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{"url":"https:\/\/www.networkpages.nl\/what-have-donald-trump-an-epidemiologist-and-a-rock-in-common\/","text":"# What have Donald Trump, an epidemiologist and a rock in common?\n\n#### How taking random decisions can help you win the elections\n\nOnce, when I was a high school student, a professor told me that one of the most fascinating aspects of science is that it usually helps you to connect things that you would never think to have connections. From a historical point of view, such connections were often related to mysticism or madness: not so many years ago you would have been accused to be an alchemist if you were claiming that the Moon has something to do with tides, not to mention what would have happened if you were admitting that our species descended from monkeys.\n\nIn this article, dear reader, I am going to show you in which way the development of your opinion during the last political issue, the spread of a virus among your acquaintances during the current pandemic, and the alignment of some particles lying inside the device from which you are reading this article are extremely comparable phenomena.\n\nPretend for a few minutes to identify yourself as Donald Trump, D. from now on. You would probably be obsessed with how to win back the voters you had a few years ago and, in doing so, you may ask yourself how the opinion of an average voter evolves as time goes by, starting from today up to the next election day. Here we can already point out a crucial aspect of D.\u2019s dilemma: he does not care about the background of their voters \u2013 what work they do, social class, interests, etc... \u2013, instead he focuses on their average social behaviors.\n\nIf D. were a Networks-scientist, he would know that in his draft there is important information that is missing: how the opinions change over time! Since otherwise he would be assuming that, as time goes by, none of the voters change their opinion, and this is not accurate enough even for our raging orange friend. If you think of yourself when you are about to vote how do you form an opinion on who to vote for? You read, you go to discussions, you follow the news, you discuss with your friends, family, and colleagues, and in the end, you take a decision. If all your friends, for example, are very enthusiastic about a candidate you may decide to vote for them. How can we understand how this very complex process of opinion formation works? In modern mathematics and physics, we often think of it as a rather random process, since there are many parameters affecting your decision.\n\nSuppose now that in D.\u2019s model each voter can have at each time step, say each day, one of two possible opinions: 0 or 1, red or blue, Republicans or Democrats, Fox News or CNN\u2013choose the couple that tickles your imagination the most. Given the opinions of all the voters today, you would like to know their opinions on the day of the elections. In other words, given the initial distribution of the opinions, you would like to know the value of all the opinions on the day of the elections. For this purpose, you need to specify how the opinions change (randomly) in time in your model, more precisely you need to write down the \u201crates\u201d, i.e. the speeds, at which each person can change their opinion at each time step. All the magic, Mr. Trump, is here: the way you tune your rates, making them less or more detailed, together with the opinion dynamics defines the model completely. The next example, which we will consider to be our reference model from now on, will provide a clear image of what I mean by the previous sentence.\n\nLet us consider now the following updating rule, describing the rates, that gives rise to a well-studied probabilistic model called the \u201cVoter Model\u201d. The interpretation of the updating rule is the following: suppose that tonight you are hanging out with a group of friends and, after drinking something, the debate focuses on the coming political issue, if you have a certain idea about who to vote and most of your friends have the opposite opinion then there is a chance that you may change your mind about it.\n\nLet us translate the latter heuristic in terms of our model. On each day (time step) pick randomly a person (colored dot) and select, again randomly, one of their neighbors\/acquaintances (trough the lines). If the opinions of the two acquaintances at that moment are equal then nothing happens, instead, if they are different, then the first-chosen person changes their mind and adopts the opinion of the randomly selected friend.\u00a0In our picture with dots and lines this corresponds to changing a red dot into a blue one and vice versa.\n\nAnimation of this updating rule\n\nIn the animation below you can experiment yourself with this updating rule. By clicking on \"Initialize weights\" some points will be colored green. How many points will be colored green can be chosen by the user, then this amount of points will be choses at random out of the five points and will be colored green. By clicking on \"Next Step\" you can see how the updating rule works. If you slide the cursor \"Timestep\" to the left you will see the steps in an animation.\n\nAgreed, but there are still several open questions. Is D. going to win the elections with this model?\u00a0Or better, is such a model somehow related to the opinion dynamics in the real world or we are just having fun with a useless toy? If not, is it possible to improve it? How? I promise you will find all the answers to these questions by the end of this article,\u00a0but before that let me justify the catchy title at the top of this page.\u00a0In particular, I want to convince you that it is possible to describe completely different phenomena with almost the same model that we have just built.\n\nAt this moment I encourage you to look back to the fascinating story of our American president and his model construction procedure. Consider again the dots mentioned before. We called them voters, but actually, the only quality that gives them this title is the understanding of what we called \u201copinion\u201d. Take now the same picture, colored dots with lines, but this time in place of opinions we attribute to the nodes one of two possible states: healthy or infected. We are only left to introduce two speeds, concerning the rate at which a person (dot) can pass from being infected to healthy and vice versa. We have successfully built an epidemic model. One last step. Did we really use in our construction that the dots have limbs, can breathe, or can type on a laptop?\n\nNot really. Indeed, no one prevents us from\u00a0interpreting\u00a0those dots as particles lying on some types of rocks -most of them used to build internal components of electronic devices- defining their magnetic dipole moments, or spin. Not familiar with that? No worries, the only relevant fact for us is that these particles appear with a feature that we can simplify by denoting it as orientation (spin). These spins can only be assigned two values: +1 for 'spin up', and -1 for 'spin down'. It is well-known that these particles have the tendency to interact with their neighbors. Or, in technical terms, they tend to\u00a0polarize.\u00a0This means that they flip the direction of their spin \u2013 from up to down, +1 or -1, or vice versa \u2013 at a certain speed. This speed depends on the number of atoms around them that have a different spin than themselves. Sounds familiar, right?\n\nAs you have noticed, I was able to describe completely different natural phenomena with almost the same model: with such a lens the diffusion of an\u00a0infection\u00a0in a group of people, the\u00a0polarization\u00a0of a piece of metal, and the spread of the\u00a0consensus\u00a0for a political party before the elections are not so far apart. Just like a good book, this model can give us different interpretations depending on the question we ask. This collection of models falls under the umbrella of a family of probabilistic models called Interacting Particle Systems (IPSs). Here, the term \u201cparticle\u201d can be interpreted as a real physical particle but it can also refer to voters and viruses. One of the most fascinating aspects of them is their impressive ductility and, besides their simplicity, they show extremely non-trivial behaviors.\n\nI am really proud of you dear reader, if you reached this point it means that you are genuinely interested in the questions posed a few lines ago about the fate of D.\u2019s strategy: is his construction useful? If not, can it be improved? Does it relate to an election dynamic? To answer them, let me briefly introduce you to some fancy concepts: remember the dots (voters, particles, infected people) mentioned before? Call them\u00a0vertices; the lines (acquaintances, particle interactions) connecting the nodes: edges; while D.\u2019s picture of dots and lines, is a\u00a0graph. Now you can tell your friends that you know what an IPS evolving on a graph is. Let us focus on the Voter Model on graphs, using the same updating rule explained before. First, we can notice that, since the graph is finite, surely sooner or later all the vertices will share the same opinion, the\u00a0consensus state. Thus, it should not be surprising that the main object of study is the so-called\u00a0consensus time, that is the expected time (in terms of the number of time steps in which we update the vertices' opinion) needed to reach the consensus state. This value mainly depends on the size of the network $N$, i.e., the number of voters. A small step back: clearly at this point, you may wonder why should we take care of the case in which\u00a0all\u00a0the citizens are voting for Trump. This is a good point, whereas such a connection is much more direct in the\u00a0infection\u00a0and\u00a0polarization\u00a0cases, in the Voter Model this is less evident and must be taken as an approximation of the real-world event that corresponds to \u201cwinning the elections\u201d.\n\nIt turns out that the\u00a0consensus time\u00a0is rather difficult to study in full generality since, as it is quite intuitive from our perspective, it strongly depends on the shape of the graph: a social network formed by a horizontal line - each voter has only two friends\/acquaintances\/neighbors - is significantly different to the complete graph, in which each voter is connected with all the other possible voters in the network. In fact, in the line graph, the number of updates of the opinions necessary to reach the consensus time is roughly $N^2$, where $N$ is the number of vertices \u2013 voters \u2013 of our graph, while in the complete graph the consensus time reduces drastically to roughly $N$. Hence a consesus is reached much faster when everyone can communicate with each other, than when playing chinese whispers.\n\nFigure: A consesus is reached much faster when everyone can communicate with each other, than when playing chinese whispers.\n\nThere are also some kinds of heterogeneous networks, i.e. more similar to the real-life ones, that have a consensus time of the order of $N^3$. What does this mean? Well, in his model, our friend D. has to do with an $N = 240$ million magnitude of eligible voters in the US. Therefore, making a mistake in the model assuming a network structure that is too different from the real one would potentially cost D. an $N^2 = (240 \\text{million})^2$ error in the computation of the consensus time!\n\nAt this moment it is wise to take a step back and visualize D.\u2019s dilemma from a different perspective. Is it just a toy model to have fun with? Or does it really have some connection with how opinions are formed? Unfortunately, the right answer is the less enthusiastic one: reality is much more complex than this. But how far are we from the right amount of complexity? Not so much, we only need to give D. the right hint. It will turn out to be: \u201cAdd more randomness\u201d. How? Do you remember the lines in D.\u2019s paper connecting the voters, the edges? Make them random. For every couple of nodes flip a coin, if the result is head then D. should draw an edge, and leave it blank otherwise. On the resulting realization of such\u00a0a random graph\u00a0run the Voter Model that we were playing with, that is implement all the opinion dynamics rules that we discussed before. The surprising result is that what comes out is much more related to a real-life election than the previous model!\n\nAnimation of the Voter Model\n\nThis animation starts with generating the graph according to the following procedure: we prescribe for each vertex the number of edges it should have, draw the corresponding 'stubs' and then randomly connect these stubs, which might lead to some self-loops and double edges. This is called the configuration model. Afterwards, the vertices are colored at random either blue or red, and then you can run the Voter Model.\u00a0By clicking on \"Next Step\" you can see how the graph is generated. If you slide the cursor \"Timestep\" to the left you will see the steps in an animation. By clicking on \"Initialize Weights\" and \"Next Step\" you can run the Voter Model and decide yourself how many vertices will be green.\n\nWhy? The intuition comes from the fact that\u00a0in this way, by adding randomness, D. can catch part\u00a0of the unpredictable behaviors regarding how groups of people change opinions on a given issue. A key idea here comes from the fact that one does not need to know all the details of the dynamics we are interested in (voting\/infection\/magnetization), but it may be sufficient to know that in a social network there are some structural properties, as communities or people with a lot of influence, which have a strong impact on these dynamics.\n\nUsing a similar argument, we could do a step further having in mind the following idea: most people, on average, may change their friends as time goes by, so why don\u2019t we try to model this behavior? The answer is that it is quite hard, but not impossible. This technique is called \u201crewiring\u201d and there is still a lot that we need to understand about it, indeed it is an extremely interesting research topic.\n\nIn conclusion, the adventure of our favorite president has a dignified ending. He only needs to arrange his dots-and-lines draft on a sheet of paper in such a way that his model is not too simple, and he could get accurate estimates of the time necessary to achieve the consensus time. Moreover, as we noticed at the beginning of the article, it should not be surprising that all the statements claimed in the last few paragraphs hold, with the right interpretation and small corrections, for the spread of a virus in a large population and the magnetization phenomena in ferromagnetic materials. It is worth remembering once again that we are still talking about models, an extreme simplification of reality that aims to give us some useful glimpse of its behavior.\n\nGoing back to D.\u2019s story, if he had an accurate model in his hands, he would probably be able to get some insights into how long it would take to win back the consensus and, perhaps, wait until then before running the next elections or even focus his election campaign on areas where it may be easier to spread his consensus. Rephrasing a famous quote: $$All models are wrong, but some of them could help you win the elections\u201d.\n\nThis line of research in NETWORKS is funded by the European Union\u2019s Horizon 2020 research and innovation programme under the Marie Sk\u0142odowska-Curie grant no. 945045. The featured image was made by Web Vectors by Vecteezy.\n\nWould you like to stay up to date whenever a new post appears on the Network Pages? Then subscribe to our mailing list, follow us on Twitter or on LinkedIn.","date":"2023-03-27 01:39:04","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 8, \"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\": 8, \"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\": 0.5226492285728455, \"perplexity\": 425.93680902353043}, \"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-2023-14\/segments\/1679296946584.94\/warc\/CC-MAIN-20230326235016-20230327025016-00129.warc.gz\"}"}
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#include <sys/types.h>
#include <sys/event.h>
#include <sys/socket.h>
#include <sys/time.h>
#include <stdio.h>
#include <stdlib.h>
#include <stdbool.h>
#include <unistd.h>
#include <string.h>
#include <errno.h>
#include <fcntl.h>
#include <syslog.h>
#include <asl.h>
#include <launch.h>
typedef struct {
aslclient client;
aslmsg logmsg;
} asl_context;
asl_context *asl_create_context(char *name, char *facility, uint32_t opts);
int asl_finalize_context(asl_context *ctx);
int asl_log_event(asl_context *ctx, uint8_t sev, char *msg);
int asl_add_output_file(asl_context *ctx, char *file);
int asl_close_output_file(asl_context *ctx, int fd);
int launchd_register();
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,111
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Q: Testing in-app purchase before going live So here is the scenario:
*
*You have already tested your in-app purchase code in the sandbox environment and everything went perfectly.
*Your app is now "Ready for Sale" and you want to make sure in-app purchases work flawlessly prior to actually letting your app hit the app store.
To try this, I downloaded a "pre-release" version from the App Store using a promo code, but when I try to purchase the unlocked version, it no longer finds the product like it did in the sandbox environment and my "transaction error" alert view will pop up saying the in-app purchase failed.
My question is this: Is the in-app purchase unavailable prior to the app going live on the App Store (in my scenario where I downloaded it early via a promo code)? For instance, when I set my availability date for Friday and the app is finally in the wild, will the in-app purchase show up and allow users to purchase it, assuming it worked in the sandbox?
Thanks for any insights! Greatly appreciated.
A: As it turns out, the app does indeed need to be available in the App Store before the in-app purchase will show up. So if you try and get a "pre-release" look like I did, it won't work until you actually release the app into the store via setting the availability date.
As soon as I let the app into the wild, in-app purchase actually worked a lot faster than it did in the sandbox environment. Very, very little lag.
Hope that helps someone in the future!
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{
"redpajama_set_name": "RedPajamaStackExchange"
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"They weren't taking electric vehicles seriously and they were just formal"
Financial Newspaper – Santiago
The British media justified its decision of character of the year, highlighting that the billionaire businessman has caused a historic change in the global automotive industry towards electric vehicles.
Elon Musk says he wants to be generous to his critics. But after years of battling an army of skeptics in the rest of the auto industry and on Wall Street, the Tesla CEO's frustration is hard to contain.
It has been almost a decade since the Model S proved that electric cars could compete with the best in style and performance, and four years since its Model 3 brought the technology to a wider market. During that time, Musk has not only had to build a market for electric vehicles almost single-handedly while fighting bankruptcy, but he has also waged a constant battle against Wall Street and held discussions with regulators.
This year has brought a vindication, as companies – from Ford to Volkswagen to Mercedes Benz – have unequivocally pledged their future to electric cars. Toyota was the last, this Tuesday, to announce an investment of US $ 35,000 million for the manufacture of these units.
"For a long time, the rest of the auto industry basically called Tesla and me stupid and fraudulent," Musk says in an interview with the Financial Times. "They said electric cars wouldn't work, you can't achieve range and performance. And even if you did that, no one would buy them."
Musk believes that climate change activists helped push manufacturers toward more sustainable technology. But he says there is a primary reason they are finally ready to go electric: "Until we started taking market share away from them in a significant way, they didn't react."
Elon Musk has been everywhere this year, and not just in a way that sets him apart as one of the most important, albeit controversial, business figures of recent times.
Armed with 66.3 million followers, he has used his hyperactive Twitter account to promote dogecoin, a cryptocurrency that initially started as a joke and whose name is a tribute to an internet meme starring a Shiba Inu dog. He has also continued to goad regulators, including the SEC, despite paying a $ 20 million fine in 2018 after the securities regulator accused him of committing securities fraud with his tweets.
Amid a stock market boom, his valuation broke the $ 1 trillion (billion) barrier this year, making Musk the world's richest person, despite shipping less than 2% of the cars. and new trucks of the world.
Yet behind the noise, speculative frenzy, and apparent rule-breaking, there is an achievement of great substance. The FT is naming Elon Musk its Person of the Year because he has sparked a historic shift in the global auto industry towards electric vehicles. Even if Tesla somehow collapsed next year – something that, unlike two years ago, no one is predicting now – Musk would have transformed one of the world's most important industries in ways that could have profound implications for governments, investors. and for the weather.
In an era often defined by new technology, Musk is claiming to be the most genuinely innovative entrepreneur of his generation.
Space bet
The results of Musk's unusual brand of taking risks and pushing boundaries have not been limited to cars. His private space company, SpaceX, brought human spaceflight back to American soil last year for the first time from the space shuttle. Its Starlink network is nearing the launch of the world's first commercial satellite broadband service (among those in Chile), and a new giant rocket that could change the economy of reaching orbit, called the Starship, is awaiting its first launch of proof.
Read the report from Diario Financiero here
We would like to thank the writer of this short article for this amazing web content
Bad Bunny premiered the music video for 'I wish you the best' in the style of The Simpsons
Bad Bunny surprised his fans with a musical gift at Christmas – La Vibra
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"redpajama_set_name": "RedPajamaCommonCrawl"
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\section{Introduction}
\label{sec:intro}
Low surface brightness (LSB) galaxies have long been known to
constitute an important segment of the galaxy population, but
challenges in identifying them in large panoramic surveys have
resulted in incomplete counts and insufficiently accurate characterization of their
intrinsic properties \citep{Sandage1984,Dalcanton1997,Impey1988,Galaz2011,Greco2018}. In
the field, LSBs are most often systems of unusually large
size and relatively inefficient star formation that populate
predominantly the faint-end of the galaxy stellar mass function. Their
extreme properties make them as challenging to observe as they are to account
for in current models of galaxy formation.
The advent of LSB-sensitive surveys and telescopes with specialized
optics such as Dragonfly \citep{Abraham2014} have rekindled interest
in LSBs and in the faint tail of the surface brightness
distribution. New cluster surveys, together with tailored re-analysis
of existing photometric data, have resulted in the identification of a
``new'' population of dwarf (i.e., $M_*<10^9\, M_\odot$) LSBs in
clusters; these systems have spheroidal morphologies, are
predominantly gas free, and are mainly red in color. Using a
conventional definition of limiting effective surface brightness
$\mu_{\rm eff} \geq 25$ mag/asec$^2$ ($g$-band) and an effective radius
$R_e >0.7$-$1.5$ kpc these systems have
become known as ``Ultra-Diffuse Galaxies'' (UDGs)
\citep{vanDokkum2015,Koda2015,Yagi2016,Munoz2015,Mihos2015,
vanderBurg2016, Venhola2017}.
Originally thought to be a distinct population, UDGs are now believed
to constitute the high-end of the size distribution of dwarf galaxies
in clusters and the field. Indeed, the $\mu_{\rm eff}>25$ mag/asec$^2$
criterion designates many ordinary LSB field galaxies as UDGs, a feature
that has often confused the discussion of whether UDGs are a ``cluster
phenomenon'' or just the faint end of the galaxy surface brightness
distribution.
Furthermore, systems of much fainter surface brightness
than any cluster UDG have long been known. For example, most
satellites of the Milky Way (MW) and Andromeda (M31) have much fainter
$\mu_{\rm eff}$ than any Dragonfly UDG in the Coma cluster.
Indeed, extreme
objects such as Crater II, And XIX, and Antlia II
\citep{Torrealba2016,McConnachie2008,Torrealba2019} have up to
$100\times$ fainter effective surface brightness than Coma Dragonly
UDGs \citep[see; e.g.,][]{Fattahi2019}.
Although UDGs are clearly not just a cluster phenomenon, this does not
mean that clusters are not special environments that may radically
transform galaxies and somehow favor or induce the formation of
UDGs. Indeed, it is well established that dwarf galaxies in clusters
span a dramatic range in sizes (Fig.~\ref{fig:obs}), from the
ultra-compact dwarfs common in Virgo \citep[UCDs,][]{Liu2015} to the
UDGs in Coma \citep{Yagi2016}, Virgo \citep{Mihos2015}, and Fornax
\citep{Venhola2017}\footnote{We have assumed a stellar mass to light
ratio $\gamma=1.96$ for $r$-band \citep{Carleton2019} and $\gamma=1$
for $V$-band to estimate stellar masses.}. Explaining this enormous
range in dwarf galaxy sizes remains a difficult challenge for galaxy
formation models.
Regarding UDGs, three questions are still hotly debated: the mass of
their surrounding halos; the origin of their large sizes; and their
relation to the environment. Halo masses estimates are necessarily
indirect, but it has been argued that the high stellar velocity
dispersions of systems like DF44 and DFX1 indicate halos as massive as
$\sim 10^{12}\, M_\odot$, suggestive of ``failed'' Milky-Way like
galaxies \citep{vanDokkum2017}. On the other hand, several other UDGs
have lower velocity dispersions, consistent with the low mass halos
expected given their stellar mass \citep{Zaritsky2017, Toloba2018}.
The globular cluster populations of UDGs are also quite diverse, and have
been used to argue for a wide range of halo masses
\citep{Beasley2016, Peng2016, Lim2018,Toloba2018}. Finally, at least
2 UDGs have velocity dispersions so small that they may be consistent
with having no dark matter halo at all
\citep{vanDokkum2018a,vanDokkum2019}. This issue, and their
interpretation, has generated a healthy discussion that is still
ongoing \citep[see e.g., ][]{Martin2018,Laporte2019,Trujillo2019}.
Given these mixed properties, there is growing consensus that
more than one formation path may be needed to explain the diversity of
UDGs.
Several UDG formation scenarios have been proposed, and may be grouped
as ``internal'' or ``external''. Internal scenarios include those
where UDGs correspond to the high-spin tail of the galaxy angular
momentum distribution \citep{Amorisco2016}, although this is
disfavoured by the spheroidal (i.e., non-disk) morphology of some
cluster UDGs \citep{Roman2017b,Eigenthaler2018}. Interestingly, however, the
simulations of \citet{Tremmel2019} find some support for the idea that
dwarf-mass halos with larger than average gas angular momenta may
explain the UDG population. Another scenario posits that most dwarfs
go through expansion and contraction phases driven by sudden outflows
of gas following episodes of active star formation \citep[see;
e.g.,][]{Navarro1996,Pontzen2012}. Dwarfs that are caught at the
expansion phase during cluster infall may be ``frozen'' in that stage,
leaving behind a dwarf of unusually low surface brightness
\citep{Chan2018,DiCintio2017}. This is a viable (albeit perhaps
contrived) scenario that awaits confirmation from detailed simulations
that include the evolution of such dwarfs in the cosmological setting
of cluster formation.
The most promising external scenario concerns the transformation
of galaxies as a consequence of the mean tidal field of the
cluster. Simulations of tidal stripping of galaxies embedded in extended dark matter
halos show that this process may lead to remnants of much lower
surface brightness than their progenitors
\citep{Penarrubia2008,Errani2015,Tomozeiu2016}. This transformation
may be aided by the loss of the gas component due to ram pressure with
the cluster's gaseous component \citep{Arraki2014,Frings2017,Yun2019}. The
final outcome are puffed-up, red, quiescent objects that resemble UDGs
\citep{Safarzadeh2017,Roman2017b}.
The formation of UDGs via tidal transformation has recently been
studied by \citet{Carleton2019}, who argue that turning a typical
spheroidal galaxy into a UDG can only be accomplished if the dark
matter halos that host field dwarfs have large constant density
``cores'', unlike the cusps expected in the cold dark matter (CDM)
scenario. This is because dwarfs in cored halos are more easily
disturbed and evolve more dramatically in surface brightness than
dwarfs in cuspy, NFW halos \citep{Navarro1996,Navarro1997}. The
\citet{Carleton2019} study, however, did not consider normal LSB
galaxies as potential UDG progenitors, an assumption that is not
clearly justified and that may compromise the general applicability of
their conclusion.
\begin{figure}
\includegraphics[width=\columnwidth]{figs/mstr_reff_obs_paper.png}
\caption{Effective radii $R_{\rm eff}$ as a function of
stellar mass $M_*$ for a compilation of observed galaxies from
the literature. Cyan and red points are used to indicate
SDSS galaxies bluer and redder than
$(g$-$r)_{\rm lim} = 0.61 + 0.052 (\log_{10}(M_*/M_\odot) -
10.0)$, respectively. Galaxies from the SPARC compilation \citep{Lelli2016} are shown with large
blue circles. The loci of GAMA early-type galaxies are shown by the grey
curves, including scatter. Green and red squares indicate
galaxies in the Fornax and Coma clusters, respectively
(references in the legend). Yellow points indicate
ultra-compact dwarfs in Virgo. UDGs are defined as in
\citet{Koda2015}: these are galaxies with effective surface
brightness ($\Sigma_{\rm eff}=(1/2)M_*/R_{\rm eff}^2$)
lower than $\Sigma_{UDG} \sim 2.9 \times 10^7 \, M_\odot$/kpc$^2$
and higher than $\Sigma_2 \sim 0.2 \times 10^7 \, M_\odot$/kpc$^2$
(or, equivalently, $\mu \sim 25$-$28$ $r$-mag/arcsec$^2$).
An additional
criterion $R_{\rm eff}>1.5$ kpc is also introduced in order
to exclude ``normal'' more compact dwarfs \citep{vanDokkum2015}. UDGs include data
from \citet{Yagi2016,Venhola2017,Liu2015,Mihos2015}.
Gray diamond-like symbols indicate Local Volume
dwarfs, with highlighted systems of lower surface
brightness than typical UDGs, such as Antlia II, Crater II,
Andromeda XIX and WLM.}
\label{fig:obs}
\end{figure}
We address these issues here by supplementing the cosmological
simulations of galaxy cluster formation from the IllustrisTNG
simulation suite \citep{Pillepich2018b} with an analytical model of
the tidal disruption of galaxies in the cluster potential. Our model
adopts the results of \citet{Errani2015} and, although it is similar
in approach to that of \citet{Carleton2019}, it also differs from it
in a few important respects. For example, we shall consider the whole
field galaxy population as potential UDG progenitors, and not just
compact early-type galaxies. In addition, we shall use the baryonic
version of the TNG simulations, and not just the dark-matter-only run
of Illustris-1. This is important, as the loss of baryons in dwarf
galaxies (which can only be followed in hydrodynamical runs) leads to
less tightly-bound systems more prone to tidal disruption
\citep{Brook2015,Sawala2015,Sawala2016}. In addition, the assembly of
the central cluster galaxy deepens the potential and may enhance tidal
stripping \citep{Donghia2010,Garrison-Kimmel2017}. Our analysis is
similar to that of \citet{Fattahi2018} (who focused on the effects of
tidal stripping on satellites of MW-like systems), but scaled up in
mass to the ``satellites'' of massive clusters.
This paper is organized as follows. Sec.~\ref{sec:method} introduces
the TNG simulations and galaxy samples and describes the tidal
evolution model applied in this work. Sec.~\ref{sec:results} presents
our main results, including the formation of UDGs and their
present-day predicted properties. In Sec.~\ref{sec:concl} we summarize
and discuss the implications of our results.
\section{Methods}
\label{sec:method}
\subsection{The simulations}
We use the cosmological hydrodynamical simulation Illustris-The Next
Generation \citep[TNG for
short;][]{Pillepich2018b,Nelson2018,Springel2018,Marinacci2018,Naiman2018,Nelson2019}. The
TNG suite of simulations includes several box sizes of varying
resolution, as well as runs with and without baryons.
In this work we use the TNG100-1 run, which is the highest resolution
box with 107 Mpc on a side that includes the treatment of baryons. The
simulations follow the evolution of $2 \times 1820^3$ DM and gas
elements from a starting redshift $z\sim 127$ until today. TNG assumes
a $\Lambda$ cold dark matter cosmology ($\Lambda$CDM) with parameters
consistent with Planck XIII results \citep{Planck2016}:
$\Omega_{\rm M}=$ $\Omega_{\rm dm} + \Omega_{\rm b} = 0.3089$,
$\Omega_{\rm b}=0.0486$, $\Omega_\Lambda = 0.6911$, Hubble constant
$H_0=67.74$ km s$^{\-1}$ Mpc$^{-1}$, and a power spectrum with
primordial index $n_s=0.9667$ normalized to $\sigma_8=0.8159$.
Gravity and baryons are followed using the {\sc arepo} code
\citep{Springel2010}. The baryonic treatment is largely based on the
code used for its predecessor simulation suite, Illustris
\citep{Vogelsberger2013, Vogelsberger2014b,Vogelsberger2014a, Genel2014,Nelson2015}, with modifications
described in \citet{Pillepich2018a}.
In brief, TNG follows self-consistently the heating and cooling of the
gas down to $10^4$ K with an effective equation of state for gas above the star formation threshold
$n_{\rm SFT} = 0.2 \; \rm cm^{-3}$. Such gas can turn into stars with a local
efficiency of $1\%$, after which stellar evolution models calculate
the mass and metallicity of the stellar particles.
Stellar feedback from winds of massive stars and supernova explosions
is coupled kinematically to the gas, with an efficiency that varies
according to the metallicity of the stars. AGN feedback including the
fast- and slow- accretion rates of gas onto supermassive black holes is
modeled with a mixture of thermal and kinetic energy deposition modes
following \citet{Weinberger2017}. Finally, ideal MHD is coupled to the
hydrodynamics to track the properties of magnetic fields
\citep{Pakmor2011, Pakmor2013}.
At the resolution level of TNG100-1, the particle masses are
$m_{\rm gas} \sim 1.4 \times 10^6$ and $m_{\rm dm}=7.5 \times 10^6$$\; \rm M_\odot$\
for baryons and dark matter, respectively. The gravitational softening
for collisionless elements like dark matter and stars is
$\epsilon \sim 740$ pc or better. The highest resolution gas
elements can have gravitational softenings as small as $\sim 185$
(comoving) pc.
Galaxies in the simulations are identified using {\sc subfind}
\citep{Springel2001,Dolag2009} on Friends-of-Friends (FoF) groups
\citep{Davis1985} identified with a linking length set to $0.2$ the
mean interparticle separation. The time evolution of galaxies and
halos is followed by using the {\sc SubLink} merger trees
\citep{rodgomez2015}.
\subsection{Simulated galaxy clusters}
\label{ssec:sample}
We select the most massive halos in the TNG100-1 simulated volume,
with virial\footnote{We adopt a virial definition where the averaged
spherical density within the virial radius, $r_{200}$, is $200$
times the critical density for closure.} mass
$M_{\rm 200} \geq 10^{14}\, M_\odot$ at $z=0$. This criterion selects 14 objects
(the most massive has $M_{200}=3.79 \times 10^{14}$$\; \rm M_\odot$) with masses
comparable to that estimated for the Virgo cluster.
We consider for our analysis all galaxies associated with the FoF groups of these
clusters that have a minimum stellar mass
$5 \times 10^7$$\; \rm M_\odot$ (or, equivalently, at least 25 stellar
particles). We also impose a similar minimum dark matter mass to
ensure that spurious self-gravitating baryonic clumps that are occasionally
identified by {\sc subfind} are excluded from our sample. A total of
$4,850$ galaxies spread over 14 clusters make up the full simulated
galaxy sample.
\begin{figure}
\includegraphics[width=\columnwidth]{figs/fit_sparcs_surfb.png}
\caption{Field galaxies from the SPARC sample
\citep{Lelli2016}. The sample may be split into a low (cyan)
and high (red) surface density sequence, each following a
slightly different $M_*$-$R_{\rm eff}$ relation, as fit by
the cyan/red lines. The fraction of galaxies in each
sequence as a function of $M_*$ is shown in the bottom
panel. We use the stellar mass of infalling galaxies into
simulated TNG clusters to assign them stellar sizes
following these two sequences (adjusted by redshift of
infall according to $(1+z_{\rm inf})^{-0.75}$, see text for
details). For comparison, the early-type sequence from GAMA
used by \citet{Carleton2019} to assign sizes is also
indicated in gray. The UDG selection criteria is
the same as in Fig.~\ref{fig:obs}.}
\label{fig:sparc}
\end{figure}
\subsection{The mass and size of simulated galaxies}
To study UDGs in the simulated galaxy population we need robust
estimates of stellar mass and half-mass radii. The TNG simulations
reproduce the galaxy stellar mass function remarkably well
\citep{Pillepich2018b}, suggesting that stellar masses are captured
correctly by the simulations. Galaxy sizes, on the other hand, are
much less robust, and are strongly affected by a number of numerical
artifacts and limitations, such as the use of a finite gravitational
softening, a limited number of particles, a relatively high threshold
for star formation, and an effective equation of state. These impose
effective minima on the size of simulated galaxies that affect dwarfs
in particular and that severely limit our ability to study UDGs
directly using simulation data.
We circumvent this limitation by assigning to each of our galaxies, at
the time of first cluster infall ($t_{\rm inf}$), a stellar
mass-dependent effective radius drawn from a realistic distribution of
galaxy sizes. Infall times are defined as the last time a galaxy
becomes a satellite of a larger system. This can be the time galaxies
join their clusters or an earlier time if they infall as part of a
galaxy group. Note that only the stellar radii are chosen that way;
stellar and dark halo masses are measured directly from the simulation
at $t_{\rm inf}$. The tidal evolution model we describe below
(Sec.~\ref{ssec:tidalmodel}) allows us to compute their final stellar mass, effective
radii and velocity dispersion at $z=0$.
Effective radii are drawn assuming that they follow the
(redshift-corrected) empirical relations for late-type galaxies (including
LSBs) taken from the SPARC compilation \citep{Lelli2016}. This
assumption is consistent with the idea that before joining a larger
system, each galaxy is a star-forming galaxy in the field.
The stellar mass vs size relation of SPARC galaxies is shown in
Fig.~\ref{fig:sparc}, assuming a mass to light ratio of $0.5$ in the
3.6$\mu$ band \citep{Lelli2016}. This relation shows considerable scatter in the
projected stellar effective radius, $R_{\rm eff}$, at fixed stellar
mass. Two groups or ``sequences'' of ``high'' and ``low'' surface
brightness may be easily identified on each side of a dividing value
of $\Sigma_{\rm lim}=(1/2)M_*/R_{\rm eff}^2= 2.9 \times 10^8\;
M_\odot$ kpc$^2$, indicated by the thick black dashed line.
The high-$\Sigma$ group dominates the massive end; the low-$\Sigma$
group contains most of the dwarfs. The fraction of galaxies that
belong to each sequence varies smoothly as a function of $M_*$, as
shown in the bottom panel Fig.~\ref{fig:sparc}. The low- and
high-$\Sigma$ sequences follow slightly different relations, which we
approximate as power laws: $\log_{10}(R_{\rm
eff}/$kpc$)= a \log_{10}(M_*/M_\odot) + b$, with
$(a_{\rm low},b_{\rm low})=(0.38,-2.94)$ and
$(a_{\rm high},b_{\rm high})=(0.39,-3.51)$ respectively. These fits
are shown in Fig.~\ref{fig:sparc} with cyan and red lines,
respectively. The rms scatter in radii for both of these relations is
similar, $\sigma_{\rm log(r)} \sim 0.15$.
The grey curves in Fig.~\ref{fig:sparc} indicate the location of
early-type galaxies from the GAMA survey, including scatter
\citep[][]{Lange2015, Liske2015}. These overlap well the regime of
high-$\Sigma$ disks, but have much smaller sizes than the low-$\Sigma$
sequence. These are the objects most likely to turn into UDGs by
tidal effects, and leaving them out, as in \citet{Carleton2019}, who
only considered GAMA spheroids, can have a strong effect on the
results.
Our modeling uses the two-sequence description of SPARC galaxies to
assign sizes to all galaxies at infall time. In practice, we proceed
as follows. Using its stellar mass at $t_{\rm inf}$, each galaxy is
randomly assigned to the low- or high-$\Sigma$ sequences by following
the SPARC fractions shown in the bottom panel of
Fig.~\ref{fig:sparc}. Once this sorting is done, we use the individual
power-law relations for the low- or high-$\Sigma$ sequence (plus
scatter) to assign an infall value of $R_{\rm eff}$ to each
galaxy. The projected effective radius is then converted into a 3D
stellar half mass radius, denoted with $r_h$ hereafter, by assuming:
$r_h = (4/3) R_{\rm eff}$.
Additionally, because the infall redshift, $z_{\rm inf}$, can be quite
early, and galaxy sizes are known to decrease with redshift, we
correct the half mass radius $R_{\rm eff}$ at infall time by
$(1 + z_{\rm inf})^{-0.75}$ \citep{Paulino-Afonso2017}. This
procedure yields a population of cluster galaxies characterized by
their infall times and stellar masses (from the simulation) as well as
their sizes (computed as described above). The evolution of the dark
halos surrounding these galaxies may be followed to $z=0$ in order to
estimate the effects of cluster tides on the stellar component, as
described below.
\subsection{Tidal evolution model}
\label{ssec:tidalmodel}
\begin{figure}
\includegraphics[width=\columnwidth]{figs/tidalmodel_errani.png}
\caption{Evolution of stellar mass (orange), half-mass radius
(green), and stellar velocity dispersion (purple) for a
galaxy embedded in a cuspy NFW halo undergoing tidal
disruption. Model taken from \citet{Errani2015}. }
\label{fig:errani}
\end{figure}
Our tidal evolution model of cluster galaxies makes use of the work of
\citet{Penarrubia2008} and \citet{Errani2015}, who showed that the
evolution of a stellar system deeply embedded in the gravitational
potential of a dark matter halo may be described in terms of a single
parameter. This parameter may be taken to be the {\it total} bound
mass fraction remaining within the original (3D) half mass radius of the
stars, $r_{h,0}$. Since each galaxy halo can be tracked through time, the total
mass lost by $z=0$ from within the initial (i.e., at infall) half-mass
radius can be computed directly from the
simulation. This mass loss can then be used to predict the changes in
stellar mass and effective radius of each individual cluster galaxy using the
effective radius assigned at infall. The mass loss estimates are robust, since
even the faintest galaxies in our sample inhabit fairly massive halos,
typically resolved with tens of thousands of dark matter
particles.
The tidal evolution model, which has been calibrated using numerical
simulations, may thus be used to predict the stellar half mass radius,
$r_h/r_{h,0}$, stellar mass, $M_*/M_{*,0}$ and velocity dispersion,
$\sigma_*/\sigma_{*,0}$, at the present day in units of their initial infall
values (``0'' subscript). The results are adequately approximated by a simple formula:
\begin{align}
g(x) = \frac{2^\alpha x^\beta}{(1 + x)^\alpha}
\label{eq:errani}
\end{align}
\noindent
where $x$ is the fraction of mass that remains bound within $r_{h,0}$
and ($\alpha$,$\beta$) are constants given in Table~\ref{tab:errani}
and taken from \citet{Errani2015}.
Fig.~\ref{fig:errani} illustrates the resulting tidal evolution tracks
when applying Eq.~\ref{eq:errani} as a function of $x$. Despite its
simplicity, these simple formulae have been shown to reproduce quite
well the tidal evolution of satellite galaxies in cosmological
hydrodynamical simulations of galaxy formation \citep[see, e.g, Fig.~3
of][]{Fattahi2018}.
According to Fig.~\ref{fig:errani}, the main effects of tidal
stripping on the stellar component of a galaxy is to gradually reduce
its stellar mass while keeping its size approximately constant. This
implies that heavily stripped galaxies will be systems of much lower
surface brightness than expected at given stellar mass. Stripping also
reduces the stellar velocity dispersion, but to a somewhat lesser
extent.
The fits using Eq.~\ref{eq:errani} have only been validated by
simulation results for the range $0.1<x<1$. This range is indicated by
thick line types in Fig.~\ref{fig:errani}. Many of our cluster
galaxies have experienced mass losses that exceed the lower bound of
that range, with a sizable fraction ($5\%$) in the range
$0.01<x<0.1$. We shall assume that the same fitting formulae apply in
that regime, although we caution that the properties of such ``extreme
tidal remnants'' should be considered as preliminary until much higher
resolution simulations are available.
Finally, it is certainly possible that some cluster galaxies may have
experienced tidal losses even more extreme than $x\sim 0.01$, the
minimum we are able to track at the resolution of the TNG100-1
run. These systems would be missed in our simulated sample, which is
based on galaxies that survive\footnote{We note that recent work
argues that current N-body simulations may substantially
underestimate the number of surviving substructure in clusters
\citep{vandenbosch2018a, vandenbosch2018b}.} to $z=0$, but they may
very well exist in true clusters. Their existence and, presumably,
their extreme properties cannot be discounted.
