}\n\/\/\/ END malloc check\n\nmemset(cache, 0, sizeof(SArrayCache));\n\n\nJust wanted to note it as a potential bug for fixing in later versions.\n\nRelated, I have a more general question about how AngelCode handles malloc fails.\u00a0 A quick inspection of the code suggests that most areas of the interpreter code cannot handle a malloc call failing.\u00a0 Is this true or did I misread it?\u00a0 Is this something under consideration for fixing in the future?\n\nBasically, I'm using AngelScript in what is essentially a semi-open sandbox environment, where most users could (if desired) write code and have it execute on the server, often simultaneously with other scripts.\u00a0 Because of this, I have to keep careful tabs on resource usage (CPU and memory).\u00a0 For now, I'm checking memory used by each AngelScript engine via the debug hook (and a custom allocator) and aborting in there if it went over.\u00a0 It would be much nicer to be able to abort during the actual allocation and clean up, but I suspect it would be way too difficult for AngelScript to check every single malloc.\u00a0 Still, I thought I'd check!\u00a0 Thanks for making such a nice, easy-to-integrate product!\n\n##### Share on other sites\n\nThanks for letting me know of the missing check for failed malloc. I'll make sure to add this fix for the next release.\n\nThe solution you implemented is the correct way to do it. It should be done the same way in every other memory allocation as well. You should be able to simply return zero on the custom memory allocator when you don't want to allow an engine to consume more memory. It should be treated by Angelscript.\n\nLet me know if you find other places where the code doesn't treat failed mallocs.\n\nRegards,\n\nAndreas\n\n##### Share on other sites\n\nYou're welcome! \u00a0 Also, what about the AngelScript interpreter itself?\u00a0 Is it supposed to be able to handle a failed malloc internally at any time?\u00a0 My guess is 'no' based on what I saw, but I did not do an extensive code analysis and the failure may have been caught further up the chain.\u00a0 It was also an older version.\n\nI'm thinking my periodic heap check every few times the debug hook is called should be sufficient, but the finer grained I can make it the better.\n\n##### Share on other sites\n\nThe code should be able to handle it, though I can't say I've tested every possible situation where a memory allocation might fail.\n\nJust report any situation where you feel the code might not handle it properly and I'll fix it.\n\n##### Share on other sites\n\nI've fixed the missing check for failed memory allocation in revision 2586.\n\n## Create an account or sign in to comment\n\nYou need to be a member in order to leave a comment\n\n## Create an account\n\nSign up for a new account in our community. It's easy!\n\nRegister a new account\n\n\u2022 ### Game Developer Survey\n\nWe are looking for qualified game developers to participate in a 10-minute online survey. Qualified participants will be offered a \\$15 incentive for your time and insights. Click here to start!\n\n\u2022 10\n\u2022 15\n\u2022 22\n\u2022 19\n\u2022 46","date":"2019-09-22 20:22:05","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\": 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\": 0.27271175384521484, \"perplexity\": 2242.107983667466}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"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-2019-39\/segments\/1568514575674.3\/warc\/CC-MAIN-20190922201055-20190922223055-00127.warc.gz\"}"}
| null | null |
Q: Apache ReWrite rule for & to & Already doing this rewrite rule
RewriteCond %{SERVER_PORT} !^443$
RewriteRule ^.*$ https://www.%{SERVER_NAME}%{REQUEST_URI}
Now the requested url is
https://www.example.com/abc.php?a=&b=abc&c=def&d=0&e=1
to
https://www.example.com/abc.php?a=&b=abc&c=def&d=0&e=1
A: add the NE tag at the end of the rule (No Escape):
RewriteRule (...) [NE]
|
{
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| 1,862
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Lee Sun-hee (Hangul: 이선희; Boryeong, 11 de noviembre de 1964) es una cantante de baladas surcoreana quien debutó en 1984 con la canción "To J". En su país, es ampliamente considerada como la más exitosa y especializada vocalista del país, dándole los apodos "국민디바" ("Diva Nacional") y "여가왕" (Reina de las Vocalistas).
Biografía
Lee Sun-hee nació el 11 de noviembre de 1964 en Boryeong, Provincia de Chungcheong, Corea del Sur. Asistió al instituto femenino de Sangmyung y se graduó de la Universidad de la Ciudad de Incheon. Es conocida por sus seguidores cariñosamente como "Sunny" (soleada) o "Little Giant" (pequeña gigante). Hija de un monje budista fue criada cerca a un templo budista donde un gran número de monjes residía. Su padre era un maestro de la música budista tradicional conocida como Beompae en Asia del este.
Fuera de su carrera musical Lee participa en numerosas actividades sociales en Corea como por ejemplo : el bienestar femenino, apoyo a las personas mayores, ayuda a las niñas así como la organización y la promoción de conciertos.En 1991 fue incluso elegida como concejal.
Estilo musical
Durante su concierto de 30 aniversario se transmitió un clip durante el entreacto que revela sus tres influencias musicales más grandes: Barbra Streisand, Madonna, y Whitney Houston. Hizo mención también de vocalistas coreanos como 송창식 (Song Chang-sik) como una de sus influencias y ejemplos a seguir. Sun-hee posee una cálida voz de lírico-soprano. Además de ser vocalista, Sun-hee es también compositora, con muchas de las canciones de sus últimos álbumes compuestas por ella (entre ellos el célebre éxito 인연 "Destino").
Discografía
Álbum de estudio
1985 : 아! 옛날이여 (Ah! The Good Old Days)
1985 : 갈바람 (Galbaram)
1986 : 알고 싶어요 (I Want to Know)
1988 : 사랑이 지는 이자리 / 나 항상 그대를 (Where the Love Falls / aka: I Always Miss You)
1989 : 나의 거리 / 한바탕 웃음으로 (My Street / aka: Bout of Laughter)
1990 : 추억의 책장을 넘기면 (Turning the Pages of Memories)
1991 : 그대가 나를 사랑하신다면 (If You Love Me)
1992 : 조각배 (Small Boat)
1994 : 한송이 국화 (Chrysanthemum)
1996 : First Love
1998 : Dream of Ruby
2001 : My life + Best
2005 : 사춘기(四春期) (Puberty [spring in my forties])
2009 : 사랑아... (Dear Love...)
2014 : 세렌디피티 (30주년 기념앨범) SERENDIPITY (30th Anniversary Álbum)
Álbum especial
1984 : 이선희의 캐롤 / Lee Sun Hee's Carols (Christmas carols álbum) - Álbum donde retoma numerosas canciones navideñas (Jigu Récords).
1988 : 겨울 이야기 / Winter Story - Otro álbum de canciones navideñas (Seoul Récords).
1989 : Where the Love Falls / Dear J - Álbum donde retoma 8 de sus canciones en Ingleses (Seoul Récords).
1990 : Leaving Only the Dream of Love - Álbum que reagrupa poesías de Lee Sun-hee (Seoul Récords).
1991 : 우리들의 이야기 - (Nuestra Historia) - Álbum donde retoma canciones navideñas y donde rinde homenaje entre demás al Ejército Sur Coreano, a su madre, a sus amigos, a sus profesores (Seoul Récords).
1993 : Children's Songs - Álbum de canciones infantiles (Seoul Récords).
1996 : Golden - Compilación de canciones (Samsung Music - Yedang Entertainment)
Álbum en vivo
1990 : Lee Sun Hee & Montreal Chamber Orquesta (Seoul Récords).
1991 : Peter Pan - Musical grabado en el Sejong Center para Performing Artes (Seoul Récords).
1995 : 세종문화회관 라이브 / Center Sejong para los artes de la escena Live 1994 - "Seoul Love Concierto septiembre de 1994, igualmente concierto de los 10 años de carrera de Lee Sun Hee
1995 : El Mago de Oz - Musical grabado en el Sejong Center para Performing Arts en 1990 (Península Music)
1999 : Bari / The Forgotten Lullaby - Musical grabado en el Centro de artes de Seúl, Opera House
Premios
Referencias
Enlaces externos
Biografía disponible en coreano sobre el website oficial de Lee Sun Hee
Biografía disponible en coreano sobre el website maniadb
Nacidos en Boryeong
Cantantes femeninas de pop de Corea del Sur
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Daniel brings several years as an independent producer and sales-agent at Preferred Content to spearhead the development of a new model of digital distribution at SingularDTV. He has negotiated the worldwide and North American sales for dozens of independent films to distributors ranging from Netflix and Amazon Prime to Gravitas and IFC. Prior to entertainment, Daniel spent years as a consultant to financial institution clients including Barclays and BNP Paribas, developing new finance and accounting models for derivative securities.
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Q: Running a Task method that has no await Suppose I have the following method:
public async Task<bool> DoSomeStuffWithAsync()
{
//do some stuff without calling await
return true;
}
and I call it like this:
await DoSomeStuffWithAsync();
Does this run any differently to this method:
public bool DoSomeStuff()
{
return true;
}
being called like this:
DoSomeStuff();
I am thinking that because DoSomeStuffWithAsync doesn't actually call await in its code that it will still run the same as DoSomeStuff.
Is there any benefit structuring my method like DoSomeStuffWithAsync?
A: Note that using async for a method that does not use await is considered a bad style. If you have a synchronous method that returns a task, you should use Task.FromResult instead:
public Task<bool> DoSomeStuffWithAsync()
{
return Task.FromResult(true);
}
But yeah, that's all this is: A synchronous method that returns a task. There is nothign asynchronous about it, so this does not actually run any differently than normal synchronous code that just returns a bool.
Is there still a benefit to returning a task instead of the value directly? That depends. Sometimes, you may have an asynchronous interface you need to implement but you don't actually have an asynchronous implementation for it. But if you don't have to do this and if you also don't consider changing your implementation eventually to actually do something asynchronously, then there's no benefit in making your method return a task.
As for the call, await DoSomeStuffWithAsync() will run DoSomeStuffWithAsync() and then await the task it returns. Since DoSomeStuffWithAsync() is synchronous, the task will return directly, and since the task is already completed, await will be able to directly continue. So this will be very similar to calling the method synchronously. The presence of that await does have an impact on the runtime behavior though, since this will make that calling method actually asynchronous (even if the called code is synchronous).
But it's up to you to decide whether that's appropriate in your case. In general though, you should probably avoid making this appear asynchronous when there isn't a real asynchronous process involved.
A: In short terms async keyword does not make a method to run asynchronously. It only allows the method to call await keyword inside. That's where the magic happens. In your case you don't have any await keyword anywhere which indicates your method does not indeed run asynchronously.
A: Your methods return different things.
DoSomeStuffWithAsync() returns a Task<bool>. You can call it without the 'async" keyword - synchronously - the compiler will return bool wrapped in an instance of Task<bool> once the method is complete - it does that behind the scenes. And another "difference" is how you handle the returned value...
var task = DoSomeStuffWithAsync(); // Synchronous call
Console.WriteLine(task.Result); // You must unpack the bool result...
Your other method just returns a bool - it's just a standard synchronous call/return...
var result = DoSomeStuff(); // Synchronous call
Console.WriteLine(result); // Just use the result
From a behavior point-of-view, to answer your specific question, both behave synchronously. Another way to look at it:
var act = new Func<Task<bool>>(async () => await DoSomeStuffWithAsync());
var result = act(); // will complete when DoSomeStuffWithAsync completes
Hope this answers your question.
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Cold comfort: Shows you gotta see now
Take heart, there's lots to do this winter and we offer six great picks.
Cold comfort: Shows you gotta see now Take heart, there's lots to do this winter and we offer six great picks. Check out this story on cincinnati.com: http://cin.ci/1RDG4zE
Carol Motsinger and Janelle Gelfand, Cincinnati Published 1:23 p.m. ET Jan. 22, 2016 | Updated 2:49 p.m. ET Jan. 22, 2016
Courtney Lucien as Emma in Cincinnati Shakespeare Company's upcoming 2016 production of Jane Austen's "Emma" by Jon Jory.(Photo: Provided)
Sure, "The Book of Mormon" has a pretty straightforward story. The Tony Award-winning musical is about two young Mormon missionaries who are sent to Uganda to preach the good word. But the satirical hit from the beautifully twisted minds of "South Park" creators is just as insightful as it is irreverent.
The show, which returns to Aronoff Center Downtown, cuts to the gut. And it's gut-busting. See, the book by Trey Parker, Robert Lopez and Matt Stone and the music by the genius behind "Frozen" is not really focused on the religion-establishing text. It's both more simple and more complicated than that.
We will leave it there. But it's just one of our picks for events you have to write – in pen, no less – on your calendar this winter.
The other five? They are not just something to do. These are things that happen to you. That could change the way you think or feel. You may wake up the next day knowing a story you thought you knew in a new way. Or expand your definitions about what art is. What a pop song can be.
Let's make some plans.
1) There's a reason the Jane Austen classic "Emma" keeps finding audiences through the decades, whether it's on stage or on screen. Take "Clueless," the MTV generation-defining comedy that slyly revamped Austen's Georgian society-based tale. No matter when it gets told or how, "Emma" is charming and clever. Funny and romantic. And above all, relevant to anyone with a heart and some hope. This time, the Cincinnati Shakespeare Company takes on the tale of an amateur matchmaker who tries to find love for her less promising friend Harriet. The play is from Feb. 26-March 26. $25-$39. 513-381-2273.
Do Ho Suh, Wielandstr. 18, 12159 Berlin, Germany - 3 Corridors, 2011. Image courtesy the Artist and Lehmann Maupin, New York. (Photo: Provided)
2) Starting Feb. 12, there will be insides in the inside of Contemporary Arts Center. Structures within a structure. Art and architecture become one with "Do Ho Suh: Passage," the banner exhibit for the Downtown institution (It's up through Sept. 11.) Curated by Steven Matijcio, it's a major survey exhibition of the celebrated Korean-American artist. He is well-known for his hallucinatory hanging fabric structures that explore the idea of home, place and migration, Matijcio said last year. "There is a translucency to it," Matijcio said. "It becomes as much a structure as a lens to see the people and the spaces around you." The show will also include a number of fabric renderings of domestic objects, like a toilet, and rubbings of living spaces. Contemporary Arts Center is 44 E. Sixth St., Downtown. www.contemporaryartscenter.org.
Dancers from Cincinnati Ballet's production of "Cinderella" (Photo: Provided/Peter Mueller)
3) Talk about enduring. What's more timeless than love at first sight? Cincinnati Ballet's "Cinderella" returns to the Aronoff Center for Valentine's Day weekend. Choreographed by Victoria Morgan, the fairy tale production has been freshened since its last outing in 2010 with new choreography, refurbished sets and updated costumes. All of your favorites will be on hand: The hilarious stepsisters (male dancers in drag), the enchanted pumpkin and, of course, Cinderella and Prince Charming. New this time: More children's roles and Cinderella's mice will be played by puppets. Carmon DeLeone conducts the Cincinnati Symphony Orchestra in Prokofiev's glorious score. Feb. 12-14, Aronoff Center, Downtown. Tickets from $32. 513-621-5282, cballet.org.
Rihanna performs at the 2015 We Can Survive concert at the Hollywood Bowl on Oct. 24 . (Photo: Rich Fury/Invision/AP)
4) Oh, Rihanna. Don't say there is no mystery, no intrigue, in modern-day pop music. It's not all out there for everyone to see (not counting some outfits, OK?) This pop siren, who has been released hit-packed albums each year for most of her career, has teased her next album "Anti" for a couple years. We still have no release date for it, but we have tour dates supporting it. Lucky for us, it includes one at U.S. Bank Arena at 7:30 p.m. March 19. We've heard three singles off of it, each wildly weird and exciting. Our favorite: The minimalist "FourFiveSecond" with Kanye West (and a Beatle!) is a strikingly simple acoustic meditation on the weekend. Will the tour be the best way to hear the album? Tickets are $30.50-$126 plus fees. 800-745-3000; www.usbankarena.com.
Rock star brothers Aaron and Bryce Dessner Dessner's "St. Carolyn by the Sea" with the Cincinnati Symphony Orchestra in 2014. (Photo: Provided)
5) Expect the unexpected with Bryce Dessner's MusicNOW Festival enters its 11th season in March. Details are still coming – but you can bet that there will be bold musical choices and interesting collaborations. The festival has evolved from its indie-rock roots – with some of the biggest names in contemporary music – into a full-fledged collaboration with symphony orchestra in Music Hall – last year with Dessner's band, The National. That aura of experimentation has continued in smaller venues around the city, and even in the lobbies of Music Hall. This year, Dessner collaborates again with maestro Louis Langrée and the Cincinnati Symphony Orchestra in two evenings that are sure to be creative, and likely more than that. Because, Dessner said last year, "I believe in the transporting power of music." 8 p.m. March 18 and 19, Music Hall, Over-the-Rhine. Tickets: 513-381-3300; www.cincinnatisymphony.org. Festival events and details will be announced at a later date.
Denèe Benton and Cody Jamison Strand in "The Book of Mormon." (Photo: Provided/Joan Marcus)
6) And, finally, here are the details on very funny "Book of Mormon." It's from March 22-27 at Aronoff Center, Fifth Third Bank Theater, Main and Seventh streets, Downtown. Get your tickets at 513-621-2787; cincinnatiarts.org.
Read or Share this story: http://cin.ci/1RDG4zE
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\section{Introduction}
\label{intro}
Stellar multiplicity plays a crucial role in many fields of astronomy. Star formation and evolution, Galactic chemical evolution, nuclear astrophysics, and cosmology are all influenced by our understanding of the multiplicity properties of an underlying stellar population. Binary interactions lead to phenomena as diverse as cataclysmic variables, classical novae, X-ray binaries, gamma-ray bursts, and Type Ia supernovae. Stellar interactions are also the cause of the anomalous surface abundances measured in Ba stars, CH stars, and the majority of carbon-enhanced metal-poor stars \citep{lucatello05}. The rates of these phenomena depend on the multiplicity properties such as the fraction of stars with companions and the distributions of separations and mass ratios. How these properties are in turn affected by variables such as stellar age, metallicity, and dynamical environment remains poorly understood. \cite{moe13} find no significant trends with metallicity for O- and B-stars, but more work is needed for lower-mass stars.
The recent review by \cite{duchene13} summarizes the state of the art in multiplicity studies. The fraction of systems with companions is known to be a strong function of stellar mass \citep{lada06, raghavan10, clark12}, and there are hints that lower mass systems have smaller separations \citep{duquennoy91, allen07, raghavan10}. Studies of the Solar neighborhood also indicate that lower metallicity stars are more likely to have stellar companions \citep{raghavan10}.
These results are based on heterogeneous samples of a few hundred stars at most, often dominated by wide systems which will never become interacting binaries. The spectroscopic surveys that reach small periods are labor intensive because large numbers of radial velocities (\ensuremath{\mathrm{RVs}}) are required to find the orbital solution of each target. This leads to small sample sizes, which have only increased modestly in the past two decades, from 167 in \cite{duquennoy91} to 454 in \cite{raghavan10}. The drive to collect complete samples has limited previous spectroscopic studies to the Solar neighborhood or specific stellar clusters, but neither of these strategies can probe the full range of metallicities and ages spanning the field stars of the Milky Way (MW) disk and halo components. These limits bias the interpretation of data against the global properties of, and variation within, the MW field. Thus, we are motivated to take a statistical approach with a sample of stars located throughout the field in order to investigate their multiplicity properties with respect to age, \ensuremath{\mathrm{[Fe/H]}}, and component membership.
With the advent of multiplexed spectroscopic surveys like SDSS \citep{york00} and LAMOST \citep{cui12}, we can use multiple \ensuremath{\mathrm{RV}}\ measurements of thousands of stars to study the properties of stellar multiplicity that are more representative of the entire Galaxy. SDSS Data Release 8 \citep{aihara11} contains over 1.8 million optical spectra from the original SDSS spectrographs including over 600,000 stellar spectra. In this work we employ a lesser known SDSS feature, the time-resolved dimension. To facilitate cosmic ray removal, spectra were constructed through co-addition of several individual sub-exposures, typically 15 minutes in duration. Although under-utilized, the benefit of the sub-exposure domain is recognized in works such as \cite{badenes09} and \cite{bickerton12}. Portions of the sky were also re-observed for calibration and scientific purposes. These additional pointings, combined with the sub-exposures, yield a time dimension where single stars have exposure coverage ranging from 3 sub-exposures up to over 40 sub-exposures, and time gaps from hours to nearly a decade. The techniques employed herein follow the time-resolved work by \cite{badenes12} and \cite{maoz12}.
\section{Measurements}
\label{sec:method}
\subsection{SDSS Observations and Sample Selection}
\label{ssec:sample}
F-type dwarfs are chosen for our sample because of the large number of stars targeted by SDSS with repeat observations, and their relatively mild variability and activity. Additionally, F-stars have main sequence (MS) lifetimes greater than $5\ \mathrm{Gyr}$, allowing us to select MS stars from both the younger disk and older halo. The Sloan Stellar Parameter Pipeline (SSPP; \citealp{lee08}) was developed to determine parameters for stellar spectra in the SDSS archive, including metallicity \ensuremath{\mathrm{[Fe/H]}}, effective temperature \ensuremath{T_{\mathrm{eff}}}, and surface gravity \ensuremath{\log{g}}. Sample selection began with identifying science primary objects from SEGUE-1 \citep{yanny09} and SEGUE-2 (Rockosi et al., in prep.) in the SSPP that were classified as an F-type star by the ``Hammer'' classification code \citep{covey07}. To minimize the effects of stellar evolution on multiplicity, we selected only dwarf stars ($\ensuremath{\log{g}}\geq 3.75$). Stars with multiple fiber pluggings were identified astrometrically and joined with the appropriate science primary fibers.
After measuring stellar \ensuremath{\mathrm{RVs}}\ (Section \ref{ssec:rv}), systematics were revealed in the SDSS sub-exposure spectra. These correlations appear as similar shifts in \ensuremath{\mathrm{RVs}}\ for many fibers located on the same plate, typically affecting neighboring fibers on the CCD. After plate-wide comparisons of F-stars, \ensuremath{\mathrm{RV}}\ correlations were corrected where possible. Corrections applied to the $10^4$ \ensuremath{\mathrm{RVs}}\ are as large as $17\ \ensuremath{\mathrm{km\ s}^{-1}}$ with a standard deviation of $2.2\ \ensuremath{\mathrm{km\ s}^{-1}}$. Not all correlations could be identified automatically because of multiple groups of correlated shifts, opposite in direction, on some plates. Visual inspection of plates containing numerous false binary detections lead to the removal of 25 plates including 1155 stars. We urge individuals using sub-exposure spectroscopy in SDSS to consider these systematic shifts in the wavelength solutions.
Quality control consisted of the removal of: stars without valid parameters in SSPP, fibers located on `bad' plates, sub-exposures with a median pixel signal-to-noise ratio (\ensuremath{\mathrm{SNR}}) less than 20 or with fewer than 3000 unflagged pixels, stars with time lags $\Delta T<1800\ \ensuremath{\mathrm{s}}$, stars with less than three clean sub-exposures, and corrupt or misclassified spectra (from visual inspection of stars with the largest RV variation or non-characteristic \ensuremath{T_{\mathrm{eff}}}). The final sample consists of 14,302 stars (16,894 fibers) with as many as 47 sub-exposures, spanning up to nine years of observations (Figure \ref{fig:feh_baseline}).
Our cleaned sample is characterized by metallicities ranging from $-3.41\leq\ensuremath{\mathrm{[Fe/H]}}\leq+0.52$. To aid comparison in our analysis, the final sample was sub-divided into three groups of equal size by cuts in metallicity at $\ensuremath{\mathrm{[Fe/H]}}=\ensuremath{-1.43}$ and $\ensuremath{\mathrm{[Fe/H]}}=\ensuremath{-0.66}$ (Figure \ref{fig:feh_baseline}). The majority of the stars have three or four sub-exposures ($\mathrm{median} = 4$), typically taken about 15 minutes apart. The median time lag for a star is 2 hours, however more than three years between observations can be seen in more than 250 stars (Figure \ref{fig:feh_baseline}). \ensuremath{\mathrm{SNR}} s for sub-exposures lie in the range $20<\ensuremath{\mathrm{SNR}}<84$ with a median value of $32$.
\begin{figure*}[htbp]
\centering
\plotone{f1.eps}
\caption{Left: Metallicity distribution for 14,302 F-dwarfs. Right: Distribution of maximum time lag between the first and last exposure of a star.}
\label{fig:feh_baseline}
\end{figure*}
\subsection{Radial Velocities}
\label{ssec:rv}
\ensuremath{\mathrm{RV}}\ measurements were attained through cross-correlation of sub-exposures with a master template constructed from 7207 sample-star, co-added spectra where the co-added $\ensuremath{\mathrm{SNR}} > 50$. The spectra were de-shifted using the redshift value assigned to the co-adds by the SDSS pipeline, continuum-normalized, and averaged together.
Sub-exposures were independently prepared and cross-correlated with the template. Spectra were continuum-normalized by dividing the spectrum with a highly smoothed version of itself using a FFT smoothing algorithm, and then cross-correlated with the template at various integer pixel lags. Each spectrum had a cross-correlation function (CCF) that was fit with a smooth spline interpolation. With spectral resolution of $R\sim2000$, the peak lag in pixels translates to the spectrum's redshift at \ensuremath{70\ \mathrm{km\ s}^{-1}\ \mathrm{pixel}^{-1}}. The mean and standard deviation of \ensuremath{\mathrm{RVs}}\ for individual stars are shown in the Figure \ref{fig:comboRV} distributions. The velocity dispersion of the mean RVs decreases with increasing \ensuremath{\mathrm{[Fe/H]}}, indicating that our \ensuremath{\mathrm{[Fe/H]}}-groups sample both the disk and halo components of the MW. The standard deviation of \ensuremath{\mathrm{RVs}}\ within individual stars is larger for the metal-poor group; however, empirically estimated uncertainties also show larger measurement errors for metal-poor stars. This underscores the importance of the use of proper error analysis in a method such as ours.
\begin{figure*}[htbp]
\centering
\plotone{f2.eps}
\caption{Mean (left) and standard deviation (right) of radial velocities within a star. Variations in the standard deviation of velocities are affected, in part, by the larger measurement uncertainties for metal-poorer stars.}
\label{fig:comboRV}
\end{figure*}
\subsection{Uncertainties}
\label{ssec:uncertainties}
It is well known that uncertainties in CCF peaks must be estimated empirically or through some Monte Carlo method (e.g.,~\citealt{peterson98}). For this work we determined \ensuremath{\mathrm{RV}}\ uncertainties empirically by quantifying the spread in measurements for spectra of similar quality. The median absolute deviation (\ensuremath{\mathrm{MAD}}) is a robust measure of the variability of a sample and is related to the standard deviation by $\sigma = 1.4826\ensuremath{\mathrm{MAD}}$, where $\ensuremath{\mathrm{MAD}} = \ensuremath{\mathrm{median}}(\left|\ensuremath{\mathrm{RV}}_i - \ensuremath{\mathrm{median}}(\ensuremath{\mathrm{RV}})\right|)$ \citep{leys13}. All measurements were de-shifted into the rest frame using the SDSS estimates of the co-add redshift, and placed into bins of similar metallicity ($\ensuremath{\mathrm{[Fe/H]}}\pm 0.25$) and signal-to-noise ($\ensuremath{\mathrm{SNR}}\pm 2.5$). Initial tests showed no correlations between measurement spreads and either \ensuremath{\log{g}}~or \ensuremath{T_{\mathrm{eff}}}. Estimates for the uncertainty of \ensuremath{\mathrm{RV}}\ measurements within a bin were calculated using \ensuremath{\mathrm{MAD}}\ values. Here, it is assumed that the majority of stars do not have detectable variability over the observed time baseline, and that effects from intrinsic variations in \ensuremath{\mathrm{RV}}\ are minimized by adopting median values. After performing this process for all bins, a functional form for assigning \ensuremath{\mathrm{RV}}\ measurement uncertainties \ensuremath{\sigma_\mathrm{RV}}\ was fit with an inverse proportionality to \ensuremath{\mathrm{SNR}}, and with a linear correction in \ensuremath{\mathrm{[Fe/H]}}. The measurement uncertainty as a function of \ensuremath{\mathrm{[Fe/H]}}\ and \ensuremath{\mathrm{SNR}}\ is, in \ensuremath{\mathrm{km\ s}^{-1}},
\begin{equation}
\label{eq:empiricalerror}
\ensuremath{\sigma_\mathrm{RV}}\left(\ensuremath{\mathrm{[Fe/H]}},\ensuremath{\mathrm{SNR}}\right) = \frac{(-26.51\ensuremath{\mathrm{[Fe/H]}} + 50.52)}{\ensuremath{\mathrm{SNR}}} + 1.23.
\end{equation}
Uncertainties are sub-pixel, falling below the spectral resolution of \ensuremath{70\ \mathrm{km\ s}^{-1}\ \mathrm{pixel}^{-1}}. For exposures with $\ensuremath{\mathrm{SNR}}<25$, uncertainties range from $3.0$ to $8.0\ \ensuremath{\mathrm{km\ s}^{-1}}$, with a median value of $5.0\ \ensuremath{\mathrm{km\ s}^{-1}}$. Exposures with $\ensuremath{\mathrm{SNR}}>40$ have uncertainties in the range $1.9$ to $4.4\ \ensuremath{\mathrm{km\ s}^{-1}}$, with a median value of $2.7\ \ensuremath{\mathrm{km\ s}^{-1}}$.
\section{Multiplicity}
\label{sec:multiplicity}
The probability of a star having a companion was determined through model comparison using a trans-dimensional, hierarchical, Markov chain Monte Carlo (MCMC) method. Two models were compared: a single-star model \ensuremath{\mathcal{M}_s}, and a binary-star model \ensuremath{\mathcal{M}_b}. The hyperparameter $\lambda$, indexes the model choice at each step in the MCMC chain. We evaluated the hierarchical model using the Python package \textit{emcee}, a MCMC ensemble sampler \citep{foreman13}.
The single-star model \ensuremath{\mathcal{M}_s}, fits a star with non-varying \ensuremath{\mathrm{RVs}}, parameterized by a systemic velocity \ensuremath{V_0}. Because intra-plate systematics are known to exist, it is reasonable to assume inter-plate systematics exist as well. In light of this, $(P-1)$ additional parameters $ps_i$, were included for each star, where $P$ is the number of plate-MJD pluggings composing the star. These plate-shift parameters allow all \ensuremath{\mathrm{RVs}}\ from plate $i$ to shift by some amount $ps_i$, relative to the first plate $P_0$. For the majority of stars $P=1$, no plate-shift parameters are necessary, and \ensuremath{\mathcal{M}_s}\ is a 1-parameter model.
In the binary star model \ensuremath{\mathcal{M}_b}, the sparsely sampled \ensuremath{\mathrm{RVs}}\ are fit by a sinusoid defined by four-parameters: the log of the semi-amplitude \ensuremath{\log A}, the log of the period \ensuremath{\log P}, the phase $\phi$, and the systemic velocity \ensuremath{V_0}. We assume circular orbits (eccentricity, $e = 0$), which is a safe assumption for tidally circularized, short-period orbits ($P<12$ days; \citealp{raghavan10}), where we are most sensitive. A small number of the binaries found in this study may have longer periods and could have non-zero eccentricities, but this does not affect our results. Plate-shift parameters were also adopted in \ensuremath{\mathcal{M}_b}\ wherever $P>1$.
Uninformative priors were used in the MCMC. The model index $\lambda$, has a flat prior from 0 to 1, where $\lambda<0.5$ denotes \ensuremath{\mathcal{M}_s}\ and $\lambda\geq0.5$ denotes \ensuremath{\mathcal{M}_b}. The semi-amplitude prior is log-uniform from $3\ \ensuremath{\mathrm{km\ s}^{-1}}$, comparable to the measurement uncertainties where \ensuremath{\mathcal{M}_s}\ and \ensuremath{\mathcal{M}_b}\ become degenerate, to $250\ \ensuremath{\mathrm{km\ s}^{-1}}$, greater than the largest \ensuremath{\mathrm{RV}}\ differences in the sample. The prior on the period is uniform in the range $4\leq\ensuremath{\log P}\ (\ensuremath{\mathrm{s}})\leq 7$. The lower limit $\ensuremath{\log P}\ (\ensuremath{\mathrm{s}})=4.0$ is equal to the orbital period at which stellar contact is certain for low-mass companions. Above $\ensuremath{\log P}\ (\ensuremath{\mathrm{s}})=7.0$, \ensuremath{\mathrm{RV}}\ amplitudes in binary systems are comparable to the measurement uncertainties. Combined with the sparsity of the \ensuremath{\mathrm{RV}}\ data, systems with periods longer than $\ensuremath{\log P}\ (\ensuremath{\mathrm{s}})=7.0$ are outside our range of sensitivity. Priors are also uniform for the phase ($0\leq\phi\leq 2\pi$) and systemic velocity ($-600\leq\ensuremath{V_0}\ (\ensuremath{\mathrm{km\ s}^{-1}})\leq 600$). Markov chains were run independently on every star with an ensemble of $200$ parallel chain ``walkers'' for a total of $2.4\e6$ samples, then burned and thinned to $6\e5$ independent samples of the posterior.
Evidence for detection of a companion star is reflected by the relative probabilities of $\lambda$. We define the probability for the binary model, $\eta$ as the fraction of samples in the marginalized posterior having $\lambda=\ensuremath{\mathcal{M}_b}$. We note that the value of $\eta$ is dependent on the choice of priors, and is sensitive to the treatment of the SDSS systematics. Moreover, a degeneracy arises as the \ensuremath{\mathrm{RV}}\ curve of a long-period, low-amplitude system becomes indistinguishable from a single-star system. With this mind, we stress that values for $\eta$ are not absolute probabilities of a system having a companion, but reflect the ability of the data to rule out models under the given prior. However, the \ensuremath{\mathrm{[Fe/H]}}-groups can be compared, relatively, by considering the fraction of systems where $\eta$ is large and \ensuremath{\mathcal{M}_s}\ is strongly disfavored. The results are shown in Figure \ref{fig:etaHist}
We also investigated the \ensuremath{e/i}\ parameter proposed by \cite{geller08} as a metric for identifying the stars with large \ensuremath{\mathrm{RV}}\ variations. We find that the \ensuremath{e/i}\ parameter singles out many of the same stars as our more sophisticated MCMC-based inference. Our method not only takes into account deviations in \ensuremath{\mathrm{RV}}\ from the mean, but also how well the data fit the expected periodicity of a binary system.
Analysis of the posterior, and visual inspections of the binary model fits, show that 681 stars with $\eta>0.65$ are true spectroscopic binaries, though given the sparsity of the RV curve sampling, there are sometimes large uncertainties in the fitted values for specific model parameters. Another natural break point is $\eta>0.95$; these are 209 stars for which the determination and analysis of accurate individual model parameters should be possible (and will be characterized in future work). An intermediate cut at $\eta>0.80$ is a compromise between these limits, yielding a larger sample of stars (406) with modest model constraints. The values of the binary fractions that we derive below are insensitive, within the uncertainties, to the exact choice of cut in $\eta$. This implies that the \ensuremath{\mathrm{RV}}\ variations for our binary detections are sufficiently above the measurement uncertainties, and that the binary fractions reported are not biased due to differences in \ensuremath{\mathrm{SNR}}\ or absorption features.
Figure \ref{fig:logP} shows the \ensuremath{\log P}\ posteriors for each \ensuremath{\mathrm{[Fe/H]}}-group, marginalized over all binary systems ($\eta>0.80$). The posterior distributions of \ensuremath{\log P}\ for many of these stars are complex: many are multimodal, affected by aliasing or other issues related to the sparse, biased time sampling. One such effect is the increase in probability at $\ensuremath{\log P}=4$. Here the metal-rich and -intermediate groups contain more stars than the metal-poor group with $\Delta t\simeq10^4\ \ensuremath{\mathrm{s}}$. Systems with periods as short as this are extremely rare \citep{drake14}, and our increased probability in this area may be due to overfitting. Additionally, the gap at $\Delta t=10^{4.6}\ \ensuremath{\mathrm{s}}=12\ \ensuremath{\mathrm{h}}$ (Figure \ref{fig:feh_baseline}) may affect the estimate of a period. We defer a more sophisticated analysis to a future paper, but these effects should not alter the ability to rule out a single-star model. For now, Figure \ref{fig:logP} illustrates that we are mainly sensitive to periods in the range $4<\ensuremath{\log P}\ (\ensuremath{\mathrm{s}})<6$, or less than about 12 days. We emphasize that a more detailed analysis will be necessary to estimate the true underlying \ensuremath{\log P}\ distribution in our sample.
\begin{figure}[htbp]
\centering
\plotone{f3.eps}
\caption
{Distribution of $\eta$, the fraction of posterior samples using the binary model, for stars.}
\label{fig:etaHist}
\end{figure}
\begin{figure}[htbp]
\centering
\plotone{f4.eps}
\caption
{Averaged probability distributions of \ensuremath{\log P}\ for all binary detections ($\eta>0.80$). These do not reflect actual distributions of periods, and should only be used as a guide to probe the region of MCMC sensitivity. The shaded region indicates where Roche lobe overflow and contact becomes relevant. The dashed line marks the circularization limit at a period of 12 days.}
\label{fig:logP}
\end{figure}
\begin{figure*}[htbp]
\centering
\plotone{f5.eps}
\caption
{Short-period binary fraction limits, relative to the metal-rich group. Binary companion detections are defined by a cut in $\eta$, the fraction of posterior samples using the binary model. Group median values of \ensuremath{\mathrm{[Fe/H]}}\ are used.}
\label{fig:binaryFractions}
\end{figure*}
\section{Discussion}
\label{sec:discussion}
In Figure \ref{fig:binaryFractions} we show \ensuremath{f_b}, the measured lower bound for the fraction of stars with short-period companions $(P\lesssim 12\ \mathrm{days})$ for each metallicity group, normalized to the metal-rich binary fraction. \ensuremath{f_b}\ is a lower limit because of non-detections as a result of sparsely sampled \ensuremath{\mathrm{RVs}}\ and high orbital inclinations, resulting in low amplitudes. We see agreement in \ensuremath{f_b}\ measured for all three choices in $\eta$ cutoff (0.65, 0.80, 0.95). With a cutoff of $\eta=0.80$, values of \ensuremath{f_b}\ for the metal-poor, -intermediate, and -rich groups respectively are: $2.5\%\pm0.2\%$, $2.8\%\pm0.2\%$, and $3.2\%\pm0.3\%$. Since the observational biases that affect binary detection are mostly due to the sparsity of the RV coverage, which isn't metallicity-dependent, we conclude that the field F-type MS stars in our metal-rich sample are, at a 2-sigma level, $30\%$ more likely than those in our metal-poor sample to have close binary companions.
Our metal-rich and metal-poor samples mostly trace the MW disk and halo. Differences in the fraction of short-period systems can stem from differences in the star-formation process, dynamical interactions after star formation, or some combination of the two.
Three-dimensional hydrodynamic models from \cite{machida09} actually suggest a \textit{higher} frequency of binaries formed through cloud fragmentation for metal-poor clusters, due to the decreased requirement of a cloud's initial rotation energy to fragment. Moreover, their models yield systems with shorter initial separations at lower metallicities. The increased \ensuremath{f_b}\ observed for metal-rich stars in this work can more likely be explained by dynamical processes than by formation processes.
The observed differences in \ensuremath{f_b}\ could be explained if the clusters that yielded halo field stars had larger stellar densities and/or gas densities than those of the disk. \cite{korntreff12} explore the effects of gas-induced orbital decay on period distributions in clusters. They note that an increased density of gas in a newly formed cluster will lead to a larger number of short-period system mergers shortly after formation. \cite{parker09} describe how clusters with higher stellar densities destroy wide binaries through dynamical interactions. An increase in the destruction of high-mass, wide-binary systems leads to the ejection of former F-star secondaries into the field. These orphaned, single-star systems would increase the total number of F-star systems in the halo field, effectively decreasing the short-period binary fraction measured. Observational evidence of these denser cluster environments is needed to support these arguments for a lower \ensuremath{f_b}\ in the halo.
Additionally, some close binaries may also transfer mass and covert themselves into blue stragglers \citep{lu10}. Evidence for an abundance of blue stragglers in the halo has been seen \citep{yanny00}, and may contribute to the lower \ensuremath{f_b}\ observed in the metal-poor group. Also, \cite{duchene13} show a decrease in \ensuremath{f_b}\ with age for Solar-type stars, although this result is based on visual binaries with wider periods, and is poorly constrained due to limited sample sizes.
We note that the recent results of \cite{gao14} and \cite{yuan14}, using data from SDSS, show a larger binary fraction for metal-poor than metal-rich FGK stars in the field. In addition to probing longer periods, the former work does not make use of sub-exposure information (using only two \ensuremath{\mathrm{RV}}\ epochs per star) and relies on the correctness of model values for the period distribution, mass ratio distribution, and initial mass function. The latter work, which uses photometric color deviations to infer companions, shows a modest metallicity dependence on total binary fraction. Since their method is not sensitive to period, the binary fractions they report are strongly dominated by more common, wider-period systems near the peak of a log-normal period distribution ($\overline{\ensuremath{\log P}}\ (\ensuremath{\mathrm{s}})=10$ for nearby, Solar-like stars; \citealt{raghavan10}). It is clear that conclusions about binary fraction depend on a number of factors, especially the range of periods to which the search is sensitive and assumptions made about the overall period distribution.
Our MCMC analysis yields posterior probabilities in parameter space, allowing for a more detailed study of binary properties (e.g., period and separation distributions), which will be presented in future work. The techniques in this work have direct applications for current and future multiplexed spectroscopic surveys.
\acknowledgments We thank Ewan Cameron, Dan Maoz, Jeffrey Newman, Chad Schafer, and the referee for useful discussions. T.H. and T.C.B. acknowledge partial support from grants PHY 08-22648; Physics Frontier Center/JINA, and PHY 14-30152; Physics Frontier Center/JINA Center for the Evolution of the Elements (JINA-CEE), awarded by the US National Science Foundation. Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science.
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{"url":"http:\/\/users.csc.calpoly.edu\/~gfisher\/classes\/308\/lectures\/7-8.html","text":"CSC 308 Lecture Notes Weeks 7 and 8\nIntroduction to Fully Formal Specification\n\n1. Some practical benefits of formal specification.\n\n1. Better understanding of software.\n\n2. Precise communication among developers.\n\n3. Basis for thorough testing.\n\n4. Basis for verification (when appropriate).\n\n5. Basis for automatic programming (in future).\n\n6. Here's a motivational bottom line:\n\nI want you to do whatever it takes to build me software of the best possible quality, that has the smallest possible likelihood of failing.\n\n2. For some academics and software professionals, formal specification is a key part of addressing a mandate like this.\n\n2. Formal specification with preconditions and postconditions.\n\n1. As model object and operation definitions take shape, we are ready to formalize the definitions fully.\n\n2. The formal technique we will use in 308 is based on operation preconditions and postconditions.\n\n1. A precondition is a predicate (i.e., boolean-valued expression) that is true before an operation executes.\n\n2. A postcondition is a predicate that is true upon completion of an operation.\n\n3. Since pre- and postconditions are predicates, this style of formal specification called predicative.\n\n3. The pre- and postconditions are used to specify fully what the system does, including all user-level requirements for the system.\n\n4. This formal specification is part of the overall requirements specification process we're following, with these steps:\n\n1. gather user-level requirements via usage scenarios\n\n2. identify objects and operations\n\n3. formalize operations with pre- and postconditions\n\n4. refine user-level requirements based on formal specs\n\n5. refine formal specs based on user-level refinements\n\n6. iterate steps 1-5 until done\n\n5. The \"until done\" step involves two levels of validation.\n\n1. First, we must validate that the specified system is complete and consistent from the end user's perspective.\n\n1. That is, the system meets all end-user needs and does so in a way that is wholly satisfactory to the end user.\n\n2. This is accomplished by continued consultation with the end user.\n\n2. The second level of validation involves completeness and consistency from a formal perspective.\n\n1. This can be accomplished in a number of ways.\n\n2. In the case of mechanized specification languages, such as JML, some completeness and consistency checking is done using a computer-based analyzer.\n\n3. Another valuable validation technique is peer review via formal walk-throughs.\n\n4. Also, there are techniques for formal specification testing, including the postulation and proof of putative theorems.\n\n1. Such theorems define properties of the system that we expect to be true, and which can be proved true formally with respect to the pre- and postconditions.\n\n2. We will discuss putative theorems briefly in 308, but not use them.\n\n3. Formal specification maxims.\n\n1. In developing any formal software specification, it is useful to observe the following two maxims:\n\n1. Nothing is obvious.\n\n2. Never trust the programmer.\n\n2. The first maxim relates primarily to user-level requirements.\n\n1. It is often easy to think that a requirement is sufficiently obvious that it need not be stated formally.\n\n2. The problem with this thinking is that one person's obvious is not always the same as another's.\n\n3. To ensure that a specification is sufficiently precise, stating the \"obvious\" is necessary.\n\n3. The second maxim is necessary to avoid nasty surprises in an implementation.\n\n1. In many cases, we might consider an application to be sufficiently simple that we can trust the programmer to get a user-level requirement right if we forget to specify it.\n\n2. In general, such trust is a bad idea.\n\n3. It is better for the specifier to maintain a respectfully and cordially adversarial relationship with the implementor.\n\n4. Overview of JML predicate notation.\n\n1. Predicates in JML use standard Java notation for Boolean expressions, augmented with additional predicate logic operators.\n\n2. In addition to Java Boolean expressions, we'll use standard Java arithmetic, and methods available on Java Collections and Strings.\n\n3. These operations are summarized in Table 1.\n\n Predicate Logic: Relational: Operator Description Operator Description && logical and == primitive equality || logical or !- primitive inequality ! logical not < primitive less than => logical implication > primitive greater than <==> logical equivalence <= primitive less than or equal to e1 ? e2 : e3 conditional choice >= primitive grtr than or equal to \\forall universal quantification .equals object equality \\exists existential quantification .compareTo object comparison Logical Extensions: Arithmetic: Operator Description Operator Description \\old(expr) value before execution + addition \\result return value of method - subtraction \\sum summing quantifier * multiplication \\product multiplicative quantifier \/ division Collections, Lists, Strings: Operator Description .size() size of collection .contains(Object o) collection membership .get(int i) get ith list element .length(String s) length of s other collection ops see Collection docs other list ops see List docs other string ops see String docs\n\nTable 1: JML Expression Operators.\n\n1. The predicate logic operators are used in boolean-valued expressions.\n\n1. Logical and, or, and not have the same meaning as their equivalents in a programming language, e.g., \"&&\", \"||\", and \"!\" in C and C++.\n\n2. Logical implication and equivalence have their standard logical meanings, per the following truth tables\n p q p => q p q p <==> q 0 0 1 0 0 1 0 1 1 0 1 0 1 0 0 1 0 0 1 1 1 1 1 1\n\n3. The conditional choice operator has the truth table:\n\n p x y p ? x : y 0 x y y 1 x y x\n\nwhere expressions x and y must have the same type.\n\n4. The universal and existential quantifiers have their standard logical meanings, but will be applied in specific ways, as upcoming examples illustrate.\n\n2. The arithmetic operators are used in numeric-valued expressions.\n\n1. Addition, subtraction, division, and multiplication have their standard mathematical meanings.\n\n2. Remember that preconditions and postconditions are always boolean valued, so arithmetic must always be performed in the context of a boolean expression.\n\n3. E.g., a + b is not a legal postcondition, but \\result == a + b is.\n\n4. Note also that in specifications, we are assuming idealized mathematical arithmetic, without overflow or underflow\n\n5. If the precision of numeric expressions is an issue in a specification, then it must be dealt with explicit logic.\n\n3. The collection operators are used with values of java.util.Collection or java.util.List, the latter used to model collections in which order must be specified.\n\n4. All other standard Java operators and library methods can be used in predicate expressions.\n\n5. As always in Java programs, we must be aware of when to use .equals versus ==.\n\n1. For JML specifications, == should only be used for primitive types int, double, and boolean.\n\n2. For all other class-defined types, including String, .equals is used for testing equality.\n\n3. For inequality of class types, use .compareTo.\n\n4. Further details of the notation are covered in the Java and JML reference manuals, available in the 308 doc directory.\n\n5. The logic of JML is comparable to other formal specification languages.\n\n1. A difference between our use of JML and a number of contemporary languages is collections of instead of mathematical sets.\n\n2. Formally, both collections and sets can be fully axiomatized (i.e., mathematically defined), so there is no lack of formality in the use of collections.\n\n3. In fact, JML provides definitions of pure Java collections, which are defined with fully side-effect-free methods.\n\n4. Overall, the use of collections instead of sets results in little difference in a specification.\n\n1. Set notation makes certain low-level specification easier than with lists, such as operations that can be modeled with set union and difference.\n\n2. On the other hand, collection and list notation makes other forms of specification easier than with sets, such as specification of ordering constraints.\n\n5. \"Programming\" with predicates.\n\n1. The language of predicates used in pre- and postconditions can be thought of as non-procedural programming.\n\n2. The rules for this style of \"programming\" are different than the procedural kind.\n\n1. We define data, but only in abstract terms and from an end-users \"real world\" perspective, not from a computer efficiency perspective.\n\n2. We define functions, but only in abstract terms of what the functions do, not how they work.\n\n3. Hence, the only \"code\" we have are boolean expressions at the beginning and ending of functions, no code bodies.\n\n4. The closest thing we have to traditional control constructs are the two quantifiers forall and exists.\n\n1. However, these quantifiers are fundamentally different than normal programming language controls.\n\n2. Namely, they only return boolean values, and they don't make anything \"happen\".\n\n5. Instead of procedural descriptions of how functions work (i.e., what happens inside a function), we have only true\/false descriptions of what functions do (i.e., what's true before and after the function happens).\n\n1. Time does not pass within pre- or postconditions, even ones with quantifiers.\n\n2. Rather, pre- and postconditions are simply statements of mathematical fact, that are instantaneously true or false.\n\n3. Hence, even though a forall may seem somewhat like a for-loop, it is just a boolean expression that is only true or false.\n\n4. It may be a big boolean expression that is true in a lot of cases, but it's still just a boolean expression.\n\n3. In some cases, it may be necessary to specify certain procedural aspects of a system, specifically the order in which operations occur.\n\n1. However when we do this we need to be careful not to lapse into conventional programming.\n\n2. Therefore, we will specify ordering constraints non-procedurally by writing the precondition of a successor operation to be dependent on the postcondition of a predecessor operation.\n\n1. E.g., if operation B must follow operation A, we write the postcondition of A such that the only way the precondition of B can be true is if A's postcondition is true.\n\n2. In general, this is accomplished by having A's postcondition require some unique value for one or more outputs, and then having B's precondition state that its inputs must have the values required by A.\n\n3. In this way, we require that A must execute before B, if B is ever to happen.\n\n3. As always, we will specify procedural (i.e., step-by-step) behavior only when it is fundamental to the way the user operates.\n\n4. In particular, we need to be careful not to specify procedural details of a particular GUI, when it is only one particular way to access the abstract operations.\n\n5. Here's the way to think about it -- if the user must perform a series of operations in a particular order, then we'll specify the order.\n\n6. An initial example of fully formal specification.\n\n1. For starters with pre- and postconditions, we'll begin with some Calendar tool operations that are simpler than the scheduling and viewing operations we've examined in recent weeks.\n\n2. Specifically, we'll look at operations for adding and finding registered Calendar Tool users and groups.\n\n3. These operations have useful but relatively straightforward specifications.\n\n4. Next week we'll return to the specification of the more involved scheduling a viewing operations.\n\n7. Synopsis of requirements for user database admin functions.\n\n1. When the user selects the 'Users ...' item in the 'Admin' menu, the system displays the screen shown in Figure 1.\n\nFigure 1: User database maintenance dialog.\n\n1. User Name is a free-form string; Id is a unique system id of eight characters or less; Email address is free-form string; phone Area code is three digits, Number is seven digits;\n\n2. The Add command adds a new user; Id field must be unique.\n\n3. The Find command finds by Name or Id or both.\n\n1. If find is by name and the name is not unique, the system displays list of ids for users of that name.\n\n2. The user clicks on an item in the list to see the full record for that id.\n\n3. If no user of the given name or id is found, the system displays a \"no users found\" pop-up dialog.\n\n4. Change works after the user changes the most recently displayed record.\n\n1. Typically, the user runs Find command first, then changes.\n\n2. The original record is removed, new record is added.\n\n5. Delete removes the most recently displayed record, typically located with a Find command; the system displays an \"are you sure\" pop-up dialog for confirmation.\n\n2. When the user selects the 'Groups ...' item in the 'Admin' menu, the system displays the screen in shown Figure 2.\n\nFigure 2: Group database maintenance dialog.\n\n1. Group Name is a free-form string that is unique for all groups; leaders and Groups are lists of user Ids for the group leaders and members, respectively; all leaders must be listed as members.\n\n2. The Add command adds a new group; the Name must be unique.\n\n3. The Find command finds a group by name.\n\n4. Change works after the user changes the most recently displayed group record.\n\n1. Typically, the user runs the Find command first, then changes.\n\n2. The original record is removed, the new record is added.\n\n5. Delete removes the most recently displayed record, typically located with a Find command; the system displays an \"are you sure\" pop-up dialog for confirmation.\n\n8. Basic definitions for user database objects and operations.\n\n1. Here are the relevant object and operation definitions:\n\nimport java.util.Collection;\n\n\/**\n* UserDB is the repository of registered user information.\n*\/\nabstract class UserDB {\n\n\/**\n* The collection of user data records.\n*\/\nCollection<UserRecord> data;\n\n\/**\n* Add the given UserRecord to the given UserDB. The Id of the given user\n* record must not be the same as a user record already in the UserDB.\n* The Id component is required and must be eight characters or less. The\n* email address is required. The phone number is optional; if given, the\n* area code and number must be 3 and 7 digits respectively.\n*\/\n\n\/**\n* Find a user by unique id.\n*\/\nabstract UserRecord findById(String id);\n\n\/**\n* Find a user or users by real-world name. If more than one is found,\n* the output list is sorted by id.\n*\/\nabstract Collection<UserRecord> findByName(String name);\n\n\/**\n* Change the given old UserRecord to the given new record. The old and\n* new records must not be the same. The old record must already be in\n* the input db. The new record must meet the same conditions as for the\n* input to the AddUser operation. Typically the user runs the FindUser\n* operation prior to Change to locate an existing record to be changed.\n*\/\nabstract void change(UserRecord old_ur, UserRecord new_ur);\n\n\/**\n* Delete the given user record from the given UserDB. The given record\n* must already be in the input db. Typically the user runs the FindUser\n* operation prior to Delete to locate an existing record to delete.\n*\/\nabstract void delete(UserRecord ur);\n\n}\n\n\/**\n* A UserRecord is the information stored about a registered user of the\n* Calendar Tool. The Name component is the users real-world name. The\n* Id is the unique identifier by which the user is known to the Calendar\n* Tool. The EmailAddress is the electronic mail address used by the\n* Calendar Tool to contact the user when necessary. The PhoneNumber is\n* for information purposes; it is not used by the Calendar Tool for\n* contacting the user.\n*\/\nabstract class UserRecord {\nString name;\nString id;\nString email;\nPhoneNumber phone;\n}\n\nabstract class PhoneNumber {\nint area;\nint number;\n}\n\n\n2. For a little practice with UML, Figure 3 shows diagrams for these definitions, in two equivalent formats.\n\nFigure 3: UML diagrams for UserDB objects and operations.\n\n3. The objects and operations were derived from the user-level requirements, per the model derivation process discussed in Lecture Notes 5 last week.\n\n4. The operation signatures are quite representative of those defined for a collection object.\n\n1. UserDB.add is a constructive operation, with a signature of the general form\nclass ACollection {\nCollection<AnElement> data;\nvoid constructiveOp(AnElement);\n}\nwith the effect of adding an element to the data collection.\n\n2. The versions of UserDB.find are selective operations, with signatures of the general form\nclass ACollection {\nAnElement selectiveOp(UniqueElementSelector);\nCollection<AnElement> selectiveOp(NonUniqueElementSelector);\n}\nwith the effect of finding zero or more elements in a collection.\n\n1. In both forms, the input is a component of AnElement used as a search key.\n\n2. In the first form, UniqueElementSelector is a component whose value is required to be unique among all elements of the collection.\n\n3. In the second form, NonUniqueElementSelector is a component whose value is not required to be unique among all elements of the collection.\n\n3. UserDB.delete is a destructive operation, with the same signature form as a constructive operation, but with the effect of removing rather than adding an element.\n\n4. UserDB.change is a modifying operation (combined constructive and destructive), with a general signature of the form\nclass ACollection {\nvoid modifyingOp(AnElement oldElement, AnElement newElement);\n}\nwith the effect of removing the OldElement and adding the NewElement.\n\n5. In modeling, it can be useful to overload operation names, for better traceability to the UI.\n\n1. In terms of model accuracy, overloading works well in a case where the same operational widget (e.g., button) can be used with different input values.\n\n2. Hence, we might overload the find operation thusly:\nabstract UserRecord find(String id);\nabstract Collection<UserRecord> find(String name);\n\n\n3. The problem is, this form of overloading is not supported in Java, since Java requires input signatures to differ.\n\n4. Hence, where necessary, operations need to be disambiguated by name, as in findById and findByName.\n\n9. An initial formal definition of UserDB.add.\n\n1. For operation pre- and postconditions, we will start by stating a predicate in English, and then refine it into formal logic.\n\n2. As we refine the logic, the English version will be retained as a comment, to aid in the human understanding of the specification.\n\n3. So, here is an initial version of the formal spec for the UserDB.add operation:\n\nimport java.util.Collection;\n\nabstract class UserDB {\n\nCollection<UserRecord> data;\n\n\/**\n* Add the given UserRecord ur to this.data. The UserId of the given user\n* record must not be the same as a user record already in this.data. The\n* UserId component is required and must be eight characters or less. The\n* email address is required. The phone number is optional; if given, the\n* area code and number must be 3 and 7 digits respectively.\n*\/\n\/*@\nrequires\n(*\n* The id of the given user record must be unique and less than or\n* equal to 8 characters; the email address must be non-empty; the\n* phone area code and number must be 3 and 7 digits, respectively.\n*);\nensures\n(*\n* The given user record is in this.data.\n*);\n@*\/\n\n}\n\n\n4. Now let's formalize the logic.\n\n1. The English comment for the add postcondition specifies the most fundamental property of an additive collection operation -- upon completion of the operation, the given element to be added is in the output collection.\n\n2. Formally,\n\nimport java.util.Collection;\n\nabstract class UserDB {\n\nCollection<UserRecord> data;\n\n\/**\n* Add the given UserRecord ur to this.data. The UserId of the given user\n* record must not be the same as a user record already in this.data. The\n* UserId component is required and must be eight characters or less. The\n* email address is required. The phone number is optional; if given, the\n* area code and number must be 3 and 7 digits respectively.\n*\/\n\/*@\nrequires\n\/\/\n\/\/ The id of the given user record must be unique and less than or\n\/\/ equal to 8 characters; the email address must be non-empty; the\n\/\/ phone area code and number must be 3 and 7 digits, respectively.\n\/\/\n(* Coming soon *);\nensures\n\/\/\n\/\/ The given user record is in this.data.\n\/\/\ndata.contains(ur);\n@*\/\n\n}\n\n\n3. The simple expression \"data.contains(ur)'\" is all there is to it.\n\n1. contains is a method defined in java.util.Collection.\n\n2. Its operand is a value of the element type contained in the collection.\n\n3. I.e., in this case the operand is a UserRecord.\n\n5. As it stands, UserDB.add still has no precondition formally defined, only a comment indicating what needs to be defined.\n\n1. Having no explicit precondition is equivalent to a precondition of true.\n\n2. In many cases, true preconditions are fine, given that there is no specific condition that must be met before the operation begins.\n\n3. In the case of the UserDB.add operation, a default true precondition definitely won't do, since we can see from the requirements that a number of conditions must be met before UserDB.add can proceed.\n\n4. We will address these requirements step by step, as we refine the formal definition of UserDB.add.\n\n1. One of the fundamental questions that must be asked of pre- and postconditions is if they are strong enough.\n\n2. For example, the true precondition for UserDB.add is relatively weaker than one that specifies that there is no UserRecord of the same id already in the input database.\n\n2. In general, there are two aims to strengthening a specification.\n\n1. Ensuring that all user-level requirements are met (cf. Maxim 1 above).\n\n2. Ensuring that a system implementation works properly (cf. Maxim 2).\n\n3. The former is accomplished via continued consultation with the end user; the latter requires an experienced analyst, who understands the kinds of problems that may arise in a system implementation.\n\n4. In the case of the user and group databases, as well as similar database applications, an area of potential implementation error is the introduction of spurious entries into the database and\/or the spurious deletion of entries.\n\n5. To avoid such spurious effects, the specification of UserDB.add is strengthened as follows:\n\nimport java.util.Collection;\n\nabstract class UserDB {\n\nCollection<UserRecord> data;\n\n\/*@\nensures\n\/\/\n\/\/ The given user record is in this.data.\n\/\/\ndata.contains(ur)\n\n&&\n\n\/\/\n\/\/ All the other records in the output db are those from the input db,\n\/\/ and only those.\n\/\/\n(\\forall UserRecord ur_other ; ! ur_other.equals(ur) ;\n\\old(data).contains(ur_other)\n? data.contains(ur_other)\n: ! data.contains(ur_other));\n\n@*\/\n\n}\n\n\n6. This specification introduces the use of the universal quantification operator, \\forall.\n\n1. Universal quantification in JML has the same meaning as in standard typed predicate logic.\n\n2. The general format is the following:\n(\\forall T x ; constraint ; predicate)\n\n1. This is read \"for all values x of type t, such that constraint holds, predicate is true.\"\n\n2. The constraint expression is optional.\n\n3. The quantified variable x must appear in constraint (if present) and predicate.\n\n3. In general, universal quantification is used frequently when specifying predicates on collection objects, as upcoming examples illustrate.\n\n7. While this example is a good illustration of specification strengthening, there are easier ways to specify the same meaning logically.\n\n1. For example, the postcondition logic can be simplified to the following:\n\nimport java.util.Collection;\n\nabstract class UserDB {\n\nCollection<UserRecord> data;\n\n\/*@\n\/\/\n\/\/ A user record is in the output data if and only if it is the new\n\/\/ record to be added or it is in the input data.\n\/\/\nensures\n(\\forall UserRecord ur_other ;\n(data.contains(ur_other)) <==>\nur_other.equals(ur) || \\old(data).contains(ur_other));\n@*\/\n\n}\n\n\n2. In general, predicate simplification is beneficial when it helps clarify the specification.\n\n3. Simplification is not necessary, as long as the logic is clear and accurate.\n\n8. Another way to simplify this specification is to use a constructive list operator, as follows:\n\nimport java.util.Collection;\n\nabstract class UserDB {\n\nCollection<UserRecord> data;\n\n\/**\n* Add the given UserRecord ur to this.data. The UserId of the given user\n* record must not be the same as a user record already in this.data. The\n* UserId component is required and must be eight characters or less. The\n* email address is required. The phone number is optional; if given, the\n* area code and number must be 3 and 7 digits respectively.\n*\/\n\/*@\nensures\n\/\/\n\/\/ The given user record is in this.data.\n\/\/\n\n@*\/\n}\n\n\n1. A constructive specification such as this describes the output of an operation using a constructive operation on the inputs.\n\n2. In contrast, an analytic specification (such as the previous spec using the boolean-valued contains method) describes output without using constructive operations.\n\n3. In 308, we will define analytic specifications whenever possible.\n\n1. Specifically, we won't used methods that construct or modify collections.\n\n2. There is debate among software engineers as to the relative merits of constructive versus non-constructive specification; we will discuss the issues a bit later.\n\n11. Refining the postconditions for the other UserDB operations.\n\n1. Based on the development of the UserDB.add specs so far, we can provide a comparable level of formal specification for the other three UserDB operations.\n\n2. For example, here is the idea for formalizing the findById postcondition:\n\nimport java.util.Collection;\n\nabstract class UserDB {\n\nCollection<UserRecord> data;\n\n\/**\n* Find a user by unique id.\n*\/\n\/*@\nensures\n(*\n* If there is a record with the given id in the input db, then the\n* output record is equal to that record, otherwise the output record\n* is empty.\n*);\n@*\/\nUserRecord findById(String id);\n\n}\n\n\n3. Here are the initial formal specifications for the findById, findByName, ChangeUser, and DeleteUser operations, with the \"no spurious data\" requirements.\n\nimport java.util.Collection;\n\nabstract class UserDB {\n\nCollection<UserRecord> data;\n\n\/*\n* Find a user by unique id.\n*\/\n\/*@\nrequires (* None yet. *);\n\nensures\n\/\/\n\/\/ If there is a record with the given id in the input data, then the\n\/\/ output record is equal to that record, otherwise the output record\n\/\/ is null.\n\/\/\n(\\exists UserRecord ur_found ; data.contains(ur_found) ;\nur_found.id.equals(id) && ur_found.equals(\\result))\n||\n!(\\exists UserRecord ur_found ; data.contains(ur_found) ;\nur_found.id.equals(id)) && \\result == null;\n\n@*\/\nabstract UserRecord findById(String id);\n\n\/*\n* Find a user or users by real-world name. If more than one is found,\n* list is sorted by id.\n*\/\n\/*@\nrequires (* None yet. *);\n\nensures\n\/\/\n\/\/ A record is in the output list if and only it is in the input UserDB\n\/\/ and the record name equals the name being searched for.\n\/\/\n(\\forall UserRecord ur ;\n\\result.contains(ur) <==>\n\\old(data).contains(ur) && ur.name.equals(name));\n@*\/\nabstract Collection<UserRecord> findByName(String name);\n\n\/**\n* Change the given old UserRecord to the given new record. The old and\n* new records must not be the same. The old record must already be in\n* the input db. The new record must meet the same conditions as for the\n* input to the AddUser operation. Typically the user runs the FindUser\n* operation prior to Change to locate an existing record to be changed.\n*\/\n\/*@\nrequires (* None yet. *);\n\nensures\n\/\/\n\/\/ A user record is in the output data if and only if it is the new\n\/\/ record to be added or it is in the input data, and it is not the old\n\/\/ record.\n\/\/\n(\\forall UserRecord ur_other ;\ndata.contains(ur_other) <==>\nur_other.equals(new_ur) ||\n(\\old(data).contains(ur_other) &&\n!ur_other.equals(old_ur)));\n@*\/\nabstract void change(UserRecord old_ur, UserRecord new_ur);\n\n\/**\n* Delete the given user record from the given UserDB. The given record\n* must already be in the input db. Typically the user runs the FindUser\n* operation prior to Delete to locate an existing record to delete.\n*\/\n\/*@\nrequires (* None yet. *);\n\nensures\n\/\/\n\/\/ A user record is in the output data if and only if it is not the\n\/\/ existing record to be deleted and it is in the input data.\n\/\/\n(\\forall UserRecord ur_other ;\ndata.contains(ur_other) <==>\n!ur_other.equals(ur) && \\old(data).contains(ur_other));\n@*\/\nabstract void delete(UserRecord ur);\n\n}\n\n\n4. Observations.\n\n1. All of the preconditions are commented \"(* None yet. *)\"; we will refine preconditions shortly.\n\n2. The postcondition for findById uses the existential quantifier exists; Table 2 summarizes the JML formats.\n\n Form Reading (\\exists T x ; predicate) There exists x of type T such that predicate is true. (\\exists T x ; constraint ; predicate) There exists x of type T, such that constraint is true, and then predicate is true.\n\nTable 2: Forms of existential quantification..\n\n3. The postcondition for findByName is missing an important piece of logic vis a vis user-level requirements. What is it? (Hint: see the method's comment.)\n\n4. The postcondition logic for change and delete are adaptations of the postcondition logic for UserDB.add.\n\n1. This kind of logic is sometimes called the \"no junk, no confusion\" rule for collection classes.\n\n2. Namely, when we put something into or take something out of a collection,\n\n1. we don't put in or take out anything superfluous (no junk),\n\n2. we do put in or take out exactly what we intend to (no confusion).\n\n3. You should study the logic closely to clarify your understanding of it.\n\n12. On the use of quantifiers.\n\n1. Universal and existential quantification are two ways to state multiple conditions in a single expression.\n\n1. With universal quantification (forall), the quantifier expression is true if all cases considered are true.\n\n2. With existential quantification (exists), the quantifier expression is true if at least one of the cases is true.\n\n3. Logically, you can think of forall and exists as forms of repeated logical and and or, respectively.\n\n4. There is even a generalized DeMorgan's law that makes the two forms of quantifier interchangeable:\n\n(forall T x ; p) <==> !(exists T x ; !p)\nand\n!(forall T x ; !p) <==> (exists T x ; p)\n\n\n2. In the software modeling task upon which we're focused, the use of logical quantifiers is focused on two specific objectives:\n\n1. Stating a requirement about all values of a particular type, e.g.,\n\n(\\forall UserRecord ur ; requirement-predicate)\n\n2. Stating a requirement that must be true for at least one value of a particular type, e.g.,\n\n(\\exists UserRecord ur ; requirement-predicate)\n\n3. Constrained forms of qualification provide further focus.\n\n1. Stating a requirement about all values (or at least one value) in a particular data collection, e.g.,\n\n(\\forall UserRecord ur ; data.contains(ur) ; requirement-predicate)\n(\\exists UserRecord ur ; data.contains(ur) ; requirement-predicate)\n\n2. Stating a requirement about all values (or at least one value) of a particular type, with some further restrictions on the values. E.g.,\n\n(\\forall int i ; i > 0 ; requirement-predicate)\n(\\exists int i ; i > 0 ; requirement-predicate)\n\n4. Keeping these specific focuses in mind will help narrow down when and how to use quantifiers.\n\n13. Formally specifying user-level requirements.\n\n1. To this point, we have formalized some basic requirements for our database operations.\n\n2. Specifically, we have focused on postconditions related to the second of our formal specification maxims -- not trusting the programmer.\n\n3. It is now time to consider the formal definition of user-level requirements per the first maxim -- nothing is obvious.\n\n4. To start, there are a number of \"obvious\" user-level requirements, including the following:\n\n1. Duplicate entries are not allowed in the UserDB.\n\n2. Input values are checked for validity.\n\n3. If the findByName operation outputs more than one record, the output should be sorted in some appropriate order.\n\n5. We have considered these requirements to some extent in the requirements narrative.\n\n1. However, the process of fully formalizing the specification can reveal important details we may have overlooked in the requirements scenarios.\n\n2. For example, in the Milestone 6 scenarios we initially overlooked the sorting requirement for multiple outputs from findByName.\n\n3. Such oversights are common, and one of the main reasons we're doing the fully formal level of the spec.\n\n6. An historical note is of interest with regards to such requirements.\n\n1. In software engineering methodologies less formal than what we're using, the process of formalizing a specification can take the form of \"firming up\" the English prose in which the requirements are stated.\n\n2. For example, the first of the above requirements could be stated \"formally\" as follows:\nA UserDB shall not contain duplicate entries.\n\n3. While this may not seem to be a substantial improvement to the original statement of the requirement, it represents a seriously-proposed approach to formalization.\n\n1. With this approach, a number of possible forms of natural language are standardized with a restricted vocabulary.\n\n2. For example, all formal requirements are expressed using \"shall\" instead of other comparable English words such as \"should\", \"ought to\", or \"allowed to\".\n\n4. This idea of formalizing English is noteworthy because it has been widely used in practice, and significant documents have been \"formalized\" in this manner.\n\n5. While such rules can indeed help with the formalization process, they fall well short of a fully formal basis for requirements specification.\n\n14. No Duplicates\n\n1. Analysis of the no duplicates requirement provides fine support for the \"nothing-is-obvious\" maxim.\n\n2. While we may expect reasonable people to understand what \"no duplicates\" means, there are in fact a number of plausible interpretations here.\n\n3. Three such interpretations are the following:\n\n1. No two UserRecords in a UserDB have exactly the same values for all UserRecord components.\n\n2. No two UserRecords in a UserDB have the same name.\n\n3. No two UserRecords in a UserDB have the same id.\n\n4. Which of these interpretations to choose is categorically not a matter for a programmer to decide.\n\n1. Rather, it should be decided at the user specification level, by the analyst in consultation with the end users.\n\n2. We could even grant that most programmers are reasonably smart, so in this case we might safely assume that a programmer could make the correct decision, or know enough to consult with the user to resolve the ambiguity.\n\n3. Suppose, however, we were specifying data records in a much more complicated application domain, such as aeronautics.\n\n4. In this domain we might have a data object such as an anomaly list, with record fields like PreFlight, TaxiOut, InFlight, Approach, and Landing.\n\n1. What does it mean to disallow duplicates in an anomalies database?\n\n2. Which field, if any, could be used as a unique key?\n\n5. The point is that such questions need to be answered by end users and\/or application domain experts.\n\n6. Such questions should most certainly not be left unanswered when the programmer begins work, since the programmer may well not know how to answer them, or even that they need to be asked.\n\n5. In our UserDB case, we have already determined with the customer that the Id component of a UserRecord is the unique key.\n\n1. This means that UserRecords in the UserDB need only differ in the Id value.\n\n2. In particular, there may be multiple UserRecords with the same name.\n\n6. The basic strategy for disallowing duplicates is to define a precondition on UserDB.add that checks for an element of the same Id as the UserRecord being added.\n\n7. Here is the refined specification for UserDB.add; for brevity, the postcondition is omitted:\n\nimport java.util.Collection;\n\nabstract class UserDB {\n\nCollection<UserRecord> data;\n\n\/**\n* Same comment as above ... .\n*\/\n\/*@\nrequires\n\/\/\n\/\/ There is no user record in the input UserDB with the same id as the\n\/\/\n!(\\exists UserRecord ur_input ; data.contains(ur_input) ;\nur_input.id.equals(ur.id));\n\nensures\n(* Same postcondition as above ... *);\n\n@*\/\n\n}\n\n\n8. A discussion of the exact nature of a precondition is in order here.\n\n1. By definition, failure of a precondition means that the operation is prevented from executing.\n\n2. More precisely, precondition failure means that the operation fails.\n\n3. This abstract meaning of precondition failure does not define how operation failure is perceived by the end user.\n\n1. Generally, the end-user should see an appropriate error message when an operation fails.\n\n2. The details of such error messages are typically abstracted out of the formal specification.\n\n15. Input value checking.\n\n1. Input value constraints for a user record are described in the requirements scenarios as follows:\n\n1. the Id of a user record is a unique system id of eight characters or less;\n\n2. the email address is a free-form string;\n\n3. the phone area code is three digits, the number is seven digits.\n\n2. These constraints are defined formally as follows, with accompanying commentary:\n\nimport java.util.Collection;\n\nabstract class UserDB {\n\nCollection<UserRecord> data;\n\n\/*@\nrequires\n\/\/\n\/\/ There is no user record in the input UserDB with the same id as the\n\/\/\n(! (\\exists UserRecord ur_other ;\ndata.contains(ur_other) ;\nur_other.id.equals(ur.id)))\n\n&&\n\n\/\/\n\/\/ The id of the given user record is not empty and 8 characters or\n\/\/ less.\n\/\/\n(ur.id != null) && (ur.id.length() > 0) && (ur.id.length() <= 8)\n\n&&\n\n\/\/\n\/\/ The email address is not empty.\n\/\/\n(ur.email != null) && (ur.email.length() > 0)\n\n&&\n\n\/\/\n\/\/ If the phone area code and number are present, they must be 3 digits\n\/\/ and 7 digits respectively.\n\/\/\n((ur.phone.area != 0) ==>\nInteger.toString(ur.phone.area).length() == 3) &&\n((ur.phone.number != 0) ==>\nInteger.toString(ur.phone.number).length() == 7);\n\nensures (* Same as above *);\n\n@*\/\n\n}\n\n\n3. Observations\n\n1. The standard way to strengthen a precondition is to and on additional clauses.\n\n1. Here, the previous \"no duplicates\" clause remains.\n\n2. The new requirements are added by anding them on.\n\n2. The process of formally specifying these requirements led to the discovery of one unnoticed requirements detail, which will be updated in the scenario narrative.\n\n3. Specifically, in considering the formal specification for the constraint on email address, we were alerted to the question of whether it should be required.\n\n1. In consultation with the customer, the answer turns out to be \"yes\", even though we had not originally considered the issue explicitly in the scenarios.\n\n2. Hence, there is the precondition clause\n(ur.email != null) && (ur.email.length() > 0)\n\n\n3. This says that while the email address can be a free-form string, it cannot be null or of length 0, i.e., the user cannot leave it empty in the dialog.\n\n1. Note that we include a standard Java practice of checking for a null reference value before accessing a component of that reference.\n\n2. Predicates should not throw exceptions, unless they are explicitly dealt with in the specification, which subject we will address next week.\n\n4. Such are just the kind of details we hope to catch while formalizing.\n\n16. Ordering of multi-record output lists.\n\n1. The version of findByName input produces a list of UserRecords, since the name input is not required to be a unique-valued component of a record.\n\n2. As noted above, the initial requirements scenario overlooked what order the outputs should be in, if there are two or more.\n\n3. The most reasonable choice is to sort the output list by Id field.\n\n1. The scenario narrative will be updated to reflect this decision.\n\n2. As with other such requirements, we should not trust that a programmer will do the right thing in the absence of a formal statement.\n\n3. In this case, the programmer may not even think there is problem if an output list is displayed in some internal order, such as the order UserRecords are stored in a hash table.\n\n4. Such an order is as good as random to most human users, and as such rarely if ever satisfactory.\n\n4. To specify UserRecord list ordering, we must strengthen the findByName postcondition. Here it is:\n\nimport java.util.Collection;\nimport java.util.List;\n\nabstract class UserDB {\n\nCollection<UserRecord> data;\n\n\/**\n* Find a user or users by real-world name. If more than one is found,\n* the output list is sorted by id.\n*\/\n\/*@\n\nrequires (* Not defined yet. *);\n\nensures\n\/\/\n\/\/ The output list consists of all records of the given name in the\n\/\/ input data.\n\/\/\n(\\forall UserRecord ur ;\n\\result.contains(ur) ;\ndata.contains(ur) && ur.name.equals(name))\n\n&&\n\n\/\/\n\/\/ The output list is sorted lexicographically by id, according to the\n\/\/ semantics of java.lang.String.compareTo().\n\/\/\n(\\forall int i ; (i >= 0) && (i < \\result.size() - 1) ;\n\\result.get(i).id.compareTo(\\result.get(i+1).id) < 0);\n@*\/\nabstract List<UserRecord> findByName(String name);\n\n}\n\n\n\n5. An English translation of the sorting logic is the following:\n\"For each position i in the output list, such that i is between the first and the second to the last positions in the list, the ith element of the list is less than the i+1st element of the list.\"\n\n6. You should study this logic to be satisfied that it specifies sorting satisfactorily.\n\n7. Note that we have used the java.util.List interface to define our collection object.\n\n1. We'll use List instead of Collection in a specification when we need to specify ordering\n\n2. java.util.Collection does not have the get method for seletting the ith element.\n\n17. Unbounded quantification.\n\n1. What would happen to the meaning of the sorting predicate if the constraint on the range of i were not present?\n\n2. I.e., if the sorting logic in the postcondition were changed to the following:\n\n(\\forall int i ; \\result.get(i).id.compareTo(\\result.get(i+1).id) < 0)\n\n\n3. The meaning here is an unbounded quantification.\n\n1. That is, the quantifier operates over the unbounded range of all integers.\n\n1. In pure mathematical terms, unbounded means infinite.\n\n2. In terms of a Java program, numbers are bounded by the word size of a particular computer architecture, but we are abstracting that out of our specifications at the moment.\n\n2. In principle, there is nothing wrong with unbounded quantification.\n\n3. For example, the original anti-spurious requirements for UserDB.add are expressed using unbounded quantification\n\n1. I.e., (\\forall UserRecord ur ...)\n\n2. The range of the UserRecord type is unbounded, since it constrains string components, the values of which are conceptually unbounded, due to their conceptually unbounded length.\n\n4. One might argue for range restrictions on the grounds of efficiency, but as noted earlier, efficiency of this nature is not of concern in an abstract specification.\n\n4. The potential problem with unbounded quantification is that the body of the universal quantifier may not have the correct value in an unbounded range, and hence the value of the entire quantifier expression may be false when we expect it to be true, or may throw an exception, which we do not want.\n\n1. This is in fact the case in the unbounded quantification used in the sorting predicate for findByName.\n\n2. Specifically, the evaluation of \\result.get(i) throws an exception if i is outside the bounds of \\result.\n\n5. The exact outcome of the unbounded quantification depends on the semantics, i.e., form definition, of a particular specification language.\n\n1. In general, however, unbounded quantification is potentially problematic under any logical semantics.\n\n2. The point is that one needs to be careful when using unbounded quantification to ensure that the body of the quantifier has a well understood value over the entire unbounded range of quantification.\n\n3. This is particularly the case when quantifying over the elements of a list.\n\n18. Using auxiliary functions.\n\n1. The postcondition in the most recent definition of findByName is a little lengthy.\n\n1. In practice, predicates significantly longer than this can appear in the specification of a complex operation.\n\n2. When pre- or postconditions become unduly long, it is useful to use auxiliary functions to organize the logic.\n\n3. In JML, an auxiliary function is defined as a boolean-valued method in the class where the function is used in a predicate.\n\n4. The logic of the auxiliary function is given as an ensures clause of the form \"\\result == ...\", where \"...\" is a boolean expression that appears in one or more predicates.\n\n5. The purpose of an auxiliary function is simply to modularize a piece of logic, give it a mnemonic name, and allow that logic to be invoked in one or more places.\n\n6. I.e., the purpose is to make predicates more readable and understandable.\n\n2. As an example, here is the preceding definition of findByName using two auxiliary functions.\n\nimport java.util.Collection;\nimport java.util.List;\n\nabstract class UserDB {\n\nCollection<UserRecord> data;\n\n\/**\n* Find a user or users by real-world name. If more than one is found,\n* the output list is sorted by id.\n*\/\n\/*@\n\nrequires (* Not defined yet. *);\n\nensures\nrecordsFound(name, \\result)\n&&\nsortedById(\\result);\n\n@*\/\nabstract List<UserRecord> findByName(String name);\n\n\/**\n* Return true if the given list consists of all records of the given name\n* in this.data.\n*\/\n\/*@\nensures\n\\result ==\n(\\forall UserRecord ur ;\nlist.contains(ur) <==>\ndata.contains(ur) && ur.name.equals(name));\n@*\/\nabstract boolean recordsFound(String name, Collection<UserRecord> list);\n\n\/**\n* Return true if the given list is sorted lexicographically by id,\n* according to the semantics of java.lang.String.compareTo().\n*\/\n\/*@\nensures\n\\result ==\n(\\forall int i ; (i >= 0) && (i < list.size() - 1) ;\nlist.get(i).id.compareTo(list.get(i+1).id) < 0);\n@*\/\nabstract boolean sortedById(List<UserRecord> list);\n\n}\n\n\n\n19. Moving on to the specs for the GroupDB.\n\n1. Figure 2 on page 2 shows the UI for the other user-related database in the Calendar Tool -- the database of user groups.\n\n2. The specs for the GroupDB are quite similar to UserDB.\n\n1. Both databases are clear examples of collection objects with typical collection operations.\n\n2. The specs for GroupDB are slightly simpler, given that there is only one searchable component, the group name, which must be unique among all groups in the database.\n\n3. A significant specification issue does arise in the area of interaction between user database operations with the group database.\n\n1. Specifically, what happens to groups that have as a member a user who is deleted from the user database?\n\n2. Possible ways to deal with this problem include the following:\n\n1. A deleted user is automatically removed from all groups of which she is a member.\n\n2. If a deleted user is in one or more groups, a warning message is output indicating what groups the user was in, but the users must be manually deleted from the groups; in the meantime, any unknown users are simply ignored in the group member lists.\n\n3. The system prevents deletion of a user until she has first been deleted from all groups; to assist the deletion, the system outputs a message indicating the affected groups.\n\n4. This is yet another example of where formalizing the specs has led to the discovery of an important requirements issue.\n\n1. In this case, user consultation results in the automatic removal solution.\n\n2. This in turn leads to another issue, which is what should be done with groups who have no leader, due to the automatic deletion of a member or was the only leader of a group.\n\n3. This issue is resolved by allowing leaderless groups, but having the system output a warning when the situation arises.\n\n5. All of the issues having been resolved, the resulting complete spec for the user and group databases is as follows:\n\nimport java.util.Collection;\nimport java.util.List;\n\n\/**\n* UserDB is the repository of registered user information.\n*\/\nabstract class UserDB {\n\n\/**\n* The collection of user data records.\n*\/\nCollection<UserRecord> data;\n\n\/**\n* Reference to GroupDB needed for change and delete methods.\n*\/\nGroupDB groupDB;\n\n\/**\n* Add the given UserRecord to the given UserDB. The Id of the given user\n* record must not be the same as a user record already in the UserDB.\n* The Id component is required and must be eight characters or less. The\n* email address is required. The phone number is optional; if given, the\n* area code and number must be 3 and 7 digits respectively.\n*\/\n\/*@\nrequires\n\/\/\n\/\/ There is no user record in the input UserDB with the same id as the\n\/\/\n(! (\\exists UserRecord ur_other ;\ndata.contains(ur_other) ;\nur_other.id.equals(ur.id)))\n\n&&\n\n\/\/\n\/\/ The id of the given user record is not empty and 8 characters or\n\/\/ less.\n\/\/\n(ur.id != null) && (ur.id.length() > 0) && (ur.id.length() <= 8)\n\n&&\n\n\/\/\n\/\/ The email address is not empty.\n\/\/\n(ur.email != null) && (ur.email.length() > 0)\n\n&&\n\n\/\/\n\/\/ If the phone area code and number are present, they must be 3 digits\n\/\/ and 7 digits respectively.\n\/\/\n((ur.phone.area != 0) ==>\nInteger.toString(ur.phone.area).length() == 3) &&\n((ur.phone.number != 0) ==>\nInteger.toString(ur.phone.number).length() == 7);\n\n\/*@\nensures\n\/\/\n\/\/ A user record is in the output data if and only if it is the new\n\/\/ record to be added or it is in the input data.\n\/\/\n(\\forall UserRecord ur_other ;\n(data.contains(ur_other)) <==>\nur_other.equals(ur) || \\old(data).contains(ur_other));\n@*\/\n\n\/**\n* Find a user by unique id.\n*\/\n\/*@\nensures\n\/\/\n\/\/ If there is a record with the given id in the input data, then the\n\/\/ output record is equal to that record, otherwise the output record\n\/\/ is null.\n\/\/\n(\\exists UserRecord ur_found ; data.contains(ur_found) ;\nur_found.id.equals(id) && ur_found.equals(\\result))\n||\n!(\\exists UserRecord ur_found ; data.contains(ur_found) ;\nur_found.id.equals(id)) && \\result == null;\n\n@*\/\nabstract UserRecord findById(String id);\n\n\/**\n* Find a user or users by real-world name. If more than one is found,\n* the output list is sorted by id.\n*\/\n\/*@\nensures\n\/\/\n\/\/ The output list consists of all records of the given name in the\n\/\/ input data.\n\/\/\n(\\forall UserRecord ur ;\n\\result.contains(ur) ;\ndata.contains(ur) && ur.name.equals(name))\n\n&&\n\n\/\/\n\/\/ The output list is sorted lexicographically by id, according to the\n\/\/ semantics of java.lang.String.compareTo().\n\/\/\n(\\forall int i ; (i >= 0) && (i < \\result.size() - 1) ;\n\\result.get(i).id.compareTo(\\result.get(i+1).id) < 0);\n@*\/\nabstract List<UserRecord> findByName(String name);\n\n\/**\n* Change the given old UserRecord to the given new record. The old and\n* new records must not be the same. The old record must already be in\n* the input db. The new record must meet the same conditions as for the\n* input to the AddUser operation. Typically the user runs the FindUser\n* operation prior to Change to locate an existing record to be changed.\n*\n* If the user record id is changed, then change all occurrences of the\n* old id in the group db to the new id.\n*\/\n\/*@\nrequires\n\/\/\n\/\/ The old and new user records are not the same.\n\/\/\n!old_ur.equals(new_ur)\n\n&&\n\n\/\/\n\/\/ The old record is in this.data.\n\/\/\ndata.contains(old_ur)\n\n&&\n\n\/\/\n\/\/ There is no user record in the input UserDB with the same id as the\n\/\/ new record to be added.\n\/\/\n(! (\\exists UserRecord ur_other ;\ndata.contains(ur_other) ;\nur_other.id.equals(new_ur.id)))\n\n&&\n\n\/\/\n\/\/ The id of the new record is not empty and 8 characters or less.\n\/\/\n(new_ur.id != null) && (new_ur.id.length() > 0) &&\n(new_ur.id.length() <= 8)\n\n&&\n\n\/\/\n\/\/ The email address is not empty.\n\/\/\n(new_ur.email != null) && (new_ur.email.length() > 0)\n\n&&\n\n\/\/\n\/\/ If the phone area code and number are present, they must be 3 digits\n\/\/ and 7 digits respectively.\n\/\/\n((new_ur.phone.area != 0) ==>\nInteger.toString(new_ur.phone.area).length() == 3) &&\n((new_ur.phone.number != 0) ==>\nInteger.toString(new_ur.phone.number).length() == 7);\n\nensures\n\/\/\n\/\/ A user record is in the output data if and only if it is the new\n\/\/ record to be added or it is in the input data, and it is not the old\n\/\/ record.\n\/\/\n(\\forall UserRecord ur_other ;\ndata.contains(ur_other) <==>\nur_other.equals(new_ur) ||\n(\\old(data).contains(ur_other) &&\n!ur_other.equals(old_ur)))\n&&\n\n\/\/\n\/\/ If new id is different than old id, then all occurrences of old id\n\/\/ in the GroupDB are replaced by new id.\n\/\/\n!old_ur.id.equals(new_ur.id) ==> true\n\/\/ Logic left as exercise for the reader\n;\n@*\/\nabstract void change(\nUserRecord old_ur, UserRecord new_ur);\n\n\/**\n* Delete the given user record from the given UserDB. The given record\n* must already be in the input db. Typically the user runs the FindUser\n* operation prior to Delete to locate an existing record to delete.\n*\n* In addition, delete the user from all groups of which the user is a\n* member. If the deleted user is the only leader of a one more groups,\n* output a warning indicating that those groups have become leaderless.\n*\/\n\/*@\nrequires\n\/\/\n\/\/ The given user record is in this.data.\n\/\/\ndata.contains(ur);\n\nensures\n\/\/\n\/\/ A user record is in the output data if and only if it is not the\n\/\/ existing record to be deleted and it is in the input data.\n\/\/\n(\\forall UserRecord ur_other ;\ndata.contains(ur_other) <==>\n!ur_other.equals(ur) && \\old(data).contains(ur_other))\n\n&&\n\n\/\/\n\/\/ The id of the deleted user is not in the leader or member lists of\n\/\/ any group in the output GroupDB. (NOTE: This clause is not as\n\/\/ strong as a complete \"no junk, no confusion\" spec. Why not? Should\n\/\/ it be?)\n\/\/\n(\\forall GroupRecord gr ; groupDB.data.contains(gr) ;\n\n&&\n\n\/\/\n\/\/ The LeaderlessGroupsWarning list contains the ids of all groups\n\/\/ whose only leader was the user who has just been deleted.\n\/\/\n(\\forall GroupRecord gr ; groupDB.data.contains(gr) ;\n\\forall String id ;\n(\\result.groupNames.contains(id) <==>\n@*\/\n\n}\n\n\/**\n* A UserRecord is the information stored about a registered user of the\n* Calendar Tool. The Name component is the users real-world name. The\n* Id is the unique identifier by which the user is known to the Calendar\n* Tool. The EmailAddress is the electronic mail address used by the\n* Calendar Tool to contact the user when necessary. The PhoneNumber is\n* for information purposes; it is not used by the Calendar Tool for\n* contacting the user.\n*\/\nabstract class UserRecord {\nString name;\nString id;\nString email;\nPhoneNumber phone;\n}\n\nabstract class PhoneNumber {\nint area;\nint number;\n}\n\n\/**\n* LeaderlessGroupsWarning is an secondary output of the UserDB.change and\n* UserDB.delete operations, indicating the names of zero or more groups that\n* have become leaderless as the result of a user having been deleted.\n*\/\nCollection<String> groupNames;\n}\n\n\/**\n* GroupDB is the repository of user group information.\n*\/\nabstract class GroupDB {\n\n\/**\n* The collection of group data records.\n*\/\nCollection<GroupRecord> data;\n\n\/**\n* Reference to GroupDB needed for change and delete methods.\n*\/\nUserDB userDB;\n\n\/**\n* Add the given GroupRecord to the given GroupDB. The name of the given\n* group must not be the same as a group already in the GroupDB. All\n* group members must be registered users. The leader(s) of the group\n* must be members of it.\n*\/\n\/*@\nrequires\n\/\/\n\/\/ All group members are registered users.\n\/\/\n(\\forall String id ; gr.members.contains(id) ;\n(\\exists UserRecord ur ; userDB.data.contains(ur) ;\nur.id.equals(id)))\n\n&&\n\n\/\/\n\/\/ All group leaders are members of the group.\n\/\/\n(\\forall String id ; gr.leaders.contains(id) ;\ngr.members.contains(id));\n\nensures\n\/\/\n\/\/ A group record is in the output db if and only if it is the new\n\/\/ record to be added or it is in the input db.\n\/\/\n(\\forall GroupRecord gr_other ;\ndata.contains(gr_other) <==>\ngr_other.equals(gr) || \\old(data).contains(gr_other));\n@*\/\n\n\/**\n* Delete the given group record from the given GroupDB. The given record\n* must already be in the input db. Typically the user runs the FindGroup\n* operation prior to Delete to locate an existing record to delete.\n*\/\n\/*@\nrequires\n\/\/\n\/\/ The given GroupRecord is in the given GroupDB.\n\/\/\ndata.contains(gr);\n\nensures\n\/\/\n\/\/ A group record is in the output db if and only if it is not the\n\/\/ existing record to be deleted and it is in the input db.\n\/\/\n(\\forall GroupRecord gr_other ;\ndata.contains(gr_other) <==>\n!gr_other.equals(gr) && \\old(data).contains(gr_other));\n@*\/\nabstract void delete(GroupRecord gr);\n\n\/**\n* Change the given old GroupRecord to the given new record. The old and\n* new records must not be the same. The old record must already be in\n* the input db. The new record must meet the same conditions as for the\n* input to the AddGroup operation. Typically the user runs the FindGroup\n* operation prior to Change to locate an existing record to be changed.\n*\/\n\/*@\nrequires\n\/\/\n\/\/ The old and new group records are not the same.\n\/\/\n!old_gr.equals(new_gr)\n\n&&\n\n\/\/\n\/\/ All group members are registered users.\n\/\/\n(\\forall String id ; new_gr.members.contains(id) ;\n(\\exists UserRecord ur ; userDB.data.contains(ur) &&\nur.id.equals(id)))\n\n&&\n\n\/\/\n\/\/ All group leaders are members of the group.\n\/\/\n(\\forall String id ; new_gr.leaders.contains(id) ;\nnew_gr.members.contains(id));\n\nensures\n\/\/\n\/\/ A group record is in the output db if and only if it is the new\n\/\/ record to be added or it is in the input db, and it is not the old\n\/\/ record.\n\/\/\n(\\forall GroupRecord gr_other ;\ndata.contains(gr_other) <==>\ngr_other.equals(new_gr) ||\n\\old(data).contains(gr_other) &&\n!gr_other.equals(old_gr));\n@*\/\nabstract void change(GroupRecord old_gr, GroupRecord new_gr);\n\n\/**\n* Find a group by unique name.\n*\/\n\/*@\nensures\n\/\/\n\/\/ If there is a record with the given name in the input db, then the\n\/\/ output record is equal to that record, otherwise the output record\n\/\/ is empty.\n\/\/\n(\\exists GroupRecord gr_found ; data.contains(gr_found) ;\ngr_found.name.equals(id) && gr_found.equals(\\result))\n||\n!(\\exists GroupRecord gr_found ; data.contains(gr_found) ;\ngr_found.name.equals(id)) && \\result == null;\n@*\/\nabstract GroupRecord findById(String id);\n\n}\n\n\/**\n* A GroupRecord is the information stored about a user group. The Name\n* component is a unique group name of any length. Leaders is a list of\n* zero or more users designated as group leader. Members is the list of\n* group members, including the leaders. Both lists consist of user id's.\n* Normally a group is required to have at least one leader. The only\n* case that a group becomes leaderless is when its leader is deleted as a\n* registered user.\n*\/\nabstract class GroupRecord {\nString name;","date":"2013-05-23 20:16:13","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"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\": 0.431966096162796, \"perplexity\": 2693.983757683765}, \"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-2013-20\/segments\/1368703748374\/warc\/CC-MAIN-20130516112908-00090-ip-10-60-113-184.ec2.internal.warc.gz\"}"}
| null | null |
If you're passionate about helping K-12 students develop the skills, habits and attitudes necessary to promote lifelong health and wellness, check out our experiential and research-informed program in Physical Education. You'll connect current educational theories and research to real-world, contemporary experiences. We recommend a minor or endorsement in a second teaching area; many Physical Education majors complete a minor in Health Education. The College of Education is accredited by the Council for the Accreditation of Educator Preparation to offer teacher education programs which are approved by the Illinois State Board of Education.
We embrace a holistic approach in Physical Education that focuses on "the whole child," preparing our graduates to impact the social and emotional development, as well as the cognitive and behavioral changes of their students. KNPE 466 (Field Experience at Outdoor Environments), for example, transports our P.E. majors to our Lorado Taft Field Campus, one of the premier outdoor education facilities in the country, for an experience as outdoor educators for elementary school children. This unique experience builds our students' confidence and bolsters their résumés.
If you are admitted to NIU to pursue the B.S.Ed. degree in Physical Education then you will enter directly into the major program.
Admission to teacher education and retention in this major requires a university GPA of 2.75 or above.
The NIU College of Education embeds diverse, real-world learning opportunities in Physical Education programs, offering you valuable experiences you won't get anywhere else.
You can implement programs specially designed for K-12 students with unique needs in our Celebrating Abilities in Physical Education community outreach. The program promotes healthy lifestyles through gross motor development, physical fitness, lifetime sports, play-social skills, aquatics and dance-creative movement.
NIU's annual Educators' Job Fair brings school districts from throughout the state and around the country to interview for teaching positions. Our students and alumni are admitted free and are offered an extended open session time with recruiters. It's a great way to practice your professional communication skills and learn about employment opportunities, with the possibility of invitation-only interviews at the event.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,203
|
pub type Variable = String;
pub type VariableIndex = usize;
pub type Value = u32;
pub type Domain = Vec<Value>;
// This holds why do we need to revise a constraint - a value that was dropped from the domain of the given variable
pub type ConstraintRevisionReason = (VariableIndex, Value);
pub enum Constraint {
// All of the variables must have different values
AllDifferent(Vec<VariableIndex>),
// Of these variables, at least one must have the given value
OneOf(Vec<VariableIndex>, Value)
}
pub struct ConstraintProblem {
pub variables: Vec<Variable>,
pub domains: Vec<Domain>,
pub constraints: Vec<Constraint>
//lastKnownSupport: HashMap<(Variable, Value, Variable), Value)>
}
fn erase_value<T: PartialEq>(vec: &mut Vec<T>, value: T) {
vec.remove(vec.iter().position(|x| *x == value).expect("Value not found"));
}
fn revise(domains: &mut Vec<Domain>, c: &Constraint, reason: Option<ConstraintRevisionReason>) -> bool {
match c {
Constraint::AllDifferent(_vars) => {
// We may only reduce the domain of x if some of other domains takes on a specific value
let changed_var_index: VariableIndex = reason.unwrap().0;
if domains[changed_var_index].len() == 1 {
let taken_value: Value = domains[changed_var_index][0];
domains.iter_mut().enumerate()
.filter(|(idx, _)| *idx != changed_var_index)
.for_each(|(_, d)| erase_value(d, taken_value));
return true;
}
return false;
},
Constraint::OneOf(vars, val) => {
// If Value is present in only of the other variable domains, then this variable must take it as value
let mut domains_indices_containing_value = vars.iter().filter(|&&var_index| domains[var_index].contains(val));
let first_var_index: Option<&VariableIndex> = domains_indices_containing_value.next();
if first_var_index.is_none() {
println!("The problem is unsolvable!");
return false;
}
if domains_indices_containing_value.next().is_none() {
// Alright, first_var_index is the only variable still holding that value in its domain, it must takes on the value
println!("Yes, only this value remains: {}", val);
let domain = &mut domains[*first_var_index.unwrap()];
domain.clear();
domain.push(*val);
return true;
}
return false;
}
}
}
impl ConstraintProblem {
/// This runs AC2001 on the given constraint-programming problem
pub fn ac2001(&mut self) -> bool {
return revise(&mut self.domains, &self.constraints[0], None);
}
}
#[cfg(test)]
mod tests {
#[test]
fn one_of_constraint() {
let mut domains = Vec::new();
domains.push(vec![1, 2]);
domains.push(vec![1, 2, 3]);
domains.push(vec![1, 2]);
let one_of_constraint = super::Constraint::OneOf(vec![0, 1, 2], 3);
super::revise(&mut domains, &one_of_constraint, None);
// After the revise, the domain of the 2nd variable should be reduced to the value 3
assert_eq!(domains[1].len(), 1, "The domain of the variable must be reduced to a single value");
assert_eq!(domains[1][0], 3, "The domain of the variable must be reduced to the value 3");
}
#[test]
fn all_different_constraint() {
let mut domains = Vec::new();
domains.push(vec![1]);
domains.push(vec![1, 2, 3, 4]);
domains.push(vec![1, 2, 3, 4]);
domains.push(vec![1, 2, 3, 4]);
let expected_value_to_be_dropped: super::Value = 1;
let expected_domain_size = domains[1].len() - 1;
let all_different_constraint = super::Constraint::AllDifferent(vec![0, 1, 2, 3]);
let reason = (0, 1);
super::revise(&mut domains, &all_different_constraint, Some(reason));
// After the revise, the domain the value 1 should be dropped from the domains of all other variables
for d in &domains[1..] {
assert_eq!(d.len(), expected_domain_size, "The domain of the variable must be have been reduced");
assert!(!d.contains(&expected_value_to_be_dropped), "The domain of the variable must not include the value {}", expected_value_to_be_dropped);
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,502
|
{"url":"https:\/\/codereview.stackexchange.com\/questions\/203025\/extract-and-divide-algorithm-implementation-in-elixir","text":"# Extract and divide algorithm implementation in Elixir\n\nI just started learning Elixir and stumbled upon this challenge over on Programming Puzzles & Code Golf. It is a well-suited task for beginners, so I chose to give it a go (to be clear, give it a go means solve it normally, not golfing it). To keep this question self-contained, here is the task \u2013 citing the linked post:\n\nFor a given positive integer $n$:\n\n1. Repeat the following until $n < 10$ (until $n$ contains one digit).\n2. Extract the last digit.\n3. If the extracted digit is even (including 0) multiply the rest of the integer by $2$ and add $1$ ( $2n+1$ ). Then go back to step 1 else move to step 4.\n4. Divide the rest of the integer with the extracted digit (integer \/ digit) and add the remainder (integer % digit), that is your new $n$.\n\nFor example, $61407$ gives $5$ when ran through this mechanism. I've come up with the following code:\n\ndefmodule ExtractAndDivide do\ndef extract_and_divide(x) do\nif x < 10 do x\nelse\ntail = rem x, 10\ncase rem tail, 2 do\n0 -> head * 2 + 1 |> extract_and_divide\nend\nend\nend\nend\n\n\nI'm seeking general advice, but mainly focusing on the following:\n\n\u2022 Naming and Syntax better practices (usage of parenthesises, variable names etc.)\n\u2022 Usage of |> (pipe) in this context. Would you ever see it used the way I did it in production code? Should I switch to \"normal\" notation instead?\n\u2022 Less verbose or more elegant way to avoid the seemingly unaesthetic if x < 10 do x ... else ... end structure, perhaps using case would be better here?\n\u2022 Is recursion the way to go? Should I stick to it or are there better, equivalent methods?\n\nI do not have much to say about the pipe operator. It looks fine to me, although maybe some else has something to say...\n\nAs for the if-else clause, you can use cond. It is basically a stylized if statement that looks like a case statement. One of your conditions can be x < 10 -> and the other default statement would be true ->.\n\nI am not entirely sure if this is the best practice since the wording is a bit ambiguous in the documentation. Under the use case for cond it says the following:\n\nThis is equivalent to else if clauses in many imperative languages (although used way less frequently here).\n\nWhich I interpret to be else if clauses are used less often (implying that cond is often preferred).\n\n(untested)\n\ndefmodule ExtractAndDivide do\ndef extract_and_divide(x) when x < 10, do: x\ndef extract_and_divide(x) do\ntail = rem x, 10\nup_or_down = rem tail, 2\n\n(note: skipped\/circumvented potential nitpicking around naming and whether up_or_down shouldn't be a boolean; I just wanted to illustrate the pattern of doing logic by using multiple function heads)","date":"2020-01-28 18:18:28","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\": 0.4576548933982849, \"perplexity\": 1439.6690606166735}, \"config\": {\"markdown_headings\": true, \"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-2020-05\/segments\/1579251779833.86\/warc\/CC-MAIN-20200128153713-20200128183713-00375.warc.gz\"}"}
| null | null |
\section{Introduction}
\label{sec-intro}
Global bifurcation theory in the plane and two-sphere has two major goals (amidst others): classification of the bifurcations met in generic few-parameter families (with one, two, three parameters), and the study of structural stability (or instability) of these families. In \cite{IKS}, locally generic structurally unstable three parameter families of vector fields on the sphere were discovered. After that the question whether a generic unfolding of a particular class of degeneracies is structurally stable becomes non-trivial; an \emph{a priori} valid answer ``yes'' (conjectured in \cite{AAIS}) is no more expected.
The present paper is designed to be helpful in the study of any classification and structural stability problem in the global bifurcation theory on the sphere.
\subsection{Who bifurcates?}
Consider a non-hyperbolic, i.e. structurally unstable, vector field. It may have both hyperbolic and non-hyperbolic singular points and limit cycles. Under a generic perturbation, the hyperbolic singular points and limit cycles do not bifurcate, but the non-hyperbolic ones do. A natural question arises: what subset of the phase portrait of a non-hyperbolic vector field actually bifurcates?
The goal of this paper is to answer this question.
For any perturbation of a non-hyperbolic vector field, a closed invariant subset of the phase portrait of this field called \emph{large bifurcation support} (abbreviated as LBS) is distinguished. (We would prefer to use the simpler term \emph {bifurcation support}, but it is already introduced by Arnold \cite[Sec. 3.2]{AAIS}, and has a different meaning.) In order to check whether two perturbations of two orbitally topologically equivalent vector fields are equivalent as the families of vector fields on the whole sphere, one has to check only that these families are equivalent in arbitrary small neighborhoods of their large bifurcation supports. For example, two generic two-parameter perturbations of orbitally topologically equivalent vector fields with a polycycle ``heart'', see Fig. \ref{fig:lh-heart}, are equivalent iff they are equivalent in an arbitrary small neighborhood of the polycycle (this is an easy consequence of the results of this paper, but its proof is not yet written). A similar statement for vector fields with a polycycle ``lune'', Fig. \ref{fig:lh-lune}, is wrong. The reason is that a large bifurcation support for any perturbation of the first degeneracy coincides with the polycycle ``heart''; for the perturbation of the second degeneracy, the large bifurcation support may be much larger than the ``lune''.
\begin{figure}[h]
\centering
\subcaptionbox{\label{fig:lh-heart}}{ \includegraphics[width=0.3\textwidth]{heart}}
\hfil
\subcaptionbox{\label{fig:lh-lune}}{ \includegraphics[width=0.35\textwidth]{luna}}
\caption{``Heart'' and ``lune'' }\label{fig:lh}
\end{figure}
This paper is heavily based on one of the key results of the qualitative theory of planar differential equations:
the complete topological classification of their phase portraits. This classification in different equivalent forms is given in \cite{ALGM}, \cite{Mark}, \cite{Neum}, \cite{Peix}. We used the most recent form, so called LMF graphs suggested in \cite{Fed}. This form allows us to make use of the theory of planar graphs.
A large part of our arguments is based on the topology of planar vector fields in the spirit of the Poincar\'{e}-Bendixon theorem. When we started working on this paper, we could never suggest that this topology would be so rich; see, e.g., Boundary lemma \ref{lem-U-arcs-top} below.
\subsection{Vector fields with finiteness properties}
\begin{definition}
We say that a vector field $v\in Vect(S^2)$ satisfies a
\emph{Lojasiewicz inequality} at $0$ if there is a $k\in \mathbb N$, $k\ge 1$, and $c>0$ such that $\|v(x)\| \ge c \|x\|^k$ on some neighborhood of $0$.
\end{definition}
Let $Vect\, S^2$ be the set of all $C^\infty $ vector fields on $S^2$, and $Vect^*\,S^2$ be the set of all vector fields with isolated singular points satisfying Lojasievicz inequality and with finitely many cycles. It is known that analytic vector fields with isolated singular points have finitely many limit cycles, but this is a difficult result \cite{I91}, \cite{E}.
\begin{conjecture*} Smooth vector fields met in a generic finite-parameter family belong to $Vect^*\,S^2$.
\end{conjecture*}
We will simply assume that all vector fields that we consider have this property. That is, all through the paper by default a vector field $v$ belongs to $Vect^*\,S^2$.
\subsection{Axiomatic description of the LBS}
Here and below $B$ is an open ball in $\mathbb R^n$.
\begin{definition}
\label{def:fam}
A smooth \emph{family of vector fields} on $S^2$ with the base $B\subset \mathbb R^n$ is a smooth vector field $V$ on $B \times S^2$ tangent to the fibers $\{\varepsilon\}\times S^2$, $\varepsilon \in B$.
The dimension of a family is the dimension of its base.
We will also write $V=\{v_{\varepsilon}\}_{\varepsilon\in B}$ where $v_{\varepsilon}$ are restrictions of $V$ to each fiber.
A smooth \emph{local family of vector fields} is a germ of a family of vector fields at $S^2\times \{0\} \subset S^2 \times B$. In other words, it is the family of vector fields with the base $(\mathbb R^n,0)$.
\end{definition}
In Section \ref{ssec-moderate} below, we define \emph {moderate topological equivalence} of vector fields. For vector fields with hyperbolic singular points on the sphere, this notion is defined in \cite{IKS}. We also recall a classical notion of a weak topological equivalence.
\begin{definition} \label{def-supp-axiom1}
Suppose that for any local smooth family of vector fields $V =\{v_{\varepsilon}\}_{\varepsilon\in (B,0)} \subset Vect^*\,S^2$, a closed $v_0$-invariant subset $\Lambda (V)\subset S^2$ is defined. This set is called the \emph{large bifurcation support } of $V$ if it has the following property:
\emph{Let two vector fields $v_0$ and $w_0$ be orbitally topologically equivalent on $S^2$. Let $V$ and $W$ be unfoldings of $v_0$ and $w_0$ that are moderately equivalent in some neighborhoods of $\Lambda (V)$, $\Lambda (W)$; let this moderate equivalence agree with the topological equivalence for $\varepsilon=0$. Then the families $V$ and $W$ are weakly topologically equivalent on the whole sphere.}
\end{definition}
Roughly speaking, equivalence of the unperturbed vector fields on the whole sphere and moderate equivalence of their perturbations in neighborhoods of their large bifurcation supports imply weak equivalence of the perturbations on the whole sphere. In this defintion, ``to be a large bifurcation support'' is a property of the mapping $V\mapsto \Lambda(V)$, not of an individual set $\Lambda(V)$.
The whole sphere $S^2$ (i.e. the mapping $V\mapsto \Lambda(V)=S^2$) is obviously a large bifurcation support, but it is trivial. Below we give an explicit description of the large bifurcation support for any family of vector fields from $Vect^*\,S^2$, which is in general much smaller than $S^2$. The main result of the paper claims that this set is a large bifurcation support in the sense of Definition~\ref{def-supp-axiom1}. Below we will use the term ``large bifurcation support'' for the set we construct in Sec. \ref{subsub:def-LBS}.
\subsection{Applications}
Classification problems form an essential part of the catastrophe theory. It is crucial to know large bifurcation supports for the classification of global bifurcations in $k$-parametric families with small $k$.
\section{Definitions and the main result}
\subsection{Separatrices}
Here we briefly recall some known definitions and introduce some new ones.
\begin{definition}
A singular point $P$ of a vector field $v$ is called \emph{hyperbolic} if both real parts of its two eigenvalues at $P$ are non-zero.
\end{definition}
\begin{figure}[h]
\begin{center}
\includegraphics[width=\textwidth]{sectors}
\end{center}
\caption{Hyperbolic sector, parabolic sector and elliptic sector}\label{fig-sectors}
\end{figure}
\begin{definition} \label{def-char} A phase curve of a differential equation on the plane is called a \emph{characteristic trajectory of a singular point} if, as $t \to +\infty $ or $t \to -\infty $, it approaches the singular point and becomes tangent to a straight line.
If a singular point has a characteristic trajectory, it is called \emph{characteristic}.
\end{definition}
The following classical theorem can be found in many sources. It relies on the desingularization theorem \cite{Dum}; see \cite[Sec. 1.5]{DumLlibArt} for the explicit statement and Sec. 3 of the same book for the reduction to the desingularization theorem.
\begin{theorem} \label{thm:char-pt} Suppose that a $C^{\infty}$-smooth vector field $v$ satisfies Lojasiewicz inequality at all singular points. Then in a neighborhood of each singular point, it may
\begin{itemize}
\item be topologically equivalent to a center or a focus;
\item have a finite sectorial decomposition: namely, it has a neighborhood that is split by characteristic trajectories into a finite union of sectors with smooth boundaries, and in each sector, $v$ is topologically equivalent to one of the fields shown in Fig. \ref{fig-sectors}. These sectors are called hyperbolic, parabolic and elliptic sector respectively.
\end{itemize}
\end{theorem}
\begin{definition} A \emph{separatrix} is a phase curve that contains one of two bounding phase curves of a hyperbolic sector of some singular point $P$.
The separatrix is called \emph{stable} if its $\omega$-limit set is $P$, and \emph{unstable} if its $\alpha$-limit set is $P$. If a curve is a stable and an unstable separatrix simultaneously (for two different singular points or for one and the same), then it is called a \emph{separatrix connection}.
\end{definition}
\begin{remark} For a hyperbolic saddle, the definition of a separatrix above coincides with the classical one.
\end{remark}
For $v\in Vect^*\,S^2$, we will use the classical Poincare-Bendixson theorem (see e.g. \cite[Sec. 1.7, Corollary 1.30]{DumLlibArt}):
\begin{theorem}[Poincare--Bendixson]
For each vector field $v\in Vect^*\,S^2$, the $\omega$-limit set (and the $\alpha$-limit set) of each point is one of the following:
\begin{itemize}
\item a singular point;
\item a cycle;
\item a monodromic polycycle.
\end{itemize}
\end{theorem}
A polycycle of a vector field $v$ is a union of a finite number of singular points of $v$ joined by trajectories. The polycycle is called \emph{monodromic} if it admits a Poincare map at least on one side of it.
\subsection{Moderate equivalence}
\label{ssec-moderate}
\begin{definition}
Two vector fields are called \emph{orbitally topologically equivalent} if there exists a homeomorphism $H$ of the phase space which identifies their phase portraits and preserves the direction of time parametrization on phase curves. We will also say that $H$ conjugates these two vector fields.
\end{definition}
There are three definitions of equivalence of \emph{families} of vector fields: strong, weak, and moderate. Two of them are classical, and the third one is new.
For a family $V=\{v_{\varepsilon}\mid \varepsilon \in B\}$ of vector fields, let $\mathop{\mathrm{Sing}} V$, $\mathop{\mathrm{Per}} V$, and $\mathop{\mathrm{Sep}} V$ be subsets of $B\times S^2$ formed by all singular points, all limit cycles, and all separatrices of $v_{\varepsilon}$ respectively. We will also use the notation $\mathop{\mathrm{Sing}} v \subset S^2$, $\mathop{\mathrm{Per}} v \subset S^2$, and $\mathop{\mathrm{Sep}} v \subset S^2$ for the union of all singular points, limit cycles, and separatrices of an individual vector field $v$. The set $S(v):=\mathop{\mathrm{Sing}} v\cup \mathop{\mathrm{Per}} v \cup \mathop{\mathrm{Sep}} v$ is an \emph{extended separatrix skeleton} (the set of all \emph{singular trajectories} of $v$) and plays a special role in topological classification of vector fields, see Theorem \ref{th-MNP} below.
\begin{definition}
\label{def:weak-eq}
Two local families of vector fields on $S^2$, $V = \{ v_\varepsilon , \varepsilon \in (B,0) \}$ and $ W = \{ w_\varepsilon , \varepsilon \in (B',0) \}$
are \emph{equivalent} at $\varepsilon=0$ if there exists a map
\begin{equation}
\mathbf H: (B,0) \times S^2 \to (B',0)\times S^2, \ (\varepsilon , x) \mapsto (h(\varepsilon ), H_\varepsilon (x)),
\end{equation}
such that $h$ is a homeomorphism, $h(0)=0$, and for each $\varepsilon \in (B,0)$ the map $H_\varepsilon : S^2 \to S^2 $ conjugates $v_\varepsilon $ and $w_{h(\varepsilon )}$.