To summarize, we model the tidal evolution of individual galaxies by
measuring the total mass at $z=0$ within the half mass radius {\it
assigned at infall} (as described in Sec.~\ref{ssec:sample}) and
comparing it with its value at infall. Eq.~\ref{eq:errani} then allows
us to predict the stellar mass, size and velocity dispersion at
present day of all cluster galaxies. We require a minimum of $10$ dark
matter particles at $z=0$ within the infall half-mass radius to
include a galaxy in our analysis.
\begin{table}
\caption{Coefficients for the tidal evolution model in
Eq.~\ref{eq:errani}, from \citet{Errani2015}.}
\setlength{\tabcolsep}{5pt}
\begin {tabular*}{5.5cm}{{l} *{4}{c} }
\hline
& & $M_{*}/M_{*, 0}$ & $\sigma/ \sigma_0$ & $r_{\rm h}/
r_{\rm h, 0}$ \\
\hline
$\alpha$ & & 3.57 & -0.68 & 1.22 \\
$\beta$ & & 2.06 & 0.26 & 0.33 \\
\hline
\end{tabular*}
\label{tab:errani}
\end{table}
\section{Results}
\label{sec:results}
\begin{figure*}
\includegraphics[width=\columnwidth]{figs/errani_mass_reff.png}
\includegraphics[width=\columnwidth]{figs/errani_mass_reff_initialdistr.png}
\caption{{\it Left:} predicted stellar mass-size relation for
simulated galaxies in$M_{\rm vir} \sim 10^{14}$$\; \rm M_\odot$ TNG clusters,
evolved according to our tidal evolution model. Galaxies assigned
originally to the low and high surface brightness sequences are
shown in cyan and red, respectively. Systems consistent with a
commonly used UDG definition are shown with large blue symbols
within the shaded box. {\it Right}: same as left panel, but for the
initial (i.e., at infall) $M_*$-$r_{\rm eff}$ relation of simulated
galaxies. The population of UDGs at $z=0$ may be divided into two
groups: galaxies that were originally 'born' consistent with the UDG
definition (``Born UDGs'', dark blue) and galaxies that were more
massive in the past and evolve through tidal effects into objects with
UDG properties today (``Tidal UDGs'', light blue).}
\label{fig:mass_size}
\end{figure*}
\subsection{UDGs in clusters}
The result of the procedure described above, applied to all 14
clusters, is shown in the left panel of Fig.~\ref{fig:mass_size}. As
in Fig.~\ref{fig:sparc}, we use cyan and red symbols to indicate
galaxies that were assigned to the low- and high- surface brightness
sequences, respectively. The red shaded region indicate the UDG
criteria, defined as in \citet{Koda2015}: $\Sigma_{\rm eff}=(1/2)M_*/R_{\rm eff}^2
< \Sigma_{\rm UDG}=2.9 \times 10^7 \, M_\odot$/kpc$^2$ (black dashed line), plus
two additional boundaries: $\Sigma_{\rm eff}>\Sigma_2 > 0.2 \times
10^7 \, M_\odot$/kpc$^2$ (i.e., the equivalent
of r-band surface brightness 28.5 mag/arcsec$^2$) and $R_{\rm
eff}>1.5$ kpc,
shown by dash-dotted red lines. The criterion
$R_{\rm eff}>1.5$ kpc is introduced in order to exclude
more compact, ``normal'' dwarfs \citep{vanDokkum2015}. On average,
$6$-$30$ galaxies fall in the UDG regime within the virial radius of
each cluster (a median of $17$ UDGs per cluster), and the numbers double if we consider the whole extent
of the cluster FoF groups, which extends slightly beyond the virial
boundaries of the cluster.
The main takeaway point is that UDGs have no trouble ``surviving'' in
the harsh tidal environment of a cluster in sizeable numbers. Indeed,
the number of simulated UDGs per cluster is in reasonable agreement
with the number of UDGs identified in Virgo (Lim et al., {\it
in prep.}) (although these authors use a slightly different selection
criteria based on the overall shape of the $M_*$-size relation)
as well as with the results of
\citet{vanderBurg2016,vanderBurg2017}, who report of order
$\sim 20$-$30$ UDGs for clusters of comparable virial mass.
We note that this result does {\it not} require that dark matter halos have
``cores'', as argued recently by \citet{Carleton2019}. Their analysis
and modeling is actually similar to ours, although they use the
dark-matter-only version of an earlier version of the Illustris
simulation suite. The main difference between our work and theirs is
that they only consider early-type galaxies from GAMA as potential
progenitors for UDGs. These galaxies are more compact than the SPARC
population we adopted here, and, hence, much more resilient to
stripping (see Fig.~\ref{fig:sparc} for a direct comparison). That
resilience is what led \citet{Carleton2019} to conclude that ``cored''
halos are needed to explain the UDG cluster population. We would argue
that our approach, which includes both low and high surface brightness
galaxies as potential UDG progenitors, samples more fairly the field
population and is, therefore, more realistic.
\subsection{Origin of UDGs}
Where do the UDGs shown in the left-hand panel of
Fig.~\ref{fig:mass_size} originate from? This may be gleaned from the
cluster galaxy properties {\it at infall}, which are shown in the
right-hand panel of the same figure. Galaxies that end up as UDGs at
$z=0$ are shown with large blue circles. Two different hues are used;
dark to denote those that were within the UDG boundaries already at
infall, and a lighter hue for those that evolved to become UDGs as a
result of tidal stripping. In what follows, we refer to these two
populations as ``born UDGs'' (B-UDGs) and ``tidal UDGs'' (T-UDGs),
respectively.
The two panels of Fig.~\ref{fig:mass_size} illustrate a couple of
interesting points. One is that many cluster UDGs are relatively
unevolved remnants of field LSB galaxies that fell into the cluster
and, presumably, lost their gas and stopped forming stars. They may
have been affected by the cluster tidal field, but only mildly. This B-UDG
population make up $\sim 50\%$ of $z=0$ UDGs in our cluster simulation. A
second point is that the remainder, T-UDGs, make up a substantial
fraction of the UDG population, and originate from a varied mixture of
galaxies which were typically much more massive at infall than today. This mixed
origin can help to explain the wide spectrum of observed UDG properties
referred to in Sec.~\ref{sec:intro}.
\begin{figure}
\includegraphics[width=\columnwidth]{figs/udgs_mass_sigma_paper.png}
\caption{Velocity dispersion-stellar mass relation
for simulated galaxies in TNG clusters after accounting for their
tidal evolution. For reference, the observed (baryonic) Tully Fisher
relation from \citet{McGaugh2012} is shown with a grey band, with rotation velocities scaled by
$\sqrt{3}$ to make them comparable to velocity dispersions. Candidate UDGs are color coded according to
their retained bound mass fraction since infall (see color
bar). UDGs that have retained most of their mass
(``born UDGs'') trace the general cluster galaxy population (shown in grey).
``Tidal UDGs'', i.e.,
those formed by significant stripping of a more massive galaxy (low
mass bound fractions---in blue), have systematically lower velocity
dispersion at given stellar mass. Data for a number of individual
UDGs from the literature are
shown in purple. While some UDGs such as Dragonfly-44
are consistent with the ``born UDG'' scenario, others such as VLSB-D
are candidates to be tidally-formed. See text for a more detailed
discussion of extreme objects such as DF2 and DF4.}
\label{fig:udgs_sigma}
\end{figure}
\subsection{UDG velocity dispersions}
One of the most controversial aspects of UDGs has been their dark
matter content. This is inferred from the line-of-sight velocity
dispersion of stellar tracers, measured through spectroscopy of
stellar absorption lines, or from the kinematics of the globular
cluster systems. Published studies report a wide range of velocity
dispersions for UDGs, from $\sigma_{los} \sim 4$ km/s in objects like DF2
and DF4 \citep[so low that it may imply little to no dark matter,
][]{vanDokkum2018a,vanDokkum2019,Danieli2019} to as high as $\sim 50$
km/s for the globular cluster population ($\sim 33$ km/s for the stellar
dispersion at $R_{\rm eff}$) of DF44, which suggests a fairly massive dark matter halo
\citep{vanDokkum2016, vanDokkum2019_DF44}.
We explore the velocity dispersion of simulated UDGs in
Fig.~\ref{fig:udgs_sigma}. Velocity dispersions are estimated from the
circular velocity at the stellar-half mass radius, assuming the
relation for dispersion-supported objects from \citet{Wolf2010}; i.e.,
$\sigma_{\rm los} = V_{\rm circ} / \sqrt{3}$. $V_{\rm circ}$ is
calculated directly from the particle data in the simulation by
measuring the dark matter and stellar mass enclosed within the evolved
($z=0$) half-mass radius of each galaxy. An additional 10\% downward
scaling is applied to all TNG velocities in order to correct for a
small systematic offset between the simulated and observed Baryonic
Tully-Fisher (BTF) relation of {\it field} galaxies \citep{McGaugh2012}.
Fig.~\ref{fig:udgs_sigma} shows the stellar mass-velocity dispersion
relation for all cluster galaxies at $z=0$ (grey points), as well as
for the UDGs (large circles, colored by the bound mass fraction
retained by each object). Reddish symbols correspond to
B-UDG systems, whose dark matter content within the stellar half mass
radius has been relatively unaffected by tides. Blue circles, on the
other hand, denote mainly T-UDGs, where tides have led to large mass
losses and, consequently, large changes in stellar mass and
velocity dispersion.
\begin{figure}
\includegraphics[width=\columnwidth]{figs/udgs_mass_metal_paper.png}
\caption{Stellar metallicity-mass relation for simulated galaxies in
TNG clusters after accounting for their tidal evolution. The color
scheme is the same as in Fig.~\ref{fig:udgs_sigma}. ``Tidal'' UDGs
are outliers in this relation, with higher metallicities at given
stellar mass than ``born UDGs" and the general cluster
population. Our results indicate that the high metallicities
measured for some UDGs \citep[see for instance
][]{Ferre2018,RuizLara2018} may be indicative of a tidal origin for
these objects.}
\label{fig:udgs_metal}
\end{figure}
Cluster galaxies relatively unaffected by tides, as well as B-UDGs,
agree well with the observed BTF (grey shaded band labeled McG12 in
Fig.~\ref{fig:udgs_sigma}, where maximum rotation velocities have been
scaled by $\sqrt{3}$ factor to turn them into $\sigma_{\rm los}$). We
note that this agreement is not guaranteed by our procedure, which
only uses the stellar mass of a galaxy at infall to assign it a
stellar half-mass radius---its velocity dispersion follows mainly from
the total dark matter contained within that radius. Simulated galaxies
are slightly offset to higher velocities at fixed stellar mass or,
equivalently, to lower $M_*$ at fixed $\sigma_{\rm los}$. This is
likely a consequence of our assumption that galaxies stop forming
stars at the time of infall, biasing galaxies to be less massive than
they should actually be.
Tidal losses can reduce both the stellar mass and the velocity
dispersion, pushing galaxies to the bottom of the relation seen in
Fig.~\ref{fig:udgs_metal}. This provides one way of distinguishing
B-UDGs from T-UDGs: at fixed stellar mass, the latter should have
significantly lower velocity dispersions than the former.
Tidally-stripped galaxies move along ``tidal tracks'' in this figure,
such as the example shown by the green solid curve in
Fig.~\ref{fig:udgs_sigma}. This curve show the loci of possible tidal remnants of a
galaxy progenitor with initial stellar mass, $M_{*,0}=5 \times 10^9 M_\odot$,
and 1D velocity dispersion $\sigma_{*,0}=50$ km/s.
Square symbols along the curve indicate remaining bound mass
fractions, spaced by successive factors of ten (i.e., $100\%$, $10\%$,
$1\%$, etc.).
For comparison, data for Coma and Virgo UDGs are shown with magenta
symbols and labeled individually in Fig.~\ref{fig:udgs_sigma}
\citep{Toloba2018,vanDokkum2018a,vanDokkum2019,Martin2018,Danieli2019,vanDokkum2019_DF44,
Makarov2015,Chilingarian2019}.
High-velocity dispersion UDGs, such as DF44, are in agreement with
simulated B-UDGs. On the other hand, the comparatively low velocity
dispersion of the Virgo VLSB-D dwarf is in better agreement with
T-UDGs, which have undergone substantial tidal
disruption. Encouragingly, the morphology of VLSB-D is very elongated,
as is its globular cluster system, strongly suggestive of ongoing
tidal disruption \citep{Toloba2018}.
One main conclusion from this comparison is that the tidal origin of some UDGs implies substantial
diversity in velocity dispersion at fixed stellar mass.
In addition, we note that the shape of the solid tidal track in
Fig.~\ref{fig:udgs_sigma} becomes shallower than the main
$M_*$-$\sigma_{\rm los}$ trend for extreme values of the mass
loss. This implies that
velocity dispersions much below $\sim 10$ km/s are highly unlikely
for systems with $M_*\sim 10^8\, M_\odot$. Such velocity dispersion is
at the high end of the estimates for DF2. Much lower
velocity dispersions, such as those reported for DF4,
seem, at first glance, difficult to reproduce in our simulations,
where galaxies form in ``cuspy'', NFW-like halos
\citep{Bose2019}. If confirmed, accommodating the lower limits
of the measured kinematics for DF2 and DF4 would be rather
challenging in our modeling. Indeed, few, if any tidal tracks originating in
the undisturbed galaxy population would leave a
remnant with such low velocity dispersion.
Several interpretations are possible for this result. One is that this
may indicate a non-tidal origin for these extreme objects; i.e., DF2
and DF4 may simply form in ways and/or halos with properties not
reproduced in our simulations. A more conservative interpretation,
however, is that the failure to reproduce systems like DF2 and DF4 is
a result of numerical limitations in the simulations, which prevent us
from including the most extremely tidally stripped systems in our
sample \citep[see ][for a similar conclusion]{Carleton2019}. Indeed,
we caution that the tidal tracks in Fig.~\ref{fig:udgs_sigma} include
a substantial extrapolation in the $x<0.1$ range, outside the regime
validated by available simulations (middle squared symbol in each
track; the extrapolated regime is shown as a thinner curve). It is
possible that larger drops in velocity dispersion may accompany
extreme tidal losses, but this t would need to be confirmed with
higher resolution simulations that extend the range probed in our
current tidal evolution model.
Finally, the extreme velocity dispersions of DF2 and DF4 may indicate
the presence of a ``core'' in the initial inner density profile of
their surrounding halos. Tidal tracks in the presence of a core differ
in shape, as indicated by the dashed green curve in
Fig.~\ref{fig:udgs_sigma}, which uses the results of the ``cored''
halo progenitor in the simulations of \citet{Errani2015}. A core
leads to much lower velocity dispersions for extreme tidal remnants,
allowing better agreement with DF2 and DF4.
It is perhaps too early to use these results as evidence for a cusp or
core in such galaxies, especially given the limited tidal track range
actually validated by simulations and the large uncertainty in the
observational estimates, but it is an intriguing issue to which we
hope to return to in future work. Low velocity dispersion UDGs such as
DF2 and DF4 are akin to the ``cold faint giant'' satellites of the MW
and M31, Crater 2 and And XIX. The kinematics, size, and stellar mass
of the latter also suggest, in the context of current $\Lambda$CDM
models of dwarf galaxy formation, that they might be the remnants of
extreme tidal stripping events \citep{Fattahi2018}. Regardless of
whether halos have cores or cusps, these extreme tidal remnants should
be relatively rare. Should many systems comparable to DF2 and DF4 be
found, or if they were found in isolation (where tidal effects would
have been negligible), this would certainly call into question this
scenario.
\begin{figure}
\includegraphics[width=\columnwidth]{figs/rad_distribution_proj_along3axis.png}
\caption{Projected number density profiles for B-UDGs and T-UDGs.
Tidally formed UDGs populate mainly in the central regions of
clusters, whereas born UDGs have a more extended spatial distribution. The overall radial distribution of UDGs is not
too different from that of all cluster galaxies, shown in grey (scaled by
$\times 1/10$). For reference, observations for UDGs in Coma are shown using magenta symbols,
scaled down by a factor of $10$ to account for the difference in virial mass.}
\label{fig:radprof}
\end{figure}
\subsection{UDG metallicities}
Like velocity dispersions, the metallicities of UDGs may also be used
to investigate their origin. In particular, T-UDGs are expected to deviate from
the mass-metallicity relation of B-UDGs, which should track the main
galaxy population. This is shown in Fig.~\ref{fig:udgs_metal}, where
it is clear that galaxies heavily affected by tidal striping (i.e.,
T-UDGs, those in bluish colors) may have, at fixed stellar
mass, higher metallicities than normal galaxies, or B-UDGs.
We note that simulations like TNG generally have difficulty
matching the average mass-metallicity relation of dwarfs \citep[see ][for a more detailed
discussion]{Nelson2018}, so the simulated metallicities in
Fig.~\ref{fig:udgs_metal} have been uniformly re-scaled to roughly match
$[Z]=-0.6$ at $M_*=10^9\, M_\odot$. This has little impact on our
conclusions, which refer mainly to the differential effect on the
metallicity of T-UDGs compared to normal galaxies(and B-UDGs)
similar stellar mass. Here, we have used the metallicity at $t_{\rm inf}$
for the simulated galaxies, but the same conclusion applies if using
the metallicities at $z=0$. For comparison, we include the present-day
mass metallicity
relations from \citet{Gallazzi2005} and \citet{Kirby2013}, assuming a
solar abundance equivalent of $Z_\odot=0.0127$, as in \citet{Schaye2015}.
Available metallicity estimates for observed UDGs are included in
Fig.~\ref{fig:udgs_metal} \citep{RuizLara2018,Ferre2018,Makarov2015,
Gu2018, Chilingarian2019,Fensch2019}
using magenta symbols, where we have transformed iron abundances
[Fe/H] into full metallicities [Z/H] when necessary following Eq.
3 in \citet{Boecker2019} and assuming [$\alpha$/Fe] = 0.25,
the median [Mg/Fe] in the \citet{Ferre2018} sample.
Combining velocity dispersion with metallicity estimates provides an
excellent diagnostic of the effect of tides. Cluster (or satellite)
galaxies that have lower $\sigma_{\rm los}$ and higher metallicity
than expected for their stellar mass are excellent candidates for
being tidal remnants, or T-UDGs. The relatively low metallicity measured for
DF2 \citep{Fensch2019} may complicate its interpretation as a tidal remnant,
although several simulated galaxies that have lost at least 55\% of their
mass within $r_{h,0}$ lie close to the mass and metallicity of DF2.
On the other hand, the relatively
high metallicities of some of the measured UDGs, Yagi275 and Yagi276
for example, bear well for our interpretation of such systems as extreme
tidal remnants, although more detailed modeling is needed to validate
this hypothesis.
\subsection{Radial distribution and orbital kinematics}
The spatial distribution and orbital kinematics of UDGs in clusters
also carry information about their origin, and it is something we
can readily study in our simulations. We explore this in
Fig.~\ref{fig:radprof}, where the left panel shows the projected
number density profile of UDGs in all our 14 clusters. Clustercentric
radii are scaled to the virial radius, and three orthogonal
projections are used for each cluster before combining them to produce
the average density profile shown by the solid black line. For comparison, the
profile corresponding to all cluster galaxies (with $M_*>5 \times 10^{7} M_\odot$,
the minimum in our sample) is shown by the grey
dotted curve, which has been rescaled down by a factor of $10$ so as to
match the central density of the UDG curve. The comparison shows that
UDGs are only slightly biased relative to the cluster population at
large, with a relatively steeper density profile.
Interestingly, B-UDGs and T-UDGs have quite different radial profiles:
the former tend to avoid the central regions of the cluster and are
overrepresented in the cluster outskirts, likely because they
typically inhabit low-mass halos that are fully disrupted
by tides in the central regions. There, only systems that were
originally fairly massive can survive the strong tidal field, leading
to a population of UDGs made up predominantly of T-UDGs. Half of all
T-UDGs are expected to lie inside one third of the virial radius,
whereas the half-number radius of B-UDGs is as large as
$\sim 0.5\, r_{200}$.
These results seem at odds with those reported by
\citet{vanderBurg2016} for the Coma cluster, where UDGs seem to avoid
the cluster center (see magenta symbols with error bars in
Fig.~\ref{fig:radprof}). Before reading too much into this
discrepancy, we note that UDGs are very difficult to identify
observationally, especially so near the cluster center, where the
intracluster light is especially bright and may compromise
detection. We plan to address this issue in more detail in future
work. We also note that UDGs in the Virgo cluster seem to follow a
cuspier distribution (Lim et al, in preparation).
\begin{figure} \includegraphics[width=\columnwidth]{figs/tinf_hist.png}
\caption{Cluster infall times of simulated galaxies. Tidal UDGs (light blue) are accreted preferentially at
early times ($t_{\rm inf} < 6$ Gyr), consistent with their
concentration near the cluster centres Born UDGs
(navy blue) have later infall times compared with both
Tidal UDGs and with the overall galaxy population (grey
histogram). }
\label{fig:tinf}
\end{figure}
The distinct radial distribution of T-UDGs and B-UDGs is also
reflected in their infall times, which are shown in
Fig.~\ref{fig:tinf}. Both populations of UDGs, as well as the cluster
population as a whole, are shown with different color histograms,
normalized to the same area. B-UDGs infall much later than T-UDGs, as
expected. They also infall later than the population as a whole, which
indicates that most early infalling B-UDGs have been tidally disrupted and
are not present at $z=0$.
T-UDGs, on the other hand, are typically more massive systems at
infall (see Fig.~\ref{fig:mass_size}), are more resilient to tides,
and may survive to the present day. In some sense, this lends support
to the idea of UDGs as ``failed massive galaxies'', whose growth has
been truncated by cluster infall and whose properties have been
radically shaped by tides. The effect, however, is modest: the average
infall halo mass of $M_*=10^8\, M_\odot$ B-UDGs is roughly
$\sim 7 \times 10^{10}\, \rm M_\odot$, compared with
$2 \times 10^{11} \, \rm M_\odot$ for T-UDGs ($\sim 10\%$ of T-UDGs
have infall virial mass $\geq 5 \times 10^{11} \, \rm M_\odot$).
The infall time distinction between UDGs has interesting implications
for stellar ages. Assuming that most galaxies stop forming stars in
earnest soon after infall (as in our model), this implies that T-UDGs
should have much older stellar populations than B-UDGs. For instance,
\citet{Ferre2018} finds intermediate to old ages ($\sim 7$ Gyr)
stellar populations in Coma UDGs, and little difference with other
``normal'' dwarfs at similar cluster-centric distances. In our
scenario, this implies that they are mainly B-UDGs.
\begin{figure} \includegraphics[width=\columnwidth]{figs/r_vr.png}
\caption{Radial velocity vs clustercentric distance $r$-$V_r$
plot for simulated galaxies. The
UDG population (Born- and Tidal-UDGs) has similar kinematics to
the overall cluster population. Dotted
curves indicate $V_{ \rm esc}$, the escape
velocity of an NFW halo with mass and concentration consistent
with the simulated clusters. For a more direct comparison, the vertical
histograms show the velocity distributions for all galaxies (gray)
and for simulated UDGs (blue) to be indistinguishable of each
other. }
\label{fig:r_vr}
\end{figure}
Since UDGs are well mixed with the general cluster population we expect
little kinematic differentiation between the two. This is shown in
Fig.~\ref{fig:r_vr}, where the phase diagram of all cluster galaxies
(scaled to virial quantities) is shown. Because the velocity dispersion is
nearly independent of radius, there is little difference in the
velocity distribution of T-UDGs, B-UDGs, and the cluster as a
whole. Most velocities sit comfortably within the escape velocity
boundaries of an NFW halo of concentration comparable to that of the
average cluster. Overall, these results seem to agree quite well with
those of \citet{Alabi2018}, who argue, based on their velocity
distribution, for a wide range of infall times for the UDG population.
\section{Summary and Conclusions}
\label{sec:concl}
We have used fourteen $\sim 10^{14}\, M_\odot$ galaxy clusters from
the Illustris TNG100-1 simulation to study the formation of cluster
UDGs in the $\Lambda$CDM cosmology. We supplement the simulations with
a tidal evolution model that allows us to track the structural
evolution of galaxies as they undergo tidal disruption in the
gravitational potential of the cluster. The model makes clear
predictions for the radial distribution and kinematics of UDGs, as
well as for the velocity dispersion, surface brightness, metallicity,
and ages of their stellar populations. We analyze the cluster
population at $z=0$ using similar UDG selection criteria as in
\citet{Koda2015}. We note that these criteria designate as UDGs a
number of field low surface brightness galaxies, which are likely
potential progenitors of cluster UDGs.
Our main finding is that cluster UDGs have primarily two different origins:
``normal'' field LSBs that are ``born'' as UDGs in the field and then
enter the cluster (B-UDGs), and a second population that is made up of
higher surface brightness galaxies transformed into UDGs by extensive
mass loss due to tides (T-UDGs). Tidal and born UDGs add up to a
sufficient number of UDGs to match observational estimates for
$M_{200} \sim 10^{14}$$\; \rm M_\odot$\ clusters \citep{vanderBurg2017}. We
conclude that there is no need to invoke ``cores'' in dark matter
halos to explain the abundance of cluster UDGs.
Our analysis also identifies diagnostics that may be used to
distinguish between these two different formation paths. The
substantial mass loss needed to tidally form UDGs turns them into
outliers of typical galaxy scaling relations, with lower velocity
dispersions and higher metallicities at fixed stellar mass. Tidal UDGs
also entered the cluster significantly earlier ($\sim 9.5$ Gyr ago) than
surviving B-UDGs, which, on average, were accreted into the cluster
$\sim 5.5$ Gyr ago. As a result, tidal UDGs dominate in the inner regions
of galaxy clusters and their stellar ages should be on average older
than those of B-UDGs.
These results are, in principle, within reach of spectroscopic
studies. The few cases with robust metallicity/age estimates ($\sim
20$ objects to date) seem to have low metallicities ($[\rm
Fe/\rm H] \leq -1.1$), often enhanced alpha-elements ($[\rm
Mg/\rm Fe] \sim 0.4$), and intermediate to old stellar populations
$\sim 7$ Gyr \citep{Makarov2015, vanDokkum2016,
Kadowaki2017,Ferre2018,RuizLara2018,Danieli2019}. These properties
are what would be expected from B-UDGs, which are in essence field
galaxies whose star formation history ceased after entering the
cluster but which they were otherwise only modestly affected by cluster tides.
Our results lend some support to the ``evolutionary'' scenario
proposed by \citet{Roman2017b} that connects young UDGs in the field,
groups and clusters. However, our simulations also predict the
formation of an extra T-UDG population made up of galaxies severely
affected by tidal stripping. The large
metallicities of $2$ UDGs from the study \citet{Ferre2018}, Yagi275 and
Yagi276, suggest a possible tidal origin for these objects, a
result that deserves future investigation.
Other examples of T-UDGs may be objects with low dark matter
component. A few candidates have been identified, such as VLSB-D
\citep{Toloba2018}, DF2 \citep{vanDokkum2018a} and DF4
\citep{vanDokkum2019}, with velocity dispersions \citep[as suggested
by their globular cluster system and stars for the case of DF2, see
][]{Danieli2019} that are too low to contain significant amounts of
dark matter. Our model identifies these systems with ``extreme tidal
remnants'' that have lost the majority of their mass (dark and
luminous) to tides. If this is true, objects like VLSB-D, DF2, and DF4
should be rare, but they may yet provide the best evidence for the
substantial transformation of galaxies through tides that our models
predict.
The existence of a T-UDG population, with its predicted older ages,
enhanced concentration at the cluster center, lower velocity
dispersions, and higher metallicities is one of the key predictions
from our study, and one that future surveys, especially those able to
identify extreme LSB galaxies against the intracluster light near the
cluster center, should be able to validate or rule out.
\section*{Acknowledgements}
We would like to thank Azadeh Fattahi, Christina Manzano-King
and Cecilia Scannapieco for
making available some of the data included in Fig.~1.
LVS acknowledges support from NASA through the HST Program AR-14583 and
from the Hellman Foundation. EWP acknowledges support from the
National Natural Science Foundation of China under Grant No. 11573002.
LH is grateful for support from NSF program AST1815978.
\bibliographystyle{mnras}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,365
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Gun Metal and Chrome with Spalted Maple Burl pen has been stabilized, sanded to 12,000 grit, buffed at 3,200 rpm's, and finally topped with a hard durable finish. Wood color and character may vary. Will combine shipping on multiple items.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,206
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package com.example.livestream;
// [START livestream_delete_input]
import com.google.cloud.video.livestream.v1.DeleteInputRequest;
import com.google.cloud.video.livestream.v1.InputName;
import com.google.cloud.video.livestream.v1.LivestreamServiceClient;
import java.io.IOException;
import java.util.concurrent.ExecutionException;
import java.util.concurrent.TimeUnit;
import java.util.concurrent.TimeoutException;
public class DeleteInput {
public static void main(String[] args) throws Exception {
// TODO(developer): Replace these variables before running the sample.
String projectId = "my-project-id";
String location = "us-central1";
String inputId = "my-input-id";
deleteInput(projectId, location, inputId);
}
public static void deleteInput(String projectId, String location, String inputId)
throws InterruptedException, ExecutionException, TimeoutException, IOException {
// Initialize client that will be used to send requests. This client only needs to be created
// once, and can be reused for multiple requests. After completing all of your requests, call
// the "close" method on the client to safely clean up any remaining background resources.
try (LivestreamServiceClient livestreamServiceClient = LivestreamServiceClient.create()) {
var deleteInputRequest =
DeleteInputRequest.newBuilder()
.setName(InputName.of(projectId, location, inputId).toString())
.build();
// First API call in a project can take up to 10 minutes.
livestreamServiceClient.deleteInputAsync(deleteInputRequest).get(10, TimeUnit.MINUTES);
System.out.println("Deleted input");
}
}
}
// [END livestream_delete_input]
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,954
|
Q: Duplicity: How to list versions of a file? Using Duplicity I can make an incremental, efficient backup to an untrusted remote storage. This works fine and I can restore files at a specific point in time (or the next backup after that).
Well now I want to know when in the history of all the backups did a specific file change.
File change timestamp and size along with the backup timestamps would be sufficient.
Any way to do that with duplicity?
EDIT: I should clarify that I want to see every change/version to that file and not only the last, e.g. when it was first introduced, when it was change. That might be at multiple backup incremental backups.
A: Since version 0.7.03 of Duplicity it is possible to list the revisions that a file has changed in:
duplicity collection-status --file-changed 'filename' 'remote-url'
A: There was an answer on the mailing list by Edgar Soldin:
no, you can currently list all backed up versions. duplicity will not tell you
when they were changed. this could however be implemented as duplicity
internally keeps track of that.
ede/duply.net
So in essence: It is not possible with vanilla duplicity now, but the data is in duplicity metadata.
A: You might try dumping the full list of files, then searching that list for the file in question:
duplicity list-current-files url > /path/to/file-list.txt
grep filename /path/to/file-list.txt
This command is poorly documented, but each line in the file has a date that appears to be last modified time.
A: YES, it is possible to do this by getting a list of all of the files in each of the snapshots that duplicity takes (i.e. the first full backup and every incremental) and then compare the file lists and look for the specific file you are concerned with.
While it will not show every change to the file unless the file changes less frequently than you make duplicity copies, it will show each state of the file at each backup point.
I wrote a shell script to do this.
A: duplicity list-current-files --time now "${DEST}" > /tmp/today
duplicity list-current-files --time 1D "${DEST}" > /tmp/yesterday
grep -Fxvf /tmp/{today,yesterday}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,167
|
One of Wolfgang Amadeus Mozart's most well-known pieces, "The Marriage of Figaro," is performed here by an unknown symphonic orchestra. The picture is one of Mozart himself. Enjoy and thanks for watching.
The Barber of Seville or the Useless Precaution (French: Le Barbier de Séville ou la Précaution inutile) is a French play by Pierre Beaumarchais, with original music by Antoine-Laurent Baudron. It was initially conceived as a comic opera, and was rejected as such in 1772 by the Comédie-Italienne. The play as it is now known was written in 1773, but, due to legal and political problems of the author, it was not performed until February 23, 1775, at the Comédie-Française in the Tuileries. It is the first play in a trilogy of which the other constituents are The Marriage of Figaro and The Guilty Mother.
Though the play was poorly received at first, Beaumarchais worked some fast editing of the script, turning it into a roaring success after three days. The play's title might be a pun on Tirso de Molina's earlier play El Burlador de Sevilla (The Trickster of Seville). Mozart wrote a set of 12 variations, K. 354, on one of Baudron's songs, "Je suis Lindor".
The Marriage of Figaro (Italian: Le nozze di Figaro, pronounced [le ˈnɔttse di ˈfiːɡaro]), K. 492, is an opera buffa (comic opera) in four acts composed in 1786 by Wolfgang Amadeus Mozart, with an Italian libretto written by Lorenzo Da Ponte. It premiered at the Burgtheater in Vienna on 1 May 1786. The opera's libretto is based on a stage comedy by Pierre Beaumarchais, La folle journée, ou le Mariage de Figaro ("The Mad Day, or The Marriage of Figaro"), which was first performed in 1784. It tells how the servants Figaro and Susanna succeed in getting married, foiling the efforts of their philandering employer Count Almaviva to seduce Susanna and teaching him a lesson in fidelity.
The Marriage of Figaro is a cornerstone of the repertoire and appears consistently among the top ten in the Operabase list of most frequently performed operas.
Beaumarchais's earlier play The Barber of Seville had already made a successful transition to opera in a version by Paisiello. Although Beaumarchais's Marriage of Figaro was at first banned in Vienna because of its licentiousness, Mozart's librettist managed to get official approval for an operatic version which eventually achieved great success.
The Marriage of Figaro (French: La Folle Journée, ou Le Mariage de Figaro ("The Mad Day, or The Marriage of Figaro")) is a comedy in five acts, written in 1778 by Pierre Beaumarchais. This play is the second in the Figaro trilogy, preceded by The Barber of Seville and followed by The Guilty Mother.
In the first play, The Barber, the story begins with a simple love triangle in which a Spanish count has fallen in love with a girl called Rosine. He disguises himself to ensure that she will love him back for his character, not his wealth. But this is all foiled when Rosine's guardian, Doctor Bartholo, who wants her hand in marriage, confines her to the house. The Count runs into an ex-servant of his (now a barber), Figaro, and pressures him into setting up a meeting between the Count and Rosine. He succeeds and the lovers are married to end the first part of the trilogy.
The Marriage was written as a sequel to The Barber. In his preface to the play, Beaumarchais says that Louis François, Prince of Conti had requested it. The play's denunciation of aristocratic privilege has been characterised as foreshadowing the French Revolution. The revolutionary leader Georges Danton said that the play "killed off the nobility"; in exile, Napoleon Bonaparte called it "the Revolution already put into action."
Marriage is a socially or ritually recognized union or legal contract between spouses.
Marriage (novel), a novel by H. G. Wells, first published in 1912.
Marriages (band), is a rock group from Los Angeles, CA.
Zhenitba (Russian: Женитьба, Zhenit'ba, Marriage) is an unfinished opera begun in 1868 by Modest Mussorgsky to his own libretto based on Nikolai Gogol's comedy Marriage. This 1842 play is a satire of courtship and cowardice, which centres on a young woman, Agafya, who is wooed by four bachelors, each with his own idiosyncrasies.
The idea to set Gogol's Marriage to music came from the advice and influence of Alexander Dargomyzhsky, who began to compose his own experimental opera, The Stone Guest, to Alexander Pushkin's tragedy just two years earlier (in 1866). Dargomyzhsky declared that the text would be set "just as it stands, so that the inner truth of the text should not be distorted", and in a manner that abolished the 'unrealistic' division between aria and recitative in favour of a continuous mode of syllabic but lyrically heightened declamation somewhere between the two.
The Marriage was a short-lived (1953–54) radio series which starred Hume Cronyn and Jessica Tandy. Based on their earlier Broadway play, The Fourposter, the series aired Sunday evenings on NBC at 7:30 p.m. Eastern Time.
New York attorney Ben Marriott (Hume Cronyn) was married to former department store fashion buyer Liz (Jessica Tandy), who struggled with her switch to domestic life. The sophisticated couple discussed such subjects as art, theater, literature and philosophy, while raising a son, Pete, and an awkward teenage daughter, Emily (Denise Alexander).
Written by Ernest Kinoy, the series had an unusual gimmick of alternating viewpoints, as Ben narrated one week and Liz the next.
The show was scheduled to move from radio to television, with Cronyn producing as well as acting in the show. However, Tandy suffered a miscarriage, and the show's debut was delayed a week. The TV series premiered in July 1954 to "warm and enthusiastic reviews." It ran through August, and abruptly ended after eight episodes.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,371
|
On the Mark Sports Bar & Grill Sports bars, lounges, & restaurants in Fayetteville
Sports bars, lounges, & restaurants
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|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,372
|
Awakenings Manchester Pre-Crónica
Manuel Gago
Clubbing Por Europa, Crónicas, Internacional
Oh, See! Fest: La Crónica
Juan Aguilera
Espacio Alternativo, Espacio Underground
Vortex triunfa con Cleric al mando de Industrial Copera
Entrevistamos a Integral Bread
JUSTIN PATTI interview [O.V.]
If you are living in the Netherlands, you should know Justin Patti. He is one of the of the most impor-tant characters of the Dutch techno scene, not only as a DJ, also because his gift for discovering new talents and organizing some of the most powerful and obscure events in Rotterdam. Justin is synonymous of dark-techno, he is the founder and leader of the Strobe crew, together with Rachid Prins and Allard Drijfhout, main responsible of the best techno parties in the city. Through this interview, he reveals in detail his professional carreer, secrets, personal experiences, his opinion about some global discussions like "digital era" vs. analog methods… besides interesting info about the Dutch trends, currently and during his beginnings. Do not miss a detail, I promise it is worth!
First to all, thank you very much for taking the time to answer these questions… Here we go!
I found out you started producing music almost ten years ago when you decided to get into the digital world… How do you consider yourself more, as a DJ or as a music producer? Which part is more developed?
I see myself more as a DJ than as a producer. You have to spend so much time in the studio and be determined to become a good producer. I've tried, but haven't got lots of time to produce music. Besides Strobe, I have a good job and love to do things that I really enjoy. One of them is spinning records (SoundCloud).
Nowadays, there is a big discussion about digital vs. analogic methods, Traktor vs. vynils… On what side are you? What do you think about this topic?
I play with Traktor, CDJ's and/or vynil. I think it's good to develop yourself as a DJ. I don't have a favorite side, it's just a different way of working the decks. To be honest I really don't care how other DJ's play, just let the music do the job and make sure the mix is fine. By the way, I do prefer the Xone mixer, but I guess that's another topic altogether… hahaha.
Currently, it is common to see you in several line-ups playing together with your colleage Rachid Prins as Patriarch Conflicts… What are the benefits to play music together? Do you feel more comfortable doing back to back than playing alone?
The benefits of playing together is that two minds become one. Rachid sometimes listens to the weirdest electronic music, which I never heard before, but (most of the time) I really love that crazy sound he brings. We might have different backgrounds musically but we still have that connection. Playing back to back with Rachid is always fun.
To be honest our sets are often unprepared so I never know what he'll be playing and I really have to anticipate and react to the tracks he's throwing at me. Of course playing solo feels a bit more comfortable because I have full control on the mixer, but we are playing as Patriarch Conflicts since 2012 so we know each other pretty well. I'm comfortable enough to know I do a proper set with Rachid.
What is the origin of that name: "Patriarch Conflicts"?
When we decided to join forces we had to make up a name. I remember we smoked some "happy cigarettes" and were just playing with the letters from our own names. We made up 'Patriarch', which comes from 'Patti' and 'Rachid', but it wasn't enough for us. That's why we decided to add a little extra to it. We grabbed the dictonary and stopped at 'Conflicts'. That's what we had back in the days (and sometimes still have) when we play back to back… conflicts. So there you go: Patriarch Conflicts.
You and your crew are working hard in Rotterdam to bring forth true underground parties. Are you happy with the results? Do you feel you are the only one doing that?
What started as a hobby is now doing business and I must say it's so much harder than I expected. When we threw parties before Strobe, as ROT at Bootleg DJ Cafe, it was a lot easier. ROT was a platform for upcoming talent and we invited friends playing there just for a few drinks. Unfortunately, Bootleg had to close its doors, so we decided to move forward and Strobe was born. It's totally different if you have multiple artists coming from other countries and have to do promo to have about a thousand of visitors. And there's plenty more to think about, lots of people do underestimate that sometimes and I deeply respect other succesfull organizations.
I must say after almost 4 years of Strobe we're really doing a decent job, the company is healthy and we've put our name not only on the streets of Rotterdam, but worldwide because of the Strobe FM Podcast series. The crowd is very loyal and they always set the right vibe at our parties. We always try to stay close to our public because they are a huge part of Strobe.
Thank god we are not the only one presenting underground parties in our city, I think it's a good thing that there are more parties like Strobe doing real undergound techno parties. It keeps techno alive!
Rotterdam is known as an industrial city, famous for the origins of hardcore… It is supposed to be an underground city with music… Do you think this is truth?
Yes! Rotterdam is underground to me and has been for decades. The gabber culture is a good example, I used to be a gabber myself and I still like to listen hardcore now and then. It brings lots of great memories back. But what is underground nowadays? What can be seen by you and me as commercial music, can be seen by someone else as underground. Maybe underground is also a feeling. By the way, we did two illegal raves with Strobe; those were pretty underground I have to say.
Can you tell us a little bit more about those raves? And can we expect another one in the near future?
The first rave was in a warehouse with local talented dj's only that all took their friends. It was a small venue for max 200 techno freaks, so we had to make a list for all the friends who were invited. The vibe was awesome; everybody knew each other or got to know each other during that 12-hour rave. The other rave was a lot bigger, an abandoned factory inhabitated by hardcore squatters but it all turned into a Strobe party with legends like Bas Mooy, Charlton, Flink and local acts. I don't know what to say about it, guess you just had to be there. We should do it again soon, but only if the time is right because it's not as easy as it looks.
Since my arrival to Rotterdam, I went to a couple of Strobe parties, and I must say they were amazing… What are your goals making parties like those? Are you thinking about expanding your events to other locations outside of Rotterdam?
The most important goal is to present the techno we personally like and share it with people who have the same feeling. Also it's important to keep Strobe financially healthy so we can keep organizing events like this. Although we're very proud that we come from Rotterdam, expanding to other cities is indeed something we've thought about. We always keep an eye out for other locations; maybe a Strobe Festival would be cool in the future. And yes, a proper abandoned venue for an illegal rave is at the very top of our wishlist, for sure.
You are going to make a great showcase with MORD Records for your next event… How hard is it to bring to Rotterdam important labels like MORD?
We started collaborating with Bas Mooy in 2011 when we did events in Bootleg DJ Cafe. I have a lot of respect for Bas since I found out about Strictly Techno. That was maybe ten years ago, I still remember that day when I saw Bas doing grocery in the supermarket and whispered to my ex-girlfriend: "Look it's Bas Mooy over there!" haha. The respect runs both ways and I must say working with Bas is always a pleasure. We are close friends now but I still look up to him. Besides an amazing artist he's also one of the nicest guys in the scene, I can always call him for a chat or advise. So to have a showcase with one of the greatest labels at the moment was not a very hard decision. Bas also wanted us to do the showcase, so everybody is happy and we're looking up to a killer event on the 10th of September in Factory 010 (link).
Thanks for answer my questions, I really hope you will continue this line of work, techno lovers like me appreciate so much your events and your music… Can you give us any info about next parties or collaborations for Strobe?
On the 5th of November we have some of the best international live acts at Factory 010 and it will be the final event for 2016. After this one we have our 4-year anniversary in January 2017 coming up. I cannot say much about it because the line up is not confirmed yet. Just follow us on Facebook and you'll find out. Pablo, thank you very much for having me and for your appreciation. It means a lot to us and it's one of the main reasons Strobe is still kicking hard.
Interviewer: Pablo Ortega
hollandjustin pattimordnetherlandsRotterdamstrobe
Rotterdam Rave Festival: la crónica
PRSPCT: The Hardcore Drum & Bass Resurrection
Bonzai Classics
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,237
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Sedegliano (Sedean en friulano) es una población de 3.914 habitantes en la provincia de Udine, en la región autónoma de Friuli-Venecia Julia.
Geografía
Demografía
Localidades de la provincia de Údine
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,683
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Here is the 4th Volume to the NBA Kings of the Crossover Series featuring former rookie of the year, Tyreke Evans of the New Orleans Pelicans. ...(Read more)
Tyreke Evans Has CRAZY Handles! NBA Kings of The Crossover Vol. 4!
Arek Kissoyan
| February 18th, 2014 | 20,289 Views | NBA Crossovers
Co-Founder / Executive Producer
#23 of Mixtapes.
Follow @Arek Kissoyan
com/wp-content/uploads/2014/01/KingofCrossover2.jpg" alt="Ballislife | Kings of The Crossover" width="600" height="125" />
Here is the 4th Volume to the NBA Kings of the Crossover Series featuring former rookie of the year, Tyreke Evans of the New Orleans Pelicans. Every Tuesday we will release a new mixtape on a new NBA player who is in the conversation for the best handles in the league and at the end of the series you, the fans, will be able to vote for who YOU think has the best handles in the league on March 11.
Kyrie Irving |
Chris Paul |
Brandon Jennings |
Jamal Crawford |
Isaiah Thomas |
Jrue Holiday |
We're starting a new series on Ballislife called NBA's Kings of The Crossover. Every Tuesday we will release a new mixtape on a new NBA player who is in the conversation for the best handles in the league and at the end of the series you, the fans, will be able to vote for who YOU think has the best handles in the league.
– Youtube.com/TheBasketballDiary
– Jamal Crawford Pro Am
– McDonald's All American
– Jordan Brand Classic
– Bobby Gross
Here is the 2nd Volume to the NBA Kings of the Crossover Series featuring the best PG in the league, Chris Paul of the LA Clippers. Every Tuesday we will release a new mixtape on a new NBA player who is in the conversation for the best handles in the league and at the end of the series you, the fans, will be able to vote for who YOU think has the best handles in the league on March 11.
Here is the 3rd Volume of the NBA Kings of the Crossover Series featuring Brandon Jennings of the Detroit Pistons. Every Tuesday we will release a new mixtape on a new NBA player who is in the conversation for the best handles in the league and at the end of the series you, the fans, will be able to vote for who YOU think has the best handles in the league on March 11.
Special thanks to: Duro Sports & Fresh Focus NYC. Check them out
This summer we got the pleasure of watching one of the illest players to grace the basketball court, the man who has shattered the NBA's record for most 4 point plays and 1 of 3 players in NBA history to score 50+ points with 3 different teams. We present you the OFFICIAL Jamal Crawford mixtape! Special thanks to Jamal Crawford, Rashad Powell, Jerome Johnson, the entire staff of the Jamal Crawford Pro Am and Hoopsmack!
Through out his life Isaiah Thomas would constantly hear people say how he wouldn't make it to the league because he was too small. Although he eventually got drafted to the league with the last pick in the 2011 NBA Draft many people believed he wouldn't be an effective player and last long in the league due to his size. Now, the shortest player in the NBA is about to enter his 3rd season in the league with his future looking as bright as ever after very impressive first 2 years in the league averaging 12.8 PPG (44.3% FG — 36.7% 3PT FG), with 4 APG and shooting 86.2% from the free throw line. At onyl 5'8″ Isaiah Thomas is living proof that size doesn't matter and is a walking inspiration to many up and coming ballers all around the world. Footage is from the Jamal Crawford Pro Am aka the best pro am of the summer.
Check out the OFFICIAL Jrue Holiday high school mixtape! Jrue was one of our all time favorite players to follow and here is why. In this video for the most part he is playing against current NBA pro's like DeMar DeRozan, Klay Thompson, Derrick Favors, Jordan Hamilton and soon to be pro Michael Snaer.
Show/Hide 0 Comments
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,828
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Населението на Еквадор през 2020 година е 17 510 643 души. При население от 13 212 742 души през 2004 година съотношението между градско и селско население е 57/43.
Възрастов състав
(2003)
0 – 14 години: 34,9% (мъже 2 430 303 / жени 2 351 166)
15 – 64 години: 60,6% (мъже 4 116 289 / жени 4 198 667)
над 65 години: 4,5% (мъже 284 082 / жени 329 727)
(2009)
0 – 14 години: 31,1% (мъже 2 312 610 / жени 2 220 378)
15 – 64 години: 62,7% (мъже 4 506 908 / жени 4 636 703)
над 65 години: 6,2% (мъже 432 144 / жени 464 358)
Коефициент на плодовитост
2003-2.99
2009-2.51
Расов състав
55% – метиси
25% – индианци
15% – бели
5% – мулати и замбо
Религия
97% – християни
90% – католици
6% – протестанти
0,2 % – православни
0,25 % – будисти
0,1 % – мюсюлмани (предимно сунити)
0,1 % – юдеи
Езици
Официален език в Еквадор е испанският.
Външни препратки
Еквадор
Еквадор
География на Еквадор
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,484
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Yet another EVE banking scandal has rocked the Reuters wire. No stranger to the EVE controversies, we get Goonswarm spymaster The Mittani's take on EVE's player-run banking troubles and offer you several new and exclusive articles on APB, Aion, and World of Warcraft. It's all in Loading... EVE Banks Debunked.
Ten Ton Hammer's very own satirical webcomic, geeked is here with a look at the age old discussion of what game stole what from what other game... and just who the heck cares. Check out geeked: The Horse is Dead, Jim today!
EVE has had its share of controversies, from dev intrigues to underhanded corporate takeovers to massive battles that turn on a single well-placed spy. The good news is you can read about all of that and more weekly in our Sins of a Solar Spymaster column. Any massive single shard sandbox must have it's dustups from time to time. But nowhere are these dustups more avoidable than in EVE's completely unsecured and unregulated banking system.
On the face of it, EVE players stockpile ISK like hotel shampoo, so why not make that ISK work for you while you're out-of-game training for long periods? If you've got 30 days of training before you can fly a battleship, why buy that battleship today? Invest your money with EIB and have money left over for that Tech 2 turret you've always wanted. Operators are standing by!
And no help was forthcoming when the head of EIB decided to cut and run. In a painfully rambling video, Cally a.k.a. Dentara Rast owns up to his lightning-fast extortion of the alleged 790 billion ISK in deposits EIB had accumulated. If the total held up (and CCP believed it was somewhat less [source]), it equates to over $100k in real money, and that's just using the CCP-approved channels. Selling ISK through direct sales websites, a.k.a. goldsellers, could net much more.
Nor are EVE's banking problems completely in the past. As recently as this January, Dynasty Banking lost some 80 billion of its equity balance when a key officer decided to pull a runner. Surely the community would learn its lesson and turn its back on the concept of an EVE bank, right? Wrong. EVE's largest player-run bank, EBank, reports that it's currently holding nearly 8.5 trillion (with a 't') in deposits from some 5,800 depositors [source]. Dynasty Banking is still taking deposits as well.
As if to reinforce The Mittani's words, EBank (the largest and most "legit" of the banks mentioned in the article segment) was at the center of EVE Online's latest banking scandal, which was peanuts compared to Ebanks total holdings and other EVE thefts. The moral of the story: stop lining the pockets of these scammers. Don't invest in player run banks, and if you already have, pull your money out now. Who cares about a run on illegitimate banks?
Why do players invest in EVE banks when story after story proclaims their untrustworthiness? Is it because players don't view virtual currency as real money (it is, folks), or is it the same old profit motive? Share your thoughts in the Loading... forum, or feel free to email me!
Instead of describing the program here, allow me to point you to the thread in our forum where you can learn all about it. Who knows? Maybe you'll be our next guru.
No quotes today, just lots of volunteer spirit!
5 new MMOG hand-crafted articles today! 29 in July! 794 in 2009!
Set for release in the first quarter of 2010, All Points Bulletin (APB) is a persistent world shooter that pits enforcers against criminals in a high octane urban playground. The game gained serious traction at E3 2009 last month thanks to an impressive demo featuring gorgeous graphics truly deserving of the HD distinction, novel gameplay, and a just-announced distribution deal with Electronic Arts. We sat down with Realtime Worlds ' Chris Dye to crack our enforcer knuckles for the first time, learning more about the unparalleled customization options and how the timeless battle between cops and robbers will play out in APB.
Flight is a huge part of Aion gameplay. Whether you choose to use it for convenience sake or strive to make it a part of your gaming strategy, at some point players will utilize this fun feature. To get you started on your life in the air, we have put together a FAQ that will tell you all you need to know to get started as a Daeva. We detail how flight works, what its limits are, and share great tips that will keep you safe in the air, and get you back onto the ground in one piece!
Aion's Gladiator is the offensive branch of the warrior archetype. This class offers a heavy plate armor wearing option with high offensive capabilities for players who want a tank that can dish out some punishment. Unlike its counterpart, the Templar, Gladiators use skills that offer more damage oriented attacks rather than defensive options. To help give players a look at what they can expect from the Gladiator, we've put together a preview of this class from Aion's 3rd beta weekend.
Metas takes another look at the WoW Guide for Comparing Raid Gear with a lot of the updated stats, like the Armor Penetration cap and Hit/Expertise as threat stats for the tank, you get to make the better choices for which epic item is an upgrade for your Fury, Arms, or Protection Warrior.
If you haven't noticed, Rogues are one of the few classes to ever learn a new weapon type in WoW's fairly long history (it's really been more than 5 years, people!). In our next WoW Rogue guide, Stow takes a look at where the axes are at for our new usage. Unfortunately... he's still looking. Find out why!
Officials confident this measure will work better than the previous "free brass knuckles and blunt instruments" period.
The real question: why would a SWAT raid panic a D.C. neighborhood?
City flush with cash, citizens tell politicians:"Urine trouble!"
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{
"redpajama_set_name": "RedPajamaC4"
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{"url":"http:\/\/www.ams.org\/mathscinet-getitem?mr=2002d:11058","text":"MathSciNet bibliographic data MR1839918 (2002d:11058) 11G05 (11F80 11G07 14G35) Breuil, Christophe; Conrad, Brian; Diamond, Fred; Taylor, Richard On the modularity of elliptic curves over \\$\\bold Q\\$$\\bold Q$: wild 3-adic exercises. J. Amer. Math. Soc. 14 (2001), no. 4, 843\u2013939 (electronic). Article\n\nFor users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews.","date":"2013-12-09 19:30:49","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 1, \"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\": 0.9774996638298035, \"perplexity\": 10726.463097080541}, \"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-2013-48\/segments\/1386163996785\/warc\/CC-MAIN-20131204133316-00042-ip-10-33-133-15.ec2.internal.warc.gz\"}"}
| null | null |
using System.Data.Entity;
using System.Linq;
using System.Threading.Tasks;
using LuckyMe.Core.Business.Draws;
using LuckyMe.Core.Data;
using LuckyMe.Core.ViewModels;
using MediatR;
namespace LuckyMe.Core.Business.Handlers.Draws
{
public class GetDrawsHandler : IAsyncRequestHandler<GetDraws,Paged<Draw>>
{
private readonly ApplicationDbContext _db;
public GetDrawsHandler(ApplicationDbContext db)
{
_db = db;
}
public async Task<Paged<Draw>> Handle(GetDraws message)
{
var basequery = _db.Draws.Where(d => d.UserId == message.UserId);
if (message.GameId != null)
{
basequery = basequery.Where(d => d.GameId == message.GameId);
}
if (message.Date != null)
{
basequery = basequery.Where(d => DbFunctions.TruncateTime(d.Date) == message.Date);
}
var total = await basequery.CountAsync();
var draws = await basequery.Include(d => d.Game).Include(d => d.User)
.OrderByDescending(d => d.Date)
.Skip((message.Page - 1) * message.ItemsPerPage)
.Take(message.ItemsPerPage)
.ToListAsync();
return new Paged<Draw>
{
Items = draws,
ItemsTotalCount = total,
ItemsPerPage = message.ItemsPerPage,
CurrentPage = message.Page
};
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 971
|
Q: VB.net replacing a byte array in a list of structures I have this structure.
Public Structure struct1
Public iNum as integer
Public previous As Byte()
Public seconds As Integer
Public nextRead As DateTime
End Structure
I then have this list of struct1
Dim lStruc1 As List(Of struct1) = New List(Of struct1)
I initialize the structure by using this
lStruct1.Add(New struct1 With {.iNum = 1, .previous = New Byte() {}, .seconds = 30, .NextRead = DateTime.Now})
lStruct1.Add(New struct1 With {.INum = 2, .previous = New Byte() {}, .seconds = 30, .NextRead = DateTime.Now})
lStruct1.Add(New struct1 With {.INum = {what ever data port}, .previous = New Byte() {}, .seconds = 30, .NextRead = DateTime.Now})
I read through the list doing this
For Each str As struct1 In lStruct1
If str.NextRead < DateTime.Now Then
' do read of data
If str.previous.SequenceEqual(bNewData) = False Then
Array.Resize(str.previous, bNewData.Length)
Array.Copy(bNewData, str.previous, bNewData.Length)
' write out new data(bNewData) to a file
End If
End If
Next
The problem is that it doesn't update the item .previous in the list lStruc1. I can change a byte in str.previous and the list item in lStruct1 will update as well. Setting str.previous to New byte() disconnects the str.previous from the list of lStruct1. With the type of structure I am using, I am not able to set an initial size as I get compiler errors.
All I need is a list of Items I am monitoring that are an array of bytes and write that list of bytes to a log when anyone of the 100's points I am monitoring changes. I am also keeping seconds as an average time that any one of these 100's of points change at different times from 10 seconds to upto 4 hours, so I keep an average of seconds of change not to continually read them.
How do I update the byte array .previous in the lStruct1 list?
I would also love to do it like this
For Each str As struct1 In lStruct1.findall(function(f) f.NextRead < datetime.now)
Next
So I don't do a check on each item. Right now I go through the entire list creating a new list and then setting lStruct1 to the new list. I know what I am doing above with .findall is almost the exact same thing as the code above, but I think it looks cleaner to do the findall way. It is not possible to do it that way because at the moment I have to create a new list of all items to update lStruct1 when complete of reading the ports that need to be read.
I know that there is an easier way to do this.
A: I fixed this by defining lStruct1 as
Dim lStruc1() as struct1
lStruct1 = new struct1() {New struct1 With {.iNum = 1, .previous = New Byte() {}, .seconds = 30, .NextRead = DateTime.Now},
New struct1 With {.iNum = 2, .previous = New Byte() {}, .seconds = 30, .NextRead = DateTime.Now},
New struct1 With {.iNum = {what ever data port}, .previous = New Byte() {}, .seconds = 30, .NextRead = DateTime.Now} }
instead of
Dim lStruc1 As List(Of struct1) = New List(Of struct1)
I then changed the looping code to this
For x as integer = 0 to lStruct1.Count - 1
If lStruct1(x).NextRead < DateTime.Now Then
' do read of data
If lStruct1(x).previous.SequenceEqual(bNewData) = False Then
lStruct1(x).previous = bNewData
dim sFilename as string = "{log port number - date and time}" + ".log"
File.WriteAllBytes(sFilename, bNewData)
End If
End If
Next
Thread.Sleep(1000)
It now works like a champ.
As for naming conventions, it is my own code, nobody but me will ever see it.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,749
|
package org.vaadin.example;
import com.vaadin.annotations.AutoGenerated;
import com.vaadin.annotations.DesignRoot;
import com.vaadin.ui.HorizontalLayout;
import com.vaadin.ui.NativeButton;
import com.vaadin.ui.Panel;
import com.vaadin.ui.VerticalLayout;
import com.vaadin.ui.declarative.Design;
/**
* !! DO NOT EDIT THIS FILE !!
*
* This class is generated by Vaadin Designer and will be overwritten.
*
* Please make a subclass with logic and additional interfaces as needed,
* e.g class LoginView extends LoginDesign implements View { … }
*/
@DesignRoot
@AutoGenerated
@SuppressWarnings("serial")
public class MainLayoutDesign extends HorizontalLayout {
protected VerticalLayout menuLayout;
protected NativeButton menuButton1;
protected NativeButton menuButton2;
protected NativeButton menuButton3;
protected Panel contentPanel;
public MainLayoutDesign() {
Design.read(this);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,187
|
CustomCreteWerks, Inc. would like to congratulate the Chicago Cubs on their World Series Victory. Chicago Cubs WIN!
Ramp Sinks by CustomCreteWerks Inc.
Selecting the countertop material to use can be a difficult choice. We want to simplify that decision for you. There are some fantastic benefits to choosing concrete for your countertops, sinks and tables and we've compiled those fantastic benefits into an infographic for you.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,132
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{"url":"https:\/\/traumstyling.com\/co1mt\/b1861f-graham-scan-convex-hull-algorithm","text":"5. Graham's Scan Given a set of points on the plane, Graham's scan computes their convex hull.The algorithm works in three phases: Find an extreme point. It is not recommended to use this algorithm when . This implementation just takes the x,y coordinates, no other libraries are needed. ... Implementations. ) Graham Scan. 2 Add X to the convex hull. C implementation of the Graham Scan convex hull algorithm. P 1 1) Find the bottom-most point by comparing y \u2026 convex-hull graham-scan-algorithm graham-scan Updated Jul 20, 2019; Python; gale31 \/ AstroSpiral Star 3 Code Issues Pull requests The Astro Spiral project presents an innovative way to compare astronomical images of the sky by building a convex spiral (modification of the Graham Scan algorithm for convex hull) according to the bright \u2026 It turns left, so the point is pushed to the stack.The same process goes on. P If you have some nails stuck on a desk randomly and you take a rubber band and stretch accross all the nails. The algorithm starts by arbitrarily partitioning the set of points PP into k<=1+n\/mk<=1+n\/m subsets(Qk)k=1,2,3...n(Qk)k=1,2,3...n with at most mm points each; notice that K=O(n\/m)K=O(n\/m). A collection of animated algorithms. Graham's scan algorithm is a method of computing the convex hull of a finite set of points in the plane with time complexity O (n log \u2061 n) O(n \\log n) O (n lo g n).The algorithm finds all vertices of the convex hull ordered along its boundary . It is named after American Mathematician Ronald Graham, who published the algorithm in 1972. This is done in O(n) time by iterating over the array of points. Bei Punkten liegt seine asymptotische Laufzeit in (\u22c5 \u2061) Beschreibung Vorbereitung. Point $(5, 5)$ is popped from the stack. Graham scan . x arthur-e \/ graham_hull.py Forked from tixxit\/hull.py. Find the lowest point by y-coordinate. I'm beginning to learn Haskell. ( ( The next post will cover Chan's algorithm. They both use a similar idea, and are implemented as a stack. Again, determining whether three points constitute a \"left turn\" or a \"right turn\" does not require computing the actual angle between the two line segments, and can actually be achieved with simple arithmetic only. This process will eventually return to the point at which it started, at which point the algorithm is completed and the stack now contains the points on the convex hull in counterclockwise order. = [5] Later D. Jiang and N. F. Stewart[6] elaborated on this and using the backward error analysis made two primary conclusions. We push these two points and the next point in the list (points $P_0, P_1$ and $P_3$ in the figure above) to the stack. Graham's scan is a method of finding the convex hull of a finite set of points in the plane with time complexity O(n log n). in a \"right turn\" (because the point What would you like to do? The intuition: For each point, it is first determined whether traveling from the two points immediately preceding these points constitutes making a left turn or a right turn; Retrieved from Wikipedia. the angle made by the line with the $x$-axis. It uses a stack to detect and remove concavities in the boundary efficiently. , Worst case time complexity of Jarvis\u2019s Algorithm is O (n^2). Dijkstra's Algorithm in Haskell. (The green color means the point is in the convex hull and red color means the point can not be in the convex hull). # let p0 be the point with minimum y-coordinate, # or the leftmost such point in case of a tie, # sort the points (except p0) according to the polar angle, # made by the line segment with x-axis in anti-clockwise direction, # if more than two points are collinear with p0, keep the farthest, d = direction(sorted_polar[i], sorted_polar[i +, An efficient way of merging two convex hulls, Check if a point lies inside a convex polygon, Determining if two consecutive line segments turn left or right, Check if any two line segments intersect given n line segments, Convex Hull Algorithms: Divide and Conquer, Determining if two consecutive segments turn left or right, http:\/\/jeffe.cs.illinois.edu\/teaching\/373\/notes\/x05-convexhull.pdf, https:\/\/www.cs.umd.edu\/class\/spring2012\/cmsc754\/Lects\/cmsc754-lects.pdf. Fortune, S. Stable maintenance of point set triangulations in two dimensions. In the late 1960s, the best algorithm for convex hull was O (n2). Graham\u2019s Scan The Graham\u2019s scan algorithm begins by choosing a point that is de\ufb01nitely on the convex hull and then iteratively adding points to the convex hull. P [5] The stated goal of the paper was not to specifically analyze the algorithm, but rather to provide a textbook example of what and how may fail due to floating-point computations in computational geometry. The execution trace of the program for the point sets given below are presented in this section. Examples. The code below uses a function ccw: ccw > 0 if three points make a counter-clockwise turn, clockwise if ccw < 0, and collinear if ccw = 0. x Many concepts and codes are referred from these articles. Berechnung im zweidimensionalen Fall. The first covered the Jarvis March and here I'll be covering the Graham Scan. All rights reserved. {\\displaystyle P_{1}=(x_{1},y_{1})} In Graham Scan, firstly the \u2026 Graham's Scan Given a set of points on the plane, Graham's scan computes their convex hull.The algorithm works in three phases: Find an extreme point. x Run the DFS-based algorithms on the following graph. y I have to implement the graham scan algorithm for convex hull but the problem is I'm not able to find a pseudo code that gives all the info. Graham\u2019s Scan The Graham\u2019s scan algorithm begins by choosing a point that is de\ufb01nitely on the convex hull and then iteratively adding points to the convex hull. In the late 1960s, the best algorithm for convex hull was O(n 2).At Bell Laboratories, they required the convex hull for about 10,000 points and they found out this O(n 2) was too slow. At around the same time of the Jarvis March, R. L. Graham was also developing an algorithm to find the convex hull of a random set of points .Unlike the Jarvis March, which is an operation, the Graham Scan is , where is the number of points and is the size for the hull. To understand the logic of Graham Scan we must undertsand what Convex Hull is: What is convex hull? Here is a brief outline of the Graham Scan algorithm: First, find the point with the lowest y-coordinate. Combinatoric problem in Haskell. At Bell Laboratories, they required the convex hull for about 10,000 points and they found out this O (n2) was too slow. x , in a \"left turn\" (because the algorithm advances to the next point Similarly it checks if the new point in the list $(5, 2)$ turns left or right from points $(0, 0)$ and $(7, 0)$. The basic concept is that we take an extreme point, sort all the other points angularly in O ( n log \u2061 n ) {\\displaystyle O(n\\log n)} , and scan angularly, with a stack in linear time to compute the convex hull. Retrieved August 23, 2018, from. [ 2 Graham's Scan algorithm will find the corner points of the convex hull. If two or more points are forming same angle, then remove all points of same angle except the farthest point from start. Active 1 month ago. An important special case, in which the points are given in the order of traversal of a simple polygon's boundary, is described later in a separate subsection. {\\displaystyle (x_{3},y_{3})} Convex hull You are encouraged to solve this task according to the task description, using any language you may know. Viewed 4k times 2. Next, the program pushes first three points from the sorted list to the stack. Second, they demonstrate that a modification of Graham scan which they call Graham-Fortune (incorporating ideas of Steven Fortune for numeric stability[7]) does overcome the problems of finite precision and inexact data \"to whatever extent it is possible to do so\". \u2212 Look at the last 3 points i Graham Scan Algorithm. Let points[0..n-1] be the input array. Call this point an Anchor point. 4. \"An Efficient Algorithm for Determining the Convex Hull of a Finite Planar Set\", \"Classroom examples of robustness problems in geometric computations\", Backward error analysis in computational geometry, https:\/\/en.wikipedia.org\/w\/index.php?title=Graham_scan&oldid=981736794, Short description is different from Wikidata, Creative Commons Attribution-ShareAlike License, This page was last edited on 4 October 2020, at 04:13. Sort the remaining points in increasing order of the angle they and the point P make with the x-axis. Sorting the points has time complexity O(n log n). However I'm still not getting a good convex hull when I'm running the program and I really don't know where to look at. Next point is $(9, 6)$. Next, point $(1, 4)$ is pushed into the stack. My graham scan implementation runs through the following steps: Parse the input from an input file. The animation was created with Matplotlib.. Computing the convex hull is a preprocessing step to many geometric algorithms and is the most important elementary problem in computational geometry, according to Steven Skiena in the Algorithm Design Manual. JavaScript Graham's Scan Convex Hull Algorithm. An implementation of Andrew's algorithm is given below in our chainHull_2D()routine. For this algorithm we will cover two similar fast 2D hull algorithms: the Graham scan, and Andrew's Monotone Chain scan. Graham scan implementation in Haskell. Algorithm check: Graham scan for convex hull (Python 2) Now I've been working on this code for the better part of two days, but somehow it still fails for some (unknown) test data. CMSC 754 Computational Geometry. 5. , Copyright \u00a9 2000\u20132017, Robert Sedgewick and Kevin Wayne. 2. Copyright \u00a9 2000\u20132017, Robert Sedgewick and Kevin Wayne. 1.Let H be the list of points on the convex hull, initialized to be empty 2.Choose p 0 to be the point with the lowest y-coordinate. Graham's Scan algorithm will find the corner points of the convex hull. 7. , Call this point P . Graham Scan algorithm for finding convex hull. Die Ermittlung der konvexen H\u00fclle von Punkten im hat als untere Schranke eine asymptotische Laufzeit von (\u2061); der Beweis erfolgt durch Reduktion auf das Sortieren von Zahlen. Last updated: Tue May 22 09:44:19 EDT 2018. Before reading this article, I recommend you to visit following two articles. 2D Convex hull exercise. Complexity. 1 (In real applications, if the coordinates are arbitrary real numbers, the function requires exact comparison of floating-point numbers, and one has to beware of numeric singularities for \"nearly\" collinear points.). ] {\\displaystyle (x_{2},y_{2})} , which is given by the expression That point is the starting point of the convex hull. {\\displaystyle [0,\\pi ]} I'm beginning to learn Haskell. I'm looking for general advice regarding the style and convention of my code, as well as best practices and ways to refactor several ugly places: Vector2D and its \u2026 {\\displaystyle {\\overrightarrow {P_{1}P_{3}}}} This is done using regex splitting. GrahamScan code in Java. ) Star 18 Fork 2 Star Code Revisions 11 Stars 18 Forks 2. ( I just can't seem to understand what data it could possibly be failing. , Since point $(1, 4)$ is the last point in the list, the algorithm terminates here. I've implemented the Graham Scan algorithm for detection of convex hull following the Real World Haskell book. {\\displaystyle P_{3}=(x_{3},y_{3})} A 2004 paper analyzed a simple incremental strategy, which can be used, in particular, for an implementation of the Graham scan. , x If there are two points with the same y value, then the point with smaller x coordinate value is considered. The procedure in Graham's scan is \u2026 This is the Graham scan algorithm in action, which is one common algorithm for computing the convex hull in 2 dimensions.. This algorithm first sorts the set of points according to their polar angle and scans the points to find the convex hull vertices. This pseudocode is adapted from Introduction to Algorithms. It uses a stack to detect and remove concavities in the boundary efficiently. If the lowest y-coordinate exists in more than one point in the set, the point with the lowest x-coordinate out of the candidates should be chosen. Program To Implement Graham Scan Algorithm To Find The Convex Hull program for student, beginner and beginners and professionals.This program help improve student basic fandament and logics.Learning a basic consept of Java program with best example. 3. Let the bottom-most point be P0. Posted by 5 years ago. Graham scan is an O(n log n) algorithm to find the convex hull of a set of points, which is exactly what this problem entails. If it turns right, we remove the item on the top of the stack and repeat this process for remaining items. y Convex Hull Algorithms Eric Eilberg Denison University Abstract This paper discusses the origins of the convex hull, and the development of algorithms designed to solve them. Consider each point in the sorted array in sequence. x Advent of Code 2018 Day 13 - Detect mine cart collisions. is removed). Embed. Dijkstra's Algorithm in Haskell. , 4. Also, this convex hull has the smallest area and the smallest perimeter of all convex polygons that contain S. 2D Hull Algorithms. \u03c0 The algorithm takes O(n log h) time, where h is the number of vertices of the output (the convex hull). Simple implementation to calculate a convex hull from a given array of x, y coordinates, the convex hull's in js I found either were a little buggy, or required dependencies on other libraries. The points in the stack are the convex hull. Viewed 2k times 1. Implementation of Graham Scan algorithm in Haskell. The procedure in Graham's scan is \u2026 Visualization : Algorithm : Find the point with the lowest y-coordinate, break ties by choosing lowest x-coordinate. Next, the set of points must be sorted in increasing order of the angle they and the point P make with the x-axis. Combinatoric problem in Haskell. JavaScript Graham's Scan Convex Hull Algorithm. (If at any stage the three points are collinear, one may opt either to discard or to report it, since in some applications it is required to find all points on the boundary of the convex hull.). The MIT Press. 2 It has the same basic properties as Graham's scan. Add P to the convex hull. P 7. This is the Graham scan algorithm in action, which is one common algorithm for computing the convex hull in 2 dimensions. x This point will be the pivot, is guaranteed to be on the hull, and is chosen to be the point with largest y coordinate. Sei = {} eine endliche Punktmenge. Then let the result be stored in the stack. We start with the most basic brute force method, Graham\u2019s Scan, progressing to the Jarvis March, then to Quick-hull and convex hulls in N-space. Graham scan is an algorithm to compute a convex hull of a given set of points in O(nlog\u2061n)time. The Graham Scan algorithm has the optimal worst-case complexity when not taken account output-sensitivity. This visualization was made within the scope of a course. ; Sort the points in order of increasing angle about the pivot. 3 x If we find any collinear points, we keep the furthest point from $P_0$ and remove all other points. 1 Find the points which form a convex hull from a set of arbitrary two dimensional points. The algorithm is asymptotically optimal (as it is proven that there is no algorithm asymptotically better), with the exception of a few problems where \u2026 \u2212 x {\\displaystyle (x_{2}-x_{1})(y_{3}-y_{1})-(y_{2}-y_{1})(x_{3}-x_{1})} I'm looking for general advice regarding the style and convention of my code, as well as best practices and ways to refactor several ugly places: Vector2D and its \u2026 6. I chose to write the implementations in C because of its execution speed, my familiarity with the language, and because I enjoy coding in it. The cosine is easily computed using the dot product, or the slope of the line may be used. First, some point (not necessarily one of the points in input) is identified which is definitely inside the convex hull. The algorithm takes O \u2026 = Last active Nov 6, 2020. The idea is to start at one extreme point in the set (I chose the bottom most point on the left edge) and sweep in a circle. Simple implementation to calculate a convex hull from a given array of x, y coordinates, the convex hull's in js I found either were a little buggy, or required dependencies on other libraries. ( The overall time complexity is therefore O(n log n), since the time to sort dominates the time to actually compute the convex hull. 2 T he first paper published in the field of computational geometry was on the construction of convex hull on the plane. , There have been numerous algorithms of varying complexity and effiency, devised to compute the Convex Hull of a set of points. The Graham scan has much better worst-case performance than the Jarvis march, but is also more complicated. \u2192 Let points [0..n-1] be the input array. Add p 0 to H since p 0 is de\ufb01nitely in the convex hull. 1 So i need to make a Convex hull using Graham scan algorithm, but i have problem, i get this kinda convex: void draw_line(Line l, Canvas& canvas) { canvas.draw_line(l.a, l.b); } double drandom(){ return rand() * 1. Graham's scan algorithm is a method of computing the convex hull of a finite set of points in the plane with time complexity O (n log \u2061 n) O(n \\log n) O (n lo g n).The algorithm finds all vertices of the convex hull ordered along its boundary . The sorted points are $[(0, 0), (7, 0), (3, 1), (5, 2), (9, 6), (3, 3), (5,5), (1, 4)]$. 2 Writing monadic Haskell to evaluate arithmetic expression . Following is Graham\u2019s algorithm. I've got an assignment in which I need to make a convex hull using Graham algorithm. Run Graham-Scan-Core algorithm to find convex hull of C 0. arthur-e \/ graham_hull.py Forked from tixxit\/hull.py. Call this point P. This step takes O(n), where n is the number of points in question. The implementation of the Graham Scan is short, but sweet. Cormen, T. H., Leiserson, C. E., Rivest, R. L., & Stein, C. (n.d.). If numeric precision is at stake, the comparison function used by the sorting algorithm can use the sign of the cross product to determine relative angles. Consider the general case when the input to the algorithm is a finite unordered set of points on a Cartesian plane. # find the point with minimum y coordinate, # in case of tie choose the point with minimun x-coordinate. If the given point belongs to the upper set, we check the angle made by the line connecting the second last point and the last point in the upper convex hull, with the line connecting the last point in the upper convex hull and the current point. convex hull by using Graham's Scan Algorithm. Run Graham-Scan-Core algorithm to find convex hull of C 0. ) All gists Back to GitHub Sign in Sign up Sign in Sign up {{ message }} Instantly share code, notes, and snippets. 1.Let H be the list of points on the convex hull, initialized to be empty 2.Choose p 0 to be the point with the lowest y-coordinate. ) This Java program submitted by Rishabh Singh. Now the stack contains the convex hull, where the points are oriented counter-clockwise and P0 is the first point. With the basics in place, we are ready to understand the Graham Scan Convex Hull algorithm. If the result is 0, the points are collinear; if it is positive, the three points constitute a \"left turn\" or counter-clockwise orientation, otherwise a \"right turn\" or clockwise orientation (for counter-clockwise numbered points). Arbitrary graham scan convex hull algorithm dimensional points algorithms: the Graham scan algorithm will find the are! A series of 3 on 2D convex hull you are encouraged to solve this task according the! # find the points based on the anti-clock wise direction from the start point this ( graham scan convex hull algorithm. R. L. Graham developed his simple and efficient algorithm in action, which is one common algorithm for convex.. Dimensional points are always graham scan convex hull algorithm the past run Graham-Scan-Core algorithm to compute a convex hull following the World! N\\Log n ) $is given below Graham-Scan-Core algorithm to compute a convex hull of a tie choose. On Foundations of computer Science Vol: graham scan convex hull algorithm the corner points of same angle, then the point with same! 09:44:19 EDT 2018 input file corner points of same angle except the farthest point from start made by the with... The above algorithm is a brief outline of the line may be used with \u201c Information Search Analysis... 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Detect and remove concavities in the sorted array in sequence ( proposed in graham scan convex hull algorithm... Array of points is the Graham scan algorithm for detection of convex hull Python. Algorithm check: Graham scan algorithm has the same graham scan convex hull algorithm, $( 1, 4 )$ is into. With \u201c Information Search and Analysis Skills \u201d and our main topic convex! Point is not part of the program sorts the set of points on set... Field of computational geometry was on the top of the convex hull in graham scan convex hull algorithm nLogn! Not require computing the convex hull ordered along its graham scan convex hull algorithm used here is a tie, the second-to-last is. Similar idea, and Andrew 's Monotone Chain algorithm used here is graham scan convex hull algorithm 's scan convex hull of 's... Sorted in increasing order graham scan convex hull algorithm increasing angle about the pivot 10 months.! 5, 5 ) \\$ 09:44:19 EDT 2018 the polar angle and scans the points has complexity! Understand what data it could possibly graham scan convex hull algorithm failing s algorithm let points [ 0.. n-1 ] be input. Be covering the Graham scan has much better worst-case performance than the Jarvis March, but also. But see if you have some nails stuck on a set of points graham scan convex hull algorithm to polar. Geometry was on the basis of their order if the next point the... Paper is our assignment with \u201c graham scan convex hull algorithm Search and Analysis Skills \u201d and our main about...\n2020 graham scan convex hull algorithm","date":"2021-06-17 06:34:32","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\": 0.4334423243999481, \"perplexity\": 739.7390999143344}, \"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-2021-25\/segments\/1623487629209.28\/warc\/CC-MAIN-20210617041347-20210617071347-00581.warc.gz\"}"}
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February 2011 Torex Auction - Session 3 - Page 3 of 18 - Moore Numismatic Auctions, Inc.
DOLLAR LOT. 1949. 1960. Both MS-63; 1961. 1962. 1967. All MS-62; 1951. 1953, SF. 1954. 1956. 1957. 1958. 1959. 1963. 1964. 1965, T.1.
3125 - 1950. ICCS Mint State-67. A superb Gem. Very light traces of blue toning.
1950. ICCS Mint State-67. A superb Gem. Very light traces of blue toning.
1950, 1951, 1952 W.L, 1954. All Extra Fine-45; 1950, 1951, 1952, W.L. All AU-50; 1953, NSF. 1955. 1956. 1957. All CH AU-55; 1953, SF, 1954.
3129 - 1950. PCGS graded Proof-Like-65. Medium toned obverse. More brilliant reverse.
1950. PCGS graded Proof-Like-65. Medium toned obverse. More brilliant reverse.
3130 - 1950. PCGS graded Proof-Like-64. Medium toned reflective fields.
1950. PCGS graded Proof-Like-64. Medium toned reflective fields.
1950. ICCS Proof-Like-64. Traces of light blue and violet toning. 1951. ICCS Proof-Like-65. Brilliant. Lot of two (2) dollars.
3132 - 1950. PCGS graded SPECIMEN-64. A brilliant dollar, with deep mirror fields.
3133 - 1950. Arnprior. ICCS Mint State-65. A brilliant gem with the faintest misting of pale blue toning.