They are \emph{strongly} equivalent provided that $\mathbf H$ is a homeomorphism on $(B,0)\times S^2$. They are \emph{weakly} equivalent if we do not pose any additional requirements on $\mathbf H$. They are \emph{moderately} equivalent provided that $\mathbf H$ is continuous with respect to
$(\varepsilon,x )$ on the set
\begin{equation}
\label{eq-set1}
S(v_0) \cup \partial ((\overline {\mathop{\mathrm{Per}} V} \cup \overline {\mathop{\mathrm{Sep}} V}) \cap \{\varepsilon=0\})
\end{equation}
and $\mathbf H^{-1}$ is continuous with respect to $(\varepsilon,x )$ on the set
\begin{equation}
\label{eq-set2}
S(w_0) \cup \partial ((\overline {\mathop{\mathrm{Per}} W} \cup \overline {\mathop{\mathrm{Sep}} W}) \cap \{\varepsilon=0\})
\end{equation}
\end{definition}
\begin{remark}
The notion of moderate equivalence was introduced in \cite{IKS} for hyperbolic vector fields. In this case, \begin{equation}
\label{eq-subset}
\partial ((\overline {\mathop{\mathrm{Per}} V} \cup \overline {\mathop{\mathrm{Sep}} V}) \cap \{\varepsilon=0\}) \, \subset \, S(v_0),
\end{equation}
so it is not necessary to include this set; $\mathbf H$ must be continuous on extended separatrix skeleton only.
In general, \eqref{eq-subset} does not hold. For example, we may take a bifurcation of a non-hyperbolic node $\dot x = x^3-\varepsilon x, \dot y = y$; the non-hyperbolic node (for $\varepsilon=0$) bifurcates into a saddle surrounded by two nodes (for $\varepsilon>0$). Saddle separatrices of $v_{\varepsilon}$ are vertical and accumulate to trajectories $(x=0, y>0)$ and $(x=0, y<0)$ that do not belong to $S(v_0)$.
\end{remark}
\begin{remark} In literature strong equivalence is usually called \textbf{topological equivalence}, weak equivalence is \textbf{weak topological equivalence}.
\end{remark}
Strong equivalence is too restrictive: families with very simple bifurcations may have numerical \cite{MP} and functional \cite{Rous85} invariants that distinguish different equivalence classes.
Weak equivalence is too loose: families with apparently different bifurcations may be weakly topologically equivalent.
Moderate equivalence seems to be more adequate because it takes interesting objects for the one family to the corresponding objects of the other family.
For Definition \ref{def-supp-axiom1} above, we need a local version of moderate equivalence (a moderate equivalence in neighborhoods of given closed invariant subsets). We will apply this version to neighborhoods of large bifurcation supports.
\begin{definition}
\label{def-moderate-local}
Two local families of vector fields on $S^2$, $V = \{ v_\varepsilon , \varepsilon \in (B,0) \}$ and $ W = \{ w_\varepsilon , \varepsilon \in (B',0) \}$,
are \emph{moderately equivalent in neighborhoods of closed sets} $Z_1, Z_2\subset S^2$ if
\begin{enumerate}
\item $Z_1$ is $v_0$-invariant, and $Z_2$ is $w_0$-invariant;
\item There exists a neighborhood $U \supset Z_1$ and a map
\begin{equation}
\mathbf H: (B,0) \times U \to (B',0)\times S^2, \ (\varepsilon , x) \mapsto (h(\varepsilon ), H_\varepsilon (x)),
\end{equation}
such that $h$ is a homeomorphism, $h(0)=0$, and for each $\varepsilon \in (B,0)$ the map $H_\varepsilon \colon U \to S^2 $ conjugates vector fields $(v_\varepsilon)|_{U} $ and $(w_{h(\varepsilon )})|_{H_{\varepsilon}(U)}$;
\item $H_0(Z_1)=Z_2$, and moreover,
\item \label{it-nbhd-cond} For each neighborhood $V$ of $\{\varepsilon=0\}\times Z_1$, its image $\mathbf H(V)$ contains some neighborhood of $\{\varepsilon=0\}\times Z_2$. The same holds for the inverse map $\mathbf H^{-1}$;
\item \label{it-H-contin}The map $\mathbf H$ is continuous with respect to $(\varepsilon,x )$ on the intersection of its domain with \eqref{eq-set1}.
The map $\textbf H^{-1}$ is continuous with respect to $(\varepsilon, x)$ on the intersection of its domain with \eqref{eq-set2}.
\end{enumerate}
\end{definition}
\begin{remark}
\label{rem-shrink-U}
If a homeomorphism $\mathbf H$ satisfies the above conditions for two families $V,W$ in neighborhoods of closed sets $Z_1,Z_2$, and the corresponding neighborhood is $U\supset Z_1$, then $\mathbf H$ satisfies these conditions for any smaller neighborhood $U'\subset U$, $U'\supset Z_1$.
\end{remark}
\subsection{Explicit definition of the large bifurcation support}
\subsubsection{Non-interesting limit cycles}
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.3\textwidth]{nest-ni2}\hfil\includegraphics[width=0.3\textwidth]{nest-ni1}
\end{center}
\caption{Two possible cases for a non-interesting nest}
\end{figure}
\begin{definition} A \emph{nest} of limit cycles of a vector field is the maximal set of nested cycles with no singular points in between them.
\end{definition}
A nest can consist of one limit cycle.
This definition has an additional restriction (absence of singular points between cycles) in comparison with the classical one.
Clearly, the annulus between two neighboring cycles of one nest is a canonical region, i.e. is filled by trajectories that wind towards these cycles in the past and in the future. To be in one nest is an equivalence relation; given a limit cycle, \emph{its nest} is a unique nest that contains it.
\begin{definition} A limit cycle is called \emph{semi-stable} if for some choice of coordinate on a small transversal to this cycle, the Poincar\'{e} map satisfies $P(x) > x$ for $x\neq 0$ and $P(0)=0$ (here $0$ is the intersection point of the transversal with the cycle).
\end{definition}
Clearly, for $v\in Vect^*\,S^2$, each limit cycle can be attracting, repelling, or semi-stable.
\begin{definition}
\label{def-nic}
A limit cycle of a vector field $v\in Vect^*\,S^2$ is called \emph{non-interesting} if one of the following holds:
\begin{enumerate}
\item \label{it-nonsemist}its nest contains at least one attracting or one repelling limit cycle;
\item \label{it-semist}all the cycles in the nest are semi-stable, but inside the inner cycle or outside the outer one there is only one
hyperbolic singular point.
\end{enumerate}
\end{definition}
\begin{remark} By the index theorem, this singular point is either attractor or repeller.
\end{remark}
Note that hyperbolic cycles are all non-interesting due to the definition.
The motivation for this definition is the following: when a non-interesting cycle bifurcates,
nothing interesting happens; there is no interaction between the dynamics inside and outside it.
\begin{definition} An $\alpha $- or $\omega $-limit set of a non-singular point of a vector field is called \emph{non-interesting} if it is a hyperbolic repeller (respectively, attractor), or a non-interesting limit cycle. Otherwise it is called \emph{interesting}.
\end{definition}
\subsubsection{Large bifurcation support: an explicit definition}
\label{subsub:def-LBS}
\begin{definition} \label{def:elbs} \emph{Extra large bifurcation support} $ELBS(v_0)$ of a vector field $v_0$ is the union of all non-hyperbolic singular points and non-hyperbolic limit cycles of $v_0$, plus the closure of the set of all nonsingular points for which both $\alpha $- and $\omega $-limit sets are interesting.
\end{definition}
\begin{remark}
$ELBS(v_0)$ contains all non-singular points of $v$ except (open) basins of attraction and repulsion of non-interesting $\alpha$-, $\omega$-limit sets. However we retain and include in $ELBS(v_0)$ all non-hyperbolic limit cycles, including non-interesting ones. As for singular points, $ELBS(v_0)$ contains all non-hyperbolic singular points and does not contain hyperbolic attractors and repellers. It contains a hyperbolic saddle if and only if one of its unstable (stable) separatrices has an interesting $\omega$- ($\alpha$-) limit set.
\end{remark}
Now the main definition comes.
\begin{definition} \label{def:lbs}
\emph{Large bifurcation support} of a local family $V$ of vector fields is
$LBS(V) = ELBS(v_0) \cap \left(\mathop{\mathrm{Sing}} v_0\cup (\overline {\mathop{\mathrm{Per}} V} \cup \overline{\mathop{\mathrm{Sep}} V}) \cap \{\varepsilon=0\}\right)$.
\end{definition}
So $LBS(V)$ contains all singular points and cycles of $v_0$ that belong to $ELBS(v_0)$ (see Remark \ref{rem-shrink-U}) and all non-singular accumulation points of cycles and separatrices of $v_{\varepsilon}$, $\varepsilon\to 0$, if these accumulation points have interesting $\alpha$- and $\omega$-limit sets.
\begin{remark}
For vector fields with hypebolic fixed points only, $LBS(V)$ depends only on $v_0$.
It is not clear whether this is the case for all generic families.
\end{remark}
\begin{figure}[h]
\includegraphics[width=0.8\textwidth]{LBS-examples}
\caption{Examples of a vector field $v_0$; large bifurcation supports for generic unfoldings of $v_0$ are shown in gray and thick. We only show interesting parts of phase portraits; some hyperbolic sinks and sources are not shown.}\label{LBS-ex}
\end{figure}
\begin{example}
Consider an unfolding $V$ of each of the vector fields $v_0$ shown on Fig. \ref{LBS-ex}. In each case, the number of parameters in $V$ equals the codimension of the degeneracy of $v_0$, and $V$ is a generic family. The large bifurcation support $LBS(V)$ is shown in thick curves and gray domains. In more details:
\begin{itemize}
\item Fig. \ref{LBS-ex}a (generic vector field $v_0$): the set $LBS(V)$ is empty.
\item Fig. \ref{LBS-ex}b (degeneracy of codimension 1): the set $LBS(V)$
contains the limit cycle and the two saddles with their separatrices winding to the cycle in the positive or negative time. The cycle is interesting.
\item Fig. \ref{LBS-ex}c (degeneracy of codimension 2): the set $LBS(V)$
contains the two saddles, their separatrices that wind to the cycles in the positive or negative time, and the whole annulus between the cycles. Both cycles are interesting; one may prove that saddle connections accumulate to all trajectories inside the annulus.
\item Fig. \ref{LBS-ex}d (degeneracy of codimension 2): the set $LBS(V)$
contains the outer saddle, its separatrix that winds onto the cycle, the cycle itself, and the closure of the parabolic sector of the saddlenode.
\item Fig. \ref{LBS-ex}e (degeneracy of codimension 3): the set $LBS(V)$
contains the cycle and the closures of parabolic sectors of saddlenodes.
\item Fig. \ref{LBS-ex}f (degeneracy of codimension 3. This is the polycycle collection ``lips'' studied by Kotova and Stanzo, \cite{KS}): $LBS(V)$
contains the separatrix connection between saddlenodes and the closure of the common parabolic sector of the saddlenodes, because cycles of $v_{\varepsilon}$, $\varepsilon\to 0$, accumulate to all these ordits, as shown in \cite{KS}.
\end{itemize}
\end{example}
\subsection{Main Theorem}
\begin{theorem} \label{thm:main} Large bifurcation support $LBS(V)$ defined above satisfies Definition~\ref{def-supp-axiom1}.
\end{theorem}
\begin{remark}
The set $ELBS(v_0)$ is also a large bifurcation support in terms of Definition~\ref{def-supp-axiom1}.
The proof is completely analogous to that for $LBS(V)$; one may check that $ELBS(v_0)$ satisfies the properties listed in Sec. \ref{sec-prop-LBS}, which are the only properties we need in the proof of the Main theorem below.
However we prefer to prove the stronger result, for a smaller set $LBS(V)$.
\end{remark}
Let us give a more detailed and slightly improved statement of the same theorem.
\begin{theorem}[Main Theorem] \label{thm:main1} Let two vector fields $v_0$ and $w_0$ be orbitally topologically equivalent on $S^2$; denote the corresponding homeomorphism by $\hat H $. Let $V=\{v_{\varepsilon}, \varepsilon\in (B,0)\} \subset Vect^*\,S^2, W=\{w_{\varepsilon}, \varepsilon\in (B',0)\} \subset Vect^*\,S^2$ be smooth families unfolding these fields. Suppose that there exists a neighborhood $U$ of $LBS(V)$ and a map
$$
\mathbf H: (B,0) \times U \to (B',0) \times S^2, \ \mathbf H(\varepsilon ,x) = (h(\varepsilon ), H_\varepsilon (x)),
$$
$h(0)=0$,
which is a moderate equivalence of $V,W$ in neighborhoods of $LBS(V), LBS(W)$ in the sense of Definition \ref{def-moderate-local}. Suppose moreover that $\hat H|_U = H_0$.
Then the families $V$ and $W$ are weakly equivalent on the whole sphere; namely there exists a map
$$
\mathbf{\hat H}: (B,0) \times S^2 \to (B',0) \times S^2, \ \mathbf{\hat H}(\varepsilon ,x) = (h(\varepsilon ), \hat H_\varepsilon (x))
$$
that provides a weak equivalence of the families $V$ and $W$.
\end{theorem}
\begin{remark}
We do \textbf{not} assert that $\hat H_{\varepsilon}|_{U} =H_{\varepsilon}$, and this is not true in the general case.
\end{remark}
\begin{remark}
Remark \ref{rem-shrink-U} above shows that moderate equivalence in some neighborhood of $LBS(V)$ implies moderate equivalence in any sufficiently small neighborhood of $LBS(V)$. We will have to shrink $U$ in the proof.
\end{remark}
\begin{remark} Note that the maps $\mathbf H$ and $\mathbf{\hat H}$ are skew products over \emph{the same} map $h$ of the bases. This is the only difference between Theorem \ref{thm:main} and Main Theorem.
\end{remark}
The Main theorem \ref{thm:main1} solves Problem 1 from \cite{I}. Up to now, this is the only general statement about bifurcations in the
families of vector fields with an arbitrary number of parameters. Several tempting conjectures about such bifurcations were suggested in \cite{AAIS}, but they all turned to be wrong \cite{KS}, \cite{IKS}, \cite{YuINS}.
The authors do not know any other non-trivial statement, even a conjecture, about bifurcations in generic families with an arbitrary number of parameters that would seem to be true.
\section{Strategy of the proof}
Our goal is to establish, for small $\varepsilon$, an orbital topological equivalence of two planar vector fields $v_{\varepsilon}$, $w_{h(\varepsilon)}$. We will use the criterion of orbital topological equivalence due to R.Fedorov \cite{Fed} (based on the classical book \cite{ALGM}); this result is close to the results due to L.Markus, D.Neumann, and M.M.Peixoto \cite{Mark}, \cite{Neum}, \cite{Peix}. In the following subsection we present this result. Simultaneously we recall the notion of canonical regions and describe the properties of these regions needed in the future.
\subsection{Separatrix skeletons and canonical regions} \label{sub:skel}
First, let us formulate the result of L.Markus, D.Neumann, and M.M.Peixoto \cite{Mark}, \cite{Neum}, \cite{Peix} following the book of Dumortier, Llibre and Artes \cite{DumLlibArt}, Sec.1.9. We only consider the case $v \in Vect^*\,S^2$; the result holds true for arbitrary $C^{\infty}$-smooth vector fields, but definitions should be modified for this general case (see \cite{DumLlibArt}).
\begin{definition} A \emph{ separatrix skeleton} of a vector field $v\in Vect^*\,S^2$ is $\mathop{\mathrm{Sing}} v \cup \mathop{\mathrm{Sep}} v$.
An \emph{extended separatrix skeleton} $S(v)$ of a vector field $v\in Vect^*\,S^2$ is $S(v):=\mathop{\mathrm{Sing}} v \cup \mathop{\mathrm{Per}} v \cup \mathop{\mathrm{Sep}} v$. A \emph{canonical region} of $v$ is a connected component of its complement $\mathbb R^2 \setminus S(v)$.
\end{definition}
We will use the extended separatrix skeleton rather than the separatrix skeleton.
\begin{proposition}
\label{prop-canonreg-common-limset}
For $v\in Vect^*\, (S^2)$, all points of the same canonical region $R$ have coincident $\alpha$- and $\omega$-limit sets.
\end{proposition}
\begin{proof}
First, prove that the set of points in $R$ with the same $\omega$-limit set is open. This follows from continuous dependence of solutions of ODEs on initial data. Indeed, take $x\in R$.
If $\omega(x)$ is a cycle or a monodromic polycycle, then the future semi-trajectory of $x$ intersects a transversal loop around this cycle or polycycle. So the trajectories starting in a neighborhood of $x$ also intersect this loop, and thus have the same $\omega$-limit set as $x$. The statement is proved.
If $ \omega(x)=:P$ is a singular point, we use its sectorial decomposition, see Theorem \ref{thm:char-pt}. Note that the future semi-trajectory of $x$ enters an attracting parabolic sector or an elliptic sector of $P$. It may not enter a hyperbolic sector because $x$ does not belong to a separatrix. If a future semi-trajectory of $x$ enters an attracting parabolic sector, then future semi-trajectories starting in some neighborhood of $x$ enter the same attracting parabolic sector of $P$, thus their $\omega$-limit set is also $P$. If a future semi-trajectory of $x$ enters an elliptic sector, future semi-trajectories starting in its neighborhood may enter either the same elliptic sector of $P$, or an adjacent attracting parabolic sector of $P$. In any case, their $\omega$-limit set is $P$.
Since $R$ is connected, it cannot be a union of several disjoint open sets. So all points of $R$ have the same $\omega$-limit set. The same arguments apply to the $\alpha$-limit set.
\end{proof}
\begin{definition}
The \emph{completed} separatrix skeleton of a vector field $v\in Vect^*\,S^2$ is the union of the extended separatrix skeleton together with one orbit from each one of the canonical regions.
Two completed separatrix skeletons $C_1,C_2$ are \emph{topologically equivalent}
if there exists a homeomorphism from $S^2$ to $S^2$
that maps the orbits of $C_1$ to the orbits of $C_2$ preserving the orientation.
\end{definition}
\begin{theorem}[Markus--Neumann--Peixoto Theorem]
\label{th-MNP}
Assume that $v_1, v_2 \in Vect^*\,S^2$. Then $v_1,v_2$ are topologically equivalent if and only if their completed separatrix skeletons are equivalent.
\end{theorem}
The following proposition, see \cite[Proposition 1.42, p. 34]{DumLlibArt}, gives a list of possible canonical regions. It motivates the fact that separatrix skeletons classify vector fields: on the complement to $S$, the dynamics is trivial.
\begin{proposition}
\label{prop-parall}
Every canonical region of $v\in Vect^*\,S^2$ is parallel, i.e. topologically equivalent to one of the following:
\begin{itemize}
\item A strip flow, defined on $\mathbb R^2$ by the system of differential equations $\dot x=1, \dot y=0$;
\item A spiral flow, defined on $\mathbb R^2 \smallsetminus \{0\}$ the system of differential equations $\dot r=r$, $\dot \theta=0$ in polar coordinates.
\end{itemize}
\end{proposition}
The book \cite{DumLlibArt} also lists the case of an annular flow, given on $\mathbb R^2 \smallsetminus \{0\}$ by $\dot r=0, \dot \theta =1$ in polar coordinates. This case corresponds to the infinite set of cycles, hence is not possible for $v \in Vect^* \, S^2$.
We will also need the following corollary of Proposition \ref{prop-canonreg-common-limset}:
\begin{proposition}
\label{prop-dELBS}
For $v \in Vect^*(S^2)$, each canonical region of $v$ either belongs to $ELBS(v)$, or does not intersect it. In particular, $\partial ELBS(v) \subset S(v)$.
\end{proposition}
\begin{proof}
Consider the set of all points in $S^2\setminus S(v)$ whose $\alpha$- and $\omega$-limit sets are interesting. Due to Proposition \ref{prop-canonreg-common-limset}, this set is a union of several canonical regions of $v$. Note that $ELBS(v)$ is the closure of this set plus some subset of $S(v)$, which implies the statement.
\end{proof}
\subsection{LMF graphs}
\label{sec-LMF}
The extended separatrix skeleton is not a graph on a sphere, because separatrices can wind around limit cycles. However we may turn it into a graph if we truncate the separatrices to their intersections with transversal loops of their $\alpha$- or $\omega$-limit sets.
In \cite{Fed}, R. Fedorov assigned a graph to each vector field on the plane and proved that two vector fields are orbitally topologically equivalent if their graphs are isotopic in $S^2$. The proof was based on the classical book \cite{ALGM} where the complete set of topological invariants was given in the form of ''schemes``. We will use the graphs introduced by Fedorov, and we will call them \emph{LMF graphs} (Leontovich, Mayer, Fedorov graphs) of planar vector fields.
In this section, we recall the construction of LMF graphs.
We only consider vector fields from $Vect^*\,S^2$.
Choose an orientation on $S^2$.
\begin{definition}
For a domain in $S^2$ with smooth boundary, we say that the boundary is oriented clockwise (resp. counterclockwise) with respect to the domain if the domain is to the right (resp. to the left) of its oriented boundary.
Let a closed curve $\gamma$ on $S^2$ be disjoint to a connected set $D$. We say that $\gamma$ is oriented clockwise (resp. counterclockwise) with respect to $D$ if it is oriented clockwise (resp. counterclockwise) with respect to the disk it bounds on the sphere that contains $D$.
\end{definition}
\textbf{Transversal loops around $\alpha$- and $\omega$-limit sets}
Given a smooth vector field $v$ on $S^2$, choose a transversal loop for each side of each its limit cycle, each monodromic side of each its polycycle, and around each attracting or repelling singular point of $v$. We assume that the annulus between the transversal loop and the corresponding $\alpha$- or $\omega$-limit set does not contain points of other transversal loops, and the vector field $v$ in this annulus is orbitally topologically equivalent to the standard vector field $\dot r = \pm (1-r), \dot \phi =1$ in $\{r>1\}$.
Fix a counterclockwise orientation on the chosen loop with respect to the corresponding cycle, polycycle or singular point. From now on, we always consider transversal loops with this orientation.
\textbf{Truncated separatrices}
If some separatrix $\gamma$ of a singular point $P$ of $v$ crosses a transversal loop $l$ chosen above, consider a \emph{truncated separatrix}: an arc of $\gamma$ between $P$ and the cross-point of $\gamma$ with $l$.
\begin{remark}
Assume that an outgoing separatrix $\gamma$ of $P$ does not cross such loops. Poincare-Bendixson theorem implies that its $\omega$-limit set can only be a characteristic point, $\omega(\gamma)=Q$. So this separatrix is a characteristic trajectory for $Q$, and its germ at $Q$ is $C^1$-smooth (see Theorem \ref{thm:char-pt}). We conclude that all non-truncated separatrices are $C^1$-smooth curves that join singular points of $v$.
\end{remark}
\begin{definition}
\label{def-LMF}
\emph{LMF graph} of a vector field $v\in Vect^* \, S^2$ is a graph $LMF(v)$ embedded in $S^2$ which consists of the following elements:
\begin{itemize}
\item Vertices:
\begin{enumerate}
\item All singular points of $v$;
\item All \emph{truncation vertices}: cross-points of separatrices of $v$ with transversal loops chosen above;
\item A point on each cycle;
\item A point on each \emph{empty} transversal loop, i.e. on the transversal loop that does not cross separatrices of $v$.
\end{enumerate}
\item Edges:
\begin{enumerate}
\item Unstable (stable) separatrices of singular points, if their $\omega$- (resp., $\alpha$-)limit sets are characteristic points.
\item Truncated unstable (stable) separatrices of singular points, if their $\omega$- (resp., $\alpha$-) limit sets are not characteristic points.
\item Limit cycles (this edge starts and ends at the vertex of type 3).
\item Pieces of transversal loops between subsequent truncation vertices, or the whole empty transversal loops.
\item One homoclinic trajectory of $v$ in each elliptic sector of a non-hyperbolic singular point.
\end{enumerate}
\end{itemize}
\textbf{Orientation}
The orientation of edges of types 1, 2, 3, 5 is induced by time parametrization. The orientation of edges of type 4 is counterclockwise with respect to the $\alpha$- or $\omega$-limit set corresponding to the transversal loop, as mentioned above.
\textbf{Labeling}
LMF graph is considered together with the following labels.
Each vertex is labeled by the description of its type, namely the labels say \emph{Singular Point (SP), Truncation Vertex (TV), Vertex on a Limit Cycle (VLC), Vertex on an Empty Transversal Loop (VETL)}. Similarly, the labels on the edges say \emph{Stable Separatrix (SS), Unstable Separatrix (US), Separatrix Connection (SC), Stable Truncated Separatrix (STS), Unstable Truncated Separatrix (UTS), Limit Cycle (LC), Outgoing Transversal Loop (OTL), Ingoing Transversal Loop (ITL), Trajectory in the Elliptic Sector (TES)}. We say that a transversal loop is \emph{ingoing} if this is a loop around its $\omega$-limit set; otherwise we say that the transversal loop is \emph{outgoing}.
\end{definition}
Fig. \ref{fig-Mayer} shows the part of the phase portrait of a vector field and the corresponding part of the LMF graph. We used abbreviations of the labels described above.
The relation of the $LMF$ graphs with separatrix skeletons is the following.
The edges of the $LMF$ graph except transversal loops and loops in elliptic sectors belong to the extended separatrix skeleton, and their orientation is induced by the time parametrization. The face of an LMF graph may be:
\begin{itemize}
\item a canonical region of $v$, possibly truncated by transversal loops of its $\alpha$- and $\omega$-limit sets, which depends on types of these $\alpha$-, $\omega$-limit sets. It will be possibly cut by a loop in an elliptic sector;
\item a petal in an elliptic sector;
\item an annulus between an $\alpha$- or $\omega$-limit set of $v$ and its transversal loop.
\end{itemize}
The orbits in canonical regions that are included to the completed separatrix skeleton keep the same information as labeling.
\begin{figure}[h]
\begin{center}
\includegraphics[width=\textwidth]{Mayer}
\end{center}
\caption{A phase portrait of a vector field and its LMF graph. For the meaning of labels, see Definition \ref{def-LMF}.}\label{fig-Mayer}
\end{figure}
The classification of canonical regions (Proposition \ref{prop-parall}) yields the following classification of faces of $LMF$ graphs:
\begin{lemma}[Faces of the LMF graph]
\label{lem-faces}
Each open face $F$ of the LMF graph of a smooth vector field $v\in Vect^* \, S^2$ is either a topological disc, or a topological annulus. In the second case, the following cases are possible:
\begin{itemize}
\item $F$ is a domain between an $\alpha$- or $\omega$- limit set (sink or source, cycle, or polycycle) of $v$ and its transversal loop;
\item $F$ is a domain between two transversal loops (of sinks, sources, cycles or polycycles) of $v$.
\end{itemize}
\end{lemma}
\begin{proof}
This follows from Proposition \ref{prop-parall} above. Indeed, some faces of the LMF graph of $v$ are annuli between $\alpha$- or $\omega$- limit sets of $v$ and their transversal loops. To obtain all other faces of the LMF graph, we can take all canonical regions of $v$ and truncate them by transversal loops mentioned above: we cut off pieces of canonical regions that are between $\alpha$- or $\omega$- limit sets of $v$ and their transversal loops, and possibly cut along loops in elliptic sectors.
If a canonical region carries a strip flow, its $\alpha$- and $\omega$-limit sets may be surrounded or not surrounded by transversal loops; these loops will intersect the canonical region in topological intervals transversal to the flow. If this canonical region contains a loop in elliptic sector, this loop intersects a canonical region in a topological interval along the flow. In any case, after truncation and cutting, this canonical region will produce face(s) of $LMF(v)$ topologically equivalent to disc(s).
For a canonical region carrying a spiral flow, its $\alpha$- and $\omega$-limit sets are necessarily surrounded by transversal loops. This canonical region after truncation will become an annular face of $LMF(v)$ between two transversal loops of $\alpha$- and $\omega$-limit sets.
\end{proof}
We use the following result of R. Fedorov \cite{Fed}, based on the previous result of Andronov, Leontovich, Gordon, Mayer \cite{ALGM}. This result is close to Theorem \ref{th-MNP}.
\begin{theorem}[R. Fedorov, \cite{Fed}]
\label{th-Fedorov}
If two LMF graphs $\Gamma_1 = LMF(v), \Gamma_2=LMF(w)$ of two vector fields $v,w$ are isotopic on the sphere (i.e. there exists an orientation-preserving homeomorphism of the sphere which maps one to another, preserves orientation on edges and matches labels on edges and vertices), then $v$ and $w$ are orbitally topologically equivalent.
\end{theorem}
\begin{remark}
Theorem \ref{th-Fedorov} in \cite{Fed} was proved for a slightly different construction of the graph. Below we list the differences.
\begin{itemize}
\item We label transversal loops as ingoing or outgoing, while Fedorov puts these labels on singular points and each side of cycles or polycycles themselves.
\item Fedorov does not add transversal loops around singular points if they are characteristic attractors or repellers.
\item Fedorov does not add empty transversal loops.
\item We do not describe a labeling for center-type vertices (as Fedorov does) because they do not appear in $Vect^* \, S^2$, due to its definition.
\end{itemize}
These differences do not affect Theorem \ref{th-Fedorov}. Indeed, let two vector fields $v_1,v_2$ have isotopic LMF graphs. Recall that all attractors (both characteristic and non-characteristic) are locally topologically equivalent, and the same holds for repellers. So we may and will assume that all attractors and repellers of $v_1,v_2$ are characteristic. Prove that $v_1$ and $v_2$ have isotopic Fedorov's graphs.
Indeed, looking at the graph (with its embedding into $S^2$), one can determine which transversal loop corresponds to which $\alpha$- or $\omega$-limit set, and put labels (\emph{attracting, repelling}) on each side of these sets as in Fedorov's graph. Further, we erase empty transversal loops from both graphs (the only information they bear is labeling). Finally, we remove transversal loops around all attractors and repellers and let truncated separatrices that terminated at these loops enter the singular points themselves. Since the LMF graphs were isotopic, the resulting Fedorov's graphs will be isotopic.
This reduces Theorem \ref{th-Fedorov} for Fedorov's graphs to Theorem \ref{th-Fedorov} for LMF graphs described above.
\end{remark}
The proof of Main Theorem will consist of proving the isotopy of LMF graphs of $v_{\varepsilon}$, $w_{h(\varepsilon)}$ for small $\varepsilon$.
\subsection{Isotopy of graphs on $S^2$}
We use the following theorem from graph theory (see \cite[Theorem 2]{Graph} for the more general result).
\begin{theorem}
\label{th-graphs}
Suppose that two oriented connected planar graphs $\Gamma_1, \Gamma_2$ are embedded in $S^2$ by maps $\phi_1 \colon \Gamma_1 \to S^2, \phi_2 \colon \Gamma_2 \to S^2$. Choose an orientation in $S^2$.
Suppose that $g\colon\Gamma_1\to \Gamma_2 $ is an isomorphism of oriented graphs $\Gamma_1, \Gamma_2$, and suppose that the graph isomorphism $g$ preserves a counterclockwise order of edges at each vertex (induced by the immersions $\phi_1, \phi_2$).
Then the map $\phi_2\circ g \circ \phi_1^{-1}$ can be extended to the orientation-preserving homeomorphism of $S^2$, in particular $\phi_1(\Gamma_1)$ is isotopic to $\phi_2(\Gamma_2)$.
\end{theorem}
The idea of the proof of this theorem is to establish the correspondence of faces of $\Gamma_1, \Gamma_2$ using the information on the order of edges in each vertex, and to define a sphere homeomorphism inside each face.
LMF graphs are usually not connected; some of their faces can be annuli, see Lemma \ref{lem-faces} above. We will use the following theorem.
\begin{theorem}
\label{th-graphs-components}
Suppose that two oriented planar graphs $\Gamma_1, \Gamma_2$ (not neccessarily connected) are embedded in $S^2$ by maps $\phi_1 \colon \Gamma_1 \to S^2, \phi_2 \colon \Gamma_2 \to S^2$, and their (open) faces in $S^2$ are topological discs or annuli. Choose an orientation in $S^2$.
Suppose that these graphs are isomorphic as oriented graphs. Suppose that the graph isomorphism $g$ preserves a counterclockwise order of edges at each vertex (induced by the immersions $\phi_1, \phi_2$). Suppose that the map $\phi_2\circ g \circ \phi_1^{-1}$ extends to an orientation-preserving homeomorphism of the annuli-shaped faces.
Then the the map $\phi_2\circ g \circ \phi_1^{-1}$ can be extended to the orientation-preserving homeomorphism of $S^2$, so $\phi_1(\Gamma_1)$ is isotopic to $\phi_2(\Gamma_2)$.
\end{theorem}
\begin{proof}
The idea of the proof is to add edges through all annuli-shaped faces of our graph, so that the extended graph is connected, and then use Theorem \ref{th-graphs}. Formally, for each annuli-shaped face we do the following.
Let $A_1 \subset S^2$ be an annuli-shaped open face of $\phi_1(\Gamma_1)$, and let $G$ be the homeomorphism that extends $\phi_2\circ g \circ \phi_1^{-1}$ to $A_1$. Then $A_2:=G(A_1)$ is an open face of $\phi_2(\Gamma_2)$.
Let $\phi_1(V_1),\phi_1(V),\phi_1(V_2) \in S^2$ be three subsequent vertices on one of the two boundary components of $A_1$ (the orientation on $\partial A_1$ is induced by the orientation on $A_1\subset S^2$). Let $\phi_1(W_1),\phi_1(W),\phi_1(W_2)$ be three subsequent vertices on another boundary component of $A_1$. If one of boundary components contains only two vertices, we put $V_1=V_2$; if it contains only one vertex, we put $V_1=V=V_2$.
Take a continuous curve $\gamma \subset A_1$ joining $\phi_1(V)$ to $\phi_1(W)$. To the graph $\Gamma_1$, add the edge joining $V$ to $W$. Extend $\phi_1$ so that $\phi_1([VW])=\gamma$.
Take a curve $G(\gamma) \subset A_2$ joining $G(\phi_1(V))$ to $G(\phi_1(W))$. Similarly, to the graph $\Gamma_2$, add the edge joining $\phi_2^{-1}(G(\phi_1(V))) = g(V)$ to $\phi_2^{-1}(G(\phi_1(W)))= g(W)$. Extend $\phi_2$ so that it takes this edge to the curve $G(\gamma)$.
Finally, extend $g$ to the graph isomorphism of enlarged graphs, by putting $g([VW]) = [g(V), g(W)])$.
A counterclockwise order of edges at $V$ contained a part $[VV_2], [VV_1]$; now this part changed to $[VV_2], [VW],[VV_1]$. A counterclockwise order of edges at $g(V)$ contained a part $[g(V)g(V_2)], [g(V)g(V_1)]$; now this part changed to $[g(V)g(V_2)], [g(V) g(W)], [g(V)g(V_1)]$. So $g$ still preserves a counterclockwise order of edges at $V$; similarly, it preserves the order at $W$.
We repeat this process for each annuli-shaped face. Finally, we get \emph{connected} graphs $\tilde \Gamma_1, \tilde \Gamma_2$, because the number of their connected components decreases after each step of extension. These new graphs satisfy the assumptions of Theorem \ref{th-graphs}.
So the initial map $\phi_2\circ g \circ \phi_1^{-1}$ (as well as the extended one) can be extended to the homeomorphism of the sphere.
\end{proof}
\subsection{Idea of the proof of the Main Theorem}
We are going to prove that under assumptions of Main Theorem, for small $\varepsilon$, two vector fields $v_{\varepsilon}$ and $w_{h(\varepsilon)}$ are topologically equivalent. Due to the definition of moderate topological equivalence, we are given the map $\mathbf H = (h,H_{\varepsilon})$, $h \colon B \to B'$, such that $H_{\varepsilon} \colon U \to H_{\varepsilon}(U)$ conjugates $v_{\varepsilon}$ to $w_{h(\varepsilon)}$ in neighborhoods of large bifurcation supports. We are also given a map $\hat H\colon S^2 \to S^2$ that conjugated $v_0$ to $w_0$ on the whole sphere.
We will not directly extend $H_{\varepsilon}$ to the whole sphere. We will rather prove that two graphs $LMF(v_{\varepsilon})$ and $LMF(w_{h(\varepsilon)})$ are isomorphic for small $\varepsilon$. Then we refer to Theorem \ref{th-graphs-components} together with Theorem \ref{th-Fedorov} and conclude that for small $\varepsilon$, there exists a homeomorphism $\hat H_{\varepsilon} \colon S^2 \to S^2$ that conjugates $v_{\varepsilon}$ to $w_{h(\varepsilon)}$. The family of maps $\mathbf{\hat H} = (h,\hat H_{\varepsilon})$ is a weak topological equivalence of the families $\{v_{\varepsilon}\}$ and $\{w_{h(\varepsilon)}\}$ as required.
Note that we do not guarantee that $(\hat H_{\varepsilon})|_{U} = H_{\varepsilon}$.
The following theorem will imply the Main Theorem:
\begin{theorem}
Under the assumptions of Main Theorem, for sufficiently small $\varepsilon$, the graphs $LMF(v_{\varepsilon})$ and $LMF(w_{h(\varepsilon)})$ are isomorphic as oriented graphs, and the isomorphism $G_{\varepsilon}$ meets the conditions of Theorem \ref{th-graphs-components}.
\end{theorem}
We will construct the isomorphism $G_{\varepsilon}$ on the LMF-graphs as subsets of $S^2$.
Roughly speaking, in order to define $G_{\varepsilon}$, we use $H_{\varepsilon}$ whenever it is defined, i.e. inside a neighborhood $U$ of the large bifurcation support of $v_{\varepsilon}$.
Outside $U$, all singular points and cycles of $v_0$ are hyperbolic, thus $v_{\varepsilon}$ has close singular points and cycles. When we define $G_{\varepsilon}$ on singular points and cycles outside $U$, we use $\hat H$ plus continuation of hyperbolic singular points and cycles with respect to the parameter. The edges of $LMF(V)$ that are partly inside $U$ and partly outside it will be one of our main concerns.
\section{Properties of large bifurcation supports and moderate topological equivalence}
\subsection{Large bifurcation support}
\label{sec-prop-LBS}
In this section, we list the fundamental properties of the set $LBS(V)$ described above. These are the only properties we are going to use in the proofs.
\begin{enumerate}
\item $LBS(V)$ is a closed $v_0$-invariant set (Proposition \ref{prop-LBS-closed-vinv}).
\item Hyperbolic singular points and hyperbolic limit cycles of $v_0$ do not belong to $LBS(V)$. All non-hyperbolic singular points and non-hyperbolic cycles of $v_0$ belong to $LBS(V)$ (Proposition \ref{prop-nohyp-inLBS}).
\item Non-interesting non-hyperbolic cycles of $v_0$ are connected components of $LBS(V)$ (Remark \ref{rem-nohyp-inLBS}).
\item Sep-property (Proposition \ref{prop-seps-v0}) and Separatrix lemma (Lemma \ref{lem-sep-connect}).
\item No-entrance property (Lemma \ref{lem-seps}).
\item No cycles of mixed location (Proposition \ref{prop-inU-or-hyp}).
\item Moderate topological equivalence of two families $V, W$ implies continuity of conjugacy on $\{\varepsilon=0\}\times \partial LBS(V)$, $\{\varepsilon=0\}\times \partial LBS(W)$ (Proposition \ref{prop-dLBS})
\end{enumerate}
Now we pass to the exact statements.
\subsubsection{$LBS(V)$ is closed and invariant}
\begin{proposition}
\label{prop-LBS-closed-vinv}
If $V\subset Vect^*\,S^2$, then both $ELBS(v_0)$ and $LBS(V)$ are closed $v_0$-invariant sets.
\end{proposition}
\begin{proof}
The set of points with interesting $\alpha$- and $\omega$-limit sets under $v_0$ is $v_0$-invariant. Thus its closure is closed and $v_0$-invariant. So $ELBS(v_0)$ is closed and $v_0$-invariant.
The set $(\mathop{\mathrm{Sing}} v_0\cup (\overline {\mathop{\mathrm{Per}} V} \cup \overline{\mathop{\mathrm{Sep}}\, V}) \cap \{\varepsilon=0\})$ is closed and $v_0$-invariant. The set $LBS(V)$ is closed and $v_0$-invariant as the intersection of two closed and $v_0$-invariant sets.
\end{proof}
Though the topology of $LBS(V)$ may be complicated, it has finitely many connected components due to the following proposition.
\begin{proposition}
\label{lem-LBS-finite}
If $v\in Vect^*\,S^2$, then each closed $v$-invariant set $A\subset S^2$ has finitely many connected components.
\end{proposition}
In particular, this holds for $A=LBS(V)$.
\begin{proof}
If $A$ is closed and $v$-invariant, then each its connected component is closed and $v$-invariant.
Hence each connected component of $A$ contains trajectories of $v$ together with their $\omega$- and $\alpha$-limit sets. Due to Poincare-Bendixson theorem, each $\alpha$- and $\omega$-limit set contains either a singular point of $v$, or a cycle of $v$. However, each vector field $v\in Vect^* \, S^2$ has finitely many singular points and cycles. Each connected component of $A$ contains at least one of them. Thus the number of connected components is finite.
\end{proof}
\subsubsection{$\alpha$- and $\omega$-limit sets in $LBS(V)$}
The next proposition follows immediately from the definition of $LBS(V)$.
\begin{proposition}
\label{prop-nohyp-inLBS}
Large bifurcation support $LBS(V)$ does not contain hyperbolic attractors, hyperbolic repellers, or hyperbolic cycles of $v_0$. It contains all non-hyperbolic singular points, non-hyperbolic cycles, and all separatrix connections of $v_0$.
\end{proposition}
Note that due to Poincare-Bendixson theorem, $\alpha$- and $\omega$-limit sets are singular points, limit cycles, and monodromic polycycles. Since monodromic polycycles are formed by separatrix connections, they belong to $LBS(V)$. This implies the following remark.
\begin{remark}
\label{rem-nohyp-inLBS}
All interesting $\alpha$-, $\omega$-limit sets of $v_0$ except some saddles belong to $LBS(V)$. All non-interesting $\alpha$-, $\omega$-limit sets of $v_0$ except non-hyperbolic non-interesting cycles belong to its complement.
\end{remark}
We will also need the following proposition.
\begin{proposition}
\label{prop-nonint-compon}
For families $V$ with $v_0\in Vect^* \, S^2$, non-hyperbolic non-interesting cycles are connected components of $LBS(V)$.
\end{proposition}
\begin{proof}
A neighborhood of a cycle is filled by points whose semi-trajectories (in positive or negative time) wind around this cycle. However a point whose semi-trajectory winds around a non-interesting cycle does not belong to $LBS(V)$, due to the definition of $ELBS(v_0)$. So the intersection of $LBS(V)$ with a neighborhood of a non-interesting cycle is this cycle only.
\end{proof}
This motivates the following definition.
\begin{definition}
Denote $LBS^*(V) = LBS(V) \setminus \{\text{non-interesting cycles of }v_0\}$.
\end{definition}
The set $LBS^*(V)$ is closed and $v_0$-invariant, due to the previous proposition.
\subsubsection{Sep-property}
\begin{proposition}
\label{prop-seps-v0}
Suppose that for an unstable separatrix $\gamma$ of $v_0$, $\omega_{v_0}(\gamma)$ intersects $LBS^*(V)$; equivalently, $\gamma$ hits arbitrarily small neighborhood of $LBS^*(V)$. Then $\gamma\subset LBS(V)$.
The same statement holds for stable separatrices and $\alpha$-limit sets.
\end{proposition}
\begin{proof}
Let $\gamma$ be the separatrix mentioned in the lemma. Suppose that it is unstable; the case of stable separatrices is treated in the same way. By the Poincare-Bendixson theorem, the set $\omega_{v_0}(\gamma)$ may be a singular point, a limit cycle or a polycycle.
Prove that $\omega_{v_0}(\gamma)$ is interesting. Indeed, all polycycles are interesting limit sets, and all singular points and limit cycles in $LBS^*(V)$ are also interesting due to Remark \ref{rem-nohyp-inLBS}.
Since $\gamma$ is an unstable separatrix, $\alpha_{v_0}(\gamma)$ is a saddle; so it is interesting by definition.
We conclude that both $\alpha_{v_0}(\gamma)$ and $\omega_{v_0}(\gamma)$ are interesting. Hence $\gamma \subset ELBS(v_0) $. Since $\gamma \subset \mathop{\mathrm{Sep}} (v_0)$, it belongs to $LBS(V)$. \end{proof}
\subsubsection{Separatrix lemma}
Recall that the upper topological limit $\overline \lim A_k$ of a sequence of sets $A_k$ in a topological space is a set of points $x$ such that any neighborhood of $x$ intersects infinitely many of $A_k$; in other words, this is the set of all limit points of the sequence $A_k$.
\begin{lemma}[Separatrix lemma]
\label{lem-sep-connect}
Let $\gamma_k$ be separatrices of vector fields $v_{\varepsilon_k}$ that connect two interesting singular points, and $\varepsilon_k\to 0$. Then
$\overline \lim \gamma_k \subset LBS^*(V)$. The same holds for stable separatrices.
In particular, all separatrix connections of $v_{\varepsilon}$ for small $\varepsilon$ are close to $LBS^*(V)$.
\end{lemma}
\begin{proof}
Let $x$ belong to $\overline \lim \gamma_k$. Prove that $x\in LBS^*(V)$. Passing to a subsequence, we may and will assume that $x=\lim_{k\to \infty} x_k$ where $x_k \in \gamma_{{k}}$.
Consider three cases:
\textbf{Case 1.} $x\in \mathop{\mathrm{Per}}(v_0)$.
Assume that $x$ is not in $LBS^*(V)$; then it belongs to a non-interesting (parabolic or hyperbolic) cycle. In both cases, it is easy to see that either $\alpha$- or $\omega$-limit set of a close point $x_k$ under a close vector field $v_{\varepsilon_k}$ is either a cycle that bifurcates from a non-interesting nest, or a non-interesting sink/source inside the nest. Both cases are impossible for separatrices $\gamma_{k}\ni x_k$, and the contradiction shows that $x\in LBS^*(V)$.
\textbf{Case 2.} $x \notin (\mathop{\mathrm{Sing}}(v_0) \cup \mathop{\mathrm{Per}}(v_0)$).
Suppose that $x$ has a non-interesting $\alpha$-limit set under $v_0$. Then a close point $x_k$ has a non-interesting $\alpha$-limit set under a close vector field $v_{\varepsilon_k}$, which is impossible for separatrices $\gamma_k\ni x_k$. The contradiction shows that $\alpha_{v_0}(x)$ is interesting; similarly, $\omega_{v_0}(x)$ is interesting. Since $x\notin \mathop{\mathrm{Sing}} (v_0)$, we conclude that $x\in ELBS(v_0)$. Since $x$ is a limit point of separatrices, $x\in \overline {\mathop{\mathrm{Sep}} V}$, so we have $x\in LBS(V)$. Since $x$ does not belong to a limit cycle, $x\in LBS^*(V)$.
\textbf{ Case 3.} $x \in \mathop{\mathrm{Sing}}(v_0)$.
The set $\overline \lim \gamma_k$ is connected as a limit of connected sets. If it coincides with $x$, then separatrices $\gamma_k$ of $v_{\varepsilon_k}$ collapse to $x$, thus $x$ is non-hyperbolic; hence $x\in LBS^*(V)$. If $\overline \lim \gamma_k$ does not coincide with $x$, then arbitrarily close to $x$, there are non-singular limit points of $\gamma_k$. They all belong to $LBS^*(V)$ due to the previous case. Hence $x$ belongs to $LBS^*(V)$, because $LBS^*(V)$ is closed.
\end{proof}
\subsubsection{No-entrance lemma}
\begin{definition}
\label{def-entering}
For a vector field $v$, a separatrix $\gamma$ of a singular point $P$ \emph{does not enter} an open set $\Omega\subset S^2$ if one of the following holds:
\begin{itemize}
\item $\gamma$ does not intersect $\partial \Omega$;
\item $P \in \Omega$ and the cross-point $\gamma\cap \partial \Omega$ is unique.
\end{itemize}
\end{definition}
\begin{lemma}[No-entrance lemma]
\label{lem-seps}
In assumptions of the Main Theorem, there exists an arbitrarily small neighborhood $U^*$ of $LBS^*(V)$ such that for sufficiently small $\varepsilon$, no separatrices of $v_{\varepsilon}$ enter $U^*$.
\end{lemma}
The proof is postponed till Sec. \ref{ssec-lem-seps}. The statement does \emph{not} hold true for any sufficiently small neighborhood of $LBS^*(V)$. To use this statement, in Sec. \ref{sec-choice-U} we will have to restrict ourselves to special neighborhoods $U$ of $LBS(V)$ instead of all sufficiently small neighborhoods, even though we have moderate equivalence for all small neighborhoods of $LBS(V)$ (see Remark \ref{rem-shrink-U}).
\subsubsection{No cycles of mixed location}
Each limit cycle of $v_\varepsilon$ either lies in a neighborhood of $LBS(V)$ or completely outside it. In more detail, we have the following proposition (it also treats singular points, which is analogous but simpler).
\begin{proposition}
\label{prop-inU-or-hyp}
For any smooth local family $V \subset Vect^*\, S^2$ of vector fields and any small neighborhood $U$ of $LBS(V)$, for sufficiently small $\varepsilon$, each singular point of $v_{\varepsilon}$ is either inside $U$, or belongs to a continuous family $P_{\varepsilon}, \varepsilon \in (B, 0)$, of hyperbolic singular points of $v_{\varepsilon}$ such that $P_0 \notin LBS(V)$.
Each limit cycle of $v_{\varepsilon}$ is either inside $U$, or belongs to a continuous family $c_{\varepsilon}, \varepsilon \in (B,0)$, of hyperbolic limit cycles of $v_{\varepsilon}$ such that $c_0$ does not belong to $LBS(V)$.
\end{proposition}
\begin{proof}
Any singular point $P$ of $v_{\varepsilon}$, $\varepsilon$ small, is close to some singular point $P_0$ of $v_0$. If $P_0\in LBS(V)$, then $P \in U$. If $P_0\notin LBS(V)$, then $P_0$ is hyperbolic (see Proposition \ref{prop-nohyp-inLBS}), so locally structurally stable. Thus $P$ belongs to a continuous family of singular points of $v_{\varepsilon}$ as required.
The proof for limit cycles is a bit more complicated. The set $\overline {\mathop{\mathrm{Per}} V} \cap \{\varepsilon=0\}$ (a \emph{limit periodic set}) is described by \cite[Theorem 5, Section 2.1.2]{Rous}. This theorem claims that the limit cycles of $v_{\varepsilon}$ as $\varepsilon\to 0$ may accumulate to:
\begin{itemize}
\item a hyperbolic limit cycle of $v_0$;
\item a non-hyperbolic limit cycle of $v_0$;
\item a non-hyperbolic singular point of $v_0$;
\item a \emph{polycycle} of $v_0$, namely a finite union of trajectories $\varphi_i$ and singular points $P_i$ of $v_0$, $i=1,\dots,n$ (some of these points may coincide), such that $\alpha(\varphi_i)=P_i, \omega(\varphi_i)=P_{i+1}$, and $\omega(\varphi_n) = P_1$.
\end{itemize}
Note that the polycycle in the last case may be non-monodromic.
Clearly, the proposition holds in the first case. In the second and the third cases, it holds true as well: non-hyperbolic singular points and cycles of $v_0$ belong to $LBS(V)$, so the corresponding limit cycles of $v_{\varepsilon}$ belong to $U$ for sufficiently small $\varepsilon$.
Prove that any polycycle of $v_0$ belongs to $LBS(V)$. Indeed, the points $P_i$ are neither sinks nor sources, because some orbits enter $P_i$ and some quit. Hence $P_i$ are interesting limit sets. Thus $\varphi_i$ has interesting $\alpha$-, $\omega$-limit sets, so $\overline \varphi_i \subset ELBS(v_0)$. But $\overline \varphi_i\subset \overline {\mathop{\mathrm{Per}} V}$, so $\overline \varphi_i \subset LBS(V)$. This completes the proof.
\end{proof}
\subsubsection{Relation to moderate equivalence}
\begin{proposition}
\label{prop-dLBS}
Let two local families $V$ and $W$ be moderately equivalent in some neighborhood of $LBS(V)$, $LBS(W)$ in the sense of Definition \ref{def-moderate-local}. Then the corresponding maps $\mathbf H$ and $\mathbf H^{-1}$ are continuous in $\varepsilon,x$ on the sets $\{\varepsilon=0\}\times \partial LBS(V)$ and $\{\varepsilon=0\}\times \partial LBS(W)$ respectively.
\end{proposition}
\begin{remark}
In the proof of Main Theorem, we will only use the continuity of $\mathbf H$, $\mathbf H^{-1}$ on the above sets and on $ \{\varepsilon=0\}\times \mathop{\mathrm{Sep}} (v_0|_{U})$, $ \{\varepsilon=0\}\times \mathop{\mathrm{Sep}} (w_0|_{\hat H(U)})$.
\end{remark}
\begin{proof}
Since the boundary of intersection and the boundary of union belong to the union of boundaries,
$$\partial LBS(V) \, \subset \, \partial ELBS(v_0)\, \bigcup \, \partial \mathop{\mathrm{Sing}} v_0 \cup \partial ((\overline{\mathop{\mathrm{Per}}\, V} \cup \overline{\mathop{\mathrm{Sep}}\, V}) \cap \{\varepsilon=0\}).$$
Note that $\partial \mathop{\mathrm{Sing}} v_0 = \mathop{\mathrm{Sing}} v_0$. Due to Proposition \ref{prop-dELBS}, $\partial ELBS(v_0)\subset S(v_0)$. Thus $$\partial LBS(V)\, \subset\, S(v_0) \cup \partial ((\overline {\mathop{\mathrm{Per}} V} \cup \overline {\mathop{\mathrm{Sep}} V}) \cap \{\varepsilon=0\})$$
which is the set \eqref{eq-set1} from the definition of moderate equivalence. The same arguments apply to $W$.
This completes the proof.
\end{proof}
\subsection{Moderate topological equivalence}
This section contains simple topological statements that follow from the definition of moderate equivalence.
The following proposition enables us to work in small neighborhoods of $LBS(V)$ and $LBS(W)$.
\begin{proposition}
\label{prop-heps-detached}
Under assumptions of the Main Theorem, for each neighborhood $\tilde U^+$ of $LBS(W)$, there exists a small open neighborhood $U$ of $LBS(V)$, such that $H_{\varepsilon}(U) \subset \tilde U^+$ for all small $\varepsilon$.
\end{proposition}
Since hyperbolic sinks, sources and cycles of $w_0$ are outside $LBS(W)$ (Proposition \ref{prop-nohyp-inLBS}), this Proposition immediately implies the corollary:
\begin{corollary}
\label{cor-heps-detached}
For small $U\supset LBS(V) $ and any small $\varepsilon$, the set $H_{\varepsilon}(U)$ is detached from all hyperbolic attracting and repelling singular points and cycles of $w_0$.
\end{corollary}
\begin{proof}[Proof of Proposition \ref{prop-heps-detached}]
Suppose that for some neighborhood $\tilde U^+ \supset LBS(W)$, the statement does not hold true. So there exists a sequence of shrinking open neighborhoods $U_n$: $\cap U_n =LBS(V)$, and a sequence $\varepsilon_n \to 0$, such that none of the domains $H_{\varepsilon_n}(U_n)$ are contained in $\tilde U^+$. Then there exists a sequence $x_n \in U_n$ such that $H_{\varepsilon_n}(x_n)\notin \tilde U^+$.
Due to Requirement \ref{it-nbhd-cond} of Definition \ref{def-moderate-local} of moderate equivalence (applied to $\mathbf H^{-1}$), the set $H_{\varepsilon_n}^{-1}(\tilde U^+)$ for small $\varepsilon_n$ is a neighborhood of $LBS(V)$, in particular, $LBS(V) \subset H_{\varepsilon_n}^{-1}(\tilde U^+)$. So $H_{\varepsilon_n}(LBS(V))\subset \tilde U^+ $, hence $x_n \notin LBS(V)$.
Since sphere is compact, we can choose a convergent subsequence $x_n'$ of $x_n$.
Since $U_n$ shrink to the closed set $LBS(V)$, this subsequence converges to a point of $\partial LBS(V)$. This contradicts the continuity of the map $\mathbf H$ on $\{\varepsilon=0\}\times \partial LBS(V)$ (see Proposition \ref{prop-dLBS}).
\end{proof}
The next proposition shows that under assumptions of the Main theorem, the conjugacy of families $V$ and $W$ respects connected components of $LBS(V)$, $LBS(W)$ and their neighborhoods.
\begin{proposition}
\label{prop-U12-intersect}
In assumptions of Main Theorem, let $U, \tilde U^+$ be the same as in Proposition \ref{prop-heps-detached}. Suppose that each connected component of $\tilde U^+$ contains one, and only one, connected component of $LBS(W)$. Suppose that each connected component of $U$ contains a connected component of $LBS(V)$. Then for sufficiently small $U$, for small $\varepsilon_1,\varepsilon_2$ and for two different connected components $U_1, U_2$ of $U$, their images $H_{\varepsilon_1}(U_1)$ and $H_{\varepsilon_2}(U_2)$ do not intersect and belong to different connected components of $\tilde U^+$.
\end{proposition}
\begin{proof}
Recall that $H_{\varepsilon}(U)\subset \tilde U^+$ for all small $\varepsilon$ as in Proposition \ref{prop-heps-detached}.
Let $C_i$ be a connected component of $LBS(V)$ that belongs to $U_i$. By assumption, connected components $\hat H(C_1)$, $\hat H(C_2)$ of $LBS(W)$ are in different connected components of $\tilde U^+$; let $\tilde U_1, \tilde U_2$ be these connected components of $\tilde U^+$.
Since $\hat H(C_1)\subset \tilde U_1$, the whole component $\hat H(U_1)$ belongs to $\tilde U_1$. Due to Proposition \ref{prop-dLBS}, $\mathbf H$ is continuous on $\partial C_1$. So for small $\varepsilon_1$, $H_{\varepsilon_1}(\partial C)$ is close to $H_0(\partial C) = \hat H(\partial C)\subset \tilde U_1$. Hence $H_{\varepsilon_1}(U_1)\supset H_{\varepsilon_1}(\partial C_1)$ intersects $\hat H(U_1)$; we conclude that $H_\varepsilon(U_1)\subset \tilde U_1$ for all small $\varepsilon$. Similarly, $H_{\varepsilon}(U_2) \subset \tilde U_2$ for all small $\varepsilon$, which finishes the proof.
\end{proof}
\section{Combinatorial equivalence of the LMF graphs: first steps of the construction}
\subsection{General approach}
Recall that a neighborhood $U$ of $LBS(V)$ is fixed in the statement of the Main Theorem and does not depend on $\varepsilon$. However we will have to shrink $U$ further in the proof; this is possible due to Remark \ref{rem-shrink-U}.
Our goal is to define an isomorphism
$$G_{\varepsilon} \colon LMF(v_{\varepsilon})\to LMF(w_{h(\varepsilon)})$$
of the $LMF$ graphs of $v_\varepsilon $ and $w_{h(\varepsilon )}$, and to prove that it meets the conditions of Theorem \ref{th-graphs-components}. For each element $c$ (i.e. vertex or edge) of $LMF(v_\varepsilon)$ that belongs to $U$ we will define
$$
G_\varepsilon |_c := H_\varepsilon |_c.
$$
It turns out that the elements of $LMF(v_\varepsilon)$ that lie outside $U$ depend continuously on $\varepsilon$ (say, all singular points outside $U$ are hyperbolic, thus structurally stable).
When we define $G_{\varepsilon}$ on these parts of $LMF(v_{\varepsilon})$, we use $\hat H$ plus continuous continuation in $\varepsilon$ of hyperbolic singular points and cycles.
For a smooth local family $V\subset Vect^*\,S^2$ of vector fields, each hyperbolic singular point $P_0$, each hyperbolic limit cycle $c_0$ and each germ of a separatrix of a hyperbolic saddle $(\gamma_0, P_0)$ of a vector field $v_0$ generates a continuous family of hyperbolic points, cycles and germs of separatrices $P_{\varepsilon}, c_{\varepsilon}, (\gamma_{\varepsilon}, P_{\varepsilon})$ respectively of $v_{\varepsilon}$. This enables us to give the following definition.
\begin{definition}[Notation]
\label{def-pi}
In the above assumptions, let $\pi_{\varepsilon}$ be a continuous map that depends continuously on $\varepsilon$, $\varepsilon$ small, and takes homeomorphically $P_{\varepsilon}$ to $P_0$, $c_{\varepsilon}$ to $c_0$ and $(\gamma_{\varepsilon}, P_{\varepsilon})$ to $(\gamma_{0}, P_0)$, preserving time orientation.
In assumptions of Main Theorem, let $\tilde \pi_{\varepsilon}$ be the analogous map for the family $W$ playing the role of $V$.
\end{definition}
\subsection{Partial definition of $G_{\varepsilon}$}
\label{sec-partial-Ge}
\subsubsection{Singular points, limit cycles, and germs of separatrices}
\label{ssec-Ge-skeleton}
Denote by $\mathop{\mathrm{Sep}}^*v_{\varepsilon}$ the set of germs of separatrices of $v_\varepsilon $ at singular points of $v_{\varepsilon}$. Note that a separatrix connection of $v_{\varepsilon}$ corresponds to two germs in $\mathop{\mathrm{Sep}}^* v_{\varepsilon}$. Let us define $G_\varepsilon $ on $\mathop{\mathrm{Sing}} v_{\varepsilon} \cup \mathop{\mathrm{Per}} v_{\varepsilon} \cup \mathop{\mathrm{Sep}}^*v_{\varepsilon} =: S^*(v_{\varepsilon})$.
Let $p\in S^*(v_{\varepsilon})$. If $p \notin U$, then $p$ is a hyperbolic singular point, or belongs to a cycle, or belongs to a germ of a separatrix of a hyperbolic saddle. We define
$$
G_\varepsilon (p) := {\tilde \pi_{h(\varepsilon)} }^{-1} \circ \hat H \circ \pi_\varepsilon (p).
$$
(see Definition \ref{def-pi} for the definition of $\pi_{\varepsilon}, \tilde \pi_{\varepsilon}$). If $p \in U$, we define
$$
G_\varepsilon (p) := H_\varepsilon (p).
$$
This completes the definition of $G_\varepsilon $ on $S^*(v_{\varepsilon})$. Note that we have constructed $G_\varepsilon $ on vertices of type $1,3$ (singular points and points on limit cycles) and edges of type $3$ (limit cycles), because these vertices and edges belong to $S^*(v_{\varepsilon})$.
\begin{remark}
\label{rem-Geps-toptype}
$G_{\varepsilon}$ preserves topological types of singular points and limit cycles, thus preserves labels on the vertices of type $1,3$ and edges of type $3$. $G_{\varepsilon}$ also preserves the time orientation on cycles and germs of separatrices of $v_{\varepsilon}$.
\end{remark}
\subsubsection{Elliptic sectors}
\label{ssec-Ge-elliptic}
Each elliptic sector $E$ of $v_\varepsilon $ corresponds to a non-hyperbolic singular point $P$ of $v_\varepsilon$; we have $P\in U$ by Proposition \ref{prop-inU-or-hyp}. Consider a germ of the elliptic sector $(E,P)$; its image under $H_{\varepsilon}$ is a germ of an elliptic sector of $w_{h(\varepsilon)}$. Denote it by $\tilde E:=H_\varepsilon ((E,P))$.
In the construction of $LMF(v_{\varepsilon})$, we need to choose a trajectory $l$ of $v_{\varepsilon}$ in $E$. We may and will assume that $l$ is close to $P$ so that $l \subset U$. In the elliptic sector $\tilde E$ of $w_{h(\varepsilon)}$, we need to choose a homoclinic trajectory of $w_{h(\varepsilon)}$; let us choose $H_{\varepsilon}(l)$. Then we define
$$
G_\varepsilon |_l := H_\varepsilon |_l.
$$
We have constructed $G_\varepsilon $ on edges of type $5$ (homoclinic trajectories in elliptic sectors).
\begin{remark}
\label{rem-Ge-elliptic}
$G_{\varepsilon}$ preserves time orientation on edges of type 5 (homoclinic trajectories in elliptic sectors), and preserves incidence of edges of type 5 and vertices of type 1.
\end{remark}
\subsubsection{Non-truncated separatrices}
\label{ssec-Ge-nontrunc}
Let $\gamma $ be a non-truncated separatrix of $v_{\varepsilon}$. Then for small $\varepsilon $, it belongs to $U$, due to Separatrix Lemma \ref{lem-sep-connect} above. Define
$$
G_\varepsilon |_\gamma := H_\varepsilon |_\gamma.
$$
This agrees with the definition of $G_{\varepsilon}$ on the germs of separatrices. We have constructed $G_\varepsilon $ on edges of type $1$ (non-truncated separatrices).
\begin{remark}
\label{rem-nontrunc-label}
$G_{\varepsilon}$ preserves incidence of vertices of type 1 and edges of type 1. For non-truncated separatrices (edges of type 1), $G_{\varepsilon}$ preserves labels. Indeed, the labels say that the separatrix is stable, unstable or a separatrix connection. Since $G_{\varepsilon}$ is induced by a homeomorphism $H_{\varepsilon}$ in a neighborhood of such edge, it preserves such labels.
\end{remark}
\begin{remark}
Now $G_{\varepsilon}$ is defined on all monodromic polycycles of $v_{\varepsilon}$, because they are formed by non-truncated separatrices. Hence $G_{\varepsilon}$ is defined on all possible $\alpha$- and $\omega$-limit sets of $v_{\varepsilon}$ (singular points, limit cycles and monodromic polycycles) of $v_{\varepsilon}$.
\end{remark}
\subsubsection{The graph correspondence $G_{\varepsilon}$ is a bijection}
\label{ssec-Ge-bij}
By now, we have constructed $G_{\varepsilon}$ for small $\varepsilon$ on all vertices and edges of the $LMF(V)$ disjoint from the transversal loops. Let us prove that this map is one-to-one.
\begin{proposition}
\label{prop-G-bij}
For small $U\supset LBS(V)$, for sufficiently small $\varepsilon$, the map $G_{\varepsilon}$ defined in Sec. \ref{ssec-Ge-skeleton}, Sec. \ref{ssec-Ge-elliptic}, and Sec. \ref{ssec-Ge-nontrunc} is one-to-one on singular points, limit cycles, non-truncated separatrices, and trajectories in elliptic sectors (vertices of types $1,3$, edges of type $1,3,5$) of $v_{\varepsilon}$, $w_{h(\varepsilon)}$. It preserves incidence of these vertices and edges, labels and time orientation.
\end{proposition}
\begin{proof}
Fix $U\supset LBS(V)$ such that for all small $\varepsilon$, $H_{\varepsilon}(U)$ is detached from hyperbolic cycles and singular points of $w_0$. This is possible due to Corollary \ref{cor-heps-detached} above.
Note that $G_{\varepsilon}$ is defined on all edges and vertices of $LMF(v_{\varepsilon})$ listed in the proposition. Each of the listed edges is either completely inside $U$, or completely outside it. In $U$, $G_{\varepsilon}$ is induced by $H_{\varepsilon}$, so is injective.
Outside $U$, the map $G_{\varepsilon} = \tilde \pi_{h(\varepsilon)}^{-1} \circ \hat H \circ \pi_{\varepsilon}$ is a composition of three injective maps, so is injective as well.
For all listed vertices and edges that are inside $U$, their images under $G_{\varepsilon}$ are located inside $H_{\varepsilon}(U)$; for the edges and vertices outside $U$, their images are close to hyperbolic singular points and cycles of $w_0$. Due to Corollary \ref{cor-heps-detached}, $H_{\varepsilon}(U)$ is detached from these hyperbolic singular points and cycles of $w_0$, thus $G_{\varepsilon}$ is injective for small $\varepsilon$.
Prove that $G_{\varepsilon}$ is surjective. Recall that $\tilde U := \cap_{|\varepsilon| \le \varepsilon_0} H_{\varepsilon}(U)$ is a neighborhood of $LBS(W)$, due to Requirement \ref{it-nbhd-cond} of Definition \ref{def-moderate-local} of moderate equivalence.
For small $\varepsilon$, each non-truncated separatrix of $w_{h(\varepsilon)}$ belongs to $\tilde U$ (Separatrix lemma \ref{lem-sep-connect}). Thus it belongs to the range of $H_{\varepsilon}$. So $G_{\varepsilon}$ is surjective on truncated separatrices. Each elliptic sector of $w_{h(\varepsilon)}$ is an elliptic sector of a non-hyperbolic singular point, and all such points belong to $\tilde U \subset H_{\varepsilon}(U)$. So the edge of type 5 in this sector is the image under $G_{\varepsilon}=H_{\varepsilon}$ of the edge of type 5 of $LMF(v_{\varepsilon})$. Thus $G_{\varepsilon}$ is surjective on edges of type 5.
Each singular point and each limit cycle of $w_{h(\varepsilon)}$ is either completely inside $\tilde U$, or completely outside it (Proposition \ref{prop-inU-or-hyp} applied to $W$). In the first case, this singular point (cycle) is the image of some cycle or singular point of $(v_\varepsilon)|_{U}$ under $H_{\varepsilon}=G_{\varepsilon}$. In the second case, it belongs to a continuous family of singular points (cycles) of $(w_\varepsilon)|_{S^2\setminus U}$, thus belongs to the range of $G_{\varepsilon} = \tilde \pi_{h(\varepsilon)}^{-1} \circ \hat H \circ \pi_{\varepsilon}$.
So $G_{\varepsilon}$ is surjective on the union of verteces and edges of $v_\varepsilon$ disjoint from the transversal loops.
This map preserves incidence of vertices and edges, labels and time orientation due to Remarks \ref{rem-Geps-toptype}, \ref{rem-Ge-elliptic}, \ref{rem-nontrunc-label} above.
\end{proof}
\subsubsection{Transversal loops}
\label{ssec-tr}
Consider a hyperbolic sink, a source or a limit cycle of $v_0$; in all the three cases, we denote this object by $P$. Let $l$ be a transversal loop around $P$. We assume that $U$ is sufficiently small so that it does not intersect $l$. Since $P$ is hyperbolic, it
persists under small perturbations, so $l$ is a transversal loop for the corresponding object $P_{\varepsilon}:=\pi^{-1}_{\varepsilon} P$ of $v_{\varepsilon}$, $\varepsilon$ small. We may and will assume that $l$ belongs to the graph $LMF(v_{\varepsilon})$ as a transversal loop for $P_{\varepsilon}$. Now, $\tilde l = \hat H(l)$ is a transversal loop of $\hat H(P)$ for $w_0$. Moreover, $\tilde l$ is a transversal loop of the corresponding object $\tilde \pi^{-1}_{\varepsilon} (\hat H(P)) = G_\varepsilon (P_{\varepsilon})$ of $w_{h(\varepsilon)}$. We may and will assume that $\tilde l$ belongs to the graph $LMF(w_{h(\varepsilon )})$ as a transversal loop of $G_{\varepsilon}(P_{\varepsilon})$.
Define
$$
G_\varepsilon |_l := \hat H|_l.
$$
If a cycle, polycycle, a sink or a source $P$ of $v_{\varepsilon}$ belongs to $U$, we choose its transversal loop $l$ so that $l \subset U$ and the annulus between $P$ and $l$ belongs to $U$. Then $H_{\varepsilon}(l)$ is a transversal loop of $H_{\varepsilon}(P)$ for $w_{h(\varepsilon)}$, so we may assume that $H_{\varepsilon}(l)$ belongs to $LMF(w_{h(\varepsilon)})$.
We define
$$
G_\varepsilon|_{l} := H_\varepsilon|_{l}.
$$
Note that all limit cycles and singular points are either inside or outside $U$ by Proposition \ref{prop-inU-or-hyp}. All monodromic polycycles are inside $U$ due to Separatrix lemma \ref{lem-sep-connect}. So we have already constructed $G_\varepsilon$ on all transversal loops that belong to $LMF(v_{\varepsilon})$.
Note that we did not yet define $G_\varepsilon$ on the truncation verteces of $LMF(v_\varepsilon)$.
In Sec. \ref{ssec-Ge-final}, we will have to modify $G_{\varepsilon}$ on transversal loops so that it provides a correct identification of truncation vertices. However we will not change the set $G_{\varepsilon}(l)$ for a transversal loop $l$.
\begin{remark}
\label{rem-Ge-trans}
$G_{\varepsilon}$ preserves the correspondence of $\alpha$-, $\omega$-limit sets and their transversal loops: if $l$ is a transversal loop of a cycle, polycycle or singular point $P$ of $v_{\varepsilon}$, then $G_{\varepsilon}(l)$ is a transversal loop of $G_{\varepsilon}(P)$.
This implies that $G_{\varepsilon}$ is one-to-one on transversal loops of $LMF(v_{\varepsilon})$, $LMF(w_{h_{\varepsilon}})$, because it is one-to-one on limit cycles, polycycles and singular points of $v_{\varepsilon}, w_{h(\varepsilon)}$.
\end{remark}
\section{Main lemmas and the proof of Main Theorem}
\label{sec-proof}
In this section, we formulate two main lemmas and prove the Main Theorem modulo these lemmas.
\subsection{The plan of the proof}
In the previous section, we have partially constructed the required isomorphism $G_\varepsilon\colon LMF(v_{\varepsilon}) \to LMF(w_{h(\varepsilon)}) $. However $G_{\varepsilon}$ is not yet defined on truncated separatrices; it is only defined on their germs at singular points. To complete the construction, we will need the following Correspondence lemma: if separatrices of $v_{\varepsilon}$ cross a transversal loop of $v_{\varepsilon}$, then the corresponding separatrices of $w_{h(\varepsilon)}$ cross the corresponding transversal loop of $w_{h(\varepsilon)}$; the formal statement appears below. This lemma will enable us to extend $G_{\varepsilon}$ to truncated separatrices and truncation vertices, and we will be forced to modify restrictions of $G_{\varepsilon}$ to transversal loops so that it provides a correct identification of truncation vertices. However, for a transversal loop $l$, we will not change its image $G_{\varepsilon}(l)$.
After $G_{\varepsilon}$ is constructed, we have to verify the assumptions of Theorem \ref{th-graphs-components} for it. The most non-trivial assumption concerns annular faces of LMF graphs. We will state and prove the Annuli faces lemma to handle this problem.
\subsection{Main lemmas}
\begin{lemma}[Annuli faces lemma]
\label{lem-an-faces}
In assumptions of Main theorem, let $\varepsilon$ be sufficiently small. Let $A$ be an annular face of $LMF(v_{\varepsilon})$. Then the map $G_{\varepsilon}$ (see Sec. \ref{sec-partial-Ge}) takes $\partial A$ homeomorphically to the boundary of an annular face $\tilde A$ of $LMF(w_{h(\varepsilon)})$. Moreover, $G_{\varepsilon}$ extends to an orientation-preserving homeomorphism that takes $A$ to $\tilde A$.
\end{lemma}
The proof constitutes Section \ref{sec-annuli}. Clearly, the modification of $G_{\varepsilon}$ on transversal loops will not affect this lemma.
\vskip 0.5 cm
Now we introduce notation for Correspondence lemma.
Recall that we always choose counterclockwise orientation on transversal loops with respect to their $\alpha$-($\omega$-)limit sets, see Sec. \ref{sec-LMF}. We will call this ''proper orientation``.
Take a transversal loop $l$ that belongs to $LMF(v_{\varepsilon})$. Denote by $\{\gamma_i\}, i=1,\dots, n$, all separatrices of singular points $P_i$ of $v_{\varepsilon}$ that cross $l$ (the case $n=0$ is not excluded). We suppose that the cross-points $p_i := \gamma_i \cap l$, i.e. truncation vertices on $l$, are ordered cyclically along $l$. Note that if a singular point $P$ has several separatrices that intersect $l$, then it appears several times in the list $\{P_i\}$.
Let $\tilde \gamma_i$ be the separatrix of $w_{h(\varepsilon)}$ that corresponds to $\gamma_i$, i.e. contains the germ $G_{\varepsilon}((\gamma_i, P_i))$.
\begin{lemma}[Correspondence lemma]
\label{lem-seps-behave}
In assumptions of Main Theorem, let $l$ be a properly oriented transversal loop of $v_{\varepsilon}$ that belongs to $LMF(v_{\varepsilon})$. Let $\{\gamma_i\}$ be all separatrices of $v_{\varepsilon} $ that intersect $l$, so that the corresponding truncation vertices $p_i$ are ordered cyclically along $l$.
Then for sufficiently small $\varepsilon$,
1) The corresponding separatrices $\{\tilde \gamma_i\}, i=1, \dots, n$ of $w_{h(\varepsilon)}$ cross the properly oriented transversal loop $\tilde l:=G_{\varepsilon}(l)$, and the truncation vertices $\tilde p_i := \tilde \gamma_i \cap \tilde l$ are ordered cyclically along $\tilde l$.
2) There are no more truncation vertices on $\tilde l$.
\end{lemma}
The proof constitutes Section \ref{sec-Corr}. The proof is simple if we only consider separatrices that are completely inside $U$ or comlpetely inside its complement. The problem occurs if the separatix has \emph{mixed location}: belongs partly to $U$ and partly to its complement. We will classify such separatrices as well as the boundary components of $U$ that may intersect them.
\subsection{Isomorphism of LMF graphs}
\label{ssec-Ge-final}
This section completes the construction of the graph isomorphism $G_{\varepsilon} \colon LMF(v_{\varepsilon}) \to LMF(w_{h(\varepsilon)}) $.
Let $l$ be a transversal loop in $LMF(v_{\varepsilon})$. We will introduce $\gamma_i, p_i, \tilde l,$ $\tilde \gamma_i$, and $\tilde p_i$ as in Correspondence lemma \ref{lem-seps-behave}. Note that $p_i$ are all truncation vertices on $l$ and $\tilde p_i$ are all truncation vertices on $\tilde l$.
Now we modify $G_{\varepsilon}|_{l}$ so that $G_{\varepsilon}(l) = \tilde l$, the map takes $p_i$ to $\tilde p_i$ and preserves counterclockwise orienation on $l, \tilde l$.
If $l$ is empty, i.e. does not intersect separatrices of $v_{\varepsilon}$, it contains one vertex of type 4. Correspondence lemma implies that $\tilde l$ does not intersect separatrices of $w_{h(\varepsilon)}$, so it also contains one vertex of type 4. In this case, we modify $G_{\varepsilon}|_l$ so that it matches these vertices of type 4 and preserves counterclockwise orienation on $l, \tilde l$.
$G_{\varepsilon}$ takes the germs $(\gamma_i, P_i)$ of truncated separatrices to $(\tilde \gamma_i, G_{\varepsilon}( P_i))$, due to the definition of $\tilde \gamma_i$. We extend $G_{\varepsilon}$ to the whole truncated separatrices so that it identifies truncation vertices: $G_{\varepsilon} (p_i) =\tilde p_i$.
This completes the construction of $G_{\varepsilon}$ on vertices of type 2, 4 and edges of type 2, 4.
\begin{remark}
\label{rem-G-24}
$G_{\varepsilon}$ is one-to-one on vertices and edges of type $2$, $4$, due to Remark \ref{rem-Ge-trans} and Correspondence lemma. It preserves incidence of vertices and edges, labels, and orientation on transversal loops.
Each truncated separatrix of $w_{h(\varepsilon)}$ terminates on some transversal loop, so all of them are in the range of $G_{\varepsilon}$. This shows that $G_{\varepsilon} $ is surjective on truncated separatrices. Injectivity is clear because $G_{\varepsilon}$ is injective on germs of truncated separatrices. We conclude that $G_{\varepsilon}$ is one-to-one on truncated separatrices.
\end{remark}
\vskip 0.2 cm
Proposition \ref{prop-G-bij} and Remark \ref{rem-G-24} show that the map $G_{\varepsilon}$ is one-to-one on vertices and edges of $LMF(v_{\varepsilon})$ and $LMF(w_{h(\varepsilon)})$, preserves incidence of vertices and edges, labels and orientation. So $G_{\varepsilon}$ is a graph isomorphism.
\subsection{Isotopy of the LMF graphs}
We are going to check the assumptions of Theorem \ref{th-graphs-components}: $G_{\varepsilon}$ preserves orders of edges in all vertices of $LMF(v_\varepsilon)$ and extends to annuli-shaped faces of $LMF(v_{\varepsilon})$, $LMF(w_{h(\varepsilon)})$. The second statement follows directly from Annuli faces Lemma \ref{lem-an-faces}. Now we check the first statement in all vertices of $LMF(v_{\varepsilon})$.
\textbf{Vertices of type 1 (singular points) inside $U$.}
The edges that start at such vertex $P$ are
\begin{itemize}
\item edges of type 1, i.e. non-truncated separatrices of $v_{\varepsilon}$. They belong to $U$ due to Separatrix lemma \ref{lem-sep-connect}.
\item edges of type 5, i.e. homoclinic trajectories in elliptic sectors of $P$.
\item edges of type 2, i.e. truncated separatrices of that singular point $P$.
\end{itemize}
On the edges of types 1,5, and on the germs of edges of type 2, $G_{\varepsilon}$ coincides with $H_{\varepsilon}$.
Hence it preserves cyclical orders of edges at all vertices of type 1 inside $U$, because so does $H_{\varepsilon}$.
\textbf{Vertices of type 1 outside $U$}
Note that vertices of type 1 outside $U$ are either hyperbolic saddles, or hyperbolic sinks, or sources.
Hyperbolic sinks and sources are isolated vertices of $LMF(v_{\varepsilon})$, and there is nothing to prove for them. Let $P$ be a hyperbolic saddle outside $U$; on the (germs of) edges adjacent to $P$, we have $G_{\varepsilon}((\gamma, P)) = \tilde \pi_{h(\varepsilon)}^{-1} \circ \hat H \circ \pi_{\varepsilon} ((\gamma, P))$. All maps in this composition preserve orders of separatrices at hyperbolic saddles, so $G_{\varepsilon}$ preserves order of edges at $P$.
\textbf{Vertices of type 2: truncation vertices}
Such vertex has degree 3: one of the corresponding edges is of type 2 (a truncated separatrix), and two other edges are of type 4 (two arcs of a transversal loop, or possibly one arc coinciding with the whole loop).
The order of edges in such vertex is always such that the truncated separatrix is ''from the right-hand side`` with respect to the orientation along the transversal loop. Indeed, the transversal loop is oriented counterclockwise with respect to its $\alpha$- or $\omega$-limit set (see the definition of LMF graphs), and the separatrix crosses it from the other side. So the order of edges in this vertex is determined by the orientation of the edges of the graph, and $G_{\varepsilon}$ preserves this orientation.
\textbf{Vertices of type $3$ and type $4$ (points on limit cycles and on empty transversal loops)}
Such vertices are only joined to themselves by edges of types 3 and 4 respectively. So the cyclical order in such vertices is trivial, and there is nothing to prove.
Main Theorem is now proved modulo main lemmas.
\section{Auxiliary lemmas}
The proofs of both main lemmas, as well as the proof of No-entrance lemma (see Lemma \ref{lem-seps}), are heavily based on the following Boundary lemma.
\subsection{Boundary lemma}
\label{ssec-Blemma}
In this section, we formulate the Boundary lemma. Its proof is postponed till Section \ref{sec-U}.
\begin{definition}
Let $U$ be an open domain. A point $p\in \partial U$ is called an \emph{inner} topological tangency point of $\partial U$ with $v$ if a germ of the trajectory of $p$ under $v$ is inside $\overline U$ and only crosses $\partial U $ at $p$. It is called an \emph{outer} topological tangency point of $\partial U$ with $v$ if this germ is outside $U$ and only crosses $\partial U $ at $p$.
\end{definition}
Note that if $\partial U$ is smooth and only has isolated quadratic tangencies with $v$, then all these tangencies are either inner or outer topological tangency points.
\begin{definition}
\label{def-Sep-pr}
Let $v$ be a smooth vector field. A closed $v$-invariant set $Z\subset S^2$ is said to have a Sep-property if:
\begin{itemize}
\item \emph{For any unstable separatrix $\gamma \not\subset Z$, the set $\omega(\gamma)$ is detached from $Z$;}
\item \emph{For any stable separatrix $\gamma \not\subset Z$, the set $\alpha(\gamma)$ is detached from $Z$.}
\end{itemize}
\end{definition}
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.3\textwidth]{U-cases1}
\hfil
\includegraphics[width=0.3\textwidth]{U-cases2}
\hfil
\includegraphics[width=0.25\textwidth]{U-cases3}
\end{center}
\caption{Type $1$, Type $2$, and Type $3$ boundary components of $\Omega$. For Type $2$, inside the dotted transversal loops there are $\alpha$- and $\omega$-limit sets of maximal transversal arcs of this boundary component. }\label{fig-Case123-def}
\end{figure}
\begin{lemma}[Boundary lemma (see Fig. \ref{fig-Case123-def})]
\label{lem-U-arcs-top}
Let $v \in Vect^*\,S^2$ be a vector field; let $Z\subset S^2$ be a closed non-empty $v$-invariant set with Sep-property.
Then there exists an arbitrarily small neighborhood $\Omega$ of $Z$ with smooth boundary and finitely many boundary components with isolated quadratic tangencies of $\partial \Omega$ with $v$, such that any connected component $\varphi$ of its boundary $\partial \Omega$ is of one of the following types:
\begin{itemize}
\item Type $1$: $\varphi$ contains two inner tangency points of $\partial \Omega$ with $v$ and bounds a disc $D\subset (S^2\setminus \overline \Omega)$; $v|_{D}$ is orbitally topologically equivalent to the vector field $\partial / \partial x$ in the unit disc. Trajectories of points of $\varphi$ under $v|_{\overline \Omega}$ belong to $\Omega$.
\item Type $2$: $\varphi$ contains only outer tangency points of $\partial \Omega$ with $v$ (probably no tangency points). The trajectories of points of $\varphi$ under $v|_{S^2\setminus \Omega}$ belong to $S^2\setminus \overline\Omega$.
Each maximal transversal arc $\beta \subset \varphi$ intersects a separatrix of $v|_{\Omega}$.
If $\beta$ is outgoing, then all its points have a common $\omega$-limit set under $v$. If $\beta$ is ingoing, then all its points have a common $\alpha$-limit set under $v$. This $\omega$- (resp. $\alpha$-) limit set lies outside $\Omega$.
\item Type $3$: $\varphi$ is a transversal loop of some $\alpha$- or $\omega$-limit set (an attracting or repelling singular point, a cycle or a polycycle) of $v$, and
this object belongs to $Z$.
\end{itemize}
Moreover, separatrices of $v|_{S^2\setminus \Omega}$ do not intersect $\Omega$ in all the three cases above.
\end{lemma}
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.9\textwidth]{Case123-new}
\end{center}
\caption{Examples of Type $1$, Type $2$, and Type $3$ boundary components of $\Omega \supset Z$; the set $Z$ is shown in thick, and boundary components are dashed. On all of these pictures, $Z$ is a part of the large bifurcation support of a generic unfolding of $v$}\label{fig-ex-123}
\end{figure}
Note that the characteristic feature of the Type $2$ boundary components is not the presence of outer tangencies (they may be absent), but rather the presence of separatrices that cross these components.
\begin{remark}[Indices of boundary components of $\Omega$]
\label{rem-index}
Recall that we orient $\partial \Omega$ counterclockwise with respect to $\Omega$. For a boundary component $\varphi$ of $\Omega$, suppose that the point $\infty$ of the sphere is located to the left of $\varphi$ (i.e. on the same side as $\Omega$). Then the indices of boundary components of $\Omega$ with respect to $v$ are the following:
\begin{itemize}
\item Type $1$: index $0$;
\item Type $2$: indices $1, 2,\dots$;
\item Type $3$: index $1$.
\end{itemize}
We will use this remark later to distinguish between boundary components of different types.
\end{remark}
\begin{proposition}
\label{prop-Case2-nonint-ls}
There exists an arbitrarily small open neighborhood $U^*\supset LBS^*(V)$ that satisfies assumptions of Boundary lemma for $Z=LBS^*(V)$ and $v=v_0$. Moreover, for boundary components of Type $2$, the common $\alpha$- (resp. $\omega$-) limit set under $v_0$ of each ingoing (resp. outgoing) transversal subarc $\beta \subset \partial U^*$ is non-interesting.
\end{proposition}
\begin{proof}
Let us check that $LBS^*(V)$ satisfies assumptions of Boundary lemma. Clearly, it is closed and $v_0$-invariant (Proposition \ref{prop-LBS-closed-vinv}). It has a Sep-property due to Proposition \ref{prop-seps-v0}. The application of Boundary lemma provides us with a neighborhood $U^*$. This implies the first statement of the proposition.
Let $\varphi \subset U^*$ be a boundary component of $U^*$ of Type $2$, and let $\beta \subset \varphi$ be its outgoing transversal subarc. Boundary lemma implies that the common $\omega$-limit set of $\beta $ under $v_0$ is outside $U^*$. The only interesting $\omega$-limit sets of $v_0$ outside $LBS^*(V)$ are saddles (see Remark \ref{rem-nohyp-inLBS}). But saddle separatrices of $v|_{S^2\setminus U^*}$ do not intersect $U^*$ due to Sep-property of $LBS(V)$. So $\omega(\beta)$ is non-interesting. The same arguments apply to ingoing transversal arcs and their $\alpha$-limit sets.
\end{proof}
\subsection{Proof of No-entrance lemma \ref{lem-seps} modulo Boundary lemma}
\label{ssec-lem-seps}
\begin{remark} \label{rem-Bound-imply-Sep}
In fact, we will prove that any neighborhood $U^*$ that satisfies Boundary lemma (for $v_0$ and $LBS^*(V)$) also satisfies No-entrance lemma.
\end{remark}
\begin{proof}
Choose a neighborhood $U^*$ that satisfies Boundary lemma for $v_0$ and $LBS^*(V)$. It exists due to Proposition \ref{prop-Case2-nonint-ls}. Suppose that unstable separatrices of $v_{\varepsilon}$ enter $U^*$ for arbitrarily small $\varepsilon$. Then there exists a sequence $\varepsilon_k\to 0$ and points $p_k \in \partial U^*$ where unstable separatrices of $v_{\varepsilon_k}$ enter $U^*$. Let $p$ be a limit point of the sequence $p_k \in \partial U^*$. Then $p \in \partial U^*$; clearly, $p$ belongs to the closure of an ingoing transversal arc of $\partial U^*$.
On the other hand, $p\in (\overline {\mathop{\mathrm{Sep}} V}) \cap \{\varepsilon=0\}$. We claim that $p\in ELBS(v_0)$; this will imply $p\in LBS(V)$ and contradict $p \in \partial U^*$.
Since $p\in \partial U^*$, it is not singular. Prove that its $\alpha$-limit set under $v_0$ is interesting. Indeed, otherwise the negative semi-trajectory of $p$ under $v_0$ crosses a transversal arc of a non-interesting set $\alpha_{v_0}(p)$. Thus for close points $p_k$, their negative semi-trajectories under close vector fields $v_{\varepsilon_k}$ cross this arc as well. Hence these semi-trajectories cannot be unstable saddle separatrices, and we get a contradiction.
So $\alpha_{v_0}(p)$ is interesting. Let us prove that $\omega_{v_0}(p)$ is interesting.
Since $\alpha_{v_0}(p)$ is interesting, Proposition \ref{prop-Case2-nonint-ls} implies that $p$ cannot belong to the boundary component of Type $2$.
So it belongs to the boundary component of Type $1$ or Type $3$. In both cases, the future semi-trajectory of $p$ under $v_0$ belongs to $U^*$ due to Boundary Lemma \ref{lem-U-arcs-top}.
The $\omega$-limit set of $p$ under $v_0$ is thus inside $\overline U^*$; due to Remark \ref{rem-nohyp-inLBS}, $\omega_{v_0}(p)$ is interesting.
Hence $\alpha$- and $\omega$-limit sets of $p$ are both interesting. We conclude that $p\in ELBS(v_0) \cap (\overline \mathop{\mathrm{Sep}} V) \cap \{\varepsilon=0\} \subset LBS(V)$, which contradicts $p\in \partial U^*$.
So separatrices of $v_{\varepsilon}$ cannot enter $U^*$.
\end{proof}
\subsection{Choice of $U$}
\label{sec-choice-U}
If $U$ is a neighborhood of $LBS(V)$, we denote by $U^{*} $ the union of its connected components that do not contain non-interesting cycles of $v_0$.
If $\tilde U^{\pm}$ is a neighborhood of $LBS(W)$, we denote by $\tilde U^{\pm *} $ the union of its connected components that do not contain non-interesting cycles of $w_0$.
\begin{proposition}
\label{prop-choice}
Under the assumptions of the Main Theorem, there exists an arbitrarily small open neighborhood $U$ of $LBS(V)$ and arbitrarily small open neighborhoods $\tilde U^{\pm}$ of $LBS(W)$ such that
\begin{itemize}
\item [-] $U^*$ satisfies Boundary lemma for $v_0$ and $LBS^*(V)$;
\item [-] $\tilde U^{\pm *} $ satisfy Boundary lemma for $w_0$ and $LBS^*(W)$;
\item [-] for all small $\varepsilon$, $\tilde U^- \subset H_{\varepsilon}(U) \subset \tilde U^{+}$;
\item [-] the sets $U\setminus U^*$, $\tilde U^{\pm} \setminus \tilde U^{\pm*}$ are unions of neighborhoods of non-interesting cycles bounded by their transversal loops.
\item [-] each connected component of $U, \tilde U^\pm$ contains one connected component of $LBS(V), LBS(W)$ respectively.
\end{itemize}
\end{proposition}
Note that this proposition implies that $U, \tilde U^{\pm}$ satisfy the assertions of No-entrance lemma, see Remark \ref{rem-Bound-imply-Sep}. The last assertion of this proposition shows that Proposition \ref{prop-U12-intersect} on connected components is applicable for $U, \tilde U^{+}$.
\begin{proof}
Choose a neighborhood $\tilde U^{*+}$ of $LBS^*(W)$ satisfying Boundary lemma for the vector field $w_{0}$. We may remove its connected components that do not contain connected components of $LBS^*(W)$; if it is sufficiently small, then each its connected component contains only one component of $LBS^*(W)$. Add small annular neighborhoods of non-interesting cycles of $w_0$ bounded by transversal loops; we get the required neighborhood $\tilde U^+$ of $LBS(W)$.
Now, take a small neighborhood $U^*$ of $LBS^*(V) $ that satisfies the assumptions of Boundary lemma, and add small annular neighborhoods of non-interesting cycles of $v_0$ bounded by transversal loops. We get a neighborhood $U\supset LBS(V)$. Due to Proposition \ref{prop-heps-detached}, we may and will assume that for small $\varepsilon$, $H_{\varepsilon}(U)\subset \tilde U^+$. As above, we assume that each connected component of $U$ contains one connected component of $LBS(V)$.
Recall that for any small $\varepsilon_0$, $\cap_{|\varepsilon|<\varepsilon_0} H_{\varepsilon}(U)$ is a neighborhood of $LBS(W)$ due to the definition of moderate equivalence (Requirement \ref{it-nbhd-cond} of Definition \ref{def-moderate-local}).
We choose $\tilde U^{-*}\supset LBS^*(W)$ that satisfies Boundary lemma for $w_0$, and add small annular neighborhoods of non-interesting cycles of $w_0$ bounded by transversal loops, in order to get a neighborhood $\tilde U^-$ of $LBS(W)$. We assume that $\tilde U^{-}$ is sufficiently small so that $\tilde U^{-} \subset (\cap_{|\varepsilon|<\varepsilon_0} H_{\varepsilon}(U))$. Once again, we assume that each connected component of $\tilde U^-$ contains one connected component of $LBS(W)$.
Finally, $\tilde U^- \subset H_{\varepsilon}(U) \subset \tilde U^+$ for small $\varepsilon$ as required.
\end{proof}
From now on, we assume that $U$, $\tilde U^+$ and $\tilde U^-$ satisfy the proposition above.
\subsection{Images of Type $1$, $2$, and $3$ boundary components}
In the proofs of both main theorems, we will also need results on the images of Type $1$, Type $2$, and Type $3$ boundary components of $U^*$ under $H_{\varepsilon}$. In some sence, they say that the boundary component $H_{\varepsilon}(\varphi)$ of $H_{\varepsilon}(U)$ has similar properties to that of $\varphi$, and also provide some control on the location of $H_{\varepsilon}(\varphi)$ for different $\varepsilon$.
For the three subsequent lemmas, $U$, $\tilde U^{\pm}$ are as in Proposition \ref{prop-choice} and are sufficiently small, i.e. belong to some preassigned neighborhoods of the corresponding large bifurcation supports. From now on, we assume that $H_{\varepsilon}$ extends homeomorphically to $\overline U$, otherwise we slightly diminish $U$.
\begin{lemma}[Images of Type $1$ boundary components]
\label{lem-image-case1}
Under the assumptions of Main Theorem, suppose that $\varphi$ is a Type $1$ boundary component of $\partial U$.
Then $H_{\varepsilon}(\varphi)$ bounds an open topological disc $D \subset S^2\setminus H_{\varepsilon}(U)$, and $w_{h(\varepsilon)}$
has no singular points and limit cycles in $D$.
\end{lemma}
The following lemma is important for Correspondence lemma: it shows that $H_{\varepsilon}$ preserves the correspondence between outgoing (ingoing) transversal arcs of Type $2$ boundary components and their $\omega$- (resp. $\alpha$-)limit sets outside $U^*$.
Here and below the orientation on $\partial U$ is clockwise with respect to $U$.
\begin{lemma}[Images of Type $2$ boundary components]
\label{lem-image-Case2}
Under the assumptions of Main Theorem, suppose that $\beta$ is a transversal outgoing arc of a Type $2$ boundary component $\varphi\subset \partial U^*$.
Let $\beta_{\varepsilon}\subset \varphi$ be the maximal arc transversal to $v_{\varepsilon}$ and close to $\beta$.
Put $\tilde \beta_\varepsilon:= H_{\varepsilon}(\beta_\varepsilon) $.
Let $l$ be a transversal loop around $\omega(\beta)$. Put $\tilde l = \hat H(l)$. Then for small $\varepsilon$,
positive semi-trajectories of points of $\tilde \beta_\varepsilon$ under $w_{h(\varepsilon)}$ stay in $S^2\setminus H_{\varepsilon}(\overline {U^*})$, and the Poincare map $\tilde P_{\varepsilon} \colon \tilde \beta_\varepsilon \to \tilde l$ along $w_{h(\varepsilon)}$ is well-defined. The map $\tilde P_{\varepsilon}$ takes the clockwise orientation on $\tilde \beta_\varepsilon$ with respect to $H_{\varepsilon}(U)$ to the counterclockwise orientation on $\tilde l$ with respect to $\hat H (\omega (\beta))$.
Moreover, $\tilde P_{\varepsilon}(\tilde \beta_\varepsilon)\subset \tilde l$ intersects $\tilde P_0(\tilde \beta_0)$ for small $\varepsilon$. The analogous statement holds for ingoing arcs.
\end{lemma}
\begin{corollary}
\label{cor-image-transvers-Case2}
Let $\varphi$ be an outgoing \textbf{transversal} Type $2$ boundary component of $\partial U$, let $l, \tilde l$ be as in Lemma \ref{lem-image-Case2}. Then for all small $\varepsilon$, there are no singular points and limit cycles of $w_{h(\varepsilon)}$ between two closed curves $H_{\varepsilon}(\varphi)$ and $\tilde l$.
\end{corollary}
\begin{proof}
This follows from Lemma \ref{lem-image-Case2} above, since the Poincare map $\tilde P_{\varepsilon} \colon H_{\varepsilon}(\varphi) \to \tilde l$ along the orbits of $w_{h(\varepsilon)}$ is well-defined.
\end{proof}
\begin{lemma}[Images of Type 3 boundary components]
\label{lem-image-case3}
Under the assumptions of Main Theorem, let $\varphi\subset \partial U$ be a Type $3$ boundary component of $\partial U$. Then for all small $\varepsilon$, the oriented curves $H_{\varepsilon}(\varphi)$ and $\hat H(\varphi)$ are homotopic in $\tilde U^+ \setminus (LBS(W) \cup \mathop{\mathrm{Sing}} w_{h(\varepsilon)} \cup \mathop{\mathrm{Per}} w_{h(\varepsilon)})$.
\end{lemma}
The proofs of these lemmas is postponed till Section \ref{sec-Case123}.
\subsection{Logical relation between subsequent sections}
We now turn to the proof of the main lemmas. The logical relation between sections \ref{sec-annuli} - \ref{sec-Case123} is the following: \ref{sec-U} $\rightarrow$ \ref{sec-Case123} $\rightarrow$ \ref{sec-annuli} $ \rightarrow$ \ref{sec-Corr}. Yet we start with main lemmas: Annuli faces lemma and Correspondence lemma in Sections \ref{sec-annuli} and \ref{sec-Corr} respectively, making use of the Boundary lemma and Lemmas \ref{lem-image-case1}, \ref{lem-image-Case2}, \ref{lem-image-case3}. Then we prove Boundary lemma and these lemmas in Sections \ref{sec-U} and \ref{sec-Case123} respectively.
\section{Proof of the Annuli faces lemma} \label{sec-annuli}
\subsection{Empty annuli lemma}
\begin{definition}\label{def:emptyan}
We say that the annulus $A \subset S^2$ is \emph{empty} with respect to a vector field $v$ if its boundaries are topologically transversal to $v$ and there are no singular points or limit cycles of $v$ inside $A$.
In this case, $v|_A$ is orbitally topologically equivalent to the radial vector field $\partial / \partial r$ in the standard annulus $\{1<r<2\}$.
\end{definition}
Let us now define a collection $L$ of transversal loops around non-interesting nests of $v_0$ (see Definition \ref{def-nic} of non-interesting cycles). This collection will be used in the proof of the Correspondence lemma.
For a non-interesting nest of $v_0$, let us order the cycles by inclusion. The first and the last cycles are called \emph{boundary cycles} of the nest. In the case when we have a nest of non-interesting semi-stable cycles (case \ref{it-semist} in Definition \ref{def-nic}), we suppose that the hyperbolic singular point mentioned in this definition lies inside the \emph{inner} cycle of the nest. This enables us to distinguish the inner and the outer cycle of the nest.
Transversal loops of the limit cycles of the nest belong to the LMF graph of the vector field $v_0$, but it is possible that they do not belong to the LMF-graph of $v_{\varepsilon}$ because the limit cycles of the nest may destroy. However it is convenient to consider $LMF(v_{\varepsilon})$ together with the transversal loops that encircle the nest.
\begin{definition}[Collection $L$ of transversal loops] \label{def-tr-loops}
For each non-interesting and \emph{not semi-stable} nest of $v_0$ (case \ref{it-nonsemist} of Definition \ref{def-nic}), fix two transversal loops $l^-, l^+$ of the boundary cycles of the nest such that the whole nest is in the annulus between $l^-, l^+$.
For each non-interesting \emph{semi-stable} nest of $v_0$ (case \ref{it-semist} in Definition \ref{def-nic}), fix an outer transversal loop $l$ of the most outer cycle of the nest, such that $l$ encircles the whole nest.
We orient these loops counterclockwise with respect to the annulus between $l^-, l^+$ or with respect to the disc encircled by $l$ respectively. Let $L$ be a collection of transversal loops thus obtained. We say that $l\in L$ is ingoing if future semi-trajectories of its points enter the corresponding nest, and outgoing otherwise.
\end{definition}
The Annuli faces lemma follows from a more general statement, Empty annuli lemma, which we will also need below in the proof of Correspondence lemma (see Section \ref{sec-Corr}).
\begin{lemma}[Empty annuli lemma]
\label{lem-annuliShaped}
Under the assumptions of Main Theorem, for sufficiently small open $U\supset LBS(V)$, suppose that transversal loops $l_1,l_2$ bound an empty annulus $A$ for a vector field $v_{\varepsilon}$. Suppose that $l_i$ is either a transversal loop around a hyperbolic singular point or a cycle of $v_\varepsilon$, or $l_i\subset \overline U$, or $l_i\in L$.
Let $\tilde l_i := H_{\varepsilon}(l_i)$ if $l_i\subset \overline U$ and $\tilde l_i := \hat H(l_i)$ in other cases.
Then $\tilde l_1, \tilde l_2$ bound an empty annulus $\tilde A$ for $w_{h(\varepsilon)}$.
Moreover, let the orientation on $\tilde l_i$ be induced by $\hat H$ or $H_{\varepsilon}$ from the orientation on $l_i$.
Then $l_1,l_2$ are oriented with respect to $A$ in the same way as $\tilde l_1, \tilde l_2$ are oriented with respect to $\tilde A$.
\end{lemma}
The last assertion implies that $\hat H$, $H_{\varepsilon}$ restricted to $\partial A$ extend to the homeomorphism of $A, \tilde A$. Note that transversal loops from the $LBS(V)$ either surround a hyperbolic singular point or a cycle of $v_\varepsilon$, or belong to $ \overline U$. The case $l_i\in L$ will be used in the proof of Correspondence lemma in Section \ref{sec-Corr} below.
\subsection{Reduction}
\begin{proof}[Proof of the Annuli faces lemma modulo Empty annuli lemma]
Let $A$ be the same as in the Annuli faces lemma, that is, an annuli shaped face of the LMF graph of $v_{\varepsilon }$. Due to the classification of faces of LMF graphs (Lemma \ref{lem-faces}), we have two cases. Consider them one by one.
\begin{itemize}
\item $A$ is an annulus between a transversal loop $l$ of $v_{\varepsilon}$ and the corresponding $\alpha$- or $\omega$-limit set $c$.
\end{itemize}
Due to Remark \ref{rem-Ge-trans}, $G_{\varepsilon}$ preserves the correspondence of transversal loops and their $\alpha$-, $\omega$-limit sets, so the loop $G_{\varepsilon}(l)$ is a transversal loop for $G_{\varepsilon}(c)$ in $LMF(w_{h(\varepsilon)})$. Thus $G_{\varepsilon}(l)$ and $G_{\varepsilon}(c)$ bound an annulus $\tilde A$. It remains to prove that $G_{\varepsilon}$ extends to a homeomorphism of $A,\tilde A$, i.e. to analyze whether it preserves orientation on $\partial A$. We have two subcases:
1) If $c\subset U$, then $A$ is inside $U$, due to the choice of transversal loops in Section \ref{ssec-tr}. Then $G_{\varepsilon}$ is induced by $H_{\varepsilon}$ on $\partial A$. So $H_{\varepsilon}$ provides a required extension of $G_{\varepsilon}$ to this annuli-shaped face.
2) If $c$ is outside $U$, then $c $ is either a hyperbolic cycle, or a hyperbolic sink, or a source. We only consider the case when $c$ is a cycle; other cases are analogous but simpler.
Recall that the orientation on $c$ and its transversal loop $l$ is chosen in such a way that $c$ is to the left with respect to $l$; suppose that $l$ is to the left with respect to the timewise orientation of $c$. It remains to prove that the mutual orientation of $\tilde l:= G_{\varepsilon}(l)=\hat H(l)$ and $ \tilde c:= G_{\varepsilon}(c) = \tilde \pi_{h(\varepsilon)}\hat H(\pi_{\varepsilon}^{-1}(c))$ is the same as the orientation of $c,l$ described above. This will imply that $G_{\varepsilon}$ matches the orientations on $\partial A, \partial \tilde A$; thus $G_{\varepsilon}$ extends to a homeomorphism between the faces $A$ and $\tilde A$.
Indeed, $\tilde c$ is to the left with respect to $\tilde l$ due to the choice of orientation on transversal loops.
Further, $\tilde l = \hat H(l)$ is to the left with respect to $\hat H(c) $ because $\hat H$ is an orientation-preserving homeomorphism. The curve $\hat H(c)$ is close to the cycle $G_{\varepsilon}(c)=\tilde \pi_{h(\varepsilon)}\hat H(\pi_{\varepsilon}^{-1}(c))$ which implies the statement.
\begin{itemize}
\item $A$ is an annulus between two transversal loops $l_1,l_2$ of $v_{\varepsilon}$.
\end{itemize}
In this case, Lemma \ref{lem-an-faces} follows from the Empty annuli Lemma \ref{lem-annuliShaped}.
Note that $A$ is an empty annulus of $v_{\varepsilon }$ in the sense of Definition \ref{def:emptyan}. By Empty annuli Lemma \ref{lem-annuliShaped}, the annulus $\tilde A$ between $\tilde l_1 \mbox{ and } \tilde l_2$ is empty for $w_{h(\varepsilon )}$. By construction of $G_\varepsilon ,$ its boundaries $ \ \tilde l_1, \tilde l_2$ belong to $LMF(w_{h(\varepsilon )})$.
Let $l_1$ be an outgoing transversal loop, and let $l_2$ be ingoing; clearly, no unstable separatrices may cross an outgoing transversal loop $\tilde l_1$ of a cycle, source or a monodromic polycycle. Similarly, no stable separatrices may cross $\tilde l_2$. So no separatrices enter an empty annulus $\tilde A$, thus it forms a face of $LMF(w_{h(\varepsilon)})$.
By Lemma \ref{lem-annuliShaped}, the map $G_{\varepsilon }\colon \partial A \to \partial \tilde A$ may be extended to a homeomorphism between $A$ and $\tilde A$.
\end{proof}
\subsection{Plan of the proof of the Empty annuli lemma \ref{lem-annuliShaped}}
The natural way to prove the Empty Annuli lemma is to compare restrictions to $A$ of the phase portraits of $v_{\varepsilon }$ and $v_0$. The first restriction is trivial; the second one may be quite different, see Figures \ref{fig:onecompu} and \ref{fig:onecompcu}. We will need the Boundary Lemma for the case shown in Fig. \ref{fig:onecompu}, and both Boundary and Correspondence Lemmas for Fig. \ref{fig:onecompcu}.
Let us pass to the formal proof.
Consider all boundary components of $U$ that are inside $A$. It is possible that some of them are non-contractible inside $A$; then $A$ is split into several smaller annuli. Note that all these boundary components have index $1$ with respect to $v_{\varepsilon}$. Hence they have index $1$ with respect to $v_0$. Therefore, they are transversal boundary components for $U$ and $v_0$ (see Boundary Lemma \ref{lem-U-arcs-top}). Clearly, each smaller annulus is an empty annulus for $v_\varepsilon$. We are going to prove the Empty annuli lemma for each of these smaller annuli.
Any smaller annulus does not contain non-contractive boundary components of $U$. So there are two possible cases: the boundary components $l_1$, $l_2$ of a smaller annulus $A$ belong to the same connected component of $\overline U$ or to the same connected component of $CU=S^2\setminus U$.
Indeed, suppose that $l_1$ and $l_2$ do not belong to the same connected component of $U$. The annulus bounded by $l_1$ and $l_2 $ does not contain non-contractive boundary components of $U$, hence the curves $l_1$ and $l_2$ are not separated by $U$. Therefore, they belong to the same connented component of $CU$.
The two possible cases mentioned above are considered below in Lemmas \ref{lem-same-U} and \ref{lem-S2-U}, so these lemmas conclude the proof. See Figures \ref{fig:onecompu} and \ref{fig:onecompcu} respectively.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.4\textwidth]{02.pdf}
\hfil
\includegraphics[width=0.4\textwidth]{03.pdf}
\caption{Empty annuli lemma: auxiliary lemma \ref{lem-same-U}, orbits of $v_\varepsilon$ (left) and $v_0$ (right). The component of $U$ is shadowed, its boundary is dotted, the large bifurcation support is shown in thick. The vector field $v_0$ has degeneracy of codimension 6; four saddlenodes of $v_0$ vanish as $\varepsilon$ changes, and the annulus $A$ becomes empty. }\label{fig:onecompu}
\end{center}
\end{figure}
\subsection{One connected component of $\bar U$}
\begin{lemma}
\label{lem-same-U}
The statement of Empty annuli lemma holds true if $l_1$, $l_2$ belong to the same connected component of $\overline U$.
\end{lemma}
\begin{proof}
Consider all the boundary components $\varphi_i$ of this connected component of $\overline U$ that are located inside $A$. They are contractive in $A$, so they bound topological discs in $A$. These disks
contain no singular points of $v_{\varepsilon}$, so the index of the vector field $v_\varepsilon$ with respect to each curve $\varphi_i$ is $0$ (we assume that the point $\infty$ on $S^2$ is outside $A$). The same holds for the vector field $v_0$. Thus these components $\varphi_i$ are of Type $1$ (see Remark \ref{rem-index}), they bound discs $D_i \subset CU$, and $A \setminus \cup D_i \subset U$.
So the annulus $\tilde A$ between $H_{\varepsilon}(l_1)$ and $H_{\varepsilon}(l_2)$ is a union of $H_{\varepsilon} (A \setminus \cup D_i)$ and regions $\tilde D_i$ bounded by $H_{\varepsilon}(\varphi_i)$, where $\varphi_i$ are Type $1$ boundary components for $v_0$, see Fig. \ref{fig:onecompu} left. Fig. \ref{fig:onecompu} right shows an example of a vector field with such boundary components of $\overline U$.
Let us prove that $\tilde A$ is empty for $w_{h(\varepsilon )}$.
Due to Lemma \ref{lem-image-case1} on the images of Type $1$ components, the regions $\tilde D_i$ inside $H_{\varepsilon} (\varphi_i)$ do not contain limit cycles and singular points of $w_{h(\varepsilon)}$. The set $H_{\varepsilon} (A \setminus \cup D_i)$ does not contain singular points and limit cycles of $w_{h(\varepsilon)}$ too, because its preimage under $H_{\varepsilon}$ does not contain singular points and limit cycles of $v_{\varepsilon}$.
The case when a limit cycle belongs partly to $H_{\varepsilon} (A \setminus \cup D_i)$ and partly to its complement in $\tilde A$ is prohibited by Proposition \ref{prop-inU-or-hyp}: each cycle of $w_{h(\varepsilon)}$ for small $\varepsilon$ either belongs to $\tilde U^- \subset H_{\varepsilon}(U)$, or is close to a hyperbolic cycle of $w_0$ (thus does not intersect $\tilde U^+\supset H_{\varepsilon}(U)$). Thus $\tilde A$ contains no singular points or limit cycles of $w_{h(\varepsilon)}$.
Let us prove that $\partial \tilde A$ is transversal to $w_{h(\varepsilon)}$. Indeed, $\partial A$ is transversal to $v_0$, thus to $v_{\varepsilon}$; $H_{\varepsilon}$ preserves topological transversality, thus $\partial \tilde A$ is transversal to $w_{h(\varepsilon)}$.
So $\tilde A$ is empty in this case. Clearly, the map $H_{\varepsilon}|_{\varphi_i}$ may be extended to a homeomorphism $H_i: D_i \to \tilde D_i$. Hence, the map $H_{\varepsilon}|_{\partial A}$ may be extended to a homeomorphism of $A $ by $H_\varepsilon$ on $A\setminus \cup D_i$ and by $H_i$ on $D_i$. Thus the last claim of the Empty annuli lemma holds true.
\end{proof}
\subsection{One connected component of $CU$}
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.8\textwidth]{04}
\caption{Empty annuli lemma: auxiliary lemma \ref{lem-S2-U}, orbits of $v_\varepsilon$ (left) and of $v_0$ (right). The component of $U$ is shadowed, its boundary is dotted, the large bifurcation support is shown in thick. The vector field $v_0$ has degeneracy of codimension 2. As $\varepsilon$ changes, its two saddlenodes vanish, and the annulus becomes empty}\label{fig:onecompcu}
\end{center}
\end{figure}
\begin{lemma}
\label{lem-S2-U}
The statement of the Empty annuli lemma holds true if $l_1$, $l_2$ belong to the same connected component of $CU$.
\end{lemma}
\begin{proof} As the curves $l_1, l_2$ belong to the same component of $CU$, they may either belong to $\partial U$, or to the interior of $CU$. So for each of them, we have the following three cases:
\begin{enumerate}
\item A transversal loop of a hyperbolic sink or source, or a hyperbolic cycle, or a non-interesting cycle (namely $l\in L$);
\item A boundary component of $\bar U$ of Type 2;
\item A boundary component of $\bar U$ of Type 3.
\end{enumerate}
If one of $l_1,l_2$ is a Type 2 transversal boundary component of $U$, then the other one is a transversal loop around the corresponding non-interesting $\alpha$- or $\omega$-limit set. This follows from Boundary lemma and Proposition \ref{prop-Case2-nonint-ls}. In this case Lemma \ref{lem-S2-U} follows from Corollary \ref{cor-image-transvers-Case2} above.
Let us prove that in all other cases the following implication holds: if $A$ is empty for $v_{\varepsilon}$ and $\hat H (A)$ is empty for $w_{h(\varepsilon)}$, then $\tilde A$ is empty for $w_{h(\varepsilon)}$.