1950. Arnprior. ICCS Mint State-65. A brilliant gem with the faintest misting of pale blue toning.
3134 - 1951. 1951 SWL. 1951 Arnprior. All three (3) dollars are ICCS Mint State-60.
1951. 1951 SWL. 1951 Arnprior. All three (3) dollars are ICCS Mint State-60.
3136 - 1951. PCGS graded Proof-Like-64. Attractive medium heavy dark golden toning.
1951. PCGS graded Proof-Like-64. Attractive medium heavy dark golden toning.
3137 - 1951. Arnprior. PCGS graded Mint State-64. Another fully brilliant and lustrous dollar.
1951. Arnprior. PCGS graded Mint State-64. Another fully brilliant and lustrous dollar.
3138 - 1951. Arnprior. PCGS graded Mint State-64. Fully brilliant and lustrous.
1951. Arnprior. PCGS graded Mint State-64. Fully brilliant and lustrous.
3139 - 1951. Arnprior. PCGS graded Mint State-64. Fully brilliant and lustrous.
3140 - 1951. Arnprior. PCGS graded Mint State-64. A fully brilliant and lustrous dollar.
1951. Arnprior. PCGS graded Mint State-64. A fully brilliant and lustrous dollar.
1951. Arnprior. Both ICCS and PCGS graded Mint State-64. Sharply struck. A fully brilliant and lustrous dollar.
3142 - 1951. Arnprior. ICCS Mint State-63. Nicely toned.
1951. Arnprior. ICCS Mint State-63. Nicely toned.
3143 - 1951. Arnprior. ICCS Mint State-63. Brilliant.
1951. Arnprior. ICCS Mint State-63. Brilliant.
3144 - 1952. Water Lines. ICCS Mint State-65. Light to medium heavy toning.
1952. Water Lines. ICCS Mint State-65. Light to medium heavy toning.
3145 - 1952. Water Lines. ICCS Mint State-65. Light to medium heavy, attractive toning. A Gem.
1952. Water Lines. ICCS Mint State-65. Light to medium heavy, attractive toning. A Gem.
3146 - 1952. Water Lines. CH-1952 Rev. 009. ICCS Mint State-65. Light to medium heavy toning.
1952. Water Lines. CH-1952 Rev. 009. ICCS Mint State-65. Light to medium heavy toning.
3147 - 1952. W.L. ICCS Mint State-64. Medium toning; 1953. NSF. Cameo. ICCS Mint State-63; 1953. S.F.
1952. W.L. 1953, NSF. Both PCGS graded Mint State-64. 1963. ICCS Proof-Like-66. Cameo. Lot of three (3) brilliant dollars.
3149 - 1952. NWL. PCGS graded Proof-Like-65. Attractive mirror fields.
1952. NWL. PCGS graded Proof-Like-65. Attractive mirror fields.
3150 - 1953. S.F. ICCS Mint State-65. 'Cameo'. A brilliant coin, with a light cameo contrast.
1953. S.F. ICCS Mint State-65. 'Cameo'. A brilliant coin, with a light cameo contrast.
3151 - 1953. S.F. NGC graded Mint State-65. Light to medium heavy golden-amber toning.
3152 - 1953. S.F. NGC graded Mint State-65. Light to medium heavy golden-amber toning.
3153 - 1953. S.F. NGC graded Mint State-65. Light golden-amber toning.
1953. S.F. NGC graded Mint State-65. Light golden-amber toning.
3154 - 1953. NSF. ICCS Mint State-65. A very lightly toned Gem.
1953. NSF. ICCS Mint State-65. A very lightly toned Gem.
3155 - 1953. SF. Short Water Lines. ICCS Proof-Like-65. Shades of gold and red.
1953. SF. Short Water Lines. ICCS Proof-Like-65. Shades of gold and red.
3156 - 1953. SF. ICCS Proof-Like-65. 'Cameo'. Deep mirror fields.
1953. SF. ICCS Proof-Like-65. 'Cameo'. Deep mirror fields.
3157 - 1954. ICCS Mint State-65. Light, multi-hued 'P.Q.' toning. A rare Gem.
1954. ICCS Mint State-65. Light, multi-hued 'P.Q.' toning. A rare Gem.
3159 - 1954. PCGS graded Proof-Like-67. 'Cameo' to 'Heavy Cameo' contrast. A brilliant Gem.
1954. PCGS graded Proof-Like-67. 'Cameo' to 'Heavy Cameo' contrast. A brilliant Gem.
3160 - 1954. ICCS Proof-Like-66. 'Heavy Cameo'. A brilliant gem Proof-Like dollar.
1954. ICCS Proof-Like-66. 'Heavy Cameo'. A brilliant gem Proof-Like dollar.
3161 - 1954. ICCS Proof-Like-65. 'Cameo'. Touches of golden toning.
1954. ICCS Proof-Like-65. 'Cameo'. Touches of golden toning.
3162 - 1954. PCGS graded Proof-Like-65. 'Cameo' contrast. Very lightly toned.
1954. PCGS graded Proof-Like-65. 'Cameo' contrast. Very lightly toned.
3163 - 1954. NGC graded Proof-Like-64. Brilliant 'Cameo' contrast.
1954. NGC graded Proof-Like-64. Brilliant 'Cameo' contrast.
3164 - 1954. ICCS Proof-Like-64. 'Cameo'. Brilliant.
1954. ICCS Proof-Like-64. 'Cameo'. Brilliant.
3165 - 1954. Proof-Like-64. 'Cameo' contrast.
3166 - 1954. ICCS Proof-Like-64; 1955. ICCS Proof-Like-65. Lot of two (2) brilliant dollars.
3167 - 1955. ICCS Mint State-65. Light to medium heavy blue and purple toning. Another gem.
3170 - 1955. 1955, Arnprior. With 'Die Breaks'. Both dollars are ICCS Mint State-60.
1955. 1955, Arnprior. With 'Die Breaks'. Both dollars are ICCS Mint State-60.
3171 - 1955. Arnprior. With 'Die Breaks'. ICCS mint State-64. Brilliant.
1955. Arnprior. With 'Die Breaks'. ICCS mint State-64. Brilliant.
3172 - 1955. Arnprior. With 'Die Breaks'. PCGS graded Mint State-63. Very lightly toned.
1955. Arnprior. With 'Die Breaks'. PCGS graded Mint State-63. Very lightly toned.
3173 - 1955. ICCS Proof-Like-66. Brilliant.
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"redpajama_set_name": "RedPajamaC4"
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News Matches Results Events. It was a book that helped me find my voice and let me know that I was not alone and that I was not the first girl to have her body violated. It marked your one-month anniversary being forcefully held in an adult prison by Commissioner Joette Katz without a crime committed or charges filed. When there was no blueprint, Maya Angelou created one — a blueprint which became mine. It reminded me that despite the heaviness of the world, you are a young person who has never asked for much. Only month later somebody told me that she's 15, I was shoked a bit but still glad that it happened. Our community is also navigating some lows: I had similar dreams at 16, and went to my prom alone.
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"redpajama_set_name": "RedPajamaC4"
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Q: Neil Turok's last exercise! In the last video of a series of lectures given by Neil Turok on GR in perimeter, he wants to give an exercise. Unfortunately (for us) the students shout, we've already done that. Although he does reveal some of the exercise he has in mind. Can any one help me and tell what the exercise probably is? (It sounds like a well known fact to prove, so you may know about it if you know GR!)
His words go : "show that the maximum proper time exprienced..., sorry? Oh you did that? Oh fine, great, you've done it! So you saw
$$\frac{\pi r_s}{2}$$
right? Ok! great haha don't show! Arright, thank you very much! Ok :)))"
A: $\pi r_s/2$ is the maximum proper time possible for an observer at the event horizon to fall to the singularity, so this is probably the exercise that Turok had in mind.
The calculation can be found in various sources, but I recommend you read No Way Back: Maximizing survival time below the Schwarzschild event horizon by Geraint F. Lewis and Juliana Kwan as this goes into the calculation in some detail.
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\section{Introduction}
Quantum measurement is a subject of importance both as a central
part of quantum theory \cite{Zurek} and for its relevance to
quantum information theory \cite{Nielsen}. At the most basic
level, the measurement is commonly thought of as a sudden
projective operation that leaves the measured system in an
eigenstate of the probed operator. It should be kept in mind,
however, that the measurement process is a physical process that
follows from a physical interaction between the measured system
and the measuring device. As a result, any realistic measurement
is carried out over a finite duration. One can therefore have the
situation where the measurement process occurs concurrently with
other physical processes. This is the situation that we consider
in this paper.
Although the analysis and results of this paper should be valid
for a variety of different physical settings, we find it useful to
formulate the analysis in terms of a specific setup. In this
paper, we use the example of a double-quantum-dot charge qubit
that is probed by a quantum point contact (QPC)
\cite{Gurvitz,Korotkov,Makhlin,Goan,Pilgram,Jordan,Johansson,Ouyang,Reuther,Clerk,Romito,Kurotani,QuantumCapacitance,Kreisbeck}.
The principle of the measurement is the fact that the current in
the QPC is sensitive to the charge state of the qubit. By
measuring the current passing through the QPC, one can infer the
state of the qubit. One limitation that arises in practical
situations is that, in order to minimize the effects of the
detector on the qubit at times when no measurement is performed
(e.g.~during the implementation of quantum gates), the
qubit-detector coupling is set to a value that is small compared
to the qubit's energy scale. As a result one must deal with some
form of weak-measurement regime. This type of weak,
charge-sensitive readout works well when the qubit is biased such
that the charge states are eigenstates of the Hamiltonian and
therefore do not mix during the measurement. In this case one can
allow the detector to probe the qubit for as long as is needed to
obtain a high signal-to-noise ratio, without having to worry about
any intrinsic qubit dynamics.
In contrast to the simple situation described above, when the
detector weakly probes the charge state of the qubit while the
Hamiltonian induces mixing dynamics between charge states, it
becomes unclear how to interpret a given measurement signal. Given
the fact that the signal typically contains a large amount of
detector-related noise and the measurement causes unavoidable
relaxation in the qubit, it might seem that this type of
measurement cannot be used to determine the initial state of the
qubit, i.e.~at the time that the experimenter decides to perform
the measurement. Indeed, there have been a number of studies
analyzing the measurement-induced decoherence and the evolution of
the qubit's state in this situation
\cite{Gurvitz,Korotkov,Makhlin,Goan,Pilgram,Jordan,Kurotani}, but
not the question of how to take the measurement data and extract
from it information about the initial state of the qubit. This
question is a key issue for qubit-state readout and is the main
subject of this paper.
We shall show below that high-fidelity measurement information can
be extracted from the measurement data, provided that additional
decoherence mechanisms are weak and the readout signal can be
monitored at a sufficiently short timescale. It turns out that not
only the measurement result, but also the measurement basis, is
determined stochastically in this case. In spite of the
uncontrollability of the measurement basis, the measurement
results contain meaningful information about the initial state of
the qubit. In order to demonstrate this statement, we show how
these results can be used to perform quantum state tomography on
the qubit. These results show that under suitable conditions and
by proper analysis of the measurement data useful information
about the state of the qubit can be extracted from this data.
This paper is organized as follows: In Sec.~II we introduce the
theoretical model for describing a charge qubit probed by a QPC.
In Sec.~III we formulate the theoretical framework for the qubit
evolution conditioned on the QPC's output signal and how
measurement information can be extracted from this signal. We also
introduce an equation that supplements the stochastic master
equation for purposes of interpreting the measured QPC signal. In
Sec.~IV we present results of numerical calculations that display
a number of interesting results in this system, most importantly
the stochastic nature of the measurement basis and the
near-independence of the measurement fidelity from the relation
between the charge basis and energy eigenbasis. In Sec.~V we
discuss how decoherence can be introduced to the problem and how
one should interpret the measurement data in the presence of
decoherence. We analyze one simple example that shows how the
measurement data recorded at later times is less `valuable' than
that recorded at earlier times. Concluding remarks are given in
Sec.~VI. Some details of the calculations are given in the
appendices.
\section{Model}
\begin{figure}[h]
\includegraphics[width=8.0cm]{QPCWMFig1.eps}
\caption{(color online) Schematic diagrams of (a) a charge qubit
with charge states $\ket{L}$ and $\ket{R}$ measured by a quantum
point contact (QPC), (b) the probability distributions of the
possible QPC current values for the two charge states of the
qubit, (c) the filled energy levels of the QPC's electrodes and
the energy barriers for tunneling between the electrodes for the
two charge states of the qubit and (d) a typical QPC current
signal. The finite widths of the probability distributions in (b)
are a result of the finite measurement time. When the distance
between the two center points, $|\overline{I}_R-\overline{I}_L|$,
is much smaller than the widths of the distributions, the QPC
performs a weak measurement on the qubit in the short interval
under consideration (i.e., the time interval for which these
probability distributions are relevant). The fast fluctuations in
(d) convey the assumption that these fluctuations have a much
shorter timescale than the qubit's precession period. In plotting
(b-d) we have assumed that $\overline{I}_R>\overline{I}_L$, which
would be the case if the qubit is defined by an extra positive
charge (e.g.~a hole) tunneling between the two wells. Taking the
opposite case, i.e.~$\overline{I}_L>\overline{I}_R$, would only
result in a few sign changes in the analysis below. The main
results are not affected by this choice.}
\end{figure}
We consider a system composed of a charge qubit capacitively
coupled to a QPC, as illustrated in Fig.~1. The qubit can be
viewed as a system where a charged particle is trapped in a
double-well potential and can occupy, and tunnel between, the
localized ground states of the two wells. We shall denote these
states by $\ket{L}$ and $\ket{R}$.
During the measurement a voltage is applied to the QPC, and a
current flows through it. Except for the discussion given in
Sec.~V, we shall assume that the QPC does not induce any qubit
decoherence except that associated with the measurement-induced
projection. Before analyzing the general case, we first consider
the case where the qubit Hamiltonian is diagonal in the charge
basis. In this case, there is no mechanism by which the states
$\ket{L}$ and $\ket{R}$ mix during the system dynamics. As a
result, if the qubit is initially in the state $\ket{L}$, the
long-time-averaged QPC current will be given by $\overline{I}_L$,
and the qubit will remain in the state $\ket{L}$. A similar
statement applies to the state $\ket{R}$ of the qubit, with
corresponding QPC current $\overline{I}_R$. The QPC current
therefore serves as an indicator of the qubit's state in the
charge basis $\{\ket{L},\ket{R}\}$, as long as the qubit
Hamiltonian does not mix the states of this basis.
On any finite timescale, there will be fluctuations in the QPC
current, and the observed value might deviate from
$\overline{I}_L$ or $\overline{I}_R$. The longer the period over
which the averaging is made, the smaller the fluctuations. One can
therefore define a measurement timescale that determines how long
one needs to wait in order to distinguish between the states
$\ket{L}$ and $\ket{R}$. The relation between this timescale and
the qubit's Hamiltonian-induced precession period separates two
measurement regimes: strong and weak qubit-detector coupling. As
mentioned above, this separation is irrelevant when the qubit
Hamiltonian is diagonal in the charge basis.
For the remainder of this paper, we analyze the general case where
the qubit Hamiltonian is not necessarily diagonal in the charge
basis. The Hamiltonian of the combined qubit+QPC system can be
expressed as
\begin{equation}
\hat{H} = \hat{H}_{\rm q} + \hat{H}_{\rm QPC} + \hat{H}_{\rm int}.
\end{equation}
In describing the qubit, we shall use the eigenbasis of the qubit
Hamiltonian $\hat{H}_{\rm q}$. Thus $\hat{H}_{\rm q}$ can be
expressed as
\begin{equation}
\hat{H}_{\rm q} = \left( \begin{array}{cc}
-\frac{E}{2} & 0 \\
0 & \frac{E}{2} \\
\end{array}
\right) = - \frac{E}{2} \hat{\sigma}_z,
\end{equation}
where $E$ is the energy splitting between the qubit's two energy
levels, and $\hat{\sigma}_z$ is the $z$-axis Pauli matrix. We
shall express the ground and excited states of the Hamiltonian as
$\ket{0}$ and $\ket{1}$, respectively. The electric dipole moment
operator $\ket{R}\bra{R} - \ket{L}\bra{L}$ can be expressed as
$\hat{\sigma}_{\bf n}$, where $\bf n$ is a vector that represents
the direction of the charge basis relative to the energy
eigenbasis:
\begin{eqnarray}
\hat{\sigma}_{\bf n} & = & \ket{R}\bra{R} - \ket{L}\bra{L}
\nonumber \\
& = & \cos\beta \; \hat{\sigma}_z + \sin\beta \; \hat{\sigma}_x,
\end{eqnarray}
where $\beta$ represents the angle between the charge basis and
the energy eigenbasis. The states of the charge basis (i.e., the
eigenstates of $\hat{\sigma}_{\bf n}$) can be expressed as
\begin{eqnarray}
\ket{R} & = & \cos\frac{\beta}{2} \ket{0} + \sin\frac{\beta}{2}
\ket{1} \nonumber
\\
\ket{L} & = & \sin\frac{\beta}{2} \ket{0} - \cos\frac{\beta}{2}
\ket{1}.
\end{eqnarray}
The QPC Hamiltonian $\hat{H}_{\rm QPC}$ can be expressed as
\begin{eqnarray}
\hat{H}_{\rm QPC} & = & \sum_m \epsilon_{S,m} \hat{a}_m^{\dagger}
\hat{a}_m + \sum_n \epsilon_{D,n} \hat{b}_n^{\dagger} \hat{b}_n
\nonumber \\ & & + \sum_{\scriptsize \begin{array}{c}\{m,n\}\in \\
{\rm tunneling} \\ {\rm channels} \end{array}} J_{m,n} \left(
\hat{b}_n^{\dagger} \hat{a}_m + \hat{a}_m^{\dagger} \hat{b}_n
\right),
\end{eqnarray}
where the operators $\hat{a}_m$ and $\hat{b}_n$, respectively,
annihilate electrons in the source and drain electrodes of the
QPC, $\hat{a}_m^{\dagger}$ and $\hat{b}_n^{\dagger}$ are their
hermitian conjugates, $\epsilon_{S,m}$ and $\epsilon_{D,n}$ are
the energies of single-particle quantum states in the source and
drain electrodes, and $J_{m,n}$ represents tunneling matrix
elements between the two electrodes. We take the qubit-QPC
interaction Hamiltonian to be of the form \cite{Gurvitz}
\begin{equation}
\hat{H}_{\rm int} = \hat{\sigma}_{\bf n} \sum_{\scriptsize
\begin{array}{c}\{m,n\}\in \\ {\rm tunneling} \\ {\rm channels}
\end{array}} \delta \! J_{m,n} \left( \hat{b}_n^{\dagger} \hat{a}_m
+ \hat{a}_m^{\dagger} \hat{b}_n \right),
\end{equation}
such that the tunneling matrix elements (and therefore the QPC
current) depend on the state of the qubit in the charge basis. The
QPC and its operation are illustrated in Figs.~1(b-d). When $J$
and $\delta \! J$ are independent of $m$ and $n$, the microscopic
degrees of freedom of the QPC are not affected by, and therefore
do not carry any information about, the state of the qubit. This
information is carried only by the total number of electrons that
have tunneled through the QPC. In the remainder of this paper, we
shall assume that this is the case. We close this section by
mentioning that the above description of the microscopic structure
of the QPC is not needed for the analysis below, provided one
accepts the description of the QPC operation explained in Sec.~III
as a phenomenological model of the detector. Indeed, the analysis
below is not restricted to the case of a charge qubit probed by a
QPC.
\section{Measurement- and Hamiltonian-induced dynamics}
In this section we introduce the theoretical framework for
analyzing the qubit state evolution and interpreting the observed
measurement data.
\subsection{Short-time evolution}
We start our analysis by considering a short time interval between
times $t$ and $t+\delta t$. For definiteness, we assume that
during this time interval a large number of electrons tunnel
through the QPC, such that it is natural to define a QPC current
$I(t)$ during this short interval (The current would be defined as
the amount of charge that has passed through the QPC divided by
$\delta t$; The small-current case will be discussed in
Sec.~IV.F). We also assume that a weak-coupling regime exists for
a properly chosen value of $\delta t$, which means that the
QPC-current probability distributions (for the states $\ket{L}$
and $\ket{R}$) are broad and almost completely overlap, as shown
in Fig.~1(b). The coherence time of the QPC and the time
resolution of the QPC's output signal are both assumed to be much
shorter than the precession period of the qubit. As a result, for
a properly chosen value of $\delta t$ one can treat the different
time intervals as independent measurement processes, with each
measurement being made in the charge basis.
As explained in Appendix A, the QPC-current probability
distribution for the qubit state $\ket{L}$ can be expressed as
\begin{equation}
P_L[I,\delta \! I,\delta t] = \sqrt{\frac{2\delta t \left( \delta
\! I \right)^2}{\pi \tau_m \left( \Delta \overline{I} \right)^2}}
\exp \left\{ - \frac{2\delta t \left(
I-\overline{I}_L\right)^2}{\tau_m
\left(\Delta\overline{I}\right)^2} \right\},
\end{equation}
and a similar expression can be derived for the qubit state
$\ket{R}$. Here we have defined a number of useful quantities:
$\delta \! I$ is the size of a finite interval of QPC currents
that we identify with a single value. One could say that with this
definition we are turning the probability distributions in
Fig.~1(b) into histograms with discrete possible values for the
current $I$. This definition will be useful for both the analytic
and numerical calculations below. The characteristic measurement
time $\tau_m$ defines the timescale needed for the QPC output to
distinguish between the states $\ket{L}$ and $\ket{R}$, and
$\Delta\overline{I}=\overline{I}_R-\overline{I}_L$, where for
definiteness we take $\overline{I}_R>\overline{I}_L$.
We can now construct matrices that describe the qubit-state
evolution depending on the observed QPC current $I(t)$. When a
given value of $I(t)$ is observed in the QPC, the density matrix
$\rho_{\rm q}(t)$ describing the quantum state of the qubit is
projected (possibly partially) according to the observed value.
This projection of the qubit's state is described by a $2 \times
2$ matrix (or propagator) that we shall call
$\hat{U}_M[I(t),\delta \! I,\delta t]$:
\begin{widetext}
\begin{equation}
\rho_{\rm q}(t+\delta t) = \frac{1}{{\rm Tr}
\left\{\hat{U}_M^{\dagger}[I(t),\delta \! I,\delta
t]\hat{U}_M[I(t),\delta \! I,\delta t] \rho_{\rm q}(t) \right\}}
\hat{U}_M[I(t),\delta \! I,\delta t] \rho_{\rm q}(t)
\hat{U}_M^{\dagger}[I(t),\delta \! I,\delta t].
\label{Eq:Density_matrix_projection}
\end{equation}
Measurement theory says that $\hat{U}_M[I(t),\delta \! I,\delta
t]$ obeys the equation \cite{Nielsen,Jacobs}
\begin{equation}
\hat{U}_M^{\dagger}[I(t),\delta \! I,\delta t]
\hat{U}_M[I(t),\delta \! I,\delta t] = P_L[I(t),\delta \! I,\delta
t] \ket{L}\bra{L} + P_R[I(t),\delta \! I,\delta t] \ket{R}\bra{R}.
\label{Eq:UMdaggerUM}
\end{equation}
Since we are keeping only the fundamentally necessary terms in the
projection dynamics, we ignore any unitary-transformation
component in $\hat{U}_M[I(t),\delta \! I,\delta t]$ that does not
affect Eq.~(\ref{Eq:UMdaggerUM}). Any such component can in
principle be determined experimentally, and it can then be
incorporated into the analysis straightforwardly; Such a component
could result if, for example, the different current states of the
QPC have different charge distributions that act as an effective
gate voltage on the charge qubit. Assuming a large overlap between
the QPC-current probability distributions, we find that the matrix
$\hat{U}_M[I(t),\delta \! I,\delta t]$ can be expressed in the
form
\begin{eqnarray}
\hat{U}_M[I(t),\delta \! I,\delta t] & = & \sqrt{P_L[I(t),\delta
\! I,\delta t]} \ket{L}\bra{L} + \sqrt{P_R[I(t),\delta \! I,\delta
t]} \ket{R}\bra{R} \nonumber \\
& = & \left( \frac{2\delta t \left( \delta \! I \right)^2}{\pi
\tau_m \left( \Delta \overline{I} \right)^2} \right)^{1/4} \left[
\exp \left\{ - \frac{\delta t \left[ I(t)-\overline{I}+\Delta
\overline{I}/2\right]^2}{\tau_m \left(\Delta\overline{I}\right)^2}
\right\} \ket{L}\bra{L} + \exp \left\{ - \frac{\delta t \left[
I(t)-\overline{I}-\Delta \overline{I}/2\right]^2}{\tau_m
\left(\Delta\overline{I}\right)^2} \right\} \ket{R}\bra{R} \right]
\nonumber \\
& \propto & \exp \left\{ -\frac{\delta t \left[ I(t) -
\overline{I} \right]}{\tau_m \Delta\overline{I}} \right\}
\ket{L}\bra{L} + \exp\left\{ \frac{\delta t \left[ I(t) -
\overline{I} \right]}{\tau_m \Delta\overline{I}} \right\}
\ket{R}\bra{R}
\nonumber \\
& \approx & 1 + \frac{\delta t \left[ I(t) - \overline{I}
\right]}{\tau_m \Delta\overline{I}} \left( \ket{R}\bra{R} -
\ket{L}\bra{L} \right) \nonumber \\
& = & 1 + \frac{\delta t \left[ I(t) - \overline{I}
\right]}{\tau_m \Delta\overline{I}} \hat{\sigma}_{\bf n},
\label{Eq:UM}
\end{eqnarray}
where we have defined
$\overline{I}=(\overline{I}_L+\overline{I}_R)/2$.
In addition to the measurement-induced evolution described by the
matrix $\hat{U}_M$ of Eq.~(\ref{Eq:UM}), the qubit Hamiltonian
induces a unitary evolution of the qubit's state. This
contribution to the qubit-state evolution (over the time interval
from $t$ to $t+\delta t$) is described by the matrix
\begin{eqnarray}
\hat{U}_H [\delta t] = \exp\left\{-i \hat{H}_{\rm q} \delta t
\right\} \approx 1 + i \frac{E}{2} \delta t \hat{\sigma}_z.
\end{eqnarray}
The matrices $\hat{U}_M[I(t),\delta \! I,\delta t]$ and $\hat{U}_H
[\delta t]$ can now be combined to give the total evolution matrix
\begin{eqnarray}
\hat{U}[I(t),\delta \! I,\delta t] & \approx &
\hat{U}_M[I(t),\delta \! I,\delta t] \times \hat{U}_H [\delta t]
\nonumber \\
& \propto & 1 + \frac{\delta t \left[ I(t) - \overline{I}
\right]}{\tau_m \Delta\overline{I}} \hat{\sigma}_{\bf n} + i
\frac{E}{2} \delta t \hat{\sigma}_z.
\label{Eq:U_combined}
\end{eqnarray}
Note that the high-resolution requirement in the QPC-current
signal, which is one of our assumptions mentioned earlier in this
section, enters in obtaining the above equation: both
$\hat{U}_M[I,\delta \! I,\delta t]$ and $\hat{U}_H [\delta t]$ are
approximately proportional to the unit matrix, with lowest-order
corrections of order $\delta t$. Therefore they commute to first
order in $\delta t$, and they can be treated as commuting
operators when $E \delta t \ll 1$, i.e.~when they both induce
small changes to the quantum state of the qubit.
When a given QPC output signal $I(t)$ [starting from the initial
time $t=0$ until the final time $t_f$] is observed, one can take
the corresponding short-time evolution matrices explained above
and use them to construct the total evolution matrix $\hat{U}_{\rm
Total}[I(t:0\rightarrow t_f),\delta \! I,\delta t]$. This matrix
describes the evolution of the qubit state from $t=0$ until
$t=t_f$, given that the current signal $I(t)$ was observed (with
discretization parameters $\delta \! I$ and $\delta t$). Using the
unit matrix as the total evolution matrix for $t=0$, we find that
\begin{equation}
\hat{U}_{\rm Total}[I(t:0\rightarrow t_f),\delta \! I,\delta t] =
\hat{U}[I(t_f-\delta t),\delta \! I,\delta t] \times \cdots \times
\hat{U}[I(0),\delta \! I,\delta t].
\label{Eq:U_Total}
\end{equation}
In Sec. III.B we shall explain how measurement information can be
extracted from the matrix $\hat{U}_{\rm Total}$.
\subsection{Extracting information from the matrix
$\hat{U}_{\rm Total}[I(t:0\rightarrow t_f), \delta \! I,\delta
t]$}
The evolution of the qubit state is obtained by taking into
account a long sequence of weak measurements and
Hamiltonian-induced unitary transformations. This long sequence of
events, however, is equivalent to a simple scenario in which a
single measurement is made on the qubit and the state of the qubit
is rotated after the measurement. In order to obtain a
quantitative description of this alternative interpretation of the
measurement data, one takes the $2 \times 2$ matrix $\hat{U}_{\rm
Total}[I(t:0\rightarrow t_f),\delta \! I,\delta t]$ and divides it
through to a polar decomposition into two parts, a measurement
matrix $\hat{U}_{\rm Meas}[I(t:0\rightarrow t_f),\delta \!
I,\delta t]$ followed by a unitary transformation $\hat{U}_{\rm
Rot}[I(t:0\rightarrow t_f),\delta \! I,\delta t]$:
\begin{equation}
\hat{U}_{\rm Total}[I(t:0\rightarrow t_f),\delta \! I,\delta t] =
\hat{U}_{\rm Rot}[I(t:0\rightarrow t_f),\delta \! I,\delta t]
\times \hat{U}_{\rm Meas}[I(t:0\rightarrow t_f),\delta \! I,\delta
t].
\end{equation}
\end{widetext}
The matrix $\hat{U}_{\rm Meas}[I(t:0\rightarrow t_f),\delta \!
I,\delta t]$ has the form
\begin{equation}
\hat{U}_{\rm Meas}[I(t:0\rightarrow t_f),\delta \! I,\delta t] =
\sqrt{P_1} \ket{\psi_1}\bra{\psi_1} + \sqrt{P_2}
\ket{\psi_2}\bra{\psi_2},
\end{equation}
where $\ket{\psi_1}$ and $\ket{\psi_2}$ are two orthogonal states
(the indices are assigned such that $P_1 \geq P_2$). In order to
characterize the measurement matrix in terms of the information it
provides about the state of the qubit, it is useful to consider
the case where one initially has no information about this state.
The qubit therefore starts in the maximally mixed state
\begin{equation}
\rho_{\rm max. \ mixed} = \left(
\begin{array}{cc}
\frac{1}{2} & 0 \\
0 & \frac{1}{2}
\end{array}
\right).
\end{equation}
After a measurement that produces the outcome corresponding to the
matrix $\hat{U}_{\rm Meas}[I(t:0\rightarrow t_f),\delta \!
I,\delta t]$ (and ignoring the presence of the rotation matrix
$\hat{U}_{\rm Rot}[I(t:0\rightarrow t_f),\delta \! I,\delta t]$
for purposes of this argument), the qubit ends up in a state
described by the density matrix
\begin{equation}
\rho_{\rm after \ Meas.} = \frac{P_1}{P_1+P_2}
\ket{\psi_1}\bra{\psi_1} + \frac{P_2}{P_1+P_2}
\ket{\psi_2}\bra{\psi_2}.
\end{equation}
This density matrix describes a statistical mixture of the states
$\ket{\psi_1}$ and $\ket{\psi_2}$, with a higher probability for
the former (except for the rare cases where $P_1=P_2$). One can
therefore say that, as a result of the measurement, one now knows
that the qubit is more likely to be in the state $\ket{\psi_1}$
than in the state $\ket{\psi_2}$, in contrast to the complete lack
of information at the initial time. The states $\ket{\psi_1}$ and
$\ket{\psi_2}$ therefore represent the measurement basis that
corresponds to the output signal $I(t)$. The measurement fidelity
is calculated as follows \cite{AshhabMeasurement}: Let us assume
that one is given a qubit that is in an unknown state, either
$\ket{\psi_1}$ or $\ket{\psi_2}$. The parameters $P_i$ are the
probabilities that the outcome defined by $I(t)$, $\delta \! I$
and $\delta t$ is obtained given that the qubit was initially in
the state $\ket{\psi_i}$. Upon observing this outcome, if one
wants to make a guess about the state of the qubit, one would
maximize the probability of making a correct guess by choosing the
state $\ket{\psi_1}$. The difference between the probability of
actually finding the state $\ket{\psi_1}$ and that of finding the
orthogonal state $\ket{\psi_2}$ is given by
\begin{equation}
F = \left|\frac{P_1-P_2}{P_1+P_2}\right|.
\end{equation}
This expression can therefore naturally be interpreted as the
measurement fidelity.
It is worth pausing here to comment on the issue of the
independence of the measurement basis from the measurement
outcome. One typically thinks of the measurement basis as being
determined by the experimenter when designing the experimental
setup and the measurement result (i.e., $+1$ or $-1$ along the
measurement axis) being determined stochastically according to the
rules of quantum mechanics based on the measurement basis used in
the experiment. The situation considered in this paper, however,
cannot be described using this simple picture. Here we have a
two-state quantum system and a measurement device that produces
one of many possible outcomes, not necessarily all providing
information in the same basis. In this case, the picture of the
measurement basis being independent of the outcome is not valid in
general. For example, it is possible in a general setting for a
certain state $\ket{\psi}$ to be a possible result of the data
analysis presented above, but not the state orthogonal to it. Such
a situation arises in the problem analyzed in
Ref.~\cite{AshhabMeasurement}, and it complicates the
interpretation of the measurement data. Fortunately, because in
this paper we are dealing with two symmetric probability
distributions [see Fig.~1(b)], we find that for every possible QPC
signal there is an `opposite' or `flipped' signal, which can be
obtained by taking the mirror image about the central line in
Fig.~1(d) [this central line is defined by
$\overline{I}=(\overline{I}_L+\overline{I}_R)/2$]. It is not
difficult to verify that if a given signal corresponds to a
certain measured state $\ket{\psi}$, the `opposite' signal will
correspond to the orthogonal state with the same values of $P_1$
and $P_2$ (see Appendix B for the full derivation). An important
consequence of the above statement is that the total probability
of finding one of two opposite signals is independent of the
initial state of the qubit. Hence, the probability of obtaining a
certain measurement basis is independent of the initial state of
the qubit. This fact leads naturally to the conceptual
interpretation that the measurement basis is determined according
to some stochastic process, and the measurement result along that
basis is made after the basis has been determined.
To summarize, the QPC's output signal can be used to derive the
matrix $\hat{U}_{\rm Total}[I(t:0\rightarrow t_f),\delta \!
I,\delta t]$. This matrix can then be used to determine the
measurement basis, the measurement result (i.e., $\pm 1$ along the
measurement axis), the fidelity (or in other words, the degree of
certainty about the obtained measurement result) and the
post-measurement rotation. Note that all the analysis above is
performed independently of the initial state of the qubit. If one
knows the initial state, one can use the evolution matrices in
order to determine the probabilities of different outcomes, as
well as the final state of the qubit after the measurement. As we
shall show below, when the measurement fidelity approaches one,
the final state can be determined even without any knowledge about
the initial state.
It is worth mentioning here that the analysis presented above can
be applied not only to qubits, but also to systems with
higher-dimensional Hilbert spaces. The main complication in the
case of higher dimensions is that the description of the results
becomes more intricate and less transparent. In this context one
can contrast the simple visualization of any qubit density matrix
as a point in the so-called Bloch sphere, whereas there is no
visually simple parametrization for a general density matrix of a
three-state system.
\subsection{Incorporating the matrix
$\hat{U}_{\rm Total}[I(t:0\rightarrow t_f), \delta \! I,\delta t]$
into the formalism of stochastic master equations}
Equation (\ref{Eq:Density_matrix_projection}) describes the
projection of the qubit's density matrix conditioned on the
observation of the QPC output $I(t)$. That equation treats the
evolution as occurring in discrete steps. Constructing a master
equation means taking Eq.~(\ref{Eq:Density_matrix_projection}) to
first order in $\delta t$, and then taking the limit $\delta
t\rightarrow 0$ in order to obtain the corresponding
continuous-time differential equation \cite{Jacobs}. One tricky
point in carrying out this procedure is the fact that the
stochastic quantity $I(t)-\overline{I}$ has fluctuations of size
$1/\sqrt{\delta t}$:
\begin{equation}
\left\langle \left(I(t)-\overline{I}\right)^2\right\rangle =
\frac{\tau_m \left( \Delta\overline{I} \right)^2}{4\delta t}.
\end{equation}
As a result, when keeping terms only to first order in $\delta t$,
one must also keep terms containing $\left(\delta t\right)^2
\times \left(I(t)-\overline{I}\right)^2$, etc. The quantity
$I(t)-\overline{I}$ can be expressed as its expectation value at
any give point in time, which is given by
$\Delta\overline{I}\langle\hat{\sigma}_{\bf n}\rangle/2$, plus a
fluctuation. It is customary to write the stochastic master
equation not in terms of the current fluctuation, but rather in
terms of a rescaled quantity, $\delta W$, whose standard deviation
is equal to $\sqrt{\delta t}$. It then follows that $\delta W$
should be defined using the relation
\begin{equation}
I(t) = \overline{I} + \frac{\Delta\overline{I}}{2}
\langle\hat{\sigma}_{\bf n}\rangle + \frac{\Delta\overline{I}
\sqrt{\tau_m}}{2 \delta t} \delta W.
\end{equation}
Substituting Eq.~(\ref{Eq:U_combined}) into
Eq.~(\ref{Eq:Density_matrix_projection}) [ignoring the subscript
$M$ here] and keeping terms to first order in $\delta t$, we find
that
\begin{widetext}
\begin{equation}
\rho_{\rm q}(t+\delta t) = \rho_{\rm q} - i \delta t \hat{H}_{\rm
q} \rho_{\rm q} + i \delta t \rho_{\rm q} \hat{H}_{\rm q} +
\frac{\delta W}{2\sqrt{\tau_m}} \left( \hat{\sigma}_{\bf n}
\rho_{\rm q} + \rho_{\rm q} \hat{\sigma}_{\bf n} - 2 \left\langle
\hat{\sigma}_{\bf n} \right\rangle \rho_{\rm q} \right) +
\frac{\delta t}{4\tau_m} \left( \hat{\sigma}_{\bf n} \rho_{\rm q}
\hat{\sigma}_{\bf n} - \rho_{\rm q} \right),
\end{equation}
where, for compactness, we have not expressed the time dependence
of $\rho_{\rm q}$ explicitly on the right-hand side [In other
words $\rho_{\rm q}$ should be interpreted as $\rho_{\rm q}(t)$].
This equation can now be rewritten in the form
\begin{equation}
\dot{\rho}_{\rm q} = - i \left[ \hat{H}_{\rm q} , \rho_{\rm q}
\right] + \frac{\xi(t)}{2\sqrt{\tau_m}} \left( \hat{\sigma}_{\bf
n} \rho_{\rm q} + \rho_{\rm q} \hat{\sigma}_{\bf n} - 2
\left\langle \hat{\sigma}_{\bf n} \right\rangle \rho_{\rm q}
\right) + \frac{1}{4\tau_m} \left( \hat{\sigma}_{\bf n} \rho_{\rm
q} \hat{\sigma}_{\bf n} - \rho_{\rm q} \right).
\label{Eq:SME}
\end{equation}
\end{widetext}
where $\xi(t)$ is obtained by using the rescaling $\xi(t)=\delta
W/\delta t$ and taking the limit $\delta t\rightarrow 0$:
\begin{eqnarray}
\left\langle \xi(t) \right\rangle & = & 0 \nonumber \\
\left\langle \xi(t) \xi(t') \right\rangle & = & \delta(t-t'),
\end{eqnarray}
where $\delta(t-t')$ is the Dirac delta function. Equation
(\ref{Eq:SME}) is the stochastic master equation for the system
under consideration.
The question now is how the matrix $\hat{U}_{\rm
Total}[I(t:0\rightarrow t_f), \delta \! I,\delta t]$ fits into the
picture of the stochastic master equation. One can derive a
stochastic equation for the matrix $\hat{U}_{\rm
Total}[I(t:0\rightarrow t_f), \delta \! I,\delta t]$ by
considering the change caused by a matrix of the form given in
Eq.~(\ref{Eq:U_combined}); note that the evolution of the matrix
$\hat{U}_{\rm Total}[I(t:0\rightarrow t_f), \delta \! I,\delta t]$
is governed by Eq.~(\ref{Eq:U_Total}). One problem that arises
here is that the matrix in the first line of
Eq.~(\ref{Eq:U_combined}) tends to zero when $\delta t \rightarrow
0$. Taking this limit therefore leads to ill-defined quantities.
In order to avoid these problems, we deal with a re-normalized
version of the matrix $\hat{U}[I(t),\delta \! I, \delta t]$, as
given in the second line of Eq.~(\ref{Eq:U_combined}); it should
be noted here that normalization of the matrix $\hat{U}_{\rm
Total}$ is irrelevant for purposes of extracting the measurement
basis and fidelity. It is now straightforward to see that an
unnormalized version of the matrix $\hat{U}_{\rm Total}$ obeys the
equation
\begin{equation}
\frac{d \hat{U}_{\rm Total}}{dt} = \left( \left[
\frac{\left\langle\hat{\sigma}_{\bf n}\right\rangle}{2\tau_m} +
\frac{\xi(t)}{2\sqrt{\tau_m}} \right] \hat{\sigma}_{\bf n} - i
\hat{H}_{\rm q} \hat{\sigma}_z \right) \hat{U}_{\rm Total}.
\label{Eq:SME_for_U}
\end{equation}
This equation supplements the stochastic master equation for
purposes of interpreting the QPC's output signal in terms of a
measurement result. In this context, one should note that what
Eq.~(\ref{Eq:SME}) gives is the qubit's density matrix as a
function of time. When $\beta\neq 0$ and the charge states mix
during the measurement, the density matrix at later times can
become completely different from that at the start of the
measurement process. Assuming that the experimenter does not have
any information about the qubit's state at $t=0$, he would be able
to use Eq.~(\ref{Eq:SME}) to determine the qubit's state at later
times, but he would not be able to cast the results in terms of
information that he has learned about the qubit's state at $t=0$.
The matrix $\hat{U}_{\rm Total}$, through the decomposition
explained in Sec.~III.B contains the information needed to make
such a statement about the measurement result. In fact, one can
say that, when taken in combination with the initial qubit density
matrix, Eq.~(\ref{Eq:SME_for_U}) replaces Eq.~(\ref{Eq:SME}): in
addition to containing the measurement information, the matrix
$\hat{U}_{\rm Total}$ can be used to calculate the density matrix
at any time.
\section{Numerical results and discussion}
We now present the results of our numerical calculations. Typical
parameters of the numerical calculations are as follows: The
discrete steps are taken to be $\delta t=0.1\pi/E$, such that one
period of the Hamiltonian-induced coherent oscillations would be
divided into 20 steps. The QPC current probability distributions
are discretized into 100 possible current values: each
distribution is Gaussian with a standard deviation $\sigma$ of 10
steps (thus there are about 50 possible values for the QPC current
with non-negligible occurrence probability). The distance between
the two center points of the current probability distributions
$\Delta \overline{I}$ determines the qubit-QPC coupling strength.
This parameter is varied when analyzing the effect of the
qubit-QPC coupling strength on the behaviour of the system. The
number of time steps in a single calculation was different for
different calculations. However, this number was always large
enough that the final qubit state reached perfect purity up to
less than one part in $10^5$, ensuring that none of our numerical
results were sensitive to the exact choice of this parameter. Each
measurement procedure is repeated up to $10^4$ times in order to
obtain accurate statistical averages. We have also used parameters
different from the typical ones given above and found that the
results are unaffected by the specific choice of these parameters.
A convenient parameter for purposes of characterizing the
qubit-QPC coupling strength is $E\tau_m/(2\pi)$, where $\tau_m$ is
the timescale needed to obtain sufficient QPC signal to read out
the state of the qubit (for the time being one can think of this
definition as applying to the case when $\beta=0$; but see below).
As the standard deviation of the QPC-output effective probability
distribution scales as $\sigma/\sqrt{N}$ when the measurement step
is repeated $N$ times, the measurement time $\tau_m$ can be
naturally defined as the product of the time step $\delta t$ and
the value of $N$ at which $2\sigma/\sqrt{N}=\Delta \! I$. $\tau_m$
is therefore given by
\begin{equation}
\tau_m = \frac{4\sigma^2 \delta t}{\Delta I^2}.
\end{equation}
We shall use the parameter $E\tau_m/(2\pi)$ for quantifying the
qubit-QPC coupling strength, and we shall use $\tau_m$ as a
characteristic measurement timescale when presenting our results
below.
\subsection{Stochastically determined measurement basis}
\begin{figure}[h]
\includegraphics[width=9.0cm]{QPCWMFig2.eps}
\caption{(color online) The spherical coordinates $\theta$ and
$\phi$ defining the stochastically determined measurement bases
obtained in simulations of the experiment under consideration
(note that, since the measurement basis is defined by an axis
extending in two opposite directions, only the upper hemisphere is
needed; any measurement axis will either lie in the plane of the
equator or have one end above the equator). Each figure contains
200 points. In each row, the qubit-QPC coupling strength is kept
fixed and $\beta$ is varied: $\beta=0$, $\pi/8$, $\pi/4$, $3\pi/8$
and $\pi/2$. In each column $\beta$ is kept fixed and the
qubit-QPC coupling strength is varied. In the top row,
$E\tau_m/(2\pi)=0.01$, which is deep in the strong-coupling
regime; The measurement basis is very close to the charge basis in
about 99\% of the runs. In the middle row, $E\tau_m/(2\pi)=0.2$,
which can be identified as the intermediate-coupling regime; The
measurement basis deviates substantially from the charge basis in
about one half of the runs. In the bottom row $E\tau_m/(2\pi)=5$,
which is in the weak-coupling regime; The measurement bases are,
in general, spread over the entire hemisphere. In generating this
figure, the initial state was taken to be the maximally mixed
state (note that the results are independent of the initial
state).}
\end{figure}
First, in Fig.~2 we show the spherical coordinates $\theta$ and
$\phi$ of a large number of (stochastically determined)
measurement bases for different qubit-detector coupling strengths
and different values of $\beta$. In the strong-coupling regime
(top row in Fig.~2), the parameter $E\tau_m/(2\pi)$ is much
smaller than one, and the measurement is completed before any
Hamiltonian-induced dynamics can occur. As a result, the
measurement basis is always the charge basis, i.e.~the natural
measurement basis for the detector under consideration [note here
that the charge basis is characterized by the direction
$(\theta,\phi)=(\beta,0)$]. As the qubit-detector coupling
strength is reduced (middle row in Fig.~2), the measurement bases
start to deviate from the charge basis, and they develop some
statistical spread. This region could be called the
intermediate-coupling regime. Deep in the weak-coupling regime
(bottom row in Fig.~2), the measurement bases are spread over all
the possible directions. The probability distribution of possible
measurement bases will be analyzed in Sec.~IV.D.
When $\beta=0$ (leftmost column in Fig.~2), the measurement basis
is always the charge basis, regardless of the qubit-QPC coupling
strength. This result is natural, since this case is the simple
one with no mixing between the states $\ket{L}$ and $\ket{R}$.
When $\beta=\pi/2$, all the possible measurement bases lie in the
$x$-$y$ plane. The reason behind this result lies in the fact that
this situation is equivalent to one where one makes a large number
of weak measurements in the $x$-$y$ plane with no
Hamiltonian-induced precession. As explained in Appendix C, the
resulting measurement bases can only be in the same plane as the
actually performed measurements, which is the $x$-$y$ plane in
this case.
The fact that the measurement basis is generally unpredictable,
and therefore uncontrollable, is a rather strange phenomenon from
a fundamental point of view. From a practical point of view, one
can wonder whether anything useful can be done with such
measurements that are performed in a stochastically determined
basis. If one absolutely requires a measurement in a given basis,
measurement results in different bases would be less useful. One
could then treat the deviation of the observed measurement basis
from the desired one as an experimental error and deal with it
accordingly. This point will be discussed further in Sec.~IV.E.
\subsection{Information acquisition rate}
\begin{figure}[h]
\includegraphics[width=7.0cm]{QPCWMFig3.eps}
\caption{(color online) The measurement fidelity as a function of
measurement duration for three different values of the angle
$\beta$ between the charge basis and the energy eigenbasis:
$\beta=0$ (red; lowest line), $\pi/4$ (green) and $\pi/2$ (blue).
Here $E\tau_m/(2\pi)=5$, i.e.~deep in the weak-coupling regime.
The fidelity increases from zero to one as the measurement
duration increases, and it is almost independent of the angle
$\beta$.}
\end{figure}
In Fig.~3, we plot the measurement fidelity as a function of
measurement duration for three different values of $\beta$ (i.e.,
the angle between the charge basis and the qubit's energy
eigenbasis), keeping all other parameters fixed. We can see that
the fidelity approaches one for long enough measurement time,
regardless of the value of the angle $\beta$. Furthermore, the
fidelity is almost independent of $\beta$. In fact, and rather
counter-intuitively, it turns out that the fidelity for $\beta\neq
0$ is higher than that for $\beta=0$. The reason behind this
phenomenon is that one can obtain a higher fidelity by performing
weak measurements along different axes rather than along the same
axis, as explained in Appendix C. This result shows that even
though more complicated analysis is needed to extract useful
measurement information when $\beta \neq 0$, the information
acquisition rate is not reduced, and in fact enhanced, by the
Hamiltonian-induced precession.
In the limit when the fidelity reaches one, the matrix
$\hat{U}_{\rm Meas}[I(t:0\rightarrow t_f),\delta \! I,\delta t]$
has the form $\ket{\psi_1}\bra{\psi_1}$. It is straightforward to
verify that any further evolution of the system will not affect
$\hat{U}_{\rm Meas}[I(t:0\rightarrow t_f),\delta \! I,\delta t]$.
In this case, further measurement only changes the final state of
the qubit, which would be given by $\hat{U}_{\rm
Rot}[I(t:0\rightarrow t_f),\delta \! I,\delta t] \ket{\psi_1}$.
It is worth mentioning here that the fidelity for the case
$\beta=0$ is very well fitted by the function
\begin{equation}
F(t) = {\rm erf} \left\{ \sqrt{\frac{t}{2\tau_m}} \right\},
\label{Eq:F_as_function_of_t}
\end{equation}
where ${\rm erf}$ stands for the error function. This result can
be obtained using the following calculation. We take the
probability distribution in
Eq.~(\ref{Eq:Probability_distribution_2}) and we replace $\delta
t$ by $t$. We now identify the portion of this distribution that
corresponds to $I>(\overline{I}_L+\overline{I}_R)/2$ as being the
probability of making a wrong inference about the state of the
qubit. The difference between the probability of making a correct
inference about the state of the qubit and that of making a wrong
inference gives Eq.~(\ref{Eq:F_as_function_of_t}). The fidelity in
Fig.~3 is also relatively well fitted by the function
\begin{equation}
F(t) \approx 1 - \exp \left\{ -\frac{t}{\tau_m} \right\},
\end{equation}
which is an indication that our definition for the measurement
time $\tau_m$ (see Appendix A) is a reasonable one.
\subsection{The information contained in the QPC's output signal}
The QPC produces an output signal that contains fast and large
fluctuations. There are therefore a very large number of possible
output signals that the QPC can display. We now ask the question
of what information is contained in a given QPC output signal. It
turns out that only a small amount of information in the signal
concerns the state of the qubit. As explained above, one can
follow a conceptually straightforward calculation in order to
extract the measurement basis, measurement result and fidelity
from the noisy signal. The signal also contains information about
how the qubit's state is rotated after the effective measurement.
It should be noted here that measurement bases and
post-measurement rotations generally come in all possible
combinations. The measurement basis, result and fidelity on the
one hand and the post-measurement rotation on the other hand can
therefore be seen as two separate pieces of information that are
contained in the QPC's output signal. The rest of the information
contained in the highly noisy signal, which is the vast majority
of information contained in the signal, concerns the QPC itself.
Thus the large fluctuations that one observes provide information
about the state of the QPC in the specific experimental run under
consideration. For purposes of reading out the state of the qubit,
this information is discarded. This situation is most clearly
demonstrated in the case $\beta=0$, where one is only interested
in the time average of the signal, discarding all the fluctuations
around the average.
\subsection{Spectral analysis of the measurement-basis probability distribution}
\begin{figure}[h]
\includegraphics[width=8.0cm]{QPCWMFig4a.eps}
\includegraphics[width=8.0cm]{QPCWMFig4b.eps}
\caption{(color online) Spectral analysis of the probability
distribution $g(\theta,\phi)$: (a) The spectral components $c_n$
for $n=0$ (red circles), 2 (black squares), 4 (green $\times$
symbols), 6 (blue triangles), 8 (purple + signs) and 10 (green
diamonds) as functions of $E\tau_m/(2\pi)$ for $\beta=\pi/8$. Note
the logarithmic scale on the $x$ axis. As $E\tau_m/(2\pi)$ is
increased, the spectral components clearly reach very stable
values by the time $E\tau_m/(2\pi)$ reaches one. These results
indicate that the value $E\tau_m/(2\pi)=5$ is sufficiently deep in
the weak-coupling regime for purposes of studying the limit $E
\tau_m/(2\pi) \rightarrow \infty$. (b) The spectral components
$c_n$ for $n=0,2,4$ and 6 [same symbol usage as in (a)] as
functions of $\beta$ in the weak-coupling regime [here we use the
value $E\tau_m/(2\pi)=5$]. As $\beta$ is increased, the
probability distribution $g(\theta,\phi)$ gradually changes from
being concentrated at the energy eigenbasis to being spread over
all directions and then to being concentrated on the equator
perpendicular to the energy eigenbasis. There seems to be one
value of $\beta$ [close to $0.3\pi$, which is close to
$\cos^{-1}(1/\sqrt{3})$] where all the spectral components with
$n\neq 0$ vanish, i.e.~$g(\theta,\phi)$ becomes completely
uniform.}
\end{figure}
We now take sets of stochastically determined measurement bases
(i.e., results of the kind presented in Fig.~2) and use them to
extract probability distributions $g(\theta,\phi)$ for the
measurement basis. The probability distributions are normalized to
unity: $\int g(\theta,\phi) d\Omega=1$ with the integral covering
the hemisphere $0\leq\theta\leq\pi/2$, $0\leq\phi\leq 2\pi$. In
particular, we are interested in these probability distributions
in the weak-coupling regime. In this regime, the probability
distributions will be symmetric about the $z$ axis,
i.e.~$g(\theta,\phi)$ will be independent of $\phi$. We use this
fact and write the probability distribution as
\begin{equation}
g(\theta,\phi) = \sum_{n=0}^{\infty} c_n Y_{n,0}(\theta,\phi),
\end{equation}
where $Y_{n,0}(\theta,\phi)$ are the usual spherical harmonics
given by
\begin{equation}
Y_{n,0}(\theta,\phi) = \sqrt{\frac{2n+1}{4\pi}} P_n(\cos\theta),
\end{equation}
and $P_n(x)$ are the Legendre Polynomials. Note that since we only
need to consider the hemisphere $0\leq\theta\leq\pi/2$, we only
need to keep even values of $n$ in the above series. The
coefficients $c_n$ are given by
\begin{equation}
c_n = 2 \int_0^{\pi/2} \!\!\! d\theta \sin \theta \int_{0}^{2\pi}
\!\! d\phi \; g(\theta,\phi) \; Y_{n,0}(\theta,\phi).
\end{equation}
In our calculations, we have a collection of $N$ stochastically
determined measurement bases. The coefficients $c_n$ can be
calculated from these data sets using the formula
\begin{equation}
c_n = \frac{2}{N} \sum_j Y_{n,0}(\theta_j,\phi_j).
\end{equation}
where the index $j$ labels the different data points.
In order to ensure that we are using parameters that are
sufficiently deep in the weak-coupling regime, we start by
performing the above spectrum analysis while varying
$E\tau_m/(2\pi)$. The results of this calculation for
$\beta=\pi/8$ are shown in Fig.~4(a). The coefficients $c_n$ seem
to be almost independent of $E \tau_m/(2\pi)$ when this parameter
is larger than one. From these results we conclude that the point
$E \tau_m/(2\pi)=5$ is sufficiently deep in the weak-coupling
regime for purposes of calculating $g(\theta,\phi)$ in the limit
$E \tau_m/(2\pi) \rightarrow \infty$.
We now calculate $c_n$ (up to $n=10$) in the weak-coupling regime
for different values of $\beta$ and plot the results in Fig.~4(b).
The probability distribution changes gradually as we go from
$\beta=0$, where the measurement basis is always along the $z$
axis (here the charge basis), to $\beta=\pi/2$, where the
measurement basis is always in the $x$-$y$ plane. There is one
point ($\beta\approx 0.3 \pi$) where $g(\theta,\phi)$ is either
approximately or exactly uniform over the entire hemisphere.
\subsection{Short-time dynamics}
We now consider the case where the measurement duration is short
enough that the measurement fidelity is much smaller than one. In
this case the state of the qubit experiences only a small amount
of projective evolution. As a result we can calculate analytically
the total effect of the projection and coherent dynamics.
Dividing the total time interval under consideration into a large
number of infinitesimal intervals, we can write the total
evolution matrix as
\begin{widetext}
\begin{eqnarray}
\hat{U}_{\rm Total} \left[ I(t:0\rightarrow t_f), \delta \! I,
\delta t \right] & \approx & \left[ 1 + \delta p(t_f)
\hat{\sigma}_{\bf n} \right] e^{i \hat{\sigma}_z E\delta t/2}
\cdots e^{i \hat{\sigma}_z E\delta t/2} \left[ 1 + \delta p(\delta
t) \hat{\sigma}_{\bf n} \right] e^{i \hat{\sigma}_z E\delta t/2}
\left[ 1 + \delta p(0) \hat{\sigma}_{\bf n} \right]
\nonumber
\\
& = & e^{i \hat{\sigma}_z E t_f/2} \left[ 1 + \delta p(t_f)
\hat{\sigma}_{{\bf n} (t_f)} \right] \times \cdots \times \left[ 1
+ \delta p(\delta t) \hat{\sigma}_{{\bf n} (\delta t)} \right]
\times \left[ 1 + \delta p(0) \hat{\sigma}_{{\bf n} (0)} \right]
\nonumber
\\
& \approx & e^{i \hat{\sigma}_z E t_f/2} \left[ 1 + \delta p(t_f)
\left\{ \sin\beta \cos(Et_f) \hat{\sigma}_x + \sin\beta \sin(Et_f)
\hat{\sigma}_y + \cos\beta \hat{\sigma}_z \right\} \right] \times
\cdots
\nonumber
\\
& & \hspace{3cm} \times \left[ 1 + \delta p(0) \left\{ \sin\beta
\cos(0) \hat{\sigma}_x + \sin\beta \sin(0) \hat{\sigma}_y +
\cos\beta \hat{\sigma}_z \right\} \right]
\nonumber
\\
& \approx & e^{i \hat{\sigma}_z E t_f/2} \left[ 1 + \int_0^{t_f}
dt \frac{\delta p(t)}{\delta t} \left\{ \sin\beta \cos(Et)
\hat{\sigma}_x + \sin\beta \sin(Et) \hat{\sigma}_y + \cos\beta
\hat{\sigma}_z \right\} \right]
\nonumber
\\
& = & e^{i \hat{\sigma}_z E t_f/2} \left[ 1 + p_x \hat{\sigma}_x +
p_y \hat{\sigma}_y + p_z \hat{\sigma}_z \right],
\end{eqnarray}
\end{widetext}
where we have defined
\begin{eqnarray}
\delta p(t) & = & \frac{ \delta t \left[ I(t) - \overline{I}
\right]}{\tau_m \Delta\overline{I}}
\nonumber
\\
{\bf n}(t) & = & \sin\beta \cos(Et) \hat{x} + \sin\beta \sin(Et)
\hat{y} + \cos\beta \hat{z}
\nonumber
\\
p_x & = & \sin\beta \int_0^{t_f} \frac{I(t) - \overline{I}}{\tau_m
\Delta\overline{I}} \cos(Et) dt
\nonumber
\\
p_y & = & \sin\beta \int_0^{t_f} \frac{I(t) - \overline{I}}{\tau_m
\Delta\overline{I}} \sin(Et) dt
\nonumber
\\
p_z & = & \cos\beta \int_0^{t_f} \frac{I(t) - \overline{I}}{\tau_m
\Delta\overline{I}} dt.
\end{eqnarray}
Taking the total measurement duration $t_f$ to be a multiple of
the qubit's free-precession period, the different measurements
strengths $p_{x,y,z}$ are different Fourier components of the
observed QPC signal. These Fourier components can be thought of as
independent quantities, leading one to think of the net result as
three independent weak measurements, one along each of the three
axes $x$, $y$ and $z$.
We have performed spectral analysis as was done in Sec.~IV.D on
the possible measurement outcomes in the short-time case analyzed
above (we do not show the full results here). In the calculations
$p_x$, $p_y$ and $p_z$ were assumed to be Gaussian-distributed
quantities with root-mean-square values that obey the
relation
\begin{equation}
p_x^{\rm rms} : p_y^{\rm rms} : p_z^{\rm rms} =
\frac{\sin\beta}{\sqrt{2}} : \frac{\sin\beta}{\sqrt{2}} :
\cos\beta.
\label{Eq:rms_ratio_of_p}
\end{equation}
The overall features of the dependence of $c_n$ on $\beta$ are
similar to those seen in Fig.~4(b), but there is no exact
agreement between the two cases. For example, the coefficient
$c_2$ is close to being a straight line all the way from $\beta=0$
to $\beta=\pi/2$ in the present case, as opposed to the curved
path that the black squares follow in Fig.~4(b). Finally, we ask
whether there is a value of $\beta$ at which all three
measurements have the same strength on average. Using
Eq.~(\ref{Eq:rms_ratio_of_p}) it is straightforward to see that
this situation occurs when $\cos\beta=1/\sqrt{3}$, i.e.~$\beta=
0.304\pi$; the point at which all measurement directions are
equally probable in Fig.~4(b). The calculations of this subsection
therefore give us a simple interpretation of the significance of
the point $\beta=0.3\pi$ in the present context.
\subsection{Alternative interpretation: Choosing the measurement
basis independently of the measurement outcome}
The idea that the measurement basis is determined stochastically
and is only specified when the experiment is completed is
conceptually unsettling. We therefore take a different approach in
this section. We now require that in order to have a meaningful
measurement, the measurement basis must be specified by the setup
and not by the stochastically determined measurement outcome.
In the strong-coupling regime, there are no conceptual problems.
In this case, the measurement is almost instantaneous, and the
measurement basis is clearly the charge basis. The measurement
fidelity is almost 100\%, up to errors caused by the finiteness of
the measurement time compared to the qubit-precession period.
These errors can be quantified straightforwardly, as we shall do
shortly.
The weak-coupling regime is the one where the conceptual problems
arise. If we take the case where the distribution of measurement
bases is essentially uniform over all direction (which seems to be
the case when $\beta\approx 0.3\pi$), we again obtain a simple,
yet somewhat surprising, result. By symmetry we can argue that all
directions are equivalent. We can therefore choose any basis as
the measurement basis. Based on this choice, we then divide all
the different possible outcomes (i.e.~the different possible QPC
output signals) into two sets, one of which corresponds to the
measurement result $+1$ and the other of which corresponds to the
result $-1$ in the chosen basis. An important question that arises
in this scenario is what the average measurement fidelity would be
under such `worst-case' conditions of having no preferred
measurement basis. In the following, we shall denote this average
measurement fidelity by the symbol $\overline{F}$ in order to
stress the difference between the results of this subsection and
those of Sec.~IV.B. For the completely uniform case, the average
fidelity $\overline{F}$ can be evaluated as follows: let us take a
spin pointing in the $+z$ direction and assume that we have
specified the $z$ axis as the one defining the measurement basis.
If we measure the spin along an axis that makes an angle $\theta$
with the $z$ axis, we obtain the result $+1$ with probability
$\cos^2(\theta/2)$ and the result $-1$ with probability
$\sin^2(\theta/2)$. By dividing the possible outcomes into two
sets (with all outcomes that correspond to states in the upper
hemisphere being interpreted as $+1$ along the $z$ axis and all
outcomes that correspond to states in the lower hemisphere being
interpreted as $-1$ along the $z$ axis) and taking the difference
between the probability of correctly obtaining the result $+1$ and
the probability of obtaining the wrong result $-1$, we find that
the average fidelity (averaged uniformly over the entire
hemisphere)
\begin{eqnarray}
\overline{F}_{{\rm uniform}, E\tau_m\rightarrow\infty} & = &
\int_0^{\pi/2} {\rm d}\theta \sin\theta \left(
\cos^2\frac{\theta}{2} - \sin^2\frac{\theta}{2} \right) \nonumber
\\
& = & \frac{1}{2}.
\end{eqnarray}
The fidelity in this `worst-case scenario' is therefore 50\%. We
emphasize again that this fidelity is independent of our choice of
measurement basis. This result again contrasts the usual intuition
where the outcomes of measurements performed in one basis provide
no information at all in an orthogonal basis. In the present case,
all measurement bases are equivalent, and one would obtain the
same (average) measurement fidelity in any one of them. This
result fits well with the result that will be explained in
Sec.~IV.G that quantum state tomography can be performed by
repeating the same measurement procedure a large number of times.
This measurement procedure gives the same amount of information in
all different bases.
As explained above, the lowest possible average fidelity is 50\%,
and it occurs when $\beta\approx 0.3\pi$. Two other values of
$\beta$ allow for simple reasoning. The obvious one is the case of
$\beta=0$, where the measurement basis is $\{ \ket{L}, \ket{R} \}$
and the fidelity is 100\% for all coupling strengths. The other
exception occurs when $\beta=\pi/2$. As can be seen from the
rightmost column of Fig.~2, the measurement basis is always in the
$x$-$y$ plane. In the weak-coupling regime, the possible
measurement bases are spread uniformly around the equator. In that
limit, one can choose any axis in the $x$-$y$ plane as defining
the measurement basis. All these bases give the same value for the
average fidelity:
\begin{eqnarray}
\overline{F}_{\beta=\pi/2,E\tau_m\rightarrow\infty} & = &
\frac{1}{2\pi} \int_0^{2\pi} {\rm d}\phi \left|
\cos^2\frac{\theta}{2} - \sin^2\frac{\theta}{2} \right| \nonumber
\\
& = & \frac{2}{\pi}.
\end{eqnarray}
Numerical results for the average fidelity for different values of
$\beta$ in the weak-coupling limit are plotted in Fig.~5. First,
in Fig.~5(a) we plot the average fidelity as a function of the
chosen measurement angle $\theta_{\rm Meas}$ for different values
of $\beta$. We find that for $\beta < 0.3\pi$ choosing the energy
eigenbasis as the measurement basis gives the highest average
fidelity, whereas for $\beta > 0.3 \pi$ choosing a basis that is
orthogonal to the energy eigenbasis gives the highest average
fidelity (note that there is an infinite number of such bases). In
Fig.~5(b) we plot the average fidelity as a function of $\beta$
for three choices of the measurement basis: $\theta_{\rm Meas}=0$,
$\theta_{\rm Meas}=\pi/2$ and the charge basis. It should be noted
here that in the weak-coupling limit there is symmetry about the
$z$ axis. As a result, any axis that is obtained by rotating the
charge basis about the Hamiltonian axis will result in the same
value of the fidelity as the charge basis. Choosing the charge
basis as the measurement basis never results in an average
fidelity that is substantially lower than the maximum possible
value.
In the intermediate coupling regime, one can choose the
measurement basis by maximizing the average fidelity. For example,
for the parameters of the middle row in Fig.~2 one can see that
the measurement basis that would maximize the fidelity deviates
slightly from the charge basis (unless $\beta=0$). One can
understand this result naturally by keeping in mind that the
qubit's state precesses about the Hamiltonian's axis while being
measured. The measurement basis is therefore approximately
obtained by taking the charge basis and rotating it by the
appropriate angle about the qubit Hamiltonian (in the opposite
sense from state precession).
\begin{figure}[h]
\includegraphics[width=8.0cm]{QPCWMFig5a.eps}
\includegraphics[width=8.0cm]{QPCWMFig5b.eps}
\caption{(color online) The average fidelity $\overline{F}$ when
defining the measurement basis independently of the specific
(stochastic) outcome in the weak-coupling regime
[$E\tau_m/(2\pi)=5$]: (a) The average fidelity as a function of
the `manually' chosen measurement angle $\theta_{\rm Meas}$ for
five different values of $\beta$: 0 (red circles), $\pi/8$ (blue
squares), $\pi/4$ (green triangles), $3\pi/8$ (black + signs) and
$\pi/2$ (cyan $\times$ symbols). Note that the results are
independent of the choice of the azimuthal angle $\phi_{\rm
Meas}$, which is also chosen manually along with $\theta_{\rm
Meas}$. One can see that for any value of $\beta$ the maximum
fidelity is obtained by choosing either $\theta_{\rm Meas}=0$ (for
$\beta<0.3\pi$) or $\theta_{\rm Meas}=\pi/2$ (for $\beta>0.3\pi$).
For $\beta=0.3$ (not shown) all values of $\theta_{\rm Meas}$ give
essentially the same value for the fidelity, namely 50\%. (b) The
average fidelity as a function of $\beta$ for three different
choices of measurement angle $\theta_{\rm Meas}$: $\theta_{\rm
Meas}=0$ (blue squares), $\theta_{\rm Meas}=\beta$ (red circles)
and $\theta_{\rm Meas}=\pi/2$ (green triangles). The choice
$\theta_{\rm Meas}=\beta$ always gives a near or true maximum in
the fidelity. However, with the exception of a single point around
$\beta=0.3\pi$, the maximum occurs at either $\theta_{\rm Meas}=0$
or $\theta_{\rm Meas}=\pi/2$.}
\end{figure}
\begin{figure}[h]
\includegraphics[width=8.0cm]{QPCWMFig6.eps}
\caption{(color online) The average fidelity $\overline{F}$ as a
function of $E\tau_m/(2\pi)$ when the measurement basis is decided
(by the experimenter) to be the charge basis. The blue open
circles correspond to the case $\beta=\pi/4$, and the red filled
circles correspond to the case $\beta=\pi/2$. The dashed lines
represent the asymptotic values of the fidelity in the
weak-coupling limit: 0.54 for the case $\beta=\pi/4$ and $2/\pi$
for the case $\beta=\pi/2$.}
\end{figure}
\begin{figure}[h]
\includegraphics[width=8.0cm]{QPCWMFig7.eps}
\caption{The rate of charge-basis fidelity deterioration with
increasing measurement time $|2\pi d\overline{F}/d(E\tau_m)|$ as a
function of $\beta$ in the strong-coupling regime. In other words,
the quantity on the $y$ axis is the absolute value of the slope of
curves similar to those shown in Fig.~6 at $E\tau_m/2\pi=0$. Each
data point is determined by calculating the fidelity
$\overline{F}$ at $E\tau_m/(2\pi)=0.06$ for a given value of
$\beta$ and then taking the ratio $[1-\overline{F}(x)]/x$, where
$x=E\tau_m/(2\pi)$.}
\end{figure}
One could perform a numerical calculation in order to analyze the
behaviour of the maximum-fidelity measurement basis as the
qubit-detector coupling strength is varied. However, here we focus
on a question that seems more experimentally relevant, namely
analyzing the measurement fidelity with the charge basis chosen as
the measurement basis. This fidelity is shown in Fig.~6 for the
two cases $\beta=\pi/4$ and $\beta=\pi/2$. In Fig.~7, we plot the
absolute value of the quantity $2\pi d\overline{F}/d(E\tau_m)$
[i.e., as in the initial slope in Fig.~6] as a function of
$\beta$. This quantity can be used to obtain an estimate for the
measurement errors associated with having a finite measurement
time.
\subsection{Small-current case}
In the above analysis, we have assumed that the QPC's output is
essentially a continuous variable with a large number of possible
values for the current at a given instant. If the QPC's current is
so small that on the timescale of monitoring the output signal
very few electrons can tunnel through the QPC, the physical
picture changes substantially (here we assume that the number of
electrons that have tunneled through the QPC can be measured
accurately). One now has a small number of possible values for the
QPC's output signal. The discretization used in our analysis,
which was done for computational convenience above, is now
provided naturally by the design of the setup. One can then follow
a similar analysis to that of Sec.~III concerning the gradual
progression of the measurement and the evolution of the quantum
state of the qubit. Note that considering the small-current case
only makes sense if the number of electrons that tunnel through
the QPC during a full qubit-precession period is small. Otherwise,
one can always choose between the small- and large-current cases
through the choice of $\delta t$. Clearly, the exact choice of
parameters used in the theoretical analysis cannot affect the
physical results.
An important difference between the cases of small and large
currents is that one expects the probability distributions shown
in Fig.~1(b) to have almost symmetric shapes for the large-current
case, whereas they almost certainly are not symmetric for the
small-current case. This point is most clearly seen by considering
the situation where at most one electron tunnels though the QPC
during the short interval under consideration. The analysis and
results in this case follow closely the analysis and results for a
switching-based detector, studied recently in
Ref.~\cite{AshhabMeasurement}. The fact that, unlike the case of a
switching-based detector, the measurement now continues after the
occurrence of a tunneling event can be incorporated into the
analysis straightforwardly. Our argument concluding that the
choice of the measurement basis is independent of the initial
state of the qubit breaks down. Perhaps more importantly, the
conceptual picture that the measurement basis is chosen first
(according to some probability distribution) and the measurement
result is then obtained in that basis becomes invalid. For
example, it is now possible for a certain state $\ket{\psi}$ to be
a possible result of the data analysis presented above, but not
the state orthogonal to it \cite{AshhabMeasurement}.
It is also worth mentioning in this context the case where the
tunneling process is coherent on the timescale of qubit
precession. In this case the QPC measures the qubit in the energy
eigenbasis, regardless of the angle $\beta$ (except when
$\beta=\pi/2$ where the QPC fails to perform any measurement on
the qubit). For further discussion of this case, see
e.g.~\cite{Makhlin}, and for a discussion of possible advantages
of weak measurement when dealing with multi-qubit circuits, see
e.g.~\cite{AshhabCoupling}.
\subsection{Quantum state tomography}
One example of a procedure where the uncontrollability of the
measurement basis can be harmless is quantum state tomography
(QST). In fact, the original idea of QST included performing
measurements in all possible bases \cite{DAriano}. We have
simulated QST by repeating the measurement procedure a large
number of times, obtaining a set of measurement results (in the
form of pre-measurement qubit states), and then minimizing the
function (see Appendix D)
\begin{equation}
{\cal T}(r,\theta,\phi) = \sum_j \left[ 1 - r \cos
\Omega(\theta,\phi,\theta_j,\phi_j) \right]^2,
\label{Eq:QST_function}
\end{equation}
where $r$, $\theta$ and $\phi$ are the spherical coordinates of a
point in the Bloch sphere; $j$ is an index labelling the different
runs of the experiment, the direction ($\theta_j,\phi_j$) defines
the qubit state obtained in a given measurement, and
$\Omega(\theta,\phi,\theta_j,\phi_j)$ is the angle between the
directions $(\theta,\phi)$ and $(\theta_j,\phi_j)$. We have chosen
several initial states covering the Bloch sphere, and the
tomography procedure always produced correctly the initial state
of the qubit. For the largest value of the qubit-detector coupling
strength that we used [see top row of Fig.~2], the procedure
became unreliable, because the vast majority of the measurements
are performed in one basis.
It is worth pausing here to ask the question of the minimum
requirements for a single measurement setting to provide full QST
information. The qubit's density matrix is characterized by three
parameters. One therefore needs the ensemble of measurements to
produce three independent variables. If the detector has four
possible outcomes (or output signals), one would obtain three
independent probabilities for three of the outcomes (with the
probability of obtaining the fourth outcome determined by
probability normalization). A four-outcome detector is therefore
sufficient, in principle, to perform full QST on a qubit. One can
generalize this argument straightforwardly to the case of larger
systems: One needs a detector with at least as many possible
outcomes as the square of the size of the Hilbert space in order
to perform QST in that Hilbert space (we do not necessarily imply
pure states here).
In Ref.~\cite{AshhabMeasurement} an `all-in-one' measurement
procedure was analyzed, where the measurement results of
identically prepared setups (using a switching-based detector) can
be used to perform full state tomography of the initial qubit
state as well as extract all relevant system parameters. One could
ask whether a similar situation occurs in the qubit-QPC system.
The answer is yes, in principle. However, because the analysis of
Ref.~\cite{AshhabMeasurement} was almost entirely analytic, it was
possible to obtain simple expressions to which the measurement
data would be fitted in order to extract all the available
parameters. In this paper, we have not been able to obtain similar
analytic expressions. As a result, it would require extensive
numerical calculations to compare the observed data with those
that would be expected for all possible sets of parameters and
then find the parameters that produce the best fit. Given the
relatively long computation times required for even the simple
numerical calculations presented in this paper, we have not
attempted to simulate the full parameter-extraction procedure.
\section{Decoherence}
So far we have assumed that there is no decoherence in the system
other than that associated with the measurement-induced
projection. In this section we briefly consider the effect of
additional decoherence channels on the measurement process.
As a first step, one can make the following observations: the
measurement has a characteristic time $\tau_m$, which determines
how long it takes for the experimenter to obtain a high-fidelity
measurement result. Decoherence introduces its own decoherence
timescales, which we collectively denote by $\tau_d$. If $\tau_m
\ll \tau_d$, decoherence should have little effect on the analysis
and results of the previous sections. If, on the other hand,
$\tau_m \gg \tau_d$, decoherence will distort the state of the
qubit before any reliable measurement result is obtained. This
situation renders the measurement almost completely useless. One
should keep in mind, however, that the above statement depends
strongly on the nature of the decoherence channel. For example, if
the energy eigenbasis coincides with the charge basis (i.e., there
is no coherent dynamics mixing the states $\ket{L}$ and $\ket{R}$)
and there is no relaxation between the states $\ket{L}$ and
$\ket{R}$, pure dephasing between these states will not hamper the
measurement, regardless of the dephasing time.
Decoherence can also be introduced to the stochastic master
equation such that Eq.~(\ref{Eq:SME}) turns into:
\begin{widetext}
\begin{equation}
\dot{\rho}_{\rm q} = - i \left[ \hat{H}_{\rm q} , \rho_{\rm q}
\right] + \frac{\xi(t)}{2\sqrt{\tau_m}} \left( \hat{\sigma}_{\bf
n} \rho_{\rm q} + \rho_{\rm q} \hat{\sigma}_{\bf n} - 2
\left\langle \hat{\sigma}_{\bf n} \right\rangle \rho_{\rm q}
\right) + \frac{1}{4\tau_m} \left( \hat{\sigma}_{\bf n} \rho_{\rm
q} \hat{\sigma}_{\bf n} - \rho_{\rm q} \right) + \sum_k \gamma_k
\left( \hat{A}_k \rho_{\rm q} \hat{A}_k^{\dagger} - \frac{1}{2}
\hat{A}_k^{\dagger} \hat{A}_k \rho_{\rm q} - \frac{1}{2} \rho_{\rm
q} \hat{A}_k^{\dagger} \hat{A}_k \right),
\label{Eq:SME_with_decoherence}
\end{equation}
\end{widetext}
where the index $k$ represents the different decoherence channels,
with $\gamma_k$ and $\hat{A}_k$ being the rate and quantum
operator that describe a given decoherence channel.
Modifying Eq.~(\ref{Eq:SME_for_U}) in order to include decoherence
is somewhat trickier. The difficulty lies in the fact that the
evolution of the density matrix is no longer of the form
\begin{equation}
\rho_{\rm q} \rightarrow \hat{U} \rho_{\rm q} \hat{U}^{\dagger},
\end{equation}
ignoring here terms in Eq.~(\ref{Eq:SME}) that were added for
normalization purposes. One way to deal with this new difficulty
is to note that the evolution of the density matrix caused by
decoherence is linear and therefore write the qubit's density
matrix in vector form, i.e.~the transpose of $(\rho_{00},
\rho_{11}, \rho_{01}, \rho_{10})$. With this rearrangement of the
matrix elements, the evolution matrices are now expressed as
$4\times 4$ matrices acting on the density matrix (which now looks
like a vector) from the left:
\begin{equation}
\rho_{\rm q} \rightarrow \hat{\cal U} \rho_{\rm q},
\end{equation}
where we now use the symbol $\hat{\cal U}$ in order to distinguish
the $4 \times 4$ matrices of this section from the $2 \times 2$
matrices that are applied symmetrically on both sides of the
density matrix.
We shall not write down the detailed expressions for the evolution
matrices $\hat{\cal U}$, because they would not add insight to the
present discussion. Instead we point out that deriving a master
equation for these matrices can be done straightforwardly by
rearranging Eq.~(\ref{Eq:SME_with_decoherence}) into vector
format. The differential equation that describes the evolution of
the density matrix $\rho_{\rm q}$ also describes the evolution of
the propagator $\hat{\cal U}_{\rm Total}$, similarly to the fact
that the Schr\"odinger equation can be applied to wave functions
or propagators. We now assume that the total evolution matrix
$\hat{\cal U}_{\rm Total}$ (including the effects of measurement,
Hamiltonian-induced dynamics and decoherence) has been obtained
using such a stochastic equation, and we ask the question of how
to extract measurement information from it. The answer is a
generalized version of the one obtained in the absence of
decoherence. The total evolution matrix can be decomposed into two
matrices: a measurement matrix followed by a trace-preserving
`quantum operation'. This quantum-operation part describes both
unitary evolution and decoherence \cite{Bengtsson}. The
measurement matrix can be used to extract the measurement basis,
result and fidelity, just as was done in the absence of
decoherence.