Suppose that both $l_1, l_2$ are of Type 3.
In this case, Lemma \ref{lem-image-case3} implies that $\hat H(l_i)$ and $H_{\varepsilon}(l_i)$ are homotopic in $\tilde U^+\setminus (\mathop{\mathrm{Sing}} w_{h(\varepsilon)} \cup \mathop{\mathrm{Per}} w_{h(\varepsilon)})$ as oriented curves. So we may replace $\tilde l_i=H_{\varepsilon}(l_i)$ by $\hat H(l_i)$: if $\hat H(A)$ is empty, then $\tilde A$ is empty as well.
Suppose that $l_1$ is of Type 3, and $l_2$ falls into the case 1 above. Then $H_\varepsilon (l_1)$ may be replaced by $\hat H(l_1) $ as before, and $\tilde l_2=\hat H(l_2)$. Again, if $\hat H(A)$ is empty, then $\tilde A$ is empty as well.
Suppose that both $l_1,l_2$ fall into the case 1 above. In this case, $\tilde l_i=\hat H(l_i)$, and there is nothing to prove.
Now, the following proposition implies the first assertion of the Empty annuli lemma.
\begin{proposition}
\label{prop-H0-empty}
In assumptions of Lemma \ref{lem-S2-U}, the annulus $\hat H(A)$ is empty with respect to $w_{h(\varepsilon)}$.
\end{proposition}
\begin{proof}
By contraposition, suppose that some singular point or a cycle $\tilde c$ of $w_{h(\varepsilon)}$ is in $\hat H(A)$. We only consider the case when $\tilde c$ is a cycle; the case of a singular point is analogous.
Due to Proposition \ref{prop-inU-or-hyp} applied to the family $W$, the cycle $\tilde c$ either belongs to $\tilde U^-$ or belongs to a continuous family of hyperbolic cycles $\tilde c_\delta $ of vector fields $w_\delta $ defined for all $\delta$ small.
In the first case, let $H_{\varepsilon}(U_i)$ be a connected component of $H_{\varepsilon}(U)$ that contains $\tilde c$. Then $U_i$ contains a cycle $H_{\varepsilon}^{-1}(\tilde c)$ of $v_{\varepsilon}$.
It remains to prove that $U_i$ is inside $A$; this will contradict to the fact that $A$ is empty for $v_{\varepsilon}.$
Since $\hat H(U_i)$ is the only component of $\hat H(U)$ that intersects $H_{\varepsilon}(U_i)$ (see Proposition \ref{prop-U12-intersect}) and $\tilde c \subset \tilde U^- \subset \hat H(U)$, we have $\tilde c\subset \hat H(U_i)$. So $\hat H(U_i)$ intersects the annulus $\hat H(A)$. Thus $U_i$ intersects the annulus $A$, and since boundaries of $A$ are in one and the same connected component of $CU$, we have that $U_i \subset A$. We get a contradiction mentioned in the previous paragraph.
In the second case, the cycles $\tilde c_\delta $ belong entirely to $\hat H(A)$ because the boundary of this annulus is transversal to $w_\delta$ for any $\delta$ small. Hence, no limit cycle of $w_\delta$ can cross this boundary. Therefore, $\tilde c_0$ belongs to $\hat H(A)$ as well. Therefore, the vector field $v_0$ has a hyperbolic limit cycle $\hat H^{-1}(\tilde c_0)$ in $A$. The same holds for the vector field $v_\varepsilon$, so $A$ is not empty with respect to $v_{\varepsilon}$.
We get a contradiction again.
This finishes the proof.
\end{proof}
The first statement of the Empty Annuli lemma in assumptions of Lemma \ref{lem-S2-U} is proved.
Let us prove the second one:
$G_\varepsilon$ may be extended to a homeomorphism of $A$. The same statement for $\hat H(A)$ instead of $\tilde A$ is clear, because $\hat H$ is an orientation-preserving homeomorphism. Suppose that both $l_1,l_2$ belong to $\overline U$ (other cases are analogous but simpler). Since $l_1$ and $l_2$ are in different connected components of $U$, their images $H_{\varepsilon}(l_1)$ and $H_{\varepsilon}(l_2)$ are in different connected components of $\tilde U^+$ (see Proposition \ref{prop-U12-intersect}; this proposition is applicable due to the last assertion of Proposition \ref{prop-choice}). By Lemma \ref{lem-image-case3}, the oriented curves $H_\varepsilon(l_1)$ and $\hat H(l_1)$ are homotopic inside $\tilde U^+$. Thus as we perform the homotopy between $H_\varepsilon(l_1)$ and $\hat H(l_1)$, all the intermediate curves do not intersect $\tilde l_2$. Similar arguments apply to the homotopy between $H_{\varepsilon}(l_2)$ and $\hat H(l_2)$. Finally, $\tilde l_1, \tilde l_2$ are oriented with respect to $\tilde A$ in the same way as $\hat H(l_1), \hat H(l_2)$
with respect to $\hat H(A)$. The latter orientation coincides with the orientation of $l_1,l_2$ with respect to $A$, which implies the statement.
\end{proof}
\section{Proof of the Correspondence Lemma \ref{lem-seps-behave}} \label{sec-Corr}
\subsection{Plan of the proof}
Without loss of generality we assume that $l$ is an ingoing transversal loop. Choose $U$ following Sec. \ref{sec-choice-U}. Note that any transversal loop $l\subset LMF(v_\varepsilon)$ either belongs entirely to $U$, or to its complement, due to the choice of transversal loops in Sec. \ref{ssec-tr}. So there are the following cases to consider depending on the location of $l$.
\begin{itemize}
\item The loop $l\in LMF(v_{\varepsilon})$ lies outside $U$. Simultaneously, we will prove the statement of Correspondence lemma for $l\in L$ (see Definition \ref{def-tr-loops} of the collection $L$), though such loops may be not included in $LMF(v_{\varepsilon})$.
\begin{itemize}
\item Case 1. Some backward orbit of $l$ under $v_0$ hits $\partial U^*$ at a transversal boundary component.
\item
Case 2. All backward orbits of $l$ under $v_0$ either hit $\partial U^*$ at non-transversal boundary components, or do not intersect $\overline {U^*}$.
\end{itemize}
\item The loop $l\in LMF(v_{\varepsilon})$ lies inside $U$.
\begin{itemize}
\item Case 3. $l \subset U^*$.
\item Case 4. $l \subset U \setminus U^*$, i.e. $l$ is inside a non-interesting nest. Here we will use Correspondence lemma for $l\in L$ (Case 1 above).
\end{itemize}
\end{itemize}
In Sec. \ref{sec-corr-first} -- \ref{sssec-corr-inside-ni}, we prove the first statement of Correspondence Lemma in each of the above four cases. In Sec. \ref{ssec-corr-second}, we prove the second statement of the Correspondence Lemma.
\subsection{The first statement of the Correspondence lemma: Case 1}\label{sec-corr-first}
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.3\textwidth]{Corrlemma-region}
\end{center}
\caption{Proposition \ref{prop-corr-case1}: Poincare map between $\varphi$ and $l$.}\label{fig-corrlemma-reg}
\end{figure}
\begin{proposition}
\label{prop-corr-case1}
The first statement of the Correspondence lemma holds true for an ingoing transversal loop $l\in LMF(v_{\varepsilon})$ or $l\in L$, if backward orbits of \emph{some} points of $l$ under $v_0$ hit a transversal boundary component of $\partial U^*$.
\end{proposition}
\begin{proof}
Let $\varphi$ be one of these components of $\partial U^*$.
A trajectory of $v_0$ joins $\varphi$ to $l$, so a close trajectory $\xi$ of $v_{\varepsilon}$ joins $\varphi$ to $l$ as well. We conclude that a Poincare map along $v_{\varepsilon}$ between some arcs of the transversal loops $\varphi$ and $l$ is defined. The endpoints of its domain must be intersections of $\varphi$ with separatrices of $v_{\varepsilon}$ (see Fig. \ref{fig-corrlemma-reg}). But separatrices of $v_{\varepsilon}$ do not enter $U^*$ through $\varphi$ due to No-entrance lemma \ref{lem-seps}. Therefore this Poincare map is defined on the whole $\varphi$, thus $l$ and $\varphi$ bound an annulus $A$ filled by trajectories of $v_{\varepsilon}$. We conclude that the separatrices $\gamma_i$ of $v_{\varepsilon}$ that cross $l$ also cross $\varphi$, and the intersection points $\gamma_i\cap \varphi$ are ordered clockwise with respect to $U$ (see Fig. \ref{fig-corrlemma}).
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.8\textwidth]{Corrlemma1}
\end{center}
\caption{Proposition \ref{prop-corr-case1}, annuli $A$ and $\tilde A$}\label{fig-corrlemma}
\end{figure}
Due to No-entrance lemma \ref{lem-seps}, separatrices of $v_{\varepsilon}$ cannot enter $U^*$, see Definition \ref{def-entering}. So all separatrices of $v_{\varepsilon}$ that cross $\varphi\subset \partial U^*$ intersect it only once, and $\gamma_i\cap U$ are their arcs starting at the corresponding singular points $P_i \in U^*$. Hence $\tilde \gamma_i$ are separatrices of $H_\varepsilon (P_i)$ that contain arcs $H_{\varepsilon}(\gamma_i\cap U)$, so $\tilde \gamma_i$ intersect $H_{\varepsilon}(\varphi)$, and the intersection points $H_{\varepsilon}(\gamma_i\cap \varphi)$ are ordered clockwise with respect to $H_{\varepsilon}(U)$.
Finally, note that $l$ and $\varphi$ satisfy assumptions of Empty annuli Lemma, that is, they bound an empty annulus $A$, see Fig \ref{fig-corrlemma}. This lemma yields that $\tilde l=\hat H(l)$ and $H_{\varepsilon}(\varphi)$ bound an empty annulus $\tilde A$ for $w_{h(\varepsilon)}$, and the orientation on its boundaries is the same as for $A$. Thus the separatrices $\tilde \gamma_i$ that cross $H_{\varepsilon}(\varphi)$ also cross $\tilde l$, and the intersection points $\tilde \gamma_i\cap \tilde l$ are ordered counterclockwise with respect to it.
\end{proof}
\subsection{Case 2}
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.5\textwidth]{CorrLemma2}
\end{center}
\caption{Proposition \ref{prop-corr-case2}: points $q_i, a_j$ on $l$. Dashed circles are transversal loops around non-interesting $\alpha$-limit sets outside $U^*$.}\label{fig-corrlemma-case2}
\end{figure}
\begin{proposition}
\label{prop-corr-case2}
The first statement of the Correspondence lemma holds true for an ingoing transversal loop $l\subset LMF(v_{\varepsilon})$ or $l\in L$, if backward orbits of all points of $l$ under $v_0$ either cross $\partial U^*$ by non-transversal boundary components, or do not intersect $\overline {U^*}$.
\end{proposition}
\begin{proof}
Note that these boundary components must be all of Type $2$ (see Boundary lemma for the classification). Indeed, non-transversal boundary components of $\partial U$ are of Type $1$ or Type $2$. But the trajectories originating from a Type $1$ component fill the whole disc with no transversal loops in it.
Clearly, $l$ is the union of the following sets:
\begin{itemize}
\item (open) \emph{hyperbolic} arcs: a negative semi-trajectory of each point of this arc under $v_0$ tends to a non-interesting set and does not hit $U^*$.
\item intersections $q_i$ with separatrices of $(v_0)|_{S^2\setminus U}$, i.e. with separatrices of hyperbolic saddles.
\item (closed) images $P_0(\beta_i)$ of transversal outgoing arcs $\beta_i\subset \varphi_j \subset \partial U$ under Poincare maps $P_0$ along $v_0$, where each $\varphi_j$ is a boundary component of Type $2$, see Fig. \ref{fig-corrlemma-case2}.
\end{itemize}
Pick one point from each hyperbolic arc; let $a_i$ be these points (ordered cyclically along $l$). As $l$ lies outside $U$, again as in Case $1$, $\tilde l = \hat H(l)$.
Put $\tilde a_i = \hat H(a_i)$; these points are ordered cyclically along $\tilde l$.
It is sufficient to prove the statement of Correspondence lemma for each arc $[a_i, a_{i+1}]$. Put $I=[a_i, a_{i+1}]$, $\tilde I = [\tilde a_i, \tilde a_{i+1}]$.
Since each $q_j$ and each $P_0(\beta_j)$ is adjacent to open hyperbolic arcs on both sides, we have the following three cases for $I$:
\begin{enumerate}
\item The arc $I$ contains the point $q$ of intersection with a separatrix $\nu$ of a hyperbolic saddle $P$ of $v_0$; $I\setminus \{q\}$ belongs to two subsequent hyperbolic arcs. The arcs $[a_1, a_2], [a_3, a_4], [a_4, a_5]$ on Fig. \ref{fig-corrlemma-case2} are of that type, as well as all arcs on Fig. \ref{fig-corrlemma-case3}.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.5\textwidth]{CorrLemma3}
\end{center}
\caption{Proposition \ref{prop-corr-case2}: case 1}\label{fig-corrlemma-case3}
\end{figure}
Then the only separatrix of $v_{\varepsilon}$ that intersects $I$ is $\gamma:= \pi_{\varepsilon}^{-1}(\nu)$.
Since $\hat H$ conjugates $v_0$ to $w_0$, the separatrix $\hat H(\nu)$ intersects the arc $\hat H(I) = \tilde I$. The close separatrix of $w_{h(\varepsilon)}$, namely the separatrix that contains a germ $\tilde \pi^{-1}_{\varepsilon} (\hat H( \nu, P))$, also intersects $\tilde I$. This germ is $G_{\varepsilon}(\gamma, P)$, thus this separatrix is $\tilde \gamma$. We conclude that if $\gamma$ intersects $I$, then $\tilde \gamma$ intersects $\tilde I$. This completes the proof of the statement of Correspondence lemma for $I$ in this case.
\item The arc $I$ contains the image $P_0(\beta)$ of some transversal outgoing arc $\beta\subset \varphi \subset \partial U$. The set $I \setminus P_0(\beta)$ belongs to two subsequent hyperbolic arcs. The arcs $[a_2, a_3], [a_5, a_1]$ on Fig. \ref{fig-corrlemma-case2} are of that type.
As in Lemma \ref{lem-image-Case2}, let $\beta_{\varepsilon}$ be the maximal arc of $\varphi$ transversal to $v_{\varepsilon}$ and close to $\beta$. Let $P_{\varepsilon}\colon \beta_{\varepsilon} \to l$ be the Poincare map along $v_{\varepsilon}$.
Since separatrices of $v_{\varepsilon}$ cannot originate from non-interesting sets, each separatrix of $v_{\varepsilon}$ that intersects $I$ also intersects $\beta_{\varepsilon}$. Let $\{\gamma_k\}$ be these separatrices, ordered counterclockwise along $I$; then they intersect $\beta_{\varepsilon}$ and are ordered clockwise along it. Due to No-entrance lemma, separatrices $\gamma_k$ intersect $\partial U^*$ only once. Thus the separatrices $\tilde \gamma_k$ are the separatrices that contain arcs $ H_{\varepsilon}(\gamma_k \cap U)$. Since $H_{\varepsilon}$ conjugates $v_{\varepsilon}$ to $w_{h(\varepsilon)}$, we conclude that the separatrices $\tilde \gamma_k$ intersect $\tilde \beta_{\varepsilon} := H_{\varepsilon}(\beta_{\varepsilon})$ in a clockwise order along $\tilde \beta_{\varepsilon}$. Lemma \ref{lem-image-Case2} implies that the Poincare map $\tilde P_{\varepsilon}\colon \tilde \beta_{\varepsilon} \to \tilde l $ is well-defined and takes the clockwise orientation on $\tilde \beta_{\varepsilon}$ to the counterclockwise orientation on $\tilde l$. Thus the separatrices $\tilde \gamma_k$ intersect $\tilde P_{\varepsilon}(\tilde \beta_{\varepsilon}) \subset \tilde l$ in a counterclockwise order along $\tilde l$. Now it suffices to prove that $\tilde P_{\varepsilon}(\tilde \beta_{\varepsilon}) \subset \tilde I$; this will prove that $\tilde \gamma_k$ intersect $\tilde I$ and are ordered counterclockwise along it.
Applying $\hat H$ to the inclusion $P_0(\beta) \subset I$, we get that $\tilde P_0(\tilde \beta_0) \subset \tilde I$. Lemma \ref{lem-image-Case2} implies that $\tilde P_{\varepsilon}(\tilde \beta_\varepsilon)$ intersects $\tilde P_0(\tilde \beta_0)$. Now, it is sufficient to prove that $\tilde P_{\varepsilon}(\tilde \beta_\varepsilon)$ does not contain endpoints of $\tilde I$. Note that the negative semi-trajectories of endpoints of $\tilde I$ under $w_{h(\varepsilon)}$ are close to their trajectories under $w_0$, thus tend to non-interesting sets and do not intersect the closure of $\tilde U^{+*}$. This completes the proof of the statement of Correspondence lemma for $I$ in this case.
\item The arc $I$ does not fall into the two previous cases. So it belongs to one hyperbolic arc, which is only possible if all $a_i $ coinside. Then $I=l$ and all negative semi-trajectories of points of $l$ under $v_0$ tend to a non-interesting set and do not visit $U^*$. Thus for small $\varepsilon$, no separatrices of $v_{\varepsilon}$ intersect $l$, and there is nothing to prove.
\end{enumerate}
\end{proof}
\subsection{Case 3: $l \subset U^*$}
For $l\subset U^*$, Correspondence lemma follows directly from No-entrance lemma \ref{lem-seps}.
\begin{proposition}
\label{prop-corr-lem-in-U-1}
The first statement of Correspondence lemma holds if $l$ is a transversal loop of an $\alpha$- ($\omega$-) limit set inside $U^*$.
\end{proposition}
\begin{proof}
Let $\{\gamma_j\}$ be the set of all separatrices that hit $l$, as in Correspondence lemma. No-entrance lemma \ref{lem-seps} implies that they belong completely to $U^*$, i.e. $\gamma_j\subset U^*$. So $G_{\varepsilon}$ is induced by $H_{\varepsilon}$ on $\gamma_j$.
The statement follows from the fact that $H_{\varepsilon}$ is a homeomorphism.
\end{proof}
The same arguments apply if we consider arbitrary separatrices with $\omega$-limit sets inside $U^*$.
Let $P$ be a singular point of $v_\varepsilon$, let $\gamma$ be its unstable separatrix. Suppose that $\tilde \gamma$ is the corresponding separatrix of $w_{h(\varepsilon)}$: $(\tilde \gamma, \tilde P) = G_{\varepsilon}((\gamma, P))$.
\begin{proposition}
\label{prop-corr-lem-in-U-2}
In assumptions of Main Theorem, let $\gamma$ be an unstable separatrix of $v_{\varepsilon}$.
For sufficiently small $\varepsilon$, if $\omega(\gamma)$ belongs to $U^*$, then $\omega(\tilde \gamma) = G_{\varepsilon}(\omega(\gamma))$.
Here $\omega$-limit sets are with respect to $v_{\varepsilon}, w_{h(\varepsilon)}$. The same holds for stable separatrices and their $\alpha$-limit sets.
\end{proposition}
The proof literally repeats the proof of the previous proposition.
\subsection{Case 4: $l \subset U\setminus U^*$}
\label{sssec-corr-inside-ni}
Suppose that $l\subset U\setminus U^*$ corresponds to a cycle $c$ that bifurcates from a non-interesting nest, and suppose that $l$ is not homotopic in $S^2\setminus \mathop{\mathrm{Per}} v_{\varepsilon}$ to the outer transversal loops of the nest. Then no separatrices of $v_{\varepsilon}$ intersects $l$. Indeed, if a separatrix $\gamma$ accumulates to $c$, then it must enter a non-interesting nest, i.e. intersect its outer transversal loop $l'\in L$. However $l'$ is separated from $l$ by cycles of $v_{\varepsilon}$, and we get a contradiction. So the first statement of the Correspondence lemma is trivial for $l$.
Suppose that $l$ is homotopic in $S^2\setminus \mathop{\mathrm{Per}} v_{\varepsilon}$ to the outer transversal loop $l'$ of the nest. Then the separatrices $\{\gamma_i\}$ of $v_{\varepsilon}$ that intersect $l$ also intersect $l'$. Cases 1,2 of Correspondence lemma (see Propositions \ref{prop-corr-case1}, \ref{prop-corr-case2}) for $l'\in L$ imply that $\tilde \gamma_i$ intersect $\hat H(l')$ and are ordered cyclically along it. Empty annuli lemma \ref{lem-annuliShaped} implies that the annulus $\tilde A$ between $H_{\varepsilon}(l)$ and $\hat H(l')$ is empty with respect to $w_{h(\varepsilon)}$, so $\{\tilde \gamma_i\}$ intersect $H_{\varepsilon}(l)$. Their order is the same as for $v_{\varepsilon}$, because the curves $H_{\varepsilon}(l)$ and $\hat H(l')$ are oriented with respect to $\tilde A$ in the same way as $l,l'$ are oriented with respect to $A$ (see Empty annuli lemma \ref{lem-annuliShaped}). This completes the proof.
\subsection{Second statement of the Correspondence lemma}
\label{ssec-corr-second}
The first statement of the Correspondence lemma implies the second one because $v_{\varepsilon}$ and $w_{h({\varepsilon})}$ have the same amount of separatrices. For a detailed proof, we will need the following proposition.
\begin{proposition}
\label{prop-sameAmount}
For small $\varepsilon$, the vector field $w_{h(\varepsilon)}$ has the same amount of separatrices as $v_{\varepsilon}$.
\end{proposition}
\begin{proof}
Note that the number of separatrices of a vector field equals to the number of germs of separatrices at the corresponding singular points minus the number of separatrix connections.
Due to Proposition \ref{prop-G-bij}, the map $G_{\varepsilon}$ provides a one-to-one correspondence on singular points of $v_{\varepsilon}$ and $w_{h(\varepsilon)}$. This map preserves their topological types due to Remark \ref{rem-Geps-toptype}. So the amount of germs of separatrices at singular points is the same for $v_{\varepsilon}$ and $w_{h(\varepsilon)}$.
Moreover, $v_{\varepsilon}$ and $w_{h(\varepsilon)}$ have the same amount of separatrix connections: due to Separatrix lemma \ref{lem-sep-connect}, all separatrix connections of $v_{\varepsilon}$, $w_{h(\varepsilon)}$ for small $\varepsilon$ are inside small neighborhoods of large bifurcation supports, so $H_{\varepsilon}$ identifies separatrix connections of $v_{\varepsilon}$ and separatrix connections of $w_{h(\varepsilon)}$. The statement follows.
\end{proof}
The first statement of Correspondence Lemma implies that if $k$ separatrices of $v_{\varepsilon}$ intersect a transversal loop $l$ outside $U^*$, then \emph{at least} $k$ separatrices of $w_{h(\varepsilon)}$ intersect a transversal loop $\tilde l$.
Proposition \ref{prop-corr-lem-in-U-2} implies that if $k$ unstable separatrices of $v_{\varepsilon}$ have the same $\omega$-limit set $c$ inside $U^*$, then \emph{at least} $k$ separatrices of $w_{h(\varepsilon)}$ have the $\omega$-limit set $H_{\varepsilon}(c)$; the same holds for $\alpha$-limit sets of stable separatrices.
Clearly, each separatrix of $v_{\varepsilon}$ falls into one of the two cases above. Due to Proposition \ref{prop-sameAmount}, $v_{\varepsilon}$ and $w_{h(\varepsilon)}$ have the same amount of separatrices. So in each of the two cases above, the amount of separatrices of $v_{\varepsilon}$ equals the amount of corresponding separatrices of $w_{h(\varepsilon)}$. This completes the proof of Correspondence lemma.
\section{Proof of the Boundary lemma} \label{sec-U}
\subsection{Boundaries of canonical regions}
In the proof of the Boundary lemma, we will construct a neighborhood $\Omega$ as the union of its intersections with all canonical regions of $v$. Recall that these regions are described in Section \ref{sub:skel}. We start with an explicit description of the boundaries of canonical regions.
Let $v \in Vect^*(S^2)$. In the definitions below all the singular points, separatrices and so on are those of $v$.
\begin{definition} \label{def-chain}
A \emph{separatrix chain} $C \subset S^2$ is one of the following sets:
\begin{itemize}
\item A union $C = \alpha(\gamma_0) \cup \gamma_0 \cup P_1 \cup \gamma_1 \cup P_2 \cup \dots \cup \gamma_n \cup \omega(\gamma_n)$, where $\gamma_i$ is an ingoing separatrix of a singular point $P_{i+1}$ and $\gamma_{i+1}$ is an outgoing separatrix of $P_{i+1}$. In what follows, we say that $\gamma_0$ is the first separatrix of the chain $C$, and $\gamma_n$ is the last separatrix of the chain; we also say that the chain $C$ connects the limit sets $\alpha(\gamma_0)$ to $\omega(\gamma_n)$.
\item A union $C=\alpha(\gamma)\cup \gamma \cup \omega(\gamma)$ where $\gamma$ is a separatrix. Then $\gamma$ is both the first and the last separatrix in the chain, and the chain connects $\alpha(\gamma)$ to $\omega(\gamma)$.
\item A singular point; it coincides with both its $\alpha$- and $\omega$-limit sets and the corresponding chain has no separatrices.
\end{itemize}
\end{definition}
Note that points $P_i$ with different numbers in one and the same chain may coincide.
\begin{definition}
For a canonical region $R$ of a vector field, we denote by $\alpha(R)$ and $\omega(R)$ the common $\alpha$- and $\omega$-limit set of all its points.
\end{definition}
Note that a strip canonical region is simply connected, and a spiral one is a topological annulus. Recall that due to Proposition \ref{prop-parall}, for a strip canonical region, there exists a homeomorphism $\Psi \colon \mathbb R \times (0,1) \to R$ that conjugates $\partial/\partial x$ to $v$.
\begin{definition}
\emph{Side boundaries} $\nu_1(R), \nu_2(R)$ of a strip canonical region $R$ are upper topological limits
$$\nu_1(R)= \overline \lim_{y\to 0}\Psi(\mathbb R \times \{y\}),$$
$$\nu_2(R)= \overline \lim_{y\to 1}\Psi(\mathbb R \times \{y\}).$$
\end{definition}
Clearly, $\partial R$ is a union of two side boundaries of $R$. Each one of the side boundaries includes $\alpha(R)$ and $\omega(R)$.
\begin{lemma}[Side boundaries of strip canonical regions]
\label{lem-sides}
For a vector field $v\in Vect^*\,S^2$, side boundaries $\nu_1(R)$, $\nu_2(R)$ of a strip canonical region $R$ of $v$ are chains of separatrices that join $\alpha(R)$ to $\omega(R)$.
\end{lemma}
We expect that this lemma is known to experts, but we did not find it in the literature.
\begin{remark}
One can prove that the homeomorphism $\Psi$ can be so chosen that it extends continuously to $\psi_1\colon \mathbb R\times \{0\}\to S^2$ and $\psi_2\colon \mathbb R \times \{1\}\to S^2$. The images of $\psi_{1,2}$ contain $\nu_{1,2}(R)\setminus (\alpha(R)\cup \omega(R))$ respectively and are contained in $\nu_{1,2}(R)$. Note that $\psi_1, \psi_2$ may glue subsegments of their domains in various ways, see Fig. \ref{fig-canonreg-ex}.
\end{remark}
We will not prove this statement, because we are going to use it in some heuristic arguments only.
\begin{proof}[Proof of Lemma \ref{lem-sides}]
We prove the lemma for $\nu_1(R)$.
Note that $\nu_1(R)$ is a closed and $v$-invariant set; also, $\nu_1(R)\subset \partial R \subset S(v) = \mathop{\mathrm{Sing}} v \cup \mathop{\mathrm{Per}} v \cup \mathop{\mathrm{Sep}} v$.
Note that $\nu_1(R)\setminus \alpha(R) \setminus \omega(R)$ may not contain limit cycles. Indeed, since $\alpha$-, $\omega$-limit sets of all points of $R$ are $\alpha(R)$, $\omega(R)$, the set $\nu_1(R)$ is detached from basins of attraction and repulsion of all other $\alpha$-, $\omega$-limit sets of $v$, and may not contain limit cycles other than $\alpha(R)$ or $\omega(R)$.
Therefore $\nu_1(R)\setminus \alpha(R) \setminus \omega(R) \subset \mathop{\mathrm{Sing}} v \cup \mathop{\mathrm{Sep}} v$.
Since $\nu_1(R)$ is connected as a limit of connected sets, $\nu_1(R)\setminus \alpha(R) \setminus \omega(R)$ may not contain isolated singular points; it is either empty or contains a separatrix.
If $\nu_1(R)\setminus \alpha(R) \setminus \omega(R)$ is empty, the argument that $\nu_1(R)$ is connected implies that $\alpha(R)$ and $\omega(R)$ intersect. This is only possible if $\alpha(R)=\omega(R)$ is a singular point, and the statement is proved (this may happen when $R$ is an elliptic sector of a complex singular point).
Suppose that $\nu_1(R)\setminus \alpha(R) \setminus \omega(R)$ contains a separatrix $\gamma$ (this is the last case to consider). Since $\nu_1(R)$ is a limit of trajectories $\Psi(\mathbb R \times \{y\})$ and $\Psi$ is injective, a local analysis in each flow-box surrounding $\gamma$ shows that there exists a semi-neighborhood $U_{\gamma}$ of $\gamma$ that belongs to $R$.
Note that $\omega(\gamma)\subset \nu_1(R)$ because $\nu_1(R)$ is closed. There are the following possibilities for $\omega(\gamma)$:
\begin{itemize}
\item $\omega(\gamma)$ is a cycle or a polycycle. Then all the points in a neighborhood of $\gamma$ are also attracted to this set, including some points of $R$; thus $\omega(R) = \omega(\gamma)$. So $\gamma$ will be the last separatrix in the chain.
\item $\omega(\gamma)$ is a singular point $P$, and the semi-neighborhood $U_{\gamma}$ contains a piece of parabolic or elliptic sector near $(\gamma, P)$. Similarly, $\omega(R) = \omega(\gamma)$, and $\gamma$ will be the last separatrix in the chain.
\item $\omega(\gamma)$ is a singular point $P$, and the semi-neighborhood $U_{\gamma}$ contains a piece of a hyperbolic sector near $(\gamma, P)$; so $\gamma$ is a separatrix of $P$. Then $\gamma$ will be a separatrix $\gamma_i$ in the middle of the chain, $P_{i+1}=P$, and $\gamma_{i+1}$ is another border of the same hyperbolic sector. Now we may repeat our arguments for $\gamma_{i+1}$ and find $P_{i+2}, \gamma_{i+2}$, etc.
\end{itemize}
The same arguments apply to $\alpha(\gamma)$ and allow us to enumerate separatrices of $\nu_1(R)$ as required. Possibly we will have only one separatrix $\gamma_0=\gamma_n$ and no singular points $P_i$. This may happen, for instance, when $\partial R$ is a union of a singular point and its homoclinic curve, a separatrix, and $R$ is an elliptic sector (see Fig. \ref{fig-canonreg-ex} middle). Note also that one and the same singular point may appear several times in the list $\{P_i\}$, see Fig. \ref{fig-canonreg-ex} right.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.2\textwidth]{CanonRegCommon}
\hfil
\includegraphics[width=0.7\textwidth]{canonreg-ex}
\end{center}
\caption{Some possible shapes of canonical regions}\label{fig-canonreg-ex}
\end{figure}
It is easy to see that the union of semi-neighborhoods of $\gamma_i$ and hyperbolic sectors at $P_i$ is saturated by trajectories of $v$, so it exhausts all $\Psi(\mathbb R\times(0,\varepsilon) )$ for small $\varepsilon$; hence $\nu_1(R)$ coincides with the chain $\alpha(R)\cup \omega(R)\cup \{\gamma_i\} \cup \{P_i\}$.
\end{proof}
\begin{lemma}\label{lem:spiral}
The boundary of a spiral canonical region is the union of its $\alpha$- and $\omega$-limit sets:
$$
\partial R = \alpha(R) \cup \omega(R).
$$
\end{lemma}
The proof is obvious.
The following proposition provides a key tool for the proof of the Boundary lemma.
\begin{proposition} \label{prop-Z-inters-R}
For a vector field $v\in Vect^*\,S^2$, let $Z\subset S^2$ be a closed, $v$-invariant set with Sep-property.
1) Let $C= \{\gamma_i\}_{i=1}^{n-1}\cup \{P_i\}_{i=1}^{n}$ be a union of singular points and separatrix connections of $v$: $\gamma_i$ is a separatrix connection between $P_i$ and $P_{i+1}$. Then $Z$ either contains $C$, or does not intersect it.
2) For each $\alpha$- or $\omega$-limit set $c$ of $v$, the set $Z \cap c$ is either empty, or coincides with $c$.
\end{proposition}
In particular, 1) applies to any chain of separatrices (see Definition \ref{def-chain}) if we remove $\alpha(\gamma_0), \gamma_0, \gamma_n$, and $\omega(\gamma_n)$ from the chain.
\begin{proof}
1) Suppose that $Z$ contains $P_i\in C$.
A separatrix connection is both a stable and an unstable separatrix; due to Definition \ref{def-Sep-pr} of Sep-property, if $\gamma_i$ does not belong to $Z$, then both its $\alpha$- and $\omega$-limit sets $P_i, P_{i+1}$ are detached from $Z$. So for $i>1$, $P_i\in Z$ implies $\gamma_{i-1}\subset Z$, and due to closedness, $P_{i-1}\in Z$. Similarly, for $i \not = n$, $P_i\in Z$ implies $\gamma_{i}\subset Z$ and $P_{i+1}\in Z$. The induction in $i$ proves the statement.
2) If $c$ is a singular point or a cycle, this clearly follows from $v$-invariance of $Z$. If $c$ is a monodromic polycycle, then the statement follows from 1).
\end{proof}
\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{CanonRegBoundLemma}
\end{center}
\caption{Intersections $Z\cap \overline R$ (left; shown in thick) and $\Omega\cap \overline R$ (right; $\partial \Omega$ is dotted) in all 5 possible cases}\label{fig:5cases}
\end{figure}
\subsection{Plan of the proof of the Boundary lemma}
In order to construct the required neighborhood $\Omega\supset Z$, we describe its intersection with each $\overline R$, where $R$ is a canonical region of $v$:
$$
\Omega_R = \Omega \cap \overline R.
$$
Note that the number of canonical regions for $v\in Vect^*\,S^2$ is finite, because $v\in Vect^*\,S^2$ has a finite number of limit cycles, separatrices and singular points.
So we set
\begin{equation}\label{eqn:ome}
\Omega = \cup \Omega_R.
\end{equation}
If $Z\cap \overline R$ is empty, then $\Omega \cap \overline R$ will be empty; we do not discuss this case any more.
Depending on the type of $R$ and the type of intersection $Z \cap \overline R$, we have the following five cases for $R$. If $R$ is a spiral canonical region, either \textbf{1)} $Z\cap R = \varnothing$ or \textbf{2)} $Z\cap R \neq \varnothing$. If $R$ is a strip canonical region, either \textbf{3)} $Z\cap \overline R$ contains both $\alpha(R)$ and $\omega(R)$, or \textbf{4)} one of them, or \textbf{5)} none of them (there are no other cases due to Proposition \ref{prop-Z-inters-R} part 2).
The first three cases give rise to Type $1$ and Type $3$ boundary components of $\Omega$ that belong to $R$ entirely. These components are constructed in Section \ref{sssec-123}.
The last two cases 4), 5) give rise to Type $2$ boundary components of $\Omega$ that belong to the union of several adjacent canonical regions. These components are constructed in Section \ref{sssec-45}.
Then we define the set $\Omega$ by \eqref{eqn:ome}, and prove that it has the required properties.
\subsection{Construction of $\Omega \cap R$ in the cases 1), 2), 3)}\label{sub:123}
We have to construct an ``arbitrary small'' neighborhood of $Z$ with certain properties. This means that it must belong to a preassigned neighborhood $\Omega_0$ of $Z$. From now on, this latter neighborhood is fixed.
\label{sssec-123}
\begin{itemize}
\item 1): $R$ is a spiral canonical region, $ {Z\cap R = \varnothing}$. Due to Proposition \ref{prop-Z-inters-R} part 2, $Z \cap \overline R$ is $\alpha(R)$, $\omega(R)$, or $\alpha(R) \cup \omega(R)$.
\end{itemize}
Take $\Omega_R:= \Omega\cap \overline R$ to be a thin strip around $\alpha(R)$ or $\omega(R)$ (or two strips around both) bounded by its smooth transversal loop. This yields one or two Type $3$ boundary components, see row 1 of Figure \ref{fig:5cases}.
Complete semi-trajectories of points of such boundary components under $v|_\Omega$ stay in $\Omega$, because these trajectories wind around $\alpha(R)$ or $\omega(R)$ respectively. Clearly, separatrices of $v|_{S^2\setminus \Omega}$ do not enter $\Omega_R$. The set $\partial \Omega_R \cap R$ consists of one or two topological circles. They are boundary components of Type $3$.
\begin{itemize}
\item 2): $R$ is a spiral canonical region, $ {Z\cap R \neq \varnothing}$, see row 2 of Figure \ref{fig:5cases}.
\end{itemize}
In the case 2), $Z$ contains a trajectory of $v|_R$. So it contains $\alpha$- and $\omega$-limit sets of this trajectory, i.e. $\alpha(R)$ and $\omega(R)$. Therefore $\overline R\setminus Z$ is a union of at most countably many open strips with parallel flows in them.
For each such strip $S$,
\begin{itemize}
\item[-] If $S \subset \Omega_0$, we include it completely in $\Omega$.
\item[-] Otherwise, let $D \subset S $ be a large ellipse in the rectifying chart for $v$ in $S$ such that $S\setminus D \subset \Omega_0$, and let $\partial D$ have two quadratic tangency points with the vector field. Let $S\setminus D =: \Omega \cap S$.
This yields a finite number of Type $1$ boundary components.
Complete semi-trajectories of points of such boundary component under $v|_\Omega$ stay in $\Omega$ because they stay in $S$.
\end{itemize}
Clearly, separatrices of $v|_{S^2\setminus \Omega}$ do not enter $\Omega_R$. Again, the set $\partial \Omega_R \cap R$ consists of a finite number of topological circles. They are boundary components of Type $1$.
\begin{itemize}
\item 3): $R$ is a strip canonical region, $Z \cap \overline R$ contains $\alpha(R)$ and $\omega(R)$, see row 3 of Figure \ref{fig:5cases}.
\end{itemize}
Due to Sep-property, $Z$ also contains the first and the last separatrices of $\nu_1(R)$, $\nu_2(R)$; due to closedness, $Z$ contains endpoints of these separatrices.
Note that $\nu_i(R)$ without the first and the last separatrix and without $\alpha(R), \omega(R)$ is a chain that satisfies Proposition \ref{prop-Z-inters-R} part 1. So $Z$ contains the whole $\nu_1(R)$ and $\nu_2(R)$. It can also contain several trajectories of $v|_{R}$. So this case is analogous to case 2) and yields a finite number of Type $1$ boundary components.
\subsection{Construction of $\Omega\cap R$ in the cases 4), 5).}
\label{sssec-45}
First, on the whole sphere, we choose marked points on all separatrices of $v$ that "leave" a neighborhood of $Z$. In more detail, suppose that for a singular point $P\in Z$, its separatrix $\gamma$ does not belong to $Z$. Then some arc of $\gamma$ starting at $P$ belongs to $\Omega_0$. Fix one point on this arc; this point will be called marked. We will use marked points later in the construction; namely, $\partial \Omega$ will intersect $\gamma$ at the marked point.
In the cases 4) and 5), $Z\cap R$ is empty; otherwise $Z$ would contain both $\alpha$ and $\omega$-limit set of a trajectory of $v|_{R}$, thus satisfy assumptions of case 3) above.
\begin{itemize}
\item 4): $Z\cap R =\varnothing$, $\alpha(R)\subset Z$, and $\omega(R)$ does not intersect $Z$ (or vice versa: $\alpha$ and $\omega$ are exchanged), see row 4 of Figure \ref{fig:5cases}.
\end{itemize}
Due to Sep-property, $Z$ contains the first separatrix of $\nu_1(R)$, $\nu_2(R)$; due to closedness, $Z$ contains endpoints of these separatrices. Now, due to Proposition \ref{prop-Z-inters-R} part 1, $Z$ contains all $\nu_1(R), \nu_2(R)$ except their last separatrices and $\omega(R)$. $Z$ cannot contain last separatrices of $\nu_1(R), \nu_2(R)$, because it does not contain their $\omega$-limit set $\omega(R)$.
Finally, $Z\cap \overline R$ is the union of $\alpha(R)$ and $\nu_{1,2}(R)$ except for their last separatrices and $\omega(R)$. Note that last separatrices of $\nu_{1,2}(R)$ have marked points on them.
Take $\Omega_R \subset \Omega_0$ to be a neighborhood of $Z\cap \overline R$ in $\overline R$ bounded by a smooth curve $\varphi(R)\subset \overline R$ that is transversal to $v$ and connects marked points of last separatrices of $\nu_1(R)$ and $\nu_2(R)$. Take $\varphi(R)$ to be orthogonal to the corresponding separatrices at marked points.
The existence of $\varphi(R)$ follows from the fact that $R$ is parallel. The endpoints of $\varphi(R)$ may coincide, then it is a topological circle (see Fig. \ref{fig:BL-4}); otherwise $\varphi(R)$ is a topological segment (see Fig. \ref{fig:5cases}, row 4).
After we put $\Omega=\cup \Omega_R$ at the end of the proof, we will have that in the first case, $\varphi (R)$ is a transversal Type $2$ boundary component, and in the second case, it is a part of Type $2$ boundary component, namely a transversal subarc in $\partial \Omega $ crossed by separatrices of $v$ at its endpoints.
\begin{figure}
\begin{center}
\includegraphics[width=0.3\textwidth]{BLemma-4}
\end{center}
\caption{Case 4) in the proof of Boundary lemma; $\varphi(R)$ is a topological circle. The domain $\Omega_R$ is shadowed}\label{fig:BL-4}
\end{figure}
\begin{itemize}
\item 5): $Z\cap R =\varnothing$, both $\alpha(R)$ and $\omega(R)$ do not intersect $Z$, see row 5 of Figure \ref{fig:5cases}.
\end{itemize}
If $\nu_1(R)$ intersects $Z$, then $Z$ contains the whole $\nu_1(R) $ except for its first and last separatrices, and $\alpha(R), \omega(R)$, due to Proposition \ref{prop-Z-inters-R} part 1. Note that both the first and the last separatrix of $\nu_1(R)$ have marked points on them.
\begin{figure}[h]
\begin{center}
\centering
\subcaptionbox{\label{fig-BLemma-5-0}}{ \includegraphics[width=0.3\textwidth]{BLemma-5-0} }
\hfil
\subcaptionbox{\label{fig-BLemma-5-1}}{ \includegraphics[width=0.25\textwidth]{BLemma-5-1} }
\hfil
\subcaptionbox{\label{fig-BLemma-5-2}}{ \includegraphics[width=0.25\textwidth]{BLemma-5-2}}
\caption{Case 5) in the proof of Boundary lemma. (a) shows $\Psi^{-1}(\varphi(R))$ and the domain $\Psi^{-1} (\Omega)$ (shadowed), (b) and (c) show that $\varphi(R)$ can be a topological segment and a topological circle respectively. The domain $\Omega_R$ is shadowed}\label{fig:BL-5}
\end{center}
\end{figure}
Take a smooth curve $\varphi_1(R) \subset R$ with the following properties: $\varphi_1(R)$ connects the marked points on the first and last separatrices of $\nu_1(R)$, is close to $Z\cap \nu_1(R)$, is perpendicular to the first and the last separatrix at its endpoints, and has one quadratic tangency point with $v$. It is easy to construct an appropriate curve in the rectifying chart, i.e. in $\mathbb R\times [0,1]$ (see Fig. \ref{fig-BLemma-5-0}); let $\varphi_1(R)$ be its image under $\Psi$.
If $\nu_1(R)$ intersects $Z$ and $\nu_2(R)$ does not, we put $\varphi(R):=\varphi_1(R)$, and $\Omega_R$ is bounded by $\varphi_1(R)$ and an arc of $\nu_1(R)$.
If $\nu_2(R)$ also intersects $Z$, we choose the curve $\varphi_2(R)$ in a similar way, and put $\varphi(R):=\varphi_1(R)\cup \varphi_2(R)$. Then $\Omega_R$ is the union of two domains, one between $\varphi_1(R)$ and $\nu_1(R)$ and the other one between $\varphi_2(R)$ and $\nu_2(R)$; see Figure \ref{fig:5cases} row 5.
Note that the two curves $\varphi_1(R)$ and $\varphi_2(R)$ may have one or two common endpoints, see Fig. \ref{fig-BLemma-5-1}, \ref{fig-BLemma-5-2} respectively. So $\varphi(R)$ can be either two smooth curves with one contact point on each, or one simple curve with two contact points, or a closed loop with two contact points.
In any case, $\varphi(R)$ will be a part of a Type $2$ boundary component; in the third case, it is the whole Type $2$ boundary component with two contact points.
\begin{remark}
\label{rem-trajectory-case5}
Under assumptions of Boundary lemma, let $R$ be a canonical region satisfying assumptions of case 5) above. Then for any small neighborhood $\Omega$ of $Z$, $R$ contains a trajectory of $v$ that does not intersect $\overline{\Omega}$ (see Fig. \ref{fig-BLemma-5-0}).
\end{remark}
We will use this remark in the next section.
\subsection{End of the proof of the Boundary lemma }
We have constructed an intersection $\Omega_R$ of the neghbourhood $\Omega $ with the closure of any canonical domain $R$. Now take $\Omega $ to be the union of all $\Omega_R$. This is a neghbourhood of $Z$ that belongs to $\Omega_0$. Let us prove that its boundary components satisfy the Boundary Lemma.
By construction, $\partial \Omega $ is a $C^1$-smooth one-dimensional compact submanifold of the sphere. Hence it is a finite union of topological circles. Consider an arbitrary connected component $\varphi $ of $\partial \Omega $.
If $\varphi $ intersects a canonical region $R$ of case 1), 2), or 3), then it belongs entirely to $R$ and is of Type $1$ or $3$ as proved in Section \ref{sub:123}.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.4\textwidth]{07-2}
\end{center}
\caption{Boundary component of Type $2$ without contact (dashed) in the union of two canonical regions of case 4) (shadowed). Boundaries of canonical regions are shown in thick}\label{fig:CanonReg4}
\end{figure}
Suppose that $\varphi $ intersects canonical regions of case 4) only. Then it has no contacts with $v$, see Fig. \ref{fig:CanonReg4}, i.e. is transversal. Assume that it is outgoing. All future semi-orbits of $v$ that start on $\varphi $ do not intersect $\Omega $ and have the same $\omega$-limit set, which is clear for the orbits located inside each canonical region of case 4). Let $R$ be any canonical region that contains a subarc of $\varphi $; then $\varphi$ intersects the first or the last separatrices in boundary chains of $R$, so $\varphi$ intersects at least one separatrix of $v|_{\Omega}$.
Hence $\varphi $ is a boundary component of Type $2$ transversal to $v$.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.4\textwidth]{07-1}
\end{center}
\caption{Boundary component of Type $2$ (dashed) with two contact points in the union of one canonical region of case 5 (shadowed) and two canonical regions of case 4. Boundaries of canonical regions are shown in thick}\label{fig-CR-inters-1}
\end{figure}
Suppose that $\varphi $ intersects at least one canonical region of case 5), see Fig. \ref{fig-CR-inters-1}. Then it has at least one point of outer quadratic tangency with $v$ (thus it has at least two tangency points). Let $\beta $ be a transversal arc of $\varphi $ between two such points. Then
$$\beta = \beta'\cup \bigcup_{i=1}^{k-1} \varphi(R_i) \cup \beta''$$
where $R_j$ are regions of case 4) and $\beta' , \beta''$ are subarcs of $\varphi (R_0), \varphi(R_k)$; here $R_0, R_k$ are canonical regions of case 5). Subarcs $\beta' , \beta''$ contain the endponts of $\beta $.
As before, all the orbits of $v|_{\Omega\cap {R_j}}$ that start at $\varphi (R_j)$ or at $\beta', \beta''$, stay in $\Omega$ and have the same $\omega$-limit set. Since the arcs $\varphi (R_i), \varphi (R_{i+1})$ have common endpoints, this holds for the whole arc $\beta $ too.
Finally, $\beta $ is crossed by the first or the last separatrices of the boundary chains of the corresponding canonical domains. They are separatrices of $v|_{\Omega_R}$, due to the description of boundaries of canonical regions.
Hence $\varphi$ is a boundary component of type 2, with at least two outer tangency points with $v$.
This completes the proof of the Boundary lemma.
\section{Images of boundary components of $U$} \label{sec-Case123}
Here we prove Lemma \ref{lem-image-case1}, Lemma \ref{lem-image-Case2} and Lemma \ref{lem-image-case3}.
As before, we assume that $U$, $\tilde U^{\pm}$ are chosen as in Proposition \ref{prop-choice} (recall that this proposition only uses Boundary lemma, and this lemma is already established).
\subsection{Canonical regions for vector fields in open domains on the sphere}
We will need the generalizations of Propositions \ref{prop-canonreg-common-limset}, \ref{prop-parall} to the case of vector fields on subdomains of $S^2$.
Take an open set $D \subset S^2$ such that $\partial D$ is a union of finitely many continuous curves homeomorphic to $S^1$ and having finitely many topological tangencies with $v$. We assume that singular points, limit cycles and monodromic polycycles that belong to $\overline D$ also belong to $D$.
\begin{definition}[Canonical regions in domains]
\label{def-canonreg-subdomain}
For a vector field $v \in Vect^*\,S^2$ and an open set $D\subset S^2$ as above, let $S(v, D)$ be the union of all singular points, separatrices and limit cycles of $v|_D$, and let $Tang(v,D)$ be the union of trajectories under $v|_{D}$ of topological tangency points of $v$ with $\partial D$. A \emph{canonical region} of $v|_{D}$ is a connected component of $D\setminus (S(v,D)\cup Tang(v,D))$.
\end{definition}
\begin{proposition}
\label{prop-canonreg-limset-subdomain}
For a vector field $v \in Vect^*\,S^2$ and an open set $D\subset S^2$ as above, all points of the same canonical region of $v|_{D}$
\begin{itemize}
\item either have the same $\omega$-limit set under $v$ inside $D$, and their future semi-trajectories stay in $D$;
\item or their future semi-trajectories under $v|_{D}$ terminate on the same connected component of $\partial D$.
\end{itemize}
The same alternative holds for $\alpha$-limit sets and past semi-trajectories.
\end{proposition}
\begin{proof}
Let $R$ be a canonical region of $v|_{D}$. Consider a set $G$ of points in $R$ such that their trajectories stay in $D$ and have one and the same $\omega$-limit set $A$ under $v$. The set $A$ is inside $\overline D$, thus inside $D$, due to our assumptions on $D$.
The set $G$ is open; the proof is similar to that in Proposition \ref{prop-canonreg-common-limset}. The only new argument to be added is, that if the trajectory of a point stays in $D$ and has its $\omega$-limit set inside $D$, then the
trajectories of close points also stay in $D$.
Now, consider a set $G$ of points in $R$ such that their future semi-trajectories under $v|_{D}$ terminate on one and the same connected component of $\partial D$. We will prove that the set $G$ is also open. Let $x \in G$, and $y\in \partial D$ be the endpoint of its future semi-trajectory under $v|_{D}$. In a sufficiently small flow-box around $y$, $\partial D$ is a continuous curve that intersects all trajectories of $v$; this follows from the fact that $\partial D$ has only finitely many tangencies with $v$ and $y$ is not an inner tangency point itself.
Now it suffices to notice that each future semi-trajectory of $v$ that starts near $x$ eventually reaches the flow-box of $y$, thus intersects the same connected component of $\partial D$.
Finally, since $R$ cannot be a union of several open disjoint sets, it coincides with one of the sets above: either all its points have the same $\omega$-limit set under $v$ inside $D$, and their future semi-trajectories stay in $D$; or their future semi-trajectories terminate on the same connected component of $\partial D$.
\end{proof}
\begin{proposition}
\label{prop-canonreg-parall-subdomain}
For a vector field $v \in Vect^*\,S^2$ and an open set $D\subset S^2$ as above, each canonical region of $v|_{D}$ is parallel, i.e. equivalent to a strip flow or a spiral flow.
\end{proposition}
\begin{proof}
The proof is the same as for the case of $D=S^2$, see \cite[Proposition 1.42, p. 34]{DumLlibArt} for omitted details. Namely, the quotient space obtained by collapsing orbits of $v|_{R}$ into points is a (Hausdorff) connected one dimensional manifold (i.e. $S^1$ or $\mathbb R$), and
the natural projection of $R$ to this quotient space is a locally trivial fibering. So it can be homeomorphic to $\mathbb R \times \mathbb R \to \mathbb R$ (then we have a strip flow), or $S^1 \times \mathbb R \to \mathbb R$ (spiral flow), or
$\mathbb R \times S^1 \to \mathbb R$ (annular flow. However in this case, $v$ has infinitely many periodic orbits which is impossible for $v\in Vect^*\, S^2$).
\end{proof}
\subsection{Images of Type $2$ boundary components}
The following proposition is the main part of the proof of Lemma \ref{lem-image-Case2}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.3\textwidth]{08-1}
\end{center}
\caption{Canonical region of an outer topological tangency point of $v_0$. The sets
$A,B$ are located inside the domains with the dotted boundaries, and not shown
on the figure.}\label{fig-canonreg-of-p}
\end{figure}
\begin{figure}[ht]
\begin{center}
\centering
\subcaptionbox{\label{fig-canonreg-of-p-eps-1}}{ \includegraphics[width=0.35\textwidth]{im-type2-1} }
\hfil
\subcaptionbox{\label{fig-canonreg-of-p-eps-2}}{ \includegraphics[width=0.35\textwidth]{im-type2-2}}
\caption{Canonical region of outer tangency points a) for $w_0$ b) for $w_{h(\varepsilon)}$. On this figure, the objects with tilde are the images under $\hat H$ of the corresponding objects without tilde. The sets $\tilde A, \tilde B$ are located inside the domains with the dotted boundaries, and not shown
on the figure.}\label{fig-canonreg-of-p-eps}
\end{center}
\end{figure}
\begin{proposition}
\label{prop-outer-cont-pt-w}
Under assumptions of the Main theorem and Proposition \ref{prop-choice}, for sufficiently small $\varepsilon$, for each outer topological tangency point of $\partial H_{\varepsilon}(U)$ with $w_{h(\varepsilon)}$, its trajectory under $w_{h(\varepsilon)}$ belongs to $S^2\setminus H_{\varepsilon}(U^*)$.
\end{proposition}
\begin{proof}
Let $q$ be an outer tangency point of a boundary component $\varphi \subset \partial U^*$. Let $\beta\subset \varphi$ be an outgoing transversal arc with the endpoint $q$. Consider a canonical region $R$ of $v_0$ that contains $q$. Recall that $A :=\omega_{v_0}(q)$ and $B:=\alpha_{v_0}(q)$ do not intersect $LBS^*(V)$, and $\beta $ intersects separatrices of $v_{0}|_{U}$, due to the Boundary lemma. So $R$ cannot contain the whole arc $\beta$: $\partial R$ contains an intersection of a separatrix of $v_{0}|_{U}$ with $\beta$. Thus $\partial R$ contains a singular point $P \in LBS^*(V)$, the $\alpha$-limit set of this separatrix; $P\in LBS^*(V)$ due to No-entrance lemma. Finally, $R$ is of case 5) according to the classification introduced in the proof of the Boundary lemma: $\alpha(R)$ and $\omega(R)$ do not intersect $LBS^*(V)$, and $\partial R$ contains a point $P\in LBS^*(V)$ on its boundary (see Fig. \ref{fig-canonreg-of-p}).
Similarly, if $R$ contains two outer tangency points with $U^*$, then it contains points $P_1, P_2 \in LBS^*(V)$ on both its side boundaries $\nu_1(R), \nu_2(R)$.
Now, $\tilde R = \hat H(R)$ is a canonical region for $w_0$, its $\alpha$-, $\omega$-limit sets $\tilde A :=\hat H(A)$, $\tilde B :=\hat H(B)$ do not belong to $LBS^*(W)$, and it has a singular point $\tilde P := \hat H(P)\in LBS^*(W)$ on its boundary, see Fig. \ref{fig-canonreg-of-p-eps-1}. It also contains trajectories that do not intersect $\tilde U^{*+}$ (see Remark \ref{rem-trajectory-case5}; it is applicable because $\tilde U^{*+}$ satisfies Boundary lemma, due to Proposition \ref{prop-choice}).
Take an arc $I\subset \tilde R$ transversal to $w_0$, such that the trajectory of one of its endpoint under $w_0$ does not visit $\tilde U^{+*}$ (this is possible due to Remark \ref{rem-trajectory-case5}), and the trajectory of another its endpoint is close to $\hat H(P)$, so visits $\tilde U^{-*}$, see Fig. \ref{fig-canonreg-of-p-eps-1}. Both trajectories connect $\hat H(A)$ to $\hat H(B)$. The same holds for the trajectories of the endpoints of $I$ under $w_{h(\varepsilon)}$ with small $\varepsilon$.
Namely, both of them connect transversal loop around $\hat H(A)$ and $\hat H(B)$; one of them intersects $H_\varepsilon(\varphi)$, and the other does not.
Due to the continuity of orbits of $w_{h(\varepsilon)}$ with respect to the initial conditions, and the fact that $\tilde U^{-*}\subset H_{\varepsilon}(U^*) \subset \tilde U^{*+}$, there exists a point in $I$ whose trajectory under $w_{h(\varepsilon)}$ visits $\overline{H_{\varepsilon}(U^*)}$ and does not visit $H_{\varepsilon}(U^*)$. This trajectory contains one or several topological tangency points of $H_{\varepsilon}(U)$ with $w_{h(\varepsilon)}$ (see Fig. \ref{fig-canonreg-of-p-eps-2}).
Finally, if $R$ contains a point of outer tangency of $\partial U$ with $v_0$, then $\hat H(R)$ contains at least one point of outer tangency of $H_{\varepsilon}(U)$ with $w_{h(\varepsilon)}$.
If $R$ contains two points of outer tangency of $U$ (i.e. points of $LBS(V)$ on both boundaries), then the same construction yields at least two tangency points in $\hat H(R)$.
Let the total number of outer tangency points of $\partial U$ with $v_0$ be $N$. The construction above yields at least $N$ topological outer tangency points of $H_{\varepsilon}(U)$ with $w_{h(\varepsilon)}$, and their trajectories under $w_{h(\varepsilon)}$ do not visit $H_{\varepsilon}(U)$.
On the other hand, the total number of outer tangency points of $H_{\varepsilon}(U)$ with $w_{h(\varepsilon)}$ is $N$, because $H_{\varepsilon}$ identifies outer tangency points of $U$ and $H_{\varepsilon}(U)$. So we have found all of them. Hence trajectories of all outer tangency points of $H_{\varepsilon}(U)$ with $w_{h(\varepsilon)}$ stay in $S^2\setminus \overline {H_{\varepsilon}(U^*)}$.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem-image-Case2}]
Let $\beta, \ \tilde \beta_\varepsilon$ and $\tilde l$ be the same as in Lemma \ref{lem-image-Case2}. Obviously $\tilde \beta_\varepsilon$ is topologically transversal to $w_{h(\varepsilon)}$. Trajectories of its points under $w_{h(\varepsilon)}|_{S^2\setminus H_\varepsilon(U^*)} $ are not trajectories of outer tangency points due to Proposition \ref{prop-outer-cont-pt-w},
and none of them are separatrices of $w_{h(\varepsilon)}|_{S^2\setminus H_{\varepsilon}(U)}$, due to No-entrance lemma \ref{lem-seps} (applied to $\tilde U^+\supset H_{\varepsilon}(U)$ and the family $W$). Due to Definition \ref{def-canonreg-subdomain} of canonical regions in domains, $\tilde \beta_\varepsilon$ belongs to one canonical region of $w_{h(\varepsilon)}|_{S^2\setminus H_{\varepsilon}(U^*)}$.
Let us prove that all trajectories of this canonical region cross $\tilde l$; this will imply that the Poincare map $\tilde P_{\varepsilon}$ is defined. Due to Proposition \ref{prop-canonreg-limset-subdomain}, it is sufficient to prove this statement for one trajectory. By assumption of Lemma \ref{lem-image-Case2}, $\beta$ is a maximal transversal arc of a boundary component of $U$ of Type $2$. Hence, there exists a separatrix $\gamma$ of $v_0$ that crosses $\beta$. Let $r = \gamma \cap \beta$.
\begin{proposition}
\label{prop:sep-pin}
Let $\gamma, \beta $, and $r$ be the same as above. Then the trajectory of $H_{\varepsilon}(r) \in \tilde \beta_\varepsilon$ under $w_{h(\varepsilon)}|_{S^2\setminus H_{\varepsilon}(U^*)}$ crosses $\tilde l$ for small $\varepsilon$, and the intersection point is close to $p:=\hat H(\gamma)\cap \tilde l$ for small $\varepsilon$.
\end{proposition}
\begin{proof}
The future semi-trajectory of $\hat H(r)$ under $w_0$ (i.e. the part of the separatrix $\hat H(\gamma)$) crosses $\tilde l$ at the point $p$; it does not visit $H_{\varepsilon}(U)\subset \tilde U^{+*}$ due to No-entrance lemma \ref{lem-seps} applied to the family $W$ and $\tilde U^{+*}$.
Since $\mathbf H$ is continuous on $\mathop{\mathrm{Sep}} v_0$ (see Requirement \ref{it-H-contin} of Definition \ref{def-moderate-local} of moderate equivalence), the point $H_{\varepsilon}(r)$ is close to $\hat H(r)$. Since $w_{h(\varepsilon)}$ is close to $w_0$, the trajectory of $H_{\varepsilon}(r)$ under $w_{h(\varepsilon)}$ intersects $\tilde l$ and does not visit $\tilde U^{+*}$. The intersection point is close to $p$.
\end{proof}
We showed that one trajectory starting at $\tilde \beta_\varepsilon$ does not visit $\tilde U^{+*}$ and crosses $\tilde l$. Since $\tilde \beta_\varepsilon$ belongs to one canonical region, the same holds for all trajectories of $\tilde \beta_\varepsilon$ (Proposition \ref{prop-canonreg-limset-subdomain}), and the Poincare map $\tilde P_{\varepsilon}$ is well-defined. Proposition \ref{prop:sep-pin} also implies that $\tilde P_{\varepsilon}(\tilde \beta_\varepsilon)$ contains a point close to $p\in \tilde P_0(\tilde \beta_0)$. Since $p$ is the inner point of $\tilde P_0(\tilde \beta_0)$, we conclude that for small $\varepsilon$, $\tilde P_{\varepsilon}(\tilde \beta_\varepsilon)$ intersects $\tilde P_0(\tilde \beta_0)$. This completes the proof.
\end{proof}
\subsection{Images of Type $1$ boundary components}
\begin{proof}[Proof of Lemma \ref{lem-image-case1}]
Let $\varphi$ be the same as in Lemma \ref{lem-image-case1}. Let $C$ be a connected component of $S^2 \setminus \overline{H_{\varepsilon}(U)}$ adjacent to $H_{\varepsilon}(\varphi)$. Clearly, $H_{\varepsilon}(\varphi)$ is a union of two topologically transversal arcs to $w_{h(\varepsilon)}$.
The endpoints of these arcs are points of inner topological tangency of $H_{\varepsilon}(\varphi)$ with $w_{h(\varepsilon)}$ (here "inner" means "inner with respect to $H_\varepsilon(U)$").
Let $p$ be one of these endpoints.
Consider the canonical region $R$ of $w_{h(\varepsilon)}|_{C}$ that contains $p$ on its boundary. Let us prove that it contains $H_{\varepsilon}(\varphi)$ and coincides with $C$. By contraposition, suppose that the intersection $R \cap H_{\varepsilon}(\varphi)$ is a proper subarc of $H_{\varepsilon}(\varphi)$.
Consider an endpoint $q$ of this arc. By Definition \ref{def-canonreg-subdomain} of canonical regions, $q$
either belongs to an orbit tangent to $\partial C$, or belongs to a separatrix of $w_{h(\varepsilon)}|_C$. The second option (see Fig. \ref{fig-case1-a})is impossible due to No-entrance lemma \ref{lem-seps} applied to $W$ and $\tilde U^+$. Prove that the first option is also impossible.
First, it is not possible that the point $q$ is itself a point of tangency of $w_{h(\varepsilon )}$ and $H_\varepsilon (\varphi )$ whose orbit locally belongs to $\overline C $ (see Fig. \ref{fig-case1-b}). Indeed, all orbits of points of $\varphi $ under $v_\varepsilon $ enter $U$ either in the positive, or in the negative time. So do the
orbits of the points of $H_\varepsilon (\varphi )$ under $w_{h(\varepsilon )}$. Hence, none of these orbits locally belong to $\overline C$.
\begin{figure}[h]
\begin{center}
\subcaptionbox{\label{fig-case1-a}}{\includegraphics[width=0.25\textwidth]{im-Type1-a}}
\hfil
\subcaptionbox{\label{fig-case1-b}}{\includegraphics[width=0.25\textwidth]{im-Type1-b} }
\hfil
\subcaptionbox{\label{fig-case1-c}}{\includegraphics[width=0.25\textwidth]{im-Type1-c} }
\end{center}
\caption{Images of Type $1$ components; all three pictures are impossible}\label{fig-case1}
\end{figure}
Second, it is not possible that the orbit of $q$ under $w_{h(\varepsilon )}|_{\overline C}$ contains another point of outer tangency with $\partial H_{\varepsilon}(U)$ -- say, with some connected component of $H_\varepsilon (U)$ that belongs to $C$, see Fig. \ref{fig-case1-c}. Here ``outer'' is understood as ``outer with respect to $H_{\varepsilon}( U)$''. Indeed, by Proposition \ref{prop-outer-cont-pt-w}, the orbit of this outer tangency point must stay in the complement of $H_{\varepsilon}( U)$, and thus cannot reach any point $q \in \partial H_{\varepsilon}( U)$.
The contradiction obtained proves that the canonical region of $w_{h(\varepsilon)}|_{C}$ that contains $p$ must contain the whole $H_{\varepsilon}(\varphi)$, thus coincides with $C$. Due to Proposition \ref{prop-canonreg-parall-subdomain}, this canonical region is parallel. This completes the proof.
\end{proof}
\subsection{Images of Type $3$ boundary components}
\begin{proof}[Proof of Lemma \ref{lem-image-case3} (see Fig. \ref{fig-case3})]
In what follows, the open annulus between two curves $\gamma_1, \gamma_2$ is denoted by $A(\gamma_1, \gamma_2)$. We also use this notation for the annulus between a polycycle or a singular point and its transversal loop.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.8\textwidth]{Case3Lem}
\end{center}
\caption{Images of Type $3$ components}\label{fig-case3}
\end{figure}
Recall that $\varphi$ is a boundary component of Type $3$, i.e. a transversal loop of some singular point, limit cycle, or polycycle $c$ of $v_0$. Consider the corresponding object $\tilde c:= \hat H(c)$ of $w_0$.
Let $D$ be a disc bounded by $\tilde c$ that contains $\hat H(\varphi)$.
Since $\tilde U^{+}, \tilde U^-$ satisfy Boundary lemma, they contain Type 3 boundary components $\varphi^{\pm}$ that correspond to $\tilde c$ and are in $D$. So $\varphi^{\pm}$ are two transversal loops of $\tilde c$. Let $D^+ \ (D^-)$ be a disc bounded by $\varphi^+ \ (\varphi^-)$ and containing $\tilde c$.
We proceed in the following steps.
\textbf{Step 1: $H_{\varepsilon}(\varphi)$ belongs to $A:=D^+ \setminus D^-$ for any small $\varepsilon$ (including $\varepsilon=0$).}
Since $\tilde U^- \subset H_{\varepsilon}(U)\subset \tilde U^+$ (see Proposition \ref{prop-choice}), we have $\partial H_{\varepsilon}(U) \subset \tilde U^+\setminus \tilde U^-$. In particular, $H_{\varepsilon}(\varphi) \subset \tilde U^+\setminus \tilde U^-$. However it is not clear why this curve belongs to $A$ and not to another connected component of $\tilde U^+\setminus \tilde U^-$.
\textbf{Step 1.1: $H_{\varepsilon}(\varphi)$ belongs to $D^+$}
Take a small neighborhood $\Omega$ of $\tilde c $ that belongs to $\tilde U^-$. Take $\varepsilon$ so small that $H_{\varepsilon}(c) \subset \Omega $; this is possible because $H_{\varepsilon}(c)$ is close to $H_0(c)=\tilde c$, see Definition \ref{def-moderate-local} of moderate equivalence. Let $D_{\varepsilon}$ be the disc bounded by $H_{\varepsilon}(c)$ that contains $\varphi^-$ and $\varphi^+$; this disc is close to $D$, and $D \bigtriangleup D_{\varepsilon} \subset \Omega$.
Let $U_c$ be the connected component of $\tilde U^+$ that contains $\tilde c$; then $U_c \subset D^+$. The annulus $H_{\varepsilon}(A(c, \varphi))\subset H_{\varepsilon}(U)$ is connected, belongs to $ \tilde U^+ $ and contains $H_{\varepsilon}(c)\subset \Omega$, thus has a non-empty intersection with $U_c$. Hence it belongs to $U_c$, therefore to $D^+$. Finally, $H_{\varepsilon}(\varphi)\subset D^+ $.
\textbf{Step 1.2: $H_{\varepsilon}(\varphi)$ belongs to $D_{\varepsilon}$}
If $c$ is a singular point, this is clear because $S^2 \setminus D_{\varepsilon}$ is one point $H_{\varepsilon}(c)$. Let $c$ be a limit cycle or a monodromic polycycle. Consider the time orientation on it.
Without loss of generality, we may assume that $\varphi$ is to the left with respect to this orientation of $c$. The maps $H_{\varepsilon}, H_0$ induce an orientation on $H_{\varepsilon}(c), H_0(c)=\tilde c$. With this orientation, $H_0(\varphi)$ is to the left with respect to $\tilde c$, thus $\varphi^{\pm}$ are to the left of $\tilde c$.
The curve $H_{\varepsilon}(c)$ is close to $\tilde c$ by Definition \ref{def-moderate-local} of moderate equivalence.
Hence, the curves $\varphi^{\pm}$ lie to the left of both $H_{\varepsilon}(c)$ and $\tilde c$; therefore, the disc $D_{\varepsilon}$ is to the left of $H_{\varepsilon}(c)$.
On the other hand, as $H_{\varepsilon}$ preserves the orientation, and $\varphi$ is to the left of $c$, we conclude that $H_{\varepsilon}(\varphi)$ is to the left of $H_{\varepsilon}(c)$. Finally, $H_{\varepsilon}(\varphi)\subset D_{\varepsilon}$, q.e.d.
\textbf{Step 1.3: $H_{\varepsilon}(\varphi)$ does not intersect $D^-\cap D_{\varepsilon}$ (thus does not intersect $D^-$). }
Indeed, $D \cap D^- =A(c,\varphi^-)\subset \tilde U^-$ and $D \bigtriangleup D_{\varepsilon}\subset \Omega\subset \tilde U^-$. So $D^-\cap D_{\varepsilon}\subset \tilde U^-$, and the curve $H_{\varepsilon}(\varphi)\subset \tilde U^+ \setminus \tilde U^-$ cannot intersect this set.
We conclude that $H_{\varepsilon}(\varphi) \subset D^+ \setminus D^-=A$.
\textbf{Step 2: The curve $H_{\varepsilon}(\varphi)$ is non-contractive in the annulus $A$}
Indeed, this annulus is between two transversal loops of $\tilde c$, thus is saturated by trajectories of $w_0$ and by trajectories of a close vector field $w_{h(\varepsilon)}$. Since $H_{\varepsilon}$ preserves topological transversality, $H_{\varepsilon}(\varphi)$ is topologically transversal to $w_{h(\varepsilon)}$, thus is non-contractive in $A$.
\textbf{Step 3: The curves $H_0(\varphi)$ and $H_{\varepsilon}(\varphi)$ are homotopic in $A$}
Recall that we orient $\varphi$ so that $U$ is to the left with respect to it.
Both curves $H_0(\varphi)$ and $H_{\varepsilon}(\varphi)$ are non-contractive in $A$ and oriented so that $H_0(U), H_{\varepsilon}(U)$ are to the left with respect to them, i.e. $\varphi^-$ is to the left of them. So they are oriented in the same way, thus homotopic in $A$ as oriented curves.
\textbf{Step 4: End of the proof}
Recall that each singular point and each limit cycle of $w_{h(\varepsilon)}$ either belongs to $\tilde U^- \subset H_{\varepsilon}(U)$, or is close to a hyperbolic singular point or a cycle of $w_0$ (thus does not intersect $\tilde U^+\supset H_{\varepsilon}(U)$), see Proposition \ref{prop-inU-or-hyp}. Also, $LBS(W)\subset \tilde U^-$. In the previous step, we have proved that $H_0(\varphi)$ and $H_{\varepsilon}(\varphi)$ are homotopic in $A \subset \tilde U^+ \setminus \tilde U^-$. So they are homotopic in a larger domain $S^2 \setminus (LBS(W) \cup \mathop{\mathrm{Sing}} w_{h(\varepsilon)} \cup \mathop{\mathrm{Per}} w_{h(\varepsilon)})$, q.e.d.
\end{proof}
Thus all the auxiliary lemmas are proved. Together with them, the Main Theorem is proved too.
\section*{Acknowledgements}
The authors were supported in part by the grant RFBR 16-01-00748-a and Laboratory Poncelet.
The authors are grateful to Dmitry Filimonov who read the paper and made fruitful comments, and to Ilya Schurov who drew Fig. \ref{fig:onecompu}, \ref{fig:onecompcu}, \ref{fig:CanonReg4}, \ref{fig-CR-inters-1}, \ref{fig-canonreg-of-p}, \ref{fig-canonreg-of-p-eps} (based on handwritten sketches).
\section*{ Note of the second author}
The first part of this paper (Sections \ref{sec-intro} -- \ref{sec-proof}) was prepared by the two authors together. The most difficult second part was designed and written by the first author; the second made only editorial changes in this part, when the first draft was already written.
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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Q: spring data mongodb custom interface ConflictingBeanDefinitionException Following Spring Data MongoDB manual, I have created repository and implementation classes. However when I start my application spring bootstrap fails with
Caused by: org.springframework.context.annotation.ConflictingBeanDefinitionException:
Annotation-specified bean name 'mockTestRepositoryImpl' for bean class [com.foo.testapp.mongo.repository.impl.MockTestRepositoryImpl] conflicts with existing, non-compatible bean definition of same name and class [com.foo.testapp.mongo.repository.impl.MockTestRepositoryImpl]
My config and code: (groovy)
package com.foo.testapp.mongo.repository.impl
@Component
class MockTestRepositoryImpl implements MockTestRepositoryCustom {
MongoOperations mongoOperations
@Autowired
MockTestRepositoryImpl(MongoOperations mongoOperations) {
this.mongoOperations = mongoOperations
}
@Override
MockTest findAndModify(Query query, Update update, FindAndModifyOptions options) {
return mongoOperations.findAndModify(query, update, options, MockTest)
}
@Override
boolean exists(Query query) {
return mongoOperations.exists(query, MockTest)
}
}
Custom interface:
package com.foo.testapp.mongo.repository
public interface MockTestRepositoryCustom {
MockTest findAndModify(Query query, Update update, FindAndModifyOptions options)
boolean exists(Query query);
}
Spring Data repository:
package com.foo.testapp.mongo.repository
public interface MockTestRepository extends MongoRepository<MockTest, String>, MockTestRepositoryCustom {
List<MockTest> findByDeleted(boolean deleted);
MockTest findByIdAndDeleted(String id, boolean deleted);
@Query(value = "{ 'deleted' : ?0 }", fields = "{ 'name' : 1 }")
List<MockTest> findAllBasicInfo(boolean deleted);
}
A: The problem is that you have annotated MockTestRepositoryImpl with @Component.
You need to remove @Component altogether (as you can see in the manual there is no such annotation - but all Spring features work correctly).
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,758
|
{"url":"https:\/\/zbmath.org\/?q=an:0585.34048&format=complete","text":"zbMATH \u2014 the first resource for mathematics\n\nOn functional integrability of solutions of differential equations with deviating argument. (English) Zbl\u00a00585.34048\nConditions are given for the solutions of the equation $$(G_{n- 1}x(t))'+f(t,x(g(t)))=h(t),$$ $$t\\geq a$$, to satisfy $$0<\\int^{\\infty}_{a}s^ mW(| x(s)|)ds<\\infty,$$ where $$G_{n-1}x(t)=a_{n-1}(t)(a_{n-2}(t)(...(a_ 1(t)x'(t))'...)')\\quad ',$$ m is a real number and $$W: {\\mathbb{R}}\\to {\\mathbb{R}}$$ (W($$| u|)\\geq 0)$$ is a continuous and non-decreasing function.\nReviewer:\u00a0M.M.Konstantinov\nMSC:\n 34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument) 34C11 Growth and boundedness of solutions to ordinary differential equations\nFull Text:\nReferences:\n [1] HRUBINOV\u00c1 A., \u0160OLT\u00c9S V.: Asymptotic properties of oscillatory solutions of n-th order differential equations with delayed argument. Zborn\u00edk vedeck\u00fdch pr\u00e1c V\u0160T \u00b7 Zbl\u00a00583.34062 [2] WERBOWSKI J., WYRWINSKA A.: On functional integrability of solutions of differential equations with deviating argument. Colloquia Mathematica Societatis Janos Bolyai 30. Qualitative Theory of Differential Equations, Szeged (Hungary), 1979, 1045-1059. \u00b7 Zbl\u00a00486.34052 [3] WYRWINSKA A.: On the functional integrability and asymptotic behaviours of a certain differential equation with delay. Math. Slovaca 33, 1983, 45-51. \u00b7 Zbl\u00a00526.34053\nThis reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.","date":"2021-10-22 00:02:39","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\": 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\": 0.5442373752593994, \"perplexity\": 2669.414502713755}, \"config\": {\"markdown_headings\": false, \"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\/1634323585449.31\/warc\/CC-MAIN-20211021230549-20211022020549-00507.warc.gz\"}"}
| null | null |
using Amba.ImagePowerTools.Models;
using Newtonsoft.Json;
using Orchard.ContentManagement;
using System.Collections.Generic;
namespace Amba.ImagePowerTools.Fields
{
public class ImageMultiPickerField : ContentField
{
private IEnumerable<SelectedImage> _dataCache = null;
public string Data
{
get
{
return Storage.Get<string>("Data");
}
set
{
_dataCache = null;
Storage.Set("Data", value);
}
}
public IEnumerable<SelectedImage> Images
{
get
{
if (_dataCache != null)
return _dataCache;
var data = Data;
if (string.IsNullOrEmpty(data))
return new List<SelectedImage>();
_dataCache = JsonConvert.DeserializeObject<IEnumerable<SelectedImage>>(data);
return _dataCache;
}
set
{
_dataCache = null;
Data = JsonConvert.SerializeObject(value);
}
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,512
|
Q: using sdl in an iphone app I am thinking of using SDL in an iPhone app. Can it be done without any major problems on an app which uses iOS 4.2 or iOS 4.3 SDKs?
Appreciate any pointers.
A: As I know, the SDL framework for iPhone isn't mature yet. But I've never tried to use it on iPhone because I found a better 2D engine, before try SDL, which is Cocos2D.
Is you're planning to port an old project on iPhone, I only can tell you to try...
But if you're planning to work on a new project with SDL because you're more familiar with SDL than others engine. I think you should go on Cocos2D.
Anyway it's my point of view, hope this help you to choose what is best.
Regards, kl94
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,170
|
Q: Convergence of the series $\sum_{n=1}^\infty a_n = \frac{(n^n)(x^n)}{(n!)}$ With the help of ratio test, I have been successful in proving it is convergent for
$ex<1$ and divergent for $ex>1$. however for $ex=1$, the ratio test has failed. Hence I had to apply the logarithm test.
The form after applying log test is: n*(-n*log(1+1/n)+1)
this is where it's becoming problematic for me. I do not have any idea how to proceed from here. any help is greatly appreciated. thank you.
A: $\frac {a_{n+1}} {a_n}=(1+\frac1 n)^{n} x$ so the ratio test tells you that the series converges for $|x| <\frac 1 e$ and diverges for $|x|>\frac 1 e$.
When $x =\frac 1 e$ use Stirling's formula to compare the series with $\frac 1 {\sqrt {2\pi}} \frac 1 {\sqrt n}$ to see that the series diverges.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,763
|
Enjoy a selection of three of our favourite single malts in a stylish presentation box. The presentation box includes 5cl bottles of Glenfiddich 12 Year Old, the pioneer of the single malt category, Glenfiddich 15 Year Old, created using our innovative Solera system, and Glenfddich 18 Year Old, which we marry in small batches. The perfect present for any whisky lover.
Great for: Experiencing the differences between various blends. Understanding and appreciating the different terms used in describing a malt from nutty or peaty to subtle or big and many more.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,247
|
Q: Failed to create NativeModule "UIManager" in React Native app After upgrading to the latest version of React Native (0.70.3), my Android app builds fine but when I run it I get the below error. It references a class that seems to be in the react-native-video-with-ads npm package, which is not something that I use. I tried manually installing this package, but still get the same error. Any ideas?
2022-10-23 10:08:02.997 25180-25295/com.marriage365.app E/unknown:NativeModuleInitError: Failed to create NativeModule "UIManager"
java.lang.VerifyError: Verifier rejected class com.brentvatne.exoplayer.ReactExoplayerViewManager: android.view.View com.brentvatne.exoplayer.ReactExoplayerViewManager.createViewInstance(com.facebook.react.uimanager.ThemedReactContext) failed to verify: android.view.View com.brentvatne.exoplayer.ReactExoplayerViewManager.createViewInstance(com.facebook.react.uimanager.ThemedReactContext): [0x4] can't resolve returned type 'Reference: android.view.View' or 'Unresolved Reference: com.brentvatne.exoplayer.ReactExoplayerView' (declaration of 'com.brentvatne.exoplayer.ReactExoplayerViewManager' appears in /data/app/~~Nq33b85Ca7DH-MTNDVGU3Q==/com.marriage365.app-mNOnqtNxaI4Mm9muOh5Rig==/base.apk!classes2.dex)
at com.brentvatne.react.ReactVideoPackage.createViewManagers(ReactVideoPackage.java:42)
at com.facebook.react.ReactInstanceManager.getOrCreateViewManagers(ReactInstanceManager.java:928)
at com.swmansion.reanimated.ReanimatedPackage.createUIManager(ReanimatedPackage.java:75)
at com.swmansion.reanimated.ReanimatedPackage.getModule(ReanimatedPackage.java:31)
at com.facebook.react.TurboReactPackage$ModuleHolderProvider.get(TurboReactPackage.java:159)
at com.facebook.react.TurboReactPackage$ModuleHolderProvider.get(TurboReactPackage.java:147)
at com.facebook.react.bridge.ModuleHolder.create(ModuleHolder.java:191)
at com.facebook.react.bridge.ModuleHolder.getModule(ModuleHolder.java:156)
at com.facebook.react.bridge.NativeModuleRegistry.getModule(NativeModuleRegistry.java:170)
at com.facebook.react.bridge.CatalystInstanceImpl.getNativeModule(CatalystInstanceImpl.java:502)
at com.facebook.react.bridge.CatalystInstanceImpl.getNativeModule(CatalystInstanceImpl.java:478)
at com.facebook.react.uimanager.UIManagerHelper.getUIManager(UIManagerHelper.java:89)
at com.facebook.react.uimanager.UIManagerHelper.getUIManager(UIManagerHelper.java:47)
at com.facebook.react.ReactInstanceManager.attachRootViewToInstance(ReactInstanceManager.java:1240)
at com.facebook.react.ReactInstanceManager.setupReactContext(ReactInstanceManager.java:1182)
at com.facebook.react.ReactInstanceManager.access$1600(ReactInstanceManager.java:136)
at com.facebook.react.ReactInstanceManager$5$2.run(ReactInstanceManager.java:1136)
at android.os.Handler.handleCallback(Handler.java:938)
at android.os.Handler.dispatchMessage(Handler.java:99)
at com.facebook.react.bridge.queue.MessageQueueThreadHandler.dispatchMessage(MessageQueueThreadHandler.java:27)
at android.os.Looper.loop(Looper.java:223)
at com.facebook.react.bridge.queue.MessageQueueThreadImpl$4.run(MessageQueueThreadImpl.java:228)
at java.lang.Thread.run(Thread.java:923)
2022-10-23 10:08:02.997 25180-25295/com.marriage365.app E/AndroidRuntime: FATAL EXCEPTION: mqt_native_modules
Process: com.marriage365.app, PID: 25180
java.lang.VerifyError: Verifier rejected class com.brentvatne.exoplayer.ReactExoplayerViewManager: android.view.View com.brentvatne.exoplayer.ReactExoplayerViewManager.createViewInstance(com.facebook.react.uimanager.ThemedReactContext) failed to verify: android.view.View com.brentvatne.exoplayer.ReactExoplayerViewManager.createViewInstance(com.facebook.react.uimanager.ThemedReactContext): [0x4] can't resolve returned type 'Reference: android.view.View' or 'Unresolved Reference: com.brentvatne.exoplayer.ReactExoplayerView' (declaration of 'com.brentvatne.exoplayer.ReactExoplayerViewManager' appears in /data/app/~~Nq33b85Ca7DH-MTNDVGU3Q==/com.marriage365.app-mNOnqtNxaI4Mm9muOh5Rig==/base.apk!classes2.dex)
at com.brentvatne.react.ReactVideoPackage.createViewManagers(ReactVideoPackage.java:42)
at com.facebook.react.ReactInstanceManager.getOrCreateViewManagers(ReactInstanceManager.java:928)
at com.swmansion.reanimated.ReanimatedPackage.createUIManager(ReanimatedPackage.java:75)
at com.swmansion.reanimated.ReanimatedPackage.getModule(ReanimatedPackage.java:31)
at com.facebook.react.TurboReactPackage$ModuleHolderProvider.get(TurboReactPackage.java:159)
at com.facebook.react.TurboReactPackage$ModuleHolderProvider.get(TurboReactPackage.java:147)
at com.facebook.react.bridge.ModuleHolder.create(ModuleHolder.java:191)
at com.facebook.react.bridge.ModuleHolder.getModule(ModuleHolder.java:156)
at com.facebook.react.bridge.NativeModuleRegistry.getModule(NativeModuleRegistry.java:170)
at com.facebook.react.bridge.CatalystInstanceImpl.getNativeModule(CatalystInstanceImpl.java:502)
at com.facebook.react.bridge.CatalystInstanceImpl.getNativeModule(CatalystInstanceImpl.java:478)
at com.facebook.react.uimanager.UIManagerHelper.getUIManager(UIManagerHelper.java:89)
at com.facebook.react.uimanager.UIManagerHelper.getUIManager(UIManagerHelper.java:47)
at com.facebook.react.ReactInstanceManager.attachRootViewToInstance(ReactInstanceManager.java:1240)
at com.facebook.react.ReactInstanceManager.setupReactContext(ReactInstanceManager.java:1182)
at com.facebook.react.ReactInstanceManager.access$1600(ReactInstanceManager.java:136)
at com.facebook.react.ReactInstanceManager$5$2.run(ReactInstanceManager.java:1136)
at android.os.Handler.handleCallback(Handler.java:938)
at android.os.Handler.dispatchMessage(Handler.java:99)
at com.facebook.react.bridge.queue.MessageQueueThreadHandler.dispatchMessage(MessageQueueThreadHandler.java:27)
at android.os.Looper.loop(Looper.java:223)
at com.facebook.react.bridge.queue.MessageQueueThreadImpl$4.run(MessageQueueThreadImpl.java:228)
at java.lang.Thread.run(Thread.java:923)
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,032
|
La Copa Paraguay 2022 fue la cuarta edición del certamen, que otorgó al campeón (o a quién correspondía) la clasificación en forma directa a la Copa Sudamericana 2023, y el derecho a disputar la Supercopa Paraguay 2022.
El club Sportivo Ameliano obtuvo su primer título tras vencer en la final al club Nacional. El Sportivo Ameliano será, además, el segundo campeón de este certamen en disputar la Supercopa de Paraguay, contra el campeón de la Primera División con mayor puntaje en la Tabla acumulada anual.
Participantes
El torneo contó con 74 equipos en su fase nacional: los 12 de la Primera División, los 16 de la Intermedia, 17 de Primera B, 12 de Primera C y 17 de UFI, donde estas tres últimas disputarán la Fase 1 en llaves de sus respectivas divisiones, en la Fase 2 se enfrentarán los ganadores de la Fase 1 con los equipos de la División Intermedia y la Fase 3 se enfrentarán los ganadores de la Fase 2 con los equipos de la División Profesional.
Equipos Clasificados a la Fase Nacional
{| cellspacing="0" style="background: #EBF5FF; border: 1px #aaa solid; border-collapse: collapse; font-size:100%;" width=80%
|- bgcolor=#006699 style="color:white;"
!width=18%|División
!width=23%|Equipo
!width=5%|
!width=64%|Vía de clasificación
|- bgcolor="#F5FAFF"
|Primera División 12 cupos
|12 de Octubre FBC (Itauguá)Cerro PorteñoGeneral Caballero (JLM)Guaireña FCGuaraníLibertadNacionalOlimpiaSol de AméricaSportivo AmelianoResistencia SCTacuary
|443444444344
|Clasificados de manera automática a la Fase Nacional (Dieciseisavos de final)
|- bgcolor="#ddeeFF"
|División Intermedia 16 cupos
|Martín LedesmaAtlético 3 de FebreroAtyrá FCSan LorenzoFernando de la MoraAtlético ColegialesSportivo LuqueñoDeportivo SantaníIndependiente FBC (CG)Guaraní (Trinidad)PastoreoRiver PlateRubio ÑuSportivo 2 de MayoSportivo IteñoSportivo Trinidense
|2434434442144444
|Clasificados de manera automática a la Fase Nacional (Fase 2)
|- bgcolor="#F5FAFF"
|Primera División B 17 cupos
|Cristóbal Colón FBC (JAS)Atlético Tembetary29 de SeptiembrePresidente HayesCristóbal Colón (Ñ)Olimpia de ItáSportivo Limpeño3 de noviembre3 de Febrero FBC (RB)Atlántida24 de Setiembre VPDeportivo RecoletaGeneral Caballero ZCSilvio PettirossiDeportivo CapiatáFulgencio YegrosGeneral Díaz
|43233241323423444
|Clasificados de manera automática a la Fase Nacional (Fase 1)
|- bgcolor="#ddeeFF"
|Primera División C 12 cupos
|Benjamín AcevalSport ColombiaDeportivo HumaitáOriental1.º de marzo (FDO)Atlético JuventudValois RivarolaDeportivo PinozáGeneral Caballero CGSport ColonialPilcomayo12 de Octubre SD
|214434124323
|Clasificados de manera automática a la Fase Nacional (Fase 1)
|- bgcolor="#F5FAFF"
|Unión del Fútbol del Interior 17 cupos
|Nacional (Yby Yaú)Guarani Unido (Liberación)Porvenir (Tobatí)Capt. Samudio (Mbocayaty)Atlético Forestal (Campo 9)Coronel Martínez (Yuty)Deportivo Juventud (S.C.D)15 de Agosto FBC (Santiago)Sol de Mayo (R.G.S.C)Obreros Unidos (Hernan.)8 de Setiembre (Areguá)1.º de Marzo (PIL) (Pilar)Deportivo Obrero (C. Bado)Sport Primavera (Curuguaty)15 de Agosto (B.A.)1.º de Mayo (Filadelfia)Dep. Alto Paraguay (F. Oli)
|11111211111311112
|Representante del I Departamento de ConcepciónRepresentante del II Departamento de San PedroRepresentante del III Departamento de CordilleraRepresentante del IV Departamento de GuairáRepresentante del V Departamento de CaaguazúRepresentante del VI Departamento de CaazapáRepresentante del VII Departamento de ItapúaRepresentante del VIII Departamento de MisionesRepresentante del IX Departamento de ParaguaríRepresentante del X Departamento de Alto ParanáRepresentante del XI Departamento CentralRepresentante del XII Departamento de ÑeembucúRepresentante del XIII Departamento de AmambayRepresentante del XIV Departamento de CanindeyúRepresentante del XV Departamento de Presidente HayesRepresentante del XVI Departamento de BoquerónRepresentante del XVII Departamento de Alto Paraguay
|}
Distribución Geográfica
Cuadro Principal
Los participantes se enfrentarán en llaves de 2 equipos en un único juego de eliminación directa en cancha neutral, de haber empate al final del período reglamentario, se ejecutarán tiros penales.
Cuadro de desarrollo
Fase 1
Esta fase se disputó del 10 de mayo al 17 de junio en 20 llaves a partido único y dos llaves triangulares, clasificaron a la fase 2 los 20 ganadores de los partidos únicos y los 2 mejores de los triangulares. En caso de empates en los partidos únicos, se definieron las series vía tiros penales.
Un total de 46 equipos disputaron esta fase: 17 equipos de la Primera B, 12 de la Primera C y 17 de la UFI.Nota:* Estos equipos jugaron en llaves triangularesClasificación UFIClasificación Primera B' Fase 2
Esta fase se disputó del 28 de junio al 21 de julio en 20 llaves a partido único, clasificaron a dieciseisavos de final los 20 ganadores. En caso de empates, se definieron las series vía tiros penales.
Un total de 20 equipos disputaron esta fase: 16 equipos de la División Intermedia, 9 de la Primera B, 6 de la Primera C y 9 de la UFI.
Octavos de final
Esta fase se disputó del 6 al 8 de septiembre en 8 llaves a partido único, clasificaron a cuartos de final los 8 ganadores. En caso de empates, se definieron las series vía tiros penales.
Un total de 16 equipos disputaron esta fase: 11 equipos de la Primera División, 2 de la División Intermedia, 2 de la Primera B y 1 de la Primera C.
Cuartos de final
Esta fase se disputó entre el 27 y 28 de septiembre en 4 llaves a partido único, clasificaron a semifinales los 4 ganadores. En caso de empates, se definieron las series vía tiros penales.
Un total de 8 equipos disputaron esta fase: 7 equipos de la Primera División y 1 de la Primera B.
Semifinales
Esta fase se disputó el 19 de octubre en 2 llaves a partido único, clasificaron a la final los 2 ganadores. Como no hubo empates, no se definieron las series vía tiros penales.
Un total de 4 equipos disputaron esta fase: 3 equipos de la Primera División y 1 de la Primera B.
Final
Los dos mejores equipos del certamen decidieron al campeón en un solo partido; al existir empate en el tiempo reglamentario, se recurrió a tiros penales. El partido se jugó el 4 de noviembre.
</center>
Anotaciones destacadas
Listado de tripletas o hat-tricks (3), póker de goles (4), manitas'' (5), o más goles, anotados por un jugador en un mismo encuentro.
Autogoles
Premios
En esta edición de la Copa Paraguay, se otorgarán los siguientes premios:
Campeón: ₲ 600.000.000
Vicecampeón: ₲ 250.000.000
Tercer Puesto: ₲ 100.000.000
Cuarto Puesto: ₲ 50.000.000
Véase también
Estadios de Paraguay
Supercopa Paraguay 2022
Apertura 2022
Clausura 2022
División Intermedia 2022
Nacional B 2022
Primera B 2022
Primera C 2022
Campeonato Paraguayo de Fútbol Femenino 2022
Referencias
Enlaces externos
Sitio oficial de la Asociación Paraguaya de Fútbol
Fútbol en 2022
Deporte en Paraguay en 2022
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,258
|
Save Rock and Roll is het vijfde studioalbum van de Amerikaanse poppunk-band Fall Out Boy. Het is hun eerste album na hun pauze van drie jaar. Aan het door Butch Walker geproduceerde album werd tussen oktober 2012 en maart 2013 gewerkt. Op 4 februari 2013 werd de leadsingle My Songs Know What You Did in the Dark (Light Em Up) uitgebracht. Het album zou in eerste instantie op 6 en 7 mei 2013 worden uitgebracht, in verband met het 10-jarig jubileum van hun eerste album Take This to Your Grave, maar werd vervroegd naar 15 april 2013.
Opname
De band had meerdere pogingen nodig om weer tot elkaar te komen op creatief gebied. Begin 2012 begonnen Patrick Stump en Pete Wentz voor het eerst in vier jaar met het schrijven van nieuw materiaal. Het verging beiden minder goed in hun persoonlijk levens en ze wilden een creatieve uitlaatklep om de emoties te uiten. De eerste schrijfsessie was echter onsuccesvol. De drie à vier nummers die zij samen schreven werden meteen opgeschort, omdat zij vonden dat het "niet juist voelde en het niet juist was". Enkele maanden later probeerden zij het opnieuw, met een nieuw uitgangspunt: de moderne versie van Fall Out Boy stond nu centraal. Het nummer Where Did the Party Go was een opsteker en deed de muzikanten beseffen dat een comeback mogelijk was, mits de band niet terug zou vallen in de muziek waarmee zij populair werden.
Zij kozen ervoor om in geheimhouding het album te schrijven zodat de band het in eigen hand had als de opnames niet zoals verwacht zouden lopen en zij het album nog konden opschorten. Hierdoor zorgden zij ervoor dat zij ook afstand namen van sociale media, om uitlatingen te voorkomen.
Tracklist
Geschreven door Andy Hurley, Patrick Stump, Joe Trohman en Pete Wentz tenzij anders vermeld.
Muziekalbum uit 2013
Rockalbum
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,790
|
{"url":"https:\/\/wwsolutionsltd.co.uk\/sportswear-shops-sbs\/33d163-ethyne-lewis-structure","text":"The notable physical and chemical properties of ethyne are listed below. Ethane is the second simplest alkane followed by methane. This Carbon has 2, 4, 6, 8, because it's sharing, right, and this Carbon has 2, 4, 6. In Part B, We Will Be Concerned With The Orbitals That Are Used To Form The Multiple Bonds. Ethane is the most important gaseous fuel. So we've got 4 valence electrons. Add them together. And this triple bond, 1, 2, 3, in the center, holds the two Carbons together\u2014bonds those together. Drawing the Lewis Structure for C 2 H 2 - Ethyne or Acetylene. Ethylene (IUPAC name: ethene) is a hydrocarbon which has the formula C 2 H 4 or H 2 C=CH 2.It is a colorless flammable gas with a faint \"sweet and musky\" odour when pure. Ethyne\/Acetylene has some irregularities in its physical properties but is a widely used chemical compound, owing to the high amount of heat it can generate. So I need to get 8 around each one. It is a hydrocarbon and the simplest alkyne . It is prepared by laboratory method using sodium propionate. CHEMICAL BONDING AND MOLECULAR STRUCTURE 103 When combining atoms share three electron pairs as in the case of two nitrogen atoms in the N 2 molecule and the two carbon atoms in the ethyne molecule, a triple bond is formed. Step 1: Find valence e- for all atoms. But this Carbon right here only has 2, 4 valence electrons. The molar mass of ethyne is equal to 26.038 grams per mole. Another use of ethylene as a monomer is in the formation of linear \u03b1-olefins. Compound: C2H2 (ethyne) Lewis Structure 3-D Structure Skeletal Structure H-C EC-H Functional Group(s) Present: Central Atom(s) Geometry: Polar Or Non-Polar. However, the molecule does exist. 2. Answers ? The carbon-carbon triple bond is only 1.20\u00c5 long. picture. And this Carbon has 2, 4, 6, 8; it has an octet. So we need to figure out something, somehow to get 8 valence electrons around this Carbon, this Carbon, and keep the Hydrogens full. Step method to draw the Lewis dot structure of C2H5OH, ethanol, ethyl alcohol. We've got Carbon, Carbon, and Hydrogen has to go on the outside. Viewing Notes: With C 2 H 2 you are going to run out of valence electrons and will have to share more than one pair of electrons between the Carbon atoms. Let's take those, put them right here. Show transcribed image text. To reproduce the Lewis structure given earlier, it is necessary to contrive a double bond (i.e., a \u03c3 bond plus a \u03c0 bond) between the two carbon atoms. i. Lewis structure represents every nonbonding valence electron with a dot, but that doesn't apply for ethyne because there are no nonbonding valence electrons. Remember that Hydrogen (H) atoms always go on the outside of a Lewis Structure. (a) Draw Lewis structures for ethane (C 2 H 6), ethylene (C 2 H 4), and acetylene (C 2 H 2). The Lewis structure for ethyne (C\u2082H\u2082) shown below is incorrect. For the HCCH Lewis structure you'll need to form a triple bond between the two carbon atoms. Understand the structure, Hybridization of C2H2. Solution. Draw Lewis diagram for the following inorganic reagents: HNO 3, H 2 SO 4, and sodium azide. Draw the Lewis structure for ethene (ethylene), H 2 CCH 2, a small hydrocarbon with a C=C double bond.. b. Looking for a Similar Assignment? ; Remember that Hydrogen (H) atoms always go on the outside of a Lewis Structure. Draw the electron dot structure of ethyne. Over here, Hydrogen, in group 1A or 1, that has 1 valence electron, but since I have two Carbons, let's multiply that by 2; and I have two Hydrogens, multiply that by 2, so that equals 8, that equals 2; I add them up, I have 10 valence electrons to work with in this Lewis structure here. Ethyne has the Lewis structure: H\u2212C\u2261C\u2212H. We have a periodic table. bent . The solution is to share three pairs of valence electrons and form a triple bond between the Carbon atoms in C2H2 . Natural sources of ethylene include both natural gas and petroleum; it is also a naturally occurring hormone in plants, in which it inhibits growth and promotes leaf fall, and in fruits, in which it promotes ripening. The Lewis structure of N\u2082O is shown on the left below with numbers on the corresponding atoms. Question: 3. *Response times vary by subject and question complexity. Example $$\\PageIndex{6}$$: Structure of Alkynes. Complete Each Part Below. Back. Hybridization of C2H2 - Acetylene (Ethyne) is sp hybridized. A mixture of ethyne and oxygen is burnt for welding. NCERT Books. of double bond present in the Lewis structure of P F 6 \u2212 is : View solution. This molecule is linear: all four atoms lie in a straight line. We also use Lewis symbols to indicate the formation of covalent bonds, which are shown in Lewis structures, drawings that describe the bonding in molecules and polyatomic ions.For example, when two chlorine atoms form a chlorine molecule, they share one pair of electrons: The two oxygen atom can both achieve a stable structure by sharing two pairs of electrons. The Ziegler-Natta polymerization of ethyleneEthylene gas is pumped under pressure into a reaction vessel, where it polymerizes under the influence of a Ziegler-Natta catalyst in the presence of a solvent. Electron dot structure of ethane of molecule is given in an image attached. In the diagram each line represents one pair of shared electrons. About this Site | Report a Problem | Comments & Suggestions, Stoichiometry: Moles, Grams, and Chemical Reactions. Ring in the new year with a Britannica Membership, https:\/\/www.britannica.com\/science\/ethylene, National Center for Biotechnology Information - PubChem - Ethylene, Washington University in St. Louis - The International Plant Growth Substances Association - Ethylene. The Lewis structure for oxygen (O2) shows the bond between two oxygen atoms. Ethyne [ACD\/Index Name] vinylene. So the formula for ethane is C2H6. 2; c. 3; d. 6; If a covalent bond were to be formed between a nitrogen atom (electronegativity 3.0) and an oxygen atom (electronegativity 3.5), which of the following statements would best describe such a bond? The structure of the ethyne molecule is illustrated below. Previous question Next \u2026 The Ethyne molecule contains a total of 3 bond(s) There are 1 non-H bond(s), 1 multiple bond(s) and 1 triple bond(s). I'm going to make these a little darker. Everything else needs 8, but Hydrogen needs 2. Draw this VSEPR structure next to the Lewis structure. We can clean it up a little bit to make it easier to see here. In Part A, We Will Be Concerned With The Shapes Of The Molecules. What's all this? a) From the list of compounds given, [Ethyne, Ethane, Methane, Ethene, Butane]. Transcript: OK, let's try the Lewis structure for C2H2: ethyne. Two, 4, 6, 8, that Carbon's fine. So we've assigned two electrons to each bond. Th\u2026 When polymerization is carried out at high pressures and temperatures, the product is called low-density polyethylene and has properties different from the high-density polyethylene formed by polymerization under Ziegler-Natta catalytic conditions (see industrial polymers). BNAT; Classes. This Carbon has 2, 4, 6, 8; it has an octet. Drawing the Lewis Structure for CH 4. asked Jul 14, 2018 in Chemistry by Anukriti bharti (38.1k points) carbon and its compounds; cbse; class-10; 0 votes. Ethylene (H2C=CH2), the simplest of the organic compounds known as alkenes, which contain carbon-carbon double bonds. Our editors will review what you\u2019ve submitted and determine whether to revise the article. c. Sketch the molecule showing the predicted bond angles. The Structure of Ethyne (Acetylene): sp Hybridization Ethyne (acetylene) is a member of a group of compounds called alkynes which all have carbon-carbon triple bonds \u2605 Propyne is another typical alkyne The arrangement of atoms around each carbon is linear with bond angles 180 o Ethyne is also called Acetylene. View solution. Carbon is in group 14, or 4A, so it has 4 valence electrons. The Lewis structure of N\u2082O is shown on the left below with numbers on the corresponding atoms. Figure 4: Constitutional Isomers of $$\\ce{C4H10}$$ The name of the compound on the left in Figure 4 is butane. An ethyne molecule is practically 2 CH molecules. You can see the Hydrogens on the outside, two Carbons in the middle. The Lewis dot structure for ethane has two C atoms single bonded to each other in the center. Bonding in acetylene. I quickly take you through how to draw the Lewis Structure of CHCH (Acetylene or ethyne). Since all the atoms are in either period 1 or 2, this molecule will adhere to the octet rule. Write Lewis symbols for the following atoms and ions : (i) S and S 2 \u2212 (ii) A l and A l + 3 (iii) H and H \u2212 View solution. tetrahedron c) What is the shape of this molecule (look only at the atoms)? It is impossible to draw a Lewis structure in which all of the atoms are surrounded by an octet of electrons. View solution. It is the simplest alkene (a hydrocarbon with carbon-carbon double bonds).. The Lewis structure for ethyne is shown as follows. Images of the chemical structure of Ethyne are given below: The 2D chemical structure image of Ethyne is also called skeletal formula, which is the standard notation for organic molecules. We'll see in a minute. That's one way to do it. Finally, the hybrid orbital concept applies well to triple-bonded groups, such as alkynes and nitriles. \u2013 How do the calculated bond orders for ethane, ethene, and ethyne compare with the bond orders predicted by the. An orbital view of the bonding in ethyne. picture. ; Remember that Hydrogen (H) atoms always go on the outside of a Lewis Structure. It is produced by heating either natural gas, especially its ethane and propane components, or petroleum to 800\u2013900 \u00b0C (1,470\u20131,650 \u00b0F), giving a mixture of gases from which the ethylene is separated. No. In this case, Hydrogen right here is already full. Lewis Structures. MOLECULES WITH MULTIPLE BONDS The Lewis Structures Of Four Molecules With Multiple Bonds Are Shown Below. Well, In order to answer your question, the hybridisation of ethyne is $sp$ How? Each C atom then has three more single bonds, each attached to an H atom. To reproduce the Lewis structure given earlier, it is necessary to contrive a double bond (i.e., a \u03c3 bond plus a \u03c0 bond) between the two carbon atoms. In longer alkyne chains, the additional carbon atoms are attached to each other by single covalent bonds. Do not add any more atoms. draw electron dot structure of ethane and butane, Figure 4 demonstrates drawing Lewis electron dot structures for these compounds. The half-life of a reaction is halved as the initial concentration of the reactant is doubled. (a) the amino acid serine: (b) urea: (c) pyruvic acid: (d) uracil: (e) carbonic acid: A compound with a molar mass of about 28 g\/mol contains 85.7% carbon and 14.3% hydrogen by mass. In addition to these compounds, ethylene and benzene combine to form ethylbenzene, which is dehydrogenated to styrene for use in the production of plastics and synthetic rubber. All right. 4.1.3 Lewis Representation of Simple Molecules (the Lewis Structures) The Lewis dot structures pr ovide a pictur e Order now and Get 10% Discount! So, first thing we need to do, count those valence electrons for C2H2. Polymerization (the repetitive joining of many small molecules into larger ones) of ethylene gives polyethylene, a polymer having many uses, particularly in the production of packaging films, wire coatings, and squeeze bottles. So here it's pretty easy. The double bond they share is denoted by the double lines joining the two atoms. \u2026 That's the Lewis structure for C2H2, ethyne. The other halogen molecules (F2, Br2, I2, and At2) form bonds like those in the chlorine molecule: one single bond between atoms and three lone pairs of electrons per atom. Note that Hydrogen only needs two valence electrons to have a full outer shell. A slurry of polyethylene, unreacted ethylene monomer, catalyst, and solvent exit the reactor. Let's move another pair in. Share this entry. This allows each halogen atom to have a noble gas electron configuration. Median response time is 34 minutes and may be longer for new subjects. Carbon has 4. Understand the structure, Hybridization of C2H2. And now Hydrogens are fine. Get a Britannica Premium subscription and gain access to exclusive content. So i've used all my valence electrons, Hydrogens are fine. In the Lewis-dot structure the valance electrons are shown by 'dot'. For example, $$\\mathrm{CH_3CH_2C\u2261CH}$$ is called 1-butyne. Share on Facebook; Share on Twitter; Share on WhatsApp; Share by Mail; \u2026 View solution. (b) What is the hybridization of the carbon atoms in each molecule? NCERT Books for Class 5; NCERT Books Class 6; NCERT Books for Class 7; NCERT Books for \u2026 [5] Choose the hydrocarbon which matches with the correct description. Ethylene is an important industrial organic chemical. Drawing the Lewis structure for CH 4 (named methane) requires only single bonds.It's one of the easier Lewis structures to draw. 200-816-9 [EINECS] 2122-48-7 [RN ] 906677 ... Aqueous Base\/Acid-Catalyzed Hydrolysis (25 deg C) [HYDROWIN v1.67]: Rate constants can NOT be estimated for this structure! Ethyne, C 2 H 2. [5] This colorless gas (lower hydrocarbons are generally gaseous in nature) is widely used as a fuel and a chemical building block. 70 More Lewis Dot Structures. Finally, the hybrid orbital concept applies well to triple-bonded groups, such as alkynes and nitriles. The simple view of the bonding in ethyne. Be on the lookout for your Britannica newsletter to get trusted stories delivered right to your inbox. So now Hydrogens are fine. It is a colorless liquid with a boiling point of 0 o C. Lewis structures (also known as Lewis dot structures or electron dot structures) are diagrams that represent the valence electrons of atoms within a molecule. But Hydrogen always goes on the outside. Note that Hydrogen only needs two valence electrons to have a full outer shell. And if you come back to our structure, now you can see these are the single bonds right here. A chemical structure of a molecule includes the arrangement of atoms and the chemical bonds that hold the atoms together. Let me explain logically what is going on here . Based on this structure, predict the H-C-H bond angle. Other names \u2013 Methyl methane, Bimethyl, Dimethyl, Eth\u2026 For CH 4 you have a total of 8 total valence electrons.. Viewing Notes: HCCH (Ethyne) can also be written as C 2 H 2. The correct Lewis structure for ethene is shown below: In the molecule ethene, both carbon atoms will be sp 2 hybridized and have one unpaired electron in a non-hybridized p orbital. The Lewis structure indicates that each Cl atom has three pairs of electrons that are not used in bonding ... Titan also contains ethane (H 3 CCH 3), acetylene (HCCH), and ammonia (NH 3). And if we count them all up, we have 2, 4, 6, 8, 10 valence electrons, which is what we had to start with, and we've written the Lewis structure for C2H2. It only needs 2 valence electrons to be full, to have a full outer shell or octet. Associate Professor of Chemistry, University of Virginia, Charlottesville. Use Coupon Code \"Newclient\" Share this: Twitter; Facebook; Like this: Like Loading... Related . Acetylene (systematic name: ethyne) is the chemical compound with the formula C 2 H 2. (By the way, do you know the names of the first two compounds given above ?) These p-orbitals will undergo parallel overlap and form one $\\sigma$ bond with bean-shaped probability areas above and below the plane of the six atoms. Not quite there yet. Lewis structure and line structure (not sure if it'll display correctly) are the same for this one: H-C\u2261C-H, with three lines connecting the C's. We could write this as, using these lines. what is the hybridization of the carbon atoms in a molecule of ethyne represented above, Carbon hybridization in Ethylene\u2014C 2 H 4. So we've only got 4 valence electrons left to work with. We have it right here. Let's put two here, let's put two here, two here, so we've used 2, 4, 6. Let us know if you have suggestions to improve this article (requires login). Drawing the Lewis Structure for C 2 H 2 - Ethyne or Acetylene. It is reactive and at lowish temperatures soon pairs up to form a more stable structure, N 2 O 4, in which all of the atoms do have a full octet. Now we need to complete the octets on the outside atoms. One point is earned for the correct Lewis structure. The structure on the right is the Lewis electron structure, or Lewis structure, for H 2 O. The first of these is the single largest use of ethylene, consuming about one-half of the annual output. If you have read the ethene page, you will expect that ethyne is going to be more complicated than this simple structure suggests. They follow the duet rule (2 electrons). Ethyne is a symmetric linear molecule, with the two carbon atoms in the center sharing a triple bond and one hydrogen on each carbon. Moreover, by sharing a bonding pair with oxygen, each hydrogen atom now has a full valence shell of two electrons. Draw electron dot structure of ethyne. So its octet is not complete. Complete the Lewis structures of these molecules by adding multiple bonds and lone pairs. Corrections? a. Starting from this structure, complete the correct structure. Remember that hydrogen atoms always go on the outside of a Lewis structure and that they only need two valence electrons for a full outer shell. The exception, of course, being the hydrogen's. So there's 2, 2. Assign two electrons to each bond. Two hydro-carbons which discharge the brown colour of Bromine solution in Carbon tetra chloride, ii. Calculate the number of valence electrons. The order of the reaction is? Hydrogen atoms only need two electrons for a full outer shell. See the answer . This problem has been solved! How to determine the hybridization of carbon in C2H2? And you notice all those valence electrons being shared in the center. 3. Updates? The simplest alkyne, ethyne (also known as acetylene), has two carbon atoms and the molecular formula of C 2 H 2.The structural formula for ethyne is . Viewing Notes: With C 2 H 2 you are going to run out of valence electrons and will have to share more than one pair of electrons between the Carbon atoms. Omissions? The carbon-carbon triple bond is only 1.20\u00c5 long. #1 = 0, #2 = +1, #3 = -1. It contains 2 carbon atoms and 6 hydrogen atoms. View solution. I quickly take you through how to draw the Lewis Structure of CHCH (Acetylene or ethyne). 1; b. a. So let's take these two right here, and we'll put them right here. Consider, for example, the structure of ethyne (common name acetylene), the simplest alkyne. The Lewis structure for ethyne, a linear molecule, is: The IUPAC nomenclature for alkynes is similar to that for alkenes except that the suffix -yne is used to indicate a triple bond in the chain. It is the simplest alkene (a hydrocarbon with carbon-carbon double bonds).. Ethylene use falls into two main categories: 1) as a monomer, from which longer carbon chains are constructed, and 2) as a starting material for other two-carbon compounds. Ethyne, which has the formula C2H2, is a nonpolar molecule. The structure of ethylene can be examined in VB terms to illustrate the use of hybridization. Note that Hydrogen only needs two valence electrons to have a full outer shell. Excess solvent is recovered from a hot water bath and recycled, and a dryer dehydrates the wet polyethylene to its final powder form. The two oxygen atom can both achieve a stable structure by sharing two pairs of electrons. Bioaccumulation Estimates from Log Kow (BCFWIN \u2026 Explain your reasoning. Each has a total of 6 valence atoms making a sum total of 12. Lewis structures? The melting point of ethylene is \u2212169.4 \u00b0C [\u2212272.9 \u00b0F], and its boiling point is \u2212103.9 \u00b0C [\u2212155.0 \u00b0F]. Two sp 2 hybridized carbon atoms can make a sigma bond by overlapping one of the three sp 2 orbitals and bond with two hydrogens each and two hydrogens make sigma bonds with each carbon by overlapping their s orbitals with the other two sp 2 orbitals. Compare with the Orbitals that are used to form enough bonds to obtain eight valence electrons, contain! The structure on the outside of a reaction is halved as the initial concentration of the carbon atoms the! Hcch ( ethyne ) 'll put them right here only has 2,,. Of linear low-density polyethylene Class 1 - 3 ; Class 4 - 5 ; Class 6 - 10 ; 6! 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Physical and chemical properties of ethyne represented above, carbon hybridization in Ethylene\u2014C 2 H 2 for each and... Everything else needs 8, but Hydrogen needs 2 shared between the two.! Lower right cell in the Lewis structure ) shows the bond between the carbon atoms C2H2! Then has three more single bonds, each attached to an H atom \u03b1-olefins have a number of applications including. Of ethyne lewis structure electrons the list of compounds given, [ ethyne, ethane, methane,,. Drawing the Lewis structure of the reactant is doubled structure by sharing a bonding pair with oxygen, each atom. Sketch the molecule showing the predicted bond angles for ethyne is equal 26.038... ( \\PageIndex { 6 } \\ ): structure of ethylene is and! Be full, to have a noble gas electron configuration carbon hybridization in Ethylene\u2014C 2 H 2 - ethyne Acetylene... Compound with the formula C 2 H 2 left to work with pairs on the.... To see here already full 6 Hydrogen atoms Log Kow ( BCFWIN \u2026 Lewis... Hydrogen needs 2 news, offers, and we 'll put them right here be full, to have full. Of polyethylene, unreacted ethylene monomer, catalyst, and information from Encyclopaedia Britannica the chemical with! Cancel out orbital concept applies well to triple-bonded groups, such as Alkynes and nitriles, course. Ethylene monomer, catalyst, and sodium azide, Stoichiometry: Moles, Grams, and chemical Reactions and. Britannica newsletter to get 8 around each one up for this email, you are to. Atom number only got 4 valence electrons of atoms and molecules, if any are. Joining the two atoms the half-life of a molecule of ethyne are below... Image of ethyne is going on here this triple bond between two atom! For new subjects each Hydrogen atom now has a full outer shell to eight. Initial concentration of the easier Lewis structures of these is the hybridization carbon...\nRobinsons Coach Holidays Isle Of Man, Fogasos En La Boca En Ingl\u00e9s, La Vigna Oregon, Il Menu, Adak Island Inn, Arkansas Pine Bluff Athletics, Kamen Rider Ryuki Ps1 Iso, The Parent 'hood Dvd Set, Greenwood Fifa 21 Sofifa, Robinsons Coach Holidays Isle Of Man,","date":"2022-08-13 18:23:06","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\": 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\": 0.5200725197792053, \"perplexity\": 3117.101396326116}, \"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-2022-33\/segments\/1659882571982.99\/warc\/CC-MAIN-20220813172349-20220813202349-00324.warc.gz\"}"}
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<h1><a href="../index.html"><span id="logoX">X</span>RegExp</a></h1>
<h1 class="subtitle">The one of a kind JavaScript regular expression library</h1>
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<h2>Table of contents</h2>
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<li><a href="#namedCapture">Named capture</a></li>
<li><a href="#inlineComments">Inline comments</a></li>
<li><a href="#modeModifier">Leading mode modifier</a></li>
<li><a href="#strictErrors">Stricter error handling</a></li>
<li><a href="#unicode">Unicode</a></li>
<li><a href="#replacementText">Replacement text</a></li>
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<h1>New syntax</h1>
<h2 id="namedCapture">Named capture</h2>
<p>XRegExp includes comprehensive support for named capture. Following are the details of XRegExp's named capture syntax:</p>
<ul>
<li>Capture: <code>(?<<em>name</em>>…)</code></li>
<li>Backreference in regex: <code>\k<<em>name</em>></code></li>
<li>Backreference in replacement text: <code>$<<em>name</em>></code></li>
<li>Backreference stored at: <code><em>result</em>.groups.<em>name</em></code></li>
<li>Backreference numbering: Sequential (i.e., left to right for both named and unnamed capturing groups)</li>
<li>Multiple groups with same name: <code>SyntaxError</code></li>
</ul>
<h3>Notes</h3>
<ul>
<li>See additional details and compare to named capture in other regex flavors here: <mark><a href="./named_capture_comparison/index.html">Named capture comparison</a></mark>.</li>
<li>JavaScript added native support for named capture in ES2018. XRegExp support predates this, and it extends this support into pre-ES2018 browsers.</li>
<li>Capture names can use a wide range of Unicode characters (see the definition of <a href="https://tc39.es/proposal-regexp-named-groups/#prod-RegExpIdentifierName"><code>RegExpIdentifierName</code></a>).</li>
</ul>
<h3>Example</h3>
<pre class="sh_javascript">const repeatedWords = XRegExp.tag('gi')`\b(?<word>[a-z]+)\s+\k<word>\b`;
// Alternatively: XRegExp('\\b(?<word>[a-z]+)\\s+\\k<word>\\b', 'gi');
// Check for repeated words
repeatedWords.test('The the test data');
// -> true
// Remove any repeated words
const withoutRepeated = XRegExp.replace('The the test data', repeatedWords, '${word}');
// -> 'The test data'
const url = XRegExp(`^(?<scheme> [^:/?]+ ) :// # aka protocol
(?<host> [^/?]+ ) # domain name/IP
(?<path> [^?]* ) \\?? # optional path
(?<query> .* ) # optional query`, 'x');
// Get the URL parts
const parts = XRegExp.exec('https://google.com/path/to/file?q=1', url);
// parts -> ['https://google.com/path/to/file?q=1', 'https', 'google.com', '/path/to/file', 'q=1']
// parts.groups.scheme -> 'https'
// parts.groups.host -> 'google.com'
// parts.groups.path -> '/path/to/file'
// parts.groups.query -> 'q=1'
// Named backreferences are available in replacement functions as properties of the last argument
XRegExp.replace('https://google.com/path/to/file?q=1', url, (match, ...args) => {
const groups = args.pop();
return match.replace(groups.host, 'xregexp.com');
});
// -> 'https://xregexp.com/path/to/file?q=1'
</pre>
<p>Regexes that use named capture work with all native methods. However, you need to use <code><a href="../api/index.html#exec">XRegExp.exec</a></code> and <code><a href="../api/index.html#replace">XRegExp.replace</a></code> for access to named backreferences, otherwise only numbered backreferences are available.</p>
<h3>Annotations</h3>
<ul>
<li><strong>Rationale:</strong> Named capture can help make regular expressions and related code self-documenting, and thereby easier to read and use.</li>
<li><strong>Compatibility:</strong> The named capture syntax is illegal in pre-ES2018 native JavaScript regular expressions and hence does not cause problems. Backreferences to undefined named groups throw a <code>SyntaxError</code>.</li>
<li><strong>Compatibility with deprecated features:</strong> XRegExp's named capture functionality does not support the <code>lastMatch</code> property of the global <code>RegExp</code> object or the <code>RegExp.prototype.compile</code> method, since those features were <a href="https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Deprecated_and_obsolete_features">deprecated</a> in JavaScript 1.5.</li>
<li><strong>Prior art:</strong> Comes from Python (feature) and .NET (syntax).</li>
</ul>
<h2 id="inlineComments">Inline comments</h2>
<p>Inline comments use the syntax <code>(?#<em>comment</em>)</code>. They are an alternative to the line comments allowed in <a href="../flags/index.html#extended">free-spacing mode</a>.</p>
<p>Comments are a do-nothing (rather than ignore-me) metasequence. This distinction is important with something like <code>\1(?#comment)2</code>, which is taken as <code>\1</code> followed by <code>2</code>, and not <code>\12</code>. However, quantifiers following comments apply to the preceeding token, so <code>x(?#comment)+</code> is equivalent to <code>x+</code>.</p>
<h3 style="margin-top:20px;">Example</h3>
<pre class="sh_javascript">const regex = XRegExp('^(?#month)\\d{1,2}/(?#day)\\d{1,2}/(?#year)(\\d{2}){1,2}', 'n');
const isDate = regex.test('04/20/2008'); // -> true
// Can still be useful when combined with free-spacing, because inline comments
// don't need to end with \n
const regex = XRegExp('^ \\d{1,2} (?#month)' +
'/ \\d{1,2} (?#day )' +
'/ (\\d{2}){1,2} (?#year )', 'nx');
</pre>
<h3>Annotations</h3>
<ul>
<li><strong>Rationale:</strong> Comments make regular expressions more readable.</li>
<li><strong>Compatibility:</strong> No known problems with this syntax; it is illegal in native JavaScript regular expressions.</li>
<li><strong>Prior art:</strong> The syntax comes from Perl. It is also available in .NET, PCRE, Python, Ruby, and Tcl, among other regular expression flavors.</li>
</ul>
<h2 id="modeModifier">Leading mode modifier</h2>
<p>A mode modifier uses the syntax <code>(?<em>imnsuxA</em>)</code>, where <code><em>imnsuxA</em></code> is any combination of XRegExp flags except <code>g</code>, <code>y</code>, or <code>d</code>. Mode modifiers provide an alternate way to enable the specified flags. XRegExp allows the use of a single mode modifier at the very beginning of a pattern only.</p>
<h3 style="margin-top:20px;">Example</h3>
<pre class="sh_javascript">const regex = XRegExp('(?im)^[a-z]+$');
regex.ignoreCase; // -> true
regex.multiline; // -> true
</pre>
<p>When creating a regex, it's okay to include flags in a mode modifier that are also provided via the separate <code>flags</code> argument. For instance, <code>XRegExp('(?s).+', 's')</code> is valid.</p>
<p>Flags <code>g</code>, <code>y</code>, and <code>d</code> cannot be included in a mode modifier, or an error is thrown. This is because <code>g</code>, <code>y</code>, and <code>d</code>, unlike all other flags, have no impact on the meaning of a regex. Rather, they change how particular methods choose to apply the regex. XRegExp methods provide e.g. <code>scope</code>, <code>sticky</code>, and <code>pos</code> arguments that allow you to use and change such functionality on a per-run rather than per-regex basis. Additionally, consider that it makes sense to apply all other flags to a particular subsection of a regex, whereas flags <code>g</code>, <code>y</code>, and <code>d</code> only make sense when applied to the regex as a whole. Allowing <code>g</code>, <code>y</code>, and <code>d</code> in a mode modifier might therefore create future compatibility problems.</p>
<p>The use of unknown flags in a mode modifier causes an error to be thrown. However, XRegExp addons can add new flags that are then automatically valid within mode modifiers.</p>
<h3>Annotations</h3>
<ul>
<li><strong>Rationale:</strong> Mode modifiers allow you to enable flags in situations where a regex pattern can be provided as a string only. They can also improve readability, since flags are read first rather than after the pattern.</li>
<li><strong>Compatibility:</strong> No known problems with this syntax; it is illegal in native JavaScript regular expressions.</li>
<li><strong>Compatibility with other regex flavors:</strong> Some regex flavors support the use of multiple mode modifiers anywhere in a pattern, and allow extended syntax for unsetting flags via <code>(?-i)</code>, simultaneously setting and unsetting flags via <code>(?i-m)</code>, and enabling flags for subpatterns only via <code>(?i:…)</code>. XRegExp does not support these extended options.</li>
<li><strong>Prior art:</strong> The syntax comes from Perl. It is also available in .NET, Java, PCRE, Python, Ruby, and Tcl, among other regular expression flavors.</li>
</ul>
<h2 id="strictErrors">Stricter error handling</h2>
<p>XRegExp makes any escaped letters or numbers a <code>SyntaxError</code> unless they form a valid and complete metasequence or backreference. This helps to catch errors early, and makes it safe for future versions of ES or XRegExp to introduce new escape sequences. It also means that octal escapes are always an error in XRegExp. ES3/5 do not allow octal escapes, but browsers support them anyway for backward compatibility, which often leads to unintended behavior.</p>
<p>XRegExp requires all backreferences, whether written as <code>\<em>n</em></code>, <code>\k<<em>n</em>></code>, or <code>\k<<em>name</em>></code>, to appear to the right of the opening parenthesis of the group they reference.</p>
<p>XRegExp never allows <code>\<em>n</em></code>-style backreferences to be followed by literal numbers. To match backreference 1 followed by a literal <code>2</code> character, you can use, e.g., <code>(a)\k<1>2</code>, <code>(?x)(a)\1 2</code>, or <code>(a)\1(?#)2</code>.</p>
<h2 id="unicode">Unicode</h2>
<p>XRegExp supports matching Unicode categories, scripts, and other properties via addon scripts. Such tokens are matched using <code>\p{…}</code>, <code>\P{…}</code>, and <code>\p{^…}</code>. See <a href="../unicode/index.html">XRegExp Unicode addons</a> for more details.</p>
<p>XRegExp additionally supports the <code>\u{N…}</code> syntax for matching individual code points. In ES6 this is supported natively, but only when using the <code>u</code> flag. XRegExp supports this syntax for code points <code>0</code>–<code>FFFF</code> even when not using the <code>u</code> flag, and it supports the complete Unicode range <code>0</code>–<code>10FFFF</code> when using <code>u</code>.</p>
<h2 id="replacementText">Replacement text</h2>
<p>XRegExp's replacement text syntax is used by the <code><a href="../api/index.html#replace">XRegExp.replace</a></code> function. It adds <code>$0</code> as a synonym of <code>$&</code> (to refer to the entire match), and adds <code>$<<em>n</em>></code> and <code>${<em>n</em>}</code> for backreferences to named and numbered capturing groups (in addition to <code>$1</code>, etc.). When the braces syntax is used for numbered backreferences, it allows numbers with three or more digits (not possible natively) and allows separating a backreference from an immediately-following digit (not always possible natively). XRegExp uses stricter replacement text error handling than native JavaScript, to help you catch errors earlier (e.g., the use of a <code>$</code> character that isn't part of a valid metasequence causes an error to be thrown).</p>
<p>Following are the special tokens that can be used in XRegExp replacement strings:</p>
<ul>
<li><code>$$</code> - Inserts a literal <code>$</code> character.</li>
<li><code>$&</code>, <code>$0</code> - Inserts the matched substring.</li>
<li><code>$`</code> - Inserts the string that precedes the matched substring (left context).</li>
<li><code>$'</code> - Inserts the string that follows the matched substring (right context).</li>
<li><code>$n</code>, <code>$nn</code> - Where n/nn are digits referencing an existing capturing group, inserts
backreference n/nn.</li>
<li><code>$<n></code>, <code>${n}</code> - Where n is a name or any number of digits that reference an existent capturing
group, inserts backreference n.</li>
</ul>
<p>XRegExp behavior for <code>$<n></code> and <code>${n}</code>:</p>
<ul>
<li>Backreference to numbered capture, if <code>n</code> is an integer. Use <code>0</code> for the entire match. Any number of leading zeros may be used.</li>
<li>Backreference to named capture <code>n</code>, if it exists. Does not overlap with numbered capture since XRegExp does not allow named capture to use a bare integer as the name.</li>
<li>If the name or number does not refer to an existing capturing group, it's an error.</li>
</ul>
<p>XRegExp behavior for <code>$n</code> and <code>$nn</code>:</p>
<ul>
<li>Backreferences without curly braces end after 1 or 2 digits. Use <code>${…}</code> for more digits.</li>
<li><code>$1</code> is an error if there are no capturing groups.</li>
<li><code>$10</code> is an error if there are less than 10 capturing groups. Use <code>${1}0</code> instead.</li>
<li><code>$01</code> is equivalent to <code>$1</code> if a capturing group exists, otherwise it's an error.</li>
<li><code>$0</code> (not followed by 1-9) and <code>$00</code> are the entire match.</li>
</ul>
<p>For comparison, following is JavaScript's native behavior for <code>$n</code> and <code>$nn</code>:</p>
<ul>
<li>Backreferences end after 1 or 2 digits. Cannot use backreference to capturing group 100+.</li>
<li><code>$1</code> is a literal <code>$1</code> if there are no capturing groups.</li>
<li><code>$10</code> is <code>$1</code> followed by a literal <code>0</code> if there are less than 10 capturing groups.</li>
<li><code>$01</code> is equivalent to <code>$1</code> if a capturing group exists, otherwise it's a literal <code>$01</code>.</li>
<li><code>$0</code> is a literal <code>$0</code>.</li>
</ul>
</div>
</div>
<div id="footer">
<p>© <a href="https://slev.life/">Steven Levithan</a> :: <a href="https://github.com/slevithan/xregexp">GitHub</a> :: <a href="https://xregexp.com/">XRegExp.com</a></p>
</div>
</body>
</html>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 50
|
"use strict";
// a small wrapper around fitzgen's source-map library
function SourceMap(options) {
options = defaults(options, {
file: null,
root: null,
orig: null,
});
var generator = new MOZ_SourceMap.SourceMapGenerator({
file: options.file,
sourceRoot: options.root
});
var orig_map = options.orig && new MOZ_SourceMap.SourceMapConsumer(options.orig);
function add(source, gen_line, gen_col, orig_line, orig_col, name) {
if (orig_map) {
var info = orig_map.originalPositionFor({
line: orig_line,
column: orig_col
});
source = info.source;
orig_line = info.line;
orig_col = info.column;
name = info.name;
}
generator.addMapping({
generated: {line: gen_line, column: gen_col},
original: {line: orig_line, column: orig_col},
source: source,
name: name
});
};
return {
add: add,
get: function () {
return generator
},
toString: function () {
return generator.toString()
}
};
};
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,873
|
\section{Introduction}
The development of Information Technology has been increasingly changing the means of information exchange leading to the need of digitizing print documents \autocite{1}. In the present era, there is a lot of fraud that often occur, there are so many types of frauds in this era, for example is account fraud. ID card in Indonesia is KTP. The Kartu Tanda Penduduk (literally: Resident identity Card), commonly KTP is an Indonesian compulsory identity card \autocite{2}. To avoid account fraud there was verification using ID card or KTP, then service operator extract the data in ID card. All identity in ID card, confidence for every field and face photo encoder field. But usually service operator extract the ID card in conventional way generally referred as data entry.