We now give a simple example illustrating the interplay between
measurement and decoherence. We assume that the Hamiltonian is
diagonal in the charge basis, and therefore there is no coherent
dynamics between the states $\ket{L}$ and $\ket{R}$. A measurement
matrix that corresponds to an outcome that occurs with probability
$P$ for the maximally mixed state and has fidelity $M$ now takes
the form
\begin{equation}
\hat{\cal U}_{\rm Meas}(P,M) = P \left(
\begin{array}{cccc}
1 \pm M & 0 & 0 & 0 \\
0 & 1 \mp M & 0 & 0 \\
0 & 0 & \sqrt{1-M^2} & 0 \\
0 & 0 & 0 & \sqrt{1-M^2} \\
\end{array}
\right),
\end{equation}
A relaxation matrix with relaxation factor $D_r$ (which can be
thought of as decaying from one to zero as time goes by) is given
by
\begin{equation}
\hat{\cal U}_{\rm relaxation}(D_r) = \left(
\begin{array}{cccc}
1 & 1-D_r & 0 & 0 \\
0 & D_r & 0 & 0 \\
0 & 0 & \sqrt{D_r} & 0 \\
0 & 0 & 0 & \sqrt{D_r} \\
\end{array}
\right),
\end{equation}
and a dephasing matrix with dephasing factor $D_p$ is given by
\begin{equation}
\hat{\cal U}_{\rm dephasing}(D_p) = \left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & D_p & 0 \\
0 & 0 & 0 & D_p \\
\end{array}
\right).
\end{equation}
If a qubit is measured, with the measurement outcome corresponding
to the matrix $\hat{\cal U}_{\rm Meas}(P,M)$, and the qubit then
relaxes and dephases according to the matrices $\hat{\cal U}_{\rm
relaxation}(D_r)$ and $\hat{\cal U}_{\rm dephasing}(D_p)$, the
total evolution matrix describing this sequence of events is given
by
\begin{widetext}
\begin{eqnarray}
\hat{\cal U}_{\rm Total} & = & \hat{\cal U}_{\rm dephasing}(D_p)
\hat{\cal U}_{\rm relaxation}(D_r) \hat{\cal U}_{\rm Meas}(P,M) \nonumber \\
& = & P \left(
\begin{array}{cccc}
1 \pm M & (1-D_r) (1 \mp M) & 0 & 0 \\
0 & D_r (1 \mp M) & 0 & 0 \\
0 & 0 & D_p \sqrt{D_r} \sqrt{1-M^2} & 0 \\
0 & 0 & 0 & D_p \sqrt{D_r} \sqrt{1-M^2} \\
\end{array}
\right).
\label{Eq:U_meas_relax_dephas}
\end{eqnarray}
This matrix will be used as a reference matrix for the scenario
that we describe below. In particular, we shall use the fact that
the measurement fidelity is given in terms of the matrix elements
in the top left corner:
\begin{equation}
F = \left| \frac{{\cal U}_{\rm Total}^{1,1}-{\cal U}_{\rm
Total}^{1,2}-{\cal U}_{\rm Total}^{2,2}}{{\cal U}_{\rm
Total}^{1,1}+{\cal U}_{\rm Total}^{1,2}+{\cal U}_{\rm
Total}^{2,2}} \right|.
\end{equation}
We now consider a qubit that is constantly being probed by the QPC
while undergoing gradual relaxation and dephasing. The
measurement-induced evolution is now expressed as
\begin{eqnarray}
\hat{U}_M \rho \hat{U}_M^{\dagger} & \propto & \left[ 1 +
\frac{\delta t \left[ I(t) - \overline{I} \right]}{\tau_m
\Delta\overline{I}} \hat{\sigma}_z \right] \left(
\begin{array}{cc}
\rho_{00} & \rho_{01} \\
\rho_{10} & \rho_{11}
\end{array}
\right) \left[ 1 + \frac{\delta t \left[ I(t) - \overline{I}
\right]}{\tau_m
\Delta\overline{I}} \hat{\sigma}_z \right] \nonumber \\
& \rightarrow & \left(
\begin{array}{cccc}
1 + G(t) + \frac{\delta t}{4\tau_m} & 0 & 0 & 0 \\
0 & 1 - G(t) + \frac{\delta t}{4\tau_m} & 0 & 0 \\
0 & 0 & - \frac{\delta t}{4\tau_m} & 0 \\
0 & 0 & 0 & - \frac{\delta t}{4\tau_m} \\
\end{array}
\right)
\left(
\begin{array}{cc}
\rho_{00} \\
\rho_{11} \\
\rho_{01} \\
\rho_{10}
\end{array}
\right),
\end{eqnarray}
where
\begin{equation}
G(t) = \frac{2 \delta t \left[ I(t) - \overline{I} \right]}{\tau_m
\Delta\overline{I}}.
\end{equation}
Using the above form for the evolution matrix and similar ones for
relaxation and dephasing, we arrive at the equation of motion for
(an unnormalized version of) the total evolution matrix:
\begin{equation}
\delta \hat{\cal U}_{\rm Total} = \delta t \left(
\begin{array}{cccc}
G(t) & \gamma_r & 0 & 0 \\
0 & -\gamma_r - G(t) & 0 & 0 \\
0 & 0 & -\frac{\gamma_r}{2}-\gamma_p-\frac{1}{2\tau_m} & 0 \\
0 & 0 & 0 & -\frac{\gamma_r}{2}-\gamma_p-\frac{1}{2\tau_m} \\
\end{array}
\right) \hat{\cal U}_{\rm Total},
\end{equation}
with the initial condition
\begin{equation}
\hat{\cal U}_{\rm Total}(t=0) = \left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right).
\end{equation}
The solution of the above differential equation is
\begin{equation}
\hat{\cal U}_{\rm Total}(t) = \left(
\begin{array}{cccc}
\tilde{G}(0,t) & \gamma_r \int_0^{t} e^{-\gamma_r t'}
\tilde{G}^{-1}(0,t') \tilde{G}(t',t) dt' & 0 & 0 \\
0 & e^{-\gamma_r t} \tilde{G}^{-1}(0,t) & 0 & 0 \\
0 & 0 & e^{-(\frac{\gamma_r}{2}+\gamma_p+\frac{1}{2\tau_m})t} & 0 \\
0 & 0 & 0 & e^{-(\frac{\gamma_r}{2}+\gamma_p+\frac{1}{2\tau_m})t} \\
\end{array}
\right),
\end{equation}
\end{widetext}
where
\begin{equation}
\tilde{G}(t_1,t_2) = \exp \left\{ \int_{t_1}^{t_2} G(t) dt
\right\}.
\end{equation}
Comparing this matrix with the one in
Eq.~(\ref{Eq:U_meas_relax_dephas}), one can see that the full
dynamics between times 0 and $t$ is equivalent to a measurement
that is followed by relaxation and dephasing operations. With some
straightforward algebra, one can extract the measurement fidelity:
\begin{eqnarray}
F & = & \left| \frac{\int_0^{t} 2 G(t') e^{-\gamma_r t'}
\tilde{G}^{-2}(0,t') dt'}{2-\int_0^{t} 2 G(t') e^{-\gamma_r t'}
\tilde{G}^{-2}(0,t') dt'} \right| \nonumber \\
& = & \left| \frac{\int_{1}^{\rm final} e^{-\gamma_r t'} d\left[
\tilde{G}^{-2}(0,t')\right]}{2+\int_{1}^{\rm final} e^{-\gamma_r
t'} d\left[ \tilde{G}^{-2}(0,t') \right]} \right|.
\label{Eq:Fidelity_with_decoherence}
\end{eqnarray}
Although the above expression looks rather complicated, one
important result can be seen relatively straightforwardly. The
only difference between this expression and the one that would be
obtained in the absence of decoherence is the factor $e^{-\gamma_r
t'}$. This factor represents the rather intuitive fact that the
parts in the QPC signal that are recorded at later times carry
less value for purposes of determining the initial qubit state
than those recorded at earlier times, with an exponentially
decaying function. What is less intuitively obvious is how the
exponential-decay function enters into the expression for the
fidelity, as can be seen from
Eq.~(\ref{Eq:Fidelity_with_decoherence}).
It is worth making a comment here regarding the `value' of the
output signal in the absence of decoherence. One might be tempted
to say that since the fidelity increase slows down as time goes by
(see Fig.~3), the `value' of the QPC's output signal decreases as
time goes by in that case as well. If that were the case, this
decrease in the `value' of the measurement signal would not be
related to decoherence. However, one can see that this is not the
case by considering the simple case $\beta=0$ in the absence of
decoherence. In that case the important quantity is the
time-averaged QPC current throughout the measurement duration,
without any weight function that decreases in time. All parts of
the signal are therefore given equal importance when extracting
the measurement result.
\begin{figure}[h]
\includegraphics[width=8.0cm]{QPCWMFig8.eps}
\caption{The maximum achievable fidelity as a function of the
ratio between the decoherence and measurement times. The red dots
are obtained using numerical simulations, and the solid line is
given by Eq.~(\ref{Eq:Fidelity_with_decoherence_estimate}).}
\end{figure}
In order to calculate the average measurement fidelity in the
presence of decoherence, one must average the measurement fidelity
over all different input states and measurement outcomes. Since
any calculation involving the above expressions seems to be a
daunting task, we use the following hand-waving calculation. In
the absence of decoherence, the measurement fidelity after
duration $t$ can be expressed as
\begin{eqnarray}
F(t) & \approx & 1 - \exp \left\{ -\frac{t}{\tau_m} \right\}
\nonumber
\\
& = & \int_0^t \frac{1}{\tau_m} e^{-t/\tau_m} dt.
\end{eqnarray}
Decoherence reduces the `value' of the later parts of the
measurement record with an exponentially decaying function (with
characteristic time $\tau_d$). An estimate for the average
fidelity in the presence of decoherence can therefore be given in
the form
\begin{eqnarray}
F(t) & \approx & \int_0^t \frac{1}{\tau_m} e^{-t/\tau_m} \times
e^{-t/\tau_d} dt.
\end{eqnarray}
Even if the measurement time is taken to infinity, where we have
$F\rightarrow 1$ in the absence of decoherence, we find that the
fidelity only reaches the value
\begin{equation}
F_{\rm max} = \frac{\tau_d}{\tau_m+\tau_d}.
\label{Eq:Fidelity_with_decoherence_estimate}
\end{equation}
Note, in particular, the simple limits that $F_{\rm max}
\rightarrow 0$ when $\tau_d \ll \tau_m$, and $F_{\rm max}
\rightarrow 1$ when $\tau_d \gg \tau_m$. We have performed
numerical simulations of the measurement process in the presence
of relaxation, and the result are shown in Fig.~8. The results
agree reasonably well with those of the hand-waving argument given
above.
\section{Conclusion}
In conclusion, we have analyzed the question of what information
can be extracted from the output signal of a detector that weakly
probes the state of a qubit when the qubit Hamiltonian induces
oscillations between the two eigenstates of the probed operator.
We have shown that the measurement basis is determined
stochastically every time the measurement is repeated. In the case
of weak qubit-detector coupling, the possible measurement bases
cover all the possible directions. The measurement basis and the
measurement result can both be extracted from the detector's
output signal. We have also shown that the information acquisition
rate is almost independent of the angle $\beta$ between the
direction defining the probed basis and that defining the qubit
Hamiltonian. In other words, given enough time, the detector will
produce a high-fidelity measurement result, regardless of the
value of $\beta$. These results show that, under suitable
conditions and by proper analysis, the detector's ability to
obtain high-fidelity information about the state of the qubit is
not affected by the apparent conflict between the measurement and
coherent-precession dynamics.
In the course of our analysis, we have introduced an equation that
supplements the stochastic master equation for purposes of
extracting measurement information from the observed signal. We
have also introduced a procedure for performing quantum state
tomography that is well suited for the case where the different
measurement bases do not have a simple relation to one another.
We have outlined the generalization of our analysis to the case
where the qubit experiences additional, non-measurement-related
decoherence. In this case, the `value' of the detector's output
signal decreases with the time at which it is recorded. We have
demonstrated this fact by analyzing the case where the measurement
is performed in the presence of relaxation.
We would like to thank D. Burgarth, J. R. Johansson, N. Lambert
and A. J. Leggett for useful discussions. This work was supported
in part by the National Security Agency (NSA), the Laboratory for
Physical Sciences (LPS), the Army Research Office (ARO) and the
National Science Foundation (NSF) grant No.~EIA-0130383. J.Q.Y.
was also supported by the National Basic Research Program of China
grant No. 2009CB929300, the National Natural Science Foundation of
China grant No. 10625416, and the MOST International Collaboration
Program grant No. 2008DFA01930.
\begin{center}
{\bf Appendix A: The QPC-current probability distribution for a
short time interval of size $\delta t$}
\end{center}
In this Appendix, we derive the expression for the probability
distribution $P[I,\delta \! I,\delta t]$ that governs the
stochastically determined values of the QPC current $I$. We use
the discretization parameters $\delta \! I$ and $\delta t$:
$\delta \! I$ defines the size of a finite interval of values for
$I$ that we identify with a single value of $I$, and $\delta t$ is
the size of the time interval under consideration. In the
derivation below, we assume that the qubit is in one of its charge
states (for definiteness $\ket{L}$) and that there is no mixing
between the charge states. We denote the average value of the
current by $\overline{I}_L$ for the state $\ket{L}$ and
$\overline{I}_R$ for the state $\ket{R}$. We are assuming that the
probability distribution can be approximated by a Gaussian
distribution function:
\begin{equation}
P_L[I,\delta \! I,\delta t] \propto \exp \left\{ - \alpha \left(
I-\overline{I}_L \right)^2 \right\}.
\label{Eq:Probability_distribution_1}
\end{equation}
In order to determine how $\delta t$ enters in the expression for
$P_L[I,\delta \! I,\delta t]$ we consider the effect of performing
$N$ measurements in a row. For sufficiently small $\delta t$, the
results of any calculation should not depend on whether we treat
these measurements as $N$ independent measurements or as a single
measurement of length $N\delta t$. Since the values of $I$
obtained in the different measurements are independent, their
average will have a standard deviation $\sigma/\sqrt{N}$, where
$\sigma$ is the standard deviation of $I$ for an interval of
length $\delta t$. In other words, the standard deviation in $I$
must be proportional to $1/\sqrt{\delta t}$. Since the coefficient
$\alpha$ in Eq.~(\ref{Eq:Probability_distribution_1}) is related
to the standard deviation $\sigma$ by $\sigma=1/\sqrt{2\alpha}$,
we find that $\alpha$ must be proportional to $\delta t$:
$\alpha=\tilde{\alpha} \delta t$.
We now introduce the measurement time $\tau_m$ by requiring that
if $\delta t$ were to reach $\tau_m$, the standard deviations (or
in other words, the widths) of the probability distributions
$P_L[I,\delta \! I,\delta t]$ and $P_R[I,\delta \! I,\delta t]$
will be equal to half the distance between their center points:
$1/\sqrt{2\tilde{\alpha} \tau_m} =
|\overline{I}_R-\overline{I}_L|/2$. Note that this definition does
not necessarily agree with other definitions of the measurement
time that appear in the literature. Assuming for definiteness that
$\overline{I}_R>\overline{I}_L$ and defining
$\Delta\overline{I}=\overline{I}_R-\overline{I}_L$, we find that
$\tilde{\alpha} = 2 / \left( \tau_m
\left[\Delta\overline{I}\right]^2 \right)$. Using the
normalization $\sum_{I} P_L[I,\delta \! I,\delta t] = \int
P_L[I,\delta \! I,\delta t] dI/(\delta \! I)=1$, we find that
\begin{equation}
P_L[I,\delta \! I,\delta t] = \sqrt{\frac{2\delta t \left( \delta
\! I \right)^2}{\pi\tau_m \left( \Delta \overline{I} \right)^2}}
\exp \left\{ - \frac{2\delta t \left( I-\overline{I}_L
\right)^2}{\tau_m \left( \Delta\overline{I} \right)^2} \right\}.
\label{Eq:Probability_distribution_2}
\end{equation}
This expression is used for deriving the matrices $U_M[I,\delta \!
I,\delta t]$ in the main text of the paper.
\begin{center}
{\bf Appendix B: The relation between a given output signal and
its `opposite'}
\end{center}
In this appendix we would like to establish the relation between
the measurement results of a certain signal and its `opposite'. If
a signal is given by $I_1(t)$, with the time $t$ running from 0 to
$t_f$, the opposite signal is given by
$I_2(t)=\overline{I}-I_1(t)$. In other words, the opposite signal
is obtained by taking the mirror image of the signal about the
central current value $\overline{I}$. We start by stating the
relation: if
\begin{widetext}
\begin{equation}
\hat{U}_{\rm Total}[I_1(t:0\rightarrow t_f),\delta \! I,\delta t]
= \hat{U}_{\rm Rot}[I_1(t:0\rightarrow t_f),\delta \! I,\delta t]
\times \hat{U}_{\rm Meas}[I_1(t:0\rightarrow t_f),\delta \!
I,\delta t],
\end{equation}
with
\begin{equation}
\hat{U}_{\rm Meas}[I_1(t:0\rightarrow t_f),\delta \! I,\delta t] =
\sqrt{P_1} \ket{\psi_1}\bra{\psi_1} + \sqrt{P_2}
\ket{\psi_2}\bra{\psi_2},
\end{equation}
then
\begin{equation}
\hat{U}_{\rm Total}[I_2(t:0\rightarrow t_f),\delta \! I,\delta t]
= \hat{U}_{\rm Rot}[I_2(t:0\rightarrow t_f),\delta \! I,\delta t]
\times \hat{U}_{\rm Meas}[I_2(t:0\rightarrow t_f),\delta \!
I,\delta t],
\end{equation}
with
\begin{equation}
\hat{U}_{\rm Meas}[I_1(t:0\rightarrow t_f),\delta \! I,\delta t] =
\sqrt{P_1} \ket{\psi_2}\bra{\psi_2} + \sqrt{P_2}
\ket{\psi_1}\bra{\psi_1},
\end{equation}
\end{widetext}
and $\hat{U}_{\rm Rot}[I_2(t:0\rightarrow t_f),\delta \! I,\delta
t]=\hat{U}_{\rm Rot}[I_1(t:0\rightarrow t_f),\delta \! I,\delta
t]$.
For $t_f=0$, both $\hat{U}_{\rm Total}[I_1(t:0\rightarrow
t_f),\delta \! I,\delta t]$ and $\hat{U}_{\rm
Total}[I_2(t:0\rightarrow t_f),\delta \! I,\delta t]$ are given by
the unit matrix, which obeys the relation given above. We now
assume that the relation holds for a given value of $t_f$, and we
try to establish that it will continue to hold at an
infinitesimally later time $t_f+\delta t$. The two changes that
can occur during this infinitesimal interval are a
measurement-induced weak projection or a Hamiltonian-induced
unitary transformation. Since $\hat{U}_{\rm
Rot}[I_2(t:0\rightarrow t_f),\delta \! I,\delta t]=\hat{U}_{\rm
Rot}[I_1(t:0\rightarrow t_f),\delta \! I,\delta t]$, application
of $\exp\{-i\hat{H}\delta t\}$ to $\hat{U}_{\rm
Total}[I_1(t:0\rightarrow t_f),\delta \! I,\delta t]$ and
$\hat{U}_{\rm Total}[I_2(t:0\rightarrow t_f),\delta \! I,\delta
t]$ clearly cannot affect the above relation between the total
evolution matrices. Demonstrating that application of the weak
projection also does not affect the relation is less
straightforward. Using Eq.~(\ref{Eq:UM}) and introducing the
(infinitesimal) quantity $q=\delta t
[I_1(t_f)-\overline{I}]/(\tau_m\Delta \overline{I})$, we find that
\begin{widetext}
\begin{eqnarray}
\hat{U}_{\rm Total}[I_1(t:0\rightarrow t_f+\delta t),\delta \!
I,\delta t] & \propto & \left( 1+q\hat{\sigma}_{\bf n} \right)
\times \hat{U}_{\rm Rot}[I_1(t:0\rightarrow t_f),\delta \!
I,\delta t] \times \left( \sqrt{P_1} \ket{\psi_2}\bra{\psi_2} +
\sqrt{P_2} \ket{\psi_1}\bra{\psi_1} \right)
\nonumber
\\
& = & \hat{U}_{\rm Rot}[I_1(t:0\rightarrow t_f),\delta \! I,\delta
t] \times \left( 1+q\hat{\sigma}_{\tilde{\bf n}} \right) \times
\left( \sqrt{P_1} \ket{\psi_2}\bra{\psi_2} + \sqrt{P_2}
\ket{\psi_1}\bra{\psi_1} \right),
\end{eqnarray}
where
\begin{equation}
\hat{\sigma}_{\tilde{\bf n}} = \hat{U}_{\rm
Rot}^{\dagger}[I_1(t:0\rightarrow t_f),\delta \! I,\delta t]
\times \left( 1+q\hat{\sigma}_{\bf n} \right) \times \hat{U}_{\rm
Rot}[I_1(t:0\rightarrow t_f),\delta \! I,\delta t].
\end{equation}
We now introduce the Pauli matrices $\tilde{\sigma}_z=
\ket{\psi_1}\bra{\psi_1} - \ket{\psi_2}\bra{\psi_2}$ and
$\tilde{\sigma}_x$, for which $q \hat{\sigma}_{\tilde{\bf
n}}=q_x\tilde{\sigma}_x + q_z\tilde{\sigma}_z$. Defining
$a_p=(\sqrt{P_1}+\sqrt{P_2})/2$ and
$a_m=(\sqrt{P_1}-\sqrt{P_2})/2$, we find that
\begin{eqnarray}
\hat{U}_{\rm Total}[I_1(t:0\rightarrow t_f+\delta t),\delta \!
I,\delta t] & \propto & \hat{U}_{\rm Rot}[I_1(t:0\rightarrow
t_f),\delta \! I,\delta t] \times \left( 1 + q_x\tilde{\sigma}_x +
q_z\tilde{\sigma}_z \right) \times \left( a_p + a_m
\tilde{\sigma}_z \right)
\nonumber
\\
& = & \hat{U}_{\rm Rot}[I_1(t:0\rightarrow t_f),\delta \! I,\delta
t] \times \left( a_p + a_p q_x \tilde{\sigma}_x + a_p q_z
\tilde{\sigma}_z + a_m \tilde{\sigma}_z - i a_m q_x
\tilde{\sigma}_y + a_m q_z \right)
\nonumber
\\
& = & \hat{U}_{\rm Rot}[I_1(t:0\rightarrow t_f),\delta \! I,\delta
t] \times \exp \{ - i a_m q_x \tilde{\sigma}_y/a_p \}
\nonumber
\\
& & \hspace{2cm} \times \left( [a_p+a_m q_z] + \left[ a_p q_x -
\frac{a_m^2 q_x}{a_p} \right] \tilde{\sigma}_x + [a_m + a_p q_z]
\tilde{\sigma}_z \right) + O(q^2)
\nonumber
\\
& = & \hat{U}_{\rm Rot}[I_1(t:0\rightarrow t_f+\delta t),\delta \!
I,\delta t] \times \hat{U}_{\rm Meas}[I_1(t:0\rightarrow
t_f+\delta t),\delta \! I,\delta t],
\end{eqnarray}
where
\begin{eqnarray}
\hat{U}_{\rm Rot}[I_1(t:0\rightarrow t_f+\delta t),\delta \!
I,\delta t] & = & \hat{U}_{\rm Rot}[I_1(t:0\rightarrow t_f),\delta
\! I,\delta t] \times \exp \{ - i a_m q_x \tilde{\sigma}_y/a_p \}
\nonumber
\\
\hat{U}_{\rm Meas}[I_1(t:0\rightarrow t_f+\delta t),\delta \!
I,\delta t] & = & a'_p + a'_{m,x} \tilde{\sigma}_x + a'_{m,z}
\tilde{\sigma}_z
\nonumber
\\
a'_p & = & a_p + a_m q_z
\nonumber
\\
a'_{m,x} & = & a_p q_x - \frac{a_m^2 q_x}{a_p}
\nonumber
\\
a'_{m,z} & = & a_m + a_p q_z.
\end{eqnarray}
\end{widetext}
We can now carry out the same derivation as above for the
`opposite' signal. The only changes are that we replace $q$ by
$-q$ (therefore replacing $q_x$ by $-q_x$ and $q_z$ by $-q_z$) and
reverse the positions of $P_1$ and $P_2$ (this latter change
changes $a_m$ into $-a_m$ and leaves $a_p$ unchanged). These
changes leave the infinitesimal rotation angle $a_m q_x/a_p$
unchanged, and we therefore find that the relation $\hat{U}_{\rm
Rot}[I_2(t:0\rightarrow t_f+\delta t),\delta \! I,\delta
t]=\hat{U}_{\rm Rot}[I_1(t:0\rightarrow t_f+\delta t),\delta \!
I,\delta t]$ holds. Similarly $a'_p$ is unaffected by the above
changes. The quantities $a'_{m,x}$ and $a'_{m,z}$, on the other
hand, change their signs. By looking at the expressions for the
$a_p$ and $a_m$ in terms of the $P_1$ and $P_2$ above, we can see
that reversing the sign of $a_m$ reverses the positions of the
symbols $P_1$ and $P_2$ in the matrix $\hat{U}_{\rm Meas}$, which
is exactly the difference between $\hat{U}_{\rm
Meas}[I_1(t:0\rightarrow t_f+\delta t),\delta \! I,\delta t]$ and
$\hat{U}_{\rm Meas}[I_2(t:0\rightarrow t_f+\delta t),\delta \!
I,\delta t]$ that was stated at the beginning of this appendix. We
have thus established the relation between the matrices
$\hat{U}_{\rm Total}[I_1(t:0\rightarrow t_f),\delta \! I,\delta
t]$ and $\hat{U}_{\rm Total}[I_2(t:0\rightarrow t_f),\delta \!
I,\delta t]$.
\begin{center}
{\bf Appendix C: The measurement fidelity for two consecutive weak
measurements}
\end{center}
In this Appendix, we show that the measurement fidelity for two
misaligned weak measurements can be higher than that obtained when
the measurement axes of the two measurements are aligned with each
other \cite{Wei}.
With no loss of generality, we take the measurement axis of the
first measurement to be the $z$ axis (i.e., the basis $\left\{
\ket{0},\ket{1}\right\}$). We assume that the measurement produces
one of two possible outcomes. The measurement matrices can
therefore be expressed as
\begin{eqnarray}
\hat{U}_{1,1} & = & \left(
\begin{array}{cc}
\sqrt{R_1 \left(1+\epsilon_1\right)} & 0 \\
0 & \sqrt{R_1 \left(1-\epsilon_1\right)}
\end{array}
\right) \nonumber
\\
\hat{U}_{1,2} & = & \left(
\begin{array}{cc}
\sqrt{1-R_1 \left(1+\epsilon_1\right)} & 0 \\
0 & \sqrt{1-R_1 \left(1-\epsilon_1\right)}
\end{array}
\right),
\end{eqnarray}
where we have neglected any coherent component in the measurement
matrix, as explained in the main text. The above matrices satisfy
the basic requirement that $\hat{U}_{1,1}^{\dagger}\hat{U}_{1,1} +
\hat{U}_{1,2}^{\dagger}\hat{U}_{1,2} = 1$. The second measurement
is performed in the basis $\left\{\cos(\theta/2) \ket{0} +
\sin(\theta/2) \ket{1},\sin(\theta/2) \ket{0} - \cos(\theta/2)
\ket{1} \right\}$. The corresponding measurement matrices are
given by
\begin{widetext}
\begin{eqnarray}
\hat{U}_{2,1} & = & \sqrt{R_2 \left(1+\epsilon_2\right)} \left(
\begin{array}{cc} \cos^2\frac{\theta}{2} & \sin\frac{\theta}{2}
\cos\frac{\theta}{2} \\
\sin\frac{\theta}{2} \cos\frac{\theta}{2} & \sin^2\frac{\theta}{2}
\end{array}
\right) + \sqrt{R_2 \left(1-\epsilon_2\right)} \left(
\begin{array}{cc} \sin^2\frac{\theta}{2} & -\sin\frac{\theta}{2}
\cos\frac{\theta}{2} \\
-\sin\frac{\theta}{2} \cos\frac{\theta}{2} &
\cos^2\frac{\theta}{2}
\end{array}
\right) \nonumber
\\
\hat{U}_{2,2} & = & \sqrt{1 - R_2 \left(1+\epsilon_2\right)}
\left(
\begin{array}{cc} \cos^2\frac{\theta}{2} & \sin\frac{\theta}{2}
\cos\frac{\theta}{2} \\
\sin\frac{\theta}{2} \cos\frac{\theta}{2} & \sin^2\frac{\theta}{2}
\end{array}
\right) + \sqrt{1 - R_2 \left(1-\epsilon_2\right)} \left(
\begin{array}{cc} \sin^2\frac{\theta}{2} & -\sin\frac{\theta}{2}
\cos\frac{\theta}{2} \\
-\sin\frac{\theta}{2} \cos\frac{\theta}{2} &
\cos^2\frac{\theta}{2}
\end{array}
\right).
\end{eqnarray}
The combined measurement has four possible measurement matrices,
each one corresponding to one of the four possible outcomes. For
example, for the outcome identified with the index 1 in both
measurements, the measurement matrix is given by
$\hat{U}_{2,1}\hat{U}_{1,1}$. In order to obtain the measurement
fidelity for this outcome, we need to consider the matrix
\begin{eqnarray}
\hat{U}_{1,1}^{\dagger} \hat{U}_{2,1}^{\dagger} \hat{U}_{2,1}
\hat{U}_{1,1} & = & \left(
\begin{array}{cc}
\sqrt{R_1 \left(1+\epsilon_1\right)} & 0 \\
0 & \sqrt{R_1 \left(1-\epsilon_1\right)}
\end{array}
\right) \left(
\begin{array}{cc}
R_2 + \epsilon_2 \cos\theta &
\frac{R_2 \epsilon_2 \sin\theta}{2} \\
\frac{R_2 \epsilon_2 \sin\theta}{2} & R_2 - \epsilon_2 \cos\theta
\end{array}
\right) \left(
\begin{array}{cc}
\sqrt{R_1 \left(1+\epsilon_1\right)} & 0 \\
0 & \sqrt{R_1 \left(1-\epsilon_1\right)}
\end{array}
\right).
\label{Eq:U1U2U2U1}
\end{eqnarray}
\end{widetext}
If we denote the eigenvalues of this matrix by $P_1$ and $P_2$
(with $P_1>P_2$), the measurement fidelity for this outcome is
given by $(P_1-P_2)/(P_1+P_2)$. For purposes of calculating the
overall fidelity of the measurement sequence, we take the
maximally mixed state,
\begin{equation}
\rho_{\rm max. \ mixed} = \left(
\begin{array}{cc}
\frac{1}{2} & 0 \\
0 & \frac{1}{2}
\end{array}
\right)
\end{equation}
as the initial state. With this initial state, the probability
that the above outcome is obtained in a given experimental run is
given by $(P_1+P_2)/2$. Thus, the overall fidelity (taking into
account all four possible outcomes) will be given by the sum of
four terms, each of which is given by $(P_1-P_2)/2$ for one of the
possible outcomes. For Eq.~(\ref{Eq:U1U2U2U1}) straightforward
algebra gives the result that
\begin{equation}
\frac{P_1-P_2}{2} = R_1 R_2 \sqrt{(1+\epsilon_1 \epsilon_2
\cos\theta)^2 - (1-\epsilon_1^2) (1-\epsilon_2^2)}.
\end{equation}
Similar expressions can be obtained for the three other possible
outcomes, resulting in the overall fidelity
\begin{eqnarray}
F & = & R_1 R_2 \sqrt{(1+\epsilon_1 \epsilon_2
\cos\theta)^2 - (1-\epsilon_1^2) (1-\epsilon_2^2)} + \nonumber \\
& & R_1' R_2 \sqrt{(1+\epsilon_1' \epsilon_2 \cos\theta)^2 -
(1-\epsilon_1'^2) (1-\epsilon_2^2)} + \nonumber \\
& & R_1 R_2' \sqrt{(1+\epsilon_1
\epsilon_2' \cos\theta)^2 - (1-\epsilon_1^2) (1-\epsilon_2'^2)} + \nonumber \\
& & R_1' R_2' \sqrt{(1+\epsilon_1' \epsilon_2' \cos\theta)^2 -
(1-\epsilon_1'^2) (1-\epsilon_2'^2)},
\end{eqnarray}
where $R_j'=1-R_j$ and $\epsilon_j'=-\epsilon_jR_j/(1-R_j)$.
The measurement fidelity can have its maximum at any point between
zero and $\pi$ (with the proper choice of the parameters $R_j$ and
$\epsilon_j$; we have verified this statement numerically). For
the special case $R_1=R_2=1/2$ and $\epsilon_1=\epsilon_2\equiv
\epsilon>0$, the overall measurement fidelity is given by
\begin{eqnarray}
F & = & \frac{1}{2} \Bigg( \sqrt{(1+\epsilon^2
\cos\theta)^2 - (1-\epsilon^2)^2} + \nonumber \\
& & \hspace{0.7cm} \sqrt{(1-\epsilon^2 \cos\theta)^2 -
(1-\epsilon^2)^2} \Bigg).
\end{eqnarray}
Differentiating this expression twice shows that it has a maximum
at $\cos\theta=0$. In fact, in the limit of weak measurement,
i.e.~$\epsilon\ll 1$, one can see easily that the above expression
can be approximated by
\begin{equation}
F \approx \epsilon \left( \left|\cos\frac{\theta}{2}\right| +
\left|\sin\frac{\theta}{2}\right| \right),
\end{equation}
which has minima at $\theta=0$ and $\theta=\pi$ ($F=\epsilon$) and
a maximum at $\theta=\pi/2$ ($F=\sqrt{2}\epsilon$). The relative
difference between the minimum and maximum fidelities is
approximately 30\%.
The result that the fidelity of two weak measurement can be
enhanced when they are not aligned with one another offers some
explanation for the result of Sec.~IV that the fidelity increases
more rapidly when the qubit Hamiltonian and the probed operator do
not commute. The qubit Hamiltonian causes a rotation in the
qubit's state while the state is being measured along a fixed
axis. This situation is, in some sense, equivalent to one where a
number of different qubit operators are sequentially probed.
A final comment that is worth making here regards the possible
bases of the combined measurement. As can be seen from the
analysis of this appendix, when the two performed measurements
correspond to directions in the $x$-$z$ plane, all the resulting
matrices will be real. As a result, all the possible effective
measurements will be in the $x$-$z$ plane as well (note that a
projector along the $y$ axis would be complex). In other words,
when successive measurements are made in one plane, they can only
result in effective measurements made in the same plane. This
result explains why in the special case $\beta=\pi/2$ all the
possible measurement bases lay in the $x$-$y$ plane; successive
measurements along the $x$ axis separated by state precession
about the $z$ axis are equivalent to a sequence of measurements
that are all performed in the $x$-$y$ plane.
\begin{center}
{\bf Appendix D: Quantum state tomography}
\end{center}
In this Appendix, we derive the function that was used to perform
quantum state tomography in Sec.~IV.
We start with the objective of performing quantum state tomography
using the available measurement data, which we assume has been
obtained using a (generally weakly coupled) QPC as explained in
the main text. We note that since the measurements in the
different experimental runs are generally performed in different
bases, we look for a procedure that treats all the measurements on
equal footing and does not have a preferred measurement basis, or
set of bases. One obvious approach to follow is to construct a
function that assigns a penalty for any deviation by the `guess'
density matrix $\rho_{\rm guess}$ from a given experimental result
(which takes the form of a measured, pure state). The guess
density matrix is characterized by the spherical coordinates
$r_{\rm guess}$, $\theta_{\rm guess}$ and $\phi_{\rm guess}$,
whereas the measured state is characterized by only the angles in
spherical coordinates $\theta_j$ and $\phi_j$ (the index $j$
denotes the different experimental runs). The function to be
minimized for purposes of state tomography will therefore have the
form
\begin{equation}
{\cal T}(r_{\rm guess},\theta_{\rm guess},\phi_{\rm guess}) =
\sum_j f(r_{\rm guess},\theta_{\rm guess},\phi_{\rm
guess},\theta_j,\phi_j).
\end{equation}
In order to proceed further, we now consider what is done in the
standard version of state tomography where the average values
$\avg{\hat{\sigma}_x}$, $\avg{\hat{\sigma}_y}$ and
$\avg{\hat{\sigma}_z}$ are measured and the density matrix $\rho =
(1 + \avg{\hat{\sigma}_x} \hat{\sigma}_x + \avg{\hat{\sigma}_y}
\hat{\sigma}_y + \avg{\hat{\sigma}_z} \hat{\sigma}_z)/2$ is
inferred from the measurement data. If we take the measurement
data for $\hat{\sigma}_z$, we know that we want the tomography
procedure to produce the result $r \cos\theta =
(N_{+1}-N_{-1})/(N_{+1}+N_{-1})$, where $N_{+1}$ and $N_{-1}$ are
the numbers of times that the measurement results $+1$ and $-1$
were obtained, respectively. We can therefore see that the
function $f(r_{\rm guess},\theta_{\rm guess},\phi_{\rm
guess},\theta_j=0,\phi_j)$ will only be a function of
$\avg{\hat{\sigma}_{z,\rm guess}} = r_{\rm guess} \cos\theta_{\rm
guess}$ (or, in other words, the projection of the vector
representing the guess density matrix along the measurement axis).
Taking this measurement data, we find that the function ${\cal T}$
has the form:
\begin{eqnarray}
{\cal T}(\avg{\hat{\sigma}_{z,\rm guess}},N_{+1},N_{-1}) & = &
N_{+1} f(\avg{\hat{\sigma}_{z,\rm guess}}) \nonumber \\ & & +
N_{-1} f(-\avg{\hat{\sigma}_{z,\rm guess}}).
\end{eqnarray}
In order to minimize the function ${\cal T}$, we take its
derivative with respect to $\avg{\hat{\sigma}_{z,\rm guess}}$:
\begin{eqnarray}
\frac{{\rm d} {\cal T}(\avg{\hat{\sigma}_{z,\rm
guess}},N_{+1},N_{-1})}{{\rm d}\avg{\hat{\sigma}_{z,\rm guess}}} &
= & N_{+1} f'(\avg{\hat{\sigma}_{z,\rm guess}}) \nonumber \\ & & -
N_{-1} f'(-\avg{\hat{\sigma}_{z,\rm guess}}).
\end{eqnarray}
If we now use the Taylor expansion
\begin{equation}
f'(\avg{\hat{\sigma}_{z,\rm guess}}) = \sum_{n=0}^{\infty} c_n
\avg{\hat{\sigma}_{z,\rm guess}}^n,
\end{equation}
we find that
\begin{widetext}
\begin{eqnarray}
\frac{{\rm d} {\cal T}(\avg{\hat{\sigma}_{z,\rm
guess}},N_{+1},N_{-1})}{{\rm d}\avg{\hat{\sigma}_{z,\rm guess}}} &
= & \sum_{n=0,2,4,...} \left\{ ( N_{+1} - N_{-1} ) c_n
\avg{\hat{\sigma}_{z,\rm guess}}^n + ( N_{+1} + N_{-1} ) c_{n+1}
\avg{\hat{\sigma}_{z,\rm guess}}^{n+1} \right\} \nonumber \\
& = & ( N_{+1} + N_{-1} ) \sum_{n=0,2,4,...} \left\{ c_n
\avg{\hat{\sigma}_{z,\rm correct}} + c_{n+1}
\avg{\hat{\sigma}_{z,\rm guess}} \right\} \avg{\hat{\sigma}_{z,\rm
guess}}^n,
\end{eqnarray}
\end{widetext}
where $\avg{\hat{\sigma}_{z,\rm correct}} =
(N_{+1}-N_{-1})/(N_{+1}+N_{-1})$. In order to ensure that the
function ${\cal T}$ has a minimum at $\avg{\hat{\sigma}_{z,\rm
guess}} = \avg{\hat{\sigma}_{z,\rm correct}}$, we must choose
$c_n=-c_{n+1}$ for every even number $n$. The simplest choice is
$c_0=-c_1=-1/2$ and $c_2=c_3=...=0$. This choice gives the
function
\begin{equation}
{\cal T}(r_{\rm guess},\theta_{\rm guess},\phi_{\rm guess}) =
\sum_j \left[ 1 - \avg{\hat{\sigma}_{z,\rm guess}} \right]^2,
\end{equation}
which, when generalized to the case where the measurement axis is
not necessarily the $z$ axis, gives Eq.~(\ref{Eq:QST_function}).