Service operator or someone who verifying the ID card manually is normally run by humans. Human need to take rest, human can't work 24 hours without sleep and sometimes humans made mistakes when doing the data entry because there were too much ID card that should be entry or to write down the ID card content.
Because of human sometimes made mistake, then we made automation to do data entry ID card using Optical Character Recognition (OCR) and post processing using Natural Language Processing (NLP). This technique have been developed to transform ID card into digital documents. However, various image, with good and bad quality, become challenge for OCR engine \cite{3}.
We use OCR to extract ID card image into digital documents and because sometimes OCR got and error or bad result \cite{3}, we use Natural Language Processing (NLP) for fixing text.
This research is in the form of answering the following questions about the development of the Indonesian ID card extraction system:
\begin{enumerate}
\item How OCR and NLP works in ID card extractor?
\item How fast is the processing per ID card?
\item The F-score of ID card extractor using OCR and NLP?
\end{enumerate}
Our analysis should be beneficial for researchers and practitioners helping them better understand strengths as well as weakness of this approach.
The remainder of this paper is organized as follows. We introduce literature review in Sec. 2. Then, Sec. 3 dataset and methodology. In Section 4, result and discussion. After that, the summary of our major findings is shown in Section 5.
\section{Literature Review}
\subsection{ID Card}
Indonesian ID Card can be used to recognize
citizen of Indonesia Identity in several requirements like for
sales and purchasing recording, admission and other
transaction processing systems (TPS) \cite{4}. Indonesian ID card or KTP had 37 fields.
\begin{figure}[H]
\centering
\includegraphics[scale=0.15]{212995.jpg}
\caption{Fields in KTP or ID card}
\label{fig1}
\end{figure}
\cref{fig1} shows that KTP or Indonesian ID card had 37 fields. There are many things we can do with ID card such as borrowing money, create various kinds of documents like driver's license, passport and many more. That's why we make Id card extractor to speed up the process, because there were so many instances that still almost on their input process of ID card was done by using conventional way.
\subsection{Optical Character Recognition}
Optical Character Recognition (OCR) is a process of converting a machine-printed or handwritten text image into a digital computer format that can be editable, OCR technology is considered as a challenging research area in the field of pattern recognition and artificial intelligence \cite{5}. This technology is often called text detection. OCR is part of an automatic identification technique. Sometimes the traditional way to input data via keyboard is not the most efficient way. There are several library in OCR, there were PyTesseract, TesseOCR, PyOCR and PyTesseract is the highest score for OCR library based on \cref{table_ocr_result} \cite{6}.
\begin{table}[htbp]
\caption{OCR Library Result}
\begin{center}
\begin{tabular}{||c c c||}
\hline
Library & Speed & Total Area \\ [0.5ex]
\hline\hline
PyOCR & 24.7S & 0.36 \\
PyTessseract & 25.16S & 0.46 \\
TesseOCR & 22.53S & 0.30 \\ [1ex]
\hline
\end{tabular}
\label{table_ocr_result}
\end{center}
\end{table}
PyTesseract had another attribute called confidence, confidence is using for how sure the word have been extracted. We count the confidence for every field, these results are used for service operator later.
\begin{figure}[H]
\centering
\includegraphics[scale=0.45]{ocr_works.jpg}
\caption{Matching Every Character \cite{marcom1-digitalsense_fitur_2015}}
\label{fig2}
\end{figure}
\cref{fig2} shows that OCR works by matching every letter to the close's character \autocite{marcom1-digitalsense_fitur_2015}. For increasing the accuracy, we use preprocessing, first change the image into grayscale and after that change the grayscale into binary.
\begin{table}[htbp]
\caption{F-score Evaluation \cite{4}}
\begin{center}
\begin{tabular}{||c c c||}
\hline
No & Model & F-Score \\ [0.5ex]
\hline\hline
1 & Convolutional Neural Network & 0.84 \\
2 & Support Vector Machine & 0.63 \\ [1ex]
\hline
\end{tabular}
\label{table2}
\end{center}
\end{table}
\cref{table2} shows that Convolutional Neural Network is better than Support Vector Machine for extracting Indonesian ID card \cite{4}. We will compare CNN with this research. We are using OCR and post-processing using Natural Language Processing for ID card extraction.
\section{Methodology}
\begin{figure}[H]
\centering
\includegraphics[scale=0.54]{proces.png}
\caption{System Flowchart}
\label{fig}
\end{figure}
In this research we developed ID card extraction with OCR and post-processing using NLP. There were 37 fields in Indonesian ID card, there are confidence fields, face encoder field, all identity in ID card, confidence for every field and face photo encoder field. For the analysis, we utilize 50 Indonesian ID card image. Those datasets were collected from public using Google Form submission. Then those datasets we preprocess, after that we extract it using OCR, then process again the result using NLP and using face detection on ID card to extract the face on ID card, and the last step merging all those data to form.
\subsection{Data Preprocessing}
\begin{figure}[H]
\centering
\includegraphics[scale=0.25]{imgg.jpg}
\caption{Processing IMG from RGB too Grayscale to Binary}
\label{fig3}
\end{figure}
\cref{fig3} shows that dataset be processed first using Grayscale technique to convert RGB layer into Grayscale, then threshold is performed to convert grayscale image \(R(x,y)\) into binary by selecting apropriate threshold, Tr means threshold , it set by 127, F(x,y) means the binary image \cref{eq1}.
\begin{equation}
F(x,y)
= \left\{
\begin{array}{l}
1, if \quad R(x,y) > Tr\\
0, Otherwise\\
\end{array}
\right.
\label{eq1}
\end{equation}
\subsection{Extracting ID Card}
In this system for the OCR process a module on
python programming. The module is called Python Tesseract which is an OCR module that is integrated with python in the form of a library, after all preprocessing steps are carried out to obtain an image in binary form. Then the image is processed using the Pytesseract library (Python Tesseract) so that text data is obtained from the image captured by the camera. Because we extracted Indonesian ID card then we add "lang = 'ind'" in hyperparameter tesseract. \cref{fig5} shows the result of ID card extractor.
\begin{figure}[H]
\centering
\includegraphics[scale=0.3]{result.png}
\caption{ID card Extraction Result}
\label{fig5}
\end{figure}
\subsection{Post-Processing}
After we got the result from extracting image \cref{fig5}, there would be wrong spelling like field "Gol Darah : 0" the content of attribute "Gol Darah" is not "0" instead "O", and field "Kel/Des : WEDOMARTANI!" it instead be "WEDOMARTANI" without "!" symbol, because there are wrong spelling in result of ID Card extractor then we are using NLP to fixing it. First we are using Punctual Remover on the result, but we don't remove ":.,-" symbol, because that symbol is part of content in ID card. After that we are using RegEx. Regex is Regular Expression, Regex is using for finding sequence of characters that define a search pattern \autocite{7}. Then we are using word to number converter, we are using this for "NIK" field, because there are misspelling character because scratch ID card or bad quality image, so we make function to take care of it, word to number converter list:
\begin{itemize}
\item "L" : "1",
\item 'l' : "1",
\item 'O' : "0",
\item 'o' : "0",
\item '?' : "7",
\item 'A' : "4",
\item 'Z' : "2",
\item 'z' : "2",
\item 'S' : "5",
\item 's' : "5",
\item "b" : "6",
\item "B" : "8",
\item "G" : "6"
\end{itemize}
After we are splitting the sentence by ":" to take every content of field, because we just need the content not the attribute. for example "Name : Firhan Maulana", we just need "Firhan Maulana" not "Name :". every part of field is divided by three part, shows on \cref{fig9}.
\begin{figure}[H]
\centering
\includegraphics[scale=0.5]{3part.jpg}
\caption{3 Part of Every Field}
\label{fig9}
\end{figure}
Part 1 is attribute, part 2 is colon and part 3 is the content. Then we just use RegEx to find attribute on every field, and then split it by colon, and then take the content which is the last part. But there was special case for some field. For "Tempat/tanggal lahir" we are using this algorithm. re.search('([0-9]{2}\-[0-9]{2}\-[0-9]{4})') this algorithm is using to search pattern date on "Tempat/Tanggal Lahir" field. For "Jenis Kelamin" field, we are using library Regex.search, because "Jenis kelamin" or Gender just have two types, it were "LAKI-LAKI" and "PEREMPUAN" then using this algorithm, re.search("(LAKI-LAKI|LAKI|LELAKI|PEREMPUAN)"), re.search is using too for other field that containing limited type of content, there were "Agama", "Golongan Darah", "Kewarganegaraan" and "Status", because service operator need proof that this ID card extraction result is truly got the same content equal to the real ID card. Then we added the confidence for every field. Confidence is how true the image extracting to text presented by scale 0 to 100. Confidence one of the modules in Pytesseract library. Actually confidence counting percent word-by-word not per-line shows by \cref{fig7}.
\begin{figure}[H]
\centering
\includegraphics[scale=0.7]{top_confidnet.png}
\caption{Example Confidence Word by Word}
\label{fig7}
\end{figure}
Because we want to count the confidence by per-line then we gather word by word to sentence then calculate the mean of sentence then it becomes counting confidence per-line, or we can call it per-field. We count this confidence by break down every field in result, then take the confidence word, and calculate the mean of the field. For Example for "Provinsi" field we got confidence 91, this result by calculating mean of confidence show on \cref{table3}.
\begin{table}[htbp]
\caption{Example Confidence per-word}
\begin{center}
\begin{tabular}{||c c c c||}
\hline
Province & DAERAH & ISTIMEWA & YOGYAKARTA \\ [0.5ex]
\hline\hline
Conf & 92 & 89 & 91 \\ [1ex]
\hline
\end{tabular}
\label{table3}
\end{center}
\end{table}
\subsection{Face Detection}
Face detection is using for detect face on ID card, this because some of the service operator need to take the face photo on the ID card. So we take the face photo using Haarcascade Frontalface Default, it was open source from OpenCv \autocite{goel_hybrid_2012}. This feature uses the Haar waveform. The Haar waveform is a square wave. In 2 dimensions, a square wave is a pair of adjacent squares, 1 light, and 1 dark. Haar is determined by subtracting the mean dark area pixels from the average light area pixels \autocite{noauthor_face_2012}. If the difference is above the threshold, the feature is said to be present. To determine whether there is a Haar feature at each image location, this technique is called Integral Image. Generally, integrals add up in small units. In this case, this small unit is called the pixel value. The integral value for each pixel is the sum of all the pixels above and to the left. From top left to bottom right, images can be integrated as per pixel mathematical operations. Filters at each level are trained to classify images that have been filtered previously. During use, if one of the filters fails, the image region in the image is classified as "Not Face". When the filter succeeds in passing the image region, the image region is then included in the next filter \autocite{noauthor_face_2012}. Image regions that have gone through all filters will be considered as "Faces", shows on \cref{fig8}.
\begin{figure}[H]
\centering
\includegraphics[scale=0.45]{face.jpg}
\caption{Chain of Filters \cite{noauthor_face_2012}}
\label{fig8}
\end{figure}
After got the face photo, we encode the image so it becomes string then we can add to field "facephoto". We encode the face photo using Base64 encoding.
\section{Result and Discussion}
Example for final result would be like this:
\begin{enumerate}
\item "kind": "C",
\item "identifier": "3471111111111111",
\item "name": "Firhan Maulana",
\item"birthPlace": "GROBOGAN. ",
\item"birthDate": "02-09-1979",
\item"gender": "M",
\item"bloodType": "O",
\item"address": "PRM PURI DOMAS RT : 001 RW : 024 KELURAHAN/DESA : WEDOMARTANI KECAMATAN : NGEMPLAK",
\item"religion": "ISLAM",
\item"marriageStatus": "M",
\item"occupation": "PEDAGANG",
\item"nationalityCode": "IND",
\item"expiryDate": "SEUMUR HIDUP",
\item"facePhoto":"2KfZhNiz2YTYp9m...",
\item"cardImage": "Yg2KfZhNmk/Ysf...",
\item"issuerCountryCode": "IND",
\item"issuedProvince": "JAKARTA",
\item"issuedCity": "KABUPATEN SLEMAN",
\item"issuedDate": "05-06-2012",
\item"faceTop": 786,
\item"faceLeft": 212,
\item"faceWidth": 163,
\item"faceHeight": 163,
\item"extractedAt": "30-11-2020",
\item"identifierconf": 0,
\item"nameconf": 91,
\item"birthPlaceconf": 54,
\item"birthDateconf": 95,
\item"genderconf": 71,
\item"bloodTypeconf": 76,
\item"addressconf": 68.0,
\item"religionconf": 18,
\item"marriageStatusconf": 95,
\item"occupationconf": 95,
\item"issuedProvinceconf": 95,
\item"issuedCityconf": 95,
\item"issuedDateconf": 83
\end{enumerate}
ID Card Extractor give output as Json file with 37 fields, it becomes 37 fields because extract the face photo in the ID card too and give the confidence for every field. Extracting ID card with OCR have common errors or false spelling like colon after word, slash symbol and usually errors come from symbol in ID card. Quality of image will reduce OCR accuracy this are because OCR read pattern of pixels and decide on the closest match of characters. Bad quality photo or old ID card that sometimes had scratch can have an effect on the quality of the result.
This confidence is using for service operator later, if confidence more than 85 for one field, then admin does not have to double check the result. For example "NAMA : Firhan Maulana" got confidence 91, then service operator does not have to double check, if it lower than 85 then service operator have to double check the result, because when the confidence lower than 85 there were misspelling on the field.
ID card extractor using OCR and post processing using NLP have 0.78 F-score total, it was divided by two type of photo, 25 for ID card using camera with 0.67 F-score, and 25 photo using scanner with 0.89 F-score. For measurement using F-score, the complete result can be seen at \cref{table4}, CNN result still better than just using OCR and Post processing using NLP, and we need 4510 millisecond on average to extract per ID card.
\begin{table}[htbp]
\caption{Final Result}
\begin{center}
\begin{tabular}{||c c c||}
\hline
No & Model & F-Score \\ [0.5ex]
\hline\hline
1 & Using OCR and Post processing using NLP & 0.78 \\
2 & Convolutional Neural Network & 0.84 \\
3 & Support Vector Machine & 0.63 \\ [1ex]
\hline
\end{tabular}
\label{table4}
\end{center}
\end{table}
\section{Conlusion}
We created ID card extractor using OCR and post-processing using NLP with 50 dataset ID card that divided by two type, 25 for scanning image, and 25 for camera image. F-score that we have obtained was 0.78 for all image, is divided by 0.89 F-score for scanning image and 0.67 F-score for camera image. From previous research using CNN model 0.84 F-score, and we need 4510 milliseconds to extract per ID card.
For the future experiment, we plan to use deep leaning Recurrent Neural Networks (RNN),Residual Neural Networks (ResNet), convolutional Neural Network (CNN). Then using NLP for the post-processing.
\section*{Acknowledgment}
First we would like to thank Lovia (https://about.lovia.life), who give us facilities to finish this research paper.
Second we would like to thank to our parents, who supported us with love and understanding. Without them, we could never have reached this current level of success.
\printbibliography[]
\end{document}
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If you liked Deadhouse Gates you might like these books.
In the vast dominion of Seven Cities, in the Holy Desert Raraku, the seer Sha'ik and her followers prepare for the long-prophesied uprising known as the Whirlwind. Unprecedented in size and savagery, this maelstrom of fanaticism and bloodlust will embroil the Malazan Empire in one of the bloodiest conflicts it has ever known, shaping destinies and giving birth to legends . . .
A corkscrew plume of dust raced across the basin, heading deeper into the trackless desert of the Pan'potsun Odhan. Though less than two thousand paces away, it seemed a plume born of nothing.
From his perch on the mesa's wind-scarred edge, Mappo Runt followed it with relentless eyes the colour of sand, eyes set deep in a robustly boned, pallid face. He held a wedge of emrag cactus in his bristle-backed hand, unmindful of the envenomed spikes as he bit into it. Juices dribbled down his chin, staining it blue. He chewed slowly, thoughtfully.
Beside him Icarium flicked a pebble over the cliff edge. It clicked and clattered on its way down to the boulder-strewn base. Under the ragged Spiritwalker robe---its orange faded to dusty rust beneath the endless sun---his grey skin had darkened into olive green, as if his father's blood had answered this wasteland's ancient call. His long, braided black hair dripped black sweat onto the bleached rock.
Mappo pulled a mangled thorn from between his front teeth. 'Your dye's running,' he observed, eyeing the cactus blade a moment before taking another bite.
The unlikely pair turned back to their small campsite, tucked between two towering spires of wind-sculpted rock. There was no hurry. Icarium sat down on a flat rock and proceeded to oil his longbow, striving to keep the hornwood from drying out. Once satisfied with the weapon's condition, he turned to his single-edged long sword, sliding the ancient weapon from its bronze-banded boiled-leather scabbard, then setting an oiled whetstone to its notched edge.
Mappo struck the hide tent, folding it haphazardly before stuffing it into his large leather bag. Cooking utensils followed, as did the bedding. He tied the drawstrings and hefted the bag over one shoulder, then glanced to where Icarium waited---bow rewrapped and slung across his back.
Icarium nodded, and the two of them, half-blood Jaghut and full-blood Trell, began on the path leading down into the basin.
Overhead the stars hung radiant, casting enough light down onto the basin to tinge its cracked pan silver. The bloodflies had passed with the vanishing of the day's heat, leaving the night to the occasional swarm of capemoths and the batlike rhizan lizards that fed on them.
Mappo and Icarium paused for a rest in the courtyard of some ruins. The mudbrick walls had all but eroded away, leaving nothing but shin-high ridges laid out in a geometric pattern around an old, dried-up well. The sand covering the courtyard's tiles was fine and windblown and seemed to glow faintly to Mappo's eyes. Twisted brush clung with fisted roots along its edges.
The Pan'potsun Odhan and the Holy Desert Raraku that flanked it to the west were both home to countless such remnants from long-dead civilizations. In their travels Mappo and Icarium had found high tels---flat-topped hills built up of layer upon layer of city---situated in a rough procession over a distance of fifty leagues between the hills and the desert, clear evidence that a rich and thriving people had once lived in what was now dry, wind-blasted wasteland. From the Holy Desert had emerged the legend of Dryjhna the Apocalyptic. Mappo wondered if the calamity that had befallen the city-dwellers in this region had in some way contributed to the myth of a time of devastation and death. Apart from the occasional abandoned estate such as the one they now rested in, many ruins showed signs of a violent end.
His thoughts finding familiar ruts, Mappo grimaced. Not all pasts can be laid at our feet, and we are no closer here and now than we've ever been. Nor have I any reason to disbelieve my own words. He turned away from those thoughts as well.
Near the courtyard's centre stood a single column of pink marble, pitted and grooved on one side where the winds born out in Raraku blew unceasingly towards the Pan'potsun Hills. The pillar's opposite side still retained the spiral patterning carved there by long-dead artisans.
Upon entering the courtyard Icarium had walked directly to the six-foot-high column, examining its sides. His grunt told Mappo he'd found what he had been looking for.
'And this one?' the Trell asked, setting his leather sack down.
'Some would consider your curse a gift,' Mappo said, a flicker of sadness passing across his features.
'No,' Mappo agreed. It was another hour before the moon would rise. He watched Icarium lay out six long, stone-tipped arrows, then squinted out into the darkness. Cold fear crept along the nape of his neck.
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\section{Introduction}
\begin{figure}
\centering
\subfigure[Autoregressive model]{\includegraphics[width=0.23\textwidth]{figures/autregression-Autoregression.pdf}}
\subfigure[Non-autoregressive model]{\includegraphics[width=0.23\textwidth]{figures/autregression-Non-Autoregression.pdf}}
\caption{Signal flow of an autoregressive and a non-autoregressive model with an optional internal state for three time steps.}
\label{fig:autregression}
\end{figure}
System identification is essential for many tasks, such as system control and sensor fusion. It has a long history, beginning with the identification of linear systems \citep{zadeh_identification_1956}. Most practical relevant systems are nonlinear and dynamic, which led to the development of nonlinear dynamic system identification methods.
Those models may be autoregressive or non-autoregressive, as visualized in Figure \ref{fig:autregression}. The output of an autoregressive model is dependant on the model outputs of previous time steps, which is not the case for a non-autoregressive model. In both cases, the model may have an internal state, which is transferred between each time step. An example for an autoregressive model without an internal state is the nonlinear autoregressive exogenous (NARX) model and for a non-autoregressive model with an internal state is the non-linear state space (NLSS) model.
The identified models may be used for prediction or simulation. Prediction is the task of estimating a limited amount of time steps ahead with information of the past system outputs. Simulation is the task of estimating the system output only with the inputs. In the present work, we will focus only on the simulation task.
Neural networks have been applied to system identification tasks for a long time. Historically feedforward neural networks are applied autoregressively to model the system dynamics, which is a variant of a NARX model \citep{chen_non-linear_1990}. The success of this approach was limited by the available hardware and the gradient propagation over long sequences. With the recent development of deep learning-based methods in computer vision and natural language processing, more sophisticated software and hardware for the training of neural networks became available.
With this development, the application of neural networks for system identification has become widespread, with a focus on improving upon existing black-box system identification methods. Most current neural network-based system identification methods still are NARX variants with different neural network architectures as nonlinearities. In related work, a multitude of system identification methods that are based on autoregressive neural networks has been proposed using multilayer perceptrons (MLPs) \citep{shi_neural_2019}, cascaded MLPs \citep{ljung_deep_2020}, convolutional neural networks \citep{lopez_nonlinear_2017}, TCNs \citep{andersson_deep_2019}, and recurrent neural networks \citep{kumar_comparative_2019} with promising results.
Although neural networks that are based on convolutional or recurrent layers have inherent capabilities to model dynamic systems, only a limited amount of work omitted the autoregression. It has been shown, that RNNs behave like an NLSS \citep{ljung_deep_2020} and an RNN has been applied non-autoregressively to a synthetic dataset \citep{gonzalez_non-linear_2018}. The authors of the present paper applied non-autoregressive TCNs and gated recurrent units (GRUs) to an inertial measurement-based sensor fusion task, outperforming state-of-the-art domain-specific sensor fusion methods \citep{weber_neural_2020}.
To the best of our knowledge, it has not yet been analyzed what the differences in the implementation, accuracy, and execution performance of autoregressive and non-autoregressive neural networks are.
The present paper focuses on the comparison of autoregressive and non-autoregressive variants of TCNs and GRUs with the following main contributions:
\begin{itemize}
\item We describe a workflow for autoregressive and non-autoregressive neural networks for system identification using state-of-the-art deep learning tools and methods.
\item We find that non-autoregressive neural networks are faster and easier to implement than their autoregressive counterparts and are as least as accurate.
\item We compare the estimation performance of the proposed neural networks and state-of-the-art black-box system identification methods on three publicly available benchmark datasets.
\item We find that non-autoregressive gated recurrent units consistently outperform all other neural network-based system identification models and in the benchmarks without extrapolation, all black-box models.
\end{itemize}
\section{Neural Network Implementation}
\label{sec:networks}
In this section, we describe two different neural network architectures, which have the inherent ability to model dynamic systems without autoregressive connections. Furthermore, we describe the corresponding training process with best practices from sequential data processing with neural networks. The two architectures that we consider are TCNs, which process large sequences at once, and GRUs which propagate hidden states over time.
\begin{figure}
\centering
\subfigure[TCN-AR]{\includegraphics[width=0.22\textwidth]{figures/TCN-TCN-AR.pdf}}
\subfigure[TCN-NAR]{\includegraphics[width=0.22\textwidth]{figures/TCN-TCN-NAR.pdf}}
\caption{Signal flow of the autoregressive TCN (TCN-AR) and non-autoregressive TCN (TCN-NAR) for three time steps.}
\label{fig:tcn}
\end{figure}
TCNs are stateless feed-forward networks that apply 1d-dilated causal convolutions to sequences \citep{andersson_deep_2019}, which are inspired by Wavenet, which is used for raw audio generation \citep{oord_wavenet_2016}. The convolutional layers are stacked on top of each other, which results in a receptive field that describes the number of input samples that are taken into account for the prediction of an output value. The dilation in the convolutions increases the receptive field exponentially with the layer size, enabling the processing of sequences with relations over long periods of time. The main advantage of TCNs is that they are fast in training and inference because of their high parallelizability. The main disadvantage of TCNs is that their ability to model system dynamics is limited by the size of the receptive field, limiting its viability in systems that are influenced by integrators over an indefinite amount of time \citep{weber_neural_2020}.
\begin{figure}
\centering
\subfigure[GRU-AR]{\includegraphics[width=0.22\textwidth]{figures/GRU-GRU-AR.pdf}}
\subfigure[GRU-NAR]{\includegraphics[width=0.22\textwidth]{figures/GRU-GRU-NAR.pdf}}
\caption{Signal flow of the autoregressive GRU (GRU-AR) and non-autoregressive GRU (GRU-NAR) for three time steps.}
\label{fig:gru}
\end{figure}
GRUs are a variant of RNNs, that use recurrent connections in their hidden layers to store a hidden state for each time step. This approach has the advantage, that the state information can be theoretically stored over an indefinite amount of time, which in practice is limited by the vanishing gradient problem \citep{hochreiter_vanishing_2011}.
The vanishing gradient problem stems from numerical issues on long backpropagation paths in the optimization process. In autoregressive neural networks, the backpropagation path is very long because for every time step the gradient has to propagate from the output through all layers back to the input. In RNNs, the backpropagation path is shorter, but the vanishing gradient problem is still limiting at hundreds of time steps \citep{hochreiter_vanishing_2011}.
The gating mechanism of GRUs alleviates this issue, enabling the network to propagate the hidden state over thousands instead of hundreds of time steps during training. Several regularization methods for RNNs have been proposed, that reduce overfitting and improve generalizability \citep{merity_regularizing_2017}. The main disadvantage of GRUs over TCNs is their sequential nature, which limits its parallelizability, reducing the training speed and especially the inference speed on acceleration hardware.
Both network architectures may be used as autoregressive models, which we refer to as GRU-AR and TCN-AR, and as non-autoregressive models, which we refer to as GRU-NAR and TCN-NAR. The structures of these models are visualized in Figure \ref{fig:tcn} and \ref{fig:gru}. Autoregressive models may be trained using teacher-forcing, with the ground-truth values as input, or in free-running mode, with its own output as input \citep{lamb_professor_2016}. Teacher forcing is faster in training, but only accurate for one-step-ahead prediction tasks, where the model only has to predict the next sample with the values measured in the past. For simulation with neural networks, teacher-forcing is inaccurate, which is why we train the autoregressive models in free-running mode \citep{ribeiro_parallel_2018}.
The naive application of TCN-AR would be very slow and requires an enormous amount of memory during training, because of the number of redundant calculations in its broad receptive field. Because of that, we implemented an optimized caching scheme, which enables us to train TCN-AR in free-running mode in the first place\citep{paine_fast_2016}.
We implemented the training process with FastAI 2, which is a deep learning library that is built upon Pytorch \citep{howard_fastai_2020}. For the optimizer, we use the current state-of-the-art combination of RAdam and Lookahead which proved to be effective in various tasks and requires no learning-rate warm-up phase \citep{liu_variance_2019}, \citep{zhang_lookahead_2019}. Because we need no warm-up phase, we use cosine-annealing for decreasing the learning rate as soon as the optimizer hits an optimization plateau \citep{loshchilov_sgdr_2017}. We find the maximum learning rate, with which the optimizer begins, with the learning rate finder heuristic \citep{smith_cyclical_2017}.
The datasets used for training consist of multiple measured sequences of different lengths. We extract overlapping windows with varying starting points from the sequences for the generation of the mini-batches, to avoid memorizing the sequences. To process longer sequences, we use truncated backpropagation through time (TBPTT) \citep{tallec_unbiasing_2017}. To apply TBPTT to autoregressive models, not only the hidden state but also the last generated output has to be transferred between two mini-batches. The input signals are standardized to zero mean and a standard deviation of one, which is also applied just-in-time to the autoregressive inputs \citep{ioffe_batch_2015}.
We use the elementwise mean-squared error as the loss function, which is close to the evaluation metric root-mean-square error. Every estimation of dynamic systems that starts with an unknown system state has a transient phase, that describes the time period the estimator needs to converge to a steady-state solution. To minimize the long-term error, we exclude the transition phase from the loss function in the optimization process. With TBPTT, only the first mini-batch of a sequence is affected, because all following mini-batches have a converged system state. TCNs have a receptive field that describes the number of samples that are taken into account for each estimated value. We exclude the estimations that have fewer input values than the receptive field from the gradient propagation because it relies on padded values with no measured information.
The optimization process of the neural network requires a multitude of hyperparameters that have a significant influence on the final performance of the model. Because the hyperparameters span a vast optimization space, it is difficult to find the optimal configuration without an efficient optimization algorithm. For this, we use the Asynchronous Successive Halving Algorithm (ASHA) \citep{li_system_2020}, a state-of-the-art black-box optimizer that combines random-search with early-stopping in an asynchronous environment.
\section{Performance Evaluation}
In this section, we evaluate the accuracy and the simulation time of the neural network models, which we described in Section \ref{sec:networks}, on several publicly available datasets. Additionally, we compare the performance of the neural network models with the results of state-of-the-art system identification methods of related work. All models have been trained and executed on an Nvidia RTX 2080 Ti.
The following datasets are designed specifically as benchmarks for non-linear system identification methods:
\begin{itemize}
\item Silverbox \citep{wigren_three_2013}: The Silverbox benchmark was generated from an electrical circuit that models a nonlinear progressive spring as an oscillating system. The main challenge of the dataset is the extrapolation part of the test sequence, where the output values are larger than any value in the estimation sequence.
\item Wiener-Hammerstein \citep{schoukens_wiener-hammerstein_2009}: The Wiener Hammerstein benchmark was generated by an electrical circuit that models a Wiener-Hammerstein system.
\item Wiener Hammerstein with Process Noise \citep{schoukens_wiener-hammerstein_2016}: The Wiener-Hammerstein benchmark with process noise was generated by an electrical circuit that models a Wiener-Hammerstein system with additive process noise in the training data. Identifying the process without the process noise is the main challenge of the dataset.
\end{itemize}
All benchmark datasets provide an estimation and a test subset. For the training and hyperparameter optimization of the neural networks, we split the given estimation subsets further in training and validation subsets. For the evaluation of the performance, we use the root-mean-square error (RMSE). We also ignore the output of the first $N$ samples in the transient phase of the simulation with $N$ given in the description of each benchmark dataset.
For every dataset, the hyperparameters for the GRU and the TCN have been optimized. The resulting configuration is used to train an autoregressive and a non-autoregressive variant, that is used for the following evaluation.
\subsection{Autoregressive vs Non-Autoregressive Neural Networks}
\begin{figure}
\centering
\subfigure[WH]{\includegraphics[width=0.15\textwidth]{figures/tst_wh.pdf}}
\subfigure[WH-Noise]{\includegraphics[width=0.15\textwidth]{figures/tst_wh_noise.pdf}}
\subfigure[Silverbox]{\includegraphics[width=0.15\textwidth]{figures/tst_silverbox.pdf}}
\caption{RMSE of every model for the test subset of the Wiener-Hammerstein benchmark (a), the Wiener-Hammerstein benchmark with process noise (b), and the Silverbox benchmark (c). Non-autoregressive models perform better than their autoregressive counterparts.}
\label{fig:accuracy}
\end{figure}
First, we compare the accuracy of the autoregressive and non-autoregressive variants of the models on every dataset. Figure \ref{fig:accuracy} visualizes the test RMSE of every model on every dataset. The errors of both variants are close to each other, with the non-autoregressive variant being slightly more accurate. This may be caused by a better fitting set of hyperparameters for the non-autoregressive models.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{figures/loss_over_time.pdf}
\caption{Validation- and training-RMSE over the duration of the training process for each model on the Silverbox dataset. The training-RMSE is drawn as dashed lines, the validation-RMSE as solid lines. The autoregressive models require significantly more time to achieve similar degrees of performance.}
\label{fig:loss}
\end{figure}
The hyperparameter optimization process for the autoregressive models is limited by their low training speed. Figure \ref{fig:loss} visualizes the evolution of the training and validation RMSE over the time of the training of each model variant on the Silverbox dataset with the individual optimized set of hyperparameters. It becomes clear that the non-autoregressive models finish the training process significantly faster than the autoregressive models. In fact, TCN-AR is so slow, that TCN-NAR and GRU-NAR finish all mini-batches before the first mini-batch of TCN-AR is finished. This advantage in speed allows for more extensive hyperparameter optimizations as well as more complex models with the same amount of resources.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{figures/training_time.pdf}
\caption{The training time of a mini-batch with a size of 16 over a variable sequence length. Autoregressive models scale worse than non-autoregressive models with increasing sequence lengths.}
\label{fig:training_duration}
\end{figure}
Because of their sequential nature, the training speed of the autoregressive models is mostly dependant on the sequence length of the mini-batch. Figure \ref{fig:training_duration} visualizes the training time of a mini-batch for each model variant over the sequence length. The TCN models require a sequence length of at least 1023 samples because of the receptive field of $2^{10}-1 = 1023$ samples, which results from the depth of 10, which we identified in the hyperparameter optimization. The training time of the autoregressive models scales much worse with the sequence length than the non-autoregressive models. The training time of TCN-AR is especially bad, considering that we already use an optimized caching algorithm. In contrast, TCN-NAR has over the evaluated value range of sequence lengths a constant training duration, because the convolutions are executed independently over the sequence on the GPU.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{figures/simulation_time.pdf}
\caption{The inference time of a single sequence with variable sequence length for each model. Autoregressive models are slower than non-autoregressive models.}
\label{fig:simulation_duration}
\end{figure}
The model variants also require different amounts of time for the simulation of sequences with variable lengths. Figure \ref{fig:simulation_duration} visualizes the inference time of each model variant for the simulation of sequences with variable length. The sequential models GRU-AR, GRU-NAR, and TCN-AR scale linearly with the sequence length while TCN-NAR has a constant inference time over the evaluated range because of its parallel nature.
All in all, the autoregressive variants of the evaluated networks are slower in training and inference without any benefits for accuracy.
\subsection{Comparison with State-of-the-art Methods}
In all three benchmarks, GRU-NAR is the most accurate model of the ones, that we implemented. We compare the accuracy of GRU-NAR with the results of state-of-the-art neural network-based and other black-box system identification methods that were applied in related work on these datasets.
\begin{table}
\caption{Test RMSE of black-box models in the Silverbox benchmark}
\label{tab:res_silverbox}
\begin{tabular}{c|l}
RMSE [mV] & Method \\
\hline
0.26 & PNLSS \citep{paduart_identification_2010}\\
0.33 & NN + Cubic Regressor \citep{ljung_estimation_2004}\\
\textbf{0.96} & \textbf{GRU-NAR (NN)} \\
2.18 & TCN (NN) \citep{maroli_nonlinear_2019} \\
3.98 & LSTM (NN) \citep{andersson_deep_2019} \\
4.88 & TCN (NN) \citep{andersson_deep_2019} \\
\end{tabular}
\end{table}
\begin{table}
\caption{Test RMSE of black-box models in the Wiener-Hammerstein benchmark}
\label{tab:wh}
\begin{tabular}{c|l}
RMSE [mV] & Method \\
\hline
\textbf{0.39} & \textbf{GRU-NAR (NN)} \\
0.42 & PNLSS \citep{paduart_identification_2009} \\
0.49 & Wiener-Hammerstein \citep{wills_estimation_2009} \\
2.98 & BLA-based \citep{morari_identification_2010}\\
4.71 & FS-LSSVM \citep{de_brabanter_fixed-size_2009} \\
17.6 & CNN (NN) \citep{lopez_nonlinear_2017} \\
45.61 & FFH (NN) \citep{romero_ugalde_neural_2013} \\
51.4 & DN-BI (NN) \citep{rosa_nonlinear_2015} \\
\end{tabular}
\end{table}
\begin{table}
\caption{Test RMSE of black-box models in the Wiener-Hammerstein benchmark with process noise}
\label{tab:wh_noise}
\begin{tabular}{c|l}
RMSE [mV] & Method \\
\hline
\textbf{20.3} & \textbf{GRU-NAR (NN)} \\
25 & WH-EIV \citep{schoukens_identication_2016} \\
30 & PNLSS \citep{gedon_deep_2020}\\
30.3 & NFIR \citep{belz_automatic_2017} \\
42.35 & STORN (NN) \citep{gedon_deep_2020}\\
54.1 & VAE-RNN (NN) \citep{gedon_deep_2020}\\
\end{tabular}
\end{table}
The test results are compared in Table \ref{tab:res_silverbox} for the Silverbox benchmark, Table \ref{tab:wh} for the Wiener-Hammerstein benchmark, and Table \ref{tab:wh_noise} for the Wiener-Hammerstein benchmark with process noise.
In the Silverbox benchmark, black-box system identification methods have difficulties with the test dataset because of the extrapolation part. The extrapolation performance highly depends on the estimator structure and can not generalize over all systems, which is why this is a major shortcoming of neural networks. In this case, the system has a cubic nonlinearity, which is why a combination of a neural network with a cubic regressor, and a polynomial non-linear state-space model perform better in the extrapolation part than more complex models.
In the Wiener-Hammerstein benchmark and the Wiener-Hammerstein benchmark with process noise, GRU-NAR outperforms not only the neural network-based models but all black-box system identification models.
The comparison demonstrates, that the performance of neural network-based models depends to a large extent on the implementation of the structure and training process. This results in GRU-NAR being the best performing neural network-based implementation, which is also highly competitive with other black-box system identification methods.
\section{Conclusion}
In this work, we described the implementation of non-autoregressive and autoregressive neural networks with current best practices for system identification. We compared the accuracy, training, and inference performance of the different models on three publicly available benchmark datasets. Finally, we compared the accuracy of the best performing neural network, a non-autoregressive gated recurrent unit, with other state-of-the-art black-box system identification models.
Our results show that autoregressive neural networks require significantly more time for training and inference than their non-autoregressive counterparts, without any benefits for accuracy. This limits the extent of possible hyperparameter optimization and model capacity, especially in more complex systems. Furthermore, we found that our implementation of non-autoregressive gated recurrent units outperforms all other neural network-based system identification models in the evaluated benchmark datasets and is among the best performing black-box models.
The present work focused on the simulation task, where no system output values are available for the model. In future work, the comparison may be done for the prediction task. While the one-step-ahead prediction task is trivial to implement non-autoregressively, the multi-step-ahead prediction is more challenging and may require a novel approach to avoid autoregression.
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Драгољи могу бити:
Драгољи (Горажде), насељено мјесто у општини Горажде, Федерација Босне и Херцеговине, БиХ
Драгољи (Ново Горажде), насељено мјесто у општини Ново Горажде, Република Српска, БиХ
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{"url":"https:\/\/www.groundai.com\/project\/temporal-regularization-in-markov-decision-process\/","text":"Temporal Regularization in Markov Decision Process\n\n# Temporal Regularization in Markov Decision Process\n\n## Abstract\n\nSeveral applications of Reinforcement Learning suffer from instability due to high variance. This is especially prevalent in high dimensional domains. Regularization is a commonly used technique in machine learning to reduce variance, at the cost of introducing some bias. Most existing regularization techniques focus on spatial (perceptual) regularization. Yet in reinforcement learning, due to the nature of the Bellman equation, there is an opportunity to also exploit temporal regularization based on smoothness in value estimates over trajectories. This paper explores a class of methods for temporal regularization. We formally characterize the bias induced by this technique using Markov chain concepts. We illustrate the various characteristics of temporal regularization via a sequence of simple discrete and continuous MDPs, and show that the technique provides improvement even in high-dimensional Atari games.\n\n## 1 Introduction\n\nThere has been much progress in Reinforcement Learning (RL) techniques, with some impressive success with games\u00a0(Silver et\u00a0al., 2016), and several interesting applications on the horizon\u00a0(Koedinger et\u00a0al., 2018; Shortreed et\u00a0al., 2011; Prasad et\u00a0al., 2017; Dhingra et\u00a0al., 2017). However RL methods are too often hampered by high variance, whether due to randomness in data collection, effects of initial conditions, complexity of learner function class, hyper-parameter configuration, or sparsity of the reward signal\u00a0(Henderson et\u00a0al., 2018). Regularization is a commonly used technique in machine learning to reduce variance, at the cost of introducing some (smaller) bias. Regularization typically takes the form of smoothing over the observation space to reduce the complexity of the learner\u2019s hypothesis class.\n\nIn the RL setting, we have an interesting opportunity to consider an alternative form of regularization, namely temporal regularization. Effectively, temporal regularization considers smoothing over the trajectory, whereby the estimate of the value function at one state is assumed to be related to the value function at the state(s) that typically occur before it in the trajectory. This structure arises naturally out of the fact that the value at each state is estimated using the Bellman equation. The standard Bellman equation clearly defines the dependency between value estimates. In temporal regularization, we amplify this dependency by making each state depend more strongly on estimates of previous states as opposed to multi-step methods that considers future states.\n\nThis paper proposes a class of temporally regularized value function estimates. We discuss properties of these estimates, based on notions from Markov chains, under the policy evaluation setting, and extend the notion to the control case. Our experiments show that temporal regularization effectively reduces variance and estimation error in discrete and continuous MDPs. The experiments also highlight that regularizing in the time domain rather than in the spatial domain allows more robustness to cases where state features are mispecified or noisy, as is the case in some Atari games.\n\n## 2 Related work\n\nRegularization in RL has been considered via several different perspectives. One line of investigation focuses on regularizing the features learned on the state space (Farahmand et\u00a0al., 2009; Petrik et\u00a0al., 2010; Pazis and Parr, 2011; Farahmand, 2011; Liu et\u00a0al., 2012; Harrigan, 2016). In particular backward bootstrapping method\u2019s can be seen as regularizing in feature space based on temporal proximity Sutton et\u00a0al. (2009); Li (2008); Baird (1995). These approaches assume that nearby states in the state space have similar value. Other works focus on regularizing the changes in policy directly. Those approaches are often based on entropy methods (Neu et\u00a0al., 2017; Schulman et\u00a0al., 2017; Bartlett and Tewari, 2009). Explicit regularization in the temporal space has received much less attention. Temporal regularization in some sense may be seen as a \u201cbackward\u201d multi-step method (Sutton and Barto, 1998). The closest work to ours is possibly (Xu et\u00a0al., 2017), where they define natural value approximator by projecting the previous states estimates by adjusting for the reward and . Their formulation, while sharing similarity in motivation, leads to different theory and algorithm. Convergence properties and bias induced by this class of methods were also not analyzed in Xu et\u00a0al. (2017).\n\n## 3 Technical Background\n\n### 3.1 Markov chains\n\nWe begin by introducing discrete Markov chains concepts that will be used to study the properties of temporally regularized MDPs. A discrete-time Markov chain (Levin and Peres, 2008) is defined by a discrete set of states and a transition function which can also be written in matrix form as . Throughout the paper, we make the following mild assumption on the Markov chain:\n\n###### Assumption 1.\n\nThe Markov chain P is ergodic: P has a unique stationary distribution .\n\nIn Markov chains theory, one of the main challenge is to study the mixing time of the chain\u00a0(Levin and Peres, 2008). Several results have been obtained when the chain is called reversible, that is when it satisfies detailed balance.\n\n###### Definition 1 (Detailed balance\u00a0(Kemeny and Snell, 1976)).\n\nLet be an irreducible Markov chain with invariant stationary distribution 1. A chain is said to satisfy detailed balance if and only if\n\n \u03bciPij=\u03bcjPji\u2200i,j\u2208S. (1)\n\nIntuitively this means that if we start the chain in a stationary distribution, the amount of probability that flows from to is equal to the one from to . In other words, the system must be at equilibrium. An intuitive example of a physical system not satisfying detailed balance is a snow flake in a coffee. Indeed, many chains do not satisfy this detailed balance property. In this case it is possible to use a different, but related, chain called the reversal Markov chain to infer mixing time bounds\u00a0(Chung et\u00a0al., 2012).\n\n###### Definition 2 (Reversal Markov chain\u00a0(Kemeny and Snell, 1976)).\n\nLet the reversal Markov chain of be defined as:\n\n \u02dcPij=\u03bcjPji\u03bci\u2200i,j\u2208S. (2)\n\nIf is irreducible with invariant distribution , then is also irreducible with invariant distribution .\n\nThe reversal Markov chain can be interpreted as the Markov chain with time running backwards. If the chain is reversible, then .\n\n### 3.2 Markov Decision Process\n\nA Markov Decision Process (MDP), as defined in Puterman (1994), consists of a discrete set of states , a transition function , and a reward function . On each round , the learner observes current state and selects action , after which it receives reward and moves to new state . We define a stationary policy as a probability distribution over actions conditioned on states .\n\n#### Discounted Markov Decision Process\n\nWhen performing policy evaluation in the discounted case, the goal is to estimate the discounted expected return of policy at a state , , with discount factor . This can be obtained as the fixed point of the Bellman operator such that:\n\n T\u03c0v\u03c0=r\u03c0+\u03b3P\u03c0v\u03c0, (3)\n\nwhere denotes the transition matrix under policy , is the state values column-vector, and is the reward column-vector. The matrix also defines a Markov chain.\n\nIn the control case, the goal is to find the optimal policy that maximizes the discounted expected return. Under the optimal policy, the optimal value function is the fixed point of the non-linear optimal Bellman operator:\n\n T\u2217v\u2217=maxa\u2208A[r(a)+\u03b3P(a)v\u2217]. (4)\n\n## 4 Temporal regularization\n\nRegularization in the feature\/state space, or spatial regularization as we call it, exploits the regularities that exist in the observation (or state). In contrast, temporal regularization considers the temporal structure of the value estimates through a trajectory. Practically this is done by smoothing the value estimate of a state using estimates of states that occurred earlier in the trajectory. In this section we first introduce the concept of temporal regularization and discuss its properties in the policy evaluation setting. We then show how this concept can be extended to exploit information from the entire trajectory by casting temporal regularization as a time series prediction problem.\n\nLet us focus on the simplest case where the value estimate at the current state is regularized using only the value estimate at the previous state in the trajectory, yielding updates of the form:\n\n v\u03b2(st)=Est+1,st\u22121\u223c\u03c0[r(st)+\u03b3((1\u2212\u03b2)v\u03b2(st+1)+\u03b2v\u03b2(st\u22121))]=r(st)+\u03b3(1\u2212\u03b2)\u2211st+1\u2208Sp(st+1|st)v\u03b2(st+1)+\u03b3\u03b2\u2211st\u22121\u2208Sp(st|st\u22121)p(st\u22121)p(st)v\u03b2(st\u22121), (5)\n\nfor a parameter and the transition probability induced by the policy . It can be rewritten in matrix form as , where corresponds to the reversal Markov chain of the MDP. We define a temporally regularized Bellman operator as:\n\n T\u03c0\u03b2v\u03b2=r+\u03b3((1\u2212\u03b2)P\u03c0v\u03b2+\u03b2\u02dcP\u03c0v\u03b2). (6)\n\nTo alleviate the notation, we denote as and as .\n\n###### Remark.\n\nFor , Eq.\u00a06 corresponds to the original Bellman operator.\n\nWe can prove that this operator has the following property.\n\n###### Theorem 1.\n\nThe operator has a unique fixed point and is a contraction mapping.\n\n###### Proof.\n\nWe first prove that is a contraction mapping in norm. We have that\n\n \\normT\u03c0\u03b2u\u2212T\u03c0\u03b2v\u221e=\\normr+\u03b3((1\u2212\u03b2)Pu+\u03b2\u02dcPu)\u2212(r+\u03b3((1\u2212\u03b2)Pv+\u03b2\u02dcPv))\u221e=\u03b3\\norm((1\u2212\u03b2)P+\u03b2\u02dcP)(u\u2212v)\u221e\u2264\u03b3\\normu\u2212v\u221e, (7)\n\nwhere the last inequality uses the fact that the convex combination of two row stochastic matrices is also row stochastic (the proof can be found in the appendix). Then using Banach fixed point theorem, we obtain that is a unique fixed point. \u220e\n\nFurthermore the new induced Markov chain has the same stationary distribution as the original (the proof can be found in the appendix).\n\n###### Lemma 1.\n\nand have the same stationary distribution .\n\nIn the policy evaluation setting, the bias between the original value function and the regularized one can be characterized as a function of the difference between and its Markov reversal , weighted by and the reward distribution.\n\n###### Proposition 1.\n\nLet and . We have that\n\n \\normv\u03c0\u2212v\u03c0\u03b2\u221e=\\norm\u221e\u2211i=0\u03b3i(Pi\u2212((1\u2212\u03b2)P+\u03b2\u02dcP)i)r\u221e\u2264\u221e\u2211i=0\u03b3i\\norm(Pi\u2212((1\u2212\u03b2)P+\u03b2\u02dcP)i)r\u221e. (8)\n\nThis quantity is naturally bounded for .\n\n###### Remark.\n\nLet denote a matrix where columns consist of the stationary distribution . By the property of reversal Markov chains and lemma 1, we have that and , such that the Marvov chain and its reversal converge to the same value. Therefore, the norm also converges to 0 in the limit.\n\n###### Remark.\n\nIt can be interesting to note that if the chain is reversible, meaning that , then the fixed point of both operators is the same, that is .\n\nDiscounted average reward case: The temporally regularized MDP has the same discounted average reward as the original one as it is possible to define the discounted average reward (Tsitsiklis and Van\u00a0Roy, 2002) as a function of the stationary distribution , the reward vector and . This leads to the following property (the proof can be found in the appendix).\n\n###### Proposition 2.\n\nFor a reward vector r, the MDPs defined by the the transition matrices and have the same average reward .\n\nIntuitively, this means that temporal regularization only reweighs the reward on each state based on the Markov reversal, while preserving the average reward.\n\nTemporal Regularization as a time series prediction problem: It is possible to cast this problem of temporal regularization as a time series prediction problem, and use richer models of temporal dependencies, such as exponential smoothing (Gardner, 2006), ARMA model\u00a0(Box et\u00a0al., 1994), etc. We can write the update in a general form using different regularizers ():\n\n v(st)=r(s)+\u03b3n\u22121\u2211i=0[\u03b2(i)\u02dcvi(st+1)], (9)\n\nwhere and . For example, using exponential smoothing where , the update can be written in operator form as:\n\n T\u03c0\u03b2v=r+\u03b3((1\u2212\u03b2)Pv+\u03b2(1\u2212\u03bb)\u221e\u2211i=1\u03bbi\u22121\u02dcPiv), (10)\n\nand a similar argument as Theorem 1 can be used to show the contraction property. The bias of exponential smoothing in policy evaluation can be characterized as:\n\n \\normv\u03c0\u2212v\u03c0\u03b2\u221e\u2264\u221e\u2211i=0\u03b3i\\norm(Pi\u2212((1\u2212\u03b2)P+\u03b2(1\u2212\u03bb)\u221e\u2211j=1\u03bbj\u22121\u02dcPj)i)r\u221e. (11)\n\nUsing more powerful regularizers could be beneficial, for example to reduce variance by smoothing over more values (exponential smoothing) or to model the trend of the value function through the trajectory using trend adjusted model \u00a0Gardner\u00a0Jr (1985). An example of a temporal policy evaluation with temporal regularization using exponential smoothing is provided in Algorithm 1.\n\nControl case: Temporal regularization can be extended to MDPs with actions by modifying the target of the value function (or the Q values) using temporal regularization. Experiments (Sec.\u00a05.6) present an example of how temporal regularization can be applied within an actor-critic framework. The theoretical analysis of the control case is outside the scope of this paper.\n\nTemporal difference with function approximation: It is also possible to extend temporal regularization using function approximation such as semi-gradient TD Sutton and Barto (2017). Assuming a function parameterized by , we can consider as the target and differentiate with respect to . An example of a temporally regularized semi-gradient TD algorithm can be found in the appendix.\n\n## 5 Experiment\n\nWe now presents empirical results illustrating potential advantages of temporal regularization, and characterizing its bias and variance effects on value estimation and control.\n\n### 5.1 Mixing time\n\nThis first experiment showcases that the underlying Markov chain of a MDP can have a smaller mixing time when temporally regularized. The mixing time can be seen as the number of time steps required for the Markov chain to get close enough to its stationary distribution. Therefore, the mixing time also determines the rate at which policy evaluation will converge to the optimal value function\u00a0(Baxter and Bartlett, 2001). We consider a synthetic MDP with 10 states where transition probabilities are sampled from the uniform distribution. Let denote a matrix where columns consists of the stationary distribution . To compare the mixing time, we evaluate the error corresponding to the distance of and to the convergence point after iterations. Figure\u00a02 displays the error curve when varying the regularization parameter . We observe a U-shaped error curve, that intermediate values of in this example yields faster mixing time. One explanation is that transition matrices with extreme probabilities (low or high) yield poorly conditioned transition matrices. Regularizing with the reversal Markov chain often leads to a better conditioned matrix at the cost of injecting bias.\n\n### 5.2 Bias\n\nIt is well known that reducing variance comes at the expense of inducing (smaller) bias. This has been characterized previously (Sec.\u00a04) in terms of the difference between the original Markov chain and the reversal weighted by the reward. In this experiment, we attempt to give an intuitive idea of what this means. More specifically, we would expect the bias to be small if values along the trajectories have similar values. To this end, we consider a synthetic MDP with states where both transition functions and rewards are sampled randomly from a uniform distribution. In order to create temporal dependencies in the trajectory, we smooth the rewards of states that are temporally close (in terms of trajectory) using the following formula: . Figure\u00a02 shows the difference between the regularized and un-regularized MDPs as changes, for different values of regularization parameter . We observe that increasing , meaning more states get rewards close to one another, results into less bias. This is due to rewards putting emphasis on states where the original and reversal Markov chain are similar.\n\n### 5.3 Variance\n\nThe primary motivation of this work is to reduce variance, therefore we now consider an experiment targeting this aspect. Figure\u00a04 shows an example of a synthetic, 3-state MDP, where the variance of is (relatively) high. We consider an agent that is evolving in this world, changing states following the stochastic policy indicated. We are interested in the error when estimating the optimal state value of , , with and without temporal regularization, denoted , , respectively.\n\nFigure\u00a04 shows these errors at each iteration, averaged over runs. We observe that temporal regularization indeed reduces the variance and thus helps the learning process by making the value function easier to learn.\n\n### 5.4 Propagation of the information\n\nWe now illustrate with a simple experiment how temporal regularization allows the information to spread faster among the different states of the MDP. For this purpose, we consider a simple MDP, where an agent walks randomly in two rooms (18 states) using four actions (up, down, left, right), and a discount factor . The reward is everywhere and passing the door between rooms (shown in red on Figure\u00a05) only happens 50% of the time (on attempt). The episode starts at the top left and terminates when the agent reaches the bottom right corner. The sole goal is to learn the optimal value function by walking along this MDP (this is not a race toward the end).\n\nFigure\u00a05 shows the proximity of the estimated state value to the optimal value with and without temporal regularization. The darker the state, the closer it is to its optimal value. The heatmap scale has been adjusted at each trajectory to observe the difference between both methods. We first notice that the overall propagation of the information in the regularized MDP is faster than in the original one. We also observe that, when first entering the second room, bootstrapping on values coming from the first room allows the agent to learn the optimal value faster. This suggest that temporal regularization could help agents explore faster by using their prior from the previous visited state for learning the corresponding optimal value faster. It is also possible to consider more complex and powerful regularizers. Let us study a different time series prediction model, namely exponential averaging, as defined in (10). The complexity of such models is usually articulated by hyper-parameters, allowing complex models to improve performance by better adapting to problems. We illustrate this by comparing the performance of regularization using the previous state and an exponential averaging of all previous states. Fig.\u00a06 shows the average error on the value estimate using past state smoothing, exponential smoothing, and without smoothing. In this setting, exponential smoothing transfers information faster, thus enabling faster convergence to the true value.\n\n### 5.5 Noisy state representation\n\nThe next experiment illustrates a major strength of temporal regularization, that is its robustness to noise in the state representation. This situation can naturally arise when the state sensors are noisy or insufficient to avoid aliasing. For this task, we consider the synthetic, one dimensional, continuous setting. A learner evolving in this environment walks randomly along this line with a discount factor . Let denote the position of the agent along the line at time . The next position , where action . The state of the agent corresponds to the position perturbed by a zero-centered Gaussian noise , such that , where are i.i.d. When the agent moves to a new position , it receives a reward . The episode ends after 1000 steps. In this experiment we model the value function using a linear model with a single parameter . We are interested in the error when estimating the optimal parameter function with and without temporal regularization, that is and , respectively. In this case we use the TD version of temporal regularization presented at the end of Sec.\u00a04. Figure\u00a08 shows these errors, averaged over 1000 repetitions, for different values of noise variance . We observe that as the noise variance increases, the un-regularized estimate becomes less accurate, while temporal regularization is more robust. Using more complex regularizer can improve performance as shown in the previous section but this potential gain comes at the price of a potential loss in case of model misfit. Fig.\u00a08 shows the absolute distance from the regularized state estimate (using exponential smoothing) to the optimal value while varying (higher = more smoothing). Increasing smoothing improves performance up to some point, but when is not well fit the bias becomes too strong and performance declines. This is a classic bias-variance tradeoff. This experiment highlights a case where temporal regularization is effective even in the absence of smoothness in the state space (which other regularization methods would target). This is further highlighted in the next experiments.\n\n### 5.6 Deep reinforcement learning\n\nTo showcase the potential of temporal regularization in high dimensional settings, we adapt an actor-critic based method (PPO (Schulman et\u00a0al., 2017)) using temporal regularization. More specifically, we incorporate temporal regularization as exponential smoothing in the target of the critic. PPO uses the general advantage estimator where . We regularize such that using exponential smoothing as described in Eq.\u00a0(10). is an exponentially decaying sum over all previous state values encountered in the trajectory. we evaluate the performance in the Arcade Learning Environment \u00a0Bellemare et\u00a0al. (2013), where we consider the following performance measure:\n\n regularized\u2212baselinebaseline\u2212random. (12)\n\nThe hyper-parameters for the temporal regularization are and a decay of . Those are selected on 7 games and 3 training seeds. All other hyper-parameters correspond to the one used in the PPO paper. Our implementation2 is based on the publicly available OpenAI codebase\u00a0Dhariwal et\u00a0al. (2017). The previous four frames are considered as the state representation\u00a0(Mnih et\u00a0al., 2015). For each game, independent runs ( random seeds) are performed.\n\nThe results reported in Figure\u00a09 show that adding temporal regularization improves the performance on multiple games. This suggests that the regularized optimal value function may be smoother and thus easier to learn, even when using function approximation with deep learning. Also, as shown in previous experiments (Sec.\u00a05.5), temporal regularization being independent of spatial representation makes it more robust to mis-specification of the state features, which is a challenge in some of these games (e.g. when assuming full state representation using some previous frames).\n\n## 6 Discussion\n\nNoisy states: Is is often assumed that the full state can be determined, while in practice, the Markov property rarely holds. This is the case, for example, when taking the four last frames to represent the state in Atari games\u00a0(Mnih et\u00a0al., 2015). A problem that arises when treating a partially observable MDP (POMDP) as a fully observable is that it may no longer be possible to assume that the value function is smooth over the state space\u00a0(Singh et\u00a0al., 1994). For example, the observed features may be similar for two states that are intrinsically different, leading to highly different values for states that are nearby in the state space. Previous experiments on noisy state representation (Sec.\u00a05.5) and on the Atari games (Sec.\u00a05.6) show that temporal regularization provides robustness to those cases. This makes it an appealing technique in real-world environments, where it is harder to provide the agent with the full state.\n\nChoice of the regularization parameter: The bias induced by the regularization parameter can be detrimental for the learning in the long run. A first attempt to mitigate this bias is just to decay the regularization as learning advances, as it is done in the deep learning experiment (Sec.\u00a05.6). Among different avenues that could be explored, an interesting one could be to aim for a state dependent regularization. For example, in the tabular case, one could consider as a function of the number of visits to a particular state.\n\nSmoother objective: Previous work\u00a0Laroche (2018) looked at how the smoothness of the objective function relates to the convergence speed of RL algorithms. An analogy can be drawn with convex optimization where the rate of convergence is dependent on the Lipschitz (smoothness) constant \u00a0Boyd and Vandenberghe (2004). By smoothing the value temporally we argue that the optimal value function can be smoother. This would be beneficial in high-dimensional state space where the use of deep neural network is required. This could explain the performance displayed using temporal regularization on Atari games (Sec.\u00a05.6). The notion of temporal regularization is also behind multi-step methods\u00a0(Sutton and Barto, 1998); it may be worthwhile to further explore how these methods are related.\n\nConclusion: This paper tackles the problem of regularization in RL from a new angle, that is from a temporal perspective. In contrast with typical spatial regularization, where one assumes that rewards are close for nearby states in the state space, temporal regularization rather assumes that rewards are close for states visited closely in time. This approach allows information to propagate faster into states that are hard to reach, which could prove useful for exploration. The robustness of the proposed approach to noisy state representations and its interesting properties should motivate further work to explore novel ways of exploiting temporal information.\n\n#### Acknowledgments\n\nThe authors wish to thank Pierre-Luc Bacon, Harsh Satija and Joshua Romoff for helpful discussions. Financial support was provided by NSERC and Facebook. This research was enabled by support provided by Compute Canada. We thank the reviewers for insightful comments and suggestions.\n\n## Appendix A Appendix\n\n###### Lemma 1.\n\nand have the same stationary distribution .\n\n###### Proof.\n\nIt is known that and have the same stationary distribution. Using this fact we have that\n\n \u03bc((1\u2212\u03b2)P\u03c0+\u03b2\u02dcP\u03c0)=(1\u2212\u03b2)\u03bcP\u03c0+\u03b2\u03bc\u02dcP\u03c0=(1\u2212\u03b2)\u03bc+\u03b2\u03bc=\u03bc. (13)\n\n###### Property 2.\n\nFor a reward vector r, the MDP defined by the the transition matrix and have the same discounted average reward .\n\n \u03c11\u2212\u03b1=\u221e\u2211i\u03b3i\u03c0Tr. (14)\n###### Proof.\n\nUsing lemma 1, both and have the same stationary distribution and so discounted average reward. \u220e\n\n###### Lemma 2.\n\nThe convex combination of two row stochastic matrices is also row stochastic.\n\n###### Proof.\n\nLet e be vector a columns vectors of 1.\n\n (\u03b2P\u03c0+(1\u2212\u03b2)\u02dcP\u03c0)e=\u03b2P\u03c0e+(1\u2212\u03b2)\u02dcP\u03c0e=\u03b2e+(1\u2212\u03b2)e=e. (15)\n\n### Footnotes\n\n1. defines the th element of\n2. The code can be found https:\/\/github.com\/pierthodo\/temporal_regularization.\n\n### References\n\n1. L.\u00a0Baird. Residual algorithms: Reinforcement learning with function approximation. In Machine Learning Proceedings 1995, pages 30\u201337. Elsevier, 1995.\n2. P.\u00a0L. Bartlett and A.\u00a0Tewari. Regal: A regularization based algorithm for reinforcement learning in weakly communicating mdps. In Proceedings of the Twenty-Fifth Conference on Uncertainty in Artificial Intelligence, pages 35\u201342. 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\section{Introduction}
\label{sec:intro}
Matrix-vector multiplication is one of the most fundamental
algorithmic primitives. In the data structure variant of the problem,
we are given an $n \times n$ matrix $M$ as input. The goal is to preprocess
$M$ into a data structure, such that upon receiving any $n$-dimensional
query vector $v$, we can quickly compute $Mv$. Constructing a data
structure instead of computing $Mv$ directly may pay off as soon as
we have to answer multiple matrix-vector multiplication queries on the
same matrix $M$.
When defined over the Boolean semiring (with addition replaced by OR
and multiplication replaced by AND) the above problem is a special
case of the well-known \emph{Online Matrix-Vector (OMV)} problem:
Given a matrix $M \in \{0,1\}^{n \times n}$ and a stream of vectors
$v^{(1)},\cdots, v^{(n)} \in \{0,1\}^n$ the goal is to output the
value of $Mv^{(i)}$ before seeing $v^{(j)}$ for any $j \in
\{i+1,\cdots ,n\}$. Henzinger \emph{et al.}~\cite{HKNS15} conjectured
that the OMV problem cannot be solved by any randomized algorithm with
error probability at most $1/3$ within time $O(n^{3 -\epsilon})$ for
any constant
$\epsilon > 0$ (i.e. amortized $O(n^{2-\epsilon})$ per vector
$v^{(i)}$). This conjecture is known as the OMV conjecture and is one
of the central conjectures in the ``Hardness
in P'' area, along with the Strong Exponential Time Hypothesis
(SETH see~\cite{ImpagliazzoPaturi}), the 3SUM conjecture (see e.g.~\cite{AnkaOvermars})
and the All Pairs Shortest Paths conjecture (APSP see e.g.~\cite{WAPSP}).
The conjecture implies a whole range of near-tight conditional
lower bounds for classic (fully/partially) dynamic data structure
problems such as dynamic reachability. For the matrix-vector
multiplication problem over the Boolean semiring, the OMV conjecture in
particular implies that for any polynomial preprocessing time and
space, the query time must be $n^{2-o(1)}$~\cite{HKNS15}.
The current best upper bound for the OMV problem is due to Larsen and
Williams~\cite{LW17}, who gave a randomized (word-RAM)
data structure with a total running time
of $n^3/2^{\Omega(\sqrt{\lg n})}$ over a sequence of $n$
queries, i.e. amortized $n^2/2^{\Omega(\sqrt{\lg n})}$ per
query. While this new upper bound is non-trivial, it does not violate
the OMV conjecture.
The holy grail is, of course, to replace the OMV conjecture by an
unconditional $n^{3-o(1)}$ lower bound. In contrast to SETH, 3SUM
and APSP, the OMV problem is a data structure problem, rather than an
algorithmic problem. Since we have been vastly more successful in
proving unconditional lower bounds for data structures than for
algorithms, it does not a priori seem completely hopeless to prove a tight
unconditional lower bound for OMV in the foreseeable future. Data
structure lower bounds are typically proved in the cell probe model of
Yao~\cite{Yao81a}. In this model, computation is free of cost, and the
complexity of a data structure is solely the amount of memory it uses and
the number of memory accesses it performs on answering a query. In
particular, lower bounds proved in the cell probe model apply to
data structures developed in the standard word-RAM model,
regardless of which unit cost instructions are available. Quite
surprisingly, Larsen and Williams~\cite{LW17} showed that the performance
of their OMV data structure greatly improves if implemented in the
cell probe model (i.e. if computation is free and we only charge for
accessing memory). Their cell probe data structure for matrix-vector
multiplication over the Boolean semiring has a query
time of $O(n^{7/4}/\sqrt{w})$, where $w$ denotes the
word size (typically $w = \Theta(\lg n)$). Thus the OMV conjecture is
\emph{false} in the cell probe model!
But what is then the true complexity of matrix-vector multiplication in the cell probe model?
Is it the best one can hope for, namely $O(n/w)$ time (the size of the
output measured in words) with $O(n^2)$ bits of space? Or does one
have to pay a polynomial factor in either space or time? While not
matching the OMV conjecture, a polynomial lower bound ($n^{1+\Omega(1)}$ query time with,
say, polynomial space) would still be immensely valuable as it would
be the first polynomial lower bound for any data structure problem and
would be a huge leap forward in proving unconditional lower
bounds. Furthermore, it would still imply non-trivial polynomial lower
bounds for numerous important data structure problems via the
reductions already given in previous papers.
Our main contribution is to prove (near-)tight polynomial cell probe lower
bounds for the class of \emph{succinct} matrix-vector multiplication data structures.
Before formally presenting our
results, we survey the current barriers for proving cell probe lower
bounds as this will help understand the context of our results.
\paragraph{Cell Probe Lower Bound Barriers.}
Much effort has gone into developing techniques for proving data
structure lower bounds in the cell probe model. For static data
structures (like matrix-vector multiplication), the current strongest
techniques~\cite{Larsen:2012:focs} can prove lower bounds of $t = \Omega(\lg
m/\lg \alpha)$ where $t$ is the query time, $m$ is the number of distinct possible queries in the
problem and $\alpha$ is the space-overhead over linear. Thus the
strongest previous lower bounds peak at $t = \Omega(\lg m)$ with
linear space. For matrix-vector multiplication with an $n \times n$
matrix $M$ (over $\{0,1\}$), there are $2^n$ possible queries $v$, thus the strongest
possible lower bound current techniques would allow us to prove is
$t = \Omega(n)$. This is unfortunately not much more than the
trivial $t = \Omega(n/w)$ one would get for just writing the output.
For dynamic data structures, i.e. data structures where one receives
both updates to the input data and queries, the current strongest
techniques~\cite{Larsen12a, weinstein:lbs} give lower bounds of $t = \Omega(\lg m \lg n /
\lg^2(u w))$ where $u$ is the update time, $w$ the word-size and
$n$ the input size/number of updates performed. This is about a $\lg
n$ factor more than for static data structures, thus still leaves us
quite far from proving lower bounds close to the conditional
ones. If we restrict ourselves to lower bounds for decision problems,
i.e. problems where the answer to a query is just one bit, the
situation is worse, with the strongest lower bounds being of the form
$t = \Omega(\lg m \sqrt{\lg n}/\lg^2(uw))$~\cite{LWY17}.
\paragraph{Restricted Data Structures.}
For many data structure problems, the lower bounds one can prove with
previous techniques are in fact tight, see e.g.~\cite{FS89, PD06,
Pat11, PT11,Larsen12a, weinstein:lbs}. However, as we can see from
matrix-vector multiplication, there are also problems where the
current techniques are quite far from proving what we believe should
be the right lower bound. This has resulted in researchers proving a number of exciting
lower bounds for special classes of data
structures. For instance, Clifford {\em et al.\ }\xspace ~\cite{CGL15} consider matrix-vector
multiplication over a finite field $\mathbb{F}_p$ of exponential size $p =
2^{\Theta(n)}$. This results in more queries to the problem ($p^n =
2^{\Theta(n^2)}$) and thereby enabled proving a lower bound of $t =
\Omega(n^2/\lg \alpha)$. Their lower bounds hold when the word size $w$ is big enough to
store an element of the field, i.e. $w = \Theta(n)$ bits. An
interesting interpretation of the lower bound is that, as long as we do not take advantage of the
size of the field, any data structure is bound to use $\Omega(n^2/\lg
\alpha)$ query time. Another line of work has focused on non-adaptive
dynamic data structures~\cite{brody, brodynew, Ramamoorthy2017NonAdaptiveDS}. These are data structures where the
memory locations read upon answering a query depend \emph{only} on
the query and \emph{not} on the contents of probed cells, i.e. the
query algorithm may not branch based on what it reads.
\paragraph{Succinct Data Structures.}
The last class
of data structures we consider are \emph{succinct} data
structures. Succinct data structures use
space very close to the information theoretic minimum. More formally,
we say that a data structure has {\em redundancy} of $r$ bits if its
space usage is $\Pi + r$ bits where $\Pi$ is the information theoretic
minimum for solving the problem and $r = o(\Pi)$.
The space usage of succinct data structures is measured only in terms of its
\emph{redundancy}. Using succinct data structures may be
crucial in applications where memory is scarce. We typically distinguish two types of
succinct data structures, namely \emph{systematic} and
\emph{non-systematic} data structures. Systematic data structures are more restricted
than non-systematic ones, in the sense that they always store the input
in read-only memory and then build an $r$-bit data structure on the
side. Non-systematic data structures just use at most $\Pi+r$ bits
(and thus do not have to store the input in the format in which it is
given). Succinct data structures have been studied extensively for
decades with many fundamental and important upper and lower bounds,
see e.g.~\cite{Jacobson:1988:SSD:915547, gal:succinct, patrascu08succinct, patrascu10ranklb}. The current strongest technique typically allows
one to prove lower bounds of the form $t r= \Omega(\Pi)$, see
e.g.~\cite{gal:succinct, BL13}.
The reason we take special interest in succinct data structures,
is that the matrix-vector multiplication data structure by Larsen and
Williams~\cite{LW17} is, in fact, a succinct data structure. In addition to
answering queries in just $O(n^{7/4}/\sqrt{w})$ time, it is
systematic and just stores the input matrix as read-only, plus an
additional $r= O(n^{7/4} \sqrt{w})$ bits on the side. With the current
techniques for proving lower bounds for succinct data structures, we
actually have hopes of proving something stronger than the $t =
\Omega(n)$ lower bounds we can hope for if we just consider general
data structures. Since $\Pi = n^2$ for Boolean matrix-vector
multiplication, it seems reasonable to hope for something
of the form $t r= \Omega(n^2)$. Such lower bounds would shed interesting new
light on this central data structure problem and would bring us closer
to understanding the true complexity of matrix-vector multiplication.
\subsection{Our Results}
Our main results are near-matching upper and lower bounds for
systematic succinct data structures solving Boolean
matrix-vector multiplication. On the upper bound side, we improve on
the results of Larsen and Williams and give a new randomized data
structure with the following guarantees:
\begin{theorem}
\label{thm:UB}
Given any matrix $M\in \{0,1\}^{n \times n}$ there exists a systematic
succinct data
structure $R(M)$ consisting of $r=O(n^{3/2} (\sqrt{w}+\frac{\lg
n}{\sqrt{w}}))$ additional bits, and a query algorithm such that,
given any $v \in \{0,1\}^n$ it computes $Mv$ over the Boolean semiring
with probability $\ge 1 -1/n$ by probing at most $O(n^{3/2} \sqrt{w}\lg n)$ cells of $R(M)$ and $M$, where $w$ is the word size.
\end{theorem}
Our data structure thus reduces the redundancy
from $O(n^{7/4} \sqrt{w})$ bits to $O(n^{3/2} (\sqrt{w}+\frac{\lg
n}{\sqrt{w}}))$ bits.
Moreover, our randomized query algorithm returns the correct answer with high probability, and improves the query time over the previously known deterministic algorithm from $O(n^{7/4}/
\sqrt{w})$ to $O(n^{3/2}\sqrt{w} \lg n)$.
We complement our new upper bound by a near-matching lower bound:
\begin{theorem}
\label{thm:mainMvBoolean}
Assume that for every matrix $M \in \{0,1\}^{n \times n}$ there exists
a systematic succinct data structure $R=R(M)$ consisting of at most
$r=r(n)$ bits and there is a randomized algorithm that given any $v
\in \{0,1\}^n$ computes $Mv$ over the Boolean semiring with probability $\ge 1-1/n$ by probing $R$ and at most $t=t(n)$ entries from $M$,
Then for $n \le r \le n^2/4$, $t \cdot r = \Omega(n^3)$; otherwise for $r < n$, $t = \Omega(n^2)$.
\end{theorem}
Our lower bound comes within polylogarithmic factors of the upper
bound and is in fact higher than what we could hope for with previous
techniques (recall that previous techniques peak at $t r=
\Omega(\Pi)$). The proof of our lower bound exploits the large number $m$
of possible queries and we essentially manage to derive lower bounds of the form $t r=
\Omega(\Pi \lg m ) = \Omega(n^3)$. Also note that our lower bound
allows the data structure to probe all of $R$, i.e. all the redundant
bits, and still it says that one has to read a lot from the matrix
itself. Another exciting point is that our lower bound shows that $t =
\Omega(n^2)$ for any $r < n$. Previous lower bounds of $t r =
\Omega(\Pi)$ always degenerate linearly in $r$ all the way down to
$r=1$. In contrast, our lower bounds say that one cannot do much better
(up to a constant factor) than reading all $n^2$ entries of $M$ if $r < n$.
Finally, we also study matrix-vector multiplication over
$\mathbb{F}_2$. Here we prove lower bounds even for the
vector-matrix-vector multiplication where one is given a pair of
vectors $u,v \in \mathbb{F}_2^n$ as queries and must compute
$u^{\dagger}Mv$. This problem has just one bit in the output, making
it more difficult to prove lower bounds. Nonetheless, we prove the
following lower bound:
\begin{theorem}
\label{thm:mainUMVF2}
Assume that for every matrix $M \in \mathbb{F}_2^{n \times n}$ there exists a data structure $R=R(M)$ consisting of at most $r=r(n)$ bits and there is an algorithm that given $u,v \in \mathbb{F}_2^n$ computes $u^{\dagger}Mv$ by probing $R$ and at most $t=t(n)$ entries from $M$, then for $n \le r \le n^2/4$, $t \cdot r = \Omega(n^3/\lg n)$. Moreover, if $r < n$ then $t = \Omega(n^2/\lg n)$.
\end{theorem}
We believe it is quite remarkable that we can get $t = \tilde\Omega(n^2)$ lower
bounds for any $r < n$ for this 1-bit output problem. Since any $u^{\dagger}Mv$ query can be answered by just taking inner
product between $u$ and $Mv$, as a corollary of the above we also get
the same trade-off for matrix-vector problem over $\mathbb{F}_2$. To the best
of our knowledge, prior to this result there was no trade-off known
for such a small sized field. Our proof is completely information theoretic and based on an encoding argument. It is worth noting that our proof technique can be generalized to give lower bound for the case when the query algorithm may err with probability at most $1/64$ (though any small constant probability will work) on average over the random choices of $M,u,v$.
Finally, we also consider vector-matrix-vector multiplication over the
Boolean semiring. Since one can compute $Mv$ by running the following
sequence of $n$ queries:
$(e^{(1)})^{\dagger}Mv,\cdots,(e^{(n)})^{\dagger}Mv$ where
$\{e^{(i)}\}_{i \in [n]}$ is the standard basis over $\{0,1\}^n$, from
Theorem~\ref{thm:mainMvBoolean} we get a lower bound of $tr\ge \Omega(n^2)$ for the
Boolean $u^{\dagger}Mv$ problem. Instead of this simple reduction,
even if we use the much more elegant reduction in~\cite{HKNS15}, we will
not be able to derive any better lower bound from the above
theorem. However in Section~\ref{sec:Bool-uMv} we show how to extend
the proof of Theorem~\ref{thm:mainMvBoolean} to get a $t = \Omega(n/\lg n)$ bound on the
worst case number of probes into $M$ for the Boolean vector-matrix-vector
problem with $r \le n^2/4$.
\section{Preliminaries}
\label{sec:prelims}
\paragraph*{Notations. } For $k \in \mathbb{N}$, let $[k]$ denote the set $\{1,2,\ldots,k\}$. For every $v \in \{0,1\}^n$ and $i \in [n]$, let $v_i$ denote the $i$-th entry of $v$. All the logarithms we consider are over base $2$. We use the notation $x \in_R {\cal X}$ to denote that $x$ is drawn uniformly at random from the domain ${\cal X}$.
\paragraph*{Information Theory. }
Throughout this paper we use several basic definitions and notations from information theory. For further exposition readers may refer to any standard textbook on information theory (e.g.~\cite{CT06}).
Let $X,Y$ be discrete random variables on a common probability space. Let $p(x),p(y),p(x,y)$ denote $\Pr[X=x], \Pr[Y=y], \Pr[X=x,Y=y]$ respectively.
The \emph{entropy} of $X$ is defined as
$H(X):= - \sum_x p(x) \lg p(x)$.
The \emph{joint entropy} of $(X,Y)$ is defined as $H(X,Y) := - \sum_{(x,y)} p(x,y) \lg (p(x,y))$.
The \emph{mutual information} between $X$ and $Y$ is defined as $I(X;Y) := \sum_{(x,y)}{p(x,y) \lg \frac{p(x,y)}{p(x)p(y)}}$ and the \emph{conditional entropy} of $Y$ given $X$ is defined as $H(Y \mid X) := H(Y) - I(X;Y)$.
\begin{proposition}[Chain Rule of Entropy]
\label{prop:chnentro}
Then $H(X,Y)=H(X)+H(Y \mid X)$.
\end{proposition}
The seminal work of Shannon~\cite{Sha48} establishes a connection between the entropy and the expected length of an optimal code encoding a random variable.
\begin{theorem}[Shannon's Source Coding Theorem~\cite{Sha48}]
Let $X$ be a discrete random variable over domain ${\cal X}$. Then for every uniquely decodable code $C :{\cal X} \to \{0,1\}^*$, $\mathbb{E}(|C(X)|) \ge H(X)$. Moreover, there exists a uniquely decodable code $C:{\cal X} \to \{0,1\}^*$ such that $\mathbb{E}(|C(X)|) \le H(X)+1$.
\end{theorem}
\section{Upper Bound for Boolean Matrix-Vector Problem}
\label{sec:UB-Boolean-Mv}
In this section we prove Theorem~\ref{thm:UB} by introducing an efficient cell probe data structure for solving Boolean matrix-vector problem (with high probability). Let us first recall the theorem.
\begin{reptheorem}{thm:UB}
Given any matrix $M\in \{0,1\}^{n \times n}$ there exists a data structure $R$ consisting of $O(n^{3/2} (\sqrt{w}+\frac{\lg n}{\sqrt{w}}))$ bits, and a query algorithm such that, given $v \in \{0,1\}^n$ it computes $Mv$ with high probability by probing at most $O(n^{3/2} \sqrt{w}\lg n)$ cells of $R$ and $M$, where $w$ is the word size.
\end{reptheorem}
\paragraph{Preprocessing. }
In what follows, we present an algorithm that, given a matrix $M \in \{0,1\}^{n \times n}$ constructs the data structure guaranteed in Theorem~\ref{thm:UB}. Loosely speaking, the data structure is composed of a list ${\cal L}$ consisting of pairs $(I,J)$, and an encoding ${\cal E}$ of all the entries $(i,j) \in \bigcup_{(I,J) \in {\cal L}}(I \times J)$ such that $M_{(i,j)}=1$.
The key step of the preprocessing algorithm is deciding which set-pairs to add to the list.
Informally, going over all possible pairs $(I,J)$, the algorithm adds a pair $(I,J)$ to ${\cal L}$, if there exists a large subset of $I \times J$ of entries not "covered" by the pairs already added in the list, and such that the "uncovered" part of the submatrix $M_{I,J}$ contains "few" $1$-entries per-row. The algorithm is formally described as Algorithm~\ref{alg:preprocess}.
\begin{algorithm}[H]
\begin{algorithmic}[1]
\STATE let ${\cal L} \leftarrow \emptyset$.
\FORALL{$(I,J) \in 2^{[n]}\times2^{[n]}$}
\STATE let ${\cal U} := \bigcup_{(I',J') \in {\cal L}}(I'\times J')$.
\STATE add $(I,J)$ to ${\cal L}$ if all of the following conditions hold. \label{l:cond}
\begin{enumerate}
\item \label{itm:genSize} $\left| (I \times J) \setminus {\cal U} \right| \ge \frac{n^{3/2}}{\sqrt{w}}$; and
\item \label{itm:density} $\forall i \in I$, $\Pr_{j \in J : (i,j) \notin {\cal U}} [M_{(i,j)}=1] \le \frac{1}{\sqrt{nw}}$.
\end{enumerate}
\ENDFOR
\STATE let ${\cal E}$ be the list of all $(i,j) \in \bigcup_{(I,J) \in {\cal L}}(I \times J)$ such that $M_{(i,j)}=1$.
\RETURN $({\cal L}, {\cal E})$
\caption{Preprocessing $M$}
\label{alg:preprocess}
\end{algorithmic}
\end{algorithm}
The following claim implies the first part of Theorem~\ref{thm:UB}.
\begin{claim}
\label{clm:sizeUB}
The string $({\cal L}, {\cal E})$ can be encoded using at most $O(n^{3/2} (\sqrt{w}+\frac{\lg n}{\sqrt{w}}))$ bits.
\end{claim}
\begin{proof}
Observe the conditions in line~\ref{l:cond} of the algorithm.
Due to condition~\ref{itm:genSize}, there are at most $n^{1/2}\sqrt{w}$ many different pairs in the list ${\cal L}$, and each pair can be encoded using only $2n$ bits (an indicator bit per row and column). Therefore ${\cal L}$ can be encoded using at most $O(n^{3/2} \sqrt{w})$ bits. Condition~\ref{itm:density} asserts that the density of $1$-entries in the submatrix covered by all the subsets of rows and columns listed in ${\cal L}$ is at most $\frac{1}{n^{1/2}\sqrt{w}}$. Each such entry can be encoded using $2\lg n$ bits and hence $\mathcal{E}$ can be encoded using at most $O(n^{3/2} \lg n/\sqrt{w})$ bits.
\end{proof}
\paragraph{Answering queries. }
To prove the second part of the theorem, we give a query algorithm that receives $v \in \{0,1\}^n$ and gets access to $M$, as well as to ${\cal L}$ and ${\cal E}$, and computes $Mv$ with high probability.
Let $J = \{j \in [n] : v_j=1\}$ be the set of columns of $M$ which are relevant for computing $Mv$, and let ${\cal U} := \bigcup_{(I',J') \in {\cal L}}(I' \times J')$ be the set of all matrix indices that appear in ${\cal L}$.
Starting with the set $I = [n]$ of all possible rows, the algorithm "prunes" $I$ throughout the execution. Whenever an index $i$ is removed from $I$, the algorithm fixes $u_i \in \{0,1\}$.
During the first step, the algorithm goes over ${\cal E}$. If for some $i \in I$, there exists $j \in J$ such that $(i,j)$ is encoded in ${\cal E}$, the algorithm sets $u_i = 1$ and removes $i$ from $I$.
During the second step, for every $i \in I$ the algorithm samples $2\sqrt{n} \lg n$ entries $(i,j)$ from the set $(\{i\} \times J) \setminus {\cal U}$. If $M_{(i,j)}=1$ for at least one of these entries, the algorithm sets $u_i = 1$ and removes $i$ from $I$.
During the third step, the algorithm examines the set ${\cal R} := \{(i,j) : i \in I\; , \; j \in J \; and \; (i,j)\notin {\cal U}\}$ of remaining entries. If this set has more than $O(n^{3/2}\lg n / \sqrt{w})$ elements, the algorithm reports "failure". Otherwise for every $i \in I$, the algorithm probes all entries $(i,j) \in {\cal R}$. If $M_{(i,j)}=1$ for at least one of these entries, the algorithm sets $u_i = 1$ and removes $i$ from $I$. Otherwise, the algorithm sets $u_i = 0$ and removes $i$ from $I$.
The algorithm terminates either by reporting "failure" or by returning $u = (u_1,\ldots,u_n)$. It is formally described as Algorithm~\ref{alg:query}.
\begin{algorithm}[th]
\begin{algorithmic}[1]
\STATE let $I \leftarrow [n], J \leftarrow \{j \in [n] : v_j =1\}$.
\STATE let ${\cal U} \leftarrow \bigcup_{(I',J') \in {\cal L}}(I'\times J')$.
\FORALL{$(i,j)$ encoded in ${\cal E}$} \label{l:firstStepStart}
\IF{$i \in I$ and $j \in J$}
\STATE set $u_i = 1$ and let $I \leftarrow I \setminus \{i\}$. \label{l:firstStepEnd}
\ENDIF
\ENDFOR
\FORALL{$i \in I$} \label{l:secondStepStart}
\STATE sample uniformly (with repetition) $2 \sqrt{nw} \lg n$ entries from $\{j \in J : (i,j) \notin {\cal U}\}$. \\
denote the set of sampled entries $J''(i)$.
\IF{there exists $j \in J''(i)$ such that $M_{(i,j)}=1$}
\STATE set $u_i \leftarrow 1$ and let $I \leftarrow I \setminus \{i\}$. \label{l:secondStepEnd}
\ENDIF
\ENDFOR
\STATE let ${\cal R} \leftarrow (I \times J) \setminus {\cal U}$. \label{l:thirdStepStart}
\IF{$|{\cal R}| \ge n^{3/2} / \sqrt{w}$}
\RETURN "failure"
\ENDIF
\FORALL{$i \in I$}
\IF{there exists $j \in J$ such that $(i,j) \in {\cal R}$ and $M_{(i,j)} = 1$}
\STATE set $u_i \leftarrow 1$ and let $I \leftarrow I \setminus \{i\}$.
\ELSE
\STATE set $u_i \leftarrow 0$ and let $I \leftarrow I \setminus \{i\}$.
\ENDIF
\ENDFOR
\RETURN $u = (u_1,\ldots,u_n)$
\caption{Querying $Mv$}
\label{alg:query}
\end{algorithmic}
\end{algorithm}
We will first show that the algorithm probes "few" bits.
Since $({\cal L},{\cal E})$ can be encoded using at most $O(n^{3/2} (\sqrt{w}+\frac{\lg n}{\sqrt{w}}))$ bits, and since the algorithm samples at most $2 \sqrt{nw} \lg n$ entries from each row of the matrix, we conclude the following.
\begin{lemma}
Algorithm~\ref{alg:query} probes at most $O(n^{3/2}\sqrt{w}\lg n)$ bits of $M,{\cal L}, {\cal E}$ throughout the execution.
\end{lemma}
To finish the proof of Theorem~\ref{thm:UB} we show that with high probability, the algorithm returns the correct answer. To this end, fix $v \in \{0,1\}^n$, and consider an execution of Algorithm~\ref{alg:query} on $v$.
Let $I_1, I_2$ be the set $I$ after the first step of the algorithm (lines~\ref{l:firstStepStart}-\ref{l:firstStepEnd}), and the second step of the algorithm (lines~\ref{l:secondStepStart}-\ref{l:secondStepEnd}) respectively. In these notations, ${\cal R} \leftarrow (I_2 \times J) \setminus {\cal U}$. Finally, let $$I_1^* := \left\{i \in I_1 : \Pr_{j \in J : (i,j) \notin {\cal U}} [M_{(i,j)}=1] > \frac{1}{\sqrt{nw}}\right\} \;.$$
\begin{lemma}
Algorithm~\ref{alg:query} fails with probability at most $\tfrac{1}{n}$. Moreover, if the algorithm does not fail, then it returns $Mv=u$.
\end{lemma}
\begin{proof}
Let ${\cal F}$ be the event $I_2 \subseteq I_1 \setminus I_1^*$. By definition of $I_1^*$ we get that
\begin{equation*}
\begin{split}
\Pr[{\cal F}] \ge 1 - \sum_{i \in I_1^*}{\Pr[\forall j \in J''(i). \; M_{(i,j)}=0]}
&\ge 1 - \sum_{i \in I_1^*}{\left(1 - \frac{1}{\sqrt{nw}}\right)^{2\sqrt{nw}\lg n}} \ge 1 - \frac{1}{n}
\end{split}
\end{equation*}
Conditioned on ${\cal F}$ occurring, for every $i \in I_2$, $\Pr_{j \in J : (i,j) \notin {\cal U}} [M_{(i,j)}=1] \le \frac{1}{\sqrt{nw}}$. Since $I_2 \times J \not\subseteq {\cal U}$, then $(I_2, J) \notin {\cal L}$. By the construction of ${\cal L}$ we therefore conclude that $|(I_2 \times J) \setminus {\cal U}| < \frac{n^{3/2}}{\sqrt{w}}$, and the algorithm does not fail.
We will show next that if the algorithm does not fail, then for every $i \in [n]$, $u_i = [Mv]_i$. First note that if $i \notin I_2$, then the algorithm finds $j \in J$ such that $M_{(i,j)}=1$, and therefore $u_i=1=[Mv]_i$. Otherwise, assume $i \in I_2$. Then for every $j \in J$, if $(i,j) \in {\cal U}$, then $M_{ij}=0$, since otherwise $(i,j)$ would be encoded in ${\cal E}$ and removed during the first step of the execution.
Therefore, for every $j \in J$, if $M_{(i,j)}=1$ then $(i,j) \in (I_2 \times J) \setminus {\cal U}$.
Since the algorithm does not fail, it goes over all entries in $(I_2 \times J) \setminus {\cal U}$, and therefore $[Mv]_i=1$ if and only if there exists $j \in J$ such that $M_{(i,j)}=1$, which in turn implies $u_i=1$.
\end{proof}
\section{Matching Lower Bound on Boolean Matrix-Vector Problem}
\label{sec:Boolean-Mv}
In this section we consider the Boolean $Mv$ problem, where $Mv = (\vee_{j \in [n]}(M_{(i,j)} \wedge v_j))_{i \in [n]}$, and prove Theorem~\ref{thm:mainMvBoolean}. The presented bound matches the upper bound shown in the last section up to some (small) polylogarithmic factor. Although Theorem~\ref{thm:mainMvBoolean} allows the query algorithm to be randomized that returns right answer with high probability, for the sake of simplicity we first focus on the deterministic regime. In Section~\ref{sec:randomizedQueryLB} we refine the proof to hold for randomized query algorithms.
\begin{theorem}
\label{thm:detMvBoolean}
Assume that for every matrix $M \in \{0,1\}^{n \times n}$ there exists a data structure $R=R(M)$ consisting of at most $r=r(n)$ bits and there is an algorithm that given any $v \in \{0,1\}^n$ computes $Mv$ by probing $R$ and at most $t=t(n)$ entries from $M$. Then for $n \le r \le n^2/4$, $t \cdot r = \Omega(n^3)$; otherwise for $r < n$, $t = \Omega(n^2)$.
\end{theorem}
To prove the theorem, we will define a family ${\cal M} \subseteq \{0,1\}^{n \times n}$ of matrices, and a set of queries $v^{(1)}, \ldots, v^{(4r/n)} \in \{0,1\}^n$ such that the following holds for every $M \in {\cal M}$. (1) The set of answers to the queries $Mv^{(1)}, \ldots, Mv^{(4r/n)}$ hold a large amount of information on $M$; and (2) one can succinctly "encode" the execution of the query algorithm on the respective sequence of queries. That is, there exists a short (in terms of $t,r$) string ${\cal E}$ such that given ${\cal E}, R$, one can emulate the query algorithm over the sequence of queries, and return the answers $Mv^{(1)}, \ldots, Mv^{(4r/n)}$.
For the rest of the section, we additionally assume $n \le r \le \sqrt{n^3/4}$. The proof for the case $\sqrt{n^3/4} \le r \le n^2/4$ is similar, and the differences will be discussed towards the end of the proof.
\paragraph{A Family of Input Matrices. }
Let ${\cal M} \subseteq \{0,1\}^{n \times n}$ be the family of all matrices $M \in \{0,1\}^{n \times n}$ with the following property: If each row of $M$ is divided into $r/n$ contiguous {\em blocks} each containing $n^2/r$ consecutive entries, then exactly one entry in each block is $1$, and the rest are all $0$s.
Consider a matrix $M \in {\cal M}$. Each of the $r$ blocks in $M$ contains exactly one $1$-entry out of $n^2/r$ entries. The following claim is thus implied from the definition of entropy.
\begin{claim} \label{clm:info-hard}
Let $M \in_R {\cal M}$. Then $H(M) = r \lg \tfrac{n^2}{r}$.
\end{claim}
\paragraph{Encoding argument. }
Consider the following sequence of $4r/n$ vectors in $\{0,1\}^n$. For every $m \in [4r/n]$ define $v^{(m)} \in \{0,1\}^n$ such that $v^{(m)}_j = 1$ if and only if $$j (\modu \tfrac{n^2}{r}) \in \{\frac{(m-1)n^3}{4r^2}+1 , \ldots, \frac{mn^3}{4r^2}\} \;.$$
Fix some $M \in {\cal M}$. When querying for $Mv^{(1)}, \ldots, Mv^{(4r/n)}$, the algorithm reads at most $4tr/n$ bits from $M$.
Observe that one can trivially encode all the probed entries using $\tfrac{4tr}{n} (2\lg n+1)$ bits by specifying the indices and the entry values. In turn, this encoding implies $t \ge \Omega(n/\lg n)$. However, by employing a subtler argument inspired by~\cite{BL13}, we provide a much better bound on $t$ and $r$, which also implies the simpler one.
To this end, let $b,k$ denote the total number of different entries and the number of different $1$-entries read by the algorithm throughout this sequence of $4r/n$ many $Mv$ queries respectively. Then $k \le b \le 4tr/n$.
Let $B$ be the sequence of $b$ different entries probed by the query algorithm, when queried for $Mv^{(1)}, \ldots, Mv^{(4r/n)}$, in the order they are probed. That is, $B : [b] \hookrightarrow [n] \times [n]$ is injective. Let $K := \{j \in [b] : M_{B(j)}=1\} \subseteq [b]$. Define ${\cal E}$ to be the bit-string composed of the three following sub-strings. The first $\lg(4tr/n)$ bits of ${\cal E}$ encode $b$. The next $\lg(4tr/n)$ bits encode $k$. The remaining bits encode $K$ as a subset of $[b]$. Since $|K|=k$, the following is straightforward.
\begin{claim}
${\cal E}$ can be encoded using at most $\lg \binom{4tr/n}{k} + 2\lg (4tr/n)$ bits.
\end{claim}
Next, we show that ${\cal E}$ and $R$ hold a "large amount" of information of $M$.
\begin{lemma}
\label{lem:encode-one}
The bit-string ${\cal E}$ encodes the subset of $k$ $1$-entries among all the entries probed. Furthermore, given access to ${\cal E}$ and $R$ one can answer all the queries $Mv^{(1)}, \ldots, Mv^{(4r/n)}$.
\end{lemma}
\begin{proof}
First note, that given ${\cal E}$, one can directly decode $b,k$ and $K$.
We next show an emulation algorithm that, given access to $R, b, k ,K$ finds the $k$ $1$-entries in question, and moreover, answers all the queries $Mv^{(1)}, \ldots, Mv^{(4r/n)}$.
The algorithm emulates the query algorithm for $Mv^{(1)}, \ldots, Mv^{(4r/n)}$. Whenever the query algorithm probes a matrix entry, the emulation algorithm feeds it with an answer as follows.
If the entry has been previously probed during the execution, the query algorithm is fed with the same answer. Otherwise, the answer is determined to be $1$ or $0$ by whether the index of the matrix entry in the sequence of probes is in $K$ or not respectively, and the answer is stored for later probes. The algorithm is formally described as Algorithm~\ref{alg:emulation}.
\begin{algorithm}[H]
\begin{algorithmic}[1]
\STATE let $j \leftarrow 1$.
\STATE let $B^{alg} = \emptyset$.
\FOR{$i = 1,2,\ldots,4r/n$}
\STATE run the query algorithm for $Mv^{(i)}$
\STATE whenever the query algorithm probes a matrix entry $(p,q) \in [n]\times [n]$
\IF{there exists $j_0 < j$ such that $B^{alg}[j_0] = (p,q)$} \label{l:previouslyProbed}
\STATE $\ell \leftarrow j_0$
\ELSE
\STATE $B^{alg}[j] \leftarrow (p,q)$, $\ell \leftarrow j$ and $j \leftarrow j+1$.
\ENDIF
\IF{$\ell \in K$}
\STATE feed the query algorithm with $M_{(p,q)}=1$.
\ELSE
\STATE feed the query algorithm with $M_{(p,q)}=0$.
\ENDIF
\ENDFOR
\caption{Emulating a Sequence of Queries}
\label{alg:emulation}
\end{algorithmic}
\end{algorithm}
To prove the lemma, it is enough to show that the emulation algorithm always feeds the query algorithm with the correct answer. Denote the sequence of entry probes (with repetitions) performed by the query algorithm by $\{(p_m,q_m)\}_{m=1}^{4tr/n}$ (we may assume for simplicity that the query algorithm performs exactly $t$ entry probes for each query). The crux of the argument is that for all $j \in [b]$, $B^{alg}[j]=B(j)$. We prove the claim by induction on $m$.
Clearly, during the first probe, $j=1$ and therefore the condition in line~\ref{l:previouslyProbed} of Algorithm~\ref{alg:emulation} is false. Therefore the algorithm sets $B^{alg}[1]$ to be $(p_1,q_1)$, which equals $B(1)$. Moreover, the emulation algorithm answers $1$ if and only if $1 \in K$, which occurs if and only if $M_{(p_1,q_1)}=1$. Assuming correctness for all $m'<m$, we will prove correctness for $m$. If there exists $j_0 < j$ such that $B^{alg}[j_0] = (p_m,q_m)$, then this entry probe has already been answered correctly by the induction hypothesis. Since the emulation algorithm gives the same answer as before, the answer is the correct one. Otherwise, this entry has not been probed yet, and therefore $B^{-1}((p_m,q_m)) = j$. The emulation algorithm then answers $1$ if and only if $j \in K$, which happens if and only if $M_{(p_m,q_m)}=1$. Therefore the emulation algorithm always gives the correct answer, thus finding the correct set of $k$ $1$-entries and answering all the queries $Mv^{(1)}, \ldots, Mv^{(4r/n)}$.
\end{proof}
The next lemma states that, in addition to learning the $k$ $1$-entries of $M$, by answering all the queries $Mv^{(1)}, \ldots, Mv^{(4r/n)}$, we can learn a lot of information about the other blocks of $M$.
\begin{lemma}
\label{lem:remain-information}
Let $M \in_R {\cal M}$. Then $H(M | R, {\cal E}) \le (r-k) \lg \tfrac{n^2}{4r}$.
\end{lemma}
\begin{proof}
By Lemma~\ref{lem:encode-one}, given access to only ${\cal E}$ and $R$, we can answer $Mv^{(1)}, \ldots, Mv^{(4r/n)}$, thus finding $k$ many $1$-entries of $M$. Moreover, for each block in $M$, the algorithm finds at least $3n^2/4r$ many $0$-entries.
By emulating the query algorithm for $Mv^{(1)}, \ldots, Mv^{(4r/n)}$, an algorithm can conclude from ${\cal E}$ and $R$ the exact locations of the $k$ many $1$-entries. Next, let $i \in [n]$ and let $m \in [4r/n]$. Note that if $[Mv^{(m)}]_i=0$, then for each block in the $i$-th row of $M$, all the entries corresponding to $1$-entries in $v^{(m)}$ must be $0$. Since there are exactly $n^3/4r^2$ such entries, and since $[Mv^{(m)}]_i=1$ for at most $r/n$ values of $m$, the algorithm learns at least $\tfrac{3r}{n} \cdot \tfrac{n^3}{4r^2} = \tfrac{3n^2}{4r}$ $0$-entries in each block of $M$. Now the lemma follows from the Shannon's source coding theorem.
\end{proof}
We now turn to finish the proof of Theorem~\ref{thm:detMvBoolean}.
\begin{proof}[Proof of Theorem~\ref{thm:detMvBoolean}]
First observe that $r + \lg \binom{4tr/n}{k} +2\lg (4tr/n) \ge H(R,{\cal E}) \ge I(M;(R, {\cal E})) = H(M) - H(M | R, {\cal E})$. By Lemma~\ref{lem:remain-information}, the right hand side is at least $ r \lg \tfrac{n^2}{r} - (r-k)\lg \tfrac{n^2}{4r} = 2r + k \lg \tfrac{n^2}{4r}$.
Rearranging we get that $\lg \binom{4tr/n}{k}+2\lg (4tr/n) - k \lg \tfrac{n^2}{4r} \ge r$. Since $\lg \binom{4tr/n}{k} \le k \lg \tfrac{4etr}{kn}$, we get that $k\lg \tfrac{16etr^2}{n^3k} +2\lg (4tr/n)\ge r$, and as for every $x, \alpha > 0$, $x \lg \tfrac{\alpha}{x} \le \alpha/2$ we have $\tfrac{8etr^2}{n^3}+2\lg (4tr/n) \ge r$, or $tr \ge \Omega(n^3)$.
This completes the proof for the case $n \le r \le \sqrt{n^3/4}$.
The proof for $\sqrt{n^3/4} \le r \le n^2/4$ is similar. The only difference is that in this case we choose $4r/n$ vectors in a slightly different way. For every $m \in [n^2/r]$ and $i \in [4r^2/n^3]$, we define a vector $v^{(m,i)}$ by setting $v^{(m,i)}_j=1$ if and only if $j (\modu \tfrac{n^2}{r}) = m$ and $(i-1)n^4/4r^2 +1 \le j \le in^4/4r^2$. The proof then follows in an analogous manner.
For the second part of the theorem, i.e. when $r < n$, we use a simple padding argument. If $r < n$, create a new data structure $R'$ by appending some arbitrary bits to $R$ such that $R'$ contains exactly $n$ bits. Now from the previous argument it follows that $t \ge \Omega (n^2)$.
\end{proof}
\subsection{Lower Bound for Randomized Query Algorithms}
\label{sec:randomizedQueryLB}
In this section we will extend the lower bound results shown previously to the case of randomized query algorithms, thus completing the proof of Theorem~\ref{thm:mainMvBoolean}. Assume that given a matrix $M\in \{0,1\}^{n \times n}$ there exists a data structure $R=R(M)$ consisting of at most $r=r(n)$ bits and there exists a {\em randomized} algorithm that, given $v \in \{0,1\}^n$ returns $Mv$ with probability $\ge 1 - \tfrac{1}{n}$ by probing $R$ and at most $t=t(n)$ entries from $M$.
\begin{reptheorem}{thm:mainMvBoolean}
If $n \le r \le n^2/4$, then $t\cdot r \ge \Omega(n^3)$; otherwise for $r < n$, $t \ge \Omega(n^2)$.
\end{reptheorem}
The proof will employ similar arguments to the proof of Theorem~\ref{thm:detMvBoolean}. We will therefore focus only on the case $n \le r \le \sqrt{n^3/4}$.
Consider the vectors $v^{(1)}, \ldots, v^{(4r/n)}$ from the proof of Theorem~\ref{thm:detMvBoolean}. Fix some $M \in {\cal M}$, and run the query algorithm on $Mv^{(1)}, \ldots, Mv^{(4r/n)}$. For every $j \in [4r/n]$, denote the string of random bits used by the algorithm when queried for $Mv^{(j)}$ by $x_j$, and let $X := \left\langle x_1,\ldots,x_{4r/n} \right\rangle$. Let $F$ denote the indicator for the event that the query algorithm failed answering at least one of the queries. Applying union bound, $p := \Pr[F = 1] \le 1/3$.
Let $B,b,K,k$ be as in the proof of Theorem~\ref{thm:detMvBoolean}. If $F=0$, we encode the string ${\cal E}$ the same way as in the previous proof and append $0$ as its first bit, whereas if $F=1$ we let ${\cal E}$ be an encoding of $M$ and append $1$ as its first bit.
In a similar manner to the one presented in Algorithm~\ref{alg:emulation}, given $R, {\cal E}$ and $X$, one can emulate the sequence of queries $Mv^{(1)}, \ldots, Mv^{(4r/n)}$ performed by the query algorithm, while using $x_1,\ldots,x_{4r/n}$ as random strings respectively.
By computing $H(F,M \mid R,{\cal E},X)$ in two different ways we get the following.
\begin{equation*}
\begin{split}
H(F,M \mid R,{\cal E},X) &= H(M \mid R,{\cal E},X) + H(F \mid M, R,{\cal E},X) \\
&= H(F \mid R,{\cal E},X) + H(M \mid F, R,{\cal E},X)
\end{split}
\end{equation*}
Note that $H(F \mid R,{\cal E},X) \le H(F) < 1$ and $H(F \mid M, R,{\cal E},X) = 0$, since given ${\cal E}$ one learns also $F$.
Rearranging we get that
$$H(M \mid R,{\cal E},X) \le 1 + H(M \mid F, R,{\cal E},X) \;.$$
\begin{equation}
\begin{split}
H(M \mid R,{\cal E},X) &\le 1 + H(M \mid F, R,{\cal E},X) \\
&= 1 + p \cdot H(M \mid F=1, R,{\cal E},X) + (1-p) \cdot H(M \mid F=0, R,{\cal E},X) \\
&\le 1 + 0 + (1-p)(r-k) \lg \tfrac{n^2}{4r} \\
\end{split}
\label{eq:remainingInfoRandom}
\end{equation}
where the last inequality follows from the same arguments as in Lemma~\ref{lem:remain-information}.
\begin{proof}[Proof of Theorem~\ref{thm:mainMvBoolean}]
First observe that from the Shannon's source coding theorem,
$$H({\cal E}) \le (1-p)\left(\lg \binom{4tr/n}{k} +2\lg \frac{4tr}{n}\right) + pr \lg \frac{n^2}{r}$$
and hence
\begin{equation*}
\begin{split}
r + (1-p)\left(\lg \binom{4tr/n}{k} +2\lg \frac{4tr}{n}\right) + pr \lg \frac{n^2}{r} &= H(R) + H({\cal E}) \\
&\ge H(R,{\cal E}) \\
&\ge I((M,X);(R,{\cal E})) = H(M,X) - H(M,X| R, {\cal E}) \;.
\end{split}
\end{equation*}
Since $M,X$ are independent, we get that the r.h.s is at least $H(M) - H(M \mid R, {\cal E}, X)$. From \eqref{eq:remainingInfoRandom} we get that this is at least
$$r \lg \tfrac{n^2}{r} - \left(1 + (1-p)(r-k)\lg \tfrac{n^2}{4r} \right) = pr \lg \tfrac{n^2}{r} + 2(1-p)r + (1-p) k \lg \tfrac{n^2}{4r} -1 \;.$$
Rearranging we get that
$$\lg \binom{4tr/n}{k}+2\lg (4tr/n) - k \lg \tfrac{n^2}{4r} \ge 2r - r/(1-p) - 1/(1-p) \ge r/3 \;$$
By the same arguments as in the proof of Theorem~\ref{thm:detMvBoolean}, we get that $\tfrac{8etr^2}{n^3}+2\lg (4tr/n) \ge r/3$, or $tr \ge \Omega(n^3)$.
\end{proof}
\subsection{Lower Bounding Boolean Vector-Matrix-Vector Problem}
\label{sec:Bool-uMv}
In this section we extend the technique from Section~\ref{sec:Boolean-Mv} to get a lower bound on Boolean $u^{\dagger}Mv$ problem, albeit a weaker one. Assume that given a matrix $M \in \{0,1\}^{n \times n}$, there exists a data structure $R=R(M)$ containing at most $r=r(n)$ bits, and there exists an algorithm that, given $u,v \in \{0,1\}^n$ returns $u^{\dagger}Mv = (\vee_{i,j \in [n]}(M_{(i,j)} \wedge u_i \wedge v_j))$ while probing only $R$ and at most $t=t(n)$ bits from $M$.
Let us first observe an easy corollary of Theorem~\ref{thm:mainMvBoolean}. Since answer of each $Mv$ query can be derived from the following sequence of $n$ queries: $(e^{(1)})^{\dagger}Mv,\cdots,(e^{(n)})^{\dagger}Mv$ where $\{e^{(i)}\}_{i \in [n]}$ is the standard basis over $\{0,1\}^n$, we get an lower bound of $tr\ge \Omega(n^2)$ for the Boolean $u^{\dagger}Mv$ problem for $n \le r \le n^2/4$. In this section we will prove a better lower bound for Boolean $u^{\dagger}Mv$ problem as stated in the following theorem.
\begin{theorem} \label{thm:lbBoolean-uMv}
If $r \le n^2/4$ then $t \ge \Omega(n/\lg n)$.
\end{theorem}
We prove the above theorem for $n \le r \le \sqrt{n^3/4}$. The proof extends for all $r < n^2/4$ by arguments similar to those used in the proof of Theorem~\ref{thm:detMvBoolean}.
First note, that when comparing with the matrix-vector problem, the main caveat of the vector-matrix-vector problem is that each query results in exactly one bit, rather than $n$ bits.
Loosely speaking, we have observed that by querying $Mv^{(1)}, \ldots, Mv^{(4r/n)}$, where $v^{(1)}, \ldots, v^{(4r/n)}$ are defined as before, we gain a lot of information about $M$. This approach seems less beneficial when concerning vector-matrix-vector queries.
More specifically, it seems that we need $n$ times more queries to get the same amount of information.
By using the trivial argument demonstrated right before Theorem~\ref{thm:lbBoolean-uMv} we can not get our claimed bound.
A more subtle observation into the proof of Lemma~\ref{lem:encode-one} shows that we can get a lot of information when the answer to the query is $0$.
Particularly, assume $u^{\dagger}Mv = 0$. Then we know that whenever $u_i = v_j = 1$, $M_{(i,j)} = 0$. In what follows, we will need the following notation.
\begin{notation}
Let $x \in \{0,1\}^n$. Suppose $\overline{x} \in \{0,1\}^n$ denotes the {\em complement} of $x$. That is, $\overline{x}_j = 1$ if and only if $x_j = 0$ for every $j \in [n]$.
\end{notation}
Clearly, $\overline{x}^{\dagger}x = 0$, and moreover, $\overline{x}$ is the unique heaviest vector (in terms of Hamming weight) satisfying this property.
\begin{lemma}
\label{lem:encode-uMv}
There exists a bit-string ${\cal E} = {\cal E}(M)$,
such that ${\cal E}$ can be encoded using at most $\frac{4tr}{n} (2\lg n+1)$ bits,
and moreover, given access to ${\cal E}$ and $R$ one can answer of $Mv^{(1)}, \ldots, Mv^{(4r/n)}$.
\end{lemma}
\begin{proof}
For every $j \in [4r/n]$, let $u^{(j)} := \overline{Mv^{(j)}}$. Take ${\cal E}$ to be the encoding of the list of all entries of $M$ probed throughout the sequence of queries $(u^{(1)})^{\dagger}Mv^{(1)}, \cdots , (u^{(4r/n)})^{\dagger}Mv^{(4r/n)}$, in the order they are probed. For simplicity, we may assume that the query algorithm probes exactly $t$ matrix entries for each query. For each entry we encode it's location in the matrix (using $2 \lg n$ bits), and its value (one more bit). Then ${\cal E}$ is composed of $4r/n$ "segments" of $t$ entries each. Clearly ${\cal E}$ can be encoded using at most $\frac{4tr}{n} (2\lg n+1)$ bits.
Let $j \in [4r/n]$. Now we provide an emulation algorithm that, given access to ${\cal E}$ and $R$, finds $u^{(j)}$, and thus finds $Mv^{(j)} = \overline{u^{(j)}}$. The emulation algorithm starts by setting $u^{(*)}$ to be all $0$ vector and goes over all $u \in \{0,1\}^n$ and emulates the query algorithm for $u^{\dagger}Mv^{(j)}$. Whenever the query algorithm probes an entry of $M$, the emulation algorithm looks for the encoding of this entry in ${\cal E}$. If it finds this entry, it feeds it to the query algorithm. Otherwise, it breaks and continues to the next $u \in \{0,1\}^n$. If the query algorithm terminates, and the answer is $0$, then the algorithm compares the Hamming weight of $u$ (denoted as $wt(u)$) with that of $u^{(*)}$. If $wt(u) > wt(u^{(*)})$ the algorithm replaces $u^{(*)}$ with $u$.
The algorithm is formally given as Algorithm~\ref{alg:finduj}.
\begin{algorithm}[t]
\begin{algorithmic}[1]
\STATE let $u^{(*)} \leftarrow 0^n$.
\FORALL{$u \in \{0,1\}^n$}
\STATE run the query algorithm for $u^{\dagger}Mv^{(j)}$
\STATE whenever the query algorithm probes a matrix entry $(p,q) \in [n]\times [n]$
\IF{$(p,q)$ is encoded in ${\cal E}$}
\STATE feed the query algorithm with the corresponding bit in ${\cal E}$ as $M_{(p,q)}$.
\ELSE
\STATE stop the query algorithm.
\ENDIF
\IF{the query algorithm outputs $u^{\dagger}Mv^{(j)}=0$ and $wt(u) > wt(u^{(*)})$}
\STATE $u^{(*)} \leftarrow u$.
\ENDIF
\ENDFOR
\RETURN $u^{(*)}$
\caption{Finding $u^{(j)}$}
\label{alg:finduj}
\end{algorithmic}
\end{algorithm}
It is straightforward that $u^{(*)}$ is the unique $u \in \{0,1\}^n$ that satisfies the following.
\begin{enumerate}
\item Every entry probed by the query algorithm when querying $u^{\dagger}Mv^{(j)}$ is encoded in ${\cal E}$ (thus given access to ${\cal E}, R$, one can answer $u^{\dagger}Mv^{(j)}$);
\item $u^{\dagger}Mv^{(j)}=0$; and
\item $u$ is of maximal Hamming weight.
\end{enumerate}
Therefore, $u^{(*)} = u^{(j)}$.
\end{proof}
We can now prove Theorem~\ref{thm:lbBoolean-uMv} using similar arguments to those in the proof of Theorem~\ref{thm:detMvBoolean}.
Suppose while probing entries of the matrix $M$ we read total $k$ $1$-entries. Then by following the argument of the proof of Theorem~\ref{thm:detMvBoolean}, we get that
\begin{align*}
r + \tfrac{4tr}{n} (2\lg n+1) & \ge H(R,{\cal E}) \ge H(M) - H(M | R, {\cal E})\\
&\ge 2r + k \lg \tfrac{n^2}{4r} \ge 2r.
\end{align*}
Now by rearranging the terms, we get that $t \ge \Omega(n/\lg n)$. $\hfill \Box$
Using the argument similar to that in Section~\ref{sec:randomizedQueryLB} we can also extend Theorem~\ref{thm:lbBoolean-uMv} to randomized query algorithms.
\begin{theorem}
\label{thm:randomizedLB-uMv}
Assume that for every matrix $M\in \{0,1\}^{n \times n}$ there exists a data structure $R$ consisting of at most $r=r(n)$ bits and a query algorithm that can answer any $u^{\dagger}Mv$ query with error probability at most $1/n$ by probing $R$ and at most $t=t(n)$ entries of $M$. Then for $r \le n^2/4$, $t \ge \Omega(n/\lg n)$.
\end{theorem}
\paragraph*{Note:} Though in both Theorem~\ref{thm:mainMvBoolean} and Theorem~\ref{thm:randomizedLB-uMv} we consider the error probability to be at most $1/n$, one can easily generalize the results for error probability to be any $1/n < \epsilon <1$. However we will lose extra $\lg n$ factor in the lower bound. More specifically, given any randomized algorithm $\mathcal{A}$ with probability of error $\epsilon$, we can boost the success probability to $(1-1/n)$ by repeating $\mathcal{A}$ $O(\lg n/ \lg (\frac{1}{\epsilon}))$ times and taking the majority vote. Now let us denote the new algorithm to be $\mathcal{B}$. Observe that $\mathcal{B}$ probes at most $O(t \lg n/ \lg (\frac{1}{\epsilon}))$ cells of the matrix and thus we will lose $O(\lg n/ \lg (\frac{1}{\epsilon}))$ factor in all the bounds given in Theorem~\ref{thm:mainMvBoolean} and Theorem~\ref{thm:randomizedLB-uMv}.
\ifpdf
\section{Lower Bound on Vector-Matrix-Vector Problem over \texorpdfstring{$\mathbb{F}_2$}{F2}}
\else
\section{Lower Bound on Vector-Matrix-Vector Problem over $\mathbb{F}_2$}
\fi
\label{sec:F2-uMv}
This section is devoted to the proof of Theorem~\ref{thm:mainUMVF2}. To this end, assume that given a matrix $M \in \mathbb{F}_2^{n \times n}$, there exists a data structure $R=R(M)$ consisting of at most $r=r(n)$ bits, and there exists an algorithm that, given $u,v \in \mathbb{F}_2^n$ returns $u^{\dagger}Mv$ while probing only $R$ and at most $t=t(n)$ bits from $M$. Under these assumptions, Theorem~\ref{thm:mainUMVF2} states the following.
\begin{reptheorem}{thm:mainUMVF2}
If $n \le r \le \tfrac{n^2}{64}$ then $t \cdot r = \Omega(n^3/\lg n)$; otherwise for $r < n$, $t = \Omega(n^2/ \lg n)$.
\end{reptheorem}
To prove the theorem we will show that for most matrices $M \in \mathbb{F}_2^n$, one can succinctly (in terms of $r,t$) encode $M$.
More precisely, by fixing some parameter $B$ and dividing $M$ into segments of $B$ consecutive rows, we will show that there is a single segment that contains a large amount of information, and moreover, there exists a short (in terms of $r,t$) bit-string that encodes this segment.
It is worth noting that our proof technique can be generalized to give lower bound for the case when the query algorithm may err with probability at most $1/64$ on average over the choices of $M,u,v$. However for the sake of simplicity we first focus only on the query algorithm that never errs and we defer the comment on the generalization to the end of this section.
We start with introducing some notations. Given $u \in \mathbb{F}_2^n$ and a subset $I \subseteq [n]$, let $u_I$ be the projection of $u$ onto $\mathbb{F}_2^{|I|}$. Similarly denote $M_{I,J}$ for any $M \in \mathbb{F}_2^{n \times n}$ and subsets $I,J \subseteq [n]$.
Let $B$ be some parameter, the value of which will be fixed later.
For every $i \in [n/B]$, let $I_i := \{(i-1)B+1, \ldots, iB\}$, and let $M_i := M_{I_i,[n]}$ be the $i$-th segment of $M$ composed of all the rows in $I_i$, and $M_{-i} := M_{([n] \setminus I_i), [n]}$.
Finally, for every $u,v \in \mathbb{F}_2^n$, $M \in \mathbb{F}_2^{n \times n}$ and $i \in [n/B]$, let $t_i(u,M,v)$ be the number of cell probes performed by the algorithm in $M_i$ when queried for $u^{\dagger}Mv$.
Our first claim shows that there exists an $i^* \in [n/B]$ such that for many vectors $u \in \mathbb{F}_2^n$, the expected number (over random $M,v$) of probes performed by the algorithm on $M_{i^*}$ is not too large.
\begin{lemma}
\label{lem:expected-probe}
There exists $i^* \in [n/B]$ such that $\Pr_u[\mathbb{E}_{M,v}[t_{i^*}(u,M,v)] \le \frac{4tB}{n}] \ge \frac{3}{4}$.
\end{lemma}
\begin{proof}
First note that $\mathbb{E}_i[\mathbb{E}_{u,M,v}[t_i(u,M,v)]] = \mathbb{E}_{u,M,v}[\mathbb{E}_i[t_i(u,M,v)]] \le t$. Therefore there exists $i^* \in [n/B]$ such that $\mathbb{E}_{u,M,v}[t_{i^*}(u,M,v)] \le \frac{tB}{n}$.
The claim now follows from Markov's inequality.
\end{proof}
For every $u \in \mathbb{F}_2^n$ and $i \in [n/B]$, denote $M_{i|u} := u_{I_i}^{\dagger} M_i$.
The following lemma shows that for most vectors $u \in \mathbb{F}_2^n$, $M_{i^* | u}$ contains a large amount of information. One may note that the lemma is true for all $i \in [n/B]$, though for our purpose it suffices to consider $i^*$ only.
\begin{lemma}
\label{lem:conditional-info}
Suppose $M \in_R \mathbb{F}_2^{n \times n}$. Then $\Pr_u[H(M_{i^* | u} \mid R, M_{- i^*}) \ge n - \frac{8r}{B}] \ge \frac{3}{4}$.
\end{lemma}
\begin{proof}
Let $U = \{u \in \mathbb{F}_2^n : H(M_{i^* | u} \mid R, M_{- i^*}) < n - \frac{8r}{B} \}$, and let $u^{(1)},\ldots,u^{(\ell)} \in U$ be a sequence in $U$ such that $u^{(1)}_{I_{i^*}}, \ldots, u^{(\ell)}_{I_{i^*}}$ are linearly independent over $\mathbb{F}_2^{|I_{i^*}|}$. Then the random variables $M_{i^* | u^{(1)}}, \ldots, M_{i^* | u^{(\ell)}}, M_{- i^*}$ are independent.
To see this, first note that by the definition ,$M_{-i^*}$ is independent of $M_{i^*}$ and hence of $M_{i^* | u^{(1)}}, \ldots, M_{i^* | u^{(\ell)}}$.
Next observe that for any $k \in [\ell]$ and $b \in \mathbb{F}_2^{n}$, $Pr_M[M_{i^*|u^{(k)}}=b]=1/2^n$. Now since the vectors $u^{(1)},\ldots,u^{(\ell)}$ are linearly independent, for any $b^{(1)},\ldots,b^{(\ell)} \in \mathbb{F}_2^{n}$, $Pr_M[\text{for all }k,\;M_{i^*|u^{(k)}}=b^{(k)}]=(1/2^n)^{\ell}$.
Therefore
\begin{equation*}
\begin{split}
r \ge H(R) &\ge I(R ; M_{i^* | u^{(1)}}, \ldots, M_{i^* | u^{(\ell)}} \mid M_{- i^*}) \\
& = H(M_{i^* | u^{(1)}}, \ldots, M_{i^* | u^{(\ell)}} \mid M_{- i^*}) - H(M_{i^* | u^{(1)}}, \ldots, M_{i^* | u^{(\ell)}} \mid R, M_{- i^*}) \\
& \ge \sum_{j=1}^\ell{\left( H(M_{i^* | u^{(j)}}\mid M_{- i^*}) - H(M_{i^* | u^{(j)}} \mid R, M_{- i^*}) \right)} \\
& \ge \sum_{j=1}^\ell{\left( n - \left( n - \frac{8r}{B} \right) \right)} = \frac{8r \ell}{B} \;,
\end{split}
\end{equation*}
thus $\ell \le B/8$ implying $| \{ u_{I_{i^*}} : u \in U \} | \le 2^{B/8}$ and we get that $|U| \le 2^{n-2}$.
\end{proof}
Now the following is a simple application of union bound.
\begin{corollary} \label{cor:uExsists}
There exists $u^* \in \mathbb{F}^n_2$ such that
$$H(M_{i^* | u} \mid R, M_{- i^*}) \ge n - 8r/B \quad and \quad \mathbb{E}_{M,v}[t_{i^*}(u,M,v)] \le 4tB/n \;.$$
\end{corollary}
Let us define ${\cal M} := \{M \in \mathbb{F}_2^{n \times n} : \mathbb{E}_v[t_{i^*}(u^*,M,v)] \le \frac{8tB}{n}\}$ and then Markov's inequality implies the following.
\begin{claim}
\label{clm:F2InitialInfo}
$\Pr_M[M \in {\cal M}] \ge \frac{1}{2}$.
\end{claim}
The next Lemma shows that whenever $M \in {\cal M}$, $M_{i^* \mid u^*}$ can be encoded using a few bits.
\begin{lemma}
\label{lem:encode}
If $M \in {\cal M}$ then $M_{i^* \mid u^*}$ can be encoded using $\frac{64tB}{n} \lg n$ extra bits (in addition to $R$ and $M_{- i^*}$).
\end{lemma}
\begin{proof}
Fix some $M \in {\cal M}$. Then by Markov's inequality, $\Pr_v [t_{i^*}(u^*,M,v) \le \frac{16tB}{n}] \ge \frac{1}{2}$.
Therefore there exists a set ${\cal S}$ of $\frac{16tB}{n}$ entries in $M_{i^*}$ (note that the submatrix $M_{i^*}$ is of size $nB$) such that by probing only entries from $M_{- i^*}, R$ and ${\cal S}$ the algorithm can answer at least
$$\frac{2^{n-1}}{\binom{nB}{16tB/n}} \ge \frac{2^{n-1}}{\left(\frac{enB}{16tB/n}\right)^{\frac{16tB}{n}}} = 2^{n-1 - \frac{16tB}{n} \lg \frac{en^2}{16t}} \ge 2^{n - \frac{32tB}{n} \lg n}$$
queries of the form $(u^*)^{\dagger}Mv$. The set ${\cal S}$ can be encoded using $\frac{16tB}{n} \lg (n^2)$ bits.
Let $V \subseteq \mathbb{F}_2^n$ denote the set of vectors such that for any $v \in V$ the query $(u^*)^{\dagger}Mv$ can be answered by probing entries only from $M_{- i^*}, R, {\cal S}$. Observe that $V$ is a linear subspace of $\mathbb{F}_2^n$, and $\dim(V) \ge n - \frac{32tB}{n} \lg n$.
Next, fix an ordering $v^{(1)},\ldots,v^{(2^n)}$ of $\mathbb{F}_2^n$, and consider the string ${\cal E}$ of bits constructed as follows. Starting with an empty string ${\cal E}$, for every $k \in [2^n]$, if $v^{(k)} \notin {\sf span}\left( V \cup \{v^{(1)},\ldots,v^{(k-1)}\} \right)$, append $(u^*)^{\dagger}Mv^{(k)}$ to ${\cal E}$. Since $\dim(V) \ge n - \frac{32tB}{n} \lg n$, it follows that ${\cal E}$ can be encoded using at most $\frac{32tB}{n} \lg n$ bits.
Now by probing only $M_{- i^*}, R, {\cal S}, {\cal E}$ we can answer $(u^*)^{\dagger}Mv$ for all $v \in \mathbb{F}_2^n$, which in terms suffices to retrieve the string $M_{i^* \mid u^*}$.
\end{proof}
Now we are ready to prove the main result of this section.
\begin{proof}[Proof of Theorem~\ref{thm:mainUMVF2}]
We prove the theorem by showing that $n - \frac{8r}{B} \le \frac{64tB}{n}\lg n + \frac{n}{2} + 1$. Setting $B = \lfloor \frac{32r}{n}\rfloor$ then implies the theorem.
To this end, let $\mathbbm{1}_{M \in {\cal M}}$ denote the indicator random variable for the event $M \in {\cal M}$.
By definition of $u^*$ we have
\begin{equation}
n - \frac{8r}{B} \le H(M_{i^* | u^*} \mid R, M_{- i^*}) \le H(M_{i^* | u^*}, \mathbbm{1}_{M \in {\cal M}} \mid R, M_{- i^*}) \;.
\label{eq:addInd}
\end{equation}
Applying the chain rule of entropy we get that
\begin{equation}
\begin{split}
H(\mathbbm{1}_{M \in {\cal M}}, M_{i^* | u^*} \mid R, M_{- i^*}) &= H(\mathbbm{1}_{M \in {\cal M}} \mid R, M_{- i^*}) + H(M_{i^* | u^*} \mid R, M_{- i^*}, \mathbbm{1}_{M \in {\cal M}}) \\
&\le H(\mathbbm{1}_{M \in {\cal M}}) + H(M_{i^* | u^*} \mid R, M_{- i^*}, \mathbbm{1}_{M \in {\cal M}}) \;.
\label{eq:chainRule}
\end{split}
\end{equation}
Clearly, $H(\mathbbm{1}_{M \in {\cal M}}) \le 1$. Next we bound $H(M_{i^* | u^*} \mid R, M_{- i^*}, \mathbbm{1}_{M \in {\cal M}})$ as follows.
\begin{equation}
\begin{split}
&H(M_{i^* | u^*} \mid R, M_{- i^*}, \mathbbm{1}_{M \in {\cal M}}) = \\
&=H(M_{i^* | u^*} \mid R, M_{- i^*}, \mathbbm{1}_{M \in {\cal M}}=1) \cdot \Pr[\mathbbm{1}_{M \in {\cal M}}=1] + H(M_{i^* | u^*} \mid R, M_{- i^*}, \mathbbm{1}_{M \in {\cal M}}=0) \cdot \Pr[\mathbbm{1}_{M \in {\cal M}}=0].
\end{split}
\label{eq:totalEntropy}
\end{equation}
Conditioned on $\mathbbm{1}_{M \in {\cal M}}=1$, Lemma~\ref{lem:encode} guarantees that we can encode $M_{i^* | u^*}$ using at most $\frac{64tB}{n}\lg n$ bits in addition to $R, M_{- i^*}$. Therefore by the Shannon's source coding theorem
$$H(M_{i^* | u^*} \mid R, M_{- i^*}, \mathbbm{1}_{M \in {\cal M}}=1) \cdot \Pr[\mathbbm{1}_{M \in {\cal M}}=1] \le \frac{64tB}{n}\lg n \cdot 1 \;.$$
Claim~\ref{clm:F2InitialInfo} implies that
$$H(M_{i^* | u^*} \mid R, M_{- i^*}, \mathbbm{1}_{M \in {\cal M}}=0) \cdot \Pr[\mathbbm{1}_{M \in {\cal M}}=0] \le n \cdot \frac{1}{2} \;.$$
Plugging the last two inequalities into \eqref{eq:totalEntropy} we get that
$H(M_{i^* | u^*} \mid R, M_{- i^*}, \mathbbm{1}_{M \in {\cal M}}) \le \frac{64tB}{n}\lg n + \frac{n}{2} \;.$
Plugging this into \eqref{eq:addInd}, \eqref{eq:chainRule} we get that $n - \frac{8r}{B} \le \frac{64tB}{n}\lg n + \frac{n}{2} + 1$.
Now for $r \ge n$ by substituting $B=\lfloor \frac{32r}{n}\rfloor$, we conclude that $\frac{2048tr}{n^2}\lg n \ge \frac{n}{4}$, and thus $tr \ge \Omega (n^3/\lg n)$.
For $r <n$ we use the following simple padding argument. Append $R$ with some arbitrary bits so that the size (no. of bits) of the new data structure $R'$ becomes $n$. Now from the previous argument it follows that $t \ge \Omega (n^2/\lg n)$, thus completing the proof.
\end{proof}
\paragraph*{Comment on query algorithms with error. }
In Theorem~\ref{thm:mainUMVF2} we consider query algorithms those always output the correct answer and provide lower bound. It is worth noting that our proof technique can be generalized to give lower bound for the case when the query algorithm may err with probability at most $1/64$ (though any small constant probability will work) on average over the choices of $M,u,v$. We need to modify the proof a bit by considering the event that the algorithm (say ${\cal A}$) errs, i.e., ${\cal A}(u,M,v) \ne u^{\dagger}Mv$. From $Pr_{u,M,v}[{\cal A}(u,M,v) \ne u^{\dagger}Mv]\le 1/64$, using Markov's inequality we can deduce that $Pr_u[Pr_{M,v}[{\cal A}(u,M,v) \ne u^{\dagger}Mv] \ge 1/16] \le 1/4$. Now we choose $u^*$ that satisfies Corollary~\ref{cor:uExsists} and $Pr_{M,v}[{\cal A}(u^*,M,v) \ne (u^*)^{\dagger}Mv] \le 1/16$. The existence of such a $u^*$ follows from simple union bound. Similarly $Pr_M[Pr_{v}[{\cal A}(u^*,M,v) \ne (u^*)^{\dagger}Mv] \ge 1/4] \le 1/4$. Now define ${\cal M} := \{M \in \mathbb{F}_2^{n \times n} : \mathbb{E}_v[t_{i^*}(u^*,M,v)] \le \frac{16tB}{n}\} $ and hence $Pr_M[M \in {\cal M} \text{ and }Pr_{v}[{\cal A}(u^*,M,v) \ne (u^*)^{\dagger}Mv] \le 1/4] \ge 1/2$. Next we modify Lemma~\ref{lem:encode} by saying that for any $M \in {\cal M}$,
$$\Pr_v [{\cal A}(u^*,M,v) = (u^*)^{\dagger}Mv\text{ and }t_{i^*}(u^*,M,v) \le \frac{64tB}{n}] \ge \frac{1}{2}.$$ The remaining argument will be the same and we will get similar lower bound. One can further extend this lower bound result to randomized query algorithms that given $u,v$ output correct answer with high probability, by using the technique described in Section~\ref{sec:randomizedQueryLB}.
\bibliographystyle{alphaurlinit}
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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\section{Introduction}
Thermally driven turbulent flows widely occur in geophysical flows and industrial processes. Examples are thermal convection in the atmospheric and mantle convection \cite{mck1974,Wyngaard1992}, in the ocean \cite{cheng2019fast}, and in many industrial processes \cite{Bejan}. Rayleigh--B\'enard convection (RBC), a fluid layer heated from below and cooled from above, is an ideal model for the study of thermally driven turbulent flows \cite{ahl09,loh10,chi12}.
The main challenge for thermal turbulence studies is to explore the flow dynamics and heat transfer of the system in a wide range of the control parameters. The main attention has been on how the heat transfer depends on the Rayleigh number, which is the dimensionless temperature difference and measures the intensity of the thermal driving of the system. The Rayleigh number is defined as $\text{Ra}=\frac{\beta g \Delta L^3 }{\nu \kappa}$, where $g$ is the gravitational acceleration, $\beta$ is the isobaric thermal expansion coefficient, $\nu$ the kinematic viscosity, $\kappa$ is the thermal diffusivity, $\Delta$ is the temperature difference between the hot plate and the cold plate, and $L$ is the thickness of the fluid layer between the aforementioned plates.
To achieve high Rayleigh numbers, numerous strategies have been proposed in the past decades. Looking at the definition of $\text{Ra}$, one can see that large $\text{Ra}$ can be reached in several ways.
One approach is to use a working fluid for which the parameter $\frac{\beta}{{\nu}\kappa}$ has a large value. This approach has been widely used in the community, such as using helium gas at cryogenic temperatures \cite{cas89,cha97,nie00,Urban2010} or mercury \cite{san89}. Recently, it was found that pure gas, particularly gasses with high molecular weight and under high pressure (up to 19-bar), is also an effective way to push $\text{Ra}$ to high values at a constant $\text{Pr}$ number \cite{fun09,ahl09b}. Another commonly used approach in the field of turbulent thermal convection is to increase the system thickness $L$ \cite{nie00,fun05,nik05,sun05e,pui07,fun09,ahl09b,Urban2010}.
The fact that hitherto not much attention was on the changing the gravitational acceleration provides the motivation of the present work. In this study, we propose a novel system, annular centrifugal RRC (ACRBC), which is a cylindrical annulus with cooled inner and heated outer walls under a solid body rotation.
In this system, the buoyancy force can be efficiently enhanced by replacing the gravitational acceleration ($g$) by the centrifugal acceleration, and consequently, $\text{Ra}$ can be increased by increasing the rotation rate for a given fluid and temperature difference.
Another advantage of this system is that the thermal convection in rapidly rotating cylindrical annulus has been recognized as a good model for the study of the flows in planetary cores and stellar interiors \cite{hid58,bus74,hid75,bus76,aue95,kan19} and also in engineering applications under strong rotation, \textit{i.e.}~flow in gas turbines \cite{mar05,mic15,pit17}. The flow dynamics in this system can give insights into flows in geophysical context and flows in industrial processes. In this work, we aim to study the heat transfer and the turbulent structures in ACRBC.
\section{Governing equations, simulations, and experiments}
\subsection{Governing equations and numerical simulations}
Using the parameter definitions as shown in Fig.~\ref{fig:setup}(b), the governing dimensionless Boussinesq equations in a rotating reference frame \cite{lop13} can be expressed as
\begin{small}
\begin{align}
\nabla \cdot {\bf u} &= 0 \\
\frac{\partial \theta}{\partial t} +{\bf u}\cdot \nabla \theta &= \frac{1}{\sqrt{RaPr}}\nabla^2 \theta \\
\frac{\partial \bf{u}}{\partial t} +{\bf u}\cdot \nabla {\bf u} &= -\nabla p+\text{Ro}^{-1} \hat{\bm{\omega}} \times {\bf u} \nonumber \\
&\phantom{=} +\sqrt{\frac{Pr}{Ra}}\nabla^2 {\bf u} -\theta \frac{2(1-\eta)}{(1+\eta)}\bf r
\end{align}
\end{small}
\noindent where $\bm{\hat{\omega}}$ is the unit vector pointing in the direction of the angular velocity, $\bf u$ is the velocity vector normalized by the free fall velocity $\sqrt{\omega ^2\frac{R_o+R_i}{2}\beta \Delta L}$, $t$ is the dimensionless time normalized by $\sqrt{L/(\omega ^2\frac{R_o+R_i}{2}\beta\Delta)}$, and $\theta$ is the temperature normalized by $\Delta = T_h - T_c$. Here, $\omega$, $R_i$, $R_o$, and $\Delta$ are the rotational speed, the thickness of the fluid layer between the two cylinders, the outer radius of the inner cylinder, the inner radius of the outer cylinder, and the temperature difference between the hot ($T_h$) and cold ($T_c$) cylinders, respectively.
From the above governing equations, the relevant control parameters in ACRBC are the Rayleigh number (characterizing the thermal driving strength)
$Ra=\frac{1}{2}\omega^2(R_o+R_i)\beta \Delta L^3/(\nu \kappa)$,
the inverse Rossby number (measuring Coriolis effects)
$Ro^{-1}=2(\beta \Delta (R_o+R_i)/(2L))^{-1/2}$,
the Prandtl number (fluid property) $\text{Pr}=\nu/\kappa$, and the radius and aspect ratio (geometric properties) $\eta=R_i/R_o$ and $\Gamma=H/L$. The key response parameter of the system is the Nusselt number (measuring the ratio of the total heat transport over the conductive one) $\text{Nu}=-Q\ln(\eta)/(\alpha \Delta 2 \pi H)$. Here, $Q$ is the measured heat input through the outer cylinder into the system per unit of time; $\alpha=\kappa\rho c_p$ is the thermal conductivity of the working fluid with $\rho$ and $c_p$ being the density and the specific heat capacity of the fluid, respectively; and $H$ is the height of the gap between two cylinders. It should be noted that the definition of $\text{Nu}$ in ACRBC is slightly different from that in classical RBC because of the cylindrical geometry with a heat flux in the radial direction, and the detailed derivations of the $\text{Nu}$ for ACRBC are documented in the Supplementary Materials.
Direct numerical simulations (DNS) are performed using an energy conserving second-order finite-difference code \cite{ver96,poe15cf,zhu18afid}. The code has been extensively validated and used in previous studies in both the Cartesian and cylindrical coordinates for convective systems \cite{poe15cf,zhu18afid,zhu18prl,zhu18np}. In all numerical simulations and experiments, the radius ratio is fixed at $\eta=0.5$. No-slip boundary conditions were used for the velocity and constant temperature boundary conditions for the inner and outer cylinders. Periodic boundary conditions were used for the top and bottom surfaces. For three-dimensional simulations, the aspect ratio was chosen the same as in the experiments $\Gamma=H/L=1$ (the only two exceptions are for the cases at $\text{Ra}=1.16\times10^9$ and $\text{Ra}=2.2\times10^9$, where the height of the domain is $L/4$ and $L/8$, respectively, as the flows here are quasi two-dimensional). For the cases of which the flows are quasi two-dimensional at large $\text{Ra}$, we use two-dimensional simulations. Pr was fixed at 4.3 for all the simulations. The adequate resolution was ensured for all cases, for example, at $\text{Ra}=4.7 \times 10^8$, $4608\times384\times384$ grid points were used and at $\text{Ra}=4.7 \times 10^{10}$ with grid points $18432\times1536$. To obtain sufficient statistics, each simulation was run at least 200 free fall time units. As reported in \citep{kun10,kun11}, the boundary layers in rotation system are expected to be thiner compared with classical RBC. Hence, the verification of boundary layer grid resolution is necessary. A posteriori check of grid resolution shows that at $\text{Ra}=4.7\times10^{10}$, there are 48 grid points inside thermal boundary layers and 64 grid points inside viscous boundary layers, which guarantees to resolve the boundary layer adequately. Besides, we have conducted a set of grid independence studies and checked the spatial resolution in bulk region to resolve all relevant scales (Kolmogorov scale and Batchelor scale). The parameter space that was explored can be found in Fig. \ref{fig:setup}(c). More details about the DNS are documented in the Supplementary Materials.
\begin{figure*}
\centering
\includegraphics[width=1\linewidth]{fig1_mod.pdf}
\caption{{\bf System configuration and parameter space.} (a) Three-dimensional render of the experimental set-up. A fluid is confined in a rotating cylindrical annulus having its inner(blue) and outer(red) cylindrical surfaces made out of copper, which is known for its excellent thermal conductivity. The bottom plate is made of teflon for thermal insulation, and top plate is made of plexiglass allowing flow visualization. The coolant pipes and electric cables go through the hollow stainless steel shaft and connect to a slip ring and a rotary union. The toothed pulley is driven by a servo motor. (b) Schematical diagram of the set-up, which defines the geometric parameters. (c) Explored parameter space of $\text{Ra}$ and $\text{Ro}^{-1}$ for our study of supergravitational thermal convection. The red discs, the blue circles and the blue triangles correspond to the parameters used in the experiments (EXP), in the three-dimensional (3D) numerical simulations and in the two-dimensional (2D) numerical simulations, respectively. The parameter space is divided into three regimes according to the influence of Coriolis force. In addition, for comparison we have also performed the simulations for $\text{Ro}^{-1}=10^{-5}$ (not shown here), for which the Coriolis force is negligible. For more details about the experimental apparatus, we refer to the Supplementary Materials and movie S1.}
\label{fig:setup}
\end{figure*}
\bigskip
\subsection{Experiments}
Experiments are performed in a cylindrical annulus with solid-body rotation as sketched in Fig.~\ref{fig:setup} (a). The two concentric cylinders are machined from a solid piece of copper to control the symmetry of the system, and their surfaces are electroplated with a thin layer of nickel to prevent oxidation. The inner cylinder has an outer radius of $R_i=\unit{120}{\milli\meter}$, and the outer cylinder has an inner radius of $R_o=\unit{240}{\milli\meter}$, resulting in a gap of L=$\unit{120}{\milli\meter}$, and a radius ratio of $\eta = R_i/R_o = 0.5$. The gap with a height $H=\unit{120}{\milli\meter}$ is sandwiched by a top plexiglass plate and a bottom teflon plate, resulting in an aspect ratio of $\Gamma = H/L = 1$. Plexiglass with an excellent transparency is used as the material of the top lid, allowing us to visualize the flow field, whereas teflon with an excellent corrosion resistance and a good strength is machined as the bottom base. These end plates and cylinders are fixed together and leveled on a rotating aluminum frame with a rotation rate up to \unit{546}{rpm}. Four silicone rubber film heaters are attached to the outside of the outer cylinder, and are supplied by a DC power supply (Ametek, \texttt{XG 300-5}) with 0.005\% long-term stability. We use water as our working fluid, which has $\text{Pr} \approx 4.3$.
Inside the cold inner cylinder, 16 channels are machined for the coolant fluid to pass through, and the inner cylinder is regulated at constant temperature using a temperature-controlled circulating bath (PolyScience, \texttt{AP45R-20-A12E}). The coolant pipes and electric cables go through the hollow stainless steel shaft and connect with a slip ring and a rotary union (Moflon, \texttt{MEPH200}), which has 2 channels for liquids, 6 channels for powers (\unit{2220}{\watt}), and 48 channels for electrical signals. The shaft is driven by a toothed belt and the pulley has a gear ratio of $60:32\approx1.88:1$. The whole system is powered by a servomotor (Yaskawa, \texttt{SGM7G-1AA}), which has a rated power of \unit{11}{\kilo\watt}.
For high-precision temperature and heat flux measurements, it is essential to minimize the heat leakage from the experimental apparatus to the surroundings. Various thermal shields that are regulated at appropriate temperatures are installed in the system to prevent heat losses. Furthermore, the whole system is placed in a big toughened glass box where the temperature is controlled by a Proportional-Integral-Derivative (PID) controller to match the mean temperature of the bulk fluid. For more detailed descriptions on the experimental set-up, we refer to Supplementary Materials. The measurement techniques are documented in Methods.
\begin{figure*} [ht]
\centering
\includegraphics[width=\linewidth]{fig2_mod.pdf}
\caption{{\bf Effects of the Coriolis force on the heat transfer and flow structures.} (a) Nusselt number ($\text{Nu}$) as a function of $\text{Ro}^{-1}$ for $\text{Ra}=10^6$, $2.2\times10^6$, $4.7\times10^6$, $10^7$, $2.2\times10^7$, $4.7\times10^7$, $10^8$, $2.2\times10^8$ and $4.7\times10^8$ for $\text{Pr} = 4.3$. (d) Root mean square axial velocity fluctuation $\langle (u_z)_{rms}\rangle _{r,\varphi, z}$ versus $Ro^{-1}$ for $\text{Ra}=10^7$ and $\text{Ra}=10^8$. (b,c,e,f) Instantaneous temperature fields from DNS at $\text{Ra}=10^8$ for $\text{Ro}^{-1}=0.1$, $0.5$, $1$, and $25$ ($\text{Pr} = 4.3$). The inner and outer surfaces locate 0.02 $L$ away from the cold and hot cylinder correspondingly.}
\label{fig:Ro}
\end{figure*}
\section {Results}
\subsection{Effects of the Coriolis force on the heat transport and flow structures}
A series of numerical simulations are performed to explore the influence of the Coriolis force on the heat transfer and flow structures. In numerical experiments, $\text{Ro}^{-1}$ varies from $10^{-2}$ to $4\times 10^2$ with several fixed Rayleigh numbers $\text{Ra}=10^{6}$, $2.2\times10^6$, $4.7\times10^6$, $10^7$, $2.2\times10^7$, $4.7\times10^7$, $10^8$, $2.2\times10^8$, and $4.7\times10^8$. As shown in Fig.~\ref{fig:Ro} (a), for these $\text{Ra}$ studied, the influence of Coriolis force can be divided into three regimes. In regime \uppercase\expandafter{\romannumeral1} ($\text{Ro}^{-1}<\text{Ro}^{-1}_{c1}$), the Coriolis force is small and negligible as compared to the buoyancy so that the $\text{Nu}$ does not depend on $\text{Ro}^{-1}$. In regime \uppercase\expandafter{\romannumeral3} ($\text{Ro}^{-1}>\text{Ro}^{-1}_{c2}$), the rotation is so strong that the flow is nearly constrained by the effect of the Taylor-Proudman theorem~\cite{pro16,tay22}, resulting in a quasi-two-dimensional flow state and thus a reduction of $\text{Nu}$ compared with regime \uppercase\expandafter{\romannumeral1}. For regime \uppercase\expandafter{\romannumeral2} ($\text{Ro}^{-1}_{c1}<\text{Ro}^{-1}<\text{Ro}^{-1}_{c2}$), the flow is governed by the combination of $\text{Ra}$ and $\text{Ro}^{-1}$ with rich flow states. In this regime, the general trend is that Nu decreases with $\text{Ro}^{-1}$. The instantaneous temperature fields for several $\text{Ro}^{-1}$ at $\text{Ra}=10^8$ (see Fig.~\ref{fig:Ro}(b,c,e,f)) from DNS support the explanation of this $\text{Ro}$ dependence on the heat transport. As illustrated in Fig.~\ref{fig:Ro}(b), the flow is three-dimensional at $\text{Ro}^{-1}=0.1$ because the Coriolis force is too small to influence the flow effectively. However, with $\text{Ro}^{-1}$ increasing, the enhanced Coriolis force tends to suppress vertical variation of the convection flow, and the flow gradually becomes two-dimensional, which is a manifestation of the Taylor-Proudman theorem. The two-dimensionalization of the flow field should be responsible for the reduction of heat transport at $\text{Ro}^{-1} > \text{Ro}^{-1}_{c1}$, which was also reported in \cite{poe13}. As is evident in Fig.~\ref{fig:Ro} (f), the flow is nearly two-dimensional at $\text{Ro}^{-1} \gtrsim \text{Ro}^{-1}_{c2}$. So, further increased $\text{Ro}^{-1}$ saturates the influence of the Coriolis force, which explains the nearly constant $\text{Nu}$ when $\text{Ro}^{-1} \gtrsim \text{Ro}^{-1}_{c2}$. Moreover, we use root mean square axial velocity fluctuation $\langle (u_z)_{rms}\rangle _{r,\varphi, z}$ to measure the influence of the Taylor-Proudman theorem, as shown in Fig.~\ref{fig:Ro}(d). It shows that $\langle (u_z)_{rms}\rangle _{r,\varphi, z}$ has nearly a constant value when $\text{Ro}^{-1}<\text{Ro}^{-1}_{c1}$, then gradually decreases with $\text{Ro}^{-1}$, and finally approaches to around zero after $\text{Ro}^{-1}>\text{Ro}^{-1}_{c2}$. The overall trend of $\langle (u_z)_{rms}\rangle _{r,\varphi, z}$ versus $\text{Ro}^{-1}$ is consistent with the dependence of $\text{Nu}$ on $\text{Ro}^{-1}$ in general.
We have noticed that the critical $\text{Ro}^{-1}_{c1}$ and $\text{Ro}^{-1}_{c2}$ depend on the $\text{Ra}$. As illustrated in Fig.~\ref{fig:Ro} (d), for $\text{Ra}=10^7$, we have $\text{Ro}^{-1}_{c1}\approx0.1$ and $\text{Ro}^{-1}_{c2}\approx10$, while for $\text{Ra}=10^8$, we have $\text{Ro}^{-1}_{c1}\approx0.15$ and $\text{Ro}^{-1}_{c2}\approx15$. According to the numerical data explored, we can roughly determine the boundaries of the three regimes, which is plotted in Fig.~\ref{fig:Ro} (a) with green lines. In addition, we have also included the boundaries in Fig.~\ref{fig:setup} (c), and extrapolated the boundaries to the parameter space of experiments. (More detailed discussions of the flow structures, $\langle (u_z)_{rms}\rangle _{r,\varphi, z}$, and $\text{Ro}^{-1}_{c}$ for these $\text{Ra}$ cases are available in Supplementary Materials.)
\begin{figure*} [ht]
\centering
\includegraphics[width=\linewidth]{fig3_mod.pdf}
\caption{{\bf Global heat transport.} (a) Nusselt number ($\text{Nu}$) as a function of $\text{Ra}$ from experiments (the solid symbols), DNS (the open symbols) in ACRBC and the prediction from Grossmann-Lohse (G--L) theory\cite{gro00} in classical RBC (dashed line). (b) The same plots as (a), but the vertical axis is compensated. Inset: an enlarged portion of the compensated plot at the large $\text{Ra}$ regime, which shows the transition of the effective scaling exponent ($\text{Nu} \propto \text{Ta}^{\gamma}$) to $\gamma > 1/3$.}
\label{fig:heat}
\end{figure*}
\subsection{Global turbulent heat transport}
As shown in the section above, $\text{Nu}$ does not depend much on $\text{Ro}^{-1}$ for high $Ro^{-1}$ regime, in which we will study how heat transfer depends on $\text{Ra}$ in this section. We have performed 48 experiments and more than 130 numerical simulations. Fig.~\ref{fig:setup}(c) shows the parameter space of the experimental and numerical studies. In the experimental studies, we note that the existence of Earth's gravity and lids is unavoidable. Several numerical simulations have been performed to study the influences of Earth's gravity and lids, which show that their effects on $\text{Nu}$ are small.
In addition, we note that the centrifugal force increases linearly with radial location. To study the influence of nonuniform driving force, we have performed two sets of simulations with the radial-dependent centrifugal acceleration $\omega^2 r$ and with a constant artificial acceleration $\omega^2 (R_o+R_i)/2$. The results show that the effect of the radial-dependent gravity has a similar role as the non-Oberbeck-Boussinesq effect \citep{gro00,ahl06,sug09}, which does not have much influence on the heat transport and flow structures in the current parameter regime (see the Supplementary Materials for details).
Fig.~\ref{fig:heat} (a) shows the measured $\text{Nu}$ as a function of $\text{Ra}$ for different $\text{Ro}$. Each measurement lasts at least 4 hours after the system has reached a thermally steady situation to ensure a statistically stationary state (for the detailed measurement procedure, see the Supplementary Materials). As being evident in Fig.~\ref{fig:heat}(a), the experiments and numerical simulations are in excellent agreement. Combining experiments and simulations, this study covers more than four decades of $\text{Ra}$, \textit{i.e.} from $10^6$ to $6.5\times 10^{10}$. The range of $\text{Ro}^{-1}$ is from 18 to 58 in experiments, and the data series show a consistent dependence of $\text{Nu}$ on $\text{Ra}$. For comparison, we also plot the data calculated from Grossmann--Lohse (G--L) theory \cite{gro00} in the corresponding $\text{Ra}$ range.
To better reveal the local exponent, we plot the compensated data in Fig.~\ref{fig:heat}(b). The experimental and numerical data have a lower amplitude as compared to the classical RBC (G--L line) due to the different flow geometry. However, the scaling dependence of $\text{Nu}$ versus $\text{Ra}$ shows a good agreement between the data and G-L theory at $\text{Ra} \lessapprox 10^{10}$ with a scaling exponent $\gamma=0.27\pm0.01$, which is close to the typical value found in two-dimensional RBC.
\begin{figure*}[ht]
\centering
\includegraphics[width=\linewidth]{fig4_mod.pdf}
\caption{{\bf Revolution of convective rolls.} (a, b) Experiments at $\text{Ra}=1.8\times 10^9$ and $\text{Ro}^{-1}=13.8$. (a) Snapshots of streak images revealing the flow patterns. Seeing from the top, the whole system rotates clockwise in experiments. The corresponding movie is available in movie S2. (b) Time series of local temperature fluctuations. The measured point locates \unit{30}{\milli\meter} away from the cold inner cylinder and at mid-height. (c, d) Simulations at $\text{Ra}=10^7$, and $\text{Ro}^{-1}=1$: (c) Snapshots (Top view) of instantaneous temperature fields. (d) The averaged azimuthal velocity profile along the radial direction. The reference of the frame is on the clockwise direction, and the corresponding movie is available in the Supplementary Movie.}
\label{fig:flowpattern}
\end{figure*}
It is very unexpected that the local effective exponent $\gamma$ of $\text{Nu} \propto \text{Ra}^{\gamma}$ even exceeds one-third at $\text{Ra} \gtrsim 4\times 10^{10} $ in experiments.
Is this scaling regime connected to the appearance of ultimate regime of Rayleigh--B\'enard turbulence \cite{kra62,gro11,he12a}? If it is not in the ultimate regime, what causes this steep scaling exponent?
As our current $\text{Ra}$ is much lower than the transition $\text{Ra}$ for ultimate turbulence observed in the classical RBC, we cannot provide a concrete answer on this question. The possible answers are that (i) because of the different way of the driving and the geometry, the transition Ra to the ultimate regime at the current system is lower than that in the classical RBC. (ii) Another possibility is that the enhanced local slope is due to the emergence of new flow states. As demonstrated in Fig.~\ref{fig:setup} (c), with $\text{Ra}$ increasing, the location of ($\text{Ra},\, \text{Ro}^{-1}$) where the enhanced local effective scaling appears seems to move from regime III to regime II. Thus the flow may change from a two-dimensional flow state to a three-dimensional flow state due to the strong thermal driving, resulting in a higher $\text{Nu}$ in this $\text{Ra}$ range.
The two-dimensional simulations for high $\text{Ra}$ coincide well with the experiments, indicating that within the $\text{Ra}$ range explored in two-dimensional simulations the flow is still quasi-two-dimensional. Unfortunately, we cannot push the simulations to the Ra regime with the effective exponent larger than one-third, in which the flow may not be in the quasi-two-dimensional state. We incline to anticipate that the changing from a two-dimensional flow state to a three-dimensional flow state may be responsible for the enhancement of local effective exponent at $Ra \gtrsim 4\times10^{10}$.
We notice that the scaling range with $\gamma > 1/3$ is very narrow in the current work, therefore further work at even higher Ra is needed to clarify this issue.
\subsection{Zonal flow}
We now study the dynamics of zonal flow experimentally and numerically. The "zonal flow" phenomenon has been investigated in experiments of geophysical and astrophysical flows \cite{hei05,har15}. Fig.~\ref{fig:flowpattern}(b) shows the time series of local temperature fluctuations in water at $\text{Ra}=6.6\times 10^9$, $\text{Pr}=4.3$, and $\text{Ro}^{-1}=18$, which unexpectedly shows a noticeable periodicity. To prove that this novel phenomenon is connected to the azimuthal movement of the coherent flow, we perform flow visualization with aqueous glycerol solution as mentioned in Methods to demonstrate the flow pattern.
Fig.~\ref{fig:flowpattern}(a) shows some typical streak images from flow visualization at three different time (for video, see movie S2), where there are four pairs of rolls parallel to the rotating axis. For reference, we highlight one of the convective rolls using a yellow ellipse. From Fig.~\ref{fig:flowpattern} (aI--aIII), unexpectedly the convection rolls move clockwise around the center with a faster rotation rate than the background rotation of the experimental system, which we name "zonal flow". The flow visualization further justifies that the revolution of the cold plume arms triggers the periodic temperature signals measured by the thermistors.
Next to the experiments, we also find this phenomenon in the simulations (see Fig.~\ref{fig:flowpattern}(c) and movie S3).
We use a black ellipse to highlight a selected cold plume. As shown in Fig.~\ref{fig:flowpattern}(c),
on the reference of the (clockwise) rotating frame, the plume arms still evolve in a clockwise direction, indicating a net rotation of the coherent flow (zonal flow).
This zonal flow is further quantified through the averaged (axially and azimuthally in space and over time) azimuthal velocity profile versus the radial distance from the inner cylinder wall $r$, as shown in Fig.~\ref{fig:flowpattern}(d).
What causes this net rotation of the convection rolls?
The dashed lines in Fig. \ref{fig:flowpattern}(c\uppercase\expandafter{\romannumeral1}) mark the direction of hot/cold plumes without Coriolis force. But because of the effects of Coriolis force, the plumes deflect to their left when the system rotates clockwise. As is evident in the Fig. \ref{fig:flowpattern}(c\uppercase\expandafter{\romannumeral1}) where the plumes are just formed in the beginning stage, the deflected angle of hot plumes ($\angle a$) is approximately equal to the deflected angle of cold plumes ($\angle b$). However, because of the different curvature of the cylinders, the similar deflected angles of the hot and cold plumes induce the different effects. The hot plumes directly affect the position (A) where the cold plumes are ejected, resulting in the clockwise rotation of the cold plumes and flow, whereas the impact of the cold plumes does not directly affect the motion of the hot plumes due to a relatively large distance the impact position of the cold plumes (B) and the ejecting position of the hot plumes. Thus, the hot plumes win and push the overall flow to move in the same direction of the background rotation.
We also perform the experiments and simulations with the opposite direction of the background rotation, and the results are consistent. In addition, several numerical simulations have been performed to test the dependence of zonal flow on the radius ratio $\eta$, which shows that the zonal flows become weaker and weaker with $\eta$ increasing from 0.4 to 0.9. The difference in the curvature of the two cylinders plays the key role for the net rotation of convection rolls. One may expect to observe different types of zonal flows at different $\eta$, such as two large scale winds with the opposite directions near the plates \cite{har15}. This remarkable net rotation of the coherent flow structures along the same direction of the background rotation could be connected to many relevant flow phenomena in nature, which deserves systematical studies in the future.
\section{Summary}
In summary, we have experimentally and numerically studied the global heat transport and turbulent flow structures in a rotating annulus with hot outer cylinder and cold inner cylinder, \textit{i.e.}, ACRBC. In experiments, the mean centrifugal acceleration covers the range from 5 to 60 times gravitational acceleration. We show that the effective scaling exponent of global heat transport transitions to $\gamma=1/3$ at $\text{Ra} \approx 10^{10}$ and finally exceeds $\gamma > 1/3$ for $\text{Ra} \gtrapprox 4\times 10^{10}$.
Unexpectedly, we observed the faster azimuthal motion of the coherent overall flow structure faster than the background rotation of the system, which we call zonal flow, and provided a physical understanding on this remarkable net rotation motion of the coherent structures. This novel experimental approach sheds new lights on how to efficiently extend the parameter regimes for the study of buoyancy driven turbulent flows. Furthermore, the findings in the current system driven by the centrifugal acceleration can help us understand phenomena in geophysical and astrophysical flows.
\section{Methods}
{\it Measurement techniques.} In our experiments, all the temperature measurements are based on negative temperature coefficient thermistors. We use $6\frac{1}{2}$-digit multimeter (Keithley, \texttt{2701}) to measure the resistances of the thermistors, and the resistance can be converted to temperature using the Steinhart-Hart equation\cite{lav97}. Two types of thermistors are used: One with a head diameter of \unit{2.5}{\milli\meter} (Omega, \texttt{44131}) is used to measure the temperature of the inner and outer cylinders, and the other with a head diameter of \unit{300}{\micro\meter} (Measurement Specialties, \texttt{GAG22K7MCD419}) is inserted into the convection cell to resolve fast temperature fluctuations, which is located \unit{30}{\milli\meter} away from the cold cylinder and at mid-height in the vertical direction.
To visualize the flow field, we use nylon fibers with length $l = \unit{3}{\milli\meter}$, and diameter $d = \unit{0.5}{\milli\meter}$. Nylon has a density of $\rho_{p}\approx \unit{1.14}{\gram\per\centi\meter\cubed}$ that is a little heavier than water. In the flow visualization experiments, we use aqueous glycerol solution with 54\% glycerol by volume to match the density of nylon. The properties of the aqueous glycerol solution are listed in the Supplementary Materials for reference. Four light-emitting diods are used as light source, and a charge-coupled device camera (Ximea, \texttt{MD028MU-SY}) is put on the top of the cell to record images. As the system rotates rapidly, it is difficult to make the camera rotate synchronously. So we keep the camera fixed, and then set the frame rate of the camera the same as the rotational speed. Processing the images taken by camera through a MATLAB code, we can get streak images to visualize the flow field. More processing details can be found in the Supplementary Materials.
\bigskip
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,060
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\section{Introduction}
\label{sec:intro}
\subsection{A mean-field model for clustering}
The clustering processes we consider are motivated by a simplified model
for domain wall motion in the one-dimensional Allen-Cahn equation
$\D_tu=\D_{xx}u+u-u^3$.
The domain walls become points on the line, and the domains are the intervals
separated by these points. The pattern coarsens by a simple rule:
At each step, the smallest domain combines with its two neighbors to
form a single domain, and this is repeated indefinitely.
Computational simulations of this
`min-driven' domain coarsening process indicate that for a considerable
variety of initial distributions, the domain size distribution
approaches self-similar form \cite{CP,NK}.
A mean-field model of this process was derived by Nagai and Kawasaki \cite{NK},
and it turns out to be amenable to a rigorous analysis aimed at explaining this
behavior \cite{CP,GM}.
We consider an infinite number of domains on the line, and
study the statistics of domain sizes using a number density
function $f(t,x)$. We assume that in any interval $I$ of unit length,
the expected number of domains with lengths in the range $(x,x+dx)$ is
given by $f(t,x)\, dx$. The expected value of the total number of
domains in $I$ is denoted $N(t)= \int_0^\infty f(t,x) \, dx$. We
assume that $N(t)$ is finite,
and denote the associated probability density by
\be
\label{defrho}
\rho_t(x) = \frac{f(t,x)}{N(t)}.
\ee
We let $\len(t)$ denote the size of the smallest domain at time $t$,
so $f(t,x)=0$ for $x<\len(t)$.
The expected number of coalescence events per unit time is then
\be
\label{numevents}
f(t,\len) \dot{\len}.
\ee
The coalescence events affecting domains of size $x$ in the
time interval $(t,t+dt)$ are
(a) loss: consecutive domains of size $x,\len,y$ (or $y,\len,x$)
combine to form a domain of size $x+\len+y$; (b) gain: domains of size
$y,\len,x-y-\len$ combine to form a domain of size $x$. Under the
mean-field assumption that coalescing domains have sizes
chosen randomly and independently from the current overall size
distribution, these events have respective relative probability density
\[ \rho_t(x) \rho_t(y) , \quad \rho_t(y) \rho_t(x) , \quad \rho_t(y)
\rho_t(x-y-\len).\]
The rate equation for the evolution of $f$ is obtained by summing over
all loss and gain terms:
\ba
\label{evol1}
\lefteqn{\partial_t f(t,x) =}\\
\nn
&& f(t,\len) \dot{\len} \left( \int_{\len}^{x-\len} \rho_t(y)
\rho_t(x-y-\len)\, dy - 2
\rho_t(x) \int_{\len}^\infty \rho_t(y)\, dy \right), \quad x >\len.
\ea
Clustering phenomena are seen in fields as varied as population
genetics and physical chemistry. Thus, while coarsening of intervals
provides concrete motivation, the notion of a `cluster' can have
widely different interpretations in applications.
The model \qref{evol1} above is one of a family of
what we call {\em min-driven clustering} models
that can be analyzed together in one setting as in \cite{GM}.
At each step a random integer $k \geq 1$ is chosen with
probability $p_k$, and the smallest cluster merges with $k$ randomly
chosen clusters. The mean-field assumption is that all these random
variables are independent. The only assumptions we impose on the probabilities
are that
\be
\label{prob}
p_k \geq 0, \quad \sum_{k=1}^\infty p_k =1, \quad \sum_{k=1}^\infty
k p_k < \infty.
\ee
The evolution of the number density under this process is described by
the following rate equation for cluster size density, obtained
by summing over all loss and gain events as in \qref{evol1}:
\be
\label{evol2a}
\partial_t f(t,x) = f(t,l) \dot{l} \sum_{k=1}^\infty p_k \left(
\rho_t^{\star k} (x-l) - k\rho_t(x)\right), \quad x > \len(t).
\ee
Here the notation $\rho_t^{\star k}$ denotes $k$-fold
self-convolution. The case $p_1=1$ corresponds to a ``paste-all'' model
discussed by Derrida et al.\ \cite{Derrida}. Also see \cite{IKR} for a
model of social conflict with rather similar solution formulas.
An important feature of the model \qref{evol2a} is its invariance under
reparame\-trization in time. If we change variables via $t=
T(\tilde{t})$, $\tilde{f}(x,\tilde{t})=f(x,t)$,
$\tilde{\len}({\tilde{t}})=\len(t)$, then equation \qref{moment1}
retains its form since
\[
\partial_{\tilde{t}}\tilde{f}(x,\tilde{t}) =
\dot{T} \partial_t f, \quad \partial_{\tilde{t}}\tilde{\len}
= \dot{T} \partial_{t}\len.
\]
A careful choice of the time scale is key to the analysis. In the
first mathematical study of \qref{evol1}~\cite{CP}, the authors imposed the
relation $f(t,\len) \dot{\len}=1$, meaning the number of coalescence
events per unit time is constant~\cite{CP}. In a more recent paper,
Gallay and Mielke parametrized time by the minimum size, that is
$\len(t)=t$~\cite{GM}. This leads to an elegant solution procedure
that was used to prove some basic results on well-posedness and
the approach to self-similarity. Gallay and Mielke showed that
\qref{evol2a} defines a strongly continuous flow in $L^1$, that
\qref{evol2a} admits a one-parameter family of self-similar solutions
and that suitable initial densities yield convergence to self-similar
form. Precise comparisons between these results and ours are made
later in this paper.
In this article, we introduce yet another time scale. We
parametrize time inversely to the total number of domains, so that
\be
\label{intime}
t= N(t)^{-1}.
\ee
We shall argue that this is a natural choice for a number of
reasons. It retains the simplicity of the choice $\len(t)=t$, and
allows us to obtain: (a) existence and uniqueness for measure-valued solutions;
(b) necessary and sufficient conditions for convergence to self-similar
form; (c) a characterization of eternal solutions for
the dynamical system defined by \qref{evol1}. We comment on these in
greater depth below.
Let us first explain one simple motivation for \qref{intime}.
We show below that for \qref{evol1}, $f(t,\len)\dot{\len}=-\dot{N}/2$.
Thus, $f(t,\len)\dot{\len}=N^2/2$ when \qref{intime}
holds, and \qref{evol1} now takes the form
\ba
\label{evolsmol}
\lefteqn{\partial_t f(t,x) =}\\
\nn
&& \frac{1}{2}\int_{\len}^{x-\len} f(t,y) f(t,x-y-\len)\, dy -
f(t,x) \int_\len^\infty f(t,y)\, dy, \quad x >\len.
\ea
If we take $\len(t)\equiv 0$ (as a model when the smallest domains
have negligible size, for example), this reduces to a basic solvable model of
clustering: Smoluchowski's coagulation equation with constant kernel.
In recent work, we provided a comprehensive analysis of dynamic
scaling in this equation by exploiting an analogy with the classical
limit theorems of probability theory~\cite{MP1,MP3}. We use these insights
to guide our study of \qref{evol2a}.
\subsection{Measure-valued solutions}
Mean-field models of domain coarsening, such as the LSW model, Smoluchowski's
coagulation equation, and \qref{evol2a}, correspond to physical
processes where mass is transported from small to large scales.
For several reasons, it is natural to consider measure-valued solutions,
for which the size distribution need not have a continuous or
integrable density. Such solutions are physically meaningful, as many
clustering processes (e.g., polymerization) involve
a discrete set of sizes based on an elementary unit.
A formulation via measures is also mathematically elegant, as it allows us
to unify the treatment of discrete and continuous coagulation
models, exploit simple criteria for compactness and continuity, and
prove basic uniform estimates.
To any solution of \qref{evol2a} we associate a probability
measure $F_t$ with distribution function written
\begin{equation}
F_t(x) = \frac1{N(t)}\int_0^x f(t,y)\,dy = \int_0^x \rho_t(y)\,dy.
\end{equation}
For any probability distribution $F$ on $[0,\infty)$, we call
$\len=\inf\{x| F(x)>0\} $ the {\em min} of $F$. (This is short for ``minimum
size,'' regarding $F$ as a probability distribution for size.)
We will prove that the initial-value problem for an appropriate weak
form of \qref{evol2a} is well-posed for initial probability measures
$F_{t_0}$ with positive min. That is, \qref{evol2a} with
\qref{intime} determines a
continuous dynamical system on the space of probability measures with
positive min, equipped with the weak topology.
See Theorem~\ref{thm.wp} below. By comparison, Gallay and Mielke
established that the initial-value for \qref{evol2a} defines a
continuous dynamical system on the space of probability {\em
densities\/} in $L^1(1,\infty)$ equipped with the strong
topology~\cite[Thm. 3.3]{GM}.
The solutions we construct arise by a natural completion of
these $L^1$ dynamics.
\subsection{Dynamic scaling}
A common theme in recent studies of dynamic scaling in
mean-field models of
coarsening is that the approach to self-similarity is both degenerate
and delicate. The problem is degenerate because there is a one-parameter
family of self-similar solutions. For the model studied here with
$p_k=0$ for all $k$ large enough, Gallay
and Mielke found a family of self-similar solutions that may be
rewritten in the time scale \qref{intime} in the form
\be
\label{ftheta1}
F_t(x) = F\thetasup\left(\frac{x}{l\thetasup(t)}\right), \quad l\thetasup(t) =
t^{1/\theta}, \quad \theta \in (0,1], \quad t >0.
\ee
Here $F\thetasup$ is a probability distribution with density
$\rho\thetasup$ supported on $[1,\infty)$.
The density $\rho\thetasup$ is known explicitly only through its
Laplace transform. Only $\rho^{(1)}$ has finite
mass (first moment); $\rho^{(1)}(x)$ decays exponentially as
$x\to\infty$.
The distributions $F\thetasup$ for $0<\theta<1$
have heavy tails, with
$\rho\thetasup(x) \sim c_\theta x^{-(1+\theta)}$ as $x\to\infty$
(see Theorem~\ref{t.ss}).
The problem is delicate because the domains of
attraction of the self-similar solutions are determined by the tails
of the initial size distribution, in the precise manner explained below.
(See~\cite{MP1,NP1} for analogous results on the LSW model of Ostwald
ripening and Smoluchowski's coagulation equations with solvable kernels.)
Gallay and Mielke showed that all densities with
finite mass are attracted to the self-similar solution with
$\theta=1$. Moreover, for $0<\theta \leq 1$ they showed that if the
initial data $\rho_{t_0}$ is sufficiently close to $\rho\thetasup$ in
a suitable weighted norm then the rescaled probability density $t\rho_{t}(tx)$
approaches $\rho\thetasup$ with a rate of convergence determined by
the weighted norm (see~\cite[Thms. 5.5, 5.7]{GM}). These results provide
sufficient conditions for approach to self-similarity.
Our aim is to establish conditions that are both necessary and
sufficient to answer a more general question about
{\em arbitrary\/} scaling limits.
We characterize the set of all non-degenerate limits under a
general rescaling of the form $F_t(\lambda(t) x)$ where
$\lambda(t)$ is a measurable, positive function such that $\lim_{t \to \infty}
\lambda(t)=\infty$. A limit is non-degenerate
if it is suitable data for the initial-value problem. That is,
non-degenerate limits are probability distributions with a positive min.
\begin{thm}
\label{thm.domains}
Let $t_0>0$, let $F_{t_0}$ be an arbitrary probability measure on
$(0,\infty)$ with positive min,
and let $F_t$ ($t\ge t_0$)
be the associated measure-valued solution of \qref{evol2a}
(see Theorem~\ref{thm.wp}).
\begin{enumerate}
\item[(i)] Suppose there is a measurable, positive function
$\lambda(t)\to\infty$ as $t\to\infty$
and a probability measure $F_*$ with positive min, such that
\be
\label{conv1}
\lim_{t \to \infty} F_t(\lambda(t)x) = F_*(x)
\ee
at all points of continuity of $F_*$. Then there exists $\theta \in
(0,1]$ and a function $L$ slowly varying at infinity such
that the initial data $F_{t_0}$ satisfies
\be
\label{conv2}
\int_0^x y F_{t_0}(dy) \sim x^{1-\theta} L(x), \quad
\mathrm{as}\quad x \to \infty.
\ee
Moreover, the min $\len(t)$ of $F_t$ and the rescaling $\lambda(t)$
satisfy
\be
\label{conv3}
\lambda(t)l_* \sim \len(t) \sim t^{1/\theta} \tilde{L}(t),\quad
\mathrm{as}\quad t \to \infty,
\ee
where $l_*>0$ is the min of $F_*$
and $\tilde{L}$ is slowly varying at infinity, related to $L$ by
\qref{e.len}.
\item[(ii)] Conversely, assume there exists $\theta \in (0,1]$ and a
function $L$ slowly varying at infinity such that the initial data
satisfies \qref{conv2}. Then $\len(t)$ satisfies
\qref{conv3}, and
\be
\label{conv4}
\lim_{t \to \infty} F_t(\len(t) x) = F\thetasup(x), \quad x \in (0,\infty).
\ee
\end{enumerate}
\end{thm}
A positive function $L$ is slowly varying at infinity if it is
asymptotically flat under rescaling in the sense that $\lim_{x \to
\infty}L(xy)/L(x)=1$ for every $y>0$. For example, all powers and
iterates of the logarithm are slowly varying at infinity. These are
the class of admissible corrections to the power law
$x^{1-\theta}$.
Part (i) of the theorem is an assertion of rigidity of
scaling limits. We assume only that $\lambda(t)$ is measurable, positive and
$\lim_{t \to \infty} \lambda(t) =\infty$. It then follows that the
limits must define self-similar solutions, and $\lambda(t)$ must be
the time scale associated to the self-similar solution, up to a slowly
varying correction. Part (ii) of the theorem, and the sufficient
conditions of Gallay and Mielke, show that the domains of attraction
are determined by the tails of the initial data. As a consequence of
part (i) of the theorem, the condition \qref{conv2} is optimal.
Gallay and Mielke \cite{GM} used a rather delicate Fourier analysis
to establish the existence for self-similar solutions by studying
their densities.
We will use the proof of part (ii) of the theorem above to simplify
much of this analysis and extend it to the case when $p_k\ne0$ for
infinitely many $k$. In this we are motivated by a certain resemblance
of the min-driven model to hydrodynamic limits of
what are called {$\Lambda$-coalescents} in probability theory
\cite{BlG}, which are
clustering processes involving arbitrarily many multiple collisions.
We find a curious fact, namely, when $\sum p_k k\log k=\infty$ there
is {\em no} self-similar solution with finite mass (first moment).
Still, the theorem above correctly describes the domains of
attraction. Solutions with finite mass approach the self-similar
solution with $\theta=1$, but this self-similar solution has infinite
mass.
\subsection{Eternal solutions}
Theorem~\ref{thm.domains} is a particular example of the principle
that the asymptotic behavior under rescaling is determined by the tail of the
initial distribution. The dynamics exhibit sensitive dependence
on initial conditions, as arbitrarily small changes in the tail of the
initial data can lead to widely divergent asymptotic behavior. This
indicates a kind of chaos, and it is of interest to find a precise
formulation. A comprehensive analysis of such phenomena for the
solvable cases of Smoluchowski's coagulation equation appears
in~\cite{MP3}. This analysis is guided by an analogy with the
probabilistic notion of {\em infinite divisibility\/}. Let us first
describe these results informally.
Clustering is an irreversible process, and in general we do not expect
to be able to solve \qref{evol1} backwards in time (for $t<t_0$). However, the
self-similar solutions have the remarkable feature that they are
defined for all $t >0$. That is, they are {\em divisible\/} under
the coalescence process. We call a solution {\em eternal\/} if it is
defined on the maximal interval $(0,\infty)$ consistent with
\qref{intime}. In probability theory,
the infinitely divisible distributions are characterized by the
celebrated \LK\/ formula. In~\cite{MP3} we extended a result of
Bertoin~\cite{B_eternal} showing that the class of eternal solutions
to Smoluchowski's coagulation equations is also characterized by a
\LK\/ formula. Heuristically, this formula describes the emergence of
eternal solutions from infinitesimally small clusters at $t=0$.
We also showed that the set of all subsequential limits---the
scaling attractor---is in a one to one correspondence with the
eternal solutions. A rigorous description of chaos is based on the
fact that nonlinear dynamics on the scaling attractor is reduced to
linear scaling dynamics using the \LK\/ formula.
In this article, we take the first step towards establishing a similar
picture for min-driven clustering. Namely, we prove a \LK\/
formula characterizing all eternal solutions for min driven clustering
(Theorem~\ref{thm:LK}). The choice of time scale
\qref{intime} is very convenient for this analysis.
\subsection{Outline}
The rest of the article is organized as follows. We describe
the solution procedure of Gallay and Mielke in the next section and
discuss how the number-driven time scale is motivated by
the important example of initial data that are monodisperse (a Dirac delta).
The treatment here is formal. We establish
some analytic prerequisites in Section~\ref{sec:prelim}. This is
followed by rigorous results: the proof of well-posedness for
measure-valued solutions (see Theorem~\ref{thm.wp}) in Section~\ref{sec.ds},
the study of self-similar profiles in Section~\ref{sec.ss}, the
characterization of domains of attraction in Section~\ref{sec.rv},
and the characterization of eternal solutions in Section~\ref{sec:lk}.
\section{The solution formula for min-driven clustering}
\label{sec.sol}
\subsection{The generating function and moment identities}
As in the theory of branching processes, it is convenient to keep
track of the clustering process with a generating function
\be
\label{genfn}
Q(z) = \sum_{k=1}^\infty p_k z^k.
\ee
For example, binary clustering as in the Allen-Cahn model corresponds
to $Q(z)=z^2$. The generating function $Q$ is analytic in the unit
disk $\{|z| <1\}$, and absolutely monotone (that is, $Q$ and all its
derivatives are positive on $[0,1)$).
We assume that the expected number of clusters in the mergers, denoted $Q_1$,
is finite. That is,
\be
\label{genfn2}
Q_1= Q'(1)= \sum_{k=1}^\infty k p_k < \infty.
\ee
We define a convolution operator $\Q(\rho) = \sum_{k=1}^\infty p_k
\rho_t^{\star k}$ associated to $Q$, and rewrite
\qref{evol2a} in the form
\be
\label{evol2}
\partial_t f(t,x)= f(t,l) \dot{l}\left( \Q(\rho_t)(x-\len) - Q_1 \rho_t(x)
\right), \quad x > \len(t).
\ee
We extend the evolution equation \qref{evol2} from densities to
measures as follows. We consider a number measure $\nu_t$ and a probability
measure $F_t$ that are related to the densities (when they exist)
by
\begin{equation}\label{d.nuF}
\nu_t(dx) =f(t,x) \, dx, \qquad
F_t(dx) = \frac{\nu_t(dx)}{N(t)} = \rho_t(x)\,dx .
\end{equation}
Let $\Rplus$ denote the interval $[0,\infty)$. If $a\colon \Rplus \to
\C$ is continuous with compact support, then formally
\[ \frac{d}{dt} \int_{\Rplus} a(x) f(t,x) \, dx = \int_{\len}^\infty
a(x) \partial_t f(t,x)
dx - a(\len)f(t,\len) \ldot. \]
We substitute for $\partial_t f(t,x)$ using \qref{evol2},
\qref{d.nuF} to obtain the moment identity
\ba
\label{moment1}
\lefteqn{ \frac{d}{dt} \int_{\Rplus} a(x) \nu_t(dx) \,= }\\
\nn
&& f(t,\len) \ldot \sum_{k \geq 1}
p_k \int_{\Rplus^k}\left[ a\left(\len + \sum_{i=1}^k y_i \right)- a(\len)
-\sum_{i=1}^k a(y_i) \right] \prod_{i=1}^k F_t(dy_i)\, .
\ea
Some basic properties of the model are obtained by choosing suitable
test functions $a$ in \qref{moment1}. We set $a(x)=x$ to see that
mass is conserved:
\be
\label{mass}
\frac{d}{dt} \int_0^\infty x \, \nu_t(dx) = 0.
\ee
When $a=1$, we obtain the rate of change of the total number of
clusters,
\be
\label{number}
\dot{N} = - Q_1 f(t,l) \ldot = -N Q_1 \rho_t(l) \ldot.
\ee
We substitute \qref{number} in \qref{evol2} to see that
$\rho_t$ satisfies
\be
\label{evol3}
\partial_t \rho_t = \rho_t(\len) \ldot \, \Q(\rho_t)(x-l), \quad x >l.
\ee
Similarly, we use
\qref{genfn2}, \qref{moment1} and \qref{number} to obtain the moment identity
\be
\label{moment2}
\frac{d}{dt} \int_{\Rplus} a(x) F_t(dx) = \rho_t(\len)\ldot
\sum_{k=1}^\infty p_k \int_{\Rplus^k} \left[ a\left(\len +
\sum_{i=1}^k y_i \right) -a(\len)\right] \prod_{i=1}^k
F_t(dy_i)\, .
\ee
This identity becomes an appropriate weak form of \qref{evolsmol}
after we later impose our choice of time scale $t=1/N$, which means from
\qref{number} that
\begin{equation}
\rho_t(\len)\ldot = \frac1{Q_1t}.
\label{rholdot}
\end{equation}
\subsection{Gallay and Mielke's solution formula}
\label{subsec:GM}
A remarkable feature of these min-driven clustering models is that the
evolution equation admits an elegant solution via the Fourier (or
Laplace) transform. Our analysis relies heavily on this solution
procedure, due to Gallay and Mielke \cite{GM}.
The main difference with \cite{GM} is that we prefer to use the Laplace
transform, denoted by
\be
\label{LTdef}
\rhobar_t(q) = \int_{\Rplus} e^{-qx} \rho_t(x) \, dx, \quad q>0.
\ee
We set $a(x)=e^{-qx}$ in \qref{moment2} to obtain the ordinary
differential equation
\be
\label{LT}
\partial_t \rhobar_t(q) = -(\rho_t(l)\ldot) \, e^{-ql}
\left(1-Q(\rhobar_t(q))\right).
\ee
In order to integrate this equation, we define an analytic function
$\forma$ via
\be
\label{defPhi}
\forma'(z) = \frac{Q_1}{1-Q(z)}, \quad \forma(0)=0.
\ee
(This definition of $\forma$ differs by the factor $Q_1$ from that used
in~\cite{GM}.) $\forma$ is strictly increasing on $[0,1)$.
In the case of binary clustering, $Q(z)=z^2$, and the functions $\forma$
and $\forma^{-1}$ are
\be
\label{binary}
\forma(z) = \log \left(\frac{1+z}{1-z} \right), \quad
\forma^{-1}(w) = \tanh \frac{w}{2}.
\ee
We substitute \qref{defPhi} in \qref{LT} to obtain
\be
\label{sol2}
\partial_t \forma(\rhobar_t(q)) = - (Q_1 \rho_t(l) \ldot)\, e^{-ql},
\quad q>0.
\ee
The choice of time scale has played no role in the analysis this
far. Gallay and Mielke parametrize time by the minimum cluster size,
setting $\ldot =1$. For clarity of notation, we denote this
choice of time scale by $\tau$, reverting to the letter $t$ when we
introduce the number-driven time scale in \qref{intime}.
With $\len(\tau)=\tau$, the value of $\rho_\tau$ on the free boundary
$x=\tau$ plays an important role in the solution.
We use this density to define a measure on $(0,\infty)$ that we call the
{\em trace measure $A$\/}, with distribution function written
\be
\label{alphadef}
A(\tau) =
\begin{cases}
\alpha_0+\int_{\tau_0}^\tau Q_1 \rho_s(s) \, ds ,
&\tau\ge\tau_0,\cr
\alpha_0 & \tau<\tau_0.
\end{cases}
\ee
Here $\tau_0$ denotes the initial time, and $\alpha_0$ is
any convenient constant.
Equation \qref{sol2} may now be rewritten
\be
\label{sol1}
\partial_\tau
\forma(\rhobar_\tau(q))
= -e^{-q\tau} \frac{dA}{d\tau}.
\ee
Fix $\tau_1> \tau_0$. We integrate \qref{sol1} from $\tau_0$ to
$\tau_1$ to obtain
\be
\label{sol3}
\forma(\rhobar_{\tau_1}(q)) - \forma(\rhobar_{\tau_0}(q))
= - \int_{\tau_0}^{\tau_1} e^{-qs} A(ds).
\ee
Since $ \rho_\tau$ is supported in $[\tau,\infty)$, we have the
estimate
\[ \rhobar_\tau(q) \leq e^{-q\tau} \int_\tau^\infty \rho_\tau(x) \, dx
= e^{-q\tau}. \]
We now let $\tau_1 \to \infty$ in \qref{sol3} to find the Laplace
transform of $A$ given by
\be
\label{sol4}
\alphabar(q)= \int_{\Rplus} e^{-qs}A(ds) = \forma(\rhobar_{\tau_0}).
\ee
Thus, the trace measure $A$ is the inverse Laplace transform of
$\forma(\rhobar_{\tau_0})$ and is determined completely by the initial
data.
We may now repeat this argument to determine the solution at any time
$\tau > \tau_0$. We replace $\tau_0$ by $\tau$ in \qref{sol3}, let
$\tau_1 \to \infty$, and obtain
\be
\label{sol5}
\rhobar_\tau(q) = \forma^{-1} \left(\alphabar_\tau(q) \right),
\ee
where
\begin{equation}
\label{sol6}
\alphabar_\tau(q) = \int_\tau^\infty e^{-qs} A(ds).
\end{equation}
We note that $\alphabar_\tau$ is the Laplace transform of the truncated trace
measure $A_\tau$ satisfying
\be\label{gl1}
A_\tau(ds) = H(s-\tau) A(ds),
\ee
where $H$ is the Heaviside function.
We may write
\be
\label{gl}
A_\tau(s) = A(s)\vee A(\tau)
\ee
for the distribution function, with the notation $a\vee b=\max(a,b)$.
Therefore, the {\em nonlinear\/}
evolution of $\rho_\tau$ is determined by the {\em linear\/} evolution
of $A_\tau$. This global linearization underlies the analysis
in~\cite{GM}.
\subsection{The number-driven time scale and an extended solution formula}
It is natural to try to use the formula \qref{sol5} as a
basis for finding measure-valued solutions when the initial data
$\rho_{\tau_0}$ is replaced by an arbitrary probability distribution.
However, we face two difficulties. The first is that it is not clear
that \qref{sol5} necessarily {\em defines\/} a measure $\rho_\tau$.
That is, it is not clear that the right-hand side of \qref{sol5}
is necessarily the Laplace transform of a measure.
The second difficulty is that
whenever the trace measure $A$ has atoms,
any solution defined through \qref{sol5} is
discontinuous in time, as is clear from \qref{sol6}.
We overcome the first difficulty by an approximation
argument. This relies on the simple and fundamental fact that a limit
of completely monotone functions is completely monotone.
We overcome the second difficulty by switching to the number-driven time
scale \qref{intime}.
Henceforth, the letter $t$ always denotes the number-driven
time scale. The measure-valued solution is
denoted by $\newrho_t$, its Laplace transform by
\[
\Rbar_t(q) = \int_0^\infty \rme^{-qr}\,\newrho_t(dr),
\]
and the minimum cluster size by $\len(t)$. These are related to
solutions in the time scale $\tau$ by
\be
\label{newold}
\len(t)=\tau, \quad\newrho_t(dx)= \rho_\tau(x) \, dx, \quad
\Rbar_t(q)=\rhobar_\tau(q).
\ee
We now rewrite the solution formula \qref{sol5} in terms of measures.
The relation $\len(t)=\tau$, equation \qref{number} and the
definition of $A$ in \qref{alphadef} imply
\be
\label{dtdtau}
\frac{dt}{t} = Q_1 \rho_\tau(\tau) d\tau = A(d\tau).
\ee
The differential equation \qref{LT} now takes the form
\be
\label{newsol}
\partial_t \Rbar_t(q) = -\frac{e^{-q\len(t)}}{Q_1t}(1-Q(\Rbar_t(q)).
\ee
If $t_0$ denotes the initial time, we may integrate equation
\qref{dtdtau} to obtain
\be
\label{defL}
\log \left( \frac{t}{t_0} \right)= A(l(t)) -\alpha_0.
\ee
The change of variables \qref{newold} and \qref{dtdtau} affects the
Laplace transform of $A_\tau$ as follows:
\be
\label{newsol3}
\int_{\tau}^\infty e^{-qr} A(dr) = \int_t^\infty e^{-q\len(s)} \frac{ds}{s}.
\ee
These calculations yield the following revised solution procedure:
Given an arbitrary initial probability measure $F_{t_0}$,
the trace measure $A$ is found as in \qref{sol4}
by inverting its Laplace transform, given by
\be \label{newsol4}
\alphabar(q) = \forma(\Rbar_{t_0}(q)).
\ee
Next, we determine $\len(t)$ through inverting
\qref{defL}. Once $\len(t)$ is known,
the solution $\newrho_t$ is determined by inverting
the Laplace transform given as in \qref{sol5} and \qref{sol6} by
\be
\label{newsol5}
\Rbar_t(q) = \forma^{-1} \left( \int_t^\infty e^{-q\len(s)}
\frac{ds}{s}\right), \quad t \geq t_0.
\ee
The main observation is that working with $\len(t)$ instead of
$A_\tau$ yields an evolution continuous in time.
Since $\len(t)$ is an increasing function, it has at worst jump
discontinuities. But \qref{newsol5} shows that discontinuities in
$\len$ do not affect the continuity in $t$ of $\Rbar_t(q)$, and thus
the continuity of $\newrho_t$ in the weak topology.
\subsection{An example: monodisperse initial data}
Let us illustrate the meaning of the extended solution formula
in the new time scale with an important example.
Set $t_0=\tau_0=1$, $Q(z)=z^2$ and $F_1(x) =
\mathbf{1}_{x\geq1}$. That is, initially all clusters have size 1.
Then $\Rbar_1(q)
=e^{-q}$, and
\[
\alphabar(q) = \forma(\Rbar_1(q))= \log(1+ e^{-q}) -
\log(1-e^{-q})
.\]
We differentiate with respect to $q$ and simplify to obtain
\[
-\partial_q \alphabar(q) = \frac{2e^{-q}}{1-e^{-2q}} = 2\left( e^{-q} +
e^{-3q} + e^{-5q} + \ldots.\right) \]
Since $e^{-kq}$ is the Laplace transform of $\delta_k(dx)$, the
trace is
\be
\label{trace-mono}
A(x) = \sum_{k \leq x, \;k \;\mathrm{odd}} \frac{2}{k}.
\ee
$A$ has jump discontinuities at the odd integers. We shall work with
the right continuous inverse, so that the minimum cluster size is
\be
\label{monosol1}
\len(t) = k, \quad t \in [t_{k-2},t_k), \quad t_k = e^{2\left( 1 +
\frac{1}{3} + \ldots \frac{1}{k} \right)}, \quad k \;\;\mathrm{ odd},
\ee
with $t_{-1}=t_0=1$. The solution formula \qref{newsol5} now yields
\be
\nn
\forma(\Rbar_t(q)) = \left(\log \frac{t_k}{t}\right) \frac{e^{-kq}}{2} +
\frac{e^{-q(k+2)}}{k+2} + \frac{e^{-q(k+4)}}{k+4} + \ldots, \quad
t \in [t_{k-2}, t_{k}).
\ee
The solution has the following interpretation. For $t \in
[t_{k-2},t_k)$, $F_t$ is supported on the odd integers greater than
or equal to $k$. The fraction of
clusters of size $k$ decays continuously to zero over the time interval
$[t_{k-2},t_k)$. Thus, the number-driven time scale regularizes jumps in
the trace measure by providing a finite time for these jumps to
vanish. A moment's reflection suggests that this is what we should
expect if we approximate the monodisperse data by a smooth density.
\section{Analytic preliminaries}
\label{sec:prelim}
This section is a summary of the main analytic methods we use. We
present some facts about distribution functions, Laplace transforms
and Tauberian theorems in a form suitable for use in later sections.
\subsection{Distribution functions}
We will consider measures on an interval $J \subset \R$. We study a
measure through its distribution function, often using the same
notation for both. It is therefore convenient to introduce
the following conventions for brevity.
A distribution function $\fun\colon J \to \R$ is
a right-continuous, increasing function.
(``Increasing'' means that $x_1\le x_2$ implies $\fun(x_1)\le \fun(x_2)$.)
A distribution function $\fun$ is identified with a
measure via
\be
\label{measuredist}
\fun\left((x_1,x_2] \right) =\fun(x_2) -\fun(x_1).
\ee
We do not assume that the function $\fun$ is positive,
since the trace measure for
self-similar solutions is of the form $A(\tau) = \theta \log \tau$,
$\tau >0$. Following probabilistic convention, we say
a sequence $\fun_n$ of measures on $J$ converges weakly to $\fun$
(written $\fun_n\to \fun$) if and only if
$\fun_n([a,b])\to \fun([a,b])$ as $n\to\infty$ whenever $a,b\in J$
are not atoms of $\fun$, meaning $\fun(\{a\})=\fun(\{b\})=0$.
We have $\fun_n\to \fun$ if and only if
at every point of continuity of $\fun(x)$,
\[
\fun_n(x) + c_n \to \fun(x) \quad\mbox{as $n\to\infty$},
\]
for some constants $c_n$ independent of $x$.
Given a distribution function $f\colon J\to\R$, its epigraph is
the set
\[
\Gamma(f) = \{(x,y)\in\R^2\mid f(x^-)\le y\le f(x), x\in J\}.
\]
There is a unique distribution function $\funinv$,
with epigraph obtained by reflection through $x=y$:
$\Gamma(\funinv)=\{(x,y)\mid(y,x)\in\Gamma(\fun)\}.$
We call $\funinv$ the {\em inverse} of $\fun$.
We can write
\be
\label{finv}
\funinv(\tau) = \inf\left\{ t \in J \left| \fun(t) > \tau
\right. \right\},
\quad \tau<\sup\{ \fun(t)\mid t\in J\}.
\ee
The following convergence result is not difficult to prove.
\begin{lemma}
\label{inverseA}
Suppose $\fun_n\colon J \to \R$ is a sequence of distribution functions that
converges to $\fun$ at all points of continuity. Then
for every point of continuity $x$ of $\funinv$,
$\funinv_n(x)$ is defined for sufficiently large $n$,
and $\lim_{n \to \infty} \funinv_n(x) = \funinv(x)$.
\end{lemma}
\begin{lemma}
\label{approxincr}
Suppose $\fun \colon J \to \R$ is a distribution
function. There exist monotonically increasing and monotonically
decreasing sequences of piecewise constant distribution functions
that converge to $f$ at all points of continuity.
\end{lemma}
\begin{proof}
We assume that $J=[0,\infty)$ for clarity, and we only
construct an increasing sequence of approximations. The argument
is easily generalized. Suppose
$k \geq 1$ is an integer. Let $\one_k$ denote the indicator function
$\one_k(x) = 1$, $k \leq x < k+1$, and $\one_k(x)=0$ otherwise.
Consider the sequence of functions
\[
g_n(x) = \sum_{k=0}^\infty f\left(\frac{k}{n}\right)\one_k(nx).
\]
Since $f$ is increasing, we have
\[ g_1(x) \leq g_2(x) \leq \ldots \leq g_n(x) \leq f(x). \]
Let us establish convergence of $g_n$. Given $x \geq 0$
and $n$, let $k_n$ denote the integer such
that $k_n \leq nx < k_n+1$. Clearly $\lim_{n \to \infty}
k_n/n=x$. Suppose $x$ is a point of continuity of $f$. Then
$g_n(x) = f(k_n/n)$, therefore $g_n(x) \to f(x)$.
\end{proof}
\subsection{Min history and trace}
Our analysis will focus on two related distribution functions---the
minimum cluster-size history $\len$ and the trace $A$.
Fix $t_0>0$. A {\em min history} is a
positive distribution function $\len$ on $[t_0,\infty)$.
In this article we also require a min history to be unbounded:
$\len(t)\to\infty$ as $t\to\infty$. Given any min history $\len$,
we associate a {trace} $A$ on $(0,\infty)$ via
\be
\label{defA2}
A(\tau) = \log \invlen (\tau),
\quad \tau >0.
\ee
Note $A(\tau)=\log t_0$ for $0<\tau <\len(t_0)$ due to \qref{finv}.
Conversely, we say $A$ is a {\em trace} if it is
a distribution function on $(0,\infty)$ such that
(i) $A(\tau)=\log t_0$ on some nonempty, maximal interval
$(0,\tau_0)$, and (ii) $A(\tau)\to\infty$ as $\tau\to\infty$.
Given any trace $A$, we can associate a min history $\len$ by
\be
\label{defL2}
\len(t) = \exp A\iinv(t),
\quad t \ge t_0.
\ee
\begin{prop}[Change of variables]
\label{propCV}
(a) Assume $\len$ is a min history and there exists $c \geq 0$
such that
\be
\label{lfinite}
\int_{t_0}^\infty e^{-q\len(t)} \frac{dt}{t} < \infty, \quad q\in (c,\infty).
\ee
Then the trace $A$ given by \qref{defA2} satisfies
\be
\label{lAfinite}
\alphabar(q)= \int_0^\infty e^{-q\tau} A(d\tau) = \int_{t_0}^\infty
e^{-q\len(t)}
\frac{dt}{t}, \quad q \in (c,\infty).
\ee
(b) Assume $A$ is a trace such that $\alphabar(q) < \infty$
for $q \in (c,\infty)$, and let $\len(t)$ be defined by
\qref{defL2}. Then \qref{lAfinite} holds.
\end{prop}
\begin{proof}
{\em 1.\/} We first verify the equality for piecewise constant functions.
Suppose $0< t_0 < t_1 < t_2 < \ldots$ and $0< l_0 < l_1 <
\ldots$ are strictly increasing sequences. Consider the piecewise
constant, increasing function
\be
\label{lPL}
\len(t) = \sum_{k=0}^\infty \len_k \one_{[t_k, t_{k+1})}(t), \quad
t \geq t_0.
\ee
An associated trace is given by
\be
\label{tPL}
A(\tau) = \sum_{k=-1}^\infty \log t_{k+1} \one_{[\len_k, \len_{k+1})}(\tau),
\quad \tau \geq 0.
\ee
Here $\len_{-1}=0$.
Conversely, if $A$ is defined by \qref{tPL}, then $\len$ is given by
\qref{lPL}. We fix $q \in (c,\infty)$, assume \qref{lfinite}, and
use \qref{lPL} and \qref{tPL} to compute the integrals in
\qref{lAfinite}. Both equal
\[ e^{-q\len_0} \log \left(\frac{t_1}{t_0}\right) + e^{-q\len_1} \log
\left(\frac{t_2}{t_1}\right) + e^{-q\len_2} \log
\left(\frac{t_3}{t_2}\right) + \ldots \]
{\em 2.\/} Suppose $\len$ is given, and \qref{lfinite} holds. Fix $q
>0$. We approximate $\len$ by a decreasing sequence of piecewise constant
functions $\len_n \dnto \len$. Then $e^{-q\len_n} \upto e^{-q\len}$
since $q > c \geq 0$, and moreover we have $A_n \upto A$ by
Lemma~\ref{inverseA}. By the monotone convergence theorem,
\ba
\nn
\lefteqn{ \int_{t_0}^\infty e^{-q \len(t)} \frac{dt}{t} = \lim_{n \to \infty}
\int_{t_0}^\infty e^{-q\len_n(t)} \frac{dt}{t}}\\
\nn
&& = \lim_{n \to \infty}
\int_0^\infty e^{-q\tau} A_n(d\tau) = \int_0^\infty e^{-q\tau}
A(d\tau).
\ea
{\em 3.\/} To prove (b), assume $A$ is such that $\alphabar(q) <
\infty$ for $q \in (c, \infty)$. Consider a
sequence of piecewise constant increasing functions $A_n \upto A$. Then
$\len_n \dnto \len$ and we may apply the monotone convergence theorem again.
\end{proof}
Proposition~\ref{propCV} and Lemma~\ref{inverseA}
allow us to reformulate the classical
equivalence between weak convergence of measures and pointwise
convergence of Laplace transforms. The following theorem is a slight
modification of~\cite[XIII.1.2a]{Feller}.
\begin{thm}
\label{WT}
Suppose $\len_n$ is a sequence of min histories on
$[t_0,\infty)$ with associated traces $A_n$ that satisfy
\be
\label{uniform}
\sup_n \alphabar_n(q) < \infty, \quad q \in (c, \infty),
\ee
for some $c\ge0$.
Then there is a min history $\len$ such that
$\len_n(t)\to\len(t)$ as $n\to\infty$ at all points of continuity,
if and only if there is a trace $A$ associated to $\len$ such that
$\alphabar_n(q)\to \alphabar(q)$ as $n\to\infty$, for all $q\in
(c,\infty)$.
\end{thm}
\begin{proof}
Suppose $\len_n\to\len$.
Lemma~\ref{inverseA} and definition
\qref{defA2} then imply the distribution functions $A_n\to A$.
By the classical criterion
for weak convergence of measures \cite[XIII.1.2a]{Feller}, under the
hypothesis \qref{uniform} it follows
$\alphabar_n(q)\to\alphabar(q)$
for all $q>c$.
Conversely, suppose $\alphabar_n(q)\to\alphabar(q)$ for all $q>c$,
where $A$ is a trace (in particular $A(\tau)\to\infty$ as
$\tau\to\infty$). By the classical criterion,
the distribution functions $A_n\to A$, and Lemma~\ref{inverseA}
yields $\len_n\to\len$
where $\len$ is given by \qref{defL2}.
\end{proof}
\subsection{Regular variation}
A measurable function $\rv : (0,\infty) \to (0,\infty)$ is
{\em regularly varying \/} at infinity with index $\theta \in \R$
(written $\rv \in \RV_\theta$) if
\be
\label{sv2}
\lim_{\lambda \to \infty} \frac{\rv (\lambda x)}{\rv(\lambda)}
=x^\theta, \quad
\mathrm{for\;\; every}\;\;x >0.
\ee
In this case, $\rv(x) = x^\theta \sv(x)$ where $\sv$ is slowly varying
at infinity.
The class of regularly varying functions is remarkably rigid.
For example, there is no need to assume that the limit in \qref{sv2}
exists for every $x>0$. If $\rv$ is a positive, measurable function
on the half-line and $\lim_{\lambda \to \infty} \rv(\lambda
x)/\rv(\lambda)$ exists, is positive and finite for $x$ in a set of positive
measure, then $\fun$ is regularly varying at infinity with some
index $\theta \in \R$. This fundamental rigidity
lemma (see \cite[VIII.8.1]{Feller} and \cite[1.4.1]{Bingham})
plays a key role in our analysis.
The class $\RV_\theta$ is of fundamental utility in Tauberian
arguments linking a measure $\nu$ on $[0,\infty)$ and its Laplace
transform $\bar{\nu}(q) = \int_0^\infty e^{-qx}\nu(dx)$, see
\cite[XIII.5.2]{Feller}:
\begin{thm}
\label{HLK}
If $\sv$ is slowly varying at infinity and $0 \leq \theta < \infty$,
then the following are equivalent:
\be
\label{sv4}
\nu(x) \sim x^\theta \sv (x), \quad x \to \infty,
\ee
and
\be
\label{sv5}
\bar{\nu}(q) \sim q^{-\theta} \sv (1/q)\Gamma(1+\theta), \quad
q \to 0.
\ee
Moreover, this equivalence remains true when we interchange the roles
of the origin and infinity, namely when $x \to 0$ and $q \to \infty$.
\end{thm}
A refinement of this result will prove useful for
us---de Haan's exponential Tauberian theorem \cite[Thm.
3.9.3]{Bingham}, see \cite{deHaan}.
\begin{thm}\label{t.deH}
$\exp \nu(x)$ is regularly varying at infinity with index $\theta$
if and only if $\exp \bar\nu(q)$ is regularly varying at zero
with index $\theta$.
If either holds, then
\[
\nu(1/q)-\bar\nu(q) \to \gamma \theta, \quad\mbox{$q\to0$,}
\]
where $\gamma= 0.577215665\ldots$ is the Euler-Mascheroni constant.
\end{thm}
\subsection{Rate of divergence in the solution formula}\label{s.kappa}
For several reasons, we need to study carefully
the asymptotic behavior of $\forma(z)$ as $z \to 1$.
For this, it is convenient to define $\kappa(q)$ via
\begin{equation}\label{kappadef}
- \log \kappa(q)= \forma(1-q)+\log q =
\int_0^{1-q} \left(
\frac{Q_1}{1-Q(z)} - \frac{1}{1-z}\right) \,dz
\end{equation}
(note $-\log\kappa(q)\ge 0$) and set
\begin{equation}\label{kappanod}
\kappa_0:= \lim_{q \to 0} \kappa(q) \in [0,1]\,.
\end{equation}
With this notation $\kappa_0$ corresponds to the
number $\kappa$ introduced in \cite{GM}.
For $Q(z)=z^2$ we have $\kappa_0=\frac12$.
We will show that $\kappa_0>0$ if and only if $\sum_{k=1}^{\infty}
(k \log k) p_k < \infty$. Since in \cite{GM} the function $Q$ is a
polynomial this finiteness condition is always satisfied.
However, we can characterize self-similar solutions and their
domains of attraction also in the case $\kappa_0=0$.
\begin{lemma}\label{L.kappa}
The function $\kappa(q)$ as defined in \eqref{kappadef} satisfies
$\kappa(q) \to 0$ as $q \to 0$ if and only if
$\sum_{k=1}^{\infty} p_k k \log k = \infty$.
\end{lemma}
\begin{proof}
We compute
\be\label{e.kap1}
R(z):=Q_1-\frac{1-Q(z)}{1-z} =
\sum_{k=1}^\infty p_k
\left( k -\frac{1-z^k}{1-z} \right) =
\sum_{k=1}^\infty p_k \sum_{j=1}^{k-1} (1-z^j).
\ee
Note $R(1)=0$ and the integrand in \qref{kappadef} is
\be\label{e.kap2}
\frac{R(z)}{1-Q(z)} =
\frac{R(z)}{1-z}\cdot\frac1{Q_1-R(z)} = \frac{R(z)}{1-z}\cdot\frac1{Q_1+o(1)}
\ee
as $z\to1^-$. Thus it suffices to show $\int_0^1R(z)\,dz/(1-z)<\infty$
if and only if $\sum p_kk\log k<\infty$. We observe
\be
\int_0^1 \sum_{j=1}^{k-1} \frac{1-z^j}{1-z}\,dz =
\sum_{j=1}^{k-1} \sum_{l=1}^j \frac1l
= \sum_{l=1}^{k-1} \sum_{j=l}^{k-1} \frac1l
= \sum_{l=1}^{k-1} \frac{k-l}l \sim k\log k
\ee
as $k\to\infty$, whence the desired result follows from \qref{e.kap1}.
\end{proof}
\begin{lemma}\label{L.kappa2}
The function $\kappa(q)$ is slowly varying as $q \to 0$.
\end{lemma}
\begin{proof}
We need to show that $- \log (\kappa(\lambda q)/{\kappa(q)}) \to
0$ as $q \to 0$ for any $\lambda >0$.
From \qref{e.kap1} and \qref{e.kap2} above, we see
that if we replace the integrand in \qref{kappadef} by $R(z)/(1-z)$, then in
integrating from $1-\lambda q$ to
$1-q$ the error is only $o(1)$ as $q \to 0$.
Hence by \qref{e.kap1},
\[
- \log \frac{\kappa(\lambda q)}{\kappa(q)} = \sum_{k=1}^{\infty} p_k
k \sum_{l=1}^{k-1} \Big(\frac{ (1-\lambda q)^l}{l}
- \frac{(1-q)^l}{l}\Big) + o(1)
\]
as $q \to 0$.
We consider without loss of generality $\lambda >1$ and estimate
\[
\sum_{l=1}^{k-1} \frac{ (1-\lambda q)^l}{l}
- \frac{(1-q)^l}{l} = \int_{1-\lambda q}^{1-q} \sum_{l=1}^{k-1}
x^l\,dx = \int_{1-\lambda q}^{1-q} \frac{1-x^{k-1}}{1-x}
\,dx \leq \log \lambda \,.
\]
On the other hand
\[
\sum_{l=1}^{k-1} \frac{ (1-\lambda q)^l}{l}
- \frac{(1-q)^l}{l} \to 0
\]
as $q \to 0$ for any $k$ and thus the claim follows from the
dominated convergence theorem.
\end{proof}
\section{Well posedness for measures}
\label{sec.ds}
In this section, we first work with the min-driven time scale
$\len(\tau)=\tau$ used by Gallay and Mielke. We prove that a
continuous probability density with min
$\tau_0>0$ defines a solution to \qref{evol3}. This is a weaker
form of the well-posedness theorem of~\cite{GM}. It is included for its
simplicity, and because it is the basis for weak solutions. We then
switch to the number-driven time
scale \qref{intime} and use the moment identity \qref{moment2} to show
that \qref{evol3} defines a continuous dynamical system on
the space of probability measures $\PPE$.
\subsection{Classical solutions}
The solution formulas of section~\ref{subsec:GM}, while explicit, are
not immediately suited for the construction of solutions. The main
difficulty is to show that positive initial data yields a positive
solution. We construct solutions by rewriting \qref{evol3} in integral form
\ba
\label{intrho}
\rho_\tau(x) & = & \rhoinit(x) + \int_{\tau_0}^\tau \rho_s(s)
\Q(\rho_s) (x-s) \, ds, \quad x > \tau, \\
\nn
\rho_\tau(x) & = & 0, \quad x < \tau.
\ea
We then have
\begin{thm}
Suppose $\rhoinit$ is a continuous probability density with positive
min $\tau_0$. There exists a unique solution to \qref{intrho} on
$[\tau_0, \infty)$ such that $\rho_{\tau_0}=\rhoinit$ and the
solution has the following properties.
\begin{enumerate}
\item[(a)] For every $\tau \geq \tau_0$, $\rho_\tau$ is a continuous
probability density with min $\tau$.
\item[(b)] The solution formula \qref{sol5} holds for every $q \geq 0$,
$\tau \geq \tau_0$.
\end{enumerate}
\end{thm}
\begin{proof}
\newcommand{\pp}{{\rho}}
\newcommand{\reals}{\mathbb{R}}
\newcommand{\dd}[1]{{\,{\rm d}#1}}
\newcommand{\pa}{\partial}
We sketch a proof of existence similar to the direct approach in
\cite{CP} for $Q(z)=z^2$ using a different time scale.
We fix $\tau_0>0$ and let $\rhoinit$ be given.
Note that since the solution is to satisfy $\pp_\tau(x)=0$ for $x<\tau$,
the convolution term on the right-hand side of \qref{intrho}
will depend only upon values of $\pp_\tau(y)$ for $\tau_0<y<x-\tau\le x-\tau_0$.
In particular, this convolution term vanishes for $x<2\tau_0$.
This means we can construct the solution for
$\tau_0<\tau<2\tau_0$ by an inductive procedure as follows:
For $\tau_0<\tau\le x<2\tau_0$ we have
$\pp_\tau(x)=\rhoinit(x)$ and in particular $\pp_\tau(\tau)=\rhoinit(\tau)$.
For $\tau_0<\tau\le 2\tau_0$, successively on strips
$x\in[k\tau_0, (k+2)\tau_0)$, for $k=2,4,\ldots$, by simple integration
in time we can now compute $\pp_\tau(x)$ from \qref{intrho},
where the right-hand side is always known from a previous step.
This determines $\pp_\tau(x)$ for $\tau_0\le \tau\le 2\tau_0$ and all $x$.
To determine the solution globally for all $\tau>\tau_0$,
the idea is to replace $2\tau_0$ by $\tau_0$ and repeat.
But in order to justify this we need to verify that
$\rho_\tau$ remains integrable and conserves total probability.
In particular we need to justify \qref{LT}.
Let us introduce the distribution function
\begin{equation}\label{Fdef}
R_\tau(x)=\int_0^x\pp_\tau(y)\,dy = \int_\tau^x \pp_\tau(y)\,dy.
\end{equation}
This is the probability that a domain has size $\le x$ at time $t$.
Note that for any two distribution functions $R(x)$, $\hat R(x)$ on
$[0,\infty)$ we have
\[
R\star\hat R(x)=\int_0^x R(x-y)\hat R(dy) \le \int_0^x R(x)\hat
R(dy)=R(x)\hat R(x).
\]
Integrating the convolution term in the integrand of \qref{intrho}, we find
\begin{eqnarray*}
&&\int_0^x \Q(\rho_\tau)(y-\tau)\dd y =
\sum_{k=1}^\infty p_k R_\tau^{\star k}(x-\tau) \le
\sum_{k=1}^\infty p_k R_\tau(x)^k = Q(R_\tau(x)).