The detailed analysis of the different possible choices of $c_n$
in the function ${\cal T}$, and the stability and reliability of
these different choices, is beyond the scope of this paper.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,068
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Tag: indie film
Breaking Up With Patriarchy in Film and TV
Post author By Sue Nador
2 Comments on Breaking Up With Patriarchy in Film and TV
Collage: Shreya Patel, Rabiya Mansoor, Window Dreams; Bonnie Anderson (top), Moxie Productions
British filmmaker Amma Asante once said: "Don't take no as a full stop, treat it like a comma." Three women indie filmmakers (Bonnie Anderson, Moxie Productions; Rabiya Mansoor and Shreya Patel, Window Dreams Productions) are doing just that. These filmmakers aren't asking for a seat at patriarchy's table. They're building a better one on their own.
Film has always been a brutal industry for everyone but women bear a disproportionate brunt of the pain. A 2021 report by Women in View found women were afforded significantly fewer film contracts than men by two major funders and far less funding. BIPOC women fared the worst. A story in World Economic Forum in March reported that while the MeToo movement uncovered gender power dynamics in Hollywood, inequities behind the scenes garnered less attention: fewer than 20% of directors and writers of the 250 highest-grossing U.S. movies were women (according to a recent study). It isn't lack of education that holds women back – a 2018 report found equal representation of women in higher education in film and television.
Indie Film Making: The Road to Freedom?
While building one's own company doesn't make systemic oppression and challenges vanish, the autonomy that comes with going indie provides film makers with the freedom to produce content that matters to them, their way, and in ways that aligned with their feminist ideals. Moxie Productions' Anderson recalled an incident where a male actor ignored her – even though she was director and producer, and looked to her male Director of Photography instead. She hasn't hired him again. For the keepers though, she provides a work experience "I wasn't finding in other places." She loves hearing how working on her set is fun. "I want everybody to have that feeling when they're on set because that's when you get the best work done."
Left to Right: Jorge Molina. Andrea Grant and Bonnie Anderson of Moxie Productions. Photos by Denise Grant.
For the co-CEOS at Window Dreams (below), going indie means you can put people and relationships first. "If there is no friendship, this doesn't exist, and it's not fun anymore," Patel said." "You don't have to be lonely at the top." The system pits marginalized creatives and producers (they are both of South Asian descent) against one another. There is often only one 'diversity' seat at the table. "Our mindset has always been, well, we'll just build our own table or we'll just make the table bigger," Mansoor said. "There could be seats for everyone."
Anderson took the leap into independent film making after years of industry experience that included lighting designer, theater director, playwright and actor. Technology has helped get hers and other women's derrieres in seats. When cameras and lighting got smaller, and editing apps became available ("Film is really all created in the editing room") she realized: "I'm tired of waiting for other people and I want to just create things for a living." She learned how to edit through YouTube videos and appointments at Apple Genuis ("They were great") to make her first film 'GPS Love': "A man falls in love with his new GPS and finds himself."
Leveraging Technology, Global Networks and Diversity
Window Dreams has been busy during this pandemic. Learning to leverage new technology has helped. The Toronto-based Mansoor had wondered whether she would ever have the opportunity to be in a writers' room with people from New York and Los Angeles. Then virtual meetings became the norm. Their documentary, 'Unity' (logline: "Love spreads faster than a virus"), had over 100 cast from almost 70 countries. While Patel slept, videos arrived from different time zones for her to edit when she woke up. 'Unity' was the closing film at the Unified Filmmakers-Munich International Film Festival last year. Their music video, 'Freedom Dance,' with Bollywood and other celebrities was directed virtually by Patel. It went viral and was reported in Rolling Stones India. "I'm retiring," Patel joked. (She is obviously decades away from retirement!)
Anderson said that if she could change one thing to support the advancement of women, she would appoint more of them "To be head of where the money is." The Women in View's On Screen Report found women give other women more breaks. They refer, for example, to the 'showrunner and producer effect'. When women occupy these positions, their teams have far greater representation of women in creative roles like writers, directors and cinematographers. When women of colour are the producers, the playing field for other women of colour is significantly more level.
A study done earlier this year, Building Inclusive Networks in the Film and Television Industry, found BIPOC women and non-binary individuals viewed networking events as vital to gaining industry access. Yet, most participants felt 'unwelcome' at industry events. Lack of diversity, micro-aggressions and cliquishness were some of the reasons cited. Participants felt greater inclusivity in networking opportunities would lead to positive outcomes: better programming for more accurate reflection of current reality; greater authenticity in the stories; fewer stereotypical or sexualized portrayals of women; on-screen reflection of the diversity of Canadians.
Left to Right: Shreya Patel and Rabiya Mansoor, Window Dreams
Even for Patel, with her expansive global network and a gift for networking and connecting – "That's where my forte is when it comes to business" – access remains a challenge. Though there's no lack of funding opportunities, finding them is a problem. Peers are generally tight-lipped for fear of competition. Mansoor and Patel feel funders need to promote these opportunities better, while supporting applicants to ensure strong submissions. They found out about the Bell Fund Slate Development Program late but hustled to submit their application. They got funding for several projects including the comedy series Layla is Relevant (which they also star in) about "A former child star and current nobody" who moves back to Sarnia (Patel's hometown) with her single mom and gamer cousin.
No Money? Carry On!
Lack of funding never stopped these women from pushing ahead on their dream projects. Anderson had pursued funding before Moxie Productions without much success. She realized: "I'm going to create a production company and make money from it. And from that money, I'll be able to create my own personal work that I want to do." She has a thriving business in educational videos (For the purpose of training doctors and other professionals) and actor and musician promotional reels. This allows her to take financial risks that help other women like her improv partner Kate Ashby. "I Just thought Kate needed her own television show," Anderson said. Talk with Kate Ashby was a talk show with a twist where prominent guests like actor Susan Coyne decided on the next guest (only revealed to Kate on camera). A new season of SNAK (four-minute lively interviews with celebrities like Sandra Shamas, Jean Yoon, Peter Mansbridge) is launching. "This is something that is dear to my heart and we celebrate Canadian talent," Anderson said.
The women at Window Dreams pursue stories about social justice knowing their payday may be far off. Years ago, while Patel was doing humanitarian work in India, she watched helplessly as poor children were waiting to be treated for terminal illnesses. She knew documentary filmmaking could shine a light on marginalized people and create change. Window Dreams' Girl Up about human trafficking in Canada took years to develop without funding. A feature film about domestic human trafficking is in the works.
Emerging Diversity, Inclusion and Equity Policies are Helping
There is reason for aspiring women indie filmmakers to be optimistic. Commitments to gender parity by publicly-funded organizations such as CBC, Canadian Media Fund, and the CRTC have increased the number of women directors in film and television. Mountains still need to be moved to increase representation of BIPOC women and all women in key industry roles like cinematography. However, women in film are increasingly leveraging the power of trust, connection and collective action. They are also helping each other out. They are helping each other out. Organizations such as Women in Film and Television (WIFT), Fem Script Lab, and Women in the Director's Chair (WIDC) offer development and networking to support women's advancement in the industry.
Anderson, Mansoor and Patel participated in the Canadian Film Centre Media Lab's Fifth Wave Initiative, a development program that integrates intersectional feminist ideals with entrepreneurship. For Anderson it was "mind blowing" to be connected with women who were rooting for her success. The enthusiasm of mentors and others to make connections to help their business thrive was invaluable.
What's their advice to help the next generation of women? Anderson would like everyone to see business differently. "It's not sales; it's building relationships." Mansoor would tell them to have "The confidence to run with an idea, knowing there are wins and losses". "Don't give up," Patel would advise. "Entrepreneurship is a long road."
We can all help make that road a little smoother.
TIFF (September 8 – 18) has a category of films 'Directed by Women'. These films deserve our support.
Publishers Note: The FitIn is a part of the Fifth Wave, a year-round program offered by CFC Media Lab and its partners to support the growth and development of women entrepreneurs in the digital media sector in southern Ontario. All enterprise founders in the Fifth Wave community are selected for both their potential and commitment toward weaving intersectional feminist ideals of equity and fairness into sustainable and scalable business growth strategies. Fifth Wave Initiative is committed to 30% participation by members of underrepresented groups. The Fifth Wave is a LiisBeth ally and content sponsor at the Lighthouse level. Applications for Cohort 5 are OPEN!
The New Future of Journalism (It's Not What You Think)
Meet three indie media mavericks who are challenging the patriarchal definition of journalism in Canada.
Moving Pictures: What We Learned from Women Filmmakers at TIFF 2019
We analyzed five female-directed films at TIFF and explore the hidden lessons for feminist entrepreneurs.
Oppression of Women Working in the Film Industry
Maureen Dowd's New York Times investigation talks to female actresses, executives and filmmakers to answer what will it take to dismantle the pervasive sexism in Hollywood.
Tags #entrepreneurialfeminist, #tiff2022, Canada, Feminism, Film, indie film, TIFF, women in film
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,299
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On behalf of The Law Offices of Dorie A. Rogers, APC posted in Divorce on Tuesday, October 30, 2018.
Divorce would be a lot easier if all spouses could cooperate and compromise to achieve swift and fair settlements. In reality, things often do not happen like that. In fact, spouses get pretty nasty sometimes, perhaps trying to hide assets or otherwise give an ex the short end of the settlement stick.
On behalf of The Law Offices of Dorie A. Rogers, APC posted in Divorce on Thursday, October 4, 2018.
Starting next year, California courts will no longer treat domestic pets solely as property in a divorce. Sponsored by Assemblyman Bill Quirk, D-Hayward, Assembly Bill 2274 will determine the continuity of a pet's care in California couples' divorce proceedings.
Getting divorced? Where will your pet live?
On behalf of The Law Offices of Dorie A. Rogers, APC on Thursday, October 4, 2018.
Is your household one of the 67 percent of homes in the U.S. that includes a pet? If so, and if you're getting divorced, issues regarding your pet may be a central focus. Just as you may have to make some important decisions as a parent regarding child custody and support, it is also critical that you and your spouse agree about pet care.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,265
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/**
* @fileoverview Ensures versioned Workbox reference docs that shouldn't be
* indexed have the appropriate <meta> tag.
*
* See https://github.com/GoogleChrome/workbox/issues/2513
*
* @author Jeff Posnick <jeffy@google.com>
*/
const gulp = require('gulp');
const replace = require('gulp-replace');
gulp.task('workbox-no-index', () => {
gulp.src(['src/content/en/tools/workbox/reference-docs/v*/*.html'])
.pipe(replace('<head>\n', '<head><meta name="robots" content="noindex">\n'))
.pipe(gulp.dest('src/content/en/tools/workbox/reference-docs'));
});
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,226
|
{"url":"http:\/\/www.taumuon.co.uk\/category\/f","text":"## Monte Carlo pricing of Exotic Options in F#\n\nI\u2019ve recently completed all exercises for the MOOC Monte Carlo methods in Finance, ran by the University of Madrid on Iversity. The course was excellent, both in its content and delivery.\n\nThe course started off by refreshing probability theory (cdf, pdf etc) and quickly moved onto generating random numbers, Brownian motion and stochastic differential equations followed by the pricing of options and calculating risk.\nThe final week\u2019s homework is on the Monte Carlo calculation of a portfolio\u2019s risk, using copulas with Student\u2019s t distribution, but as that week\u2019s homework is still running I\u2019m blogging about the penultimate week\u2019s homework.\n\nThis homework is about pricing path dependent options, and demonstrating how using a control variate can provide a better result for the same number of iterations (better meaning less error, i.e. with a smaller standard deviation). The homework further shows that when pricing a barrier\noption, using an up-and-out option as the control variate produces better results than using a European option as the control variate.\n\nThe course delivery was extremely slick. The videos were filmed specially, this wasn\u2019t a rehashing of a lecture in front of students (as seems to happen in some other online courses). The lectures were short, and to the point, managing to cram a lot of information in with zero fluff. I really appreciated this as I was watching on a daily commute.\n\nThe assessments were by multiple choice question, but with only one attempt, so it did end up actually feeling like a challenge. The questions were a mixture of maths and coding, with the balance moving more towards coding as the weeks went on.\nThe code was delivered in Matlab, which I had to get reacquainted to after a decade break. The main downside I found is that I ended up doing a fair bit of copy-paste coding, so completed some of the homeworks in .NET instead. I personally found the coding easy, and found the maths more challenging (in a good way).\n\nNow, for the code \u2013 the F# code is available here: https:\/\/gist.github.com\/taumuon\/11302896\n\nApart from the Monte Carlo code, the fact that in F# all functions are in the curried form means that composing the payoff functions using partial function application is beautiful. This would be lambda hell to replicate in C#.\n\nIf this course runs again I\u2019d wholeheartedly recommend it.\n\n## MOOCs\n\nI\u2019ve just enrolled on Iversity\u2019s Monte Carlo Methods in Finance course, and have converted some of week 1\u2019s Matlab demo code over to F# and Deedle: https:\/\/gist.github.com\/taumuon\/8602365\n\nI spent the latter half of last year diving into various online courses. I completed Coursera\u2019s Mathematical Methods for Quantitative Finance, and also followed along with a number of other courses to varying levels of completeness.\n\nI completed three weeks assignments of Udacity\u2019s Cuda programming course.\u00a0 Week three was painful due to an error in the reference code, and week 4 was crashing due to a memory exception. I was using a GPU emulator in Ubuntu, and decided that it would be easier with real hardware. I watched the remaining videos and found the parallel algorithms explanations useful.\n\nI completed the first two weeks of Coursera\u2019s Scientific Computing. These were maths exercises, which I enjoyed and that\u2019s what inspired me to do the Maths Methods for Quant Finance course. The Matlab exercises I was planning to do in F#, but left the course when other attendees were complaining that to complete the homework to the correct numerical accuracy you needed the exact version of Matlab the instructor was using, and they were unable to use Gnu Octave.\n\nIt is great that there\u2019s so much free high standard material available. The fixed timescale nature of the course is a bit annoying \u2013 if work or life gets in the way one week it may make it impossible to catch up with the remainder of the course. I may get around to trying a course again next time it comes around though.\n\n## RunKeeper Visualisations\n\nMy last post described how I\u2019m using the Twitter API to receive tweets off the live stream.\n\nSince then, I\u2019ve used the API to filter for tweets containing the #runkeeper hashtag, and used that to scrape the user\u2019s activity from the RunKeeper site (including the GPS points from the user\u2019s exercise). I\u2019ve stored that information in a MongoDB, which has allowed me to do some simple visualisation:\n\nThe above video (best played at 720p) shows activities plotted against time of day (the sun overhead in the video indicates midday for that region).\n\nI haven\u2019t charted this to confirm, but to my eyes it looks like amount of exercise activity peaks around sunrise and sunset, with almost none at night time (which isn\u2019t really a surprise).\n\nFor the curious, this is what the colour of the dots indicate:\n\nlet\u00a0createSphere\u00a0(latitude:float)\u00a0(longitude:float)\u00a0(activityType:string)\u00a0=\nlet\u00a0color\u00a0=\u00a0match\u00a0activityType\u00a0with\n|\u00a0\"RUN\"\u00a0->\u00a0\"Red\"\n|\u00a0\"WALK\"\u00a0->\u00a0\"Green\"\n|\u00a0\"BIKE\"\u00a0->\u00a0\"Blue\"\n|\u00a0\"MOUNTAINBIKE\"\u00a0->\u00a0\"Orange\"\n|\u00a0\"SKATE\"\u00a0->\u00a0\"Yellow\"\n|\u00a0\"ROWING\"\u00a0->\u00a0\"Cyan\"\n|\u00a0\"HIKE\"\u00a0->\u00a0\"Brown\"\n|\u00a0\"OTHER\"\u00a0->\u00a0\"Black\"\n|\u00a0_\u00a0->\u00a0\"Black\"\nString.Format(sphere,\u00a0latitude,\u00a0longitude,\u00a0color)\n\nOn a local scale, the actual GPS traces are of interest:\n\nActivities are present in the more-populated regions of the UK. The blue and orange traces indicating cycling and mountain biking activities are clearly visible.\n\nNOTE: if you think the map looks weird, it doesn\u2019t have the usual aspect ratio \u2013 I\u2019m not using a Mercator projection do plot the points, but am simply plotting the longitude and latitude linearly.\n\nOn an even smaller scale, landmarks around London are visible:\n\nThe GPS data contains altitude information, so there are more interesting visualisations that could be done, including generating contour plots. Also, the above only contains three days worth of data \u2013 with a larger data set it would be possible to plot to determine whether activities peak on a weekend etc.\n\n### Implementation\n\n#### Acknowledgements:\n\nThe 3D plot of the globe was drawn using POV-Ray. The 3D globe model was from grabcad, with the converted to a POV-Ray model using PoseRay.\n\nThe UK outline was obtained from the NOAA.\n\nThe code was implemented in F# (which was a pleasure to get back to after the C++ I\u2019ve been doing recently). I did try the MongoDB.FSharp library to store my data records, but they failed to deserialise from the database. In any case, I wanted more control over the data types saved (I wanted to store the GPS data as GeoJson3DGeographicCoordinates, with the first point stored separately as a GeoJson2DGeographicCoordinates with an index on this value). I could have created .NET classes and used the BSON serializer, but it seemed more effort than writing directly to the DB (and this is about the only time I\u2019ve seen the benefit of C# implicit conversions, but I can live without them in F#).\n\nWhy use the Twitter API, and why not scrape the RunKeeper site directly? That\u2019s because of times \u2013 the RunKeeper website displays the activity time, but it is displayed in local time, and it\u2019s not clear whether that\u2019s in the user\u2019s timezone, or the timezone of the activity. It seems cleaner to instead assume that the tweet has been posted as soon as the activity is finished and store that time as UTC (this assumption may of course not be true, but the results seem realistic).\n\n### Show me the Code!\n\nThe code\u2019s not in a bad shape, but I would like to tidy it up a little before releasing it into the world. I\u2019m busy with other things at the moment, but if I get much interest I can go ahead and do that\u2026\n\n## Sipping from the twitter stream\n\nI\u2019ve been playing around with connecting to twitter\u2019s streaming API, and displaying a live stream of tweets returned.\nTo do this, I was originally using DotNetOpenAuth, along with the HttpClient, which worked fine for the sample stream, but would return authentication errors for the filtered stream. I looked at the HTML message in fiddler2, and the oauth parameters weren\u2019t ordered, which twitter requires. Instead, I\u2019m using TwitterDoodle, which uses HttpClient.\nThe C#5\/.NET4.5 async\/await code is quite elegant \u2013 it\u2019s not CPU intensive so it\u2019s fine running on the UI thread, without blocking. My first instinct prior to C#5 would have been to use RX for this, but now if I\u2019m doing something simple I\u2019d stick to async\/await, only using RX if doing something more complex like buffering or batching.\n\n\u00a0\u00a0\u00a0\u00a0private\u00a0async\u00a0void\u00a0ProcessTweets()\u00a0\u00a0\u00a0\u00a0{\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0using\u00a0(var\u00a0t\u00a0=\u00a0new\u00a0TwitterConnector())\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0{\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0var\u00a0response\u00a0=\u00a0await\u00a0t.GetSampleFirehoseConnection();\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0var\u00a0res\u00a0=\u00a0await\u00a0response.Content.ReadAsStreamAsync();\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0using\u00a0(var\u00a0streamReader\u00a0=\u00a0new\u00a0StreamReader(res,\u00a0Encoding.UTF8))\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0{\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\/\/\u00a0streamReader.EndOfStream\u00a0can\u00a0block,\u00a0so\u00a0instead\u00a0check\u00a0for\u00a0null\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0while\u00a0(!cts.IsCancellationRequested)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0{\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0var\u00a0r\u00a0=\u00a0await\u00a0streamReader.ReadLineAsync();\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if\u00a0(r\u00a0==\u00a0null)\u00a0{\u00a0return;\u00a0}\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0ProcessTweetText(r);\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}\u00a0\u00a0\u00a0\u00a0} \u00a0\u00a0\u00a0\u00a0private\u00a0void\u00a0ProcessTweetText(string\u00a0r)\u00a0\u00a0\u00a0\u00a0{\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if\u00a0(!string.IsNullOrEmpty(r))\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0{\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0var\u00a0tweetJToken\u00a0=\u00a0JsonConvert.DeserializeObject<dynamic>(r);\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0var\u00a0tweetObj\u00a0=\u00a0tweetJToken[\"text\"];\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if\u00a0(tweetObj\u00a0!=\u00a0null)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0{\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0var\u00a0tweetText\u00a0=\u00a0tweetObj.ToString();\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0viewModel.Items.Add(tweetText);\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}\u00a0\u00a0\u00a0\u00a0}\n\nThe equivalent F# async code obviously looks quite similar, with the added goodness of Type Providers. Time dependent, I am planning to do some more analysis of tweets which would be a good fit for F#.\n\ntype\u00a0tweetProvider\u00a0=\u00a0JsonProvider<\"SampleTweet.json\",\u00a0SampleList=true> type\u00a0MainWindowViewModel()\u00a0=\u00a0\u00a0inherit\u00a0ViewModelBase()\u00a0\u00a0let\u00a0items\u00a0=\u00a0new\u00a0ObservableCollection<string>()\u00a0\u00a0member\u00a0x.Items\u00a0\u00a0\u00a0\u00a0with\u00a0get\u00a0()\u00a0=\u00a0items\u00a0\u00a0member\u00a0x.ProcessTweet\u00a0tweet\u00a0=\u00a0\u00a0\u00a0\u00a0let\u00a0tweetParsed\u00a0=\u00a0tweetProvider.Parse(tweet)\u00a0\u00a0\u00a0\u00a0match\u00a0tweetParsed.Text\u00a0with\u00a0\u00a0\u00a0\u00a0|\u00a0Some(v)\u00a0->\u00a0\u00a0x.Items.Add\u00a0v\u00a0\u00a0\u00a0\u00a0|\u00a0None\u00a0->\u00a0()\u00a0\u00a0member\u00a0x.ProcessTweets\u00a0=\u00a0\u00a0\u00a0\u00a0let\u00a0a\u00a0=\u00a0async\u00a0{\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0use\u00a0t\u00a0=\u00a0new\u00a0TwitterConnector()\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0let!\u00a0response\u00a0=\u00a0t.GetSampleFirehoseConnection()\u00a0|>\u00a0Async.AwaitTask\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0let!\u00a0result\u00a0=\u00a0response.Content.ReadAsStreamAsync()\u00a0|>\u00a0Async.AwaitTask\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0use\u00a0streamReader\u00a0=\u00a0new\u00a0StreamReader(result,\u00a0Encoding.UTF8)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0let\u00a0rec\u00a0processTweetsAsync\u00a0(s:StreamReader)\u00a0=\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0async\u00a0{\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0let!\u00a0r\u00a0=\u00a0s.ReadLineAsync()\u00a0|>\u00a0Async.AwaitTask\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\/\/\u00a0streamReader.EndOfStream\u00a0can\u00a0block,\u00a0so\u00a0instead\u00a0check\u00a0for\u00a0null\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if\u00a0r\u00a0<>\u00a0null\u00a0then\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0x.ProcessTweet\u00a0r\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0return!\u00a0processTweetsAsync\u00a0s\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0do!\u00a0processTweetsAsync\u00a0streamReader\u00a0\u00a0\u00a0\u00a0}\u00a0\u00a0\u00a0\u00a0a\u00a0|>\u00a0Async.StartImmediate\u00a0\u00a0member\u00a0x.GoCommand\u00a0=\u00a0\u00a0\u00a0\u00a0\u00a0new\u00a0RelayCommand\u00a0((fun\u00a0canExecute\u00a0->\u00a0true),\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(fun\u00a0action\u00a0->\u00a0x.ProcessTweets))\n\n## Option pricing in F# using the Explicit Finite Difference method\n\nIt\u2019s been a little while since I\u2019ve coded any F#, so I\u2019ve done a little coding kata to polish off the rust.\n\nI\u2019ve converted the explicit finite difference option pricing example from (Paul Wilmott Introduces Quantitative Finance).\n\nI followed similar steps as those I performed in my previous blog post on the binomial tree option pricing; I converted the VBA code into very similar looking imperative F# code, before refactoring out the loops into recursive calls.\n\nIt was a useful exercise as converting the algorithm made me actually focus on the mechanics of how it worked, much more than simply translating VBA code into imperative F#. It\u2019d be interesting to hear in the comments whether people find it easy to read.\n\nThe code is available https:\/\/gist.github.com\/taumuon\/4999749.\n\n## Binomial Tree Option Pricing in F#\n\nThis post will look at implementing the binomial tree algorithm in an imperative style, and refactoring it to be more functional.\n\nThe wikipedia page on the binomial options pricing model provides pseudo-code for the pricing of an American Put option.\n\nThat code can be implemented almost identically in F#.\n\nand is called as follows:\n\nFunctional programmers have probably died a little on the inside on seeing that code, but one of the good thing about F# being multi-paradigm is that it\u2019s easy to take code of this form and implement it, and test it for correctness before refactoring.\n\nI did actually look at whether the results of the code were as expected at this point, but I\u2019ll cover this later in this blog post.\n\nThe first thing to do is to change the initial values V to be created more functionally:\n\nNow, the main algorithm consists of two parts, an outer loop where each \u2018level\u2019 of the tree is calculated stepping up towards the root, using the results of the previous level\u2019s nodes (the nodes nearest to the leaves). The inner loop iterates over all of the nodes at a given level calculating the option\u2019s value at that node.\n\nFirst off, the outer loop can be replaced with a recursive method:\n\nThe next change is quite minor \u2013 it\u2019s for each pass of the inner loop to create a new array, instead of mutating the vi array:\n\nThis change now means that there\u2019s no need for any mutable state, so the initial values can be created as a sequence instead of an array, and the inner loop can be written in a functional style:\n\nI find this code style is clearer to understand than following the array mutations to see what each part of the code is doing and how those parts interact.\n\nIf I was implementing this in C# I\u2019d probably change the variable names to be more descriptive (e.g. strikePrice, riskFreeRate etc) but as one of the key advantages of F# is its succinctness it makes more sense to leave the well-known symbols in. Anyone implementing the algorithm with knowledge of the domain would know what the symbols mean without the more verbose names.\n\n#### Validating the results\n\nThe calculated option value with the requested parameters is 4.486090. Running the EqutyOptions example in the .NET port of QuantLib, QLNet (http:\/\/sourceforge.net\/projects\/qlnet), with the same parameters gives the following results:\n\nThe figures are in the same ballpark, but the difference is curious.\n\nChapter 45 of GPU GEMS 2\u00a0 creates the initial values in this way:\n\nThis gives a result of 4.486375 which still doesn\u2019t match QLNet, but is nearer.\n\nThe result of running the Cox version of Bubo\u2019s code from here http:\/\/cs.hubfs.net\/topic\/None\/59053 also equals 4.486375, which matches the results of using this initialisation function. As an aside, I like how that examples shows how F# partial function application allows the variations on the algorithm to be composed, and much more succinctly than using OO.\n\nBubo\u2019s Tian model does match QuantLib\u2019s result exactly. I\u2019ll have to dig a bit deeper to figure out which variations on the binomial tree model these different examples are using.\n\nThe code is available here: https:\/\/github.com\/taumuon\/FSharp-Binomial-Tree\n\nThis blog post will look at using the continuation monad to sum the nodes in a tree, and then look at a case where a more complex monad is needed.\n\nThe book Real World Functional Programming gave an example of an unbalanced tree, which was deeply nested such that recursing over the tree to sum the values would blow the stack:\n\nThe solution provided was to sum the tree using continuation passing:\n\nThis is now able to sum the tree without crashing, but the code doesn\u2019t flow as well as in the original sumTree method. However, monads can come to the rescue again, in this case the continuation monad:\n\nThe code flow now is easier to understand. The continuation passing style and continuation monad turns the code flow inside out, which isn\u2019t a big deal, but means that the func to perform on the result needs to be passed in:\n\n### Parallel sum\n\nThis is all well and good, but to exploit multicore machines, I want to take advantage of the parallelism in the tree summation algorithm, by summing the different branches of the nodes using tasks. A na\u00efve implementation to do this may be:\n\nThe problem with this is that it runs slower than the sequential case for a large balanced tree \u2013if there are only 4 cores in the system it\u2019s counter-productive to create thousands of tasks. The current depth of the tree should be passed around, and tasks spawned until traversal reaches that depth:\n\nThis is now faster for a large balanced tree, but for the deeply nested unbalanced tree it throws a StackOverflowException. Note that sumTreeParallel coped with arbitrarily nested trees, as the work to be done no longer kept being pushed onto the stack.\n\nMore state is being passed around (the current depth), which is external to the main business logic \u2013 it does feel like the state monad could be used here. First though, this is fixed up so that it has the efficiency of the parallel summation, but can cope with arbitrarily deeply nested trees:\n\nUrghhhh. The code is definitely more convoluted now \u2013 I played with using the state monad to pass the depth around, but it\u2019s not tail recursive. I thought that it might be possible to make use of the continuation monad here, but the problem is, is that it really needs aspects of both the state and continuation monads. I\u2019m not sure how monad combinations work in F#, but I thought I\u2019d throw this out here to see if anyone shows me the light. The post The Mother of All Monads does say that all other monads could be implemented in terms of the continuation monad, but I couldn\u2019t see an example of anyone implementing the state monad in terms of the continuation monad.\n\n## State Monad in C# and F#\n\nMy colleagues and I recently have been trying to gain a deeper understanding of monads, other than using LINQ and RX. Watching all the channel9 videos on monads, the concept on the surface seemed quite simple, but we were felt that we were somehow missing some key point. After reading various blog posts, the penny is finally dropping.\n\nSome post I read somewhere (I can\u2019t remember where) said that it\u2019s easier to understand the individual monads before trying to understand the whole generalised concept. Mike Hadlow\u2019s blog series (http:\/\/mikehadlow.blogspot.co.uk\/2011\/01\/monads-in-c1-introduction.html) really explain well the identity and maybe monads (which we already understood, which is just as well as we\u2019re using the maybe monad in our codebase). Unfortunately he didn\u2019t explain the state monad.\n\nThe only information we could find on the state monad in C# was Brian Beckman\u2019s Channel 9 video: http:\/\/channel9.msdn.com\/Shows\/Going+Deep\/Brian-Beckman-The-Zen-of-Expressing-State-The-State-Monad\n\nWe found Brian Beckman\u2019s example overly complex, there\u2019s quite a lot going on in that video (and the code needed de-obfuscating to read more like idiomatic C#), but it\u2019s probably a good second or third example once the basics are understood. I wanted to look at a simpler example, and found this article: http:\/\/ertes.de\/articles\/monads.html#section-6 the explanation of the get and put methods show pretty clearly how the state monad works, and you should probably read and understand that section before carrying on here, as I\u2019m going to jump straight on to show how the random number generator example looks in F# and C#.\n\n### C# implementation\n\nFirst off, here\u2019s an OO random number generator:\n\nIt is easy to add three random numbers together using an instance of the generator.\n\nWe can make NextShort purely functional, so that calling it has no side-effects, by passing the current state in, and returning the new state back out, along with the result:\n\nTo add three numbers, the state has to be kept around, and passed to each of the calls in turn. This obfuscates the main business logic.\n\nJumping straight to the punch line, using a state monad means that the boilerplate state management can be hidden away so that the algorithm reads as clearly as it did before:\n\nThis is a trivial example, but hopefully it makes the point. The rest of this blog post will show the implementation, along with the same in F#.\n\nFirst off, we have a state class, which holds a Func that given a state, returns a tuple of the state and value. The F# example doesn\u2019t bother with the State class, and I could have made the C# version simply use the Func everywhere, but it seemed clearer to me to have the wrapper object just to get a bit of context.\n\nI created a non-generic State class as somewhere convenient to put the Get and Set methods:\n\nThe meat of the monad lives in extension methods in a StateEx class:\n\nand SelectMany is implemented in terms of Bind the same as in Mike Hadlow\u2019s example for the Identity and Maybe monads:\n\nThe computation to compose can be expressed in the GetRandom method:\n\nThis returns an operation (a Func in the State container) that invokes NextShort with the old state, and returns a computation with the new state and value.\n\nThe resulting composed operation using LINQ is described above, but this is what it looks like desugared:\n\n### F# implementation\n\nThe code for the random number generator and composed operation look similar to the C#:\n\nThis has the same problem of explicitly passing the state around. I can ruin the surprise again and show how using monads hides the explicit state passing:\n\nThis code makes use of F#\u2019s support for monads via Computation Expressions. The code for the state builder is from here: http:\/\/www.navision-blog.de\/2009\/10\/23\/using-monads-in-fsharp-part-i-the-state-monad\/\n\nThe Haskell example also had the >> operator, but of course F# uses that as the forward composition operator J\n\ngetRandom looks pretty much identical to the Haskell example:\n\nThe composed version using the StateBuilder is shown above, but here\u2019s the desugared version:\n\nThe F# version is much clearer to read than the C#, in part due to the type inference meaning that the code has less pointless fluff. Having functions live in the module and not as part of a class aids that too (can simply call setState instead of State.Set in the C#)\n\nFollowing through the simpler example really helped my understanding, and hopefully will see examples in the future where it will be sensible to use the state monad in my coding.\n\n## Visualising Sound\n\nThis is the first in a series of blog posts about visualising sound, in a similar way to an oscilloscope. The geek in me thought it\u2019d be a fun thing to do, as well investigate different technological approaches to implementing the app (interesting as it\u2019s a real-time performant app).\n\nAn oscilloscope has a single screen, which refreshes on a given time period, displaying a number of traces. The user can control that time period, as well as the gain (the y axis scale).\n\nThe top graph is essentially the same view as an oscilloscope containing a single trace, and a fixed time period (further blog posts may investigate varying time periods).\n\nThe bottom graph is a sliding window with a longer time period \u2013 this is the advantage of implementing an oscilloscope in code, we can create charts that aren\u2019t really feasible in a classic CRT oscilloscope.\n\nThis series of blog posts will investigate implementing this screen using F# idioms, as well as using the Reactive Extension for .NET (RX Framework), and TPL Dataflow.