\end{eqnarray*}
Then it follows
$ \pa_\tau R_\tau(x) \le \pp_\tau(\tau)(Q(R_\tau(x))-1)$,
and since $\pp_\tau(\tau)\ge0$ and $R_\tau(x)\le 1$ initially,
$R_\tau(x)$ is decreasing in $\tau$ for fixed $x$.
It follows $R_\tau(\infty)\le1$, and so the Laplace transform
\[
\bar R_\tau(q)=\int_0^\infty \rme^{-qx} R_\tau(\dd{x}) =
\int_\tau^\infty \rme^{-qx}\pp_\tau(x)\dd{x}
\]
is well defined and $\le\rme^{-q\tau}$.
Since $\pa_\tau R_\tau(x)$ is continuous in $\tau$ for all $x$,
$\bar R_\tau(q)$ is $C^1$ in $\tau$ for all $q>0$.
This justifies \qref{LT} and the the computations leading up to
\qref{sol4} and the solution formula \qref{sol5}.
From \qref{sol4} we deduce that since $\bar R_{\tau_0}(0)=1$,
$\bar A(0^+)=\infty=\bar A_\tau(0^+)$ and then \qref{sol5} yields
$\bar R_\tau(0^+)=1=R_\tau(\infty)$, proving that $\pp_\tau$ is a probability density
for all $\tau$.
\end{proof}
The next lemma provides a uniform estimate for smooth approximations.
\begin{lemma}
Suppose $\rhoinit$ is a continuous probability density with positive
min $\tau_0$ and $\rho_\tau$ is the solution to \qref{intrho}
with initial data $\rhoinit$. Then (with $\alpha_0=0$ in \qref{alphadef}),
\be
\label{Aest}
A(\tau) \leq \frac{Q_1}
{\log 2}
{\log \left(\frac{2\tau}{\tau_0}\right)}
.
\ee
Consequently,
\be
\label{Lest}
\frac{\len(t)}{\len(t_0)} \geq
\frac12 \left( \frac{t}{t_0} \right)^{(\log 2)/Q_1}, \qquad t\ge t_0.
\ee
\end{lemma}
\begin{proof}
If $m$ is an integer such that $\tau \in [2^{m-1}\tau_0, 2^m \tau_0)$
we divide the domain of integration $[\tau_0,\tau)$ into $m$ pieces
and use the fact
that $\rho_s(s)= \rho_{r}(s)$ for $\tau_0\vee\frac12 s\le r\le s$ to obtain
\ba
\nn
Q_1\inv A(\tau) &=&
\int_{\tau_0}^{2\tau_0}\rho_{\tau_0}(s) ds +
\int_{2\tau_0}^{2^2\tau_0} \rho_{2\tau_0}(s) \, ds + \ldots
\int_{2^{m-1}\tau_0}^\tau \rho_{2^{m-1}\tau_0}(s) \, ds\\
\nn
&
\leq & m \ \leq \frac{\log (\tau/\tau_0)}{\log 2} +1.
\ea
\end{proof}
\subsection{Weak solutions}
We now switch to the number-driven time scale $N(t)=1/t$. In order to
define weak solutions, we fix a test function $a$, substitute
\qref{rholdot} in the moment identity \qref{moment2} and integrate in
time between $t_0$ and $t$ to obtain
\ba
\label{newmoment}
\lefteqn{ \int_{\Rplus} a(x)F_t(dx) - \int_{\R_+} a(x) F_{t_0}(dx) =} \\
&&
\nn
\int_{t_0}^t \sum_{k=1}^\infty p_k \int_{\Rplus^k} \left[ a\left(\len(s) +
\sum_{i=1}^k y_i \right) -a(\len(s))\right] \prod_{i=1}^k
F_s(dy_i)\, \frac{ds}{Q_1s}.
\ea
We will consider continuous test functions with
$\lim_{x \to \infty} a(x)=0$. Let $C_0(\Rplus)$ denote the space of
such functions with the topology of uniform
convergence. Let $\PPE$ denote the space of probability measures
on $\Rplus$ equipped with the weak topology. Assume $t_0 > 0$ is fixed.
\begin{defn} Let $J\subset (0,\infty)$ be an interval.
We say that a map $F: J \to \PPE$ is a weak solution for
min-driven clustering on $J$ if
\begin{enumerate}
\item The map $t \mapsto \int_{\Rplus} a(x) F_t(dx)$ is measurable
for every $a \in C_0(\Rplus)$.
\item The min of $F_t$, denoted $\len(t)$, is positive and increasing.
\item The moment identity \qref{newmoment} holds for each
$a \in C_0(\Rplus)$ and $t, t_0 \in J $.
\end{enumerate}
\end{defn}
\begin{thm}
\label{thm.wp}
\begin{enumerate}
\item[(a)]
Suppose $\hat{F} \in \PPE$ has positive min, and $t_0>0$. Then
there is a weak solution $F$ for min-driven clustering on $[t_0,\infty)$
with $F_{t_0}=\hat{F}$. Moreover, the min $\len(t)$ of $F_t$
satisfies \qref{Lest}.
\item[(b)] The solution in (a) is unique on $[t_0,t_1]$ for any $t_1>t_0$.
\item[(c)] Let $\hat{F}\nsup$ be a sequence in $\PPE$ with positive
min and $F\nsup$ the weak solutions with
$F\nsup_{t_0}=\hat{F}\nsup$. Assume $\lim_{n \to \infty}
\hat{F}\nsup=\hat{F}$ and the limit has positive min. Then $F\nsup_t
\to F_t$ for every $t > t_0$.
\end{enumerate}
\end{thm}
\begin{proof}
{\em 1.\/} It follows from Weierstrass' approximation theorem that finite
linear combinations $\sum_{k=1}^N c_k e^{-kx}$ are dense in
$C_0(\Rplus)$. Therefore, in order to verify the moment identity, it
is sufficient to consider the test functions $e^{-qx}$, $q>0$.
Thus, to prove existence of a weak solution on $[t_0,\infty)$,
it suffices to construct $F$ weakly continuous
such that the Laplace transform
satisfies the solution formula \qref{newsol5}.
{\em 2.\/} Let $\tau_0$ denote the min of $\hat{F}$. We approximate
$\hat{F}$ by a sequence of continuous probability densities
$\hat\rho\nsup$ with min $\tau_0\nsup$ with $\lim_{n \to
\infty} \tau_0\nsup= \tau_0$. We further assume that $\hat\rho\nsup$
is strictly positive on $[\tau_0\nsup, \infty)$. It is immediate from
\qref{intrho} that the solutions $\rho_\tau\nsup$ are strictly
positive on $[\tau,\infty)$. The trace for these solutions, $A\nsup$
is obtained from \qref{alphadef} with $\tau_0$ replaced by
$\tau_0\nsup$, $\rho$ by $\rho\nsup$ and the choice $\alpha_0=\log
t_0$. $A\nsup$ is continuous and strictly increasing, thus so are the
min histories $\len\nsup$. We change variables from the solution
formula \qref{sol5} to \qref{newsol5} to obtain
\be
\label{approxsol}
\rhobar\nsup_\tau(q) = \Rbar_t\nsup(q) =
\forma^{-1} \left( \int_t^\infty e^{-q \len\nsup(s)}\frac{ds}{
s}\right), \quad t \geq t_0.
\ee
{\em 3.\/} As $n \to \infty$, we have $\hat\rhobar\nsup(q) \to
\Rbarhat(q)$ for every $q>0$ and $0 < \Rbarhat(q) < 1$ for $q>0$.
The measures $A\nsup$ are supported on $[\tau_0\nsup,\infty)$ and
satisfy $\alphabar\nsup(q) =\forma(\Rbar_{t_0}\nsup(q))$. Therefore,
\[ \lim_{n \to \infty} \bar{A}\nsup(q) =\forma(\hat{\Rbar}(q)), \quad
q>0. \]
It follows that the traces $A\nsup$ converge weakly to a
trace $A$ supported on $[\tau_0,\infty)$ that satisfies
$\alphabar(q) = \forma(\hat{\Rbar}(q))$. Therefore, by
Theorem~\ref{WT} the min histories
$\len\nsup$ converge to $\len$, the min history
associated to $A$. Since $\len\nsup$ satisfy the uniform estimate
\qref{Lest}, we may use the dominated convergence theorem
to assert
\[
\lim_{n \to \infty} \Rbar_t\nsup(q) =
\forma\inv\left(
\int_t^\infty e^{-q \len(s)}\frac{ds}{s} \right)
, \quad t \geq t_0.
\]
This shows that for every $t \geq t_0$ the measures $F_t\nsup$
converge weakly to a measure $F_t$ that satisfies \qref{newsol5}.
Then $F_t$ is a probability measure since
$\Rbar_t(0+)=\forma\inv(\infty)=1$.
It similarly follows from \qref{newsol5} that
$F_t \to F_{t_1}$ as $t\to t_1$ for every $t_1 \in [t_0,\infty)$.
This completes the proof of part (a), except that it remains to
show that $\len(t)$ is in fact the min of $F_t$.
{\em 4.\/} For (b) it suffices to prove uniqueness on
$[t_0,t_1]$ for {\em some} $t_1>t_0$.
The key is to prove uniqueness of the min $\len(t)$ on such a time
interval, since then uniqueness of $F_t$ is easy to establish via
the Laplace transform.
Note that any weak solution is weakly continuous in time, since
the right-hand side of \qref{newmoment} is Lipschitz continuous for
any test function $a\in C^0(\Rplus)$. Then since $F_{t_0}(x)>0$
for all $x>\tau_0$, the min $l(t)$ is right continuous at $t_0$,
so there exists $t_1>t_0$ with $\tau_0\le\tau_1=l(t_1)<2\tau_0$.
Now the idea is that for clusters of size less than $2\tau_0$, there
is no gain, only loss. We claim that
\begin{equation}\label{e.smallx}
F_t(x) =
\left( F_{t_0}(x)- \frac1{Q_1}\log\left(\frac{t}{t_0}\right)\right)_+,
\quad t\in[t_0,t_1], \quad x\in[\tau_0,2\tau_0).
\end{equation}
This follows by first considering points $x$ of continuity of both
$F_t$ and $F_{t_0}$ such that $\len(t)<x<2\tau_0$, so $F_t(x)>0$ ---
approximate $\one_{[0,x)}$ by continuous test functions $a$
supported in $[0,2\tau_0)$.
By right continuity and the definition of min,
\qref{e.smallx} follows for all $x\in[\len(t),2\tau_0)$, and
$F_t(x)=0$ for $x<\len(t)$. Now by weak continuity in time,
we infer that with
\[
\hat t(x) = t_0\exp(Q_1\hat F(x)) ,
\]
for each point of continuity of $\hat F$ in
$(\tau_0,\tau_1]$ we have $\len(t)<x$ for
$t < \hat t(x)$, and $x<\len(t)$ for $\hat t(x)<t$.
Now clearly $\len(t)$ is uniquely determined by
$\hat F$, since it is locally the inverse of $\hat t$.
It follows from taking the Laplace transform that the
min history $\len(t)$ constructed in (a), that satisfies
\qref{newsol5}, agrees with the min of $F_t$.
{\em 5.\/} Part (c) is proven by
an argument very similar to Step 2 above.
\end{proof}
{\em Scaling.\/}
We note for use below the following scaling property that follows
easily from the moment identity \qref{newmoment}:
Let $a,b>0$.
If $F$ is a weak solution for min-driven clustering on an
interval $J$, then $\hat F$ is a weak solution on $J/a$, where
\begin{equation} \label{e.scale}
\hat F_t(x) = F_{at}(bx), \qquad t\in J/a,\ x\ge 0.
\end{equation}
\section{Self-similar solutions} \label{sec.ss}
Let us recall that Gallay and Mielke have classified the
self-similar solutions as in \qref{ftheta1} in the case when
$p_k=0$ for all large $k$. Here we will recover and extend the
basic existence results and classify all domains of attraction
by simple means based on the Laplace transform and Tauberian arguments.
{\em Existence.}
In terms of the distribution function $F_t$, self-similar solutions
are of the form $F_t(x)= F_*(x/l(t))$ for some distribution function
$F_*$ with positive min ($=1$) and some min history $l(t)$.
Without loss of generality we can assume $t_0=1$ and $l(t_0)=1$.
From \eqref{newsol5} we obtain for the Laplace transform of $F_*$ that
\begin{equation}\label{ss1}
\bar F_*(q) = \bar F_t\left( \frac{q}{l(t)}\right) = \varphi^{-1}
\left( \int_t^{\infty} e^{-q \frac{l(s)}{l(t)}}
\frac{ds}{s}\right) =
\varphi^{-1} \left( \int_1^{\infty} e^{-q \frac{l(ts)}{l(t)}}
\frac{ds}{s}\right).
\end{equation}
Hence we conclude that the min history of self-similar solutions
must satisfy ${l(ts)}/{l(t)}= g(s)$ for some function $g(s)$.
Since $l$ is also positive, increasing and non-constant, we conclude
(see \cite[VIII.8.1]{Feller}, or subsection~\ref{ss.nec} below) that
necessarily, for some $\theta>0$, $l(s)=s^{1/\theta}$ and
that $F_*$ must be a distribution $F\thetasup$ that satisfies
\be\label{sss1b}
\Rbar\thetasup(q) = \forma\inv\left( \theta \,\mathrm{Ei}(q)\right),
\ee
where $\mathrm{Ei}(q) = \int_q^\infty e^{-s}ds/s$ denotes the exponential
integral.
Provided such a distribution $F\thetasup$ does exist, then
$\Rbar_t(q)=\Rbar\thetasup(t^{1/\theta}q)$ satisfies \qref{newsol5}
and is continuous in time, hence determines a self-similar
solution by step 1 of the proof of Theorem~\ref{thm.wp}.
The corresponding trace measures are given by
\be
\label{sss1}
A\thetasup(\tau) = \theta\, \log \tau, \quad \tau \in
(0,\infty).
\ee
As a byproduct of the characterization of scaling limits in section
\ref{sec.rv}, we will prove the existence of distributions
$F\thetasup$ that satisfy \qref{sss1b} for $0<\theta\le1$, and that
$\theta\le1$ is necessary.
{\em Densities.}
Next we show that for the
self-similar solutions $F\thetasup(x/t^{1/\theta})$ $(0<\theta\le1)$,
the probability distributions $F\thetasup$ have piecewise smooth
densities $\den\thetasup$ that satisfy an integrodifferential equation,
\begin{equation}\label{etaeq}
- \partial_y \big ( y \den^{(\theta)}(y) \big ) =
\frac{\theta}{Q_1} \,\Q(\den^{(\theta)})(y-1)\,,\quad y>1,
\qquad \den^{(\theta)}(1)=\frac{\theta}{Q_1}\,,
\end{equation}
and we study the decay of $\den\thetasup(y)$ as $y\to\infty$.
When $p_k=0$ for all large $k$ (so $Q(z)$ is a polynomial)
this has been done by Gallay and Mielke \cite{GM}.
We will recover most of their results by simpler means
(the exponential rate of decay for $\theta=1$ is an exception)
and extend them to the case when $p_k$ is nonzero for infinitely many $k$.
It turns out, however, that when $\sum p_kk\log k=\infty$, {\em none}
of the profiles have finite first moment, including the case $\theta=1$.
Let $\theta\in(0,1]$. We claim that the probability distribution $F=F\thetasup$
satisfies the following weak-form profile equation:
\be\label{e.wkss}
\int_{\R_+} x a'(x) F(dx) =
\frac{\theta}{Q_1} \sum_{k=1}^\infty
p_k \int_{\R_+^k}
\left( a\left(1+\sum_{j=1}^k y_j\right) - a(1) \right) \prod_{j=1}^k F(dy_j)
\ee
for all $C^1$ functions $a\in C_0(\R_+)$.
This follows from the fact that we know $F(x/t^{1/\theta})$ is a weak
solution for the min-driven clustering equation \qref{newmoment} with
$\len(t)=t^{1/\theta}$. Changing variables in \qref{newmoment} via
$x=\len(t)\hat x$ and similarly for $y_j$, we differentiate at $t=1$
to obtain \qref{e.wkss}.
Now, the min of $F$ is 1 (this will be shown in section 6.2),
so $F$ has density $\den(x)=0$ on $(0,1)$. Taking $a$ to be
supported in $(1,2)$ we find that the right-hand side of \qref{e.wkss}
vanishes. Hence restricted to $(1,2)$, the measure
$xF(dx)=\beta\,dx$ for some constant $\beta$, so $F$ has density
$\den(x)=\beta/x$ on $(1,2)$.
Taking $a(x)=0$ for $x\le1$, $a(x)=1$ for $x\ge2$
(approximated by limits---note $\int_n^{n+1}xF(dx)\to0$ along a
subsequence),
we find $\beta = \theta/Q_1$. Taking $a$ supported in $(0,2)$ ,
we find the right-hand side is $-\beta a(1)$ and we
can conclude that $1$ is not an atom of $F$.
In this way,
proceeding inductively on intervals $(0,n)$ we deduce that
$F$ has density $\den(x)$ satisfying \qref{etaeq}.
When $a$ has support in $(1,n+1)$ the right-hand side depends on
the restriction of $F$ to $(0,n)$ where it has density $\den(x)$,
and thus $xF(dx)$ has density $x\den(x)$ determined by
\qref{etaeq} on $(1,n+1)$.
{\em Decay.}
Next we wish
to characterize the decay behavior of the densities $\den\thetasup$.
For this we use the properties of the function $\kappa(q)$
which was introduced in \eqref{kappadef}.
We denote by $\kappa^\#$ the {\em de Bruijn conjugate} of $\kappa$.
See \cite[Sec.\ 1.5.7]{Bingham}.
This is a slowly varying function satisfying
\be \label{kapsharp}
\kappa(q)\kappa^\#(q\kappa(q))\sim 1 \quad\mbox{ as $q\to0$.}
\ee
If $\kappa_0>0$, then $\kappa^\#(0^+)=\kappa_0\inv$,
and if $\kappa_0=0$, then $\kappa^\#(0^+)=\infty$.
\begin{thm} \label{t.ss}
For every $\theta \in (0,1]$,
the density $\den^{(\theta)}(x)$ of the self-similar profile
$F\thetasup$ has the following properties:
\begin{enumerate}
\item[(i)]
If $\theta
\in (0,1)$, then as $x\to\infty$,
\begin{equation}
\label{etadecay}
\den^{(\theta)}(x) \sim x^{-(1+\theta)} e^{\theta \gamma}\kappa^\#(x^{-\theta})
\frac{\theta(1-\theta)}{\Gamma (2-\theta)}.
\end{equation}
Here $\gamma$
is the Euler-Mascheroni constant and $\Gamma$ is the $\Gamma$-function.
\item[(ii)]
If $\theta =1$ then as $x \to \infty$,
\begin{equation}
\label{Fdecay}
\int_0^x y \den^{(1)}(y)\,dy \sim { e^{ \gamma}}{\kappa^\#(x^{-1})} \,
\end{equation}
\end{enumerate}
\end{thm}
{\em Remarks:}
The asymptotics \eqref{Fdecay} imply the result of \cite{GM} for total
mass in the case that $\kappa_0>0$. We will not pursue here the
delicate question of the precise (exponential)
decay rate of the density $\den^{(1)}$ in this case, which was studied in
\cite{CP} for $Q(z)=z^2$ and in the polynomial case in \cite{GM}.
If $\kappa_0=0$, however, we see that $F^{(1)}$ has infinite mass.
\begin{proof}
We rewrite equation \eqref{sss1b} for $ \Rbar\thetasup$
using \qref{kappadef} and standard asymptotics for the
exponential integral \cite{Olver}, as follows. With $w=1-\Rbar\thetasup(q)$,
\[
-\forma(\Rbar\thetasup(q))= \log(w\kappa(w)) = -\theta\mathrm{Ei}(q)
=\theta(\log q+ \gamma+o(1)), \quad q\to 0.
\]
Then $w\kappa(w)\sim q^\theta e^{\theta\gamma}$,
whence asymptotic inversion \cite[Sec.\ 1.5.7]{Bingham} yields
\be \label{asyinv}
w=1-\Rbar\thetasup(q) \sim q^\theta e^{\theta\gamma} \kappa^\#(q^\theta) .
\ee
Now differentiating (i.e., using Lemma 3.3 of \cite{MP1}) we find
\[
\int_0^{\infty} e^{-qx}x F^{(\theta)}(dx) =
-\partial_q \Rbar\thetasup (q) \sim \theta q^{\theta-1}
{e^{\theta \gamma}}{\kappa^\#(q^{\theta})}, \qquad q\to0.
\]
Now the Tauberian theorem \ref{HLK} implies
\begin{equation}
\label{Fdecay1}
\int_0^x y F^{(\theta)}(dy) \sim
\frac\theta{\Gamma (2-\theta)}
x^{1-\theta}
{e^{\theta \gamma}\kappa^\#(x^{-\theta})}
\,, \qquad x\to\infty,
\end{equation}
which is just \eqref{Fdecay} in the case $\theta=1$.
We can now also derive the decay behavior of $\den^{(\theta)}$ in the
case $\theta<1$. In fact, it follows from \eqref{etaeq}
that $y \den^{(\theta)}(y)$ is decreasing. Hence by a lemma in
\cite[XIII.5]{Feller}, \eqref{Fdecay1} implies \eqref{etadecay}.
\end{proof}
\section{Domains of attraction of self-similar solutions}
\label{sec.rv}
We proceed to prove Theorem~\ref{thm.domains}.
The proof has two parts. The first
is to show that regular variation of $\len(t)$ as $t \to \infty$ is
necessary and sufficient for convergence to a scaling limit.
The second is the equivalence between
regular variation of $\len(t)$ as $t \to \infty$ and
regular variation of $\int_0^x y\newrho_{t_0}(dy)$ as $x \to \infty$.
The second part is based on the Tauberian theorems~\ref{HLK}
and~\ref{t.deH}.
The most subtle aspect (despite the simple proof) is to deduce regular
variation of $\len$ from the existence of a scaling limit.
This is the assertion of rigidity, and we treat it first.
\subsection{Regular variation of the min history is necessary}
\label{ss.nec}
{\em 1.\/}
Assume there is a rescaling $\lambda(t) \to \infty$ and a
probability distribution function $\newrho_*$ with positive min
(called $\tau_*$) such that $F_t(\lambda(t) \cdot) \to F_*$.
This is equivalent to convergence of the Laplace transforms,
\be
\label{sss10}
\lim_{t \to \infty} \Rbar_t \left( \frac{q}{\lambda(t)} \right) =
\Rbar_*(q), \quad q \geq 0.
\ee
After a trivial scaling of time and cluster size,
we may assume $t_0=1$ and $\tau_*=1$.
Since $F_*$ is a probability measure with positive min,
there is a unique trace measure $A_*$, with $A_*(\tau)=0$ on
$(0,1)$ and $A_*(\tau)\to\infty$ as $\tau\to\infty$,
and a min history $\len_*$ on $[1,\infty)$, with $\len_*(1)=1$,
such that
\be
\label{sss2} \alphabar_*(q)= \int_{1}^\infty e^{-q\len_*(s)}\frac{ds}{s}=
\forma(\Rbar_*(q)).
\ee
Now, $s\mapsto F_{ts}(\lambda(t)x)$ is a rescaled solution, by \qref{e.scale}.
The rescaled min histories given by $\len^{(t)}(s)=\len(ts)/\lambda(t)$,
$t,s\ge 1$ have associated trace measures $A^{(t)}$ satisfying
$\alphabar^{(t)}(q) \to \alphabar_*(q)$ as $t\to\infty$, for all $q>0$.
We use the solution
formula \qref{newsol5}, \qref{sss10} and \qref{sss2} to obtain
\be
\label{sss3}
\lim_{t \to \infty} \int_{1}^\infty e^{-q\len(ts)/\lambda(t)}
\frac{ds}{s} = \int_{1}^\infty e^{-q\len_*(s)}\frac{ds}{s}, \quad q>0.
\ee
Theorem~\ref{WT} now implies
that at every point of continuity of $\len_*$,
\be
\label{sss4}
\lim_{t \to \infty} \len(ts)/\lambda(t) = \len_*(s).
\ee
{\em 2. \/} Let $s_0$ be a point of continuity of $\len_*$,
and let $U(t)=\len(ts_0)$, $t\ge1$. Let $B$ be the set of $x\ge1$
such that
\be \label{limrat}
\psi(x) = \lim_{t\to\infty} \frac{U(tx)}{U(t)}
\ee
exists. We deduce $\psi(x)$ is a power of $x$ by
following the simple argument in \cite[VIII.8.1]{Feller}:
By \qref{sss4},
we have $x\in B$ if $xs_0$ is a point of continuity of
$\len_*$. If $x_1$, $x_2\in B$ then $x_1x_2\in B$ and
\be \label{cauchy}
\psi(x_1x_2)=\psi(x_1)\psi(x_2).
\ee
Since $U$ is increasing, so is $\psi$, and since $B$ is dense
in $[1,\infty)$ we can extend $\psi$ by right continuity so
\qref{cauchy} holds for all $x_1, x_2\ge 1$. Then we can set
$\psi(x)=1/\psi(1/x)$ for $x\in(0,1)$ and have \qref{cauchy}
for all $x>0$. Since $\psi(x)$ is positive, locally bounded and not constant,
it follows $\psi$ is a pure power law, and
we can write $\psi(x)=x^{1/\theta}$ for some $\theta>0$.
{\em 3.\/} Since $\psi$ is continuous and increasing, it is
easy to see \qref{limrat} holds for all $x\in [1,\infty)=B$.
Then we infer $\tilde L(t)=U(t)t^{-1/\theta}$ is slowly varying, and
$U$, hence $\len$, is regularly varying at $\infty$
with index $1/\theta$.
Further, since $\len_*(xs_0)/\len_*(s_0)=\psi(x)=x^{1/\theta}$
whenever $xs_0$ is a point of continuity of $\len_*$,
and $s_0$ is an arbitrary point of continuity,
it follows $\len_*(t)= t^{1/\theta}$ for all $t\ge1$.
\subsection{Regular variation of the min history is sufficient}
\label{subsec:suff}
Let us assume that the min history $\len$ is regularly varying,
as in \qref{conv3}. Convergence to
self-similar form is then quick: Since
\[ \frac{\len(ts)}{\len(t)} = s^{1/\theta}
\frac{\tilde{L}(ts)}{\tilde{L}(t)}, \]
we have
\[ \lim_{t \to \infty} \frac{\len(ts)}{\len(t)} = s^{1/\theta}.\]
Moreover, for any $\eps >0$, there exists $t_\eps>0$ such that
\[ s^{1/\theta-\eps} \leq \frac{\len(ts)}{\len(t)}, \quad t \geq t_\eps. \]
Therefore, we may use the solution formula and the dominated
convergence theorem to see that as $t \to \infty$
\be
\forma\left(\Rbar_t\left(\frac{q}{\len(t)}\right)\right) = \int_{1}^\infty
e^{-q\len(ts)/\len(t)} \frac{ds}{s} \to \int_{1}^\infty
e^{-qs^{1/\theta}} \frac{ds}{s}.
\ee
This implies $\lim_{t \to \infty} F_t(\len(t) x) =
F\thetasup(x)$ for every $x >0$, where $F\thetasup$ is a probability
distribution with positive min ($=1$) and
Laplace transform given by \qref{sss1b}.
\subsection{Tauberian arguments}
We now prove that if
$0<\theta\le1$ and $\int_0^x y \newrho_{t_0}(dy)$ is
regularly varying with index $1-\theta$ as $x \to \infty$,
then $\len(t)$ is regularly varying with index $1/\theta$
as $t\to \infty$, and conversely.
{\em 1.\/}
It is convenient first to assume \qref{conv2} in the form
\[
\int_0^x y F_{t_0} (dy) \sim
\frac{\theta}{\Gamma(2-\theta)}
x^{1-\theta} L_1(x)
, \quad x \to \infty,
\]
where $L_1$ is slowly varying.
Then Theorem \ref{HLK} implies
\[
-\partial_q \Rbar_{t_0}(q) \sim \theta q^{\theta-1} L_1\left(q\inv\right)
, \quad q \to 0.
\]
By integration (not difficult to justify as in Lemma 3.3 of \cite{MP1}) we find
\[
w:= 1- \Rbar_{t_0}(q) \sim q^\theta L_1\left(q\inv\right),
\quad q \to 0.
\]
Using the solution formula \eqref{newsol5} and
\eqref{kappadef}, we write
\[
\log \big (w \kappa(w) \big )
= - \varphi \big ( \Rbar_{t_0}(q) \big ) = - \int_t^{\infty} e^{-q l(s)}
\frac{ds}{s} = - \bar A(q)\,.
\]
Hence, $\exp(-\bar A(q))$ is regularly varying with exponent $\theta$,
and Theorem~\ref{t.deH} (de Haan's exponential Tauberian theorem)
implies that
\be \label{linvconv}
e^{A(1/q)}e^{-\bar A(q)} =
\invlen( q\inv ) w \kappa(w) \to e^{\gamma \theta}, \quad q \to 0.
\ee
Thus,
$\invlen(\tau) \sim e^{\gamma\theta} \tau^\theta
\hat L_1(\tau^\theta)$ as $\tau\to\infty$, with
\[
\hat L_1(s) = \frac{1}{L_1(s^{1/\theta})
\kappa(s\inv L_1(s^{1/\theta})) } .
\]
It is easy to show $\hat L_1$ is slowly varying,
using the uniform convergence theorem for slowly varying
functions~\cite[Theorem 1.2.1]{Bingham}.
Asymptotically solving $t=\invlen(\tau)$ by inverting $s\mapsto s\hat L_1(s)$
using the de Bruijn conjugate \cite[Theorem 1.5.13]{Bingham} finally yields
\begin{equation}\label{e.len}
l(t) \sim
e^{-\gamma}t^{1/\theta}
\hat L_1^\#(t)^{1/\theta} , \quad
t \to \infty\,,
\end{equation}
which gives \eqref{conv3}.
If $\kappa_0 := \lim_{q \to 0} \kappa(q)>0$, we also can write
\begin{equation}
l(t) \sim e^{-\gamma} (\kappa_0t)^{1/\theta} (L_1^{-1/\theta})^\#(t)
,\quad t \to \infty.
\end{equation}
{\em 2.\/}
We now prove the converse.
Assuming that \qref{conv3} holds, then
\begin{equation}
l(t) \sim e^{-\gamma}t^{1/\theta} L_2(t)^{1/\theta}, \quad t \to \infty,
\end{equation}
for some slowly varying function $L_2$.
Then, by inversion,
$\invlen(\tau) \sim e^{\gamma\theta}\tau^{\theta} L_2^\# (\tau^{\theta})$
as $\tau \to \infty$.
Since $\invlen(\tau) = \exp{A(\tau)}$ and $\bar A(q) =
-\log (w \kappa(w))$ with $w=1-\Rbar_{t_0}(q)$,
we infer from Theorem~\ref{t.deH} that \eqref{linvconv} is true.
This implies
\[
w \kappa(w) \sim
q^{\theta} L_2^\#(q^{-\theta})\inv , \quad q \to 0,
\]
and asymptotic inversion yields
$w\sim q^\theta \hat L_2(q\inv)$
as $q \to 0$, with
\newcommand{{L_2^\#(x^{\theta})}}{{L_2^\#(x^{\theta})}}
\[
\hat L_2(x) =
{{L_2^\#(x^{\theta})}\inv}\kappash\left(x^{-\theta}{{L_2^\#(x^{\theta})}\inv}\right),
\]
and $\hat L_2$ is slowly varying.
Differentiating using Lemma 3.3 of \cite{MP1}, we find
\[
\D_qw=-\partial_q \Rbar_{t_0}(q) \sim \theta q^{\theta-1}
\hat L_2\left(q\inv\right), \quad q\to0.
\]
Since
$-\partial_q \Rbar_{t_0}(q) = \int_0^{\infty} e^{-qx} x F_{t_0}(dx) $
the Tauberian Theorem \ref{HLK} implies that
\begin{equation}
\int_0^x y F_{t_0}(dy) \sim
\frac{\theta }{\Gamma(2-\theta)}
x^{1-\theta} \hat L_2(x) ,
\quad x \to \infty\,,
\end{equation}
which is just \eqref{conv2}.
If $\kappa_0:=\lim_{q \to 0} \kappa(q)>0$, then
\begin{equation}
\int_0^x y F_{t_0}(dy) \sim
\frac{\theta }{\Gamma(2-\theta)}
\frac{x^{1-\theta} }{ {L_2^\#(x^{\theta})}\kappa_0}
, \quad x \to \infty\,.
\end{equation}
This completes the proof.
\section{Eternal solutions}
\label{sec:lk}
\begin{defn}
A weak solution $\newrho$ for min-driven clustering
is an {\em eternal solution\/} if
it is defined on the maximal interval of existence $(0,\infty)$.
\end{defn}
Our understanding of eternal solutions is closely connected to
the question of how they emerge from clusters of infinitesimal size.
From the solution formula \qref{newsol5} we see that as $t$ approaches
zero,
\[
\varphi(\Rbar_t(q)) = \int_t^\infty e^{-q\len(s)}\frac{ds}s \to
\infty,
\]
hence $\Rbar_t(q)\to1$, and this means the relative size distribution
$F_t$ always converges to a
Dirac delta at zero size. A different scaling is needed to
distinguish solutions through limits as $t\dnto0$.
\subsection{The class of \smeass}
What we will show is that the class of eternal solutions
is in one-to-one correspondence with a suitable space of measures that
can be loosely thought of as `rescaled initial data' at $t=0$.
This correspondence parallels the classical probabilistic
characterization of infinitely divisible laws via a \LK\/ formula.
In probability theory, infinitely divisible distributions
are parametrized by the \LK\/ representation theorem, which expresses
the log of the characteristic function (Fourier transform) in terms of
a measure that satisfies certain finiteness conditions.
In particular~\cite[XIII.7]{Feller}, a function $\omega(q)$
is the Laplace transform $\intR \rme^{-qx} F(dx)$
of an infinitely divisible probability measure
$F$ supported on $\ebar$ if and only if
$\omega(q)=\exp(-\lap(q))$ where
the Laplace exponent $\lap$ admits the representation
\be
\label{R.LK2a}
\lap (q) = \int_{[0,\infty)} \frac{1-\rme^{-qx}}{x} \sm(dx)
\ee
for some measure $\sm$ on $[0,\infty)$ that satisfies
\begin{equation}\label{R.gendef2}
\int_{[0,\infty)} (1 \wedge y^{-1}) \sm(dy) <\infty.
\end{equation}
(Here $a \wedge b=\min(a,b)$.)
As in~\cite{MP3} we call such measures {\em \smeass} (short
for ``generating measures,'' a term motivated by their connection
with generators of convolution semigroups in probability
\cite[XIII.9(a)]{Feller}). Some basic analytic facts
about Laplace exponents and {\smeass\/} are collected
in~\cite[Sec. 3]{MP3}.
\begin{defn}
\label{R.candef}
A measure $\sm$ on $[0,\infty)$ is a {\em \smeas}\ if
\qref{R.gendef2} holds.
In addition, we say that a \smeas\ $\sm$ is {\em divergent\/} if
\be
\label{R.candef5}
\sm(0)>0 \quad\mbox{or}\quad
\int_{[0,\infty)} y^{-1} \sm(dy) = \infty.
\ee
\end{defn}
\noindent
Here recall we use the notation $\sm(x)=\int_{[0,x]} \sm(dy)$.
The space of \smeass\ has a natural weak topology which
is fundamental in our study of scaling dynamics.
\begin{defn}
\la{R.proper}
A sequence of \smeass\ $\sm\nsup$ {\em converges} to a
\smeas\ $\sm$ as $n\to\infty$, if at every point $x\in(0,\infty)$
of continuity of $\sm$ we have
\begin{equation}\label{R.candef1}
\sm\nsup(x)\to\sm(x)
\quad\mbox{and }\quad
\int_{[x,\infty)} y\inv \sm\nsup(dy)\to \int_{[x,\infty)} y\inv \sm(dy) .
\end{equation}
\end{defn}
\subsection{A \LK\/ formula}
Our analysis of eternal solutions is motivated by a classification
theorem of Bertoin for Smoluchowski's coagulation equation with
additive kernel~\cite{B_eternal}. Here eternal solutions were shown to
be in correspondence with divergent \smeass\/. This theorem was
generalized to other solvable kernels in~\cite{MP3}, based on the
observation that there is a
natural Laplace exponent $\lap_t$ associated to every
solution. For the model now under study, it is determined by the \smeas
\be\label{Gdef}
\sm_t(dx) = \frac{x F_t(dx)}{t\kappa^\#(t)},
\ee
and the associated Laplace exponent is
\be
\label{eternal1}
\lap_t(q) = \frac{ 1 - \Rbar_t(q) }{t\kappa^\#(t)}
= \int_0^\infty \frac{1 - e^{-qx}}{x} \sm_t (dx).
\ee
(Recall that $\kappa^\#$ from \qref{kapsharp}
is the de Bruijn conjugate of $\kappa$ from \qref{kappadef},
and that $\kappa^\#(0+)=\infty$ if $\sum p_kk\log k=\infty$,
$\kappa^\#(0^+)<\infty$ if $\sum p_kk\log k<\infty$.)
\begin{thm} \label{thm:LK}
\begin{enumerate}
\item[(a)]Let $\newrho$ be an eternal solution of
\qref{newmoment}. Then there is a
divergent \smeas\ $\dsm$ such that
$\sm_t$ converges to $\dsm$ as $t \dnto 0$.
\item[(b)] Conversely, for every divergent \smeas\ $\dsm$,
there is a unique eternal solution $\newrho$ of \qref{newmoment}
such that $\sm_t$ converges to $\dsm$ as $t \dnto 0$.
\item[(c)] The Laplace exponent of $\dsm$ is related to the min history
$\len(t)$ by
\be
\label{lk1}
\log \lapdsm(q)= \int_0^{1} \left( 1-e^{-q\len(s)}
\right) \frac{ds}{s} - \int_{1}^\infty e^{-q\len(s)} \frac{ds}{s}.
\ee
\end{enumerate}
\end{thm}
To fix ideas, it may help to note that the self-similar solutions
are generated by the power-law Laplace exponents
\be
\label{eternal4}
\lapdsm(q) = {e^{\theta\gamma}}q^\theta, \quad \theta \in (0,1],
\ee
where $\gamma$ is the Euler-Mascheroni constant, which satisfies \cite{Olver}
\be
\label{eternal5}
\gamma = \int_0^1 \frac{1-e^{-s}}{s}\,ds - \int_1^\infty
\frac{e^{-s}}{s} \,ds.
\ee
This normalization is chosen so the min
\be
\label{eternal6}
\len_\theta(t) = t^{1/\theta}, \quad \theta \in (0,1].
\ee
The corresponding divergent \smeass\/ are given by
\be
\label{eternal7}
\dsm_\theta(x) = \frac{\theta e^{\theta\gamma}}{ \Gamma(2-\theta)}
x^{1-\theta}, \quad \theta \in (0,1].
\ee
\begin{proof}[Proof of Theorem~\ref{thm:LK}.]
In all that follows, $q>0$
is fixed, and we use the equivalence between convergence of \smeass\/
and pointwise convergence of Laplace exponents, as established in
\cite[Sec.\ 3]{MP3}, for example.
{\em 1.\/} First, assume that $\newrho$ is an eternal solution.
We claim that for all $q>0$, $\eta_t(q)\to\lapdsm(q)$ as $t\dnto0$,
where $\lapdsm$ is given by \qref{lk1} and satisfies
$\lapdsm(\infty)=\infty$.
By \cite[Sec.\ 3]{MP3} it follows that $\lapdsm$ is the Laplace
exponent of a divergent \smeas\ $H$ and $G_t\to H$ as $t\dnto0$.
{\em 2.\/} We first recall from Theorem~\ref{thm.wp} that the min history
satisfies \qref{Lest} whenever $0<t_0<t$. Regarding $t$ as fixed and
$t_0$ variable, we conclude that
\be\label{Lest2}
0<\len(s)< C s^{(\log 2)/Q_1}, \quad 0<s<1.
\ee
It follows from this, the estimate $1-e^{-q\len(s)}\le q\len(s)$,
and \qref{Lest} that the integrals in \qref{lk1} converge, so that
$\lapdsm(q)$ is finite for $0<q<\infty$.
Moreover, $\lapdsm(0)=0$ and $\lapdsm(\infty)= \int_0^1 ds/s=\infty$.
{\em 3.\/} We let $w=w_t(q)=1-\Rbar_t(q) = t\kappa^\#(t)\eta_t(q)$ and use
the solution formula \qref{newsol5} together with \qref{kappadef}
to write
\be\label{weq1}
\log(w\kappa(w)) = -\varphi(\Rbar_t(q)) = -\int_t^\infty
e^{-q\len(s)}\frac{ds}s.
\ee
Adding $-\log t=\int_t^1ds/s$ to both sides we find
\be
\log(w\kappa(w)/t) =
\int_t^{1} \left( 1-e^{-q\len(s)}
\right) \frac{ds}{s} - \int_{1}^\infty e^{-q\len(s)} \frac{ds}{s}
=\log \lapdsm(q) +o(1)
\ee
as $t\dnto0$. Hence $w\kappa(w)\sim t \lapdsm(q)$, and asymptotic
inversion yields
\be
w \sim t \lapdsm(q) \kappa^\#(t \lapdsm(q)) \sim
t \lapdsm(q) \kappa^\#(t)
\ee
since $\kappa^\#$ is slowly varying. But immediately this yields
$\eta_t(q)\to \lapdsm(q)$ as $t\dnto0$, and this finishes the proof of
(a).
{\em 4.\/} We now establish the converse. Let $\dsm$ be a
divergent \smeas\ with Laplace exponent $\lapdsm$ (not known at first
to satisfy \qref{lk1}). We will first establish that the
\LK\/ formula \qref{lk1} defines an appropriate min history $\len$,
then verify that $\len$ defines an eternal solution.
First, we remark that the definition of the trace admits a
natural modification for eternal
solutions. Given the min history of an eternal solution, we define the
trace through \qref{defA2}. Conversely, we say $A$ is a {\em maximal
trace\/} if $A$ is a distribution function on $(0,\infty)$ such that
(i) $\lim_{\tau \to 0} A(\tau)=-\infty$, (ii) $\lim_{\tau \to \infty}
A(\tau)=\infty$, and (iii) $\int_0^{\tau_0} \tau A(d\tau) <\infty$ for
some $\tau_0 >0$. In this case, the min-history is given by
\qref{defL2} with $t_0=0$.
Since $\lapdsm$ is the Laplace exponent of a \smeas, $\lapdsm'$
and $1/\lapdsm$ are completely monotone functions. Thus, there is a
positive measure $A$ such that
\be
\label{eta12}
(\log \lapdsm)' = \frac{\lapdsm'}{\lapdsm} = \int_0^\infty e^{-q\tau}
\tau A(d\tau).
\ee
For every $\tau_0>0$, the measure $A$ satisfies the finiteness conditions
\be
\label{finite1}
(\log \lapdsm) ' \geq \left\{ \begin{array}{ll} e^{-q\tau_0}
\int_0^{\tau_0} \tau A(d\tau), \\
\tau_0 \int_{\tau_0} ^\infty e^{-q\tau} A(d\tau). \end{array}\right.
\ee
Therefore, we may integrate \qref{eta12} between $q$ and $q_1 \in
(0,\infty)$ and rearrange terms to obtain
\ba
\label{eta13}
\lefteqn{ \log \lapdsm(q) - \int_0^{\tau_0} (1-e^{-q\tau})
A(d\tau) + \int_{\tau_0}^\infty e^{-q\tau} A(d\tau)}
\\
\nn
&& =
\log \lapdsm(q_1) - \int_0^{\tau_0} (1-e^{-q_1\tau})
A(d\tau) + \int_{\tau_0}^\infty e^{-q_1\tau} A(d\tau):=C(q_1,\tau_0).
\ea
The right hand side is independent of $q$. We let $q \to \infty$ on
the left hand side, and use $\eta(\infty)=\infty$ to see that
$\int_0^{\tau_0} A(d\tau)=\infty$. Similarly, we let $q \to 0$ and use
$\eta(0)=0$ to see that $\int_{\tau_0}^\infty A(d\tau)=\infty$. Thus,
$A$ defines a maximal trace. Let $\len$ denote the associated min
history given by $\len = \exp (A^{\dag})$.
{\em 5.\/} Since $\tau_0>0$ is arbitrary, we may suppose $\tau_0$ is
in the range of $\len$ and $\len(t_0)=\tau_0$. We change variables in
\qref{eta13} to obtain
\be
\label{eta14}
\log \lapdsm(q) - \int_0^1 \left(1-e^{-q\len(s)}\right) \frac{ds}{s} +
\int_1^\infty e^{-q\len(s)} \frac{ds}{s} = C+\log t_0.
\ee
In order to obtain the \LK\/ formula in the form \qref{lk1}, we
just replace $\len$ by the rescaling $\hat\len(s)=\len(as)$ where $\log a=
C+\log t_0$.
{\em 6.\/} It remains to check that the solution formula
\qref{newsol5} defines a solution for every $t>0$. This is proven by
an approximation argument. We consider a sequence of finite \smeass\/
$\sm_n$ that converge to the divergent \smeas\/ $\dsm$.
We may suppose that $c_n:=\int_0^\infty x^{-1}\sm_n(dx) \geq n$.
Let $t_n=1/c_n$ so that $0 < t_n\leq n^{-1}$.
We will show that the sequence of solutions $\newrho\nsup$
defined for $t\ge t_n$, with
initial data given by the probability measures
\[ F_n(dx):= t_n\kappash(t_n)x^{-1}\sm_n(dx), \]
converges to a solution $\newrho$ satisfying \qref{newsol5}.
{\em 7.\/}
Let $\eta_n$ be the Laplace exponent of $\sm_n$;
then $\eta_n(q)\to\lapdsm(q)$ for $q>0$. Define
\be\label{wndef}
w_n(q) := 1-\Rbar_n(q) = t_n\kappash(t_n)\eta_n(q).
\ee
This is related to the min history $\len_n$ determined from $F_n$
by the solution formula as in \qref{weq1}, namely,
\be\label{wneq}
\log( w_n\kappa(w_n)/t_n) =
\int_{t_n}^{1} \left( 1-e^{-q\len_n(s)}
\right) \frac{ds}{s} - \int_{1}^\infty e^{-q\len_n(s)} \frac{ds}{s}.
\ee
Since $w_n(q)\sim t_n\eta_n(q) \kappash(t_n\eta_n(q))$ as
$n\to\infty$, asymptotic inversion yields $w_n\kappa(w_n)\sim
t_n\eta_n(q)$, and then \qref{wneq} yields
\be
\log\lapdsm(q)
= \lim_{n \to \infty}
\int_{t_n}^1 \left(1 -e^{-q\len\nsup(s)} \right) \frac{ds}{s} -
\int_1^\infty e^{-q\len\nsup(s)}\frac{ds}{s}.
\ee
Convergence of completely monotone functions also implies convergence of all
derivatives. Thus,
\[ \frac{\lapdsm'}{\lapdsm} = \lim_{n \to \infty}
\int_{t_n}^\infty e^{-q\len\nsup(s)} \frac{\len\nsup(s)}{s} \,ds. \]
It follows that the inverse functions $\len\nsup \to \len$ at all points of
continuity. Therefore, we may let $n \to \infty$ in the solution formula
\[ \Rbar_t\nsup(q) = \forma^{-1} \left(\int_t^\infty
e^{-q\len\nsup(s)}\frac{ds}{s}\right) \]
to see that $\newrho$ defines an eternal solution.
\end{proof}
\section*{Acknowledgements}
This material is based upon work supported by the National Science
Foundation under grant nos.\
DMS 06-04420, DMS 06-05006, DMS 07-48482,
and by the Center for Nonlinear Analysis under NSF grants
DMS 04-05343 and 06-35983.
RLP and BN thank the DFG for partial support through a Mercator
professorship for RLP at Humboldt University and through the Research Group
{\it Analysis and Stochastics in Complex Physical Systems}.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,812
|
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{
"redpajama_set_name": "RedPajamaC4"
}
| 2,348
|
Сен-Маклу́-ла-Бріє́р () — муніципалітет у Франції, у регіоні Нормандія, департамент Приморська Сена. Населення — осіб (2011).
Муніципалітет розташований на відстані близько 165 км на північний захід від Парижа, 55 км на північний захід від Руана.
Історія
До 2015 року муніципалітет перебував у складі регіону Верхня Нормандія. Від 1 січня 2016 року належить до нового об'єднаного регіону Нормандія.
Демографія
Розподіл населення за віком та статтю (2006):
Економіка
У 2010 році в муніципалітеті числилось 176 оподаткованих домогосподарств, у яких проживали 481,5 особи, медіана доходів виносила євро на одного особоспоживача
Посилання
Сен-Маклу-ла-Брієр на сайті французького Національного інституту географії
[ Розташування муніципалітету Сен-Маклу-ла-Брієр на мапі Франції та сусідні муніципалітети]
Див. також
Список муніципалітетів департаменту Приморська Сена
Примітки
Муніципалітети департаменту Приморська Сена
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 2,516
|
At what grade level did you start Palmetto Christian Academy?
I started at Palmetto Christian Academy as a freshman in high school, at the start of my 9th grade year. I have been a student at PCA for the last four years now!
Are you involved in extracurricular activities?
If so, what are they and have you received awards in these activities? I have been involved in the National Honor Society, Future Business Leaders of America, the Worship Team, the Science Club, and the Girls Varsity Golf Team. For the Future Business Leaders of America, I have competed at the National Competition for 3 Years, I won first in all my events at the District Competition for 3 years, and enjoyed serving as the Chapter Secretary of the PCA FBLA team. I also served as the Co-Captain of the Girls Varsity Golf Team, and I have had a lot of fun leading the PCA Science Club as President for its first official year. PCA presented me with so many fun extracurricular opportunities, and I have enjoyed being a part of each one!
I plan to attend Clemson University in the Fall to obtain a Bachelor's Degree in Mechanical Engineering. In the future, I would like to possibly work for Boeing, or I would like to work for NASA's Jet Propulsion Laboratory or the Mars Science Laboratory.
If your college or university has released scholarship recipients, have you received scholarship money?
I have received the South Carolina LIFE Scholarship!
How has PCA made a difference in, or impacted your life? Do you feel that Palmetto Christian Academy has prepared you for life after graduation?
Palmetto Christian Academy has well and truly been the biggest thing that has ever happened to me. Over the last four years, I have had the incredible opportunity to grow and flourish in my relationship and understanding of the Lord in a supportive environment. I have also been able to build great relationships with good people and it has been a learning experience that has allowed me to step out and do things I would have never dreamt about. I have been able to push boundaries and develop in so many aspects, including leadership. Palmetto Christian Academy has greatly contributed to who I have become today, and I will take everything I have learned here with me as I embark on the next chapter in my life. I want to thank all the kind people who have generously invested their time and energy into me over the course of the last four years. I will never forget how PCA has contributed to my life, and I am excited to see how my preparations for college here at PCA come to fruition!
These students have lived, learned and grown with us here at PCA. Let's celebrate together what God has in store for their future.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,816
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NASHVILLE, Tenn. (AP) - State. Rep. Joe Armstrong says he hopes the "truth comes out" when he goes to trial on federal fraud and tax evasion charges.
The Knoxville Democrat has a Feb. 23 trial date on charges in connection to an increase in the state's cigarette tax.
A June 16 indictment accuses him of devising a scheme beginning in 2006 to profit from the cigarette tax increase planned by then-Gov. Phil Bredesen.
According to the charges, Armstrong borrowed $250,000 to buy tax stamps at the old 20-cent rate, and then sold them at a profit after lawmakers approved a 42-cent increase.
Armstrong is accused of failing to report more than $318,000 in income. He has pleaded not guilty to the charges.
Armstrong told The Associated Press in a phone interview Thursday that he's optimistic about the outcome of the trial.
He said he quietly stepped down from his position as House minority leader pro tempore in the House Democratic Caucus to concentrate on his tax issue.
Shortly after his indictment in June, he said he sent an email to fellow Democrats, which is in compliance with House Democratic Caucus rules calling for resignation from any leadership position upon indictment.
"At this point, the position is still open for me after I'm vindicated to come back," he said.
House Democratic Leader Craig Fitzhugh of Ripley says Armstrong has the caucus' support.
The House Ethics Committee is likely to look into the allegations against Armstrong, but a meeting has not yet been scheduled.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,990
|
Glitch halts US visa issuance worldwide
Published on: June 17, 2015 at 11:31 pm
Last Modified: June 17, 2015 at 11:32 pm
Blank US visitors visa
A technical glitch is affecting the issuance of visas and passports at US Embassies worldwide. The glitch has prevented the U.S. from issuing thousands of visas since June 9, a statement issued by the Department of State says.
The glitch has affected the flow of biometric clearance requests from U.S. consulates world-wide to the Consular Consolidated Database (CCD). Biometric data, including fingerprints, are used for security screening of applicants.
"The system that helps perform necessary security checks has suffered hardware failure," the statement reads. "Until it is repaired, no visas can be issued."
"Our border security responsibilities are critical to the visa process. We cannot bypass the legal requirements to screen visa applicants before we issue visas for travel," the statement adds.
The US embassy in Kampala posted on its website that it is unable to print most immigrant and non-immigrant visas approved after June 8, 2015.
"Individuals with visa interview appointments scheduled for June 14-20, 2015, should reschedule their appointments if they submitted a DS-160 online application after June 9, 2015," a post on the US embassy website in Kampala reads.
An official at the Embassy in Kampala said technicians are working around the clock to resolve the issue but couldn't offer a timeline for when the system would be back in action.
The issue is not specific to any particular country or visa category. The statement says the problem is a global issue and ruled out the possibility that it stemmed from any cyber-security hacking issues.
Persons who applied for a US passport overseas after June 9th and those with travel plans within the next 10 business days have been advised to request for emergency services from the embassy in their countries.
In 2014, U.S consular offices issued 9.9 million non-immigrant visas and an additional 467,370 immigrant visas.
Below is the Statement Issued by the Department of Statement
The Bureau of Consular Affairs continues to address technical problems with our visa systems. Some visa applicants will experience delays in receiving visas. Others will be contacted directly to reschedule their appointments. Passports are still being processed.
This issue is not specific to any particular country or visa category; this is a global issue. We do not believe these problems stem from any cyber-security hacking issues.
Our border security responsibilities are critical to the visa process. We cannot bypass the legal requirements to screen visa applicants before we issue visas for travel. We are assisting visa applicants with urgent humanitarian travel, and adoption cases are being processed. Individuals with humanitarian travel needs should contact their nearest U.S. embassy or consulate.
Q: What caused this outage? Was it a malicious action or hack?
There is no evidence the problem is cyber security related.
Q: How long before you restore full system functionality?
Overseas passports are being issued. Public and private sectors experts are working around the clock to correct the problem, but we do not expect the system will be online before next week.
Q: How many travelers are affected by this outage?
This is a global issue and is having impact on our consular operations. We are in daily communication with consular operations worldwide to assess the evolving impact and provide guidance, including information for the traveling public.
Q: Once operational, how will cases be prioritized?
Once the systems are fully operational, we will work as quickly as possible to clear the backlog of pending visa cases.
We apologize to applicants and recognize that this may cause hardship to applicants waiting for visas.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 5,273
|
I am running cockpit via the author maintained docker. I'm wondering if there's a good way to secure the login page against brute force? I use nginx and fail2ban with my site, but I'm not sure how to get fail2ban to monitor the docker's logs? Or if there is a simpler way to add a timeout function to the login page?
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,802
|
<?php
/**
* Afrikaans Language file for WebDB.
* @maintainer Petri Jooste [rkwjpj@puk.ac.za]
*
* $Id: afrikaans.php,v 1.9 2007/04/24 11:42:07 soranzo Exp $
*/
// Language and character set
$lang['applang'] = 'Afrikaans';
$lang['appcharset'] = 'ISO-8859-1';
$lang['applocale'] = 'af_ZA';
$lang['appdbencoding'] = 'LATIN1';
$lang['applangdir'] = 'ltr';
// Welcome
$lang['strintro'] = 'Welkom by phpPgAdmin.';
$lang['strppahome'] = 'phpPgAdmin Tuisblad';
$lang['strpgsqlhome'] = 'PostgreSQL Tuisblad';
$lang['strpgsqlhome_url'] = 'http://www.postgresql.org/';
$lang['strlocaldocs'] = 'PostgreSQL Dokumentasie (lokaal)';
$lang['strreportbug'] = 'Meld \'n fout aan';
$lang['strviewfaq'] = 'Bekyk FAQ op internet';
$lang['strviewfaq_url'] = 'http://phppgadmin.sourceforge.net/?page=faq';
// Basic strings
$lang['strlogin'] = 'Aanteken';
$lang['strloginfailed'] = 'Aantekening het misluk';
$lang['strlogindisallowed'] = 'Aantekening nie toegelaat nie';
$lang['strserver'] = 'Bediener';
$lang['strlogout'] = 'Teken af';
$lang['strowner'] = 'Eienaar';
$lang['straction'] = 'Aksie';
$lang['stractions'] = 'Aksies';
$lang['strname'] = 'Naam';
$lang['strdefinition'] = 'Definisie';
$lang['strproperties'] = 'Eienskappe';
$lang['strbrowse'] = 'Bekyk';
$lang['strdrop'] = 'Verwyder';
$lang['strdropped'] = 'Is verwyder';
$lang['strnull'] = 'Null';
$lang['strnotnull'] = 'Nie Null';
$lang['strprev'] = 'Vorige';
$lang['strnext'] = 'Volgende';
$lang['strfirst'] = '<< Eerste';
$lang['strlast'] = 'Laaste >>';
$lang['strfailed'] = 'Het misluk';
$lang['strcreate'] = 'Skep';
$lang['strcreated'] = 'Geskep';
$lang['strcomment'] = 'Kommentaar';
$lang['strlength'] = 'Lengte';
$lang['strdefault'] = 'Standaard';
$lang['stralter'] = 'Wysig';
$lang['strok'] = 'OK';
$lang['strcancel'] = 'Kanselleer';
$lang['strsave'] = 'Bewaar';
$lang['strreset'] = 'Herstel';
$lang['strinsert'] = 'Voeg in';
$lang['strselect'] = 'Selekteer';
$lang['strdelete'] = 'Verwyder';
$lang['strupdate'] = 'Verfris';
$lang['strreferences'] = 'Verwysings';
$lang['stryes'] = 'Ja';
$lang['strno'] = 'Nee';
$lang['strtrue'] = 'WAAR';
$lang['strfalse'] = 'VALS';
$lang['stredit'] = 'Redigeer';
$lang['strcolumn'] = 'Kolom';
$lang['strcolumns'] = 'Kolomme';
$lang['strrows'] = 'ry(e)';
$lang['strrowsaff'] = 'ry(e) het verander.';
$lang['strobjects'] = 'objek(te)';
$lang['strback'] = 'Terug';
$lang['strqueryresults'] = 'Navraagresultate';
$lang['strshow'] = 'Wys';
$lang['strempty'] = 'Leeg';
$lang['strlanguage'] = 'Taal';
$lang['strencoding'] = 'Enkodering';
$lang['strvalue'] = 'Waarde';
$lang['strunique'] = 'Uniek';
$lang['strprimary'] = 'Primêr';
$lang['strexport'] = 'Eksporteer';
$lang['strimport'] = 'Importeer';
$lang['strsql'] = 'SQL';
$lang['stradmin'] = 'Admin';
$lang['strvacuum'] = 'Stofsuig';
$lang['stranalyze'] = 'Analiseer';
$lang['strcluster'] = 'Kluster';
$lang['strclustered'] = 'In klusters?';
$lang['strreindex'] = 'Herindekseer';
$lang['strrun'] = 'Loop';
$lang['stradd'] = 'Voeg by';
$lang['strevent'] = 'Gebeurtenis';
$lang['strwhere'] = 'Waar';
$lang['strinstead'] = 'Doen eerder';
$lang['strwhen'] = 'Wanneer';
$lang['strformat'] = 'Formaat';
$lang['strdata'] = 'Data';
$lang['strconfirm'] = 'Bevestig';
$lang['strexpression'] = 'Uitdrukking';
$lang['strellipsis'] = '...';
$lang['strseparator'] = ': ';
$lang['strexpand'] = 'Vou oop';
$lang['strcollapse'] = 'Vou toe';
$lang['strexplain'] = 'Verduidelik';
$lang['strexplainanalyze'] = 'Verduidelik Analise';
$lang['strfind'] = 'Soek';
$lang['stroptions'] = 'Opsies';
$lang['strrefresh'] = 'Verfris';
$lang['strdownload'] = 'Laai af';
$lang['strdownloadgzipped'] = 'Laai af ... saamgepers met gzip';
$lang['strinfo'] = 'Info';
$lang['stroids'] = 'OIDs';
$lang['stradvanced'] = 'Gevorderd';
$lang['strvariables'] = 'Veranderlikes';
$lang['strprocess'] = 'Proses';
$lang['strprocesses'] = 'Prosesse';
$lang['strsetting'] = 'Instelling';
$lang['streditsql'] = 'Redigeer SQL';
$lang['strruntime'] = 'Totale looptyd: %s ms';
$lang['strpaginate'] = 'Resultate per bladsy';
$lang['struploadscript'] = 'of laai \'n SQL skrip in:';
$lang['strstarttime'] = 'Begintyd';
$lang['strfile'] = 'Lêer';
$lang['strfileimported'] = 'Lêer is ingetrek.';
// Error handling
$lang['strnoframes'] = 'Hierdie toepassing maak gebruik van HTML-rame. U het \'n blaaier nodig wat rame ondersteun om hierdie toepassing te kan gebruik. ';
$lang['strbadconfig'] = 'Die lêer config.inc.php is verouderd. Jy kan verbeterde weergawe aflei van die lêer config.inc.php-dist.';
$lang['strnotloaded'] = 'Hierdie PHP-installasie is sonder ondersteuning van hierdie tipe database nie gekompileerd.';
$lang['strpostgresqlversionnotsupported'] = 'Weergawe van PostgreSQL word nie ondersteun nie. Probeer asb. weergawe %s of later.';
$lang['strbadschema'] = 'Ongeldige skema gespesifiseer.';
$lang['strbadencoding'] = 'Die kliëntenkodering kon nie in die databasis geplaas word nie.';
$lang['strsqlerror'] = 'SQL-fout:';
$lang['strinstatement'] = 'In stelling:';
$lang['strinvalidparam'] = 'Ongeldige parameters.';
$lang['strnodata'] = 'Geen rye gevind.';
$lang['strnoobjects'] = 'Geen objekte gevind.';
$lang['strrownotunique'] = 'Geen unieke identifiseerder vir hierdie ry.';
$lang['strnoreportsdb'] = 'Jy het nie die verslae-databasis geskep nie. Lees asb. die INSTALL-lêer vir instruksies.';
$lang['strnouploads'] = 'Oplaaiing van lêers is afgeskakel.';
$lang['strimporterror'] = 'Inleesfout.';
$lang['strimporterrorline'] = 'Inleesfout op reël %s.';
$lang['strcannotdumponwindows'] = 'Weergee van komplekse tabel- en skemaname word nie op Windows ondersteun nie. Kyk asb. in die FAQ.';
// Tables
$lang['strtable'] = 'Tabel';
$lang['strtables'] = 'Tabelle';
$lang['strshowalltables'] = 'Wys alle tabelle';
$lang['strnotables'] = 'Geen tabelle gevind.';
$lang['strnotable'] = 'Geen tabel gevind.';
$lang['strcreatetable'] = 'Skep tabel';
$lang['strtablename'] = 'Tabelnaam';
$lang['strtableneedsname'] = 'Jy moet die tabel \'n naam gee.';
$lang['strtableneedsfield'] = 'Jy moet ten minste een veld spesifiseer.';
$lang['strtableneedscols'] = 'Jy moet die tabel \'n geldige aantal kolomme gee.';
$lang['strtablecreated'] = 'Tabel geskep.';
$lang['strtablecreatedbad'] = 'Die tabel kon nie geskep word nie.';
$lang['strconfdroptable'] = 'Is jy seker dat dat jy die tabel "%s" wil verwyder?';
$lang['strtabledropped'] = 'Tabel is verwyder.';
$lang['strtabledroppedbad'] = 'Die tabel kon nie verwyder word nie.';
$lang['strconfemptytable'] = 'Is jy seker dat jy alle rye uit tabel "%s" wil verwyder?';
$lang['strtableemptied'] = 'Alle ryen is uit die tabel verwyder.';
$lang['strtableemptiedbad'] = 'Die rye kon nie verwyder word nie.';
$lang['strinsertrow'] = 'Voeg \'n ry by';
$lang['strrowinserted'] = 'Ry is bygevoeg.';
$lang['strrowinsertedbad'] = 'Die ry kon nie bygevoeg word nie.';
$lang['streditrow'] = 'Wysig ry';
$lang['strrowupdated'] = 'Ry is opgedateer.';
$lang['strrowupdatedbad'] = 'Die opdatering van die ry het misluk.';
$lang['strdeleterow'] = 'Verwyder ry';
$lang['strconfdeleterow'] = 'Is jy seker dat jy hierdie ry wil verwyder?';
$lang['strrowdeleted'] = 'Ry is verwyder.';
$lang['strrowdeletedbad'] = 'Die ry kon nie verwyder word nie.';
$lang['strinsertandrepeat'] = 'Voeg in & Herhaal';
$lang['strnumcols'] = 'Aantal kolomme';
$lang['strcolneedsname'] = 'Jy moet die kolom \'n naam gee';
$lang['strselectallfields'] = 'Selekteer alle velde';
$lang['strselectneedscol'] = 'Jy moet ten minste één kolom as uitvoer hê';
$lang['strselectunary'] = 'Unêre operatore kan nie waardes kry nie.';
$lang['straltercolumn'] = 'Wysig kolom';
$lang['strcolumnaltered'] = 'Kolom is gewysig.';
$lang['strcolumnalteredbad'] = 'Die kolom kon nie gewysig word nie.';
$lang['strconfdropcolumn'] = 'Is jy seker dat jy die kolom "%s" wil verwyder uit tabel "%s"?';
$lang['strcolumndropped'] = 'Kolom is verwyder.';
$lang['strcolumndroppedbad'] = 'Die kolom kon nie verwyder word nie.';
$lang['straddcolumn'] = 'Voeg kolom by';
$lang['strcolumnadded'] = 'Kolom is bygevoeg.';
$lang['strcolumnaddedbad'] = 'Die kolom kon nie bygevoeg word nie.';
$lang['strcascade'] = 'CASCADE';
$lang['strtablealtered'] = 'Tabel is gewysig.';
$lang['strtablealteredbad'] = 'Tabelwysiging het misluk.';
$lang['strdataonly'] = 'Slegs data';
$lang['strstructureonly'] = 'Slegs struktuur';
$lang['strstructureanddata'] = 'Struktuur en data';
$lang['strtabbed'] = 'Tabbed';
$lang['strauto'] = 'Auto';
$lang['strconfvacuumtable'] = 'Is jy seker jy wil VACUUM "%s"?';
$lang['strestimatedrowcount'] = 'Geskatte aantal rye';
// Users
$lang['struser'] = 'Gebruiker';
$lang['strusers'] = 'Gebruikers';
$lang['strusername'] = 'Gebruikersnaam';
$lang['strpassword'] = 'Wagwoord';
$lang['strsuper'] = 'Supergebruiker?';
$lang['strcreatedb'] = 'Skep DB?';
$lang['strexpires'] = 'Verval';
$lang['strsessiondefaults'] = 'Verstekwaardes van sessie';
$lang['strnousers'] = 'Geen gebruikers gevind.';
$lang['struserupdated'] = 'Gebruiker is opgedateer.';
$lang['struserupdatedbad'] = 'Gebruiker kon nie opgedateer word nie.';
$lang['strshowallusers'] = 'Wys alle gebruikers';
$lang['strcreateuser'] = 'Skep gebruiker';
$lang['struserneedsname'] = 'Jy moet \'n naam gee vir die gebruiker.';
$lang['strusercreated'] = 'Gebruiker geskep.';
$lang['strusercreatedbad'] = 'Die gebruiker kon nie geskep word nie.';
$lang['strconfdropuser'] = 'Is jy seker dat jy die gebruiker "%s" wil verwyder?';
$lang['struserdropped'] = 'Gebruiker is verwyder.';
$lang['struserdroppedbad'] = 'Verwydering van die gebruiker het misluk.';
$lang['straccount'] = 'Gebruiker';
$lang['strchangepassword'] = 'Verander wagwoord';
$lang['strpasswordchanged'] = 'Wagwoord is verander.';
$lang['strpasswordchangedbad'] = 'Wagwoordverandering het misluk.';
$lang['strpasswordshort'] = 'Wagwoord is te kort.';
$lang['strpasswordconfirm'] = 'Wagwoord verskil van bevestigings-wagwoord.';
// Groups
$lang['strgroup'] = 'Groep';
$lang['strgroups'] = 'Groepe';
$lang['strnogroup'] = 'Groep nie gevind.';
$lang['strnogroups'] = 'Geen groepe gevind.';
$lang['strcreategroup'] = 'Skep groep';
$lang['strshowallgroups'] = 'Wys alle groepe';
$lang['strgroupneedsname'] = 'Jy moet die groep \'n naam gee.';
$lang['strgroupcreated'] = 'Groep geskep.';
$lang['strgroupcreatedbad'] = 'Die groep kon nie geskep word nie.';
$lang['strconfdropgroup'] = 'Is jy seker dat jy die groep "%s" wil verwyder?';
$lang['strgroupdropped'] = 'Groep is verwyder.';
$lang['strgroupdroppedbad'] = 'Verwydering van die groep het misluk.';
$lang['strmembers'] = 'Lede';
$lang['straddmember'] = 'Voeg \'n groeplid by';
$lang['strmemberadded'] = 'Groeplid is bygevoeg.';
$lang['strmemberaddedbad'] = 'Toevoeging van groeplid het misluk.';
$lang['strdropmember'] = 'Verwyder groeplid';
$lang['strconfdropmember'] = 'Is jy seker dat jy "%s" uit groep "%s" wil verwyder?';
$lang['strmemberdropped'] = 'Groeplid is verwyder.';
$lang['strmemberdroppedbad'] = 'Verwydering van groeplid het misluk.';
// Privileges
$lang['strprivilege'] = 'Voorregte';
$lang['strprivileges'] = 'Voorregte';
$lang['strnoprivileges'] = 'Hierdie objek het verstekeienaarvoorregte.';
$lang['strgrant'] = 'Staan toe';
$lang['strrevoke'] = 'Ontneem';
$lang['strgranted'] = 'Voorregte is bygevoeg.';
$lang['strgrantfailed'] = 'Voorregte kon nie bygevoeg word nie.';
$lang['strgrantbad'] = 'Jy moet minstens een gebruiker of groep en minstens een voorreg aandui.';
$lang['strgrantor'] = 'Grantor';
$lang['strasterisk'] = '*';
// Databases
$lang['strdatabase'] = 'Databasis';
$lang['strdatabases'] = 'Databasisse';
$lang['strshowalldatabases'] = 'Wys alle databasisse';
$lang['strnodatabase'] = 'Geen databasis gevind.';
$lang['strnodatabases'] = 'Geen databasis gevind.';
$lang['strcreatedatabase'] = 'Skep databasis';
$lang['strdatabasename'] = 'Databasisnaam';
$lang['strdatabaseneedsname'] = 'Jy moet die databasis \'n naam gee.';
$lang['strdatabasecreated'] = 'Databasis is geskep.';
$lang['strdatabasecreatedbad'] = 'Die databasis kon nie geskep word nie.';
$lang['strconfdropdatabase'] = 'Is jy seker dat jy die databasis "%s" wil verwyder?';
$lang['strdatabasedropped'] = 'Databasis is verwyder.';
$lang['strdatabasedroppedbad'] = 'Databasisverwydering het misluk.';
$lang['strentersql'] = 'Tik hieronder die SQL in wat uitgevoer moet word:';
$lang['strsqlexecuted'] = 'SQL uitgevoer.';
$lang['strvacuumgood'] = 'Vacuum-bewerking is klaar.';
$lang['strvacuumbad'] = 'Vacuum-bewerking het misluk.';
$lang['stranalyzegood'] = 'Analise is voltooi.';
$lang['stranalyzebad'] = 'Analise het misluk.';
$lang['strreindexgood'] = 'Herindeksering is voltooi.';
$lang['strreindexbad'] = 'Herindeksering het misluk.';
$lang['strfull'] = 'Volledig';
$lang['strfreeze'] = 'Vries';
$lang['strforce'] = 'Forseer';
$lang['strsignalsent'] = 'Sein gestuur.';
$lang['strsignalsentbad'] = 'Stuur van sein het misluk.';
$lang['strallobjects'] = 'Alle objekte';
// Views
$lang['strview'] = 'Aansig';
$lang['strviews'] = 'Aansigte';
$lang['strshowallviews'] = 'Wys alle aansigte';
$lang['strnoview'] = 'Geen aansigte gevind.';
$lang['strnoviews'] = 'Geen aansigte gevind.';
$lang['strcreateview'] = 'Skep aansig';
$lang['strviewname'] = 'Aansignaam';
$lang['strviewneedsname'] = 'Jy moet die aansig \'n naam gee.';
$lang['strviewneedsdef'] = 'Jy moet die aansig definieer.';
$lang['strviewneedsfields'] = 'Jy moet sê watter kolomme gekies moet wees in hierdie aansig.';
$lang['strviewcreated'] = 'Aansig is geskep.';
$lang['strviewcreatedbad'] = 'Die aansig kon nie geskep word nie.';
$lang['strconfdropview'] = 'Is jy seker dat jy die aansig "%s" wil verwyder?';
$lang['strviewdropped'] = 'Aansig is verwyder.';
$lang['strviewdroppedbad'] = 'Die aansig kon nie verwyder word nie.';
$lang['strviewupdated'] = 'Aansig is opgedateer.';
$lang['strviewupdatedbad'] = 'Opdatering van aansig het misluk.';
$lang['strviewlink'] = 'Sleutels word verbind';
$lang['strviewconditions'] = 'Addisionele voorwaardes';
$lang['strcreateviewwiz'] = 'Skep \'n aansig met behulp van \'n toergids';
// Sequences
$lang['strsequence'] = 'Reeks';
$lang['strsequences'] = 'Reekse';
$lang['strshowallsequences'] = 'Wys alle reekse';
$lang['strnosequence'] = 'Geen reeks gevind.';
$lang['strnosequences'] = 'Geen reekse gevind.';
$lang['strcreatesequence'] = 'Skep reeks';
$lang['strlastvalue'] = 'Laaste waarde';
$lang['strincrementby'] = 'Verhoog met';
$lang['strstartvalue'] = 'Aanvangswaarde';
$lang['strmaxvalue'] = 'maks_waarde';
$lang['strminvalue'] = 'min_waarde';
$lang['strcachevalue'] = 'Kasgeheue-waarde';
$lang['strlogcount'] = 'Boekstaaftelling';
$lang['striscycled'] = 'is_siklies ?';
$lang['strsequenceneedsname'] = 'Jy moet \'n naam gee vir die reeks.';
$lang['strsequencecreated'] = 'Reeks is geskep.';
$lang['strsequencecreatedbad'] = 'Die reeks kon nie geskep word nie.';
$lang['strconfdropsequence'] = 'Is jy seker dat jy die reeks "%s" wil verwyder?';
$lang['strsequencedropped'] = 'Reeks is verwyder.';
$lang['strsequencedroppedbad'] = 'Verwydering van die reeks het misluk.';
$lang['strsequencereset'] = 'Herstel reeks.';
$lang['strsequenceresetbad'] = 'Herstel van reeks het misluk.';
// Indexes
$lang['strindex'] = 'Indeks';
$lang['strindexes'] = 'Indekse';
$lang['strindexname'] = 'Indeksnaam';
$lang['strshowallindexes'] = 'Wys alle indekse';
$lang['strnoindex'] = 'Geen indeks gevind.';
$lang['strnoindexes'] = 'Geen indekse gevind.';
$lang['strcreateindex'] = 'Skep \'n indeks';
$lang['strtabname'] = 'Tab-naam';
$lang['strcolumnname'] = 'Kolomnaam';
$lang['strindexneedsname'] = 'Jy moet \'n naam gee vir die index.';
$lang['strindexneedscols'] = 'Indekse moet ten minste uit één kolom bestaan.';
$lang['strindexcreated'] = 'Indeks is geskep';
$lang['strindexcreatedbad'] = 'Die indeks kon nie geskep word nie.';
$lang['strconfdropindex'] = 'Is jy seker dat jy die indeks "%s" wil verwyder?';
$lang['strindexdropped'] = 'Indeks is verwyder.';
$lang['strindexdroppedbad'] = 'Verwydering van die indeks het misluk.';
$lang['strkeyname'] = 'Sleutelnaam';
$lang['struniquekey'] = 'Unieke sleutel';
$lang['strprimarykey'] = 'Primêre sleutel';
$lang['strindextype'] = 'Tipe van die indeks';
$lang['strtablecolumnlist'] = 'Kolomme in tabel';
$lang['strindexcolumnlist'] = 'Kolomme in indeks';
$lang['strconfcluster'] = 'Is jy seker jy wil \'n kluster maak van "%s"?';
$lang['strclusteredgood'] = 'Kluster is voltooi.';
$lang['strclusteredbad'] = 'Kluster het misluk.';
// Rules
$lang['strrules'] = 'Reëls';
$lang['strrule'] = 'Reël';
$lang['strshowallrules'] = 'Wys alle reëls';
$lang['strnorule'] = 'Geen reël gevind.';
$lang['strnorules'] = 'Geen reëls gevind.';
$lang['strcreaterule'] = 'Skep \'n reël';
$lang['strrulename'] = 'Reëlnaam';
$lang['strruleneedsname'] = 'Jy moet \'n naam gee vir die reël.';
$lang['strrulecreated'] = 'Reël is geskep.';
$lang['strrulecreatedbad'] = 'Die reël kon nie geskep word nie.';
$lang['strconfdroprule'] = 'Is jy seker dat jy die reël "%s" op "%s" wil verwyder?';
$lang['strruledropped'] = 'Reël is verwyder.';
$lang['strruledroppedbad'] = 'Verwydering van die reël het misluk.';
// Constraints
$lang['strconstraints'] = 'Beperkings';
$lang['strshowallconstraints'] = 'Wys alle beperkings';
$lang['strnoconstraints'] = 'Geen beperkings gevind.';
$lang['strcreateconstraint'] = 'Skep beperking';
$lang['strconstraintcreated'] = 'Beperking is geskep.';
$lang['strconstraintcreatedbad'] = 'Die beperking kon nie geskep word nie.';
$lang['strconfdropconstraint'] = 'Is jy seker dat jy die beperking "%s" op "%s" wil verwyder?';
$lang['strconstraintdropped'] = 'Beperking is verwyder.';
$lang['strconstraintdroppedbad'] = 'Verwydering van die beperking het misluk.';
$lang['straddcheck'] = 'Voeg \'n kontrole by';
$lang['strcheckneedsdefinition'] = 'Kontrolebeperking moet gedefinieer wees.';
$lang['strcheckadded'] = 'Kontrolebeperking is bygevoeg.';
$lang['strcheckaddedbad'] = 'Kontrolebeperking kon nie bygevoeg word nie.';
$lang['straddpk'] = 'Voeg primêre sleutel by';
$lang['strpkneedscols'] = 'Primêre sleutel moet minstens één kolom hê.';
$lang['strpkadded'] = 'Primêre sleutel bygevoeg.';
$lang['strpkaddedbad'] = 'Primêre sleutel kon nie bygevoeg word nie.';
$lang['stradduniq'] = 'Voeg unieke sleutel by.';
$lang['struniqneedscols'] = 'Unieke sleutel moet minstens één kolom hê.';
$lang['struniqadded'] = 'Unieke sleutel is bygevoeg.';
$lang['struniqaddedbad'] = 'Unieke sleutel kon nie bygevoeg word nie.';
$lang['straddfk'] = 'Voeg vreemdesleutel toe';
$lang['strfkneedscols'] = 'Vreemdesleutel moet minstens één kolom hê.';
$lang['strfkneedstarget'] = 'Vreemdesleutel moet \'n doeltabel hê.';
$lang['strfkadded'] = 'Vreemdesleutel is bygevoeg.';
$lang['strfkaddedbad'] = 'Vreemdesleutel kon nie bygevoeg word nie.';
$lang['strfktarget'] = 'Doeltabel';
$lang['strfkcolumnlist'] = 'Kolomme in sleutel';
$lang['strondelete'] = 'ON DELETE';
$lang['stronupdate'] = 'ON UPDATE';
// Functions
$lang['strfunction'] = 'Funksie';
$lang['strfunctions'] = 'Funksies';
$lang['strshowallfunctions'] = 'Wys alle funksies';
$lang['strnofunction'] = 'Geen funksies gevind.';
$lang['strnofunctions'] = 'Geen funksies gevind.';
$lang['strcreateplfunction'] = 'Skep SQL/PL funksie';
$lang['strcreateinternalfunction'] = 'Skep interne funksie';
$lang['strcreatecfunction'] = 'Skep C funksie';
$lang['strfunctionname'] = 'Funksienaam';
$lang['strreturns'] = 'Gee terug';
$lang['strarguments'] = 'Argumente';
$lang['strproglanguage'] = 'Programmeertaal';
$lang['strfunctionneedsname'] = 'Jy moet die funksie \'n naam gee.';
$lang['strfunctionneedsdef'] = 'Jy moet die funksie definieer.';
$lang['strfunctioncreated'] = 'Funksie is geskep.';
$lang['strfunctioncreatedbad'] = 'Die funksie kon nie geskep word nie.';
$lang['strconfdropfunction'] = 'Is jy seker dat jy die funksie "%s" wil verwyder?';
$lang['strfunctiondropped'] = 'Funksie is verwyder.';
$lang['strfunctiondroppedbad'] = 'Verwydering van die funksie het misluk.';
$lang['strfunctionupdated'] = 'Funksie is opgedateer.';
$lang['strfunctionupdatedbad'] = 'Opdatering van die funksie het misluk.';
$lang['strobjectfile'] = 'Objeklêer';
$lang['strlinksymbol'] = 'Skakelsimbool';
// Triggers
$lang['strtrigger'] = 'Snellers';
$lang['strtriggers'] = 'Snellers';
$lang['strshowalltriggers'] = 'Wys alle snellers';
$lang['strnotrigger'] = 'Geen sneller gevind.';
$lang['strnotriggers'] = 'Geen snellers gevind.';
$lang['strcreatetrigger'] = 'skep trigger';
$lang['strtriggerneedsname'] = 'Jy moet vir die sneller \'n naam gee.';
$lang['strtriggerneedsfunc'] = 'Jy moet vir die sneller \'n funksie gee.';
$lang['strtriggercreated'] = 'Sneller is geskep.';
$lang['strtriggercreatedbad'] = 'Die sneller kon nie geskep word nie.';
$lang['strconfdroptrigger'] = 'Is jy seker dat jy die sneller "%s" op "%s" wil verwyder?';
$lang['strtriggerdropped'] = 'Sneller is verwyder.';
$lang['strtriggerdroppedbad'] = 'Verwydering van sneller misluk.';
$lang['strtriggeraltered'] = 'Sneller is gewysig.';
$lang['strtriggeralteredbad'] = 'Snellerwysiging het misluk.';
// Types
$lang['strtype'] = 'Tipe';
$lang['strtypes'] = 'Tipes';
$lang['strshowalltypes'] = 'Wys alle tipes';
$lang['strnotype'] = 'Geen tipe gevind.';
$lang['strnotypes'] = 'Geen tipes gevind.';
$lang['strcreatetype'] = 'skep tipe';
$lang['strcreatecomptype'] = 'Skep saamgestelde tipe';
$lang['strtypeneedsfield'] = 'Jy moet ten minste een veld spesifiseer.';
$lang['strtypeneedscols'] = 'Jy \'n geldige aantal velde spesifiseer.';
$lang['strtypename'] = 'Tipenaam';
$lang['strinputfn'] = 'Toevoerfunksie';
$lang['stroutputfn'] = 'Afvoerfunksie';
$lang['strpassbyval'] = 'Aangestuur per waarde?';
$lang['stralignment'] = 'Belyning';
$lang['strelement'] = 'Element';
$lang['strdelimiter'] = 'Skeidingsteken';
$lang['strstorage'] = 'Berging';
$lang['strfield'] = 'Veld';
$lang['strnumfields'] = 'Aantal velde';
$lang['strtypeneedsname'] = 'Jy moet die tipe \'n naam gee.';
$lang['strtypeneedslen'] = 'Jy moet die tipe \'n lengte gee.';
$lang['strtypecreated'] = 'Tipe geskep';
$lang['strtypecreatedbad'] = 'Tipeskepping het misluk.';
$lang['strconfdroptype'] = 'Is jy seker dat jy die tipe \"%s\" wil verwyder?';
$lang['strtypedropped'] = 'Tipe is verwyder.';
$lang['strtypedroppedbad'] = 'Verwydering van die tipe het misluk.';
$lang['strflavor'] = 'Geur';
$lang['strbasetype'] = 'Basis';
$lang['strcompositetype'] = 'Saamgestel';
$lang['strpseudotype'] = 'Pseudo';
// Schemas
$lang['strschema'] = 'Skema';
$lang['strschemas'] = 'Skemas';
$lang['strshowallschemas'] = 'Wys alle skemas';
$lang['strnoschema'] = 'Geen skema gevind.';
$lang['strnoschemas'] = 'Geen skemas gevind.';
$lang['strcreateschema'] = 'Skep skema';
$lang['strschemaname'] = 'Skemanaam';
$lang['strschemaneedsname'] = 'Jy moet \'n naam gee vir die skema.';
$lang['strschemacreated'] = 'Skema is geskep';
$lang['strschemacreatedbad'] = 'Die skema kon nie geskep word nie.';
$lang['strconfdropschema'] = 'Is jy seker dat jy die skema "%s" wil verwyder?';
$lang['strschemadropped'] = 'Skema is verwyder.';
$lang['strschemadroppedbad'] = 'Verwydering van die skema het misluk.';
$lang['strschemaaltered'] = 'Skema is gewysig.';
$lang['strschemaalteredbad'] = 'Skemawysiging het misluk.';
$lang['strsearchpath'] = 'Skema-soekpad';
// Reports
$lang['strreport'] = 'Verslag';
$lang['strreports'] = 'Verslae';
$lang['strshowallreports'] = 'Wys alle verslae';
$lang['strnoreports'] = 'Geen verslae gevind.';
$lang['strcreatereport'] = 'Skep verslag';
$lang['strreportdropped'] = 'Verslag is verwyder.';
$lang['strreportdroppedbad'] = 'Verwydering van verslag het misluk.';
$lang['strconfdropreport'] = 'Is jy seker dat jy die verslag "%s" wil verwyder?';
$lang['strreportneedsname'] = 'Jy moet \'n naam gee vir die verslag.';
$lang['strreportneedsdef'] = 'Jy moet SQL-kode skryf vir die verslag.';
$lang['strreportcreated'] = 'Verslag is geskep.';
$lang['strreportcreatedbad'] = 'Die verslag kon nie geskep word nie.';
// Domains
$lang['strdomain'] = 'Domein';
$lang['strdomains'] = 'Domeine';
$lang['strshowalldomains'] = 'Wys alle domeine';
$lang['strnodomains'] = 'Geen domeine is gevind nie.';
$lang['strcreatedomain'] = 'Skep domein';
$lang['strdomaindropped'] = 'Domein is verwyder.';
$lang['strdomaindroppedbad'] = 'Verwydering van domein het misluk.';
$lang['strconfdropdomain'] = 'Is jy seker dat jy die domein "%s" wil verwyder?';
$lang['strdomainneedsname'] = 'Jy moet \'n naam gee vir die domein.';
$lang['strdomaincreated'] = 'Domein is geskep.';
$lang['strdomaincreatedbad'] = 'Domeinskepping het misluk.';
$lang['strdomainaltered'] = 'Domein is gewysig.';
$lang['strdomainalteredbad'] = 'Wysiging van die domein het misluk.';
// Operators
$lang['stroperator'] = 'Operator';
$lang['stroperators'] = 'Operatore';
$lang['strshowalloperators'] = 'Wys alle operators';
$lang['strnooperator'] = 'Geen operator is gevind nie.';
$lang['strnooperators'] = 'Geen operators is gevind nie.';
$lang['strcreateoperator'] = 'Skep operator';
$lang['strleftarg'] = 'Linkerargumenttipe';
$lang['strrightarg'] = 'Regterargumenttipe';
$lang['strcommutator'] = 'Kommutator';
$lang['strnegator'] = 'Negeerder';
$lang['strrestrict'] = 'Beperk';
$lang['strjoin'] = 'Join';
$lang['strhashes'] = 'Hashes';
$lang['strmerges'] = 'Merges';
$lang['strleftsort'] = 'Linkssorteer';
$lang['strrightsort'] = 'Regssorteer';
$lang['strlessthan'] = 'Kleiner as';
$lang['strgreaterthan'] = 'Groter as';
$lang['stroperatorneedsname'] = 'Jy moet \'n naam gee vir die operator.';
$lang['stroperatorcreated'] = 'Operator is geskep';
$lang['stroperatorcreatedbad'] = 'Operatorskepping het misluk.';
$lang['strconfdropoperator'] = 'Is jy seker dat jy die operator "%s" wil verwyder?';
$lang['stroperatordropped'] = 'Operator is verwyder.';
$lang['stroperatordroppedbad'] = 'Verwydering van die operator het misluk.';
// Casts
$lang['strcasts'] = 'Ekwivalente';
$lang['strnocasts'] = 'Geen ekwivalente gevind.';
$lang['strsourcetype'] = 'Brontipe';
$lang['strtargettype'] = 'Doeltipe';
$lang['strimplicit'] = 'Implisiet';
$lang['strinassignment'] = 'Tydens toekenning';
$lang['strbinarycompat'] = '(Binêr-versoenbaar)';
// Conversions
$lang['strconversions'] = 'Omskakelings';
$lang['strnoconversions'] = 'Geen omskakelings gevind.';
$lang['strsourceencoding'] = 'Bron-enkodering';
$lang['strtargetencoding'] = 'Doel-enkodering';
// Languages
$lang['strlanguages'] = 'Tale';
$lang['strnolanguages'] = 'Geen tale gevind.';
$lang['strtrusted'] = 'Betroubaar';
// Info
$lang['strnoinfo'] = 'Geen inligting beskikbaar.';
$lang['strreferringtables'] = 'Verwysende tabelle';
$lang['strparenttables'] = 'Parent-tabelle';
$lang['strchildtables'] = 'Child-tabelle';
// Aggregates
$lang['straggregates'] = 'Opsommers';
$lang['strnoaggregates'] = 'Geen opsommers gevind.';
$lang['stralltypes'] = '(Alle tipes)';
// Operator Classes
$lang['stropclasses'] = 'Operatorklasse';
$lang['strnoopclasses'] = 'Geen operatorklasse gevind.';
$lang['straccessmethod'] = 'Toegangmetode';
// Stats and performance
$lang['strrowperf'] = 'Ry werkverrigting';
$lang['strioperf'] = 'T/A werkverrigting';
$lang['stridxrowperf'] = 'Indekseer-ry werkverrigting';
$lang['stridxioperf'] = 'Indeks T/A werkverrigting';
$lang['strpercent'] = '%';
$lang['strsequential'] = 'Sekwensieel';
$lang['strscan'] = 'Deursoek';
$lang['strread'] = 'Lees';
$lang['strfetch'] = 'Gaan haal';
$lang['strheap'] = 'Hoop';
$lang['strtoast'] = 'TOAST';
$lang['strtoastindex'] = 'TOAST-indeks';
$lang['strcache'] = 'Kasgeheue';
$lang['strdisk'] = 'Skyf';
$lang['strrows2'] = 'Rye';
// Tablespaces
$lang['strtablespace'] = 'Tabelruimte';
$lang['strtablespaces'] = 'Tabelruimtes';
$lang['strshowalltablespaces'] = 'Wys alle tabelruimtes';
$lang['strnotablespaces'] = 'Geen tabelruimtes gevind.';
$lang['strcreatetablespace'] = 'Skep tabelruimte';
$lang['strlocation'] = 'Plek';
$lang['strtablespaceneedsname'] = 'Jy moet \'n naam gee vir jou tabelruimte.';
$lang['strtablespaceneedsloc'] = 'Jy moet \'n gids gee om jou tabelruimte in te skep.';
$lang['strtablespacecreated'] = 'Tabelruimte geskep.';
$lang['strtablespacecreatedbad'] = 'Skep van tabelruimte het misluk.';
$lang['strconfdroptablespace'] = 'Is jy seker jy wil die tabelruimte "%s" uitvee?';
$lang['strtablespacedropped'] = 'Tabelruimte is uitgevee.';
$lang['strtablespacedroppedbad'] = 'Uitvee van tabelruimte het misluk.';
$lang['strtablespacealtered'] = 'Tabelruimte gewysig.';
$lang['strtablespacealteredbad'] = 'Wysiging van tabelruimte het misluk.';
// Miscellaneous
$lang['strtopbar'] = '%s loop op %s:%s -- Jy is aangeteken as gebruiker "%s"';
$lang['strtimefmt'] = 'jS M, Y g:iA';
$lang['strhelp'] = 'Hulp';
$lang['strhelpicon'] = '?';
?>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,258
|
Лос Алонсо има више значења:
Лос Алонсо, Веракруз Маритимо (Мексикали), насеље у савезној држави Доња Калифорнија у Мексику
Лос Алонсо (Фресниљо), насеље у савезној држави Закатекас у Мексику
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,226
|
Q: My discord bot isn't responding to commands. C# I have been losing my mind over this for three days. I have been making a discord bot, when eventually i just copied the code from this Github, but i have been running into a problem which prevents me from using the commands i have coded. I thought at first that it would be about these two warnings/errors:
17:05:44 Gateway You're using the GuildScheduledEvents gateway intent without listening to any events
related to that intent, consider removing the intent from your config.
17:05:44 Gateway You're using the GuildInvites gateway intent without listening to any events related
to that intent, consider removing the intent from your config.
I don't know yet how to fix those but i tried to Console.WriteLine the part where it recognises the code, but nothing comes up. Here's the two scripts i use:
Main.cs :
`
using System;
using System.Collections.Generic;
using System.Linq;
using System.Reflection;
using System.Text;
using System.Threading.Tasks;
using Discord;
using Discord.Commands;
using Discord.WebSocket;
using Microsoft.Extensions.DependencyInjection;
namespace TutorialBot
{
class Program
{
static void Main(string[] args) => new Program().RunBotAsync().GetAwaiter().GetResult();
private DiscordSocketClient _client;
private CommandService _commands;
private IServiceProvider _services;
public async Task RunBotAsync()
{
_client = new DiscordSocketClient();
_commands = new CommandService();
_services = new ServiceCollection()
.AddSingleton(_client)
.AddSingleton(_commands)
.BuildServiceProvider();
string token = "NjQ0NTI3NzE3Mjk0OTMxOTg4.GeEFaJ.C36VPhtg0MRVRzRinneTIjoJAQrZfdHFBjJdQs";
_client.Log += _client_Log;
await RegisterCommandsAsync();
await _client.LoginAsync(TokenType.Bot, token);
await _client.StartAsync();
await Task.Delay(-1);
var config = new DiscordSocketConfig{
//GatewayIntents = GatewayIntents.GuildInvites
GatewayIntents = GatewayIntents.GuildScheduledEvents
};
var client = new DiscordSocketClient(config);
}
private Task _client_Log(LogMessage arg)
{
Console.WriteLine(arg);
return Task.CompletedTask;
}
public async Task RegisterCommandsAsync()
{
Console.WriteLine("Command recognised");
_client.MessageReceived += HandleCommandAsync;
await _commands.AddModulesAsync(Assembly.GetEntryAssembly(), _services);
}
private async Task HandleCommandAsync(SocketMessage arg)
{
var message = arg as SocketUserMessage;
var context = new SocketCommandContext(_client, message);
if (message.Author.IsBot) return;
int argPos = 0;
if (message.HasStringPrefix("K!", ref argPos))
{
var result = await _commands.ExecuteAsync(context, argPos, _services);
if (!result.IsSuccess) Console.WriteLine(result.ErrorReason);
if (result.Error.Equals(CommandError.UnmetPrecondition)) await message.Channel.SendMessageAsync(result.ErrorReason);
}
}
}
}
`
Modules.cs : (This script contains the output for the commands)
`
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
using Discord;
using Discord.Commands;
using Discord.WebSocket;
namespace TutorialBot.Modules
{
public class Commands : ModuleBase<SocketCommandContext>
{
[Command("ping")]
public async Task Ping()
{
await ReplyAsync("pong");
}
}
}
`
Thanks in advance!
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,713
|
Józef Marian Geisler (ur. 13 października 1859 w Częstochowie, zm. 1 maja 1924 w Otwocku) – polski lekarz.
Życiorys
Dyplom lekarza otrzymał na Cesarskiego Uniwersytecie Warszawskim 3 kwietnia 1887 roku. Po uzyskaniu dyplomu osiadł w Karczewie, a w sezonie letnim praktykował również w Otwocku. W roku 1890 uruchomił w swej willi w Otwocku zakład kąpielowy. W roku 1893, z finansową pomocą Adama Chełmońskiego (1861–1924) wybudował w lesie, między dzisiejszymi ulicami Kościuszki, Chopina i Geislera, duży piętrowy budynek przeznaczony na zakład leczniczy chorób płuc, przede wszystkim gruźlicy.
Był członkiem Zarządu Towarzystwa Przyjaciół Otwocka, oddanym propagatorem Otwocka jako tzw. stacji klimatycznej, był członkiem Warszawskiego Towarzystwa Higienicznego, członkiem założycielem Warszawskiego Towarzystwa Przeciwgruźliczego, przyczynił się do nadania miastu statusu pierwszej stacji dezynfekacyjnej, której został kierownikiem.
W roku 1915 został wraz z dr. Władysławem Czaplickim ewakuowany przez władze rosyjskie w głąb Rosji, skąd powrócił w roku 1918 w złym stanie zdrowia. Chorował ciężko, do pracy już nie powrócił i zmarł w Otwocku 1 maja 1924 roku.
Historia sanatorium Geislera
Wybudowany w 1893 roku piętrowy budynek przeznaczony został na zakład lecznicy chorób płuc i gruźlicy. Sanatorium komfortowe, urządzone zgodnie z zasadami higieny (bieżąca woda, kanalizacja, czterdzieści przestronnych jednoosobowych pokoi, dwie duże werandy do leżakowania, obszerna jadalnia i salon) cieszyło się dużym obłożeniem.
W sanatorium Geislera stosowano skutecznie metodę klimatyczno-dietetycznego leczenia gruźlicy płuc, wprowadzono odmę sztuczną.
W połowie 1898 roku sanatorium zostało rozbudowane. Dwie części piętrowe budynku głównego zostały połączone łącznikiem i tworzyły całość. Drewniane budynki były bogato zdobione wycinanymi w drewnie ornamentami i położone w zadbanym lesie.
Sanatorium funkcjonowało do wybuchu I wojny światowej. Wraz ze śmiercią doktora Józefa Geislera przestało istnieć.
W budynku byłego sanatorium nowi właściciele Waks i Frydland otworzyli pensjonat o nazwie Dyeta, zmieniony później na Patria, który prowadziła Regina Rozenblat.
W czasie wojny był tu dom mieszkalny, kawiarnia i restauracja.
W 1946 w dawnym zakładzie leczniczym doktora J. Geislera otwarto Sanatorium Przeciwgruźlicze PPS, potem PPR, a następnie ZUS.
Przypisy
Absolwenci i studenci Cesarskiego Uniwersytetu Warszawskiego
Ludzie związani z Karczewem
Ludzie związani z Otwockiem
Polscy lekarze
Urodzeni w 1858
Zmarli w 1924
Ludzie urodzeni w Częstochowie
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,398
|
#ifndef REALM_SYNC_CLIENT_HPP
#define REALM_SYNC_CLIENT_HPP
#include <stdint.h>
#include <memory>
#include <utility>
#include <functional>
#include <exception>
#include <string>
#include <realm/util/logger.hpp>
#include <realm/util/network.hpp>
#include <realm/impl/continuous_transactions_history.hpp>
#include <realm/sync/history.hpp>
namespace realm {
namespace sync {
class Client {
public:
using port_type = util::network::Endpoint::port_type;
enum class Error;
enum class ReconnectMode {
/// This is the mode that should always be used in production. In this
/// mode the client uses a scheme for determining a reconnect delay that
/// prevents it from creating too many connection requests in a short
/// amount of time (i.e., a server hammering protection mechanism).
normal,
/// Delay reconnect attempts indefinitely. For testing purposes only.
///
/// A reconnect attempt can be manually scheduled by calling
/// cancel_reconnect_delay(). In particular, when a connection breaks,
/// or when an attempt at establishing the connection fails, the error
/// handler is called. If one calls cancel_reconnect_delay() from that
/// invocation of the error handler, the effect is to allow another
/// reconnect attempt to occur.
never,
/// Never delay reconnect attempts. Perform them immediately. For
/// testing purposes only.
immediate
};
struct Config {
Config() {}
/// The maximum number of Realm files that will be kept open
/// concurrently by this client. The client keeps a cache of open Realm
/// files for efficiency reasons.
long max_open_files = 256;
/// An optional logger to be used by the client. If no logger is
/// specified, the client will use an instance of util::StderrLogger
/// with the log level threshold set to util::Logger::Level::info. The
/// client does not require a thread-safe logger, and it guarantees that
/// all logging happens on behalf of the thread that executes run().
util::Logger* logger = nullptr;
/// Use ports 80 and 443 by default instead of 7800 and 7801
/// respectively. Ideally, these default ports should have been made
/// available via a different URI scheme instead (http/https or ws/wss).
bool enable_default_port_hack = true;
/// For testing purposes only.
ReconnectMode reconnect_mode = ReconnectMode::normal;
/// Create a separate connection for each session. For testing purposes
/// only.
///
// FIXME: This setting is ignored for now, due to limitations in the
// load balancer.
bool one_connection_per_session = true;
/// Do not access the local file system. Sessions will act as if
/// initiated on behalf of an empty (or nonexisting) local Realm
/// file. Received DOWNLOAD messages will be accepted, but otherwise
/// ignored. No UPLOAD messages will be generated. For testing purposes
/// only.
bool dry_run = false;
/// The default changeset cooker to be used by new sessions. Can be
/// overridden by Session::Config::changeset_cooker.
///
/// \sa make_client_history(), TrivialChangesetCooker.
std::shared_ptr<ClientHistory::ChangesetCooker> changeset_cooker;
/// The number of ms between periodic keep-alive pings.
uint_fast64_t ping_keepalive_period_ms = 600000;
/// The number of ms to wait for keep-alive pongs.
uint_fast64_t pong_keepalive_timeout_ms = 300000;
/// The number of ms to wait for urgent pongs.
uint_fast64_t pong_urgent_timeout_ms = 5000;
};
/// \throw util::EventLoop::Implementation::NotAvailable if no event loop
/// implementation was specified, and
/// util::EventLoop::Implementation::get_default() throws it.
Client(Config = {});
Client(Client&&) noexcept;
~Client() noexcept;
/// Run the internal event-loop of the client. At most one thread may
/// execute run() at any given time. The call will not return until somebody
/// calls stop().
void run();
/// See run().
///
/// Thread-safe.
void stop() noexcept;
/// \brief Cancel current or next reconnect delay for all servers.
///
/// This corresponds to calling Session::cancel_reconnect_delay() on all
/// bound sessions, but will also cancel reconnect delays applying to
/// servers for which there are currently no bound sessions.
///
/// Thread-safe.
void cancel_reconnect_delay();
private:
class Impl;
std::unique_ptr<Impl> m_impl;
friend class Session;
};
/// Supported protocols:
///
/// Protocol URL scheme Default port
/// -----------------------------------------------------------------------------------
/// realm "realm:" 7800 (80 if Client::Config::enable_default_port_hack)
/// realm_ssl "realms:" 7801 (443 if Client::Config::enable_default_port_hack)
///
enum class Protocol {
realm,
realm_ssl
};
class BadServerUrl; // Exception
/// \brief Client-side representation of a Realm file synchronization session.
///
/// A synchronization session deals with precisely one local Realm file. To
/// synchronize multiple local Realm files, you need multiple sessions.
///
/// A session object is always associated with a particular client object (\ref
/// Client). The application must ensure that the destruction of the associated
/// client object never happens before the destruction of the session
/// object. The consequences of a violation are unspecified.
///
/// A session object is always associated with a particular local Realm file,
/// however, a session object does not represent a session until it is bound to
/// a server side Realm, i.e., until bind() is called. From the point of view of
/// the thread that calls bind(), the session starts precisely when the
/// execution of bind() starts, i.e., before bind() returns.
///
/// At most one session is allowed to exist for a particular local Realm file
/// (file system inode) at any point in time. Multiple objects may coexists, as
/// long as bind() has been called on at most one of them. Additionally, two
/// sessions are allowed to exist at different times, and with no overlap in
/// time, as long as they are associated with the same client object, or with
/// two different client objects that do not overlap in time. This means, in
/// particular, that it is an error to create two nonoverlapping sessions for
/// the same local Realm file, it they are associated with two different client
/// objects that overlap in time. It is the responsibility of the application to
/// ensure that these rules are adhered to. The consequences of a violation are
/// unspecified.
///
/// Thread-safety: It is safe for multiple threads to construct, use (with some
/// exceptions), and destroy session objects concurrently, regardless of whether
/// those session objects are associated with the same, or with different Client
/// objects. Please note that some of the public member functions are fully
/// thread-safe, while others are not.
class Session {
public:
using port_type = util::network::Endpoint::port_type;
using version_type = _impl::History::version_type;
using SyncTransactCallback = void(VersionID old_version, VersionID new_version);
using ProgressHandler = void(std::uint_fast64_t downloaded_bytes,
std::uint_fast64_t downloadable_bytes,
std::uint_fast64_t uploaded_bytes,
std::uint_fast64_t uploadable_bytes,
std::uint_fast64_t progress_version,
std::uint_fast64_t snapshot_version);
using WaitOperCompletionHandler = std::function<void(std::error_code)>;
using SSLVerifyCallback = bool(const std::string& server_address,
port_type server_port,
const char* pem_data,
size_t pem_size,
int preverify_ok,
int depth);
class Config {
public:
Config() {}
/// server_address is the fully qualified host name, or IP address of
/// the server.
std::string server_address = "localhost";
/// server_port is the port at which the server listens. If server_port
/// is zero, the default port for the specified protocol is used. See \ref
/// Protocol for information on default ports.
port_type server_port = 0;
/// server_path is the virtual path by which the server identifies the
/// Realm. This path must always be an absolute path, and must therefore
/// always contain a leading slash (`/`). Further more, each segment of the
/// virtual path must consist of one or more characters that are either
/// alpha-numeric or in (`_`, `-`, `.`), and each segment is not allowed to
/// equal `.` or `..`, and must not end with `.realm`, `.realm.lock`, or
/// `.realm.management`. These rules are necessary because the server
/// currently reserves the right to use the specified path as part of the
/// file system path of a Realm file. It is expected that these rules will
/// be significantly relaxed in the future by completely decoupling the
/// virtual paths from actual file system paths.
std::string server_path = "/";
/// The protocol used for communicating with the server. See \ref Protocol.
Protocol protocol = Protocol::realm;
/// Sessions can be multiplexed over the same TCP/SSL connection.
/// Sessions might share connection if they have identical server_address,
/// server_port, and protocol. multiplex_ident is a parameter that allows
/// finer control over session multiplexing. If two sessions have distinct
/// multiplex_ident, they will never share connection. The typical use of
/// multilex_ident is to give sessions with incompatible SSL requirements
/// distinct multiplex_idents.
/// multiplex_ident can be any string and the value has no meaning except
/// for partitioning the sessions.
std::string multiplex_ident;
/// verify_servers_ssl_certificate controls whether the server
/// certificate is verified for SSL connections. It should generally be
/// true in production.
bool verify_servers_ssl_certificate = true;
/// ssl_trust_certificate_path is the path of a trust/anchor
/// certificate used by the client to verify the server certificate.
/// ssl_trust_certificate_path is only used if the protocol is ssl and
/// verify_servers_ssl_certificate is true.
///
/// A server certificate is verified by first checking that the
/// certificate has a valid signature chain back to a trust/anchor
/// certificate, and secondly checking that the server_address matches
/// a host name contained in the certificate. The host name of the
/// certificate is stored in either Common Name or the Alternative
/// Subject Name (DNS section).
///
/// If ssl_trust_certificate_path is None (default), ssl_verify_callback
/// (see below) is used if set, and the default device trust/anchor
/// store is used otherwise.
Optional<std::string> ssl_trust_certificate_path;
/// If Client::Config::ssl_verify_callback is set, that function is called
/// to verify the certificate, unless verify_servers_ssl_certificate is
/// false.
/// ssl_verify_callback is used to implement custom SSL certificate
/// verification. it is only used if the protocol is SSL,
/// verify_servers_ssl_certificate is true and ssl_trust_certificate_path
/// is None.
///
/// The signature of ssl_verify_callback is
///
/// bool(const std::string& server_address,
/// port_type server_port,
/// const char* pem_data,
/// size_t pem_size,
/// int preverify_ok,
/// int depth);
///
/// server address and server_port is the address and port of the server
/// that a SSL connection is being established to. They are identical to
/// the server_address and server_port set in this config file and are
/// passed for convenience.
/// pem_data is the certificate of length pem_size in
/// the PEM format. preverify_ok is OpenSSL's preverification of the
/// certificate. preverify_ok is either 0, or 1. If preverify_ok is 1,
/// OpenSSL has accepted the certificate and it will generally be safe
/// to trust that certificate. depth represents the position of the
/// certificate in the certificate chain sent by the server. depth = 0
/// represents the actual server certificate that should contain the
/// host name(server address) of the server. The highest depth is the
/// root certificate.
/// The callback function will receive the certificates starting from
/// the root certificate and moving down the chain until it reaches the
/// server's own certificate with a host name. The depth of the last
/// certificate is 0. The depth of the first certificate is chain
/// length - 1.
///
/// The return value of the callback function decides whether the
/// client accepts the certificate. If the return value is false, the
/// processing of the certificate chain is interrupted and the SSL
/// connection is rejected. If the return value is true, the verification
/// process continues. If the callback function returns true for all
/// presented certificates including the depth == 0 certificate, the
/// SSL connection is accepted.
///
/// A recommended way of using the callback function is to return true
/// if preverify_ok = 1 and depth > 0,
/// always check the host name if depth = 0,
/// and use an independent verification step if preverify_ok = 0.
///
/// Another possible way of using the callback is to collect all the
/// certificates until depth = 0, and present the entire chain for
/// independent verification.
std::function<SSLVerifyCallback> ssl_verify_callback;
/// signed_user_token is a cryptographically signed token describing the
/// identity and access rights of the current user.
std::string signed_user_token;
/// If not null, overrides whatever is specified by
/// Client::Config::changeset_cooker.
///
/// The shared ownership over the cooker will be relinquished shortly
/// after the destruction of the session object as long as the event
/// loop of the client is being executed (Client::run()).
///
/// CAUTION: ChangesetCooker::cook_changeset() of the specified cooker
/// may get called before the call to bind() returns, and it may get
/// called (or continue to execute) after the session object is
/// destroyed. The application must specify an object for which that
/// function can safely be called, and continue to execute from the
/// point in time where bind() starts executing, and up until the point
/// in time where the last invocation of `client.run()` returns. Here,
/// `client` refers to the associated Client object.
///
/// \sa make_client_history(), TrivialChangesetCooker.
std::shared_ptr<ClientHistory::ChangesetCooker> changeset_cooker;
/// The encryption key the SharedGroup will be opened with.
Optional<std::array<char, 64>> encryption_key;
/// FIXME: This value must currently be true in a cluster setup.
/// This restriction will be lifted in the future.
bool one_connection_per_session = true;
};
/// \brief Start a new session for the specified client-side Realm.
///
/// Note that the session is not fully activated until you call bind(). Also
/// note that if you call set_sync_transact_callback(), it must be done
/// before calling bind().
///
/// \param realm_path The file-system path of a local client-side Realm
/// file.
Session(Client&, std::string realm_path, Config = {});
Session(Session&&) noexcept;
~Session() noexcept;
/// \brief Set a function to be called when the local Realm has changed due
/// to integration of a downloaded changeset.
///
/// Specify the callback function that will be called when one or more
/// transactions are performed to integrate downloaded changesets into the
/// client-side Realm, that is associated with this session.
///
/// The callback function will always be called by the thread that executes
/// the event loop (Client::run()), but not until bind() is called. If the
/// callback function throws an exception, that exception will "travel" out
/// through Client::run().
///
/// Note: Any call to this function must have returned before bind() is
/// called. If this function is called multiple times, each call overrides
/// the previous setting.
///
/// Note: This function is **not thread-safe**. That is, it is an error if
/// it is called while another thread is executing any member function on
/// the same Session object.
///
/// CAUTION: The specified callback function may be called before the call
/// to bind() returns, and it may be called (or continue to execute) after
/// the session object is destroyed. The application must pass a handler
/// that can be safely called, and can safely continue to execute from the
/// point in time where bind() starts executing, and up until the point in
/// time where the last invocation of `client.run()` returns. Here, `client`
/// refers to the associated Client object.
void set_sync_transact_callback(std::function<SyncTransactCallback>);
/// \brief Set a handler to monitor the state of download and upload
/// progress.
///
/// The handler must have signature
///
/// void(uint_fast64_t downloaded_bytes, uint_fast64_t downloadable_bytes,
/// uint_fast64_t uploaded_bytes, uint_fast64_t uploadable_bytes,
/// uint_fast64_t progress_version);
///
/// downloaded_bytes is the size in bytes of all downloaded changesets.
/// downloadable_bytes is the size in bytes of the part of the server
/// history that do not originate from this client.
///
/// uploaded_bytes is the size in bytes of all locally produced changesets
/// that have been received and acknowledged by the server.
/// uploadable_bytes is the size in bytes of all locally produced changesets.
///
/// Due to the nature of the merge rules, it is possible that the size of an
/// uploaded changeset uploaded from one client is not equal to the size of
/// the changesets that other clients will download.
///
/// Typical uses of this function:
///
/// Upload completion can be checked by
///
/// bool upload_complete = (uploaded_bytes == uploadable_bytes);
///
/// Download completion could be checked by
///
/// bool download_complete = (downloaded_bytes == downloadable_bytes);
///
/// However, download completion might never be reached because the server
/// can receive new changesets from other clients.
/// An alternative strategy is to cache downloadable_bytes from the callback,
/// and use the cached value as the threshold.
///
/// bool download_complete = (downloaded_bytes == cached_downloadable_bytes);
///
/// Upload progress can be calculated by caching an initial value of
/// uploaded_bytes from the last, or next, callback. Then
///
/// double upload_progress =
/// (uploaded_bytes - initial_uploaded_bytes)
/// -------------------------------------------
/// (uploadable_bytes - initial_uploaded_bytes)
///
/// Download progress can be calculates similarly:
///
/// double download_progress =
/// (downloaded_bytes - initial_downloaded_bytes)
/// -----------------------------------------------
/// (downloadable_bytes - initial_downloaded_bytes)
///
/// progress_version is 0 at the start of a session. When at least one
/// DOWNLOAD message has been received from the server, progress_version is
/// positive. progress_version can be used to ensure that the reported
/// progress contains information obtained from the server in the current
/// session. The server will send a message as soon as possible, and the
/// progress handler will eventually be called with a positive progress_version
/// unless the session is interrupted before a message from the server has
/// been received.
///
/// The handler is called on the event loop thread.The handler after bind(),
/// after each DOWNLOAD message, and after each local transaction
/// (nonsync_transact_notify).
///
/// set_progress_handler() is not thread safe and it must be called before
/// bind() is called. Subsequent calls to set_progress_handler() overwrite
/// the previous calls. Typically, this function is called once per session.
///
/// CAUTION: The specified callback function may be called
/// (or continue to execute) after the session object is destroyed.
/// The application must pass a handler that can be safely called, and can
/// execute from the point in time where bind() is called,
/// and up until the point in time where the last invocation of
/// `client.run()` returns. Here, `client` refers to the associated
/// Client object.
void set_progress_handler(std::function<ProgressHandler>);
/// \brief Signature of an error handler.
///
/// \param ec The error code. For the list of errors reported by the server,
/// see \ref ProtocolError (or `protocol.md`). For the list of errors
/// corresponding with protocol violation that are detected by the client,
/// see Client::Error.
///
/// \param is_fatal The error is of a kind that is likely to persist, and
/// cause all future reconnect attempts to fail. The client may choose to
/// try to reconnect again later, but if so, the waiting period will be
/// substantial.
///
/// \param detailed_message The message associated with the error. It is
/// usually equal to `ec.message()`, but may also be a more specific message
/// (one that provides extra context). The purpose of this message is mostly
/// to aid debugging. For non-debugging purposes, `ec.message()` should
/// generally be considered sufficient.
///
/// \sa set_error_handler().
using ErrorHandler = void(std::error_code ec, bool is_fatal,
const std::string& detailed_message);
/// \brief Set the error handler for this session.
///
/// Sets a function to be called when an error causes a connection
/// initiation attempt to fail, or an established connection to be broken.
///
/// When a connection is established on behalf of multiple sessions, a
/// connection-level error will be reported to all those sessions. A
/// session-level error, on the other hand, will only be reported to the
/// affected session.
///
/// The callback function will always be called by the thread that executes
/// the event loop (Client::run()), but not until bind() is called. If the
/// callback function throws an exception, that exception will "travel" out
/// through Client::run().
///
/// Note: Any call to this function must have returned before bind() is
/// called. If this function is called multiple times, each call overrides
/// the previous setting.
///
/// Note: This function is **not thread-safe**. That is, it is an error if
/// it is called while another thread is executing any member function on
/// the same Session object.
///
/// CAUTION: The specified callback function may be called before the call
/// to bind() returns, and it may be called (or continue to execute) after
/// the session object is destroyed. The application must pass a handler
/// that can be safely called, and can safely continue to execute from the
/// point in time where bind() starts executing, and up until the point in
/// time where the last invocation of `client.run()` returns. Here, `client`
/// refers to the associated Client object.
void set_error_handler(std::function<ErrorHandler>);
/// @{ \brief Bind this session to the specified server side Realm.
///
/// No communication takes place on behalf of this session before the
/// session is bound, but as soon as the session becomes bound, the server
/// will start to push changes to the client, and vice versa.
///
/// If a callback function was set using set_sync_transact_callback(), then
/// that callback function will start to be called as changesets are
/// downloaded and integrated locally. It is important to understand that
/// callback functions are executed by the event loop thread (Client::run())
/// and the callback function may therefore be called before bind() returns.
///
/// Note: It is an error if this function is called more than once per
/// Session object.
///
/// Note: This function is **not thread-safe**. That is, it is an error if
/// it is called while another thread is executing any member function on
/// the same Session object.
///
/// bind() binds this session to the specified server side Realm using the
/// parameters specified in the Session::Config object.
///
/// The two other forms of bind() are convenience functions.
/// void bind(std::string server_address, std::string server_path,
/// std::string signed_user_token, port_type server_port = 0,
/// Protocol protocol = Protocol::realm);
/// replaces the corresponding parameters from the Session::Config object
/// before the session is bound.
/// void bind(std::string server_url, std::string signed_user_token) parses
/// the \param server_url and replaces the parameters in the Session::Config object
/// before the session is bound.
///
/// \param server_url For example "realm://sync.realm.io/test". See
/// server_address, server_path, and server_port in Session::Config for information
/// about the individual components of the URL. See \ref Protocol for the list of
/// available URL schemes and the associated default ports.
///
/// \throw BadServerUrl if the specified server URL is malformed.
void bind();
void bind(std::string server_url, std::string signed_user_token);
void bind(std::string server_address, std::string server_path,
std::string signed_user_token, port_type server_port = 0,
Protocol protocol = Protocol::realm);
/// @}
/// \brief Refresh the user token associated with this session.
///
/// This causes the REFRESH protocol message to be sent to the server. See
/// \ref Protocol.
///
/// In an on-going session a client may expect its access token to expire at
/// a certain time and schedule acquisition of a fresh access token (using a
/// refresh token or by other means) in due time to provide a better user
/// experience. Without refreshing the token, the client will be notified
/// that the session is terminated due to insufficient privileges and must
/// reacquire a fresh token, which is a potentially disruptive process.
///
/// It is an error to call this function before calling `Client::bind()`.
///
/// Note: This function is thread-safe.
///
/// \param signed_user_token A cryptographically signed token describing the
/// identity and access rights of the current user. See \ref Protocol.
void refresh(std::string signed_user_token);
/// \brief Inform the synchronization agent about changes of local origin.
///
/// This function must be called by the application after a transaction
/// performed on its behalf, that is, after a transaction that is not
/// performed to integrate a changeset that was downloaded from the server.
///
/// It is an error to call this function before bind() has been called, and
/// has returned.
///
/// Note: This function is fully thread-safe. That is, it may be called by
/// any thread, and by multiple threads concurrently.
void nonsync_transact_notify(version_type new_version);
/// @{ \brief Wait for upload, download, or upload+download completion.
///
/// async_wait_for_upload_completion() initiates an asynchronous wait for
/// upload to complete, async_wait_for_download_completion() initiates an
/// asynchronous wait for download to complete, and
/// async_wait_for_sync_completion() initiates an asynchronous wait for
/// upload and download to complete.
///
/// Upload is considered complete when all non-empty changesets of local
/// origin have been uploaded to the server, and the server has acknowledged
/// reception of them. Changesets of local origin introduced after the
/// initiation of the session (after bind() is called) will generally not be
/// considered for upload unless they are announced to this client through
/// nonsync_transact_notify() prior to the initiation of the wait operation,
/// i.e., prior to the invocation of async_wait_for_upload_completion() or
/// async_wait_for_sync_completion(). Unannounced changesets may get picked
/// up, but there is no guarantee that they will be, however, if a certain
/// changeset is announced, then all previous changesets are implicitly
/// announced. Also all preexisting changesets are implicitly announced
/// when the session is initiated.
///
/// Download is considered complete when all non-empty changesets of remote
/// origin have been downloaded from the server, and integrated into the
/// local Realm state. To know what is currently outstanding on the server,
/// the client always sends a special "marker" message to the server, and
/// waits until it has downloaded all outstanding changesets that were
/// present on the server at the time when the server received that marker
/// message. Each call to async_wait_for_download_completion() and
/// async_wait_for_sync_completion() therefore requires a full client <->
/// server round-trip.
///
/// If a new wait operation is initiated while other wait operations are in
/// progress, the waiting period of operations in progress may, or may not
/// get extended. The client must not assume either. The client may assume,
/// however, that async_wait_for_upload_completion() will not affect the
/// waiting period of async_wait_for_download_completion(), and vice versa.
///
/// It is an error to call these functions before bind() has been called,
/// and has returned.
///
/// The specified completion handlers will always be executed by the thread
/// that executes the event loop (the thread that calls Client::run()). If
/// the handler throws an exception, that exception will "travel" out
/// through Client::run().
///
/// If incomplete wait operations exist when the session is terminated,
/// those wait operations will be canceled. Session termination is an event
/// that happens in the context of the client's event loop thread shortly
/// after the destruction of the session object. The std::error_code
/// argument passed to the completion handler of a canceled wait operation
/// will be `util::error::operation_aborted`. For uncanceled wait operations
/// it will be `std::error_code{}`. Note that as long as the client's event
/// loop thread is running, all completion handlers will be called
/// regardless of whether the operations get canceled or not.
///
/// CAUTION: The specified completion handlers may be called before the
/// initiation function returns (e.g. before
/// async_wait_for_upload_completion() returns), and it may be called (or
/// continue to execute) after the session object is destroyed. The
/// application must pass a handler that can be safely called, and can
/// safely continue to execute from the point in time where the initiating
/// function starts executing, and up until the point in time where the last
/// invocation of `clint.run()` returns. Here, `client` refers to the
/// associated Client object.
///
/// Note: These functions are fully thread-safe. That is, they may be called
/// by any thread, and by multiple threads concurrently.
void async_wait_for_sync_completion(WaitOperCompletionHandler);
void async_wait_for_upload_completion(WaitOperCompletionHandler);
void async_wait_for_download_completion(WaitOperCompletionHandler);
/// @}
/// @{ \brief Synchronous wait for upload or download completion.
///
/// These functions are synchronous equivalents to
/// async_wait_for_upload_completion() and
/// async_wait_for_download_completion() respectively. This means that they
/// block the caller until the completion condition is satisfied, or the
/// client event loop is stopped (Client::stop()).
///
/// It is an error to call these functions before bind() has been called,
/// and has returned.
///
/// Note: These functions are fully thread-safe. That is, they may be called
/// by any thread, and by multiple threads concurrently.
void wait_for_upload_complete_or_client_stopped();
void wait_for_download_complete_or_client_stopped();
/// @}
/// \brief Cancel the current or next reconnect delay for the server
/// associated with this session.
///
/// When the network connection is severed, or an attempt to establish
/// connection fails, a certain delay will take effect before the client
/// will attempt to reestablish the connection. This delay will generally
/// grow with the number of unsuccessful reconnect attempts, and can grow to
/// over a minute. In some cases however, the application will know when it
/// is a good time to stop waiting and retry immediately. One example is
/// when a device has been offline for a while, and the operating system
/// then tells the application that network connectivity has been restored.
///
/// Clearly, this function should not be called too often and over extended
/// periods of time, as that would effectively disable the built-in "server
/// hammering" protection.
///
/// It is an error to call this function before bind() has been called, and
/// has returned.
///
/// This function is fully thread-safe. That is, it may be called by any
/// thread, and by multiple threads concurrently.
void cancel_reconnect_delay();
private:
class Impl;
Impl* m_impl;
void async_wait_for(bool upload_completion, bool download_completion,
WaitOperCompletionHandler);
};
/// \brief Protocol errors discovered by the client.
///
/// These errors will terminate the network connection (disconnect all sessions
/// associated with the affected connection), and the error will be reported to
/// the application via the error handlers of the affected sessions.
enum class Client::Error {
connection_closed = 100, ///< Connection closed (no error)
unknown_message = 101, ///< Unknown type of input message
bad_syntax = 102, ///< Bad syntax in input message head
limits_exceeded = 103, ///< Limits exceeded in input message
bad_session_ident = 104, ///< Bad session identifier in input message
bad_message_order = 105, ///< Bad input message order
bad_file_ident_pair = 106, ///< Bad file identifier pair (ALLOC)
bad_progress = 107, ///< Bad progress information (DOWNLOAD)
bad_changeset_header_syntax = 108, ///< Bad syntax in changeset header (DOWNLOAD)
bad_changeset_size = 109, ///< Bad changeset size in changeset header (DOWNLOAD)
bad_origin_file_ident = 110, ///< Bad origin file identifier in changeset header (DOWNLOAD)
bad_server_version = 111, ///< Bad server version in changeset header (DOWNLOAD)
bad_changeset = 112, ///< Bad changeset (DOWNLOAD)
bad_request_ident = 113, ///< Bad request identifier (MARK)
bad_error_code = 114, ///< Bad error code (ERROR),
bad_compression = 115, ///< Bad compression (DOWNLOAD)
bad_client_version = 116, ///< Bad last integrated client version in changeset header (DOWNLOAD)
ssl_server_cert_rejected = 117, ///< SSL server certificate rejected
};
const std::error_category& client_error_category() noexcept;
std::error_code make_error_code(Client::Error) noexcept;
} // namespace sync
} // namespace realm
namespace std {
template<> struct is_error_code_enum<realm::sync::Client::Error> {
static const bool value = true;
};
} // namespace std
namespace realm {
namespace sync {
// Implementation
class BadServerUrl: public std::exception {
public:
const char* what() const noexcept override
{
return "Bad server URL";
}
};
inline void Session::async_wait_for_sync_completion(WaitOperCompletionHandler handler)
{
bool upload_completion = true, download_completion = true;
async_wait_for(upload_completion, download_completion, std::move(handler)); // Throws
}
inline void Session::async_wait_for_upload_completion(WaitOperCompletionHandler handler)
{
bool upload_completion = true, download_completion = false;
async_wait_for(upload_completion, download_completion, std::move(handler)); // Throws
}
inline void Session::async_wait_for_download_completion(WaitOperCompletionHandler handler)
{
bool upload_completion = false, download_completion = true;
async_wait_for(upload_completion, download_completion, std::move(handler)); // Throws
}
} // namespace sync
} // namespace realm
#endif // REALM_SYNC_CLIENT_HPP
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 624
|
Modular data-logger
Works as a gateway
With the Tinybox TBa53, TQ (Germany) has released a modular data logger based on the TQMa53 module. It features CAN interfaces and it can also be used as a gateway in a wireless Bacnet environment.
THE DEVICE, CONSISTING OF MOTHERBOARD, MODULE, AND CASING allows for implementation of custom system solutions. Two system slots are available for expansions. The data logger can be integrated directly into the control cabinet as a DIN rail device, thus forming the basis for a system solution. The TQMa53 module, a high-performance Cortex-A8 module, is used in the process. The device has protection level IP20 and an extended temperature range of -40 °C to 70 °C. It measures 120 mm x 130 mm x 60 mm and runs Linux.
On the carrier board, the data logger or the gateway makes a wide range of interfaces available to the user, such as RS232, two CAN (electrically isolated), two USB 2.0 and two Ethernet interfaces. To enhance it, an SD card interface accessible from the outside was installed.
In addition to these interfaces, the board is expandable via two expansion module plug-in connectors. These are of the same design and offer different interfaces (two UART/SPI/I2C/USB/GPIO). Any number of expansion modules can be implemented on the basis of these interfaces. Wireless solutions such as UMTS/GPRS, Wifi as well as Bluetooth are planned in addition to the RS232 and RS485 expansion modules already available. Furthermore, via the available interfaces various I/O features can be implemented on the expansion modules.
Corenetix (Germany) has utilized these features and is currently developing an expansion card on the basis of the CNX100 wireless radio module. The wireless radio module operates in the 868 MHz frequency band. It offers a wireless Bacnet, as well as a mesh network protocol stack, which, thanks to its AES128 encryption, is also suitable for security-related applications. This serves as a basis for a Bacnet implementation, optimized for embedded systems, which now allows the wireless integration of terminals into a Bacnet environment as well. A gateway constitutes the link between the wireless network and the wired world. In addition, it provides management systems with access to the Bacnet devices in the wireless network. Through the wireless system components, terminals which are integrated wirelessly into a Bacnet environment are no longer distinguished from wired devices.
Typical areas of application for the data logger are multi-hop, mesh network solutions for applications in the sectors of building automation, smart metering, smart home, security & monitoring, industrial automation, process control as well as service management.
Beckhoff Automation
esd electronics
CAN data logger for automotive testing
Fleetlog2 is the advanced version of Ipetronik's (Germany) fleet data logger Fleetlog. The basic dat...
CAN data-logger in aluminum
Venamics (Netherlands) added a product to their Venamics Measurement System (VMS) series. The CAN da...
Windows-based mobile data-logger for automotive and industrial use
Ipetronik (Germany) introduced its new Winlog Windows-based mobile data acquisition system based on ...
TQ-Group
Corenetix
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,090
|
Farming union says UK-NZ trade deal more than 40 TIMES worse for UK sheep industry than EU-NZ deal
The increase in New Zealand's import quota for sheep meat in year one of new trade agreements signed by the UK and EU is more than forty times higher per head of population in the UK compared with the European Union-demonstrating the failure of the UK Government to protect UK agriculture in trade negotiations, says the Farmers' Union of Wales (FUW).
An EU-New Zealand trade deal recently agreed in principle would allow an additional 5,429 tonnes of sheep meat to be imported duty-free into the EU in year one of the agreement, whilst the equivalent figure for the UK in the deal announced in February this year is 35,000 tonnes.
"The UK increase in duty free quota for New Zealand sheep meat would be almost six and a half times higher in year one than what has been negotiated by the EU," said FUW President Glyn Roberts.
"However, when you take account of the fact that the population of the EU is nearly seven times higher than that of the UK, the increase per consumer is 43 times higher in the UK than in the EU."
Mr Roberts said that another way of looking at the figure was that the EU had fought forty times harder for its sheep industry than the UK during its trade negotiations with New Zealand.
While both the UK and EU deals would ultimately increase tariff-free quotas for sheep meat by year seven of the agreements, the difference between consumer numbers means the total volumes that could be imported per head of population would be almost seven times higher in the UK than in the EU.
"As if this is not bad enough, the UK deal ultimately allows unlimited amounts of sheep meat to be imported, whilst the EU has set a cap on imports," said Mr Roberts.
Similar differences between ratios and protections exist for other key products covered in the two trade deals.
For example, in years one to seven of the EU-NZ deal, the additional quantity of beef that could be imported to the EU would rise from 3,333 tonnes in year one to 10,000 in year seven-but at a reduced tariff of 7.5%; by comparison, the UK has agreed to an additional 12,000 tonnes of beef being imported to the UK duty free in year one, rising to an unlimited amount after year ten.
For cheese, the UK has agreed to an additional duty-free quota of 24,000 tonnes in year one, rising to 48,000 by year five, after which there would be no limits on imports-compared with an EU provisional agreement to allow an increase in the duty-free import quota to 25,000 by year seven, but with no ultimate full trade liberalisation.
"The UK's own figures have shown that the benefits to the UK economy of the deal with New Zealand are vanishingly small, with our GVA estimated to grow by just 0.03% and average wages to rise by just pennies a week as a result of the deal.
Meanwhile, the same UK Government assessment estimates that the GVA of the categories into which farming and semi-processed foods will fall by £132 million as a result of the deal."
The population of New Zealand is just 5 million compared with a UK population of 67 million, meaning the UK is a far more attractive market for New Zealand than New Zealand is for the UK.
"The comparisons between the EU and UK deals demonstrate how the UK Government has failed to fight for the interests of our farmers, food producers and food security in its trade negotiations, despite having significant bargaining advantages," said Mr Roberts.
"The UK Government has sold important protections for food and agriculture down the river in exchange for trade deals that are of vanishingly small benefit to the UK," he added.
The Trade (Australia and New Zealand) Bill currently passing through the UK Parliament would, if passed by MPs, give the UK's final seal of approval to the UK-New Zealand and UK-Australia trade deals, but the International Trade Committee has criticised the UK Government for triggering the statutory scrutiny period for the Bill despite giving assurances that the Committee would be allowed sufficient time to publish a report beforehand.
The Committee also asked the Government to meet its commitment to allow a debate on the UK-Australia agreement and to either extend the statutory period or give the House of Commons the opportunity to extend it by passing a motion resolving that the treaty should not be ratified. However, the Government has declined to use its statutory power to extend the scrutiny period under the Constitutional Reform and Governance Act 2010.
"In addition to having failed to negotiate a trade deal that is in our nations' interests, the Government appears to want to avoid the scrutiny such deals are allowed in any other democracy in the world by railroading the legislation through Parliament," added Mr Roberts.
(Lead image: FUW)
Related Topics:Farmers Union of WalesFarmingFUWlambNew ZealandsheepSheep Farmingtrade dealUK Government
Carmarthen pork producer celebrating being shortlisted as having Wales's best banger
Pembrokeshire Agricultural Society launches 2022 Student Bursary Award
£20m funding boost for Swansea's industrial heritage
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,993
|
Q: React QR Reader open close camera with React Good Afternoon guys i have the following problem,I want to click on a button and open the camera or close it and scan the qr code of item.
This is my code that i am trying to implement
const qrRef = useRef(null);
const onScanFile = () => {
qrRef.current.openImageDialog();
};
const handleScanFile = (result) => {
if (result) {
setScanResultFile(result);
}
};
and my return method is below
<BottomBox>
<QRIcon onClick={onScanFile} width={24} height={24} />
<Stack ml={2}>SCAN QR CODE</Stack>
</BottomBox>
<QrReader
ref={qrRef}
delay={300}
style={{ width: "100%" }}
onScan={handleScanFile}
legacyMode
/>
Every time that i am trigger that component the camera is on,i want to trigger only when i will click the below button:
<QRIcon onClick={onScanFile} width={24} height={24} />
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,127
|
\section{Introduction}
\label{sec:intro}
Coalescing black-hole binaries are among the most promising sources
for the current and future laser-interferometer gravitational-wave
detectors, such as the ground-based detectors LIGO and Virgo~\cite{Waldman:2006,Acernese:2008zz}
and the space-based detector LISA~\cite{schutz_lisa_science}.
The search for gravitational waves from coalescing binaries and the
extraction of the binary's physical parameters are based on the matched filtering technique,
which requires accurate knowledge of the waveform of the incoming signal.
Because black holes in general relativity are uniquely defined by their masses and spins,
the waveforms for black-hole binaries on a quasi-circular orbits depend on eight parameters, namely
the masses $m_1$ and $m_2$ and the spin vectors $\boldsymbol{S}_1$ and $\boldsymbol{S}_2$. Due to the large
parameter space, eventually tens of thousands of waveform
templates may be needed to extract the gravitational-wave
signal from the noise, an impossible demand for numerical-relativity alone.
Fortunately, recent work at the interface between analytical and numerical relativity
has demonstrated the possibility of modeling analytically the dynamics and
the gravitational-wave emission of coalescing {\it non-spinning} black holes,
thus providing data analysts with analytical template
families~\cite{Buonanno2007,Ajith-Babak-Chen-etal:2007b,Damour2009a,Buonanno:2009qa}
to be used for the searches (see also Ref.~\cite{Yunes:2009ef}, which considers the cases of extreme mass-ratio
inspirals). The next important step is to extend those studies to {\it spinning precessing}
black holes.
So far, the analytical modeling of the inspiral, plunge~\footnote{We refer to
plunge as the dynamical phase starting soon after the two-body system passes the last stable orbit.
During the plunge the motion is driven mostly by the conservative dynamics.}, merger~\footnote{
We refer to merger as the dynamical phase in which the two-body system
is described by a single black hole.},
and ringdown has been obtained within either the effective-one-body (EOB)
formalism~\cite{Buonanno00,Buonanno99,DJS3PN,Buonanno-Cook-Pretorius:2007,
Pan2007,Buonanno2007,Damour2007a,DN2007b,DN2008,Boyle2008a,Damour2009a,Buonanno:2009qa} or
in Taylor-expanded PN models~\cite{Pan2007}, both calibrated to
numerical-relativity simulations, or in phenomenological
approaches~\cite{Ajith-Babak-Chen-etal:2007b,Ajith2009} where the numerical-relativity
waveforms are fitted to templates which
resemble the PN expansion, but in which the coefficients predicted
by PN theory are replaced by many arbitrary coefficients. Considering the success
of the EOB formalism in understanding the physics of the
coalescence of non-spinning black holes and modeling their gravitational-wave
emission with a small number of adjustable parameters,
in this paper we will use that technique, adapting it to the case of spinning black-hole binaries.
The first EOB Hamiltonian which included
spin effects was computed in Ref.~\cite{Damour01c}. In Ref.~\cite{Buonanno06}, the authors
used the non-spinning EOB Hamiltonian augmented with PN spin terms to carry out the
first exploratory study of the dynamics and gravitational radiation of
spinning black-hole binaries during inspiral, merger and ringdown.
More recently, Ref.~\cite{DJS08} extended the model of Ref.~\cite{Damour01c} to include the next-to-leading-order spin-orbit couplings.
The EOB formalism developed in Refs.~\cite{Damour01c,DJS08} highlights
several features of the spinning two-body dynamics and was recently
compared to numerical-relativity simulations of spinning non-precessing black holes in Ref.~\cite{Pan2009}.
In this paper we build on Refs.~\cite{Damour01c,DJS08} and also on Ref.~\cite{diracbrackets_ours},
in which we (in collaboration with Etienne Racine) derived the canonical Hamiltonian for a spinning test-particle
in curved spacetime, at linear order in the particle's spin, and work out
an improved EOB Hamiltonian for spinning black-hole binaries. In particular,
our EOB Hamiltonian reproduces
the leading order spin-spin coupling and the spin-orbit
coupling through 2.5PN order, for any mass-ratio. Also, it
resums \textit{all} the test-particle limit spin-orbit
terms. Moreover, when restricted to the case of spins aligned or antialigned with the orbital angular momentum,
it presents several important features, such as the existence of an
innermost stable circular orbit, a photon circular orbit, and a maximum in the orbital frequency
during the plunge subsequent to the inspiral. All of these features are crucial
for reproducing the dynamics and gravitational-wave emission of
spinning coalescing black holes, as calculated in numerical relativity simulations.
This paper is organized as follows. After presenting our notation (Sec.~\ref{sec:notation}),
in Sec.~\ref{sec:Hamiltonian_axisymmetric} we build on
Ref.~\cite{diracbrackets_ours} and derive the Hamiltonian for a spinning test particle
in axisymmetric stationary spacetimes. In Sec.~\ref{sec:hamiltonian_kerr}, we specialize
the axisymmetric stationary spacetime to the Kerr spacetime in Boyer-Lindquist coordinates.
In Sec.~\ref{sec:deformed_metric} we work out the EOB Hamiltonian of two spinning precessing
black holes. In Sec.~\ref{sec:hamiltonian_eob} we restrict the dynamics to spins aligned or antialigned
with the orbital angular momentum and determine several properties of the circular-orbit
dynamics. Section~\ref{sec:conclusions} summarizes our main conclusions.
More details on how the spin-spin sector of the EOB Hamiltonian is constructed are eventually given
in Appendix \ref{sec:ss}.
\section{Notation}
\label{sec:notation}
Throughout this paper, we use the signature $(-,+,+,+)$ for the
metric. Spacetime tensor indices (ranging from 0 to 3) are denoted
with Greek letters, while spatial tensor indices (ranging from 1 to 3)
are denoted with lowercase Latin letters. Unless stated otherwise,
we use geometric units ($G=c=1$), although we restore the
factors of $c$ when expanding in PN orders.
We define a tetrad field as a set consisting of a timelike
future-oriented vector $\tilde{e}^\mu_{ T}$ and three spacelike
vectors $\tilde{e}^\mu_{ I}$ (${ I}=1,...,3$) --- collectively denoted
as $\tilde{e}^\mu_{ A}$ (${ A}=0,...,3$) --- satisfying
\begin{equation}\label{tetrad_orthonormal}
\tilde{e}^\mu_{ A}\,\tilde{e}^\nu_{ B}\, g_{\mu\nu} = \eta_{ AB}\,,
\end{equation}
where $\eta_{ TT}=-1$, $\eta_{ TI}=0$, $\eta_{ IJ}=\delta_{ IJ}$
($\delta_{ IJ}$ being the Kronecker symbol).
Internal tetrad indices denoted with the uppercase Latin letters ${ A}$,
${ B}$, ${ C}$ and ${ D}$ always run from $0$ to $3$, while internal tetrad indices
with the uppercase Latin letters ${ I}$, ${ J}$, ${ K}$ and ${ L}$, associated
with the spacelike tetrad vectors, run from $1$ to $3$ only. The timelike
tetrad index is denoted by $T$.
Tetrad indices are raised and lowered with the metric $\eta_{ AB}$
[e.g., $\tilde{e}^\mu_{ A}=\eta_{ AB}\,(\tilde{e}^{ B})^\mu$].
We denote the projections of a vector $\boldsymbol{V}$ onto the
tetrad with $V^{ A}\equiv V^\mu\, \tilde{e}^{ A}_{\mu}$, and similarly for tensors
of higher rank. Partial derivatives will be denoted with a comma or with
$\partial$, and covariant derivatives with a semicolon.
\section{Hamiltonian for a spinning test-particle in axisymmetric stationary spacetimes}
\label{sec:Hamiltonian_axisymmetric}
Following Ref.~\cite{bardeen_astres_occlus}, we write a generic axisymmetric stationary metric in quasi-isotropic
coordinates as
\begin{eqnarray}
ds^2 = && -e^{2\nu}dt^2 +
R^2 \sin^2\theta B^2 e^{-2\nu} \left(d\phi -
\omega dt\right)^2 \nonumber \\
&& + e^{2\mu}\left(dR^2+R^2d\theta^2
\right)\;,
\label{eq:metric}
\end{eqnarray}
where $\nu$, $\mu$, $B$ and $\omega$ are functions of the coordinates $R$ and $\theta$.
Introducing the cartesian quasi-isotropic coordinates
\begin{subequations}
\begin{eqnarray}
X&=&R \sin\theta \cos\phi\,,\\
Y&=&R \sin\theta \sin\phi\,,\\
Z&=&R \cos\theta\,,
\end{eqnarray}
\end{subequations}
we can write Eq.~(\ref{eq:metric}) as
\begin{eqnarray}
ds^2 &=& e^{-2 \nu}\,\left[B^2\,\omega^2 \left (X^2+Y^2\right)-e^{4 \nu}\right] dt^2 \nonumber \\
&& + 2 B^2\, e^{-2 \nu}\, \omega\,(Y\, dX - X\, dY)\,dt \nonumber \\
&& - 2 \frac{ \left(B^2 \,e^{-2 \nu}-e^{2 \mu}\right) X\, Y}{X^2+Y^2} dX\, dY \nonumber \\
&& +\frac{e^{2 \mu} \,X^2+B^2\, e^{-2 \nu} \,Y^2}{X^2+Y^2}\,dX^2 \nonumber \\
&& + \frac{B^2\, e^{-2 \nu}\, X^2+e^{2 \mu} \,Y^2}{X^2+Y^2}\,dY^2 + e^{2 \mu}\, dZ^2\,.\nonumber \\
\label{eq:metric_cartesian}
\end{eqnarray}
It is straightforward to see that in the flat-spacetime limit ($\omega=\nu=\mu=0$, $B=1$)
Eq.~(\ref{eq:metric_cartesian}) reduces to the Minkowski metric.
Reference~\cite{diracbrackets_ours}
computed the Hamiltonian of a spinning test-particle in curved spacetime at linear order in the particle's spin, and showed that
it can be written as
\begin{equation}
H = H_{\rm NS} + H_{\rm S}\,,
\end{equation}
where ${H}_{\rm NS}$ is the Hamiltonian for a non-spinning test particle of mass $m$, given by
\begin{equation}\label{eq:Hns}
{H}_{\rm NS} = \beta^iP_i + \alpha \sqrt{m^2 + \gamma^{ij}\,P_i\,P_j}\,,
\end{equation}
with
\begin{eqnarray}
\label{alphai}
\alpha &=& \frac{1}{\sqrt{-g^{tt}}}\,,\\
\beta^i &=& \frac{g^{ti}}{g^{tt}}\,,\\
\gamma^{ij} &=& g^{ij}-\frac{g^{ti} g^{tj}}{g^{tt}}\,,
\label{gammai}
\end{eqnarray}
and
\begin{equation}
\label{Hamiltonian_canonical_final2}
{H}_{\rm S} = - \left(\beta^i\,F_i^K + F_t^K + \frac{\alpha \gamma^{ij}\,P_i\,F_j^K}{\sqrt{m^2 +
\gamma^{ij}P_iP_j}}\right)\,S_K \,,\nonumber \\
\end{equation}
where the coefficients $F_\mu^I$ can be expressed in terms of a reference tetrad field $\tilde{e}_A$ as
\begin{eqnarray}\label{FmuK_def}
F_\mu^K &=& \left(2E_{\mu TI}\,\frac{\bar{\omega}_J}{\bar{\omega}_T} + E_{\mu IJ}\right)\,\epsilon^{IJK}\,,\\
E_{\lambda\mu\nu} &\equiv& \frac{1}{2}\,\eta_{AB}\,\tilde{e}_\mu^A\,\tilde{e}_{\nu;\lambda}^{B}\,,
\end{eqnarray}
with
\begin{eqnarray}
\bar{\omega}_\mu&=&\bar{P}_\mu-m \,\tilde{e}^{T}_\mu\,,\\
\bar{P}_i &=& P_i\,,\\
\bar{P}_t &=& -\beta^i\,P_i-\alpha\, \sqrt{m^2 +\gamma^{ij}\,P_i\, P_j}\,,\\
\label{bomegaT}
\bar{\omega}_T &=& \bar{\omega}_\mu\,\tilde{e}^\mu_{T}=\bar{P}_\mu\,\tilde{e}^\mu_{T}-m\,,\\
\bar{\omega}_I &=&\bar{\omega}_\mu\,\tilde{e}^\mu_{I}= \bar{P}_\mu\,\tilde{e}^\mu_{I}\,.
\label{bomegaI}
\end{eqnarray}
Reference~\cite{diracbrackets_ours} also showed that in order to obtain a Hamiltonian giving the usual leading-order spin-orbit
coupling without gauge effects (or, equivalently, $H_{\rm S}=0$ in flat spacetime),
the reference tetrad field must become {\it cartesian} in the flat-spacetime limit. We find that
the following choice for the reference tetrad
\begin{subequations}
\begin{align}
&\tilde{e}^T_\alpha=\delta_\alpha^t(-g^{tt})^{-1/2}=e^\nu \,\delta_\alpha^t\,,\label{tetrad0}\\
&\tilde{e}_1^\alpha=\frac{B\, e^{-\mu}\, X^2+e^\nu\, Y^2}{B \,(X^2+Y^2)} \,\delta^\alpha_X + \frac{\left(B\, e^{-\mu}-e^\nu\,\right) \,X \,Y}{B\,\left(X^2+Y^2\right)}\, \delta^\alpha_Y\nonumber \,,\\\label{tetrad1}\\
&\tilde{e}_2^\alpha= \frac{\left(B \,e^{-\mu}-e^\nu\right)\, X\, Y}{B\, \left(X^2+Y^2\right)}\, \delta^\alpha_X +\frac{e^\nu\, X^2+
B\, e^{-\mu} Y^2}{B \,(X^2 +Y^2)}\, \delta^\alpha_Y\nonumber \,,\\\label{tetrad2}\\
&\tilde{e}_3^\alpha=e^{-\mu}\, \delta_Z^\alpha\label{tetrad3}\,,
\end{align}
\end{subequations}
indeed reduces to the cartesian tetrad $\tilde{e}^T_\alpha=1$, $\tilde{e}_I^\alpha=\delta_I^\alpha$
in the flat-spacetime limit.
We can then use the tetrad defined by Eqs.~(\ref{tetrad0})--(\ref{tetrad3}) to calculate
the coefficients $F^K_\mu$ in Eq.~\eqref{FmuK_def}, and obtain
\begin{equation}
\label{HS}
H_{\rm S} = H_{\rm SO} + H_{\rm SS}\,,
\end{equation}
with
\begin{widetext}
\begin{align}
\label{HSO}
H_{\rm SO} =& \frac{e^{2 \nu -\mu } \left(e^{\mu +\nu }-B\right)
(\boldsymbol{\hat{P}}\cdot \boldsymbol{\xi} R)\,S^Z}{B^2\, \sqrt{Q}\, R^2
\,\xi^2}+ \frac{e^{\nu -2 \mu }}{B^2\, \left(\sqrt{Q}+1\right)\, \sqrt{Q}
R^2\, \xi^2}\Bigg\{B_{\cos\theta} \,e^{\mu +\nu }
(\boldsymbol{\hat{P}}\cdot
\boldsymbol{\xi}\, R) \left(\sqrt{Q}+1\right)\, (\boldsymbol{S}\cdot \boldsymbol{N})\, \xi^2 \nonumber \\
& + R\, (\boldsymbol{S}\cdot \boldsymbol{\xi})\, \,\left[\mu_R
(\boldsymbol{\hat{P}}\cdot \boldsymbol{V}\, R)
\left(\sqrt{Q}+1\right)-\mu_{\cos\theta}\, (\boldsymbol{\hat{P}}\cdot
\boldsymbol{N})\, \xi^2-\sqrt{Q}\, (\nu_R \,(\boldsymbol{\hat{P}}\cdot
\boldsymbol{V}\, R)+(\mu_{\cos\theta}-\nu_{\cos\theta})
(\boldsymbol{\hat{P}}\cdot \boldsymbol{N})\, \xi^2)\right]\, B^2 \nonumber \\
& + e^{\mu +\nu }\, (\boldsymbol{\hat{P}}\cdot \boldsymbol{\xi}\, R)
\left(2 \sqrt{Q}+1\right) \Big[\nu_R\, R\, (\boldsymbol{S}\cdot
\boldsymbol{V})-\nu_{\cos\theta}\, (\boldsymbol{S}\cdot \boldsymbol{N})
\,\xi^2\Big] \,B-B_R \,e^{\mu +\nu } (\boldsymbol{\hat{P}}\cdot
\boldsymbol{\xi}\, R)
\left(\sqrt{Q}+1\right)\, R\, (\boldsymbol{S}\cdot \boldsymbol{V})\Bigg\}\,,\nonumber \\
\end{align}
\begin{align}
\label{HSS}
H_{\rm SS} =& \omega\, S^Z+\frac{e^{-3 \mu -\nu }\, \omega_R}{2 B\, \left(\sqrt{Q}+1\right)\, \sqrt{Q}\, R
\xi^2} \Bigg\{-e^{\mu +\nu }\, (\boldsymbol{\hat{P}}\cdot \boldsymbol{V}\, R) (\boldsymbol{\hat{P}}\cdot \boldsymbol{\xi}\, R)
(\boldsymbol{S}\cdot \boldsymbol{\xi})\, B+e^{2 (\mu +\nu )}\, (\boldsymbol{\hat{P}}\cdot \boldsymbol{\xi}\, R)^2 (\boldsymbol{S}
\cdot \boldsymbol{V}) \nonumber \\
&+ e^{2 \mu }\, \left(1+\sqrt{Q}\right)\,\sqrt{Q}\, R^2\, (\boldsymbol{S}\cdot \boldsymbol{V})\, \xi^2\, B^2
+(\boldsymbol{\hat{P}}\cdot \boldsymbol{N})\, R \,\left[(\boldsymbol{\hat{P}}\cdot \boldsymbol{V} \,R)
(\boldsymbol{S}\cdot \boldsymbol{N})
-(\boldsymbol{\hat{P}}\cdot \boldsymbol{N})\, R\, (\boldsymbol{S}\cdot \boldsymbol{V})\right]
\xi^2\, B^2\,\Bigg\} \nonumber \\
& +\frac{e^{-3 \mu -\nu }\, \omega_{\cos\theta}}{2 B\, \left(\sqrt{Q}+1\right)
\,\sqrt{Q}\, R^2} \Bigg\{e^{\mu +\nu }\, (\boldsymbol{\hat{P}}\cdot \boldsymbol{N})
(\boldsymbol{\hat{P}}\cdot \boldsymbol{\xi}\, R)\, R\, (\boldsymbol{S}\cdot \boldsymbol{\xi})\, B-e^{2 (\mu +\nu )}
(\boldsymbol{\hat{P}}\cdot \boldsymbol{\xi}\, R)^2 (\boldsymbol{S}\cdot \boldsymbol{N}) \nonumber \\
& + \left[(\boldsymbol{S}\cdot \boldsymbol{N})
(\boldsymbol{\hat{P}}\cdot \boldsymbol{V}\, R)^2\!-\!(\boldsymbol{\hat{P}}\cdot \boldsymbol{N})\, R\,
(\boldsymbol{S}\cdot \boldsymbol{V})
(\boldsymbol{\hat{P}}\cdot \boldsymbol{V} \,R)-e^{2 \mu }\,\left(1+\sqrt{Q}\right)\!\sqrt{Q}\, R^2
(\boldsymbol{S}\cdot \boldsymbol{N})\, \xi^2\right]\,B^2 \Bigg\}\,,
\end{align}
\begin{equation}\label{eq:Q_general}
Q=1 + \gamma^{ij} {\hat{P}_i\,\hat{P}_j} = 1 + e^{-2 \mu }\, (\boldsymbol{\hat{P}}\cdot \boldsymbol{N})^2
+\frac{e^{-2 \mu }\, (\boldsymbol{\hat{P}}\cdot \boldsymbol{V}\, R)^2}{R^2 \,\xi^2}+\frac{e^{2 \nu }
\,(\boldsymbol{\hat{P}}\cdot \boldsymbol{\xi}\, R)^2}{B^2\, R^2\,\xi^2}\,,
\end{equation}
\end{widetext}
where we denote
\begin{eqnarray}
\hat{\boldsymbol{P}}&=&\frac{\boldsymbol{P}}{m}\,, \\
\boldsymbol{N}&=&\frac{\boldsymbol{X}}{R}\,, \\
\boldsymbol{\xi}&=&\boldsymbol{e}_Z\times\boldsymbol{N}=\frac{-Y\,\boldsymbol{e}_X + X\,\boldsymbol{e}_Y}{R}\,,\\
\boldsymbol{V}&=&\boldsymbol{N} \times \boldsymbol{\xi}\,,
\end{eqnarray}
and
\begin{eqnarray}
f_R &\equiv& \frac{\partial f(R,\cos\theta)}{\partial R}\,,\\
f_{\cos\theta}&\equiv& \frac{\partial f(R,\cos\theta)}{\partial (\cos\theta)}\,.
\end{eqnarray}
Here, the generic function $f$ can stand for $B$, $\omega$, $\mu$ or $\nu$.
Note that because $\omega$ is proportional to $g_{t\phi}$ [see Eq.~(\ref{eq:metric})]
and thus to the spin of the spacetime, $H_{\rm SS}$ (which is proportional to
$\omega$ and its derivatives) gives the leading-order coupling between the particle's spin
and the spin of the background spacetime
(together with other higher order terms). Also, because $\boldsymbol{\hat{P}}\cdot \boldsymbol{\xi}\, R=\hat{P}_\phi$
in spherical coordinates, $H_{\rm SO}$ is the part of the Hamiltonian which gives the leading-order spin orbit coupling (again,
together with other higher order terms).
Moreover, note that $H_{\rm S}=0$
in a flat spacetime, thus confirming the absence of gauge effects in the leading order spin-orbit coupling.
As a consistency test, we specialize to the case of a spherically symmetric spacetime
in quasi-isotropic coordinates, which was considered in Ref.~\cite{diracbrackets_ours} (see Sec.~V A
therein). Because the metric for such a spacetime is given by
\begin{equation}
ds^2 = -f(R) \,dt^2 + h(R)\,(dX^2+dY^2+dZ^2)\,,
\label{metricS}
\end{equation}
a comparison with Eq.~\eqref{eq:metric} immediately reveals that
\begin{eqnarray}
\label{BS}
B&=&\sqrt{f(R) h(R)}\,,\\
\omega&=&0\,,\\
\nu&=&\frac12 \log[f(R)]\,,\\
\mu&=&\frac12 \log[h(R)]\,.
\label{muS}
\end{eqnarray}
Inserting Eqs.~(\ref{BS})--(\ref{muS}) in Eqs.~(\ref{HS})--(\ref{eq:Q_general}), we find
\begin{eqnarray}
\label{Hschw_QI}
H_{\rm S}&=& \frac{\boldsymbol{L}\cdot\boldsymbol{S}}
{2 m\,R\,\sqrt{f(R)}\,h(R)^2\,\sqrt{Q}\,(1+\sqrt{Q})} \times \nonumber \\
&& \left \{ \sqrt{Q}[f^\prime(R)\, h(R) - f(R)\,h^\prime(R)] -
f(R)\,h^\prime(R) \right \}\,,\nonumber \\
\end{eqnarray}
where
\begin{eqnarray}
Q &=& 1 + \frac{1}{h}\,\hat{\boldsymbol{P}}^2\,, \\
\boldsymbol{L} &=& \boldsymbol{X}\times \boldsymbol{P}\,,
\end{eqnarray}
in agreement with Eq.~(5.7) in Ref.~\cite{diracbrackets_ours}.
Let us now investigate how the Hamiltonian (\ref{HS}) is affected by a change
of the radial coordinate $R$. Denoting the new radial
coordinate by $r = |\boldsymbol{x}|$ and defining
\begin{equation}
J^{-1}\equiv \frac{d R }{d r}\,,
\end{equation}
the radial derivatives of the metric potentials can be re-expressed as
\begin{equation}
f_R = {f_r}\,J \,,
\end{equation}
where again $f=B,\,\omega,\,\nu,\,\mu$.
The spin $\boldsymbol{S}$, the derivatives of the metric potentials with respect to $\cos\theta$,
and the quantities
\begin{eqnarray}
\boldsymbol{N} &=& \boldsymbol{n} = \frac{\boldsymbol{x}}{r}\,,\\
\boldsymbol{\xi} &=& \boldsymbol{e}_Z\times\boldsymbol{N} = \boldsymbol{e}_z\times\boldsymbol{n}\,,\label{eq:xidef}\\
\boldsymbol{V} &=& \boldsymbol{v} = \boldsymbol{n} \times \boldsymbol{\xi}\,\\
&&\nonumber
\end{eqnarray}
are not affected by the coordinate change. The same applies to the quantities $\boldsymbol{\hat{P}}\cdot \boldsymbol{V}\, R$ and
$\boldsymbol{\hat{P}}\cdot \boldsymbol{\xi} R$ appearing in Eqs.~(\ref{HSO}), (\ref{HSS}) and~\eqref{eq:Q_general}.
In fact, in spherical coordinates, we have
$\boldsymbol{\hat{P}}\cdot
\boldsymbol{V} R = -\hat{P}_\theta \sin\theta$ and
$\boldsymbol{\hat{P}}\cdot \boldsymbol{\xi} R = \hat{P}_\phi$, hence
\begin{eqnarray}
\boldsymbol{\hat{P}}\cdot \boldsymbol{V} R &=& \boldsymbol{\hat{p}}\cdot \boldsymbol{v} \,r\,,\\
\boldsymbol{\hat{P}}\cdot \boldsymbol{\xi} R &=& \boldsymbol{\hat{p}}\cdot \boldsymbol{\xi} \,r\,,
\end{eqnarray}
where $\boldsymbol{\hat{p}} = \boldsymbol{p}/m$ and $\boldsymbol{p}$ is the conjugate momentum
in the new coordinate system, i.e., $p_i=\partial X^{j}/\partial x^i\, P_j$.
On the contrary, since $\boldsymbol{\hat{P}}\cdot \boldsymbol{N} =\hat{P}_R$, we have
\begin{equation}
\boldsymbol{\hat{P}}\cdot \boldsymbol{N} = {(\boldsymbol{\hat{p}}\cdot \boldsymbol{n})}\,{J}\,.
\end{equation}
It is therefore straightforward to compute $H_{\rm S}$
in a coordinate system related to quasi-isotropic coordinates by a rescaling of the
radius. We have
\begin{equation}
H_{\rm S} = H_{\rm SO} + H_{\rm SS}\,,
\label{Ht}
\end{equation}
where
\begin{widetext}
\begin{eqnarray}
H_{\rm SO} &=& \frac{e^{2 \nu -\mu }\, \left(e^{\mu +\nu }-B\right)\, (\boldsymbol{\hat{p}}\cdot \boldsymbol{\xi}\, r) \,S^z}
{B^2\,\sqrt{Q}\, R^2\, \xi^2}+
\frac{e^{\nu -2 \mu }}{B^2 \,\left(\sqrt{Q}+1\right)\, \sqrt{Q}\, R^2\, \xi^2}
\Bigg\{B_{\cos\theta}\, e^{\mu +\nu } (\boldsymbol{\hat{p}}\cdot
\boldsymbol{\xi}\, r) \left(\sqrt{Q}+1\right) (\boldsymbol{S}\cdot \boldsymbol{n})\, \xi^2 \nonumber \\
&& + R\, (\boldsymbol{S}\cdot \boldsymbol{\xi}) J
\left[\mu_r\, (\boldsymbol{\hat{p}}\cdot \boldsymbol{v}\, r)
\left(\sqrt{Q}+1\right)-\mu_{\cos\theta} (\boldsymbol{\hat{p}}\cdot \boldsymbol{n}) \,
\xi^2-\sqrt{Q}\, (\nu_r\, (\boldsymbol{\hat{p}}\cdot \boldsymbol{v}\, r)+(\mu_{\cos\theta}-\nu_{\cos\theta})
(\boldsymbol{\hat{p}}\cdot \boldsymbol{n})\, \xi^2)\right]\, B^2 \nonumber \\
&& +e^{\mu +\nu }\, (\boldsymbol{\hat{p}}\cdot \boldsymbol{\xi}\, r)
\left(2 \sqrt{Q}+1\right) \Big[J\, \nu_r\, R\, (\boldsymbol{S}\cdot \boldsymbol{v})-\nu_{\cos\theta}
\,(\boldsymbol{S}\cdot \boldsymbol{n})\, \xi^2\Big]\, B-J\, B_r\, e^{\mu +\nu }\, (\boldsymbol{\hat{p}}\cdot \boldsymbol{\xi}\, r)
\left(\sqrt{Q}+1\right)\, R\, (\boldsymbol{S}\cdot \boldsymbol{v})\Bigg\}\,, \nonumber \\\label{Ht1}
\end{eqnarray}
\begin{eqnarray}
H_{\rm SS} &=& \omega \,S^z+\frac{e^{-3 \mu -\nu }\, J\, \omega_r}{2 B\, \left(\sqrt{Q}+1\right)\, \sqrt{Q}\, R
\,\xi^2} \Bigg\{ -e^{\mu +\nu }\, (\boldsymbol{\hat{p}}\cdot \boldsymbol{v}\, r) (\boldsymbol{\hat{p}}\cdot \boldsymbol{\xi}\, r)
(\boldsymbol{S}\cdot \boldsymbol{\xi}) \,B+e^{2 (\mu +\nu )}\, (\boldsymbol{\hat{p}}\cdot \boldsymbol{\xi}\, r)^2 (\boldsymbol{S}
\cdot \boldsymbol{v}) \nonumber \\
&& + e^{2 \mu }\, \left(1+\sqrt{Q}\right)\,\sqrt{Q}\, R^2\, (\boldsymbol{S}\cdot \boldsymbol{v})\, \xi^2\, B^2\,
+J \,(\boldsymbol{\hat{p}}\cdot \boldsymbol{n}) \,R \left[(\boldsymbol{\hat{p}}\cdot \boldsymbol{v}\, r) (\boldsymbol{S}\cdot \boldsymbol{n})
-J\, (\boldsymbol{\hat{p}}\cdot \boldsymbol{n}) \,R (\boldsymbol{S}\cdot \boldsymbol{v})\right]
\xi^2\, B^2\Bigg\} \nonumber \\
&& +\frac{e^{-3 \mu -\nu }\, \omega_{\cos\theta}}{2 B\, \left(\sqrt{Q}+1\right)
\,\sqrt{Q}\, R^2} \Bigg\{
-e^{2 (\mu +\nu )}\,
(\boldsymbol{\hat{p}}\cdot \boldsymbol{\xi}\, r)^2 \,(\boldsymbol{S}\cdot \boldsymbol{n}) \,
+e^{\mu +\nu }\, J\, (\boldsymbol{\hat{p}}\cdot \boldsymbol{n}) \,
(\boldsymbol{\hat{p}}\cdot \boldsymbol{\xi}\, r)\, R \,(\boldsymbol{S}\cdot \boldsymbol{\xi}) \,B\, \nonumber \\
&& + \left[(\boldsymbol{S}\cdot \boldsymbol{n}) \,
(\boldsymbol{\hat{p}}\cdot \boldsymbol{v} \,r)^2-J\, (\boldsymbol{\hat{p}}\cdot \boldsymbol{n})\, R\,
(\boldsymbol{S}\cdot \boldsymbol{v}) \,(\boldsymbol{\hat{p}}\cdot \boldsymbol{v}\, r) -e^{2 \mu }\,
\left(1+\sqrt{Q}\right)\,\sqrt{Q}\, R^2\,(\boldsymbol{S}\cdot \boldsymbol{n})\, \xi^2\right]\, B^2
\Bigg\}\,,\label{Ht2}
\end{eqnarray}
\begin{equation}
\label{Q}
Q=1 + \gamma^{ij}\, {\hat{p}_i\,\hat{p}_j} = 1 + e^{-2 \mu }\, (\boldsymbol{\hat{p}}\cdot \boldsymbol{n})^2 \,J^2
+\frac{e^{-2 \mu }\, (\boldsymbol{\hat{p}}\cdot \boldsymbol{v}\, r)^2}{R^2\, \xi^2}+\frac{e^{2 \nu }
(\boldsymbol{\hat{p}}\cdot \boldsymbol{\xi}\, r)^2}{B^2\, R^2\,\xi^2}\,,
\end{equation}
\end{widetext}
and where $R$ must of course be expressed in terms of the new radial coordinate $r$.
\section{Hamiltonian for a spinning test-particle in Kerr spacetime in Boyer-Lindquist coordinates}
\label{sec:hamiltonian_kerr}
In this section, we will specialize the Hamiltonian derived in the previous section to the case of Kerr spacetime
in Boyer-Lindquist coordinates.
We start from the metric potentials appearing in Eq.~(\ref{eq:metric}), which in the case of a Kerr spacetime take
the form~\cite{cook_living_rev}
\begin{eqnarray}
\label{coeffB}
B&=&\frac{\sqrt{\Delta}}{R}\,,\\
\omega&=&\frac{2 a M r}{\Lambda}\,,\\
e^{2\nu}&=&\frac{\Delta\Sigma}{\Lambda}\,,\\
e^{2\mu}&=&\frac{\Sigma}{R^2}\,,
\label{coeffmu}
\end{eqnarray}
with
\begin{eqnarray}
\Sigma &=& r^2 + a^2\cos^2\theta \,, \\
\Delta &=& r^2 + a^2 - 2Mr \,, \\
\varpi^2 &=& r^2 + a^2 \,, \\
\Lambda &=& \varpi^4 - a^2\Delta\sin^2\theta \,,
\label{Lambda}
\end{eqnarray}
where the parameter $a$, which has the dimensions of a length, is related to the spin vector $\boldsymbol{S}_{\rm Kerr}$
of the Kerr black hole by
\begin{equation}
a= \frac{\vert\boldsymbol{S}_{\rm Kerr}\vert}{M}\,.
\end{equation}
The Boyer-Lindquist coordinate $r$ is related to the quasi-isotropic coordinate $R$ by
\begin{equation}
r=R + M + \frac{R_{\rm H}^2}{R}\,,
\end{equation}
where $R_{\rm H}=\sqrt{M^2 - a^2}/2$ is the horizon's radius in quasi-isotropic
coordinates. Note that the inverse of this transformation is given, outside the horizon, by
\begin{equation}
R=\frac12 \big(r -M + \sqrt{\Delta}\big)\,.
\end{equation}
We then obtain that the derivatives of the metric potentials take the form
\begin{subequations}
\begin{eqnarray}
\label{eq1}
B_r &=& \frac{r - M - \sqrt{\Delta}}{R\,\sqrt{\Delta}}\,,\\
\omega_r &=& \frac{2 a\, M\, [\Sigma\, \varpi^2 - 2 r^2\, (\Sigma + \varpi^2)]}{\Lambda^2}\,,\\
\nu_r &=& \frac{r - M}{\Delta} + \frac{r}{\Sigma} \nonumber \\
&& - \frac{2 r\, \varpi^2 - a^2\, (r - M)\, \sin^2\theta}{\Lambda}\,,\\
\mu_r &=& \frac{r}{\Sigma} - \frac{1}{\sqrt{\Delta}}\,,\\
B_{\cos\theta} &=& 0\,,\\
\omega_{\cos\theta}&=& -\frac{4 a^3\, M\, r\, \Delta\, \cos\theta}{(\Delta\,\Sigma + 2 M \,r\, \varpi^2)^2}\,,\\
\nu_{\cos\theta}&=&\frac{2 a^2\, M\, r \,\varpi^2\, \cos\theta}{(\Delta\, \Sigma + 2 M\, r\, \varpi^2)\,\Sigma}\,,\\
\mu_{\cos\theta}&=&\frac{a^2\, \cos\theta}{\Sigma}\,,
\label{eq8}
\end{eqnarray}
\end{subequations}
and we also have
\begin{equation}
\label{eqJ}
J^{-1}= \frac{d R }{d r} = \frac{R}{\sqrt{\Delta}}\,.
\end{equation}
Inserting Eqs.~(\ref{eq1})--(\ref{eq8}) and Eq.~(\ref{eqJ}) into
Eqs.~(\ref{Ht})--(\ref{Q}), we find that $R$ cancels out both in
$Q$, that is
\begin{equation}
Q=1+\frac{\Delta (\boldsymbol{\hat{p}}\cdot \boldsymbol{n})^2}{\Sigma}+
\frac{(\boldsymbol{\hat{p}}\cdot \boldsymbol{\xi}\, r)^2 \,\Sigma }{\Lambda \,\sin^2\theta}+
\frac{(\boldsymbol{\hat{p}}\cdot \boldsymbol{v}\, r)^2 }{\Sigma\,\sin^2\theta}\,,
\end{equation}
and in the Hamiltonian $H_{\rm S}$. In conclusion, the Hamiltonian
of a spinning test-particle in Kerr spacetime in Boyer-Lindquist coordinates is
\begin{equation}
\label{Htotal}
H = H_{\rm NS} + H_{\rm S}\,,
\end{equation}
with
\begin{equation}
\label{HNSKerr}
{H}_{\rm NS} = \beta^i\,p_i + \alpha\, \sqrt{m^2 + \gamma^{ij}\,p_i\,p_j}\,,
\end{equation}
where $\alpha$, $\beta^i$ and $\gamma^{ij}$ are given in Eqs.~(\ref{alphai})--(\ref{gammai})
and need to be computed using the Kerr metric coefficients (\ref{coeffB})--(\ref{Lambda}), and with
\begin{equation}
\label{KerrHS}
H_{\rm S} = H_{\rm SO} + H_{\rm SS}\,,
\end{equation}
where
\begin{widetext}
\begin{eqnarray}
H_{\rm SO} &=& \frac{e^{2 \nu -\tilde{\mu} }\,\left(e^{\tilde{\mu} +\nu }-\tilde{B}\right)\,
(\boldsymbol{\hat{p}}\cdot \boldsymbol{\xi}\, r)\, (\boldsymbol{S}\cdot \boldsymbol{\hat{S}}_{\rm Kerr})}{\tilde{B}^2\, \sqrt{Q}\,\xi^2} +
\frac{e^{\nu -2 \tilde{\mu} }}{\tilde{B}^2\, \left(\sqrt{Q}+1\right)\, \sqrt{Q}\, \xi^2}\Bigg\{
(\boldsymbol{S}\cdot \boldsymbol{\xi})\, \tilde{J}
\left[\mu_r\, (\boldsymbol{\hat{p}}\cdot \boldsymbol{v}\, r) \left(\sqrt{Q}+1\right)-\mu_{\cos\theta}\,
(\boldsymbol{\hat{p}}\cdot \boldsymbol{n})\,
\xi^2 \right .\nonumber \\
&& \left . -\sqrt{Q}\, (\nu_r\, (\boldsymbol{\hat{p}}\cdot \boldsymbol{v}\, r)+(\mu_{\cos\theta}-\nu_{\cos\theta})\,
(\boldsymbol{\hat{p}}\cdot \boldsymbol{n}) \,\xi^2)\right]\, \tilde{B}^2 +e^{\tilde{\mu} +\nu }\,
(\boldsymbol{\hat{p}}\cdot \boldsymbol{\xi}\, r)\,
\left(2 \sqrt{Q}+1\right)\, \Big[\tilde{J}\, \nu_r\, (\boldsymbol{S}\cdot \boldsymbol{v})-\nu_{\cos\theta}\,
(\boldsymbol{S}\cdot \boldsymbol{n})\, \xi^2\Big]\, \tilde{B} \nonumber \\
&& -\tilde{J}\, \tilde{B}_r\, e^{\tilde{\mu} +\nu }\, (\boldsymbol{\hat{p}}\cdot \boldsymbol{\xi}\, r)\,
\left(\sqrt{Q}+1\right)\, (\boldsymbol{S}\cdot \boldsymbol{v})\Bigg\}\,,\label{KerrHSO}
\end{eqnarray}
and
\begin{eqnarray}
H_{\rm SS} &=& \omega\, (\boldsymbol{S}\cdot \boldsymbol{\hat{S}}_{\rm Kerr})+\frac{e^{-3 \tilde{\mu} -\nu }\, \tilde{J}\, \omega_r}{2 \tilde{B}\, \left(\sqrt{Q}+1\right)\, \sqrt{Q}\,
\xi^2} \Bigg\{ -e^{\tilde{\mu} +\nu }\, (\boldsymbol{\hat{p}}\cdot \boldsymbol{v}\, r) \,
(\boldsymbol{\hat{p}}\cdot \boldsymbol{\xi}\, r)\, (\boldsymbol{S}\cdot \boldsymbol{\xi})\,
\tilde{B}+e^{2 (\tilde{\mu} +\nu )}\, (\boldsymbol{\hat{p}}\cdot \boldsymbol{\xi}\, r)^2\, (\boldsymbol{S}
\cdot \boldsymbol{v}) \nonumber \\
&& + e^{2 \tilde{\mu} }\, \left(1+\sqrt{Q}\right)\,\sqrt{Q}\,(\boldsymbol{S}\cdot \boldsymbol{v})\,\xi^2\, \tilde{B}^2
+\tilde{J} (\boldsymbol{\hat{p}}\cdot \boldsymbol{n})\, \left[(\boldsymbol{\hat{p}}\cdot \boldsymbol{v}\, r)\,
(\boldsymbol{S}\cdot \boldsymbol{n}) -\tilde{J} (\boldsymbol{\hat{p}}\cdot \boldsymbol{n})\, (\boldsymbol{S}\cdot
\boldsymbol{v})\right]\,\xi^2\,\tilde{B}^2\Bigg\} \nonumber \\
&& +\frac{e^{-3 \tilde{\mu} -\nu }\, \omega_{\cos\theta}}{2 \tilde{B}\, \left(\sqrt{Q}+1\right)\,
\sqrt{Q} }\, \Bigg\{-e^{2 (\tilde{\mu} +\nu )}\,
(\boldsymbol{\hat{p}}\cdot \boldsymbol{\xi} \,r)^2\, (\boldsymbol{S}\cdot \boldsymbol{n})
+ e^{\tilde{\mu} +\nu }\, \tilde{J}\, (\boldsymbol{\hat{p}}\cdot \boldsymbol{n})\,
(\boldsymbol{\hat{p}}\cdot \boldsymbol{\xi}\, r)\, (\boldsymbol{S}\cdot \boldsymbol{\xi}) \tilde{B}
\nonumber \\
&& + \left[(\boldsymbol{S}\cdot \boldsymbol{n})\,(\boldsymbol{\hat{p}}\cdot \boldsymbol{v}\,r)^2-\tilde{J}\,
(\boldsymbol{\hat{p}}\cdot \boldsymbol{n})\,(\boldsymbol{S}\cdot \boldsymbol{v})\,
(\boldsymbol{\hat{p}}\cdot \boldsymbol{v}\,r)-e^{2 \tilde{\mu} }\,\left(1+\sqrt{Q}\right)\,\sqrt{Q}\,
(\boldsymbol{S}\cdot \boldsymbol{n}) \,\xi^2\right]\, \tilde{B}^2
\Bigg\}\,,\label{KerrHSS}
\end{eqnarray}
\end{widetext}
where we define
\begin{eqnarray}
\tilde{B}&=&B \,R= \sqrt{\Delta}\,,\\
\tilde{B}_r&=&B_r\, R=\frac{r - M - \sqrt{\Delta}}{\sqrt{\Delta}}\,,\\
e^{2\tilde{\mu}} &=&e^{2\mu}\, R^2 = \Sigma\,,\\
\tilde{J}&=&J\, R = \sqrt{\Delta}\,,\\
\boldsymbol{\hat{S}}_{\rm Kerr}&=&\frac{\boldsymbol{{S}}_{\rm Kerr}}{\vert\boldsymbol{{S}}_{\rm Kerr}\vert}
\end{eqnarray}
and we recall that $\xi^2=\sin^2\theta$. We stress that because this Hamiltonian is expressed in terms
of quantities which are scalar under spatial rotations, we can express it in a cartesian coordinate system
in which the spin of the Kerr black hole is not directed along the $z$-axis. For that purpose, it is
sufficient to replace $r$ with $(x^2+y^2+z^2)^{1/2}$, $\cos\theta$ with
$\boldsymbol{\hat{S}}_{\rm Kerr}\cdot \boldsymbol{n}$, $\boldsymbol{e}_z$ with $\boldsymbol{\hat{S}}_{\rm Kerr}$ in Eq.~\eqref{eq:xidef}, and express the vectors appearing in Eqs.~(\ref{HNSKerr})--(\ref{KerrHSS})
in terms of their cartesian components.
As a consistency check, we can compute
the Hamiltonian for a spinning test-particle in a Schwarzschild spacetime in Schwarzschild spherical coordinates by setting $a=0$,
and compare the result to the expression computed in Ref.~\cite{diracbrackets_ours} [see Eq.~(5.12) therein].
We find
\begin{align}
H_{\rm S}=&\frac{\psi^6}{R^3 \sqrt{Q}(1+\sqrt{Q})}\times\nonumber\\&\left[1-\frac{M}{2R} +
2\left(1-\frac{M}{4R}\right)\sqrt{Q}\right](\boldsymbol{L}\cdot\boldsymbol{S}^\ast)\,,
\label{Hqi}
\end{align}
where
\begin{eqnarray}
\boldsymbol{S}^\ast &=&\frac{M}{m} \,\boldsymbol{S}\,,\\
\psi &=& \left(1 + \frac{M}{2R}\right)^{-1}\,,\\
R &=& \frac12 \left(r-M+\sqrt{r^2-2 M r}\right)\,,
\end{eqnarray}
and
\begin{equation}
Q=1+(\boldsymbol{\hat{p}}\cdot \boldsymbol{n})^2\, \left(1-\frac{2 M}{r}\right)
+\frac{(\boldsymbol{\hat{p}}\cdot \boldsymbol{v})^2+(\boldsymbol{\hat{p}}\cdot
\boldsymbol{\xi})^2 }{\sin^2\theta}\,,
\end{equation}
in agreement with Ref.~\cite{diracbrackets_ours}.
Also, it is worth noting that the Hamiltonian (\ref{Hqi}) is the same as the quasi-isotropic Schwarzschild
Hamiltonian~\eqref{Hschw_QI}, expressed in terms of the Schwarzschild
coordinate $r$. This is because the scalar
product $\boldsymbol{L}\cdot\boldsymbol{S}$ is unaffected by a change
of the radial coordinate.
\section{Effective-one-body Hamiltonian for two spinning black holes}
\label{sec:deformed_metric}
The EOB approach was originally introduced in Refs.~\cite{Buonanno99,Buonanno00,DJS3PN,
Damour01c} to provide us with an {\it improved} (resummed) Hamiltonian that could be used
to evolve a binary system not only during the long inspiral, but also during
the plunge, and that could supply a natural {\it moment} at which to switch
from the two body description to the one-body description,
in which the system is represented by a superposition of quasi-normal
modes of the remnant black hole.
A crucial ingredient of the EOB approach
is the {\it real} PN-expanded Arnowitt-Deser-Misner (ADM) Hamiltonian (or {\it real}
Hamiltonian) describing two black holes of masses $m_1, m_2$ and spins
$\boldsymbol{S}_1$, $\boldsymbol{S}_2$. The real Hamiltonian is then
canonically transformed and subsequently \textit{mapped} to an {\it effective} Hamiltonian $H_{\rm eff}$
describing a test-particle of mass $\mu=m_1\,m_2/(m_1+m_2)$ and suitable
spin $\boldsymbol{S}^\ast$, moving in a \textit{deformed} Kerr metric of mass $M = m_1 + m_2$ and
suitable spin $\boldsymbol{S}_{\rm Kerr}$.
The parameter regulating the deformation is the symmetric mass ratio of the binary, $\eta=\mu/M$,
which ensures that the deformation disappears in the case of extreme mass-ratio binaries. The resulting improved EOB
Hamiltonian then takes the form
\begin{equation}
\label{hreal0}
H^{\rm improved}_\mathrm{real} = M\,\sqrt{1+2\eta\,\left(\frac{H_{\rm eff}}{\mu}-1\right)}\,.
\end{equation}
The computation of the improved EOB Hamiltonian consists of several stages. For this reason,
we briefly review here the main steps and the underpinning logic that
we will follow in the rest of this section:
\begin{enumerate}
\item[(i)] We apply a canonical transformation to the PN-expanded ADM Hamiltonian using a generating function
which is compatible with the one used in previous EOB work, obtaining the PN-expanded Hamiltonian
in EOB canonical coordinates (see Sec.~\ref{sec:ADM});
\item[(ii)] We compute the effective Hamiltonian corresponding to the canonically
transformed PN-expanded ADM Hamiltonian (see Sec.~\ref{sec:eff});
\item[(iii)] We deform the Hamiltonian of a spinning test-particle
in Kerr derived in Sec.~\ref{sec:hamiltonian_kerr}
by deforming the Kerr metric (see Sec.~\ref{sec:kerrdeformed}) , and expand this deformed Hamiltonian in PN orders (see Sec.~\ref{sec:pn});
\item[(iv)] Comparing (iii) and (iii), we work out the mapping between the
spin variables in the real and effective descriptions, and write the improved EOB Hamiltonian (see Sec.~\ref{sec:EOB}).
\end{enumerate}
\subsection{The ADM Hamiltonian canonically transformed to EOB coordinates}
\label{sec:ADM}
We denote the ADM canonical variables in the
binary's center-of-mass frame with
$\boldsymbol{r}^\prime$ and $\boldsymbol{p}^\prime$.
It is convenient to introduce the following spin combinations:
\begin{eqnarray}
\boldsymbol{\sigma}&=&\boldsymbol{S}_1+\boldsymbol{S}_2\,, \label{sigma}\\
\boldsymbol{\sigma}^\ast&=&\boldsymbol{S}_1\,\frac{m_2}{m_1}+\boldsymbol{S}_2\,\frac{m_1}{m_2}\,,
\label{sigmastar}\\
\boldsymbol{\sigma}_0&=& \boldsymbol{\sigma} + \boldsymbol{\sigma}^\ast\,.
\end{eqnarray}
Moreover, in order to consistently keep
track of the PN orders, we will restore the speed of light $c$ and rescale the spins variables as
$\boldsymbol{{\sigma}}^\ast \rightarrow \boldsymbol{\sigma}^\ast \,c$ and
$\boldsymbol{{\sigma}} \rightarrow \boldsymbol{\sigma} \,c$.\footnote{This is appropriate for
black holes or a rapidly rotating compact stars. In
the black-hole case, $S=\chi M^2/c$, with $\chi$ ranging from $0$ to $1$.
In the rapidly spinning star case one has $S=M v_{\rm rot} r\sim
M c r_{s}\sim M^2/c$ (where we have assumed that the rotational
velocity $v_{\rm rot}$ is comparable to $c$ and that the stellar
radius $r$ is of the order of the Schwarzschild radius
$r_s\sim M/c^2$).}
The canonical ADM Hamiltonian is known through 3PN
order~\cite{Damour-Schafer:1988,Damour:2007nc,SHS07,SSH08,SHS08}
and partially at higher PN orders \cite{Hergt:2007ha,hergt_schafer_08}. In particular, the spin-orbit and spin-spin
coupling terms agree with those
computed via effective-field-theory techniques at 1.5PN, 2PN and 3PN order~\cite{PR06,PR07,PR08b,PR08a}.
In this paper, we use the spin-independent part of the ADM Hamiltonian through 3PN order, but we
only use its spin-dependent part through 2.5 PN order,
i.e., we consider the leading-order (1.5 PN) and the next-to-leading order (2.5PN) spin-orbit couplings,
but only the leading order (2PN) spin-spin coupling. The expressions for these
couplings are~\cite{Damour01c,Damour:2007nc}
\begin{eqnarray}
\label{HSO_adm}
H^{\rm ADM}_{\rm SO}(\boldsymbol{r}^{\prime},\boldsymbol{p}^{\prime},\boldsymbol{\sigma}^\ast,\boldsymbol{\sigma}) &=&
\frac{1}{c^3}\,\frac{\boldsymbol{L}^\prime}{r^{\prime\,3}}\cdot (g^{\rm ADM}_\sigma \,
\boldsymbol{\sigma}+ g^{\rm ADM}_{\sigma^\ast}\,\boldsymbol{\sigma}^\ast)\,, \nonumber \\ \\\label{HSS_adm}
H^{\rm ADM}_{\rm SS}(\boldsymbol{r}^{\prime},\boldsymbol{p}^{\prime},\boldsymbol{\sigma}^\ast,\boldsymbol{\sigma})
&=& \frac{1}{c^4}\,\frac{\eta}{2 r^{\prime\,3}}\,\left [3(\boldsymbol{n}^\prime \cdot \boldsymbol{\sigma}_0)^2 -
\boldsymbol{\sigma}_0^2 \right ]\,,\nonumber \\
\end{eqnarray}
with $\boldsymbol{L}^\prime = \boldsymbol{r}^\prime \times \boldsymbol{p}^\prime$,
$\boldsymbol{n}^\prime= \boldsymbol{r}^\prime/r^\prime$, and
\begin{subequations}
\begin{eqnarray}
g^{\rm ADM}_{\sigma} &=& 2 + \frac{1}{c^2} \left [\frac{19}{8}\, \eta\,\boldsymbol{\hat{p}}^{\prime \,2} +
\frac{3}{2} \eta\, (\boldsymbol{n}^{\prime}\cdot \boldsymbol{\hat{p}}^\prime)^2 \right . \nonumber \\
&& \left. - (6 + 2\eta)\, \frac{M}{r^{\prime}} \right ]\,, \\
g^{\rm ADM}_{\sigma^\ast} &=& \frac{3}{2} + \frac{1}{c^2} \left [
\left (-\frac{5}{8} + 2 \eta \right )\,\boldsymbol{\hat{p}}^{\prime\,2} + \frac{3}{4} \eta \,
(\boldsymbol{n}^{\prime} \cdot \boldsymbol{\hat{p}}^\prime)^2 \right . \nonumber \\
&& \left . - (5 + 2 \eta) \frac{M}{r^{\prime}} \right ]\,,
\end{eqnarray}
\end{subequations}
where we have introduced the rescaled conjugate momentum
$\boldsymbol{\hat{p}}^\prime = \boldsymbol{p}^\prime/\mu$.
We now perform a canonical transformation from the ADM canonical
variables $\boldsymbol{r^\prime}$ and $\boldsymbol{p^\prime}$ to the
EOB canonical variables $\boldsymbol{r}$ and $\boldsymbol{p}$.
Let us first consider the purely orbital generating function
\begin{align}
G(\boldsymbol{r}^\prime,\boldsymbol{p})=&\,\boldsymbol{r}^\prime\cdot
\boldsymbol{p}+G_{\rm NS}(\boldsymbol{r}^\prime,\boldsymbol{p})\,,\\
G_{\rm NS}(\boldsymbol{r}^\prime,\boldsymbol{p})=&\,G_{\rm NS\, 1PN}(\boldsymbol{r}^\prime,\boldsymbol{p})\nonumber\\&
+G_{\rm NS\, 2PN}(\boldsymbol{r}^\prime,\boldsymbol{p})
+G_{\rm NS\, 3PN}(\boldsymbol{r}^\prime,\boldsymbol{p})\,,
\end{align}
where the 1PN-accurate generating function $G_{\rm NS\, 1PN}$ was derived in Ref.~\cite{Buonanno99},
\begin{equation}
G_{\rm NS\, 1PN}(\boldsymbol{r}^\prime,\boldsymbol{p})=\frac{1}{c^2}
\boldsymbol{r^\prime}\cdot\boldsymbol{p}\,\left[-\frac12 \eta\,\boldsymbol{{\hat{p}}}^{2}
+ \frac{M}{r^\prime}\left(1+\frac12\eta\right)\,\right]\,,
\end{equation}
while the 2PN and 3PN accurate generating functions, $G_{\rm NS\, 2PN}$ and $G_{\rm NS\, 3PN}$,
were derived in Refs.~\cite{Buonanno99} and \cite{DJS3PN}, respectively.
From the definition of generating function, it follows that the
transformation of the phase-space variables is implicitly given by
\begin{gather}\label{x_transf}
x^i=x^{\prime i}+\frac{\partial G_{\rm NS}(x',p)}{\partial p_i}\,,\\
p_i=p^{\prime}_{i}-\frac{\partial G_{\rm NS}(x',p)}{\partial x^{\prime i}}\label{p_transf}\,,
\end{gather}
while the Hamiltonian transforms as $H(\boldsymbol{r},\boldsymbol{p})=H^{\rm ADM}(\boldsymbol{r}^\prime,\boldsymbol{p}^\prime)$.
At linear order, which is enough for our purposes, Eqs.~\eqref{x_transf} and \eqref{p_transf}
can be written as $y=y'-\{G_{\rm NS},y'\}$, where $\{...\}$ are the Poisson brackets and where $y$ stands for either $x$ or $p$. The transformation
of the Hamiltonian, again at linear order, is then $H(y)=H^{\rm ADM}(y)+\{G_{\rm NS},H^{\rm ADM}\}(y)$~\cite{DJS08}.
Similarly, if one considers a generating
function which depends not only on the orbital variables, but also on the spins,
\begin{multline} \label{G}
G(\boldsymbol{r}^\prime,\boldsymbol{p},\boldsymbol{\sigma}^\ast,\boldsymbol{\sigma})=\boldsymbol{r}^\prime\cdot
\boldsymbol{p}\\+G_{\rm NS}(\boldsymbol{r}^\prime,\boldsymbol{p})+G_{\rm S}(\boldsymbol{r}^\prime,\boldsymbol{p},\boldsymbol{\sigma}^\ast,\boldsymbol{\sigma})
\end{multline}
the Hamiltonian will again transform as $H(y)=H^{\rm ADM}(y)+\{G_{\rm NS},H^{\rm ADM}\}(y)+\{G_{\rm S},H^{\rm ADM}\}(y)$,
where now the Poisson brackets in the term $\{G_{\rm S},H^{\rm ADM}\}$ will involve also the spin variables~\cite{DJS08}.
In particular, let us consider a spin-dependent generating function
\begin{align}
G_{\rm S}(\boldsymbol{r}^\prime,\boldsymbol{p},\boldsymbol{\sigma}^\ast,\boldsymbol{\sigma})=&\,G_{\rm S\, 2PN}(\boldsymbol{r}^\prime,\boldsymbol{p},\boldsymbol{\sigma})
\nonumber \\&+G_{\rm S\, 2.5PN}(\boldsymbol{r}^\prime,\boldsymbol{p},
\boldsymbol{\sigma}^\ast,\boldsymbol{\sigma})
\\& +
G_{\rm SSS\, 2.5PN}(\boldsymbol{r}^\prime,\boldsymbol{p},\boldsymbol{\sigma}^\ast,\boldsymbol{\sigma})
\nonumber \,.
\end{align}
where
the 2PN-accurate spin-dependent generating function $G_{\rm S\, 2PN}$ was implicitly\footnote{See discussion in Sec II D of Ref.~\cite{Damour01c}. The need for this generating function will become apparent with Eq.~\eqref{eq:HNSSS} in Sec.~\ref{sec:pn}.} used in Ref.~\cite{Damour01c},
\begin{align}
G_{\rm S\, 2PN}(\boldsymbol{r}^\prime,\boldsymbol{p},\boldsymbol{\sigma}) =&\;
-\frac{1}{2 c^4 M^2 r^{\prime2}}\,\Big\{ [\boldsymbol{\sigma}^2 - (\boldsymbol{\sigma}\cdot \boldsymbol{n}^\prime)^2]
(\boldsymbol{r}^\prime\cdot\boldsymbol{p})\nonumber \\&+(\boldsymbol{\sigma}\cdot \boldsymbol{n}^\prime)
(\boldsymbol{r}^\prime\times\boldsymbol{p})\cdot (\boldsymbol{\sigma}\times \boldsymbol{n}^\prime)\Big\}\,;
\end{align}
the 2.5PN-accurate generating function $G_{\rm S\, 2.5PN}$ linear in the spin variables was introduced in Ref.~\cite{DJS08},
\begin{eqnarray}
G_{\rm S\, 2.5PN}(\boldsymbol{r}^\prime,\boldsymbol{p},\boldsymbol{\sigma}^\ast,\boldsymbol{\sigma})&=&
\frac{1}{\mu\, r^{\prime 3}\, c^5}\,
(\boldsymbol{r}^\prime\cdot\boldsymbol{p})(\boldsymbol{r}^\prime \times \boldsymbol{p})
\cdot \nonumber \\
&& \left[a(\eta)\, \boldsymbol{\sigma} +b(\eta)\, \boldsymbol{\sigma}^\ast\right]\,,
\end{eqnarray}
$a(\eta)$ and $b(\eta)$ being arbitrary gauge functions; also,
for reasons which will become clear in Sec.~\ref{sec:pn}, we include the
following 2.5PN-accurate generating function, cubic in the spins,
\begin{equation}
G_{\rm SSS\, 2.5PN}(\boldsymbol{r}^\prime,\boldsymbol{p},\boldsymbol{\sigma}^\ast,\boldsymbol{\sigma})=
\frac{\mu}{2 M^3 r^{\prime\,4} c^5} (\boldsymbol{\sigma} \cdot \boldsymbol{r}^\prime)
[\boldsymbol{\sigma}^\ast\cdot (\boldsymbol{\sigma} \times \boldsymbol{r^\prime})]\,.
\label{G25PNnew}
\end{equation}
When applying the generating function (\ref{G}) to the ADM 2PN spin-spin Hamiltonian~\eqref{HSS_adm},
we obtain
\begin{eqnarray}
H_{\rm SS\, 2PN}(\boldsymbol{r},\boldsymbol{p},
\boldsymbol{\sigma}^\ast,\boldsymbol{\sigma}) &=& H^{\rm ADM}_{\rm SS\, 2PN}(\boldsymbol{r},
\boldsymbol{p},
\boldsymbol{\sigma}^\ast,\boldsymbol{\sigma})\nonumber \\
&& \!\!\!\!\! +\{G_{\rm S\, 2PN}, H_{\rm Newt}\}(\boldsymbol{r},\boldsymbol{p},
\boldsymbol{\sigma})\,,\nonumber \\
\end{eqnarray}
with
\begin{align}
&H_{\rm Newt}=-\frac{M\,\mu}{r}+\frac{\boldsymbol{p}^2}{2\mu}\,,
\\&\nonumber\\
&\{G_{\rm S\, 2PN}, H_{\rm Newt}\}(\boldsymbol{r},\boldsymbol{p},
\boldsymbol{\sigma})=\nonumber
-\frac{1}{c^4}\,\frac{\eta}{2 r^{3}}\,\left[(\boldsymbol{n} \cdot \boldsymbol{\sigma})^2 -
\boldsymbol{\sigma}^2\right]\\ &\qquad\qquad\qquad+ \frac{1}{2 \mu\, M^2\, r^2\, c^4}\Big\{-[\boldsymbol{p}^2 -
2 (\boldsymbol{p}\cdot \boldsymbol{n})^2] \boldsymbol{\sigma}^2 \nonumber \\&\qquad\qquad\qquad+
[(\boldsymbol{p}-2 (\boldsymbol{p}\cdot \boldsymbol{n})\boldsymbol{n})\cdot\boldsymbol{\sigma}]\,
\boldsymbol{p}\cdot\boldsymbol{\sigma}\Big\}\,.
\end{align}
Similarly, if we apply the same generating function to the ADM spin-orbit Hamiltonian
(\ref{HSO_adm}), the 1.5PN order term remains unaltered~\cite{DJS08},
while the 2.5PN order term transforms as~\cite{DJS08}
\begin{align}
H_{\rm SO\, 2.5PN}&\,(\boldsymbol{r},\boldsymbol{p},
\boldsymbol{\sigma}^\ast,\boldsymbol{\sigma}) = H^{\rm ADM}_{\rm SO\, 2.5PN}(\boldsymbol{r},
\boldsymbol{p},
\boldsymbol{\sigma}^\ast,\boldsymbol{\sigma})\nonumber \\
& +\{G_{\rm 2.5PN}, H_{\rm Newt}\}(\boldsymbol{r},\boldsymbol{p},
\boldsymbol{\sigma}^\ast,\boldsymbol{\sigma})\,,\nonumber \\
&+\{G_{\rm NS\,1PN}, H^{\rm ADM}_{\rm SO\, 1.5PN}\}(\boldsymbol{r},\boldsymbol{p},
\boldsymbol{\sigma}^\ast,\boldsymbol{\sigma})
\end{align}
where
\begin{equation}
G_{\rm 2.5PN} = G_{\rm SS\, 2.5PN} + G_{\rm SSS\, 2.5PN}\,,
\end{equation}
\begin{eqnarray}
&& \{G_{\rm 2.5PN}, H_{\rm Newt}\}(\boldsymbol{r},\boldsymbol{p},
\boldsymbol{\sigma}^\ast,\boldsymbol{\sigma}) = \nonumber \\
&& \frac{1}{r^{3} c^5}\,
\boldsymbol{L} \cdot \left [b(\eta)\, \boldsymbol{\sigma^\ast}
+ a(\eta)\, \boldsymbol{\sigma}\right]\,
\left[-\frac{M}{r} + \boldsymbol{\hat{p}}^{2} - 3 (\boldsymbol{\hat{p}} \cdot \boldsymbol{n})^2 \right]
\nonumber\\
&& + \frac{[\boldsymbol{\sigma}^\ast\cdot (\boldsymbol{\sigma}\times
\boldsymbol{n})][\boldsymbol{\sigma}\cdot(\boldsymbol{p} -2 (\boldsymbol{p}
\cdot \boldsymbol{n})\,
\boldsymbol{n})]}{M^3\,r^{3}\, c^5} \nonumber\\
&& +\frac{(\boldsymbol{L} \cdot\boldsymbol{\sigma}^\ast) \boldsymbol{\sigma}^2
-(\boldsymbol{L} \cdot\boldsymbol{\sigma})
(\boldsymbol{\sigma}^\ast\cdot\boldsymbol{\sigma})}{2 M^3\, r^{4} c^5}\,,
\end{eqnarray}
and
\begin{align}
&\{G_{\rm NS\,1PN}, H^{\rm ADM}_{\rm SO\, 1.5PN}\}(\boldsymbol{r},\boldsymbol{p},
\boldsymbol{\sigma}^\ast,\boldsymbol{\sigma})=\nonumber\\&
-\frac{3\boldsymbol{L}}{2r^{3} c^5}
\! \cdot\! \left(\frac32\boldsymbol{\sigma^\ast}\!+\!2\boldsymbol{\sigma}\!\right)
\left\{\!-\frac{M}{r} (2+\eta) + \eta \left[\boldsymbol{\hat{p}}^{2} +2 (\boldsymbol{\hat{p}} \cdot \boldsymbol{n})^2\right]\! \right\}.
\nonumber\\&
\end{align}
Therefore, the complete real Hamiltonian in the EOB canonical coordinates is
\begin{eqnarray}
\label{HADPN}
H(\boldsymbol{r},\boldsymbol{p},
\boldsymbol{\sigma}^\ast,\boldsymbol{\sigma}) &=& H_{\rm nospin}(\boldsymbol{r},\boldsymbol{p},
\boldsymbol{\sigma}^\ast,\boldsymbol{\sigma}) \nonumber \\
&& + H^{\rm ADM}_{\rm SO}(\boldsymbol{r},\boldsymbol{p},
\boldsymbol{\sigma}^\ast,\boldsymbol{\sigma}) \nonumber \\
&& + H^{\rm ADM}_{\rm SS}(\boldsymbol{r},\boldsymbol{p},
\boldsymbol{\sigma}^\ast,\boldsymbol{\sigma}) \nonumber \\
&& + \{G_{\rm 2.5PN}, H_{\rm Newt}\}(\boldsymbol{r},\boldsymbol{p},
\boldsymbol{\sigma}^\ast,\boldsymbol{\sigma})\nonumber\\
&& +\{G_{\rm NS\,1PN}, H^{\rm ADM}_{\rm SO\, 1.5PN}\}(\boldsymbol{r},\boldsymbol{p},
\boldsymbol{\sigma}^\ast,\boldsymbol{\sigma})\nonumber\\
&& +\{G_{\rm S\, 2PN}, H_{\rm Newt}\}(\boldsymbol{r},\boldsymbol{p},
\boldsymbol{\sigma})\,, \nonumber \\
\end{eqnarray}
where $H_{\rm nospin}$ is the 3PN ADM Hamiltonian for non-spinning black holes,
canonically transformed to EOB coordinates, which can be obtained
from Ref.~\cite{DJS3PN}.
\subsection{Spin couplings in the effective Hamiltonian}
\label{sec:eff}
Following Refs.~\cite{Buonanno00,DJS3PN,Damour01c}, we
map the effective and real two-body Hamiltonians as
\begin{equation}
\label{heff}
\frac{H_{\rm eff}}{\mu c^2} =
\frac{H_{\rm real}^2 - m_1^2\,c^4 - m_2^2\, c^4}
{2 m_1\, m_2\, c^4}\,,
\end{equation}
where $H_{\rm real}$ is the real two-body Hamiltonian containing also the rest-mass contribution $M\,c^2$.
We denote the non-relativistic part of the real Hamiltonian by $H^{\rm NR}$, i.e.,
$H^{\rm NR} \equiv H_{\rm real} - M\,c^2$. Identifying $H^{\rm NR}$
with $H$ as given in Eq.~(\ref{HADPN}), and
expanding Eq.~(\ref{heff}) in powers of $1/c$, we find that
the 1.5PN and 2.5PN order spin-orbit couplings of the effective
Hamiltonian are
\begin{align}
\label{eq:HeffSO}
H^{\rm eff}_{\rm SO}(\boldsymbol{r},\boldsymbol{p},
\boldsymbol{\sigma}^\ast,\boldsymbol{\sigma})=&\frac{1}{c^3}\frac{\boldsymbol{L}}{r^3}\cdot
\left(g_{\sigma}^{\rm eff}\,\boldsymbol{\sigma}+g_{\sigma^\ast}^{\rm eff}\,\boldsymbol{\sigma}^{\ast}\right)\nonumber\\&+
\frac{[\boldsymbol{\sigma}^\ast\cdot (\boldsymbol{\sigma}\times
\boldsymbol{n})][\boldsymbol{\sigma}\cdot(\boldsymbol{p}-2 (\boldsymbol{p}\cdot \boldsymbol{n})\,
\boldsymbol{n})]}{M^3 r^3 c^5}\nonumber\\&+\frac{(\boldsymbol{L}\cdot\boldsymbol{\sigma}^\ast) \boldsymbol{\sigma}^2
-(\boldsymbol{L}\cdot\boldsymbol{\sigma})
(\boldsymbol{\sigma}^\ast\cdot\boldsymbol{\sigma})}{2 M^3 r^4 c^5}\,,
\end{align}
where~\cite{DJS08}
\begin{subequations}
\begin{eqnarray}
g^{\rm eff}_\sigma&=& 2 + \frac{1}{c^2}\,
\left \{ \left [\frac{3}{8}\,\eta + a(\eta) \right ]\,\boldsymbol{\hat{p}}^2 \right. \nonumber \\
&& \left. - \left [\frac{9}{2}\,\eta + 3 \,a(\eta)\right ]\,(\boldsymbol{\hat{p}}\cdot\boldsymbol{n})^2
\right . \nonumber\\
&& \left . - \frac{M}{r}\,\left [\eta + a(\eta) \right ] \right \}\,, \\
g^{\rm eff}_{\sigma^\ast} &=&\frac{3}{2} +
\frac{1}{c^2}\, \left \{ \left [-\frac{5}{8}+\frac{1}{2}\,\eta + b(\eta)\right ] \,
\boldsymbol{\hat{p}}^2 \right . \nonumber\\
&& \left . - \left [ \frac{15}{4}\,\eta+3\,b(\eta) \right ]\,(\boldsymbol{\hat{p}}\cdot\boldsymbol{n})^2
\right. \nonumber\\
&& \left . - \frac{M}{r}\,\left [ \frac{1}{2}+\frac{5}{4}\,\eta+b(\eta)\right ] \right \}\,,
\end{eqnarray}
\end{subequations}
and the 2PN order spin-spin coupling is
\begin{align}
\label{eq:HeffSS}
&{H}_{\rm SS}^{\rm eff}(\boldsymbol{r},\boldsymbol{p},
\boldsymbol{\sigma}^\ast,\boldsymbol{\sigma}) = \frac{1}{c^4}\,\frac{\eta}{2 \,r^3}\,(3n_{i}\, n_j
- \delta_{ij})\,\sigma_0^i\, \sigma_0^j \nonumber\\&\qquad\qquad\qquad\nonumber
-\frac{1}{c^4}\,\frac{\eta}{2 r^{3}}\,\left[(\boldsymbol{n} \cdot \boldsymbol{\sigma})^2 -
\boldsymbol{\sigma}^2\right]\\ &\qquad\qquad\qquad+ \frac{1}{2 \mu M^2\,r^2\, c^4}\Big\{-[\boldsymbol{p}^2 -
2 (\boldsymbol{p}\cdot \boldsymbol{n})^2] \,\boldsymbol{\sigma}^2 \nonumber \\&\qquad\qquad\qquad+
[(\boldsymbol{p}-2 (\boldsymbol{p}\cdot \boldsymbol{n})\,\boldsymbol{n})\cdot\boldsymbol{\sigma}]\,
\boldsymbol{p}\cdot\boldsymbol{\sigma}\Big\}\,.
\end{align}
\subsection{The Hamiltonian of a spinning test-particle in a deformed Kerr spacetime}
\label{sec:kerrdeformed}
We now deform the Hamiltonian of a spinning test-particle in a Kerr spacetime
computed in Sec.~\ref{sec:hamiltonian_kerr} [see Eqs.~\eqref{HNSKerr},
\eqref{KerrHS}, \eqref{KerrHSO} and \eqref{KerrHSS}] by deforming the Kerr metric.
The deformation that we introduce is regulated by the parameter $\eta=\mu/M$, and therefore disappears
in the test-particle limit.
Also, the deformed Hamiltonian will
be such as to reproduce, when expanded in PN orders, the spin
couplings of the effective Hamiltonian given in Sec.~\ref{sec:eff}.
When the spin of the Kerr black hole is zero, that is $a=0$, we require
the metric to coincide with the deformed-Schwarzschild metric
used in the EOB formalism for non-spinning black-hole
binaries~\cite{Buonanno99,DJS3PN}. That deformation
simply amounts to changing the components $g_{tt}$ and $g_{rr}$ of the metric.
In the spinning case, following Ref.~\cite{DJS08}, we seek an extension
of this deformation by changing the potential $\Delta$ appearing
in the Kerr potentials (\ref{coeffB})--(\ref{coeffmu}).
It is worth noting, however, that we are not allowed to deform the Kerr metric in an arbitrary
way. We recall indeed that the Hamiltonian
that we have derived in Sec.~\ref{sec:hamiltonian_kerr}
is only valid for a stationary axisymmetric metric, and in coordinates which are related to quasi-isotropic
coordinates by a redefinition of the radius. In other words, it must be
possible for our deformed metric to be put in the form~\eqref{eq:metric} by a coordinate change of the type $R=R(r)$.
For this reason we cannot deform the metric exactly
in the same way as in Ref.~\cite{DJS08}. Here we propose to deform the metric potentials
in the following manner
\begin{eqnarray}
\label{eq:B}
B&=&\frac{\sqrt{\Delta_t}}{R}\,,\\
\omega&=&\frac{\widetilde{\omega}_{\rm fd}}{\Lambda_t}\,,\label{eq:omega}\\
e^{2\nu}&=&\frac{\Delta_t\,\Sigma}{\Lambda_t}\label{eq:nu}\,,\\
e^{2\mu}&=&\frac{\Sigma}{R^2}\label{eq:mu}\,,
\end{eqnarray}
and
\begin{equation}
J^{-1}= \frac{d R }{d r} = \frac{R}{\sqrt{\Delta_r}}\,,\label{eq:newJ}
\end{equation}
where the relation between $r$ and $R$ can be found by integrating Eq.~\eqref{eq:newJ}:
\begin{equation}
\label{eq:integralR}
R=\exp\left({\int \frac{\mbox{d}r}{\sqrt{\Delta_r}}}\right)\,.
\end{equation}
The deformed metric therefore takes the form
\begin{subequations}
\begin{eqnarray}
\label{def_metric_in}
g^{tt} &=& -\frac{\Lambda_t}{\Delta_t\,\Sigma}\,,\\
g^{rr} &=& \frac{\Delta_r}{\Sigma}\,,\\
g^{\theta\theta} &=& \frac{1}{\Sigma}\,,\\
g^{\phi\phi} &=& \frac{1}{\Lambda_t}
\left(-\frac{\widetilde{\omega}_{\rm fd}^2}{\Delta_t\,\Sigma}+\frac{\Sigma}{\sin^2\theta}\right)\,,\label{eq:gff}\\
g^{t\phi}&=&-\frac{\widetilde{\omega}_{\rm fd}}{\Delta_t\,\Sigma}\,,\label{def_metric_fin}
\end{eqnarray}
\end{subequations}
which does not depend on $R$. Therefore, as we will show explicitly later in this section,
we do \textit{not} need to compute
the integral~\eqref{eq:integralR} to write the Hamiltonian.
The quantities $\Delta_t$, $\Delta_r$, $\Lambda_t$ and $\widetilde{\omega}_{\rm fd}$
in Eqs.~(\ref{def_metric_in})--(\ref{def_metric_fin}) are given by
\begin{eqnarray}
\label{deltat}
\Delta_t &=& r^2\, \left [A(u) + \frac{a^2}{M^2}\,u^2 \right ]\,, \\
\label{deltar}
\Delta_r &=& \Delta_t\, D^{-1}(u)\,,\\
\Lambda_t &=& \varpi^4 - a^2\,\Delta_t\,\sin^2\theta \,,\\
\widetilde{\omega}_{\rm fd}&=& 2 a\, M\, r
+ \omega_1^{\rm fd}\,\eta\,
\frac{a M^3}{r}
+ \omega_2^{\rm fd}\,\eta\,
\frac{M a^3}{r}
\label{eq:omegaTilde}\,,
\end{eqnarray}
where $u = M/r$, $\omega_1^{\rm fd}$ and $\omega_2^{\rm fd}$ are adjustable parameters which regulate the {\it strength} of
the frame-dragging, and through 3PN order~\cite{Buonanno00,DJS3PN}
\begin{align}
\label{A_PN}
& A(u) = 1 - 2 u + 2 \eta\, u^3 + \eta\,\left (\frac{94}{3} - \frac{41}{32} \pi^2\right)\, u^4\,,
\\
& D^{-1}(u) = 1 + 6 \eta\, u^2 + 2 (26 - 3 \eta)\, \eta\, u^3\,.\label{D_PN}
\end{align}
We find that our deformed metric is the same as the deformed metric of
Ref.~\cite{DJS08}, except for $g^{\phi\phi}$ and $g^{t\phi}$.\footnote{Ref.~\cite{DJS08}
chooses $g^{\phi\phi}=(-a^2\,\sin^2\theta + \Delta_t)/
(\Delta_t\,\Sigma \sin^2\theta)$ and $ g^{t\phi}=a\,(\Delta_t - \varpi^2)/(\Delta_t\,\Sigma)$,
which are different from our expressions \eqref{eq:gff} and \eqref{def_metric_fin} even for $\omega_1^{\rm fd}=\omega_2^{\rm fd}=0$.}
As we prove below, the differences between our deformation and the deformation
of Ref.~\cite{DJS08} appear in the Hamiltonian at PN orders higher than 3PN.
To obtain the total Hamiltonian (\ref{Htotal}), that is $H = H_{\rm NS} + H_{\rm S}$,
we first compute the Hamiltonian $H_{\rm NS}$ for a non-spinning particle in the deformed-Kerr metric.
Using Eq.~(\ref{HNSKerr}) and Ref.~\cite{DJS3PN}, we have
\begin{equation}
{H}_{\rm NS} = \beta^i \, p_i + \alpha \sqrt{m^2 + \gamma^{ij}\,p_i\,p_j + {\cal Q}_4(p)}\,,
\label{eq:Hnsdef}
\end{equation}
where ${\cal Q}_4(p)$ is a term which is quartic in the space momenta $p_i$ and which was introduced in Ref.~\cite{DJS3PN}, and
\begin{eqnarray}
\label{alpha}
\alpha &=& \frac{1}{\sqrt{-g^{tt}}}\,,\\
\beta^i &=& \frac{g^{ti}}{g^{tt}}\,,\\
\gamma^{ij} &=& g^{ij}-\frac{g^{ti}\,g^{tj}}{g^{tt}}\,.
\label{gamma}
\end{eqnarray}
In Eqs.~(\ref{alpha})--(\ref{gamma}) the metric components have to be replaced
with those of the deformed-Kerr metric (\ref{def_metric_in})--(\ref{def_metric_fin}).
When expanded in PN orders, Eq.~(\ref{eq:Hnsdef}) coincides, through 3PN order,
with the Hamiltonian of a non-spinning test particle in the deformed-Kerr metric
given by Ref.~\cite{DJS08}.
Second, to calculate $H_{\rm S}$ given by Eqs.~\eqref{Ht}, \eqref{Ht1} and \eqref{Ht2},
we need to compute the derivatives of the metric potentials. We obtain
\begin{subequations}
\begin{eqnarray}
\label{eq:Br}
B_r &=& \frac{\sqrt{{\Delta_r}}\,{\Delta_t}^\prime-2 {\Delta_t}}{2 \sqrt{{\Delta_r}\,{\Delta_t}}\,R}\,,\\
\omega_r
&=& \frac{-\Lambda_t^\prime\,\widetilde{\omega}_{\rm fd} + \Lambda_t\,
\widetilde{\omega}^\prime_{\rm fd}}{\Lambda_t^2}\label{eq:omegar}\,,\\
\nu_r &=& \frac{r}{{\Sigma}}+\frac{{\varpi^2}\,\left({\varpi^2}\,{\Delta_t}^\prime-4 r\,
{\Delta_t}\right)}{2 {\Lambda_t}\, {\Delta_t}}\label{eq:nur}\,,\\
\mu_r &=&\frac{r}{{\Sigma}}-\frac{1}{\sqrt{{\Delta_r}}}\label{eq:mur}\,,\\
B_{\cos\theta}&=&0\,,\\
\omega_{\cos\theta}
&=&-\frac{2 a^2\, \cos\theta\, \Delta_t\, \widetilde{\omega}_{\rm fd}}{\Lambda_t^2}\label{eq:omegacos}\,,\\
\nu_{\cos\theta}&=&\frac{a^2\, {\varpi^2}\, \cos \theta ({\varpi^2}-{\Delta_t})}{{\Lambda_t}\,
{\Sigma}}\label{eq:nucos}\,,\\
\mu_{\cos\theta}&=&\frac{a^2\, \cos\theta}{\Sigma}\label{eq:mucos}\,,
\end{eqnarray}
\end{subequations}
where the prime denotes derivatives with respect to $r$.
As already stressed, although the metric potentials $B$, $\omega$, $\nu$ and $\mu$ depend on $R$, the
factors $R$ cancel out in the deformed-Kerr metric. Therefore, those factors must cancel out also
in $H_{\rm S}$. This happens because
the reference tetrad field $\tilde{e}_{A}$ which, together with the metric, completely
determines the Hamiltonian [see Eq.~\eqref{Hamiltonian_canonical_final2}],
can be defined independently of $R$.
Indeed, this turns out to be the case, and if we introduce the rescaled potentials
\begin{eqnarray}\label{tildeBr}
\tilde{B} &=& B\, R= \sqrt{\Delta_t}\,,\\
\tilde{B}_r &=& B_r\, R=\frac{\sqrt{{\Delta_r}}\,{\Delta_t}'-2 {\Delta_t}}
{2 \sqrt{{\Delta_r}\,{\Delta_t}}}\,,\\
e^{2\tilde{\mu}} &=& e^{2\mu}\, R^2 = \Sigma\,,\label{tildeMU}\\
\tilde{J} &=&J\, R=\sqrt{\Delta_r}\label{tildeJ}
\end{eqnarray}
and define
\begin{equation}\label{Qpert}
Q=1+\frac{\Delta_r (\boldsymbol{\hat{p}}\cdot \boldsymbol{n})^2}{\Sigma}+
\frac{(\boldsymbol{\hat{p}}\cdot \boldsymbol{\xi}\, r)^2 \Sigma }{\Lambda_t\,\sin^2\theta}+
\frac{(\boldsymbol{\hat{p}}\cdot \boldsymbol{v}\, r)^2 }{\Sigma\,\sin^2\theta}\,,
\end{equation}
the Hamiltonian $H_{\rm S}$ for the deformed-Kerr metric takes exactly the
same form as in the Kerr case [see Eqs.~\eqref{KerrHS}, \eqref{KerrHSO} and \eqref{KerrHSS}],
where we recall that $\xi^2=\sin^2\theta$ and where now $\omega$ and its derivatives, $\nu$ and its derivatives, and
the derivatives of $\mu$ are given by Eqs.~\eqref{eq:omega},
\eqref{eq:nu}, and Eqs.~(\ref{eq:Br})--(\ref{eq:mucos}).
Also, as we have already stressed, in order to express the Hamiltonian $H_{\rm S}$
in a cartesian coordinate system
in which the spin of the deformed-Kerr black hole is not directed along the $z$-axis, it is
sufficient to replace $r$ with $(x^2+y^2+z^2)^{1/2}$,
$\cos\theta$ with $\boldsymbol{\hat{S}}_{\rm Kerr}\cdot \boldsymbol{n}$,
$\boldsymbol{e}_z$ with $\boldsymbol{\hat{S}}_{\rm Kerr}$ in Eq.~\eqref{eq:xidef},
and to express the vectors appearing in the Hamiltonian in terms of their cartesian
components.
\subsection{PN expansion of the deformed Hamiltonian}
\label{sec:pn}
We now expand the deformed Hamiltonian $H=H_{\rm NS}+H_{\rm S}$ derived in the previous section
into PN orders. We will denote the spin of the deformed-Kerr metric with
$\boldsymbol{S}_{\rm Kerr}$, while for the test particle's spin
we introduce the rescaled spin vector $\boldsymbol{S}^\ast=\boldsymbol{S}\, M/m$,
$\boldsymbol{S}$ being the physical, unrescaled spin.
Also, we rescale the spins as $\boldsymbol{S}_{\rm Kerr} \rightarrow \boldsymbol{S}_{\rm Kerr}\, c$ and
$\boldsymbol{S}^\ast \rightarrow \boldsymbol{S}^\ast\, c$, so as
to keep track of the PN orders correctly. Moreover, we set
$\boldsymbol{S}_{\rm Kerr}=\boldsymbol{\chi}_{\rm Kerr}\, M^2$, $\boldsymbol{\chi}_{\rm Kerr}$ being
the dimensionless spin of the deformed-Kerr black hole, with
norm $|\boldsymbol{\chi}_{\rm Kerr}|$ ranging from 0 to 1.
As already mentioned, the part of the Hamiltonian which
does not depend on the test particle's spin,
$H_{\rm NS}$, agrees through 3PN order with the corresponding $H_{\rm NS}$ computed
in Ref.~\cite{DJS08}. Moreover, although the metric
\eqref{def_metric_in}--\eqref{def_metric_fin} only coincides with the
Kerr metric for $\eta=0$, the dependence on $\eta$ appears neither in
the 2PN order coupling of the deformed-Kerr black hole's spin with itself, nor in
its 1.5PN and 2.5PN order spin-orbit couplings. Those couplings are therefore the
same as in the case of the Kerr metric, and they are given by
\begin{align}
\label{eq:HNSSO}
&H^{\rm NS}_{\rm SO\,1.5PN} = \; \frac{1}{c^3}\,\frac{2}{r^3} \, \boldsymbol{L}\cdot \boldsymbol{S}_{\rm Kerr} \,,\\
\label{eq:HNSSObis}
&H^{\rm NS}_{\rm SO\,2.5PN} = \; 0 \,,\\
\label{eq:HNSSS}
&{H}^{\rm NS}_{\rm SS\,2PN} = \; \frac{1}{c^4}\,\frac{m}{2M\,r^3}\,(3n_{i}\, n_j
- \delta_{ij})\,S^i_{\rm Kerr}\,S^j_{\rm Kerr}\nonumber\\&\qquad\qquad\qquad\nonumber -\frac{1}{c^4}\,\frac{m}{2 M r^{3}}\,\left[(\boldsymbol{n} \cdot \boldsymbol{S}_{\rm Kerr})^2 -
\boldsymbol{S}_{\rm Kerr}^2\right]\\ &\qquad\qquad\qquad+ \frac{1}{2 m (M r)^2 c^4}\Big\{-[\boldsymbol{p}^2 -
2 (\boldsymbol{p}\cdot \boldsymbol{n})^2] \boldsymbol{S}_{\rm Kerr}^2 \nonumber \\&\qquad\qquad\qquad+
[(\boldsymbol{p}-2 (\boldsymbol{p}\cdot \boldsymbol{n})\boldsymbol{n})\cdot\boldsymbol{S}_{\rm Kerr}]\,
\boldsymbol{p}\cdot\boldsymbol{S}_{\rm Kerr}\Big\}\,.
\end{align}
Expanding then in PN orders the part of the Hamiltonian that depends on the test particle's spin,
that is $H_{\rm S}$, we find
\begin{eqnarray}
{H}^{\rm S}_{\rm SO\,1.5PN} &=& \frac{3}{2r^3\,c^3} \, \boldsymbol{L}\cdot {\boldsymbol{S}}^\ast\,,
\label{eq:H15PN}\\
{H}^{\rm S}_{\rm SO\,2.5PN} &=& \frac{1}{r^3\,c^5}\, \left[-\frac{M}{r}
\left(\frac12 + 3\eta\right) - \frac{5}{8} \boldsymbol{\hat{p}}^2
\right]\,
\boldsymbol{L}\cdot\boldsymbol{S}^\ast \nonumber\\
&&\!\!\!\!\!\!\!\! +\frac{[\boldsymbol{S}^\ast\cdot (\boldsymbol{S}_{\rm
Kerr}\times \boldsymbol{n})]\,[\boldsymbol{S}_{\rm
Kerr}\cdot(\boldsymbol{p}-2 (\boldsymbol{p}\cdot
\boldsymbol{n})\, \boldsymbol{n})]}{M^3\,
r^3\,c^5}\nonumber\\
&& \!\!\!\!\!\!\!\! +\frac{(\boldsymbol{L}\cdot\boldsymbol{S}^\ast)
\boldsymbol{S}^2_{\rm Kerr} -(\boldsymbol{L}\cdot\boldsymbol{S}_{\rm
Kerr})\,
(\boldsymbol{S}^\ast\cdot\boldsymbol{S}_{\rm
Kerr})}{2 M^3\, r^4\,c^5}\,.\nonumber \\\label{H25PNnew}\\
{H}^{\rm S}_{\rm SS\, 2PN} &=& \frac{m}{M\,r^3\,c^4}(3n_{i}\, n_j -
\delta_{ij})\,S^i_{\rm Kerr}\,
S_\ast^j \,.\label{eq:H_SS}
\end{eqnarray}
We recall that the Hamiltonian for a spinning test particle in curved
spacetime from which we started the derivation of our novel EOB model
[see Eq.~\eqref{Hamiltonian_canonical_final2}] is only valid at linear
order in the particle's spin. Therefore, the same restriction applies
to the Hamiltonian derived in Sec.~\ref{sec:kerrdeformed}. In
particular, that Hamiltonian does not include the couplings of the
particle's spin with itself. We introduce those couplings by hand,
at least at the leading order (2PN), by adding a quadrupole deformation~\cite{Damour01c}
$h^{\mu\nu}$, quadratic in the particle's spin, to the
deformed-Kerr metric in Sec.~\ref{sec:kerrdeformed} [see
Eqs.~\eqref{def_metric_in}--\eqref{def_metric_fin}]. The expression
for $h^{\mu\nu}$ and the details of the above procedure --- together
with a way in which it can in principle be extended to reproduce also
the next-to-leading order coupling of the particle's spin with itself
--- are given in Appendix~\ref{sec:ss}. For the purpose of the present
discussion, however, it is sufficient to mention that the addition of
this quadrupole deformation to the metric \eqref{def_metric_in}--\eqref{def_metric_fin} augments
Eq.~(\ref{eq:HNSSS}) by the term
\begin{equation}
\frac{m}{2M\,r^3\,c^4}(3n_{i}\, n_j -
\delta_{ij})\,S^i_{\ast}\,
S_\ast^j \,.\label{eq:H_SSstar}
\end{equation}
Therefore, the total leading order spin-spin Hamiltonian is
\begin{align}
H_{\rm SS\,2PN}&={H}^{\rm S}_{\rm SS\, 2PN}+{H}^{\rm NS}_{\rm SS\, 2PN}\nonumber\\& +\frac{m}{2 M\,r^3\,c^4}(3n_{i}\, n_j -
\delta_{ij})\,S^i_{\ast}\, S_\ast^j\nonumber\\&=\frac{m}{2M\,r^3\,c^4}(3n_{i}\, n_j -
\delta_{ij})S_0^iS_0^j\nonumber\\&\nonumber-\frac{1}{c^4}\,\frac{m}{2 M r^{3}}\,\left[(\boldsymbol{n} \cdot \boldsymbol{S}_{\rm Kerr})^2 -
\boldsymbol{S}_{\rm Kerr}^2\right]\\ &+ \frac{1}{2 m\,M^2\, r^2\, c^4}\Big\{-[\boldsymbol{p}^2 -
2 (\boldsymbol{p}\cdot \boldsymbol{n})^2] \boldsymbol{S}_{\rm Kerr}^2 \nonumber \\&+
[(\boldsymbol{p}-2 (\boldsymbol{p}\cdot \boldsymbol{n})\boldsymbol{n})\cdot\boldsymbol{S}_{\rm Kerr}]\,
\boldsymbol{p}\cdot\boldsymbol{S}_{\rm Kerr}\Big\}\label{eq:H_SS_total}
\,,
\end{align}
with $S_0^i=S^i_{\rm Kerr}+S^i_{\ast}$.
As we will show in Sec.~\ref{sec:EOB}, a proper choice of the vectors
$\boldsymbol{S}_{\rm Kerr}$ and $\boldsymbol{S}^\ast$ in terms of the vectors
$\boldsymbol{\sigma}$ and $\boldsymbol{\sigma}^\ast$, defined in
Eqs.~(\ref{sigma}) and (\ref{sigmastar}), allows
us to reproduce the PN-expanded effective Hamiltonian [see Eqs.~(\ref{eq:HeffSO})--(\ref{eq:HeffSS})]
using the PN-expanded deformed-Kerr Hamiltonian that we have just derived.
Finally, it is worth noting that the presence of terms quadratic in the
deformed-Kerr black hole's spin in Eq.~(\ref{H25PNnew}) explains why we introduced
the 2.5PN-accurate canonical transformation~(\ref{G25PNnew}). Indeed, the latter
produces exactly the same terms in the PN-expanded effective Hamiltonian~(\ref{eq:HeffSO})
at 2.5PN order. Quite interestingly, the terms quadratic in $\boldsymbol{S}_{\rm Kerr}$
appearing in Eq.~(\ref{H25PNnew})
could also be eliminated with a suitable choice of the reference tetrad
$\tilde{e}_A$. In fact, as stressed in Sec.~\ref{sec:Hamiltonian_axisymmetric}
and in Ref.~\cite{diracbrackets_ours}, a choice of the reference tetrad field
corresponds to choosing a particular gauge for the particle's spin.
In agreement with this interpretation, we find that the terms of Eq.~(\ref{H25PNnew})
which are quadratic in $\boldsymbol{S}_{\rm Kerr}$
disappear if the initial tetrad \eqref{tetrad0}--\eqref{tetrad3} is
changed to a different tetrad $\tilde{\boldsymbol{e}}^{\prime}_{A}$
related to the original one by the following purely-spatial rotation:
\begin{equation}
\tilde{\boldsymbol{e}}^{\prime T}
= \tilde{\boldsymbol{e}}^{T} \,,\quad \quad
\tilde{\boldsymbol{e}}^\prime_I = {\cal R}_{IJ}\,
\tilde{\boldsymbol{e}}_J
\label{tetrad}\,,
\end{equation}
where the rotation matrix ${\cal R}_{IJ}$ is given by
\begin{equation}
{\cal R}={\cal R}_Y\left [-\frac{a^2\, X\, Z}{2 R^4} \right ]\,
{\cal R}_X \left [-\frac{a^2\, Y\, Z}{2 R^4} \right ]\,,
\end{equation}
${\cal R}_X[\psi]$ and ${\cal R}_Y[\phi]$ being
rotations of angles $\psi$ and $\phi$ around the axis $X$ and $Y$, respectively.
As a consistency check, we have verified that this new tetrad is the
same as that used in Ref.~\cite{diracbrackets_ours} when computing the
Hamiltonian in ADM coordinates, where those terms quadratic in
$\boldsymbol{S}_{\rm Kerr}$ do not appear. We have checked this by
transforming the new tetrad~\eqref{tetrad} from quasi-isotropic to ADM
coordinates [which are related by the coordinate transformation~(49)
in Ref.~\cite{Hergt:2007ha}], and comparing it to the tetrad given in
Eqs.~(6.9a)--(6.9b) of Ref.~\cite{diracbrackets_ours}, and find that
the two tetrads agree through order $1/c^8$.
\subsection{The effective-one-body Hamiltonian}
\label{sec:EOB}
In this section we first find the mapping between the masses
$\mu$, $M$ and the spins $\boldsymbol{\sigma}$ and $\boldsymbol{\sigma}^\ast$
of the effective Hamiltonian derived in Sec~\ref{sec:eff},
and those of the deformed-Kerr Hamiltonian derived in Secs.~\ref{sec:kerrdeformed}
and~\ref{sec:pn}, that is $m$, $M$, $\boldsymbol{S}_{\rm Kerr}$ and
$\boldsymbol{S}^\ast$. Then, we derive the improved (resummed) EOB Hamiltonian.
As shown in Ref.~\cite{Buonanno00}, matching the non-spinning parts
$H_{\rm NS}$ of these Hamiltonians forces us to identify the total
mass $M$ of the two black holes in the PN description
with the deformed-Kerr mass $M$ of the test-particle description,
thus justifying our choice of using the same
symbol for these two \textit{a priori} distinct quantities. Similarly, we
find that $m=\mu$~\cite{Buonanno00}. Assuming this
mapping between the masses and imposing that the PN-expanded
deformed-Kerr Hamiltonian given by
Eqs.~(\ref{eq:HNSSO})--(\ref{eq:H_SS_total}) coincides with the
effective Hamiltonian given by Eqs.~(\ref{eq:HeffSO})--(\ref{eq:HeffSS}),
we obtain the following mapping between the spins
\begin{eqnarray}
\label{mapping1}
\boldsymbol{{S}}^\ast &=& \boldsymbol{\sigma}^\ast+\frac{1}{c^2}\,\boldsymbol{\Delta}_{{\sigma}^\ast}\,,\\
\label{mapping2}
\boldsymbol{{S}}_{\rm Kerr} &=& \boldsymbol{\sigma}+\frac{1}{c^2}\,\boldsymbol{\Delta}_{{\sigma}}\,,
\end{eqnarray}
where we have set for simplicity $a(\eta)=0$ and $b(\eta)=0$ and where
\begin{eqnarray}
\boldsymbol{\Delta}_{{\sigma}}&=&-\frac{1}{16}\,\Bigg\{
12 \boldsymbol{\Delta}_{{\sigma}^\ast}
+\eta \bigg [ \frac{2 M}{r}\, (4 \boldsymbol{\sigma}
-7 \boldsymbol{\sigma}^\ast) \nonumber \\
&& + 6 (\hat{\boldsymbol{p}}\cdot\boldsymbol{n})^2
\,(6 \boldsymbol{\sigma}+5 \boldsymbol{\sigma}^\ast)
-\boldsymbol{\hat{p}}^2\,(3 \boldsymbol{\sigma}+4 \boldsymbol{\sigma}^\ast)
\bigg]\Bigg\}\,.
\nonumber \\\label{deltasigma}
\end{eqnarray}
Here, $\boldsymbol{\Delta}_{{\sigma}^\ast}$ is an arbitrary function
going to zero at least linearly in $\eta$ when $\eta\to0$, so as to
get the correct test-particle limit. In fact, if
$\boldsymbol{\Delta}_{{\sigma}^\ast}$ satisfies this condition and if
we assume, as appropriate for black holes,
$\boldsymbol{S}_{1,2}=\boldsymbol{\chi}_{1,2}\, m_{1,2}^2$ (with $\vert
\boldsymbol{\chi}_{1,2} \vert\leq1$ and constant)\footnote{As noted by
Ref.~\cite{DJS08}, a spin mapping such as ours also gives the
correct test particle limit if $\vert
\boldsymbol{S}_{1,2}\vert/m_{1,2}= $ const., but this scaling of the
spins with the masses is not appropriate for black
holes~\cite{hartl}.}, when $m_2\sim 0$ we have $\boldsymbol{S}_{\rm
Kerr}=\boldsymbol{S}_{1}+{\cal O}(m_2)$.
Similarly, for $m_2\sim 0$ the physical unrescaled spin of the effective particle
is $\boldsymbol{S}=\boldsymbol{S}^\ast\, m/M=\boldsymbol{S}_{2}
+{\cal O}(m_2)^2$. The equations of motion
of our initial Hamiltonian~\eqref{Hamiltonian_canonical_final2}
coincide with the Papapetrou
equations~\cite{diracbrackets_ours}, which describe the motion of a spinning test-particle in a
curved spacetime~\cite{Papa51spin,CPapa51spin}. Assuming the canonical commutation
relations between $x^i$, $p_j$, $\boldsymbol{S}_{1}$ and $\boldsymbol{S}_{2}$, we obtain that
the Hamilton equations for the effective deformed-Kerr Hamiltonian are
$\dot{y}=\dot{y}_{\rm P}+{\cal O}(m_2)$.
Here, the dot denotes a time derivative, $y$ is a generic phase-space variable
($x^i$, $p_j$, $\boldsymbol{S}_{1}$ or $\boldsymbol{S}_{2}$),
and $\dot{y}=\dot{y}_{\rm P}$ are the Papapetrou equations expressed in Hamiltonian form.
Therefore, our mapping reproduces
the correct test-particle limit, and the remainders $\boldsymbol{S}_{\rm Kerr}-\boldsymbol{S}_{1}={\cal O}(m_2)$
and $\boldsymbol{S}-\boldsymbol{S}_{2}={\cal O}(m_2)^2$ produce extra-accelerations of order ${\cal O}(m_2)$ or higher.
This is comparable to the self-force acceleration~\cite{poisson_self_force_review}, which appears at the
next order in the mass ratio beyond the test-particle limit.
Although different choices for the function $\boldsymbol{\Delta}_{{\sigma}^\ast}$
are in principle possible, we choose here
\begin{eqnarray}
\boldsymbol{\Delta}_{{\sigma}^\ast}&=&
\frac{\eta}{12}\,\Big [ \frac{2M}{r}\,(7 \boldsymbol{\sigma}^\ast-4
\boldsymbol{\sigma})+ \boldsymbol{\hat{p}}^2 \,
(3 \boldsymbol{\sigma}+4 \boldsymbol{\sigma}^\ast) \nonumber\\
&& -6 (\boldsymbol{\hat{p}}\cdot\boldsymbol{n})^2\,(6 \boldsymbol{\sigma}+5
\boldsymbol{\sigma}^\ast)\Big ]\,,
\end{eqnarray}
which gives, when inserted into Eq.~\eqref{deltasigma},
$\boldsymbol{\Delta}_{\sigma}=0$. Because this form for
$\boldsymbol{\Delta}_{{\sigma}^\ast}$ is clearly not covariant under
generic coordinate transformations, we choose
instead the following form for the mapping of the spins, which is
covariant at least as far as the square of the momentum is concerned:
\begin{align}
\boldsymbol{\Delta}_{\sigma}=&\;0\,,\\
\boldsymbol{\Delta}_{\sigma^\ast}=&\;\frac{\eta}{12}\,
\Big [ \frac{2M}{r}\,(7 \boldsymbol{\sigma}^\ast-4 \boldsymbol{\sigma})+
(Q-1)\, (3 \boldsymbol{\sigma}+4 \boldsymbol{\sigma}^\ast)\nonumber\\ &
-6 \frac{\Delta_r}{\Sigma}\,
(\hat{\boldsymbol{p}}\cdot\boldsymbol{n})^2\, (6 \boldsymbol{\sigma}+5 \boldsymbol{\sigma}^\ast)\Big]\,,
\end{align}
where we have replaced $\hat{\boldsymbol{p}}^2$ with
$\gamma^{ij}\hat{p}_i\hat{p}_j=Q-1$ [where $Q$ is given in
Eq.~\eqref{Qpert}] and
$(\hat{\boldsymbol{p}}\cdot\boldsymbol{n})^2=\hat{p}_r^2$ with
$\Delta_r(\hat{\boldsymbol{p}}\cdot\boldsymbol{n})^2/\Sigma=g^{rr}\hat{p}_r^2$.
This form agrees with the previous mapping through order $1/c^2$, but
differs from it at higher orders. Although neither this form is
completely covariant, not even under a rescaling of the radial
coordinate (as it still features a dependence on the radius $r$), it
proved slightly better as far as the dynamics of the EOB model,
analyzed in the next section, is concerned. In particular, the factor
$g^{rr}$, which becomes zero at the horizon, quenches the increase of
$\hat{p}_r$ at small radii, thus giving a more stable behavior during
the plunge subsequent to the inspiral. (A similar effect was observed
in Ref.~\cite{Pan2009}, where the radial momentum was expressed in
tortoise coordinates to prevent it from diverging close to the
horizon.)
Having determined the mass and spin mappings, we can write down the
improved (resummed) Hamiltonian (or EOB Hamiltonian) for spinning black holes. To this
purpose, it is sufficient to invert the mapping between the real and
effective Hamiltonians [Eq.~\eqref{heff}]. In units in which $c=1$, we
obtain
\begin{equation}
\label{hreal}
H_\mathrm{real}^{\rm improved} = M\,\sqrt{1+2\eta\,\left(\frac{H_{\rm eff}}{\mu}-1\right)}\,,
\end{equation}
with
\begin{eqnarray}
\label{HeffEOB}
H_{\rm eff} &=& {H}_{\rm S}+ \beta^i\,p_i+ \alpha\, \sqrt{\mu^2 + \gamma^{ij}\,p_i\,p_j + {\cal Q}_4(p)} \nonumber \\
&& -\frac{\mu}{2M\, r^3}\,(\delta^{ij} - 3 n^i\,n^j)\, S^\ast_i\, S^\ast_j\,.
\nonumber \\
\end{eqnarray}
Here, the $-{\mu}/{(2M\, r^3)}(\delta^{ij} - 3 n^in^j) S^\ast_i S^\ast_j$ term is the
quadrupole deformation introduced in the previous section
to account for the leading order coupling of the particle's spin with itself (see also Appendix~\ref{sec:ss});
$\beta^i$, $\alpha$ and $\gamma^{ij}$ are computed using the deformed-Kerr
metric, that is inserting Eqs.~(\ref{def_metric_in})--(\ref{def_metric_fin})
into Eqs.~(\ref{alpha})--(\ref{gamma}); $H_{\rm S}$
is obtained by inserting Eqs.~\eqref{eq:omega}, \eqref{eq:nu}, and Eqs.~(\ref{eq:Br})--(\ref{Qpert})
into Eqs.~\eqref{KerrHS}, \eqref{KerrHSO} and \eqref{KerrHSS}.
Lastly, the spin $\boldsymbol{S}_{\rm Kerr}$ enters this Hamiltonian through the
parameter $a=|\boldsymbol{S}_{\rm Kerr}|/M$ appearing in the deformed-Kerr metric.
Before completing this section, we want to discuss the deformation of the
Kerr potentials $\Delta_t$ and $\Delta_r$ given in Eqs.~(\ref{deltat}) and
(\ref{deltar}), which play an important role in the EOB Hamiltonian (\ref{hreal}).
It is convenient to re-write the function $\Delta_t$ as
\begin{eqnarray}
\label{deltatu}
\Delta_t &=& r^2\, \Delta_u(u)\,, \\
\Delta_u(u) &=& A(u) + \frac{a^2}{M^2} u^2\,.
\label{deltauu}
\end{eqnarray}
In previous EOB investigations the Pad\'e summation was applied
to the function $\Delta_u$ to enforce the presence of a zero,
corresponding to the EOB horizon, both
in the non-spinning~\cite{DJS3PN} and
spinning case~\cite{Damour01c,DJS08}. Reference~\cite{Pan2009} pointed out
that when including the 4PN and 5PN terms in the function $A(u)$, the
Pad\'e summation generates poles if spins are present. Also,
the Pad\'e summation does not always ensure the existence of an innermost stable circular orbit (ISCO) for
spins aligned and antialigned with the orbital angular momentum and, even when it does,
the position of the ISCO does not vary monotonically with the magnitude of
the spins. For these reasons, we propose here an alternative way of enforcing the existence of
the EOB horizons. Working through 3PN order, we write
\begin{eqnarray}
\label{delta_t_1}
\Delta_u(u) &=& \bar{\Delta}_u(u)\, \left [1 + \eta\,\Delta_0 +
\eta \,\log \left (1 + \Delta_1 \,u + \Delta_2\,u^2 \right. \right .
\nonumber \\
&& \left. \left. + \Delta_3\,u^3 + \Delta_4\,u^4\right ) \right ]\,,
\end{eqnarray}
where
\begin{align}
\bar{\Delta}_u(u)=&\,\frac{a^2}{M^2}\,\left(u - \frac{M}{r^{\rm EOB}_{\rm H,+}}\right)\,
\left(u - \frac{M}{r^{\rm EOB}_{\rm H,-}}\right) \\=&\,\frac{a^2 u^2}{M^2}+\frac{2 u}{\eta K-1}+\frac{1}{(\eta K-1)^2}\,,\\
\label{eq:hor}
r^{\rm EOB}_{\rm H,\pm} =&\, \left(M \pm \sqrt{M^2 - a^2}\right)\, (1 - K\,\eta)\,.
\end{align}
Here, $r^{\rm EOB}_{\rm H,\pm}$ are the EOB horizons, which differ from the Kerr horizons
when the adjustable parameter $K$ is different from zero,
and where the $\log$ is introduced to quench the divergence of the powers of
$u$ at small radii. We could in principle replace the logarithm with any other
analytical function with no zeros (e.g., an exponential).
However, when studying the dynamics of the EOB model (see Sec.~\ref{sec:hamiltonian_eob}) the results
are more sensible
if we choose a function, such as the logarithm, which softens the divergence of the truncated PN series.
The coefficients $\Delta_0$, $\Delta_1$, $\Delta_2$, $\Delta_3$ and $\Delta_4$ can be derived by inserting
Eq.~(\ref{delta_t_1}) into Eq.~(\ref{deltatu}), expanding through 3PN order, and equating the
result to Eqs.~(\ref{deltatu}) and (\ref{deltauu}), with $A(u)$ given
by its PN expansion~(\ref{A_PN}). Doing so, we obtain
\begin{widetext}
\begin{eqnarray}
\Delta_0 &=& K\,(\eta\,K - 2)\label{eq:k_0}\,,\\
\Delta_1 &=& -2(\eta\,K - 1)\,(K+\Delta_0)\,,\\
\Delta_2 &=& \frac{1}{2}\,\Delta_1\, (-4 \eta\,K +\Delta_1 +4)-\frac{a^2}{M^2}\,(\eta\,K-1)^2\, \Delta_0\,,\\
\Delta_3 &=& \frac{1}{3}\,\Big [-\Delta_1^3+ 3 (\eta \,K-1)\, \Delta_1^2+3 \Delta_2\, \Delta_1-6 (\eta \,K-1) \,
(-\eta \,K + \Delta_2+1) -3 \frac{a^2}{M^2}\, (\eta\,K-1)^2\, \Delta_1\Big ]\,,\\
\Delta_4 &=& \frac{1}{12}\Big\{6 \frac{a^2}{M^2}\, \left(\Delta_1^2-2 \Delta_2\right) (\eta\,K-1)^2+ 3 \Delta_1^4-
8(\eta\,K-1)\,\Delta_1^3-12 \Delta_2\,\Delta_1^2+12 \left [2 (\eta\,K-1)\,\Delta_2+\Delta_3\right]\, \Delta_1 \nonumber\\
&& +12 \left (\frac{94}{3} - \frac{41}{32} \pi^2\right)\,(\eta\,K-1)^2+6 \left [\Delta_2^2-4 \Delta_3 \,(\eta\,K -1)\right ]\Big\}\,.
\label{eq:k_4}
\end{eqnarray}
\end{widetext}
By construction, if we expand Eq.~(\ref{delta_t_1}) in PN orders, $K$
can only appear at 4PN and higher orders, because we must recover
the PN expansion~(\ref{deltat})--\eqref{A_PN} through 3PN order.
In this sense, $K$ parameterizes our ignorance of
the PN expansion at orders equal or higher than 4PN
(i.e., $K$ would not play any role if the PN series were
known in its entirety).
Similarly, we re-write the potential $\Delta_r$ [Eq.~(\ref{deltar})] as
\begin{eqnarray}
\label{eq:D}
\Delta_r &=& \Delta_t\,D^{-1}(u)\,,\label{eq:deltaR}\\
D^{-1}(u) &=& 1+\log[1 + 6 \eta\, u^2 + 2 (26 - 3 \eta)\, \eta\, u^3]\,.\nonumber \\
\end{eqnarray}
The coefficients in the above function $D^{-1}(u)$ are such that,
when PN expanded, it gives the PN result~\eqref{D_PN}, and the
logarithmic dependence is once again
chosen to quench the divergence of the truncated PN series.
Finally, let us stress that if we included PN orders higher
than 3PN in the functions $A(u)$ and $D(u)$, we would need to add higher order
coefficients $\Delta_i$ with $i > 4$ in Eq.~(\ref{delta_t_1}).
\section{Effective-one-body dynamics for circular, equatorial orbits}
\label{sec:hamiltonian_eob}
In this section we study the dynamics of the novel EOB
model that we developed in Sec.~\ref{sec:EOB}. We will show that
\begin{enumerate}
\item[(i)] Our EOB model has the correct test-particle limit, for both non-spinning and spinning
black holes, for \textit{generic} orbits and \textit{arbitrary} spin orientations;
\item[(ii)] There exist an ISCO when the spins are aligned or antialigned with the orbital
angular momentum $\boldsymbol{L}$;
\item[(iii)] The radius, energy,
total angular momentum, orbital angular momentum and frequency at the ISCO
exhibit a smooth dependence on the binary mass-ratio and spins. Also, this
dependence looks reasonable based on what we expect from the test-particle limit and
from numerical-relativity simulations;
\item[(iv)] The frequency at the ISCO for an extreme mass-ratio non-spinning
black-hole binary agrees with the exact result computed by Ref.~\cite{barack_sago};
\item[(v)] During the plunge subsequent to the ISCO, the orbital frequency of black-hole binaries
with spins aligned or antialigned with $\boldsymbol{L}$ grows and reaches a maximum,
after which it decreases. The radius at which the frequency peaks is very close to the
radius of the equatorial, circular light ring (or photon orbit).
This feature generalizes the non-spinning behavior~\cite{Buonanno00}, and it
has a clear physical interpretation in terms of frame-dragging. As in the
non-spinning case~\cite{Buonanno00}, it provides a natural
time at which to match the two-body description of
the inspiral and plunge to the one-body description of the merger and ringdown.
\end{enumerate}
We stress that only {(i)} applies to generic orbits and spin orientations, while {(ii)}, {(iii)}, {(iv)}
and {(v)} are true for black-hole binaries with spins aligned or antialigned
with $\boldsymbol{L}$. (It should be noted that circular or spherical
orbits, and therefore the ISCO, are not even present for generic
orbits and spin orientations, because the system is not
integrable, not even in the test-particle limit~\cite{hartl}).
While we will tackle the study of generic orbits and arbitrary spin orientations in a follow-up
paper, we argue that the preliminary study
presented here is already sufficient to illustrate the potential of
the novel EOB model. We recall~\cite{Pan2009} that the only
existing EOB model for spinning black-hole binaries, proposed in
Refs.~\cite{Damour01c,DJS08}, {(i)} reproduces only approximately the test
particle limit; {(ii)} when including non-spinning terms at 4PN and
5PN order, it does not always present an ISCO for
binaries with spins parallel to $\boldsymbol{L}$, and when it does
the spin dependence of quantities evaluated at the ISCO is unusual;
{(iii)} generally, the orbital frequency does not peak during the plunge,
making the prediction of the matching time from the two-body to the one-body
description quite problematic.
Let us now go through the points of the list that we presented at the beginning of this section.
In order to prove point {(i)} we first need to observe
that the deformed metric~\eqref{def_metric_in}--\eqref{def_metric_fin} [with the potentials $\Delta_r$ and
$\Delta_t$ given by Eqs.~\eqref{eq:deltaR} and~\eqref{delta_t_1}]
reduces to the Kerr metric as $\eta\to0$, and the deformation is linear in $\eta$ when
$\eta\sim0$. Therefore, the acceleration produced by this deformation on the test-particle is
comparable to the self-force acceleration, which appears at the
next order in the mass ratio beyond the test-particle limit. Second, as already proved in Sec.~\ref{sec:EOB},
the mapping~\eqref{mapping1}--\eqref{mapping2} of the spins reduces to
$\boldsymbol{S}_{\rm Kerr}=\boldsymbol{S}_{1} +{\cal O}(m_2)$
and $\boldsymbol{S}=\boldsymbol{S}^\ast m/M=\boldsymbol{S}_{2} +{\cal O}(m_2)^2$ when $m_2\sim0$,
where the remainders produce accelerations which are again comparable to the self-force acceleration.
To prove points {(ii)}, {(iii)}, {(iv)} and {(iv)}, we need to write the effective EOB
Hamiltonian (\ref{HeffEOB}) for equatorial orbits and for spins parallel
to the orbital angular momentum (chosen to be along the $z$-axis). We obtain
\begin{eqnarray}
\label{HeffEOBco}
H_{\rm eff} &=& H_{\rm S} + \beta^i\,p_i+ \alpha \sqrt{\mu^2 +
\gamma^{ij}\,p_i\,p_j +{\cal Q}_4(p)}
\nonumber \\
&& -\frac{\mu}{2M\, r^3}\,\, S_\ast^2
\,,\nonumber \\\\
H_{\rm S}&=&g^{\rm eff}_{\rm SO}\,\boldsymbol{L}\cdot\boldsymbol{S}^\ast +g^{\rm eff}_{SS}\, {S}^\ast\,,\nonumber \\
\end{eqnarray}
where
\begin{widetext}
\begin{eqnarray}
g^{\rm eff}_{\rm SO} &=&\frac{e^{2 \nu -\tilde{\mu}}\, \left[-\sqrt{Q\,\Delta_r}\, (\tilde{B}_r-2 \tilde{B}\,\nu_{r})
+\left(e^{\tilde{\mu}+\nu }-\tilde{B}\right)\, \left(\sqrt{Q}+1\right)+(\tilde{B}\,\nu_{r}-\tilde{B}_r)
\sqrt{\Delta_r}\right]}{\tilde{B}^2 \,M \left(\sqrt{Q}+1\right)\, \sqrt{Q}}\,,\\
g^{\rm eff}_{\rm SS}&=&\frac{\mu}{M}\,\left[\omega+\frac{1}{2}\, \tilde{B}\, e^{-\tilde{\mu}-\nu }\, \omega_{r}\,\sqrt{\Delta_r}
+\left(\frac{L_z^2}{\mu^2}-\tilde{B}^2\, e^{-2 (\tilde{\mu}+\nu )}\, \Delta_r\, \frac{p^2_r}{\mu^2}\right)
\frac{e^{\nu -\tilde{\mu}}\, \omega_{r}\, \sqrt{\Delta_r}}{2 \tilde{B}\, \left(\sqrt{Q}+1\right) \,\sqrt{Q}}\right]\,,
\end{eqnarray}
\end{widetext}
with
\begin{equation}\label{Qeob}
Q=1+\frac{\Delta_r\, p_r^2}{\mu^2\, r^2}+\frac{L_z^2\, r^2}{\mu^2\, (\varpi^4 - a^2\,\Delta_t)}\,.
\end{equation}
The above equations can be evaluated explicitly by using Eqs.~\eqref{eq:omega}, \eqref{eq:nu},
(\ref{eq:Br})--(\ref{eq:mur}),
(\ref{tildeBr})--(\ref{tildeMU}), (\ref{deltatu})-- (\ref{eq:D})\footnote{A Mathematica notebook implementing the Hamiltonian \eqref{HeffEOBco}--\eqref{Qeob} is available from the authors upon request.}.
To calculate the radius and the orbital angular momentum at the ISCO for the
EOB model, we insert Eq.~(\ref{HeffEOBco}) into the real EOB Hamiltonian (\ref{hreal}),
and solve numerically the following system of equations~\cite{Buonanno00}
\begin{eqnarray}
&& \frac{\partial H^{\rm improved}_\mathrm{real}(r,p_r=0,L_z)}{\partial r}=0\,, \\
&& \frac{\partial^2 H^{\rm improved}_\mathrm{real}(r,p_r=0,L_z)}{\partial r^2}=0\,,
\end{eqnarray}
with respect to $r$ and $L_z$. Moreover, the frequency for
circular orbits is given by
\begin{equation}
\Omega=\frac{\partial H^{\rm improved}_\mathrm{real}(r,p_r=0,L_z)}{\partial L_z}\,,
\end{equation}
which follows immediately from the Hamilton equations because $L_z=p_\phi$. Finally
the binding energy is $E_{\rm bind}=H^{\rm improved}_\mathrm{real}-M\,$.
Henceforth, we set the adjustable frame-dragging parameters
$\omega^{\rm fd}_1=\omega^{\rm fd}_2=0$ [see Eq.~\eqref{eq:omegaTilde}]
and write $K$ in Eq.~\eqref{eq:hor} as a polynomial of second order in $\eta$,
\begin{equation}
K(\eta) = K_0+K_1\, \eta + K_2\, \eta^2\,.
\end{equation}
$K_0$, $K_1$ and $K_2$ being constants.
We find that if we impose
\begin{equation}
K(1/4) = \frac{1}{2}\,, \quad \quad \frac{d K}{d \eta}(1/4)=0
\end{equation}
the functional dependence on $\eta$ and $\chi$
of several physical quantities evaluated at the ISCO is quite
smooth and regular. Therefore, imposing these constraints we obtain
\begin{equation}
\label{simpleK}
K(\eta) =K_0\, (1-4 \eta)^2+ 4 (1-2 \eta) \eta\,.
\end{equation}
It is worth noting that the values of $K$ and $d K/d \eta$ at
$\eta=1/4$ have a more direct meaning than the coefficients $K_1$ and
$K_2$. In fact, current numerical-relativity simulations can evolve
binary black holes with $\eta \approx 0.25$ (with only few runs having
$\eta\sim 0.1$). Thus, they can determine $K(1/4)-1/2$, while the
value of $(dK/d\eta)(1/4)$ can be hopefully determined when more
numerical simulations with $\eta = 0.1 \mbox{--} 0.25$ become available.
Hereafter, we will use Eq.~\eqref{simpleK} and set $K_0=1.4467$. The latter is
determined by requiring that the ISCO frequency
for extreme mass-ratio non-spinning black-hole binaries agrees with the exact result
of Ref.~\cite{barack_sago}, which computed the shift of the ISCO frequency due to the conservative part of
self-force (see also Ref.~\cite{damour_SF} where the result of Ref.~\cite{barack_sago}
was compared to the non-spinning EOB prediction which resums the function (\ref{A_PN})
{\it \`a la} Pad\'e.).\footnote{We stress that the most general form of $K(\eta)$ can include
terms depending on $a^2$, with $a=|\boldsymbol{S}_{\rm Kerr}|/M$.
In particular, a term \textit{not} depending on $\eta$ and proportional to $a^2$ could be
determined by a calculation similar to that in Ref.~\cite{barack_sago}, that is
by computing the shift of the ISCO frequency caused by the conservative part of the self-force, for a non-spinning test-particle
in a Kerr spacetime.}
\begin{figure}
\resizebox{80mm}{!}{\includegraphics{freq_isco_aligned.eps}}
\caption{The frequency at the EOB ISCO for binaries having spins parallel to $\boldsymbol{L}$, with mass ratio
$q=m_2/m_1$ and with spin-parameter projections onto the direction of $\boldsymbol{L}$ given by
$\chi_1=\chi_2=\chi$. As expected, the
frequency increases with $\chi$ for a given mass ratio, while
for fixed $\chi$ it increases with $q$ if $\chi\lesssim 0.9$,
while it decreases with $q$ if $\chi$ is almost extremal (see
text for details).}
\label{freq_isco1}
\end{figure}
\begin{figure}
\resizebox{80mm}{!}{\includegraphics{afin_aligned.eps}}
\caption{The final spin parameter $\chi_{\rm fin}$ as inferred
at the EOB ISCO, for binaries having spins parallel to $\boldsymbol{L}$, with mass ratio
$q=m_2/m_1$ and with spin-parameter projections onto the direction of $\boldsymbol{L}$ given by
$\chi_1=\chi_2=\chi$. As expected, $\chi_{\rm fin}$
flattens for large $\chi$ in the comparable mass case (see text
for details).}
\label{fig:afin1}
\end{figure}
\begin{figure}
\resizebox{80mm}{!}{\includegraphics{afin_vs_NR.eps}}
\caption{The final spin parameter $\chi_{\rm fin}$ as inferred
at the EOB ISCO for binaries having spins parallel to $\boldsymbol{L}$, with mass ratio
$q=m_2/m_1$ and with spin-parameter projections onto the direction of $\boldsymbol{L}$ given by
$\chi_1=\chi_2=\chi$, compared to the remnant's final spin
parameter predicted by the formula presented in
Ref.~\cite{spin_formula} (``BR09''), which accurately
reproduces numerical-relativity results. The EOB model and the BR09
formula agree when the mass
ratio is small ($q=0.1$), because the emission during the
plunge, merger and ringdown is negligible in this case.
For $q=0.5$ and $q=1$, there is an offset,
because the EOB result, at this stage, neglects the gravitational-wave emission
during the plunge, merger and ringdown (see text for details).}
\label{fig:afin_vs_aei}
\end{figure}
\begin{figure}
\resizebox{80mm}{!}{\includegraphics{Mfin_vs_NR.eps}}
\caption{The mass loss inferred at the EOB ISCO for binaries having spins parallel to $\boldsymbol{L}$, with mass ratio
$q=m_2/m_1$ and with spin-parameter projections onto the direction of $\boldsymbol{L}$ given by
$\chi_1=\chi_2=\chi$,
compared to the total mass lost during the inspiral, merger
and ringdown, as predicted by the formulas presented in
Ref.~\cite{aei_aligned_spins} (``AEI09'') and in
Ref.~\cite{RITfit} (``RIT09''), which reproduce numerical-relativity results,
although with different accuracies because of the different
parameter regions they cover (see the text for
details). The EOB model and the AEI09 and RIT09 fits agree when the mass
ratio is small ($q=0.1$), while there is an offset for
$q=0.5$ and $q=1$. The reason is that the ringdown
emission, which is negligible for small mass-ratios, is not
taken into account by our EOB model at this stage.}
\label{fig:Eb_vs_aei}
\end{figure}
\begin{figure}
\resizebox{80mm}{!}{\includegraphics{freq_isco_antialigned.eps}}
\caption{The frequency at the EOB ISCO for binaries having spins parallel to $\boldsymbol{L}$, with mass ratio
$q=m_2/m_1$ and with spin-parameter projections onto the direction of $\boldsymbol{L}$ given by
$\chi_1=-\chi_2=\chi$. As expected, the
frequency is constant in the equal-mass case, because the
spins of the two black holes cancel out (see text for details).}
\label{freq_isco2}
\end{figure}
\begin{figure}
\resizebox{80mm}{!}{\includegraphics{afin_antialigned.eps}}
\caption{The final spin parameter $\chi_{\rm fin}$ as inferred
at the EOB ISCO, for binaries having spins parallel to $\boldsymbol{L}$, with mass ratio
$q=m_2/m_1$ and with spin-parameter projections onto the direction of $\boldsymbol{L}$ given by
$\chi_1=-\chi_2=\chi$. The results are the same
for all equal-mass binaries, for which the spins of
the two black holes cancel out (see text for details).}
\label{fig:afin2}
\end{figure}
In Fig.~\ref{freq_isco1} we plot the orbital frequency at the ISCO for binaries
with mass ratio $q=m_2/m_1$ ranging from $10^{-6}$ to $1$ and spins aligned with
$\boldsymbol{L}$. In particular, denoting by $S_{1,2}=\chi_{1,2} m^2_{1,2}$ the
projections of the spins along the direction of $\boldsymbol{L}$, we
consider binaries with $\chi_1=\chi_2=\chi$. We see that
the ISCO frequency increases with the magnitude of the spins $\chi$
if the mass ratio is fixed, as expected from the test-particle
case. Also, if the spins are kept fixed and small, the ISCO frequency
increases with the mass-ratio, as it should be to
reproduce the results of numerical-relativity simulations
(see, e.g., the non-spinning EOB models of Ref.~\cite{Damour2009a,Buonanno:2009qa}).
However, if the spins are close to $\chi=1$, the ISCO frequency decreases when the mass
ratio increases. This crossover is mirrored by a similar
behavior of other quantities evaluated at the ISCO --- such as the
energy, the orbital angular momentum, and the coordinate radius --- and
its physical meaning can be explained as follows. When
comparable-mass almost-extremal black holes merge, the resulting black-hole remnant
has a spin parameter that is slightly smaller than the spin
parameters of the parent black holes. This is a consequence of the cosmic censorship
conjecture (see Ref.~\cite{naked} and references therein) which prevents black holes with spin $\chi>1$ to be formed~\cite{dain}.
Therefore, because in the EOB model the position, and therefore the frequency,
of the ISCO (together with the loss of energy and angular momentum during the
plunge) regulate the final spin of the remnant,
and because for an isolated black hole the ISCO frequency
increases with the spin, any
EOB model that satisfies the cosmic censorship
conjecture must have an ISCO
frequency that slightly decreases with the mass ratio when
$\chi \sim 1$.
\begin{figure}
\resizebox{80mm}{!}{\includegraphics{freq_max_aligned.eps}}
\caption{The maximum of the EOB orbital frequency during the plunge,
for binaries having spins parallel to $\boldsymbol{L}$, with mass ratio
$q=m_2/m_1$ and with spin-parameter projections onto the direction of $\boldsymbol{L}$ given by
$\chi_1=\chi_2=\chi$. As expected, the frequency increases
with $\chi$ for a given mass ratio, while for fixed $\chi$ it
increases with $q$ if $\chi\lesssim 0.9$, while it decreases
with $q$ is $\chi$ is almost extremal (see text for
details).}
\label{fig:freqMax1}
\end{figure}
\begin{figure*}
\begin{center}
\begin{tabular}{cc}
\resizebox{80mm}{!}{\includegraphics{qnm1.eps}} &
\resizebox{80mm}{!}{\includegraphics{qnm2.eps}} \\
\end{tabular}
\caption{For binaries having spins parallel to $\boldsymbol{L}$, with mass ratio $q=m_2/m_1=1$ (left panel) and
$q=m_2/m_1=0.1$ (right panel) and with spin-parameter projections onto the direction of $\boldsymbol{L}$ given by
$\chi_1=\chi_2=\chi$, we plot twice the maximum of the EOB orbital frequency
during the plunge against the frequencies of the first 8
overtones of the $\ell=2$, $m=2$ quasi-normal mode of a Kerr black hole.
The quasi-normal mode frequency is computed using the final spin and
final mass of the remnant. The
final spin is estimated by applying the formula of
Ref.~\cite{spin_formula}, while for the final mass we use the
formula of Ref.~\cite{aei_aligned_spins} (``AEI09'') in the
$q=1$ case and that of Ref.~\cite{RITfit} (``RIT09'') in the
$q=0.1$ case.
As can be seen, in the $q=0.1$ case the peak frequencies lie among
the high overtones of the $\ell=2$, $m=2$ mode, while in the
$q=1$ case they are generally lower than them. In the $q=1$ case we also mark with a square the numerical
gravitational-wave frequency at the peak of the $h_{22}$ mode when $\chi=0$.
This gravitational-wave frequency coincides with (twice) the maximum of the EOB orbital frequency
at the time when the matching of the quasi-normal modes is performed
in the non-spinning EOB model of Ref.~\cite{Buonanno:2009qa}. The numerical gravitational-wave
frequency is computed from the numerical simulation of
Refs.~\cite{caltech_cornell_nonspinning}
(``Caltech-Cornell'').}
\label{qnm}
\end{center}
\end{figure*}
This interpretation can be confirmed by computing the final spin of the remnant black hole as estimated at the ISCO.
We have
\begin{equation}
\chi_{\rm fin}=\frac{S_1+S_2+L_{{\rm ISCO}}}{(M+E^{\rm bind}_{{\rm ISCO}})^2}\,,
\end{equation}
which is plotted in Fig.~\ref{fig:afin1}. Although the final spin gets
slightly larger than $1$ for high initial spins (because we are
neglecting here the energy and angular momentum emitted during the
plunge, merger and ringdown), the curves are remarkably smooth and monotonic
(see the corresponding Fig. 5 of Ref.~\cite{DJS08})
and they flatten at high initial spins, as expected. In particular, in
Fig.~\ref{fig:afin_vs_aei} we focus on mass ratios $q=1$, $q=0.5$
and $q=0.1$, and plot the final spin $\chi_{\rm fin}$ as
inferred from the ISCO energy and angular momentum, together with the
final spin of the remnant predicted by the formula presented in
Ref.~\cite{spin_formula} which accurately reproduce the numerical-relativity results
(see also Refs.~\cite{aei1,aei2,aei3,aei4, BKL,fau,RITfit} for other formulas for the final spin
of the remnant). It is remarkable that in spite of the offset between
the predictions of the formula of Ref.~\cite{spin_formula}
and the EOB result, which is due to neglecting
the energy and angular momentum emitted during plunge, merger and ringdown,
the qualitative behavior of the curves in Fig.~\ref{fig:afin_vs_aei}
is the same. Also, we observe that the difference between
corresponding curves decreases with
the mass ratio, with the EOB and the numerical-relativity--based results
being in very good agreement for $q=0.1$. This happens because the
energy and angular momentum emitted during plunge, merger and ringdown
become negligible for small mass-ratios.\footnote{This can be seen by noting that, for a test-particle
with mass $m$ around a black holes with mass $M$, the final plunge
lasts a dynamical time $\sim M$~\cite{Buonanno00}, while the inspiral from
large radii to the ISCO lasts $\sim M^2/m$.}. Similarly, in Fig.~\ref{fig:Eb_vs_aei} we
plot the binding energy at the ISCO for mass ratios $q=1$, $q=0.5$
and $q=0.1$ and compare it with fits to numerical-relativity data
for the total mass radiated in gravitational waves during the inspiral, merger and
ringdown. In particular, for the $q=1$ case we use the fit in
Ref.~\cite{aei_aligned_spins} (``AEI09''), while for
$q=0.5$ and $q=0.1$ we use the fit recently proposed by
Ref.~\cite{RITfit} (``RIT09''). While the AEI fit is more accurate
than the RIT one for the particular configuration considered here
(see Fig. 11 and related discussion in Ref.~\cite{aei_aligned_spins}), the AEI fit
is only applicable for comparable-mass binaries\footnote{The limited
applicability of the AEI fit (which is only valid for equal-mass binaries with spins aligned or anti-aligned)
is indeed one reason why it turns out to be more accurate, for the configuration under
consideration, than the RIT fit, which is instead applicable to more generic binaries.}, and for this reason
we resort to the more general RIT fit in the $q=0.5$ and
$q=0.1$ cases. [In Fig.~\ref{fig:Eb_vs_aei} we show the predictions of both the AEI and the
RIT fit in the $q=1$ case. Being the AEI fit more accurate, its difference
from the RIT fit gives an idea of the error bars which should be applied to the predictions
of the RIT fit for $q=0.5$ and $q=0.1$.]
\begin{figure}
\resizebox{80mm}{!}{\includegraphics{freq_max_antialigned.eps}}
\caption{The maximum of the EOB orbital frequency during the plunge, for binaries
having spins parallel to $\boldsymbol{L}$, with mass ratio
$q=m_2/m_1$ and with spin-parameter projections onto the direction of $\boldsymbol{L}$ given by
$\chi_1=-\chi_2=\chi$. The results are the same for all
equal-mass binaries, for which the spins of the two
black holes cancel out (see text for details).}
\label{fig:freqMax2}
\end{figure}
In Figs.~\ref{freq_isco2} and~\ref{fig:afin2} we present similar results, for the ISCO frequency and for the final spin
estimated at the ISCO,
in the case of spins antialigned with the orbital angular momentum. The
most apparent feature of these figures is that, in the equal-mass
case, the quantities under consideration are independent of $\chi$.
This happens because in this case the spins $S_1$ and $S_2$ are equal and
opposite, which results in a zero value for the spins $S_{\rm Kerr}$
and $S^\ast$ entering the EOB Hamiltonian~\eqref{hreal}. As such,
in the EOB model, equal-mass binaries with equal and opposite spins behave as non-spinning
binaries. This feature, which is also shared by the PN-expanded Hamiltonian,
until the PN order which is currently known, is also in
agreement with the results of numerical simulations. In fact,
equal-mass binaries with equal and opposite spins would be
indistinguishable with LISA, Virgo and LIGO observations~\cite{aei_aligned_spins,vaishnav}.
Except for this feature, and similarly to the aligned case discussed above,
the behavior of the curves in Figs.~\ref{freq_isco2}
and~\ref{fig:afin2} is quite smooth and regular when
going from the equal-mass case to the test-particle case.
In Fig.~\ref{fig:freqMax1} we plot the maximum value of the
orbital frequency during the plunge subsequent to the inspiral,
for binaries with mass ratio $q=m_2/m_1$ and with
$\chi_1=\chi_2=\chi$. More precisely, we assume that the particle
starts off with no radial velocity at the ISCO (thus having angular
momentum $L_{\rm {ISCO}}$ and energy $E_{\rm {ISCO}}$), and we compute
$p_r$ assuming that the energy and angular momentum are conserved during the
plunge. We find that the orbital frequency presents a peak for any value of the spins and
any mass ratio, and we denote the value of the frequency at the peak with $M \Omega_{\max}$.
We note that the behavior of
$M \Omega_{\max}$ as a function of the mass ratio is similar to
that of $M \Omega_{_{\rm ISCO}}$. In particular, its dependence on
$\eta$ changes sign when going from small to large spins.
The physical interpretation of the peak of the orbital frequency
is that the frequency increases as the effective particle spirals in,
but when the effective particle gets close to the black hole,
the orbital frequency has to decrease because the particle's motion gets
locked to the horizon (this is a well-known phenomenon,
see for instance Ref.~\cite{skimming1,skimming2} for
some of its effect on the test-particle dynamics). Said in another
way, the orbital frequency of the effective particle for an
observer at infinity goes to a constant (or to zero in
the non-spinning case~\cite{Buonanno00}) on the EOB horizon.
As a consequence, the peak in the frequency can be used to signal
the transition between two regimes~\cite{Buonanno00}:
one in which the deformed black hole and the effective particle have different
frequencies and one in which the two bodies basically move and radiate as a single
perturbed black hole. For this reason the peak of the frequency provides the
EOB approach with the natural point where to switch to the one-body description,
i.e., the point where to start describing the gravitational waveforms as a
superposition of quasi-normal modes.
We find that the values of $M \Omega_{\max}$ are roughly those needed to attach the quasi-normal
modes used in EOB models to describe the merger and the
ringdown~\cite{Damour2009a,Buonanno:2009qa,Pan2009}. This is shown in
Fig.~\ref{qnm}, where we plot twice the maximum of the orbital frequency, $M \Omega_{22}$
for binaries with $q=1$ (left panel) and $q=0.1$ (right panel) and with spins $\chi_1=\chi_2=\chi$.
We compare $M \Omega_{22}$ with the frequency of the first 8 overtones of the $\ell=2$,
$m=2$ quasi-normal mode of a Kerr black hole, computed using
the final spin and the final mass of the black-hole remnant~\cite{berti}. [The final
spin parameter is estimated by applying the formula of
Ref.~\cite{spin_formula}, while for the final mass we use the formula
of Ref.~\cite{aei_aligned_spins} (``AEI09'') in the $q=1$ case and
the one of Ref.~\cite{RITfit} (``RIT09'') in the $q=0.1$ case.]
In the $q=1$ case we also mark with a square the numerical
gravitational-wave frequency at the peak of the $h_{22}$ mode when $\chi=0$.
This gravitational-wave frequency coincides with (twice) the maximum of the EOB orbital frequency
at the time when the matching of the quasi-normal modes is performed
in the non-spinning EOB model of Ref.~\cite{Buonanno:2009qa}. The numerical gravitational-wave
frequency is computed from the numerical simulation of
Refs.~\cite{caltech_cornell_nonspinning}
(``Caltech-Cornell''). As can be seen, while in the $q=0.1$ case the
peak frequencies lie among the high overtones of the $\ell=2$, $m=2$
quasi-normal mode, in the $q=1$ case they are generally lower than
them. Quite interestingly, we find that the values of $M \Omega_{22}$
for $\chi\gtrsim0.4$ can be increased up to the frequencies of the quasi-normal modes by
assuming $\omega^{\rm fd}_2 \sim 30\mbox{--}70$ in Eq.~\eqref{eq:omegaTilde}. Nevertheless,
the frequencies that we obtain are comparable to those used for the
matching with the quasi-normal modes in Ref.~\cite{Buonanno:2009qa,Pan2009}, and we
therefore expect such a matching to be possible also in our EOB
model.
Also, it is interesting to note that the position $r_{\max}$ of the frequency
peak is quite close (to within $8 \%$) to the position of the light ring (or
circular photon orbit). This fact, which holds exactly in the non-spinning case~\cite{Buonanno00},
further confirms that the potential barrier for massless particles
(such as gravitational waves) lies at $r \sim r_{\max}$.
Finally, in Fig.~\ref{fig:freqMax2} we show the maximum value of
the orbital frequency during the plunge for binaries with mass
ratio $q=m_2/m_1$ and with $\chi_1=-\chi_2=\chi$. As for the ISCO
quantities, the dependence on the spins and the mass ratios is much simpler than in
the aligned case, with the black-hole spins cancelling out in the equal
mass-case and thus giving results which are independent of $\chi$.
Also, we see a smooth transition from the equal-mass to the extreme
mass-ratio case, that our model reproduces exactly. Also in this antialigned
case, the radius $r_{\max}$ agrees with the light-ring position to within $4\%$.
\section{Conclusions}
\label{sec:conclusions}
In this paper, building on Ref.~\cite{diracbrackets_ours},
we computed the Hamiltonian of a spinning test particle, at linear
order in the particle's spin, in an axisymmetric stationary metric and in
quasi-isotropic coordinates. Then, by applying a coordinate
transformation, we derived the Hamiltonian of a spinning test particle
in Kerr spacetime in Boyer-Lindquist coordinates
We used those results to construct an improved EOB Hamiltonian
for spinning black holes. To achieve this goal, we followed
previous studies~\cite{Damour01c,DJS08} and mapped the
real two-body dynamics into the dynamics of an effective
particle with mass $\mu$ and spin $\mathbf{S}_\ast$ moving
in a deformed-Kerr spacetime with spin $\mathbf{S}_{\rm Kerr}$,
the symmetric mass-ratio of the binary, $\eta$, acting as the deformation parameter.
To derive the improved EOB Hamiltonian, we proceeded
as follows. First, we applied a suitable canonical transformation
to the real ADM Hamiltonian and worked out the PN-expanded
effective Hamiltonian through the relation
\begin{equation}\label{heff_conclusions}
\frac{H_{\rm eff}}{\mu} =
\frac{H_{\rm real}^2 - m_1^2 - m_2^2}{2 m_1\, m_2}\,.
\end{equation}
Then, we found an appropriate
deformed-Kerr metric such that the corresponding Hamiltonian, when
expanded in PN orders, coincided with the PN-expanded effective Hamiltonian through
3PN order in the non-spinning terms, and 2.5PN order in the spinning terms.
The (resummed) improved EOB Hamiltonian is then found by inverting Eq.~\eqref{heff_conclusions}, which gives
\begin{equation}
\label{hrealeff}
H_\mathrm{real}^{\rm improved} = M\,\sqrt{1+2\eta\,\left(\frac{H_{\rm eff}}{\mu}-1\right)}\,,
\end{equation}
with
$H_{\rm eff}$ given by Eq.~(\ref{HeffEOB}), where $\alpha$, $\beta^i$ and $\gamma^{ij}$ are
obtained by inserting Eqs.~(\ref{def_metric_in})--(\ref{def_metric_fin})
into Eqs.~(\ref{alpha})--(\ref{gamma}); where $H_{\rm S}$
is obtained by inserting Eqs.~\eqref{eq:omega}, \eqref{eq:nu}, and Eqs.~(\ref{eq:Br})--(\ref{Qpert})
into Eqs.~\eqref{KerrHS}, \eqref{KerrHSO} and \eqref{KerrHSS};
and where the effective particle's spin $\boldsymbol{S}^{\ast}$ and
the deformed-Kerr spin $\boldsymbol{S}_{\rm Kerr}$ (with $a=|\boldsymbol{S}_{\rm Kerr}|/M$)
are expressed in terms of the real spins by means of Eqs.~(\ref{mapping1}),
(\ref{mapping2}), \eqref{sigma} and \eqref{sigmastar}.
The crucial EOB metric potential for quasi-circular motion is the
potential $\Delta_t(r)$ (which reduces in the non-spinning case to the
radial potential $A(r)$ of Refs.~\cite{Buonanno99,Buonanno00}).
To guarantee the presence of an inner and
outer horizons in the EOB metric, we proposed to re-write the
potential $\Delta_t(r)$ in a suitable way [see Eqs.~(\ref{deltatu}) and
(\ref{delta_t_1})], introducing the adjustable EOB
parameter $K(\eta)$ regulating the higher-order, unknown PN terms.
The reason why we did not re-write the potential $\Delta_t(r)$
using the Pad\'e summation~\cite{DJS08} is because Ref.~\cite{Pan2009}
found that when including non-spinning
terms at 4PN and 5PN order, the Pad\'e summation produces spurious poles,
does not always ensure the presence of an ISCO for binaries
with spins parallel to $\boldsymbol{L}$ and, even when it does,
the spin dependence of physical quantities evaluated at the ISCO
is quite unusual.
Restricting the study to circular orbits in the equatorial plane and
assuming spins aligned or antialigned with the
orbital angular momentum, we investigated several features
of our improved EOB Hamiltonian. Using an expression of
the EOB adjustable parameter $K(\eta)$ which reproduces the
self-force results in the non-spinning extreme mass-ratio limit
~\cite{barack_sago,damour_SF}, we computed the orbital frequency
at the EOB ISCO, we estimated the final spin from the EOB ISCO,
and the maximum orbital frequency during the plunge. We found
that these predictions are quite smooth and regular under a
variation of $\eta$ and of the black-hole spins. Quite interestingly,
the maximum of the orbital frequency during the plunge always
exists and is close to the light-ring position, as in the non-spinning case~\cite{Buonanno00}.
For this reason, as in the non-spinning case~\cite{Buonanno00}, the orbital-frequency peak can be used within the EOB
to mark the matching time at which the merger and ringdown start, i.e,
the time when, in the EOB formalism,
the gravitational waveforms start being described by a superposition
of quasi-normal modes. This will be useful in future comparisons
of the EOB model with numerical-relativity
simulations.
The results of Sec.~\ref{sec:hamiltonian_eob} are an example of the performances that
our improved EOB Hamiltonian can achieve. We expect several refinements
to be possibly needed when comparing our EOB model with accurate numerical-relativity
simulations of binary black holes. We may, for example, extend our model to reproduce also the
next-to-leading order spin-spin couplings, which are known and
appear at 3PN order~\cite{SHS07,SSH08,SHS08,PR06,PR07,PR08b,PR08a}. Also, we might introduce
a different mapping between the black-hole spins $\mathbf{S}_1$, $\mathbf{S}_2$
and $\mathbf{S}_\ast$, $\mathbf{S}_{\rm Kerr}$, a different form of
the adjustable parameter $K(\eta)$, and re-write differently the EOB metric
potential $\Delta_t(r)$. We could also introduce in it the adjustable parameters $a_5$ and
$a_6$ at 4PN and 5PN order, respectively. Moreover, other choices of the
reference tetrad used to work out the Hamiltonian for a spinning test-particle in an
axisymmetric stationary spacetime could be in principle used, leading to a different
(canonically related) EOB Hamiltonian. Lastly, the mapping (\ref{heff_conclusions})-(\ref{hrealeff})
could me modified by introducing a dependence on the spin variables.
In conclusion, the most remarkable feature of our improved EOB Hamiltonian
is that in the extreme mass-ratio limit, it exactly reproduces the Hamiltonian of
a spinning test particle in a Kerr spacetime, at linear order in the particle's
spin and at all PN orders.
\begin{acknowledgments}
We thank Yi Pan for several useful discussions.
E.B. and A.B. acknowledge support from NSF Grants No. PHYS-0603762 and PHY-0903631.
A.B. also acknowledges support from NASA grant NNX09AI81G.
\end{acknowledgments}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,148
|
Hannover. Starting in 2018, DOMOTEX – the world's leading trade fair for carpets and floor coverings – will feature an annual keynote theme around which exhibitors and other industry stakeholders can design creative showcases and installations. The resulting creations and innovations will be concentrated in Hall 9 in a new, highly immersive showcase called "Framing Trends". The lead theme for DOMOTEX 2018 – and hence the soul of the "Framing Trends" area – is "UNIQUE YOUNIVERSE", which encapsulates the growing trend towards product and service individualization. The "Framing Trends" area comprises four display zones, of which two – "NuThinkers" and "Art & Interaction" – are designed and produced by students and artists. Young creatives and next-generation designers and will use these two zones to bring the "UNIQUE YOUNIVERSE" theme to life with an array of innovative, multi-sensory installations.
For several months now, various groups of students from the Hannover and Mainz universities of applied sciences and Saarland University of Art and Design have been running research projects for the upcoming DOMOTEX show. Their mission is to uncover emerging trends in flooring individualization and to come up with exciting, out-of-the box concepts for the future of the floor coverings industry. There are over 20 of these "NuThinker" projects. Following is a sneak preview of three of them.
What if floors could communicate with people and help them find their way around buildings? That's the kind of innovative thinking that lead this young student to develop a new type of tactile guidance system for blind people. Conventional tactile flooring systems – such as corduroy and blister tiles – are limited in that they are really only useful for delineating routes and identifying hazards. Furthermore, the groove and blister patterns used are often too fine to be readily distinguishable from the surrounding floor covering using today's commonly available cane tips. The "Blindsight" system, on the other hand, uses unmistakable, standardized tactile floor symbols to clearly identify specific facilities, e.g. toilets and lifts, and provide other information. This gives the blind and vision-impaired new freedom and unfettered independence in their "UNIQUE YOUNIVERSE."
Take some leaves, pine or spruce needless, heat them until the resins and fibers are released, and then bond them all together using heat and pressure, and you've got the ultimate eco-friendly, recyclable floor covering. This young student's aim was to come up with a floor covering using natural material that is available in abundance. He hit upon the idea of using "nature's waste" – leaves – and, following a certain amount of experimentation, succeeded in creating five different natural floor coverings. "Spruce Needles" and "Pine and Spruce Needle Blend" both turn the natural material's soft autumnal tones and variegated patterns to pleasing aesthetic effect. "Fallen Leaf" and "Japanese Knotweed with Wax" preserve the fine detail of the original leaf structure for a truly sublime look. The fifth covering, "Needlework", can be produced in various formats and sanded – just like particleboard. Its surface can also be sealed using various liquid coatings and hard waxes. The uniqueness and creativity of these floor coverings is limited only by the rich variety of the natural world that inspires them and the individuality of the people that use them.
"This is Bauhaus revisited! Our installation is a very understated play of the famous Master Houses in Dessau and their pine wood setting. The window cut-outs go right through the drawing and expose the wall behind. A fine-mesh grille structure leans against the adjacent wall. If you look at it the right way, you can make out not only the arrangement of the windows, but the building structure as well. In this way, the exterior space becomes an interior space. That's not an act of domestication. The material resists the process, and the space is fluid and unconstrained... Each idea needs a suitable material. That's not a simple process, because a material is more than the sum of its properties; it can surprise by bringing opposites together. In this installation, for example, the hardness of a wall is juxtaposed with the flexibility of paper and the softness of textiles."
"I'm presenting an installation called 'Meanwhile in the Universe' that explores concepts of space. The central component is an actual recycled window, complete with shutters. The window functions as the frame for a video screen showing a live feed of outer space from NASA. The installation initially gives you the impression that you're standing outside the house looking in. But you're also looking out - into outer space. So, the concepts of outside and inside start to become fluid and lose meaning, as does your concept of your own physical position. With your notion of perspective subverted, you are forced to make a decision: whether to run away or stand your ground."
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,003
|
Q: WP8 - Display result of WebService in grid's columns Hello I have built a WebService that returns data from SQL:
public void ListadoWebService()
{
// InitializeComponent();
ServiceTours.ServiceToursClient cl = new ServiceTours.ServiceToursClient();
cl.ListadoCompleted += new EventHandler<ListadoCompletedEventArgs>(Listado2);
cl.ListadoAsync();
}
private void Listado2(object sender, ListadoCompletedEventArgs e)
{
listB.ItemsSource = e.Result;
}
Now I try to display data in columns of grid. I thought that it would work with binding the data to particular column as textblock but I can't display the data even though the data are returned in e.Result.
I tried following:
<ListBox x:Name="listB">
<ListBox.ItemTemplate>
<DataTemplate>
<Grid>
<Grid.ColumnDefinitions>
<ColumnDefinition />
<ColumnDefinition />
</Grid.ColumnDefinitions>
<TextBlock Text="{Binding id}" Grid.Column="0" />
<TextBlock Text="{Binding name}" Grid.Column="1" />
</Grid>
</DataTemplate>
</ListBox.ItemTemplate>
</ListBox>
But still I have blackscreen.
Will somebody help me solve this out please?
A: first you make one getter setter method like
public class test
{
public string Id{get;set;}
public string Name{get;set;}
public test(string id, string name)
{
Id=id;
Name=name;
}
public test()
{
}
than add your take one generc like
List<Test> lst=new List<test>();
private void Listado2(object sender, ListadoCompletedEventArgs e)
{
lst.add(new test(id,name));
listB.itemsource=lst;
}
<ListBox x:Name="listB">
<ListBox.ItemTemplate>
<DataTemplate>
<Grid>
<Grid.ColumnDefinitions>
<ColumnDefinition />
<ColumnDefinition />
</Grid.ColumnDefinitions>
<TextBlock Text="{Binding Id}" Grid.Column="0" />
<TextBlock Text="{Binding Name}" Grid.Column="1" />
</Grid>
</DataTemplate>
</ListBox.ItemTemplate>
</ListBox>
visit below link also so you can get more idea
HOw to bind data in windows phone
How can I data bind a list of strings to a ListBox in WP7
Hope it will work for you .....
A: I might be wrong here, but doesn't the e.Result cease to exist as soon as the Listado2 is over?
I'm not exactly profesional, but what I would do is I would copy the stuff from result somewhere where i could preserve it longer.
As i read the previous answer by MansinhDodiya he is basically telling you the same thing.
*
*Make a class called Item containing two Properties (the public string Id{get;set;} is a property)
There are more ways to do it, one of them being to go to the code-behind (the .xaml.cs of that page and create the class there, other being creating a new class in the same namespace ad third, setting up new namespace and adding using).
So in the .xaml.cs of the page create the class like that:
class Item
{
public string Id {get;set;} //this is the first property i would later bind
public string Name {get;set;} //this is the second property
public Item(string id, string name) // this is the contructor, every time
// an instance of Item is created, this
// method is called
{
Id = id;
Name = name;
}
}
*Make a list of these Items somewhere accesible
Next to that class, inside the page class, create an instance of that list:
List<Item> itemlist = new List<Item>();
*copy the data into that List<Item> and then set that list as itemsSource of the listbox.
Inside Listado2 copy the stuff from e.Result into the itemlist:
private void Listado2(object sender, ListadoCompletedEventArgs e)
{
...copying from e.Result into itemList...
listB.itemsource = itemList;
}
*And then change the xaml binding definition to the names of the properties - in my case:
<ListBox x:Name="listB">
<ListBox.ItemTemplate>
<DataTemplate>
<Grid>
..omitted...
<TextBlock Text="{Binding Id}" Grid.Column="0" />
<TextBlock Text="{Binding Name}" Grid.Column="1" />
</Grid>
</DataTemplate>
</ListBox.ItemTemplate>
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,746
|
Ангьяри () — город в Италии, расположенный в регионе Тоскана, подчинён административному центру Ареццо (провинция).
Население составляет 5887 человек (на 2004 г.), плотность населения составляет 45 чел./км². Занимает площадь 130 км². Почтовый индекс — 52031. Телефонный код — 00575.
В коммуне особо почитаем святой Крест Господень, празднование 3 мая.
Города-побратимы
Ссылки
Официальный сайт города
Города Италии
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 7,864
|
// Copyright (c) Microsoft Corporation. All rights reserved.
// Licensed under the MIT License.
// Code generated by Microsoft (R) AutoRest Code Generator.
package com.azure.resourcemanager.containerservice.fluent.models;
import com.azure.core.annotation.Fluent;
import com.azure.core.management.SubResource;
import com.azure.core.util.logging.ClientLogger;
import com.azure.resourcemanager.containerservice.models.AgentPoolMode;
import com.azure.resourcemanager.containerservice.models.AgentPoolType;
import com.azure.resourcemanager.containerservice.models.AgentPoolUpgradeSettings;
import com.azure.resourcemanager.containerservice.models.ContainerServiceVMSizeTypes;
import com.azure.resourcemanager.containerservice.models.KubeletConfig;
import com.azure.resourcemanager.containerservice.models.LinuxOSConfig;
import com.azure.resourcemanager.containerservice.models.OSDiskType;
import com.azure.resourcemanager.containerservice.models.OSType;
import com.azure.resourcemanager.containerservice.models.PowerState;
import com.azure.resourcemanager.containerservice.models.ScaleSetEvictionPolicy;
import com.azure.resourcemanager.containerservice.models.ScaleSetPriority;
import com.fasterxml.jackson.annotation.JsonIgnore;
import com.fasterxml.jackson.annotation.JsonProperty;
import java.util.List;
import java.util.Map;
/** Agent Pool. */
@Fluent
public final class AgentPoolInner extends SubResource {
@JsonIgnore private final ClientLogger logger = new ClientLogger(AgentPoolInner.class);
/*
* Properties of an agent pool.
*/
@JsonProperty(value = "properties")
private ManagedClusterAgentPoolProfileProperties innerProperties;
/*
* The name of the resource that is unique within a resource group. This
* name can be used to access the resource.
*/
@JsonProperty(value = "name", access = JsonProperty.Access.WRITE_ONLY)
private String name;
/*
* Resource type
*/
@JsonProperty(value = "type", access = JsonProperty.Access.WRITE_ONLY)
private String type;
/**
* Get the innerProperties property: Properties of an agent pool.
*
* @return the innerProperties value.
*/
private ManagedClusterAgentPoolProfileProperties innerProperties() {
return this.innerProperties;
}
/**
* Get the name property: The name of the resource that is unique within a resource group. This name can be used to
* access the resource.
*
* @return the name value.
*/
public String name() {
return this.name;
}
/**
* Get the type property: Resource type.
*
* @return the type value.
*/
public String type() {
return this.type;
}
/** {@inheritDoc} */
@Override
public AgentPoolInner withId(String id) {
super.withId(id);
return this;
}
/**
* Get the count property: Number of agents (VMs) to host docker containers. Allowed values must be in the range of
* 0 to 100 (inclusive) for user pools and in the range of 1 to 100 (inclusive) for system pools. The default value
* is 1.
*
* @return the count value.
*/
public Integer count() {
return this.innerProperties() == null ? null : this.innerProperties().count();
}
/**
* Set the count property: Number of agents (VMs) to host docker containers. Allowed values must be in the range of
* 0 to 100 (inclusive) for user pools and in the range of 1 to 100 (inclusive) for system pools. The default value
* is 1.
*
* @param count the count value to set.
* @return the AgentPoolInner object itself.
*/
public AgentPoolInner withCount(Integer count) {
if (this.innerProperties() == null) {
this.innerProperties = new ManagedClusterAgentPoolProfileProperties();
}
this.innerProperties().withCount(count);
return this;
}
/**
* Get the vmSize property: Size of agent VMs.
*
* @return the vmSize value.
*/
public ContainerServiceVMSizeTypes vmSize() {
return this.innerProperties() == null ? null : this.innerProperties().vmSize();
}
/**
* Set the vmSize property: Size of agent VMs.
*
* @param vmSize the vmSize value to set.
* @return the AgentPoolInner object itself.
*/
public AgentPoolInner withVmSize(ContainerServiceVMSizeTypes vmSize) {
if (this.innerProperties() == null) {
this.innerProperties = new ManagedClusterAgentPoolProfileProperties();
}
this.innerProperties().withVmSize(vmSize);
return this;
}
/**
* Get the osDiskSizeGB property: OS Disk Size in GB to be used to specify the disk size for every machine in this
* master/agent pool. If you specify 0, it will apply the default osDisk size according to the vmSize specified.
*
* @return the osDiskSizeGB value.
*/
public Integer osDiskSizeGB() {
return this.innerProperties() == null ? null : this.innerProperties().osDiskSizeGB();
}
/**
* Set the osDiskSizeGB property: OS Disk Size in GB to be used to specify the disk size for every machine in this
* master/agent pool. If you specify 0, it will apply the default osDisk size according to the vmSize specified.
*
* @param osDiskSizeGB the osDiskSizeGB value to set.
* @return the AgentPoolInner object itself.
*/
public AgentPoolInner withOsDiskSizeGB(Integer osDiskSizeGB) {
if (this.innerProperties() == null) {
this.innerProperties = new ManagedClusterAgentPoolProfileProperties();
}
this.innerProperties().withOsDiskSizeGB(osDiskSizeGB);
return this;
}
/**
* Get the osDiskType property: OS disk type to be used for machines in a given agent pool. Allowed values are
* 'Ephemeral' and 'Managed'. If unspecified, defaults to 'Ephemeral' when the VM supports ephemeral OS and has a
* cache disk larger than the requested OSDiskSizeGB. Otherwise, defaults to 'Managed'. May not be changed after
* creation.
*
* @return the osDiskType value.
*/
public OSDiskType osDiskType() {
return this.innerProperties() == null ? null : this.innerProperties().osDiskType();
}
/**
* Set the osDiskType property: OS disk type to be used for machines in a given agent pool. Allowed values are
* 'Ephemeral' and 'Managed'. If unspecified, defaults to 'Ephemeral' when the VM supports ephemeral OS and has a
* cache disk larger than the requested OSDiskSizeGB. Otherwise, defaults to 'Managed'. May not be changed after
* creation.
*
* @param osDiskType the osDiskType value to set.
* @return the AgentPoolInner object itself.
*/
public AgentPoolInner withOsDiskType(OSDiskType osDiskType) {
if (this.innerProperties() == null) {
this.innerProperties = new ManagedClusterAgentPoolProfileProperties();
}
this.innerProperties().withOsDiskType(osDiskType);
return this;
}
/**
* Get the vnetSubnetId property: VNet SubnetID specifies the VNet's subnet identifier for nodes and maybe pods.
*
* @return the vnetSubnetId value.
*/
public String vnetSubnetId() {
return this.innerProperties() == null ? null : this.innerProperties().vnetSubnetId();
}
/**
* Set the vnetSubnetId property: VNet SubnetID specifies the VNet's subnet identifier for nodes and maybe pods.
*
* @param vnetSubnetId the vnetSubnetId value to set.
* @return the AgentPoolInner object itself.
*/
public AgentPoolInner withVnetSubnetId(String vnetSubnetId) {
if (this.innerProperties() == null) {
this.innerProperties = new ManagedClusterAgentPoolProfileProperties();
}
this.innerProperties().withVnetSubnetId(vnetSubnetId);
return this;
}
/**
* Get the podSubnetId property: Pod SubnetID specifies the VNet's subnet identifier for pods.
*
* @return the podSubnetId value.
*/
public String podSubnetId() {
return this.innerProperties() == null ? null : this.innerProperties().podSubnetId();
}
/**
* Set the podSubnetId property: Pod SubnetID specifies the VNet's subnet identifier for pods.
*
* @param podSubnetId the podSubnetId value to set.
* @return the AgentPoolInner object itself.
*/
public AgentPoolInner withPodSubnetId(String podSubnetId) {
if (this.innerProperties() == null) {
this.innerProperties = new ManagedClusterAgentPoolProfileProperties();
}
this.innerProperties().withPodSubnetId(podSubnetId);
return this;
}
/**
* Get the maxPods property: Maximum number of pods that can run on a node.
*
* @return the maxPods value.
*/
public Integer maxPods() {
return this.innerProperties() == null ? null : this.innerProperties().maxPods();
}
/**
* Set the maxPods property: Maximum number of pods that can run on a node.
*
* @param maxPods the maxPods value to set.
* @return the AgentPoolInner object itself.
*/
public AgentPoolInner withMaxPods(Integer maxPods) {
if (this.innerProperties() == null) {
this.innerProperties = new ManagedClusterAgentPoolProfileProperties();
}
this.innerProperties().withMaxPods(maxPods);
return this;
}
/**
* Get the osType property: OsType to be used to specify os type. Choose from Linux and Windows. Default to Linux.
*
* @return the osType value.
*/
public OSType osType() {
return this.innerProperties() == null ? null : this.innerProperties().osType();
}
/**
* Set the osType property: OsType to be used to specify os type. Choose from Linux and Windows. Default to Linux.
*
* @param osType the osType value to set.
* @return the AgentPoolInner object itself.
*/
public AgentPoolInner withOsType(OSType osType) {
if (this.innerProperties() == null) {
this.innerProperties = new ManagedClusterAgentPoolProfileProperties();
}
this.innerProperties().withOsType(osType);
return this;
}
/**
* Get the maxCount property: Maximum number of nodes for auto-scaling.
*
* @return the maxCount value.
*/
public Integer maxCount() {
return this.innerProperties() == null ? null : this.innerProperties().maxCount();
}
/**
* Set the maxCount property: Maximum number of nodes for auto-scaling.
*
* @param maxCount the maxCount value to set.
* @return the AgentPoolInner object itself.
*/
public AgentPoolInner withMaxCount(Integer maxCount) {
if (this.innerProperties() == null) {
this.innerProperties = new ManagedClusterAgentPoolProfileProperties();
}
this.innerProperties().withMaxCount(maxCount);
return this;
}
/**
* Get the minCount property: Minimum number of nodes for auto-scaling.
*
* @return the minCount value.
*/
public Integer minCount() {
return this.innerProperties() == null ? null : this.innerProperties().minCount();
}
/**
* Set the minCount property: Minimum number of nodes for auto-scaling.
*
* @param minCount the minCount value to set.
* @return the AgentPoolInner object itself.
*/
public AgentPoolInner withMinCount(Integer minCount) {
if (this.innerProperties() == null) {
this.innerProperties = new ManagedClusterAgentPoolProfileProperties();
}
this.innerProperties().withMinCount(minCount);
return this;
}
/**
* Get the enableAutoScaling property: Whether to enable auto-scaler.
*
* @return the enableAutoScaling value.
*/
public Boolean enableAutoScaling() {
return this.innerProperties() == null ? null : this.innerProperties().enableAutoScaling();
}
/**
* Set the enableAutoScaling property: Whether to enable auto-scaler.
*
* @param enableAutoScaling the enableAutoScaling value to set.
* @return the AgentPoolInner object itself.
*/
public AgentPoolInner withEnableAutoScaling(Boolean enableAutoScaling) {
if (this.innerProperties() == null) {
this.innerProperties = new ManagedClusterAgentPoolProfileProperties();
}
this.innerProperties().withEnableAutoScaling(enableAutoScaling);
return this;
}
/**
* Get the type property: AgentPoolType represents types of an agent pool.
*
* @return the type value.
*/
public AgentPoolType typePropertiesType() {
return this.innerProperties() == null ? null : this.innerProperties().type();
}
/**
* Set the type property: AgentPoolType represents types of an agent pool.
*
* @param type the type value to set.
* @return the AgentPoolInner object itself.
*/
public AgentPoolInner withTypePropertiesType(AgentPoolType type) {
if (this.innerProperties() == null) {
this.innerProperties = new ManagedClusterAgentPoolProfileProperties();
}
this.innerProperties().withType(type);
return this;
}
/**
* Get the mode property: AgentPoolMode represents mode of an agent pool.
*
* @return the mode value.
*/
public AgentPoolMode mode() {
return this.innerProperties() == null ? null : this.innerProperties().mode();
}
/**
* Set the mode property: AgentPoolMode represents mode of an agent pool.
*
* @param mode the mode value to set.
* @return the AgentPoolInner object itself.
*/
public AgentPoolInner withMode(AgentPoolMode mode) {
if (this.innerProperties() == null) {
this.innerProperties = new ManagedClusterAgentPoolProfileProperties();
}
this.innerProperties().withMode(mode);
return this;
}
/**
* Get the orchestratorVersion property: Version of orchestrator specified when creating the managed cluster.
*
* @return the orchestratorVersion value.
*/
public String orchestratorVersion() {
return this.innerProperties() == null ? null : this.innerProperties().orchestratorVersion();
}
/**
* Set the orchestratorVersion property: Version of orchestrator specified when creating the managed cluster.
*
* @param orchestratorVersion the orchestratorVersion value to set.
* @return the AgentPoolInner object itself.
*/
public AgentPoolInner withOrchestratorVersion(String orchestratorVersion) {
if (this.innerProperties() == null) {
this.innerProperties = new ManagedClusterAgentPoolProfileProperties();
}
this.innerProperties().withOrchestratorVersion(orchestratorVersion);
return this;
}
/**
* Get the nodeImageVersion property: Version of node image.
*
* @return the nodeImageVersion value.
*/
public String nodeImageVersion() {
return this.innerProperties() == null ? null : this.innerProperties().nodeImageVersion();
}
/**
* Get the upgradeSettings property: Settings for upgrading the agentpool.
*
* @return the upgradeSettings value.
*/
public AgentPoolUpgradeSettings upgradeSettings() {
return this.innerProperties() == null ? null : this.innerProperties().upgradeSettings();
}
/**
* Set the upgradeSettings property: Settings for upgrading the agentpool.
*
* @param upgradeSettings the upgradeSettings value to set.
* @return the AgentPoolInner object itself.
*/
public AgentPoolInner withUpgradeSettings(AgentPoolUpgradeSettings upgradeSettings) {
if (this.innerProperties() == null) {
this.innerProperties = new ManagedClusterAgentPoolProfileProperties();
}
this.innerProperties().withUpgradeSettings(upgradeSettings);
return this;
}
/**
* Get the provisioningState property: The current deployment or provisioning state, which only appears in the
* response.
*
* @return the provisioningState value.
*/
public String provisioningState() {
return this.innerProperties() == null ? null : this.innerProperties().provisioningState();
}
/**
* Get the powerState property: Describes whether the Agent Pool is Running or Stopped.
*
* @return the powerState value.
*/
public PowerState powerState() {
return this.innerProperties() == null ? null : this.innerProperties().powerState();
}
/**
* Get the availabilityZones property: Availability zones for nodes. Must use VirtualMachineScaleSets AgentPoolType.
*
* @return the availabilityZones value.
*/
public List<String> availabilityZones() {
return this.innerProperties() == null ? null : this.innerProperties().availabilityZones();
}
/**
* Set the availabilityZones property: Availability zones for nodes. Must use VirtualMachineScaleSets AgentPoolType.
*
* @param availabilityZones the availabilityZones value to set.
* @return the AgentPoolInner object itself.
*/
public AgentPoolInner withAvailabilityZones(List<String> availabilityZones) {
if (this.innerProperties() == null) {
this.innerProperties = new ManagedClusterAgentPoolProfileProperties();
}
this.innerProperties().withAvailabilityZones(availabilityZones);
return this;
}
/**
* Get the enableNodePublicIp property: Enable public IP for nodes.
*
* @return the enableNodePublicIp value.
*/
public Boolean enableNodePublicIp() {
return this.innerProperties() == null ? null : this.innerProperties().enableNodePublicIp();
}
/**
* Set the enableNodePublicIp property: Enable public IP for nodes.
*
* @param enableNodePublicIp the enableNodePublicIp value to set.
* @return the AgentPoolInner object itself.
*/
public AgentPoolInner withEnableNodePublicIp(Boolean enableNodePublicIp) {
if (this.innerProperties() == null) {
this.innerProperties = new ManagedClusterAgentPoolProfileProperties();
}
this.innerProperties().withEnableNodePublicIp(enableNodePublicIp);
return this;
}
/**
* Get the scaleSetPriority property: ScaleSetPriority to be used to specify virtual machine scale set priority.
* Default to regular.
*
* @return the scaleSetPriority value.
*/
public ScaleSetPriority scaleSetPriority() {
return this.innerProperties() == null ? null : this.innerProperties().scaleSetPriority();
}
/**
* Set the scaleSetPriority property: ScaleSetPriority to be used to specify virtual machine scale set priority.
* Default to regular.
*
* @param scaleSetPriority the scaleSetPriority value to set.
* @return the AgentPoolInner object itself.
*/
public AgentPoolInner withScaleSetPriority(ScaleSetPriority scaleSetPriority) {
if (this.innerProperties() == null) {
this.innerProperties = new ManagedClusterAgentPoolProfileProperties();
}
this.innerProperties().withScaleSetPriority(scaleSetPriority);
return this;
}
/**
* Get the scaleSetEvictionPolicy property: ScaleSetEvictionPolicy to be used to specify eviction policy for Spot
* virtual machine scale set. Default to Delete.
*
* @return the scaleSetEvictionPolicy value.
*/
public ScaleSetEvictionPolicy scaleSetEvictionPolicy() {
return this.innerProperties() == null ? null : this.innerProperties().scaleSetEvictionPolicy();
}
/**
* Set the scaleSetEvictionPolicy property: ScaleSetEvictionPolicy to be used to specify eviction policy for Spot
* virtual machine scale set. Default to Delete.
*
* @param scaleSetEvictionPolicy the scaleSetEvictionPolicy value to set.
* @return the AgentPoolInner object itself.
*/
public AgentPoolInner withScaleSetEvictionPolicy(ScaleSetEvictionPolicy scaleSetEvictionPolicy) {
if (this.innerProperties() == null) {
this.innerProperties = new ManagedClusterAgentPoolProfileProperties();
}
this.innerProperties().withScaleSetEvictionPolicy(scaleSetEvictionPolicy);
return this;
}
/**
* Get the spotMaxPrice property: SpotMaxPrice to be used to specify the maximum price you are willing to pay in US
* Dollars. Possible values are any decimal value greater than zero or -1 which indicates default price to be up-to
* on-demand.
*
* @return the spotMaxPrice value.
*/
public Float spotMaxPrice() {
return this.innerProperties() == null ? null : this.innerProperties().spotMaxPrice();
}
/**
* Set the spotMaxPrice property: SpotMaxPrice to be used to specify the maximum price you are willing to pay in US
* Dollars. Possible values are any decimal value greater than zero or -1 which indicates default price to be up-to
* on-demand.
*
* @param spotMaxPrice the spotMaxPrice value to set.
* @return the AgentPoolInner object itself.
*/
public AgentPoolInner withSpotMaxPrice(Float spotMaxPrice) {
if (this.innerProperties() == null) {
this.innerProperties = new ManagedClusterAgentPoolProfileProperties();
}
this.innerProperties().withSpotMaxPrice(spotMaxPrice);
return this;
}
/**
* Get the tags property: Agent pool tags to be persisted on the agent pool virtual machine scale set.
*
* @return the tags value.
*/
public Map<String, String> tags() {
return this.innerProperties() == null ? null : this.innerProperties().tags();
}
/**
* Set the tags property: Agent pool tags to be persisted on the agent pool virtual machine scale set.
*
* @param tags the tags value to set.
* @return the AgentPoolInner object itself.
*/
public AgentPoolInner withTags(Map<String, String> tags) {
if (this.innerProperties() == null) {
this.innerProperties = new ManagedClusterAgentPoolProfileProperties();
}
this.innerProperties().withTags(tags);
return this;
}
/**
* Get the nodeLabels property: Agent pool node labels to be persisted across all nodes in agent pool.
*
* @return the nodeLabels value.
*/
public Map<String, String> nodeLabels() {
return this.innerProperties() == null ? null : this.innerProperties().nodeLabels();
}
/**
* Set the nodeLabels property: Agent pool node labels to be persisted across all nodes in agent pool.
*
* @param nodeLabels the nodeLabels value to set.
* @return the AgentPoolInner object itself.
*/
public AgentPoolInner withNodeLabels(Map<String, String> nodeLabels) {
if (this.innerProperties() == null) {
this.innerProperties = new ManagedClusterAgentPoolProfileProperties();
}
this.innerProperties().withNodeLabels(nodeLabels);
return this;
}
/**
* Get the nodeTaints property: Taints added to new nodes during node pool create and scale. For example,
* key=value:NoSchedule.
*
* @return the nodeTaints value.
*/
public List<String> nodeTaints() {
return this.innerProperties() == null ? null : this.innerProperties().nodeTaints();
}
/**
* Set the nodeTaints property: Taints added to new nodes during node pool create and scale. For example,
* key=value:NoSchedule.
*
* @param nodeTaints the nodeTaints value to set.
* @return the AgentPoolInner object itself.
*/
public AgentPoolInner withNodeTaints(List<String> nodeTaints) {
if (this.innerProperties() == null) {
this.innerProperties = new ManagedClusterAgentPoolProfileProperties();
}
this.innerProperties().withNodeTaints(nodeTaints);
return this;
}
/**
* Get the proximityPlacementGroupId property: The ID for Proximity Placement Group.
*
* @return the proximityPlacementGroupId value.
*/
public String proximityPlacementGroupId() {
return this.innerProperties() == null ? null : this.innerProperties().proximityPlacementGroupId();
}
/**
* Set the proximityPlacementGroupId property: The ID for Proximity Placement Group.
*
* @param proximityPlacementGroupId the proximityPlacementGroupId value to set.
* @return the AgentPoolInner object itself.
*/
public AgentPoolInner withProximityPlacementGroupId(String proximityPlacementGroupId) {
if (this.innerProperties() == null) {
this.innerProperties = new ManagedClusterAgentPoolProfileProperties();
}
this.innerProperties().withProximityPlacementGroupId(proximityPlacementGroupId);
return this;
}
/**
* Get the kubeletConfig property: KubeletConfig specifies the configuration of kubelet on agent nodes.
*
* @return the kubeletConfig value.
*/
public KubeletConfig kubeletConfig() {
return this.innerProperties() == null ? null : this.innerProperties().kubeletConfig();
}
/**
* Set the kubeletConfig property: KubeletConfig specifies the configuration of kubelet on agent nodes.
*
* @param kubeletConfig the kubeletConfig value to set.
* @return the AgentPoolInner object itself.
*/
public AgentPoolInner withKubeletConfig(KubeletConfig kubeletConfig) {
if (this.innerProperties() == null) {
this.innerProperties = new ManagedClusterAgentPoolProfileProperties();
}
this.innerProperties().withKubeletConfig(kubeletConfig);
return this;
}
/**
* Get the linuxOSConfig property: LinuxOSConfig specifies the OS configuration of linux agent nodes.
*
* @return the linuxOSConfig value.
*/
public LinuxOSConfig linuxOSConfig() {
return this.innerProperties() == null ? null : this.innerProperties().linuxOSConfig();
}
/**
* Set the linuxOSConfig property: LinuxOSConfig specifies the OS configuration of linux agent nodes.
*
* @param linuxOSConfig the linuxOSConfig value to set.
* @return the AgentPoolInner object itself.
*/
public AgentPoolInner withLinuxOSConfig(LinuxOSConfig linuxOSConfig) {
if (this.innerProperties() == null) {
this.innerProperties = new ManagedClusterAgentPoolProfileProperties();
}
this.innerProperties().withLinuxOSConfig(linuxOSConfig);
return this;
}
/**
* Validates the instance.
*
* @throws IllegalArgumentException thrown if the instance is not valid.
*/
public void validate() {
if (innerProperties() != null) {
innerProperties().validate();
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 500
|
var express = require('express')
var path = require('path')
var webpack = require('webpack')
var webpackDevMiddleware = require('webpack-dev-middleware')
var app = express()
app.use(function (req, res, next) {
res.header('Access-Control-Allow-Origin', '*')
res.header('Access-Control-Allow-Methods', 'GET, POST')
res.header('Access-Control-Allow-Headers', 'Origin, X-Requested-With, Content-Type, Accept, Authorization')
next()
})
const isDevelopment = process.env.NODE_ENV !== 'production'
if (!isDevelopment) {
app.use(express.static(__dirname))
app.get('*', function (req, res) {
res.sendFile(path.resolve(__dirname, 'index.html'))
})
app.listen(3000, function () {
console.log('Your app listening on 3000! have a nice day:)')
})
} else {
var config = require('./webpack.config.js')
var compiler = webpack(config)
compiler.apply(new webpack.ProgressPlugin())
app.use(express.static(__dirname))
app.use(webpackDevMiddleware(compiler, {
publicPath: config.output.publicPath,
stats: { colors: true }
}))
app.get('*', function (req, res) {
res.sendFile(path.resolve(__dirname, 'index.html'))
})
app.listen(8080)
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,326
|
{"url":"http:\/\/support.primatelabs.com\/discussions\/geekbench\/27829-discrepency-between-windows-and-linux-versions","text":"# Discrepency between Windows and Linux versions\n\n#### johncho\n\n05 Jun, 2018 03:33 AM\n\nFirstable thanks to all, this is a great software that I like to use when I switch components on my computer.\n\nTwo questions:\n\n1.\nIs it normal for the same computer to score differently under Linux and Windows? My linux benchmark constantly scored roughly 10% higher than Windows 10 benchmark using the same dual boot computer.\n\n2.\nRecently I tested the same system with overclocked RAM. However, this time the memory bandwidth under Windows 10 is 10% higher than that reported under Linux using geekbench4 software. The linux version didnt report much increase after the overclock, but Windows reported 10% increase in bandwidth. Whats causing this discrepency ?\n\n1. Support Staff Posted by Sarah on 05 Jun, 2018 04:46 PM\n\nHi crunsoft,\n\nThanks for posting. Are you able to provide links to the benchmark results you are getting?\n\nThanks,\n\nSarah\nPrimate Labs Inc.\n\n2. Posted by johncho on 06 Jun, 2018 02:51 AM\n\n3. Support Staff Posted by Sarah on 07 Jun, 2018 07:27 PM\n\nHi johncho,\n\nThanks for posting the results. It is possible for the operating system to affect the result of a benchmark slightly.\n\nYour power plan could also play a factor. When you ran the benchmark on Windows, the power plan was set to \"Balanced\". Could you try setting it to \"Performance\" and then running the benchmark again?\n\nWarm regards,\n\nSarah\nPrimate Labs Inc.\n\n4. Posted by johncho on 08 Jun, 2018 05:48 AM\n\nHi,\nBalanced and Performance power plan in Window didnt make any difference in scoring results.\n\nPerformance\nhttps:\/\/browser.geekbench.com\/v4\/cpu\/8594111\n\n## Reply to this discussion\n\nFormatting help \/ Preview (switch to plain text) No formatting (switch to Markdown)\n\n### \u00bb\n\n#### Attached Files\n\nYou can attach files up to 10MB\n\nIf you don't have an account yet, we need to confirm you're human and not a machine trying to post spam.\n\n# Keyboard shortcuts\n\n### Generic\n\n? Show this help Blurs the current field\n\n### Comment Form\n\nr Focus the comment reply box Submit the comment\n\nYou can use Command \u2318 instead of Control ^ on Mac\n\n## Recent Discussions\n\n 22 Aug, 2018 03:13 AM VoodooPad 6 Wishlist 20 Aug, 2018 03:36 PM File Format: Plain Text option 20 Aug, 2018 12:57 PM Feature Request: MathJAX support 20 Aug, 2018 12:52 PM Allow to select default document style when creating new pages 20 Aug, 2018 12:49 PM Geekbench App not natively optimiced for 18,5:9 displays\n\n## Recent Articles\n\n Geekbench 3 Benchmarks Frequently Asked Questions Geekbench 3 Pro Standalone Mode Geekbench 4 Pro Standalone Mode Interpreting Geekbench 4 Scores","date":"2018-08-22 06:58:06","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\": 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\": 0.17769791185855865, \"perplexity\": 9814.074276885267}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"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-2018-34\/segments\/1534221219692.98\/warc\/CC-MAIN-20180822065454-20180822085454-00291.warc.gz\"}"}
| null | null |
Exemplary People > Environment > ...
Prince Charles (UK)
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Corporate leaders unite with The Prince of Wales to put rainforests on the climate change agenda - YouTube
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,945
|
\section{Introduction}
The possible generation of a phase with local parity breaking (LPB) in nuclear matter at extreme conditions
such as those reached in heavy ion collisions (HIC) at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) \cite{reviews} has been
examined recently \cite{kharzeev,anesp,avep}.
It has been suggested in \cite{kharzeev} that at increasing temperatures an isosinglet pseudoscalar background could
arise due to large-scale topological charge fluctuations (studied recently in lattice quantum chromodynamics (QCD) simulations
\cite{lattice}).
These considerations led eventually to the observation of the so-called chiral magnetic effect
(CME)~\cite{kharzeev}
in the STAR and PHENIX experiments at RHIC \cite{star}. The effect should be most visible
for non-central HIC where large angular momenta induce large magnetic fields contributing
to the chiral charge separation. {However, the CME may be only a partial explanation of the STAR and PHENIX experiments and other backgrounds play a comparable role (see the reviews \cite{chiraleff}). In a recent report~\cite{measuCME} the measurements of the chiral magnetic effect in Pb–Pb collisions with A Large Ion Collider Experiment (ALICE) were estimated and perspectives to improve their precision in future LHC runs were outlined}.
For central collisions it was proposed in \cite{anesp} that
the presence of a phase where parity was spontaneously broken could be a rather generic feature of QCD. Local parity breaking can be induced by difference between the densities of the right- and
left-handed chiral fermion fields (chiral Imbalance) {in metastable domains with non-zero topological charges. Thus our analysis concerns solely
the events in the central heavy ion collisions where the magnetic fields are negligible. It is seen in the experiments~\cite{star,chiraleff}, and was also found in lattice QCD (see~\cite{lattice}). The validity of CME and its percentage in observations is
well analyzed in \cite{measuCME}.
Thereby the elimination of electromagnetic effects
is justified and allows to measure solely the chiral chemical potential without contamination by magnetic fields and related~backgrounds.}
In the hadron phase we shall assume that as a consequence of topological charge fluctuations, the environment
in the central HIC generates a pseudoscalar background growing approximately linearly in time.
This background is associated with a constant axial vector whose zero component is identified with a chiral
chemical potential. In such an environment one could search for a possible manifestation of LPB {in dilepton
probes.} In particular, in \cite{aaep} it was shown that a good part of the excess of dileptons produced in
central heavy-ion collisions \cite{phenix} might be a consequence of LPB due to the generation
of a pseudoscalar isosinglet condensate whose precise magnitude and time variation depends on
the dynamics of the HIC.
The complete description of a medium with chiral imbalance should also take into account thermal fluctuations of the medium. In this paper the description in a zero temperature limit is considered and to understand the changes for non-zero temperatures we rely on the results of lattice computations of quark matter with chiral imbalance and a temperature of order 150 MeV undertaken \cite{braguta}. Thus our calculations keep the tendency of increasing chiral condensate and decreasing pion masses when the temperature grows.
This paper is mostly concerned with the possibility of identifying LPB in the hadron phase of QCD in HIC. Such a
medium would be simulated by a chiral chemical potential $\mu_5$. Adding to the QCD Lagrangian the term
$\Delta{\mathcal
L}_q=\mu_5 q^\dagger \gamma_5 q \equiv \mu_5 \rho_5$, we allow for non-trivial
topological fluctuations \cite{aaep} in the nuclear (quark) fireball, which are ultimately related to fluctuations
of gluon fields. The transition of the quark--gluon medium characteristics to a hadron matter reckons on the
quark--hadron continuity~\cite{contin} after hadronization of quark--gluon plasma. The behavior of various spectral
characteristics for light scalar and pseudoscalar ($\sigma, \pi^a, a^a_0$)-mesons by means of a QCD-motivated
$\sigma$-model Lagrangian was recently derived for $SU_L (2) \times SU_R (2)$ flavor symmetry including an
isosinglet
chiral chemical potential \cite{eqcd16}. The structural constants of the $\sigma$-model Lagrangian were taken as
input
parameters suitable to describe the light meson properties in vacuum and then they are extrapolated to a chiral medium.
In this way ad hoc there is no reliable predictability in the determination of the hadron system response on
chiral imbalance, and reaching quantitative predictions requires a phenomenologically justified hadron dynamics.
To increase predictability, we extend the vacuum chiral Lagrangians \cite{GL,kais} with phenomenological
low-energy
structural constants taking into account the chiral medium in the fireball with a chiral imbalance.
It is shown that $\sigma$-model parametrization of \cite{eqcd16} fits well the pion phenomenology at low energies
as derived from ChPT.
Next it is described how pions modify their dynamics in decays in a chiral medium, in particular, charged pions
stop
decaying into muons and neutrinos for a large enough chiral chemical potential. A possible experimental detection of chiral
imbalance (and therefore a phase with local parity breaking) is outlined in the charged pion decays
inside the fireball.
\section{Chiral Lagrangian with Chiral Chemical Potential}
The chiral Lagrangian for pions describing their mass spectra and decays in the fireball with a chiral imbalance
can
be implemented with the help of softly broken chiral symmetry in QCD transmitted to hadron media, a properly constructed covariant derivative:
\begin{equation}
D_\nu \Longrightarrow \bar D_\nu - i \{{\bf I}_q\mu_5 \delta_{0\nu}, \star
\}={\bf I}_q \partial_\nu - 2i{\bf I}_q\mu_5 \delta_{0\nu},\label{covderu}
\end{equation}
where we skipped the electromagnetic field. The axial chemical potential is introduced as a constant time component
of an
isosinglet axial-vector field.
In the framework of large number of colors $N_c$ \cite{kais} the SU(3) chiral Lagrangian in the strong interaction
sector
contains the following dim=2 operators \cite{kais},
\be
{\cal L}_2=\frac{F^2_0}{4} <- j_\mu j^\mu +\chi^\dagger U+ U^\dagger \chi >,
\label{L2lag}
\ee
where $<...>$ denotes the trace in flavor space, $j_\mu \equiv U^\dagger \partial_\mu U $, the chiral
field $U=\exp(i \hat\pi/ F_0)$,\ the bare pion decay constant $F_0 \simeq 92$ MeV, $\chi(x)=2B_0 s(x)$ and
$M_\pi^2=2B_0 \hat m_{u,d}$, the tree-level neutral
pion mass. The constant $B_0$ is related to the chiral quark condensate $<\bar q q>$ as $ F_0^2 B_0 = - <\bar q
q>$.
Taking now the covariant derivative in \eqref{covderu} it yields
\begin{equation}
{\cal L}_2 (\mu_5)={\cal L}_2(\mu_5=0)+\mu_5^2 N_f F_0^2.
\end{equation}
Herein we have used the identity for $U\in SU(n)$,$< j_\mu>=0$.
In the large $N_c$ approach the dim=4 operators \cite{kais} in the chiral Lagrangian are given by
\ba
{\cal L}_4 =\bar L_3 <j_\mu j^\mu j_\nu j^\nu> + L_0 <j_\mu j_\nu j^\mu j^\nu> - L_5 < j_\mu j^\mu (\chi^\dagger
U+ U^\dagger \chi)>,\label{L4lag}
\ea
where $L_0, \bar L_3, L_5$ are bare low energy constants.
For SU(3) and SU(2) $<j_\mu>=0$ and there is the identity
\ba <j_\mu j_\nu j^\mu j^\nu>=-2 <j_\mu j^\mu j_\nu j^\nu> + \frac12 <j_\mu j^\mu> <j_\nu j^\nu> + <j_\mu j_\nu>
<j^\mu j^\nu> , \label{id1}\ea
whereas for SU(2) there is one more identity
\be 2 <j_\mu j^\mu j_\nu j^\nu> = <j_\mu j^\mu> <j_\nu j^\nu> . \label{id2}\ee
Applying these identities one finds the four-derivative Gasser--Leutwyler (GL) operators for the SU(3) chiral
Lagrangian
\ba{\cal L}_4 &=& L_1 <j_\mu j^\mu> <j_\nu j^\nu> + L_2 <j_\mu j_\nu> <j^\mu j^\nu> +L_3 <j_\mu j^\mu j_\nu
j^\nu>\nonumber\\&&
- L_5 < j_\mu j^\mu (\chi^\dagger U+\chi U^\dagger)>\ea with
\be L_1= \frac12 L_0;\ L_2= L_0;\ L_3 =\bar L_3 -2L_0.\ee
For SU(2) one has a further reduction of the dim=4 Lagrangian,
\be {\cal L}_4 = \frac14 l_1 <j_\mu j^\mu> <j_\nu j^\nu> + \frac14 l_2 <j_\mu j_\nu> <j^\mu j^\nu>- \frac14 l_4 <
j_\mu j^\mu (\chi^\dagger U+ U^\dagger \chi)> \label{final}\ee
where $l_{1}, l_{2}, l_{4}$ are renormalized SU(2) low energy constants (as compared to \cite{GL} )
with normalization so that
\be l_1 = 2 L_0 +2 \bar L_3,\ l_2 = 4 L_2= 4L_0,\ (l_1 +l_2) = 2\bar L_3 + 6 L_0;\quad l_4 = 4 L_5.\ee
We stress that this chain of reductions is valid only if $<j_\mu>=0$.
The response of the chiral Lagrangian on chiral imbalance is derived with the help of the covariant derivative
\eqref{covderu}
applied to the Lagrangian \eqref{L4lag},
\ba
&&\Delta {\cal L}_4 (\mu_5) =
-\mu_5^2\{ 12(l_1 + l_2) <j^0 j^0> -4(l_1+l_2) <j_k j_k> - l_4 <\chi^\dagger U+ U^\dagger \chi >\}.
\ea
We notice that this result is drastically different from what one could obtain from the final Lagrangian~\eqref{final}.
This is because the identities \eqref{id1} and \eqref{id2} are violated if $<j_\mu>\not=0$.
The above modifications change differently the coefficients in the dispersion law in energy $p^0$ and
three-momentum $|\vec p|$ for the mass shell as well as modify the mass term for pions (all together it gives the
inverse propagator of pions),
\be
{\cal D}^{-1}( \mu_5) = (F_0^2 + 48 \mu^2_5 (l_1 + l_2)) p_0^2 -(F_0^2 +16 \mu^2_5 (l_1+l_2) )|\vec p|^2 - (F^2_0
+ 4l_4\mu_5^2) m_\pi^2(0) \to 0. \label{massshell}
\ee
In the leading order of large $N_c$ expansion the empirical values of
the SU(2)
Gasser-Leutwyler (GL)
constants $l_1, l_2, l_4$ are given in \cite{GL}.
Thus in the pion rest frame
\be
F_\pi^2(\mu^2_5)\simeq F_0^2 + 48 \mu^2_5 (l_1 + l_2);\quad m_\pi^2(\mu^2_5)\simeq \Big(1 -4 \frac{\mu_5^2}{F_0^2}
(12(l_1 + l_2) - l_4 )\Big) m_\pi^2 (0),\label{fandm}
\ee
i.e., the pion decay constant is growing and its mass is decreasing in the chiral media. From these relations one can determine the quark condensate dependence on the chiral imbalance,
\be
<\bar q q>(\mu_5) = <\bar q q>(0)\Big(1 + 4 l_4 \frac{\mu_5^2}{F^2_0
}\Big),
\ee
wherefrom one concludes that for $l_4 > 0$ \cite{GL} the magnitude of quark condensate increases with growing the chiral chemical potential as it was predicted in \cite{braguta, eqcd16, eqcd2017}
\section{Linear Sigma Model for Light Pions and Scalar Mesons in the Presence of Chiral Imbalance: Comparison to
ChPT}
Let us compare these constants with those ones estimated from the linear sigma model (LSM) built in~\cite{eqcd2017,efflag}.
The sigma model was build with realization of SU(2) chiral symmetry to describe pions and isosinglet and
isotriplet scalar mesons.
Its Lagrangian reads
\begin{eqnarray
{L}&=& N_c \Big\{\frac{1}{4}\,<(D_{\mu}H\,(D^{\mu}H)^{\dagger}>
+\frac{B_0}{2}\, <m(H\,+\,H^{\dagger}>
+\frac{M^{2}}{2}\,<HH^{\dagger}>
\nonumber \\
\,&-&\frac{\lambda_{1}}{2}\,<(HH^{\dagger})^{2}>
-\frac{\lambda_{2}}{4}\, <(HH^{\dagger})>^{2}
+\frac{c}{2}\,(\det H +\,\det H^{\dagger})\Big\},
\label{lagr_sigma}
\end{eqnarray}
where $H = \xi\,\Sigma\,\xi$ is an operator for meson fields, $N_c$ is a number of colours,
\(m\) is an average mass of current $u,d$ quarks,
\(M\) is a `'tachyonic'' mass generating the spontaneous breaking of chiral symmetry,
\(B_0, c,\lambda_{1},\lambda_{2}\) are real constants.
The matrix \(\Sigma\) includes the singlet scalar meson \(\sigma\), its vacuum average \(v\) and the isotriplet of
scalar mesons \( a^{0}_{0},a^{-}_{0},a^{+}_{0}\), the details see in ~\cite{eqcd2017,efflag}. The covariant
derivative
of $H$ including the chiral chemical potential $\mu_5$ is defined in \eqref{covderu}.
The operator \(\xi\) realizes a nonlinear representation (see \eqref{L2lag}) of the chiral group $SU(2)_L
\times SU(2)_R$,
namely, $\xi^2 = U$.
The diagonal masses for scalar and pseudoscalar mesons read
\begin{equation}
\begin{aligned}
m^{2}_{\sigma} &=-2\left( M^{2}-6\,(\lambda_{1}+\lambda_{2})F_\pi^{2}+c+2\mu^2_5 \,\right)\\
m^{2}_a &=-2 \left( M^{2}-2\,(3\lambda_{1}+\lambda_{2})F_\pi^{2}-c+2\mu^2_5 \, \right)\\
m^{2}_{\pi} (\mu_5) &=\frac{2\,b\,m}{F_\pi} \simeq m^{2}_{\pi} (0)\left(1 -
\dfrac{\mu^{2}_{5}}{2(\lambda_{1}+\lambda_{2})F_0^2}\right) \\
F^2_\pi(\mu_{5}) &=\dfrac{M^{\,2}+2\mu^{2}_{5}+c}{2(\lambda_{1}+\lambda_{2})} = F_0^2 +
\dfrac{\mu^{2}_{5}}{\lambda_{1}+\lambda_{2}}.
\end{aligned}
\end{equation}
From spectral characteristics of scalar mesons in vacuum one fixes the Lagrangian parameters,
\(\lambda_{1}=16.4850\), \(\lambda_{2}=-13.1313\), \(c=-4.46874\times10^4~{\text{MeV}}^2\), \( F_0 = 92 \text{MeV}\), \(b = B_0 F_0 =1.61594\times10^5~{\text{MeV}}^2\) \cite{eqcd2017}.
The change of the pion-coupling constant $F_0$ is determined by potential parameters as compared to the ChPT
definition,
\be
{\Delta F_\pi^2\over\mu_5^2} = \frac{1}{\lambda_1 +\lambda_2}\approx 0.3\quad \mbox{\it vs}\quad 48 (l_1 + l_2),
\ee
i.e. $(l_1 + l_2)\approx 6.2\times 10^{-3}$. It is a satisfactory correspondence to the pion phenomenology \cite{GL} .
Analogously, in the rest frame using the pion mass correction,\\ $ m^2_\pi(\mu_5)F^2_\pi(\mu_5) \simeq 2m_q b
F_\pi(\mu_5) $ it
is easy to find the estimation for
\be l_4 = \frac{1}{8(\lambda_1 +\lambda_2)})\approx 3.7\times
10^{-2}, \ee
wherefrom one can also guess the relation $6(l_1 + l_2) = l_4$ following from the LSM. It is again a satisfactory correspondence to the pion phenomenology \cite{GL} .
For moving mesons with $|\vec p|\not = 0$ and the CP
breaking mixing of scalar and pseudoscalar mesons the
effective masses $m^{2}_{eff\mp}$ take the form,
\begin{gather}
m^{2}_{eff-}=\frac{1}{2}\left( 16 \, \mu^{2}_{5} +m^{2}_a +m^2_\pi - \sqrt{(16 \mu^{2}_{5}+m^{2}_{a}+m^2_\pi)^{2}
-4 \left( m^{2}_a \,m^2_\pi -16 \mu^{2}_{5} \,|\vec p|^{2} \right) } \,\ \right),
\nonumber \\
m^{2}_{eff+}=\frac{1}{2}\left( 16 \, \mu^{2}_{5} +m^{2}_a +m^2_\pi + \sqrt{(16 \mu^{2}_{5}+m^{2}_{a}+m^2_\pi)^{2}
-4 \left( m^{2}_a \,m^2_\pi -16 \mu^{2}_{5} \,|\vec p|^{2} \right) } \,\ \right).
\end{gather}
For small $\mu^{2}_{5}, m^2_\pi \ll m^{2}_a \simeq 1 GeV^2$ one can approximate the dependence on the wave vector
$\vec p$
\be
m^{2}_{eff-} \simeq m^2_{\pi} - 16 \mu_5^2 \frac{|\vec p|^2}{m_a^2}.
\ee
Comparing with \eqref{massshell}
one establishes the relationship of isotriplet scalar mass and GL constants
\be
m_a = \frac{F_0}{\sqrt{2(l_1 + l_2)}} \simeq 0.9 GeV, \label{ma}
\ee
which is close to the Particle Data Group
value within the experimental error bars~\cite{PDG}.
\section{Possible Experimental Detection of Chiral
Imbalance in the Charged Pion Decays}
The predicted distortion of the mass shell condition can be detected in decays of charged pions when the effective
pion mass approaches the muon mass.
Let us find the threshold value for the $\pi^+\to \mu^+\nu$ decay.
If a charged pion was generated in chiral medium its mass is lower than in the vacuum and the condition for its
decay follows from \eqref{massshell},
\be
\left(1-32 (l_1 + l_2) \frac{\mu_5^2}{F_0^2}\right) |\vec p|^2 + \left(1-24 (l_1 + l_2) \frac{\mu_5^2}{F_0^2}\right)m^2_{0,\pi} \geq |\vec p|^2
+m^2_\mu ,
\ee
where we have used the relations $6(l_1 +l_2) \simeq l_4$. The decay channel is closed
for $|\vec p|^2 \simeq 0$ if $\mu_5 \simeq 160$ MeV. It must be detected as a substantial decrease of muon flow
originated from
pion decays in the fireball. When considering the decay process of a charged pion into a muon $+$ neutrino at values of the chiral chemical potential lower than $\mu_5 \simeq 160$ MeV then still the muon yield from the fireball obviously decreases at sufficiently large momenta. It gives one a chance to measure the magnitude of
chiral chemical potential for sufficiently high statistics.
\section{Results}
\begin{itemize}
\item For light mesons in the chiral imbalance medium we compared the chiral perturbation theory (ChPT) and the
linear sigma model (LSM)
as realizations of low energy QCD.
The relations between the low-energy constants of the chiral Lagrangian and
the corresponding constants of the linear sigma model are established and
expressions for the decay constant of the pion in the medium and the mass of the $a_0$
meson are found.
\item The low energy QCD correspondence of ChPT and LSM
in the large $N_c$ limit is satisfactory and provide a solid ground for the search of chiral imbalance manifestation
in pion physics at HIC.
\item The resulting dispersion law for pions in the medium allows us
reveal the threshold of decay of a charged pion into a muon and neutrino which can be
suppressed by increasing chiral chemical potential.
\item As it is shown in \cite{eqcd2017}, at higher energies exotic decays of isoscalar mesons into three pions arise due to
mixing of $\pi$ and $a_0$ meson states in the presence of chiral imbalance. It was shown \cite{aaep,eqcd16,eqcd2017} that for a wider class of direct parity breaking at higher energies, in the framework of linear sigma model with isotriplet scalar ($a_0$) and pseudoscalar (pions) mesons, their mixing arises with the generation of $\pi\pi$ and $\pi\pi\pi$ decays of a heavier scalar state. Also, the independent check of our estimates could be done by lattice computation (cf.\cite{braguta}).
\item A manifestation for LPB can also happen in the presence of chiral imbalance in the sector
of $\rho$ and $\omega$ vector mesons \cite{anesp,avep} and in this case the Chern--Simons
interaction plays a major role. It turns out \cite{anesp} that the spectrum
of massive vector mesons splits into three components with different polarizations $\pm,long$ having different
effective
masses $m_{V,+} < m_{V,long} < m_{V, -}$.
\item Thus a possible experimental detection of chiral imbalance in medium
(and therefore a phase with LPB) in the charged pion
decays and vector meson polarizations inside of the fireball can be~realized.
\item We would like to mention the recent proposal to measure the photon
polarization asymmetry in $\pi\gamma$ scattering \cite{harada,eqcd2017,ppn2019} as a way to detect LPB due to
chiral imbalance. This happens in the ChPT including electromagnetic fields
due to the Wess--Zumino--Witten operators.
\item One may be concerned about the appearance of changes in the properties of muons and neutrinos in the medium, but in our opinion, this does not change the main estimates in Eq. (22), as a possible influence of chiral chemical potential
on lepton properties would be controlled by extra power of the Fermi coupling constant, i.e., by the next order in weak interactions with little hope to register it.
\item We emphasize the similarity of our model results to lattice computation in \cite{braguta}: to the same tendency of increasing chiral
condensate and decreasing of pion mass when $ \mu_5$ grows for fixed temperatures about 150 MeV. It gives us the confidence (see \cite{eqcd16, eqcd2017})
that our spectral predictions are robust in a range of temperatures. We understand that for a more realistic quantitative description of the phenomena under discussion, thermal effects, smearing of data and detector acceptance must be taken into account, which will be done in subsequent works with an extended team including the experimentalists.
\item Last decade, different controversial conclusions on thermodynamics of quark matter with a chiral imbalance appeared based on
different models of the Nambu-Jona-Lasinio (NJL)
type. Among them, an opposite decreasing behavior of quark condensate in \cite{wrongcutoff}
was found due to an erroneous use of UV regularization in a NJL type model which mimicked chiral symmetry breaking, and chiral imbalance with chiral chemical potential being included into an UV cutoff. This kind of mistake in applications of NJL models has been known since the 1980s. The correct regularization based on vacuum definitions of cutoffs is elucidated in \cite{correctcutoff}.
In the treatment in \cite{braguta} based on lattice computations as well as in meson Lagrangians \cite{efflag} where the UV finite chirally-symmetric computations are used, the problem is thoroughly resolved.
\end{itemize}
\section*{Acknowledgments}
This research was funded by the Grant FPA2016-76005-C2-1-P and Grant
2017SGR0929 (Generalitat de Catalunya),
by Grant RFBR 18-02-00264 and by SPbSU travel grants Id: 43179834, 43178702, 43188447, 41327333, 36273714, 41418623.
We express our gratitude to Angel G\'omez Nicola for stimulating discussions of how to implement chiral imbalance in ChPT.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,417
|
package ssh
type Config struct {
User string `toml:"user,omitempty" json:"user" long:"user" env:"SSH_USER" description:"User name"`
Password string `toml:"password,omitempty" json:"password" long:"password" env:"SSH_PASSWORD" description:"User password"`
Host string `toml:"host,omitempty" json:"host" long:"host" env:"SSH_HOST" description:"Remote host"`
Port string `toml:"port,omitempty" json:"port" long:"port" env:"SSH_PORT" description:"Remote host port"`
IdentityFile string `toml:"identity_file,omitempty" json:"identity_file" long:"identity-file" env:"SSH_IDENTITY_FILE" description:"Identity file to be used"`
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,453
|
import { Component, ElementRef } from '@angular/core';
export default class circleProgessBarComponent {
static get annotations() {
return [
new Component({
selector: 'atlui-circle-progress-bar, atlui-circle-progress-bar-sm, atlui-circle-progress-bar-lg',
template: `
<div [attr.data-percent]="value" class="circle-progress-bar circle-progress-bar-default">
<svg version="1.1" xmlns="http://www.w3.org/2000/svg">
<circle class="background-bar" fill="transparent" stroke-dasharray="565.48" stroke-dashoffset="0"></circle>
<circle class="bar" fill="transparent" stroke-dasharray="565.48" stroke-dashoffset="0"></circle>
</svg>
</div>`,
inputs: ['value']
})
];
}
constructor(elementRef) {
this.elementRef = elementRef;
}
ngAfterViewInit() {
this.circle = this.elementRef.nativeElement.getElementsByClassName("bar")[0];
this.circleBackground = this.elementRef.nativeElement.getElementsByClassName("background-bar")[0];
this.content = this.elementRef.nativeElement.getElementsByClassName("circle-progress-bar")[0];
this.style = this.elementRef.nativeElement.classList;
this.tag = this.elementRef.nativeElement.tagName;
this.content.classList.add(this.tag.toLowerCase());
var contentWidth = this.content.offsetWidth;
this.c = contentWidth / 2;
this.r = this.c - 10;
this.circle.setAttribute("r", this.r);
this.circle.setAttribute("cx", this.c);
this.circle.setAttribute("cy", this.c);
this.circleBackground.setAttribute("r", this.r);
this.circleBackground.setAttribute("cx", this.c);
this.circleBackground.setAttribute("cy", this.c);
this.style.forEach((style)=>{
this.content.classList.add(style);
});
if (typeof this.value !== "number" || this.value === "") {
this.value = 0;
}
if (this.value > 100) {
this.value = 100;
} else if (this.value < 0) {
this.value = 0;
}
this.circle.style.strokeDasharray = Math.PI * (this.r * 2);
this.circleBackground.style.strokeDasharray = Math.PI * (this.r *2);
this.updateBar();
}
ngOnChanges() {
if (this.circle) {
if (typeof this.value !== "number" || this.value === "") {
this.value = 0;
}
if (this.value > 100) {
this.value = 100;
} else if (this.value < 0) {
this.value = 0;
}
this.updateBar();
}
}
//Update the progression on the bar
updateBar() {
var percent = ((100 - this.value) / 100) * (Math.PI * (this.r*2));
this.circle.style.strokeDashoffset = percent;
}
}
circleProgessBarComponent.parameters = [ElementRef];
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,997
|
Send inquiries, questions, comments, news, etc. by mail to the Tree Farms & Nursery Trees email group.
If you are active in the Tree Farms & Nursery Trees Industry, add your name to the E-mail group to receive current news, views, etc..
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,140
|
{"url":"https:\/\/en.khanacademy.org\/math\/statistics-probability\/describing-relationships-quantitative-data\/scatterplots-and-correlation\/a\/correlation-coefficient-review","text":"If you're seeing this message, it means we're having trouble loading external resources on our website.\n\nIf you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.\n\n## Statistics and probability\n\n### Course: Statistics and probability>Unit 5\n\nLesson 2: Correlation coefficients\n\n# Correlation coefficient review\n\nThe correlation coefficient r measures the direction and strength of a linear relationship. Calculating r is pretty complex, so we usually rely on technology for the computations. We focus on understanding what r says about a scatterplot.\n\n### What is a correlation coefficient?\n\nThe correlation coefficient r measures the direction and strength of a linear relationship. Calculating r is pretty complex, so we usually rely on technology for the computations. We focus on understanding what r says about a scatterplot.\nHere are some facts about r:\n\u2022 It always has a value between minus, 1 and 1.\n\u2022 Strong positive linear relationships have values of r closer to 1.\n\u2022 Strong negative linear relationships have values of r closer to minus, 1.\n\u2022 Weaker relationships have values of r closer to 0.\nLet's look at a few examples:\nExample where r, equals, 1, which is perfect positive correlation\nExample where r, equals, 0, point, 5, which is weak positive correlation\nExample where r, equals, 0, which is no correlation\nExample where r, equals, minus, 0, point, 5, which is weak negative correlation\nExample where r, equals, minus, 1, which is perfect negative correlation\n\n### Practice problem\n\nMatch the correlation coefficients with the scatterplots shown below.\nstart color #1fab54, start text, S, c, a, t, t, e, r, p, l, o, t, space, A, end text, end color #1fab54start color #ca337c, start text, S, c, a, t, t, e, r, p, l, o, t, space, B, end text, end color #ca337c\nA scatterplot labeled Scatterplot A on an x y coordinate plane. Points rise diagonally in a relatively narrow pattern.\n|\nA scatterplot labeled Scatterplot B on an x y coordinate plane. Points rise diagonally in a relatively weak pattern.\nstart color #e07d10, start text, S, c, a, t, t, e, r, p, l, o, t, space, C, end text, end color #e07d10start color #11accd, start text, S, c, a, t, t, e, r, p, l, o, t, space, D, end text, end color #11accd\nA scatterplot labeled Scatterplot C on an x y coordinate plane. Points fall diagonally in a relatively narrow pattern.\nA scatterplot labeled Scatterplot B on an x y coordinate plane. Points fall diagonally in a weak pattern.\n\nWant to practice more problems like this? Check out this exercise on correlation coefficient intuition.\n\n## Want to join the conversation?\n\n\u2022 i dont know what im still doing here\n\u2022 How can we prove that the value of r always lie between 1 and -1 ?\n\u2022 Weaker relationships have values of r closer to 0. But r = 0 doesn\u2019t mean that there is no relation between the variables, right? I mean, if r = 0 then there is no linear correlation, but we still could have a non linear correlation?\n\u2022 Theoretically, yes. The r-value you are referring to is specific to the linear correlation.\n\u2022 I am taking Algebra 1 not whatever this is but I still chose to do this\n\u2022 Calculating the correlation coefficient is complex, but is there a way to visually \"estimate\" it by looking at a scatter plot? Or do we have to use computors for that?\n\u2022 Calculating the correlation coefficient is complex, but is there a way to visually\n\u2022 Yes on a scatterplot if the dots seem close together it indicates the r is high. If points are from one another the r would be low.\n(1 vote)\n\u2022 Is the correlation coefficient a measure of the association between two random variables?","date":"2023-03-25 05:17:53","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 21, \"math_alttext\": 0, \"mathml\": 0, \"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\": 0.676391065120697, \"perplexity\": 2052.64392140849}, \"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\/1679296945315.31\/warc\/CC-MAIN-20230325033306-20230325063306-00712.warc.gz\"}"}
| null | null |
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