\n\nThere are further things that could be implemented in future blog posts which may be interesting to see how the varying approaches look like:\n\n\u00b7 Trigger, or Trigger and hold: Only refresh the oscilloscope view once a certain trigger level has been reached. This is interesting as may want to include a certain number of immediately prior to the point where the trigger was set.\n\n\u00b7 Log many traces.\n\n\u00b7 Spectrum analyser\/FFT.\n\n\u00b7 Digital filtering.\n\n\u00b7 Comparing traces, or more complicated math channels.\n\n\u00b7 Heatmap.\n\n\u00b7 Adjustable time offset (delay) \u2013 useful on the oscilloscope view to centre a waveform on the screen, or for when comparing two or more channels output.\n\n### F# Implementation\n\nThis blog post covers an F# implementation of the microphone, using asynchronous workflows. The graphing is being done by a nightly build download of Dynamic Data Display (D3). I\u2019m really impressed with the performance, but it is quite difficult to customise.\n\nThe implementation of this is pretty simple; I\u2019ll start from the detail out.\n\nThe inner loop of the program is an asynchronous workflow that reads from the buffer and returns a sequence:\n\nNote the slightly-strange looking code:\n\nF# has an Async.AwaitWaitHandle() method, which unfortunately only waits on a single handle. We want to wait on both handles so that we get notified when the buffer is full every 4096 instead of every 8192 bytes. With 4 bytes per sample, and a sample rate off 44KHz, this is equivalent to getting notified at an approximate rate of 40 times per second instead of 20 times per second.\n\nI could have implemented Async.AwaitAnyWaitHandle() taking an array of WaitHandles, but looking at the code in the F# PowerPack, the code was quite complex. So, the code instead creates a new future to do the waiting and let us know which WaitHandle was set (this does mean that we\u2019ve got the minor overhead of scheduling a new task to run on the task pool).\n\nThe Async.StartImmediate method ensures that the ProcessStream call is marshalled back onto the UI thread. It may be worth in the future looking at doing more of the data processing on a dedicated thread, leaving the UI thread free for drawing and user input.\n\nThe convertByteArrayToSequence is simple, it just iterates over the buffer in 4 byte chunks, and converts the values to floats, which it yields in the sequence:\n\nThe ProcessStream method looks like this:\n\nFor completeness, this is the Seq.sample module extension:\n\nThe nastiest bit of the code in ProcessStream is probably towards the end, where the windowedUnderlyingData is manipulated to ensure that the window only contains 1000 samples. It would be nice to do this in a non-imperative way, using Seq.windowed, but the problem is, is that the sequence we\u2019re operating on is only the result of one buffer read operation, whereas windows etc. should operate over the whole data stream, and the sequences can\u2019t be combined into a single sequence using yield! as they\u2019re all generated asynchronously. Similarly, the buffer takes non-overlapping windows over the input sequence, and without taking a buffer over the whole stream, it may miss samples off the end of the sequence portions. Tomas Petricek has written a library, AsyncSeq, http:\/\/tomasp.net\/blog\/async-sequences.aspx which I may investigate in a later post.\n\nThe alternative to this would be to implement mailbox processors, having different agents for the buffering, windowing and sampling operations. I did start investigating this, but didn\u2019t feel happy with them being pull-based (they don\u2019t return data unless asked). I could have set them up to act more like a dataflow network, but it does seem to go against their intended use of managing state between different threads. I may revisit this in a future blog post.\n\nI feel that even though F#\u2019s does have nice features which helped to quickly implement the app, RX would probably be a better fit. I guess I won\u2019t know until I implement it in RX and compare the differences.\n\n## F# Flocking (boids)\n\nA couple of years ago I was blogging about some simple boid behaviour I had implemented in C# (here). I came across this paper which had some interesting ideas I wanted to try out: \"Self-organised complex aerial displays of thousands of starlings: a model\" by H. Hildenbrandt, C. Carere, C-K. Hemelrijk.\n\nAs I\u2019m getting more into F#, I thought this would be a simple reasonable sized application to create to get a feel for how F# works on a slightly larger project (though definitely not large-scale).\n\nBefore taking a look at the code, I\u2019ll quickly describe which parts of the paper I implemented.\n\nThe video generated is available on http:\/\/www.youtube.com\/watch?v=eil5K7Ir3i8\n\nI haven\u2019t implemented all of the ideas in the paper, so my simple application doesn\u2019t exhibit the same realistic flocking behaviour, but it does have more realistic behaviour than my earlier efforts. The interesting behaviours were the desire to not stray too far from the roost, the attempt to maintain a steady cruise speed, and to use the nearest seven topological neighbours for cohesion and alignment. I implemented the simulation in 2D for simplicity.\n\nI hardcoded a perception radius instead of implementing the continual adjustment of the perception radius for seven topological neighbours. I also have totally omitted the flight dynamic simulation (no gravity, no banked turns). The paper discusses that the banked turns gives a further air of realism to the simulation.\n\n### The coding\n\nThis was a real pleasure to implement in F# \u2013 a lot less time was thinking about application design, classes and their interactions, the logic in the paper was easy to transfer directly to the keyboard. F# is very compact, so the core logic can be seen on just a couple of screens.\n\nI didn\u2019t feel the lack of intellisense too much when developing the core algorithm, I did miss the ability to refactor functions and navigate to usages. I especially missed intellisense when developing the WPF visualisation part of the app, when interacting with .NET objects; I did miss the ability for Visual Studio to automatically add the required namespaces. I must have being spoilt by Visual Studio and Resharper for too long!\n\nThe actual WPF application wasn\u2019t such a good fit for F# \u2013 there\u2019s no generated code behind file for the XAML, and I feel that using F# would be painful in a WPF or Silverlight application (but just for the view, it should be OK for the ViewModels down).\n\nI implemented a F# Vector type, which can be specified with units of measure (or none). I used units of measure throughout \u2013 this was really powerful, and did let me quickly find a few bugs in the implementation.\n\n(NOTE: I still haven\u2019t found a code websnippet tool I\u2019m happy with \u2013 you need to click in each of the regions below and scroll down and right to see the whole code). Alternately, the zipped up solution can be downloaded from http:\/\/www.taumuon.co.uk\/jabuka\/FSharpFlock.zip\n\nVector3D.fs:\n\nmodule Vector3Dtype Vector3D<[<Measure>] 'u>(x : float<'u>, y : float<'u>, z : float<'u>) = static member Zero() = Vector3D<_>(0.0<_>, 0.0<_>, 0.0<_>) member v.X = x member v.Y = y member v.Z = z static member (+) (lhs:Vector3D<_>, rhs:Vector3D<_>) = Vector3D(lhs.X + rhs.X, lhs.Y + rhs.Y, lhs.Z + rhs.Z) static member (-) (lhs:Vector3D<_>, rhs:Vector3D<_>) = Vector3D(lhs.X - rhs.X, lhs.Y - rhs.Y, lhs.Z - rhs.Z) static member (*) (v:Vector3D<_>, a:float<_>) = Vector3D(v.X * a, v.Y * a, v.Z * a) static member (*) (a:float<_>, v:Vector3D<_>) = Vector3D(a * v.X, a * v.Y, a * v.Z) static member (\/) (v:Vector3D<_>, a) = Vector3D(v.X \/ a, v.Y \/ a, v.Z \/ a) member v.DotProduct(rhs:Vector3D<_>) = (v.X * rhs.X) + (v.Y * rhs.Y) + (v.Z * rhs.Z) member v.magnitude = sqrt(v.DotProduct(v)) * 1.0<_> member lhs.CrossProduct(rhs:Vector3D<_>) = Vector3D((lhs.Y * rhs.Z - lhs.Z * rhs.Y) * 1.0<_>, (-lhs.X * rhs.Z + lhs.Z * rhs.X) * 1.0<_>, (lhs.X * rhs.Y - lhs.Y * rhs.X) * 1.0<_>) member v.normalise = let magnitude = float v.magnitude Vector3D<_>((v.X \/ magnitude), (v.Y \/ magnitude), (v.Z \/ magnitude))let sumVectors(vectors : Vector3D<_>[]) = let initial = Vector3D<_>(0.0<_>, 0.0<_>, 0.0<_>) Array.fold (+) initial vectors\n\nBoidUtils.fs:\n\nmodule BoidUtilsopen Microsoft.FSharp.Mathopen Vector3Dopen SIlet radiusRoost = 150.0<m>let hardRadius = 2.0<m> \/\/ 0.2<m>let mass = 0.08<kg>let timeStep = 0.005<s>let relaxationTime = 0.05<s>let cruiseSpeed = 20.0<m\/s>let horizontalRoostWeighting = 0.01<N\/m>let weightingAlignment = 0.5<kg * s^-2>let weightingCohesion = 1.0<kg s^-2>let weightingSeparation = 2.0let perceptionRadius = 50.0<m>type BodyAxes = { Forward:Vector3D<1>; Side:Vector3D<1>; Up:Vector3D<1> }type Boid = { Position:Vector3D<m>; Speed:float<m \/ s>; Orientation:BodyAxes; }\/\/ All parameterless functions are evaluated once, just on module opening, so pass random in.let InitialiseRandomPosition(rand:System.Random) = Vector3D<m>((300.0 * (rand.NextDouble()-0.5)) * 1.0<m>, ((300.0 * (rand.NextDouble()-0.5)) * 1.0<m>), 0.0<m>)let InitialiseRandomVelocity(rand:System.Random) = Vector3D<m\/s>((100.0 * (-0.5 + rand.NextDouble()) * 1.0<m\/s>), (100.0 * (-0.5 + rand.NextDouble())) * 1.0<m\/s>, 0.0<m\/s>)let InitialiseRandomOrientation(rand:System.Random) = {Forward=Vector3D(0.0, 1.0, 0.0); Side=Vector3D(1.0, 0.0, 0.0); Up=Vector3D(0.0, 0.0, 1.0)}let setOrientation(oldOrientation:BodyAxes, velocity:Vector3D<m\/s>) = let normalisedVelocity = velocity.normalise let y = normalisedVelocity.CrossProduct(Vector3D<m \/ s>(0.0<m\/s>, 0.0<m\/s>, 1.0<m\/s>)) {oldOrientation with Forward=normalisedVelocity * 1.0<m^-1 s>; Side=y*1.0<m^-2 s^2>}let calculateCruiseSpeedForce (boid:Boid) = (mass \/ relaxationTime) * (cruiseSpeed - boid.Speed) * boid.Orientation.Forwardlet calculateRoostForce (boid:Boid) = let horizontalPosition = Vector3D(boid.Position.X, boid.Position.Y, 0.0<_>) let distanceFromOrigin = horizontalPosition.magnitude match (distanceFromOrigin) with | _ when distanceFromOrigin < radiusRoost -> Vector3D<N>(0.0<N>, 0.0<N>, 0.0<N>) | _ -> let normalRoostingArea = horizontalPosition.normalise let d = boid.Orientation.Forward.DotProduct normalRoostingArea let distanceFromRoost = distanceFromOrigin - radiusRoost let orientationRoostDotProduct = boid.Orientation.Side.DotProduct normalRoostingArea let weightingFactor = match (orientationRoostDotProduct) with | _ when orientationRoostDotProduct > 0.0<m> -> -1.0 | _ -> 1.0 weightingFactor * (radiusRoost * horizontalRoostWeighting * (0.5 + (0.5<m^-1> * d)) * (boid.Orientation.Side))let findDistanceBetweenBoids(boid:Boid, other:Boid) = (boid.Position - other.Position).magnitudelet findNearestNeighbours(boid:Boid, boids:Boid[]) = let sortedByDistance = boids |> Array.sortBy(fun other -> findDistanceBetweenBoids(boid, other)) Array.sub sortedByDistance 0 7let findAverageForwardDirectionDifference(boid:Boid, boids:Boid[]) = let differences = boids |> Array.map (fun i -> 1.0<m> * (i.Orientation.Forward - boid.Orientation.Forward)) let sumDifferences = sumVectors(differences) (1.0 \/ (float sumDifferences.magnitude)) * sumDifferenceslet calculateAlignmentForce(boid:Boid, nearest:Boid[]) = let averageDifference = findAverageForwardDirectionDifference(boid, nearest) weightingAlignment * averageDifferencelet findAveragePosition(boid:Boid, boids:Boid[]) = let positions = boids |> Array.map (fun i -> i.Position) let sumPositions = sumVectors(positions) (1.0 \/ float boids.Length) * sumPositionslet findNeighboursInRadius(boid:Boid, boids:Boid[], radius:float<m>) = boids |> Array.filter(fun other -> other <> boid && findDistanceBetweenBoids(boid, other) <= radius)let calculateCentrality(boid:Boid, boids:Boid[]) = let separations = boids |> Array.map(fun i -> (i.Position - boid.Position).normalise) let sumSeparations = sumVectors(separations) let count = boids.Length match (count) with | 0 -> 1.0 | _ -> (1.0 \/ float count) * (sumSeparations.magnitude \/ 1.0<m>)let calculateCohesionForce(boid:Boid, nearest:Boid[], boidsInPerceptionRadius:Boid[]) = let boidsOutsideHardRadius = nearest |> Array.filter(fun i -> abs ((boid.Position - i.Position).magnitude) > hardRadius) let centrality = calculateCentrality(boid, boidsInPerceptionRadius) let averagePosition = findAveragePosition(boid, nearest) centrality * weightingCohesion * (averagePosition - boid.Position)let calculateSeparationForce(boid:Boid, boidsInPerceptionRadius:Boid[]) = let nearest = boidsInPerceptionRadius let separations = nearest |> Array.map(fun i -> i.Position - boid.Position) let sigma = 1.8 let forcesToNeighbours = separations |> Array.map(fun i -> let magnitude = i.magnitude let multiplier = match (magnitude) with | _ when magnitude < hardRadius -> 1.0 | _ -> System.Math.Exp(-((magnitude - hardRadius)*(magnitude - hardRadius)\/1.0<m^2>) \/ (sigma * sigma)) multiplier * magnitude * (i.normalise) * 1.0<kg * m^-1 * s^-2>) let sumForces = sumVectors(forcesToNeighbours) match (nearest.Length) with | _ when (nearest.Length) = 0 -> Vector3D<N>.Zero() | _ -> (-weightingSeparation \/ float nearest.Length) * sumForceslet calculateSocialForce(boid:Boid, boids:Boid[]) = let nearest = findNearestNeighbours(boid, boids) let boidsInPerceptionRadius = findNeighboursInRadius(boid, boids, perceptionRadius) calculateAlignmentForce(boid, nearest) + calculateCohesionForce(boid, nearest, boidsInPerceptionRadius) + calculateSeparationForce(boid, boidsInPerceptionRadius)let calculateForce (boid:Boid, boids:Boid[]) = (boid |> calculateRoostForce) + (boid |> calculateCruiseSpeedForce) + (calculateSocialForce(boid, boids))let iterateBoid (boid:Boid, boids:Boid[]) = let originalPosition = boid.Position let originalVelocity = boid.Speed * boid.Orientation.Forward let force = calculateForce(boid, boids) let acceleration = force\/mass let velocity = originalVelocity + (acceleration * timeStep) let position = originalPosition + (velocity * timeStep) let newOrientation = setOrientation(boid.Orientation, velocity) {Position=position;Speed=velocity.magnitude;Orientation=newOrientation}\n\n\n\nProgram.fs:\n\nmodule Boidsopen SIopen Vector3Dopen BoidUtilsopen Systemopen System.IOopen System.Windowsopen System.Windows.Threadingopen System.Windows.Controlsopen System.Windows.Shapesopen System.Windows.Mediaopen System.Windows.Media.Imaginglet window = Application.LoadComponent(new System.Uri(\"\/FSharpFlock;component\/MainWindow.xaml\", System.UriKind.Relative)) :?> Windowlet rand = new System.Random()let viewWidth = 480.0let viewHeight = 360.0let mutable frameCount = 0let mutable boids = [|for i in 0 .. 300 -> let position = InitialiseRandomPosition(rand) let velocity = InitialiseRandomVelocity(rand) let tempOrientation = {Forward=Vector3D(0.0, 1.0, 0.0); Side=Vector3D(1.0, 0.0, 0.0); Up=Vector3D(0.0, 0.0, 1.0)} let orientation = setOrientation(tempOrientation, velocity) {Position = position; Speed=velocity.magnitude; Orientation=orientation} |];;let updateBoids(boids) = boids |> Array.map (fun boid -> iterateBoid(boid, boids))let (?) (fe:FrameworkElement) name : 'T = fe.FindName(name) :?> 'Tlet GetRotationForBoid(boid:Boid) = let forward = boid.Orientation.Forward let angleRadians = Math.Atan2(forward.X, forward.Y) let angleDegrees = angleRadians * 180.0 \/ Math.PI let rotateTransform = new Media.RotateTransform(angleDegrees, 0.0, 0.0) rotateTransformlet color = Media.Color.FromArgb(64uy, 255uy, 0uy, 0uy)let saveFrame(canvas : Canvas) = let size = new Size(viewWidth, viewHeight) canvas.Measure(size) canvas.Arrange(new Rect(size)) let renderTargetBitmap = new RenderTargetBitmap(int viewWidth, int viewHeight, 96.0, 96.0, PixelFormats.Pbgra32) let sourceBrush = new VisualBrush(canvas) renderTargetBitmap.Render(canvas) let bitmapFrame = BitmapFrame.Create(renderTargetBitmap) let jpegEncoder = new JpegBitmapEncoder() jpegEncoder.Frames.Add(bitmapFrame) let filename = System.String.Format(\"C:Anim{0:0000}.jpg\", frameCount) use stream = new FileStream(filename, FileMode.CreateNew) jpegEncoder.Save(stream) let createBoidGraphics(boid:Boid) = let obj : Polygon = new Polygon() obj.Points <- new Media.PointCollection( [|new Point(-10.0, -5.0); new Point(0.0, 0.0); new Point(-10.0, 5.0); new Point(-5.0, 0.0)|] ) obj.RenderTransform <- GetRotationForBoid(boid) obj.Fill <- new Media.SolidColorBrush(color) obj.Stroke <- Media.Brushes.Black obj.StrokeThickness <- 1.0 objlet drawBoids() = let win : Window = window let canvas : Canvas = win?canvas canvas.Children.Clear() for i = 0 to boids.Length - 1 do let graphicalBoid = createBoidGraphics(boids.[i]) let unitlessPosition = boids.[i].Position * 1.0<m^-1> System.Windows.Controls.Canvas.SetTop(graphicalBoid, (viewHeight \/ 2.0) + (unitlessPosition.X)) System.Windows.Controls.Canvas.SetLeft(graphicalBoid, (viewWidth \/ 2.0) + (unitlessPosition.Y)) canvas.Children.Add(graphicalBoid) |> ignore \/\/ saveFrame(canvas) frameCount <- frameCount + 1let timer = new DispatcherTimer();timer.Tick.Add(fun evArgs -> [0..5] |> Seq.iter(fun x -> boids <- boids |> updateBoids) drawBoids() )timer.Interval <- TimeSpan.FromMilliseconds(1.0);let setup(win: Window) = win.Loaded.Add(fun _ -> timer.Start())[<STAThread>][<EntryPoint>]let main args = setup window (new Application()).Run(window) |> ignore 0\n\nMainWindow.xaml:\n\n<Window xmlns=\"http:\/\/schemas.microsoft.com\/winfx\/2006\/xaml\/presentation\" xmlns:x=\"http:\/\/schemas.microsoft.com\/winfx\/2006\/xaml\" Title=\"F# Flock\" Height=\"360\" Width=\"480\"> <Canvas Name=\"canvas\" HorizontalAlignment=\"Stretch\" VerticalAlignment=\"Stretch\" Background=\"LightBlue\"\/><\/Window>\n\nThe rest of the code I hope is simple to read and understand. Coding this up I had some of the most fun programming for a while. With F#, I found I was able to concentrate on the core of the algorithm and very quickly see results.\n\nFor this type of project, very algorithmic, F# was a perfect fit. I\u2019d like to get a feel for how to implement F# on a large scale project, and to see how it feels to expose the functionality via an object oriented layer.","date":"2019-02-23 11:45:31","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.2795703411102295, \"perplexity\": 2318.5538194659116}, \"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-2019-09\/segments\/1550249500704.80\/warc\/CC-MAIN-20190223102155-20190223124155-00115.warc.gz\"}"}
| null | null |
In fact, wintry weather provided such a boost to UK clothing and footwear sales last month that they helped lift total sales to a five-year December high.
So why is it that the outlook for the sector in 2010 seems so bleak – perhaps even as challenging as 2009?
A look behind the headlines shows the results were undeniably flattered by strong comparisons with last December's dire results, when sales fell 3.3% as the economic downturn reached fever pitch.
But fear seems to have played a major role too, with shoppers undoubtedly worried about what the year ahead might hold. Many will have boosted December's numbers by bringing forward planned purchases ahead of the VAT rise on 1 January.
And even without a crystal ball it's possible to predict with some certainty that the spectre of higher taxes in the year ahead is set to pressure consumer spending even further. No matter which party takes office in the upcoming General Election, the bill for the banking crisis in the UK is sure to lead to tax hikes.
There also remains the prospect of rising fuel prices later in the year, a likely freeze on public sector pay, uncertainties on job prospects and a continued tight credit market – making for an environment where consumers will hardly be awash with cash.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,920
|
\section{Introduction}
In this paper we use the theory of Kolyvagin systems to
prove (most of) a conjecture of Darmon from \cite{darmon}.
In \cite[Conjecture 4.1]{gross}, inspired by work of the first author and
Tate \cite{mazurtate}, and of Hayes \cite{hayes}, Gross conjectured a
``refined class number formula'' for abelian extensions $K/k$ of global fields.
Attached to this extension (and some chosen auxiliary data)
there is a generalized Stickelberger element $\theta_{K/k} \in \mathbf{Z}[G]$, where $G := \mathrm{Gal}(K/k)$,
with the property that for every complex-valued character $\chi$ of $G$,
$\chi(\theta_{K/k})$ is essentially $L(K/k,\chi,0)$ (modified by the chosen auxiliary data).
Gross' conjectural formula is a congruence for $\theta_{K/k}$, modulo a certain specified
power of the augmentation ideal of $\mathbf{Z}[G]$, in terms of a regulator that Gross defined.
In a very special case, Darmon formulated an analogue of Gross' conjecture involving
first derivatives of $L$-functions at $s=0$.
Suppose $F$ is a real quadratic field, and $K_n := F(\boldsymbol{\mu}_n)$
is the extension of $F$ generated by $n$-th roots of unity,
with $n$ prime to the conductor of $F/\mathbf{Q}$.
Darmon defined a Stickelberger-type
element $\theta_n' \in K_n^\times \otimes \mathbf{Z}[\mathrm{Gal}(K_n/F)]$, interpolating
the first derivatives $L'(\chi\omega_F,0)$, where $\omega_F$ is
the quadratic character attached to $F/\mathbf{Q}$ and $\chi$ runs through even Dirichlet characters
of conductor $n$.
Darmon conjectured that $\theta_n'$ is congruent, modulo a specified
power of the augmentation ideal, to a regulator that he defined.
See \S\ref{statement} and Conjecture \ref{dconj} below for a precise statement.
Our main result is a proof of Darmon's conjecture ``away from the $2$-part''.
In other words, we prove that the difference of the two sides of Darmon's
conjectured congruence is an element of $2$-power order.
The idea of our proof is a simple application of the results proven
in \cite{kolysys}. For every odd prime $p$ we show that although
neither the left-hand side nor the right-hand of Darmon's
conjectured congruence (as $n$ varies) is a ``Kolyvagin system'' as defined in \cite{kolysys},
each side is {\em almost} a Kolyvagin system; moreover, both sides fail to be
Kolyvagin systems in precisely the same way. That is, we show that the left-hand side
and right-hand side form what we call in this paper {\em pre-Kolyvagin systems}
in the sense that they each satisfy the specific set of local and global
compatibility relations given in Definition \ref{preksdef} below.
It seems that pre-Kolyvagin systems are what tend to occur ``in nature'',
while Kolyvagin systems satisfy a cleaner set of axioms.
We show that the two concepts are equivalent,
by constructing (see Proposition \ref{preksks}) a natural transformation $\mathcal{T}$ that turns
pre-Kolyvagin systems into Kolyvagin systems and has the properties that:
\begin{itemize}
\item $\mathcal{T}$ does not change the term associated to $n=1$, and
\item $\mathcal{T}$ is an isomorphism from the $\Z_p$-module of pre-Kolyvagin systems
to the $\Z_p$-module of Kolyvagin systems.
\end{itemize}
Since it was proved in \cite{kolysys} that (in this situation) the
space of Kolyvagin systems is a free $\Z_p$-module of rank one,
it follows that if two pre-Kolyvagin systems agree when $n=1$,
then they agree for every $n$. In the case $n= 1$, Darmon's congruence
follows from classical formulas for $L'(\omega_F,0)$, so
we deduce that (the $p$-part, for every odd prime $p$ of)
Darmon's conjectured congruence formula holds for all $n$.
Darmon's conjecture begs for a generalization.
A naive generalization, even just to the case where
$F$ is a real abelian extension of $\mathbf{Q}$, is unsuccessful
because the definition of Darmon's regulator
does not extend to the case where $[F:\mathbf{Q}] > 2$.
In a forthcoming paper we will use the ideas and conjectures of
\cite{RS} to show how both Gross' and Darmon's conjectures are special cases of a much more
general conjecture. In the current paper we treat only Darmon's conjecture because it can be
presented and proved in a very concrete and explicit manner.
The paper is organized as follows. In \S\ref{nota} we describe our setting and notation, and
in \S\ref{statement} we state Darmon's conjecture and our main result (Theorem \ref{mainthm}).
In \S\ref{augq} we recall some work of Hales \cite{hales}
on quotients of powers of augmentation ideals, that will enable us to translate the
definition of Kolyvagin system given in \cite{kolysys} into a form that will be more
useful for our purposes here. In \S\ref{kssect} we give the definition of a Kolyvagin system
(for the Galois representation $\Z_p(1) \otimes \omega_F$).
In \S\ref{prekssect} we define pre-Kolyvagin system, and give an isomorphism between the
space of pre-Kolyvagin systems and the space of Kolyvagin systems.
In \S\ref{cupks} (resp., \S\ref{rpks}) we show that the ``Stickelberger'' side
(resp., regulator side) of Darmon's formula is a pre-Kolyvagin system as $n$ varies.
Finally, in \S\ref{pfsect} we combine the results of the previous sections to
prove Theorem \ref{mainthm}.
\section{Setting and notation}
\label{nota}
Fix once and for all a real quadratic field $F$, and let $f$
be the conductor of $F/\mathbf{Q}$. Let $\omega = \omega_F$ be the quadratic Dirichlet
character associated to $F/\mathbf{Q}$, and
$\tau$ the nontrivial element of $\mathrm{Gal}(F/\mathbf{Q})$.
If $M$ is a $\mathrm{Gal}(F/\mathbf{Q})$-module, we let $M^-$ be the subgroup of elements of
$M$ on which $\tau$ acts as $-1$.
Throughout this paper $\ell$ will always denote a prime number.
Let $\mathcal{N}$ denote the set of squarefree positive integers prime to $f$. If
$n \in\mathcal{N}$ let $n_+$ be the product of all primes dividing $n$ that split in $F/\mathbf{Q}$,
and $r(n) \in \mathbf{Z}_{\ge 0}$ the number of prime divisors of $n_+$:
\begin{align*}
n_+ &:= \prod_{{\ell\mid n},{\omega(\ell) = 1}} \ell, \\
r(n) &:= \#\{\ell : \ell \mid n_+\} = \#\{\ell : \text{$\ell \mid n$ and $\ell$ splits in $F$}\}.
\end{align*}
For every $n \in \mathcal{N}$ let $\boldsymbol{\mu}_n$ be the
Galois module of $n$-th roots of unity in $\bar{\mathbf{Q}}$, define
$$
\Gamma_n := \mathrm{Gal}(F(\boldsymbol{\mu}_n)/F) \cong \mathrm{Gal}(\mathbf{Q}(\boldsymbol{\mu}_n)/\mathbf{Q}) \cong (\mathbf{Z}/n\mathbf{Z})^\times,
$$
and let $I_n$ denote the augmentation ideal of $\mathbf{Z}[\Gamma_n]$, which is generated over
$\mathbf{Z}$ by $\{\gamma-1 : \gamma \in \Gamma_n\}$.
There is a natural isomorphism
\begin{equation}
\label{caug}
\Gamma_n \cong I_n/I_n^2
\end{equation}
defined by sending $\gamma \in \Gamma_n$ to $\gamma-1 \pmod{I_n^2}$.
If $m \mid n$ then we can view $\Gamma_m$ either as the quotient $\mathrm{Gal}(F(\boldsymbol{\mu}_m)/F)$
of $\Gamma_n$, or as the subgroup $\mathrm{Gal}(F(\boldsymbol{\mu}_n)/F(\boldsymbol{\mu}_{n/m}))$. With the latter
identification we have
$$
\Gamma_n = \prod_{\ell \mid n} \Gamma_\ell, \quad I_n/I_n^2 = \bigoplus_{\ell\mid n}I_\ell/I_\ell^2
$$
the product and the sum taken over primes $\ell$ dividing $n$.
We will usually write the group operation in multiplicative groups such as $F^\times$
with standard multiplicative notation (for example, with identity element $1$). However,
when dealing with ``mixed'' groups such as $F^\times \otimes I_n^r/I_n^{r+1}$,
we will write the operation additively and use $0$ for the identity element.
Fix an embedding $\bar\mathbf{Q} \hookrightarrow \mathbf{C}$.
\section{Statement of the conjecture}
\label{statement}
In this section we state our modified version of Darmon's conjecture
(mostly following \cite{darmon}) and our main result (Theorem \ref{mainthm}).
If $n \in \mathcal{N}$, let $\zeta_n \in \boldsymbol{\mu}_n$ be the
inverse image of $e^{2 \pi i /n}$ under the chosen embedding $\bar\mathbf{Q} \hookrightarrow \mathbf{C}$,
and define the cyclotomic unit
$$
\alpha_n :=
\prod_{\gamma \in \mathrm{Gal}(\mathbf{Q}(\boldsymbol{\mu}_{nf})/\mathbf{Q}(\boldsymbol{\mu}_n))}\gamma(\zeta_{nf}-1)^{\omega_F(\gamma)}
\quad\in~ F(\boldsymbol{\mu}_n)^\times
$$
and the ``first derivative $\theta$-element''
$$
\theta'_n =
\sum_{\gamma \in \Gamma_n}\gamma(\alpha_n) \otimes \gamma
\quad\in~ F(\boldsymbol{\mu}_n)^\times \otimes \mathbf{Z}[\Gamma_n].
$$
\begin{rem}
The element $\theta'_n$ is an ``$L$-function derivative evaluator'' in the sense that
for every even character $\chi : \Gamma_n \to \mathbf{C}^\times$, classical formulas give
$$
(\log|\cdot| \otimes \chi)(\theta'_n)
:= \sum_{\gamma\in\Gamma_n}\chi(\gamma)\log|\gamma(\alpha_n)| = -2L'_n(0,\omega_F\chi)
$$
where $L_n(s,\omega_F\chi)$ is the Dirichlet $L$-function with Euler factors
at primes dividing $n$ removed, and $|\cdot|$ is the absolute value corresponding to
our chosen embedding $\bar{\mathbf{Q}} \hookrightarrow \mathbf{C}$.
\end{rem}
Suppose $n \in \mathcal{N}$.
Let $X_n$ be the group of divisors of $F$ supported above $n\infty$, and
let $\mathcal{E}_n := \mathcal{O}_F[1/n]^\times$, the group of $n$-units of $F$.
We will write the action of $\mathbf{Z}[\Gamma_n]$ on $\mathcal{E}_n$ additively,
so in particular $(1-\tau)\mathcal{E}_n = \{\epsilon/\epsilon^\tau : \epsilon \in \mathcal{E}_n\}$.
Let $\lambda_0 \in X_n$ be the archimedean place of $F$ corresponding
to our chosen embedding $\bar\mathbf{Q} \hookrightarrow \mathbf{C}$.
\begin{lem}
\label{freer+1}
Suppose $n \in \mathcal{N}$, and let $r = r(n)$.
\begin{enumerate}
\item
We have
$X_{n}^- = X_{n_+}^-$, $\mathcal{E}_{n}^- = \mathcal{E}_{n_+}^-$, and
$(1-\tau)\mathcal{E}_{n} = (1-\tau)\mathcal{E}_{n_+}$.
\item
The group $(1-\tau)\mathcal{E}_n$ is a free abelian group of rank $r+1$,
and is a subgroup of finite index in $\mathcal{E}_n^-$.
\item
The group $X_n^-$ is a free abelian group of rank $r+1$. If
$n_+ = \prod_{i=1}^r \ell_i$, and $\ell_i = \lambda_i\lambda_i^\tau$, then
$\{\lambda_0-\lambda_0^\tau,\lambda_1-\lambda_1^\tau,\ldots,\lambda_r-\lambda_r^\tau,\}$
is a basis of $X_n^-$.
\end{enumerate}
\end{lem}
\begin{proof}
The only part that is not clear is that $(1-\tau)\mathcal{E}_n$ is torsion-free, i.e.,
$-1 \notin (1-\tau)\mathcal{E}_n$. Let $d > 1$ be a squarefree integer such that
$F = \mathbf{Q}(\sqrt{d})$. If $x^\tau = -x$, then $x/\sqrt{d} \in \mathbf{Q}$, so $x$
is not a unit at the primes dividing $d$. Since $n$ is
prime to $d$, we cannot have $x \in \mathcal{E}_n$.
\end{proof}
\begin{defn}
A {\em standard} $\mathbf{Z}$-basis of $X_n^-$ is a basis of the form described
in Lemma \ref{freer+1}(iii).
Given a standard basis of $X_n^-$, a $\mathbf{Z}$-basis $\{\epsilon_0, \ldots, \epsilon_r\}$
of $(1-\tau)\mathcal{E}_n$ will be called {\em oriented}
if the (regulator) determinant of the logarithmic embedding
$$
(1-\tau)\mathcal{E}_n \longrightarrow X_n^- \otimes \mathbf{R}, \quad \epsilon \mapsto
\sum_{\lambda\mid n_+\infty}\log|\epsilon|_{\lambda}\cdot\lambda
$$
with respect to the two bases is positive. Concretely, this regulator
is the determinant of the matrix whose entry in row $i$ and column $j$ is
$\log|\epsilon_j|_{\lambda_i}$.
\end{defn}
\begin{rem}
Choosing a standard basis of $X_n^-$ is equivalent to ordering the prime
divisors $\ell_i$ of $n_+$ and choosing one prime of $F$ above each $\ell_i$.
Any basis of $(1-\tau)\mathcal{E}_n$ can be oriented either by reordering the
basis, or inverting one of the basis elements.
\end{rem}
\begin{defn}
Suppose $n \in \mathcal{N}$ and $\lambda$ is a prime of $F$ dividing $n_+$.
Define a homomorphism
$$
\gr{\lambda}{n} : F^\times \longrightarrow I_n/I_n^2
$$
by
$$
\grx{\lambda}{n}{x} = [x, F_\lambda(\boldsymbol{\mu}_n)/F_\lambda]-1 \pmod{I_n^2}
$$
where $[x, F_\lambda(\boldsymbol{\mu}_n)/F_\lambda] \in \Gamma_n$ is the local Artin symbol.
Note that if $\mathrm{ord}_\lambda(x) = 0$, then $[x, F_\lambda(\boldsymbol{\mu}_n)/F_\lambda]$
belongs to the inertia group $\Gamma_\ell \subset \Gamma_n$, so
$\grx{\lambda}{n}{x} = \grx{\lambda}{\ell}{x} \in I_\ell/I_\ell^2$
and $\grx{\lambda}{n/\ell}{x} = 0$. In general, if $d \mid n$ then
$$\grx{\lambda}{n}{x} = \grx{\lambda}{d}{x} + \grx{\lambda}{n/d}{x}
\in I_d/I_d^2 \oplus I_{n/d}/I_{n/d}^2 = I_n/I_n^2.$$
\end{defn}
\begin{defn}
\label{Rdef}
(See \cite[p.\ 308]{darmon}.)
Suppose $n \in \mathcal{N}$, and let $r = r(n)$.
Choose a standard basis $\{\lambda_0-\lambda_0^\tau,\ldots,\lambda_r-\lambda_r^\tau\}$
of $X_n^-$ and an oriented basis
$\{\epsilon_0, \ldots, \epsilon_r\}$ of $(1-\tau)\mathcal{E}_n$, and define
the regulator $R_n \in \mathcal{E}_n^- \otimes I_n^r/I_n^{r+1}$ by
$$
R_n := \left|
\begin{array}{ccccccc}
\epsilon_0 & \epsilon_1 & \cdots & \epsilon_r \\
\grx{\lambda_1}{n}{\epsilon_0} & \grx{\lambda_1}{n}{\epsilon_1} & \cdots & \grx{\lambda_1}{n}{\epsilon_r} \\
\vdots & \vdots && \vdots\\
\grx{\lambda_r}{n}{\epsilon_0} & \grx{\lambda_r}{n}{\epsilon_1} & \cdots & \grx{\lambda_r}{n}{\epsilon_r}
\end{array}
\right| \quad \in (1-\tau)\mathcal{E}_n \otimes I_n^r/I_n^{r+1}.
$$
This determinant, and the ones that follow below, are meant to be evaluated
by expanding by minors along the top row, i.e.,
\begin{equation}
\label{minors}
R_n := \sum_{j=0}^r (-1)^{j}\epsilon_j \otimes \det(A_{1j})
\end{equation}
where $A_{1j}$ is the $r \times r$ matrix (with entries in $I_n/I_n^2$)
obtained by removing the first row and
$j$-th column of the matrix above.
\end{defn}
Note that this definition of $R_n$ does not depend on the choice of $\mathbf{Z}$-bases.
The possible ambiguity of $\pm1$ is removed by requiring that the basis
of $(1-\tau)\mathcal{E}_n$ be oriented.
Let $h_n$ denote the ``$n$-class number'' of $F$, i.e., the order of the ideal class
group $\mathrm{Pic}(\mathcal{O}_F[1/n])$. For the rest of this section we write simply $r$ instead of $r(n)$.
\begin{thm}[(Darmon {\cite[Theorem 4.2]{darmon}})]
For every $n \in \mathcal{N}$, we have
$$\theta'_n \in F(\boldsymbol{\mu}_n)^\times \otimes I_n^r.$$
\end{thm}
For $n \in \mathcal{N}$, let $\tilde\theta'_n$ denote the image of $\theta'_n$
in $F(\boldsymbol{\mu}_n)^\times \otimes I_n^r/I_n^{r+1}$. Let $s$ be the number of prime divisors
of $n/n_+$; we continue to denote by $r$ the number of prime factors of $n_+$.
The following is a slightly modified version of Darmon's ``leading term'' conjecture
\cite[Conjecture 4.3]{darmon}.
\begin{conj}
\label{dconj}
For every $n \in \mathcal{N}$, we have
$$
\tilde\theta'_n = -2^s h_n R_n \quad
\text{in $(F(\boldsymbol{\mu}_n)^\times/\{\pm1\}) \otimes I_n^r/I_n^{r+1}$}.
$$
\end{conj}
The main theorem of this paper is the following.
\begin{thm}
\label{mainthm}
For every $n \in \mathcal{N}$, we have
$$
\tilde\theta'_n = -2^s h_n R_n \quad
\text{in $F(\boldsymbol{\mu}_n)^\times \otimes I_n^r/I_n^{r+1} \otimes \mathbf{Z}[1/2]$}.
$$
\end{thm}
In other words, the $p$-part of Conjecture \ref{dconj} holds for every odd prime
$p$; in still other words, $\tilde\theta'_n + 2^s h_n R_n$ has $2$-power order
in $F(\boldsymbol{\mu}_n)^\times \otimes I_n^r/I_n^{r+1}$.
A key step in the proof of Theorem \ref{mainthm} is the following observation.
\begin{prop}[(Darmon {\cite[Theorem 4.5(1)]{darmon}})]
\label{n=1}
Conjecture \ref{dconj} holds if $n=1$.
\end{prop}
\begin{proof}
When $n=1$ we have $r=0$, $I_n^r/I_n^{r+1} = \mathbf{Z}$,
$
\tilde\theta'_1 = \theta'_1 = \alpha_1 \in \mathcal{O}_F^\times,
$
and $R_1 = \epsilon/\epsilon^\tau$, where $\epsilon$ is a generator of
$\mathcal{O}_F^\times/\{\pm1\}$ and $|\epsilon/\epsilon^\tau| = |\epsilon|^2 > 1$
at our specified archimedean place.
Dirichlet's analytic class number formula shows that
$$
-\frac{1}{2}\log|\alpha_1| = L'(0,\omega_F) = h_F\log|\epsilon|
= \frac{1}{2}h_F \log|\epsilon/\epsilon^\tau |
$$
where $h_F = h_1$ is the class number of $F$.
Hence $\alpha_1 = \pm(\epsilon/\epsilon^\tau)^{-h_F}$ in $\mathcal{O}_F^\times$.
\end{proof}
\begin{rems}
(i) In Darmon's formulation \cite[Conjecture 4.3]{darmon},
the regulator $R_n$ was defined with respect to a basis of $\mathcal{E}_n^-/\{\pm1\}$
instead of $(1-\tau)\mathcal{E}_n$, and
there was an extra factor of $2$ on the right-hand side.
This agrees with Conjecture \ref{dconj} if and only if $[\mathcal{E}_n^-:\pm(1-\tau)\mathcal{E}_n] = 2$,
i.e., if and only if $-1 \notin \mathbf{N}_{F/\mathbf{Q}}\mathcal{E}_n$.
(ii) The ambiguity of $\pm1$ in Conjecture \ref{dconj} is necessary.
Namely, even when $n=1$,
we may only have $\tilde\theta'_1 = h_1R_1$ in $\mathcal{O}_F^\times/\{\pm1\}$.
Since $\alpha_1$ is always positive (it is a norm from a CM field to $F$),
the proof of Proposition \ref{n=1} shows that $\tilde\theta'_1 \ne -h_1R_1$
in $F^\times$ when $h_F$ is odd and $\mathcal{O}_F^\times$ has a unit of norm $-1$.
Note that in this case $\tilde\theta'_1$ and $-h_1R_1$ differ (multiplicatively)
by an element
of order $2$ in $F^\times$, so the discrepancy disappears when we tensor with $\mathbf{Z}[1/2]$.
\end{rems}
\section{Augmentation quotients}
\label{augq}
\begin{defn}
Suppose $n \in \mathcal{N}$, and let $r = r(n)$.
Let $\I_n \subset I_n^r/I_n^{r+1}$ be the (cyclic) subgroup generated by monomials
$\prod_{\ell\mid n_+}(\gamma_\ell-1)$ with $\gamma_\ell \in \Gamma_\ell$.
Let $\mathcal{I}^{\mathrm{old}}_n \subset I_n^r/I_n^{r+1}$ be the subgroup generated by monomials
$\prod_{i=1}^r(\gamma_i - 1)$ where each $\gamma_i \in \Gamma_{\ell_i}$ for
some $\ell_i$ dividing $n$, and $\{\ell_1,\ldots,\ell_r\} \ne \{\ell: \ell \mid n_+\}$
(i.e., either one of the $\ell_i$ divides $n/n_+$, or $\ell_i = \ell_j$ for some $i \ne j$).
If $n = d_1d_2$ then there is a natural identification
$\I_n = \mathcal{I}^{\mathrm{new}}_{d_1}\mathcal{I}^{\mathrm{new}}_{d_2} \subset I_{n}^{r}/I_{n}^{r+1}$, and if $n = \ell$ is prime then
$\mathcal{I}^{\mathrm{new}}_\ell = I_\ell/I_\ell^2$ and $\mathcal{I}^{\mathrm{old}}_\ell = 0$.
\end{defn}
If $d \mid n$, let
$$
\pi_d : \mathbf{Z}[\Gamma_n] \twoheadrightarrow \mathbf{Z}[\Gamma_d] \hookrightarrow \mathbf{Z}[\Gamma_n]
$$
denote the composition of the natural maps.
We also write $\pi_d$ for the induced map on $I_n^k/I_n^{k+1}$ for $k \ge 0$.
The following proposition is based on work of Hales \cite{hales}.
\begin{prop}
\label{crit}
Suppose $n \in\mathcal{N}$, and $r=r(n)$.
Then:
\begin{enumerate}
\item
$I_n^r/I_n^{r+1} = \I_n \oplus \J_n$.
\item
If $d \mid n_+$ and $d > 1$, then $\pi_{n/d}(\I_n) = 0$
and $\pi_{n/d}(I_n^r/I_n^{r+1}) \subset \J_n$.
\item
$\I_n = \{v \in I_n^r/I_n^{r+1} : \text{$\pi_{n/\ell}(v) = 0$ for every $\ell$ dividing $n_+$}\}$.
\item
The map $\otimes_{\ell\mid n_+}\Gamma_\ell \to \I_n$ defined by
$\otimes_{\ell\mid n_+} \gamma_\ell \mapsto \prod_{\ell\mid n_+}(\gamma_\ell-1)$
is an isomorphism.
\end{enumerate}
\end{prop}
\begin{proof}
Let $A_n$ be the polynomial ring $\mathbf{Z}[Y_\ell : \ell \mid n]$ with one variable
$Y_\ell$ for each prime $\ell$ dividing $n$. Fix a generator $\sigma_\ell$ of
$\Gamma_\ell$ for every $\ell$ dividing $n$, and define a map $A_n \to \mathbf{Z}[\Gamma_n]$
by sending $Y_\ell \mapsto \sigma_\ell-1$.
By Corollary 2 of \cite{hales}, this map induces an isomorphism from the
homogeneous degree-$r$ part of $A_n/(J_n+J_n')$ to
$I_n^r/I_n^{r+1}$, where $J_n$ is the ideal of $A_n$ generated by
$\{(\ell-1)Y_\ell : \ell \mid n\}$, and $J_n'$ is the ideal generated by certain other explicit
homogeneous relations (see \cite[Lemma 2]{hales}).
The only fact we need about these ``extra'' relations is:
\begin{equation}
\label{star}
\text{\em if $f \in J_n'$, then every monomial that occurs in $f$ is divisible by
the square of some $Y_\ell$.}
\end{equation}
Note that $\I_n$ is the image in $I_n^r/I_n^{r+1}$ of the subgroup of
$A_n/(J_n+J_n')$ generated by $\mathbf{Y}_n$, where $\mathbf{Y}_n := \prod_{\ell\mid n_+}Y_\ell$.
Similarly, $\J_n$ is the image of the subgroup generated by all other
monomials of degree $r$.
By \eqref{star}, $\mathbf{Y}_n$ does not occur in any of the relations in $J_n'$,
and assertion (i) follows.
Assertion (ii) is clear, since $\pi_{n/d}$ kills those monomials
that include $(\gamma - 1)$ with $\gamma \in \Gamma_\ell$ for $\ell$ dividing $d$,
and leaves the other monomials unchanged.
Fix $v \in I_n^r/I_n^{r+1}$.
If $v \in \I_n$ and $\ell \mid n_+$, then $\pi_{n/\ell}(v) = 0$ by (ii).
Conversely, suppose that $\pi_{n/\ell}(v) = 0$
for every $\ell$ dividing $n_+$. Choose $f \in A_n$ homogeneous of degree $r$
representing $v$, and suppose $f$ has the minimum number of monomials among
all representatives of $v$. We will show that $\mathbf{Y}_n \mid f$, and hence
$v \in \I_n$.
Fix a prime $\ell$ dividing $n_+$. The map
$\pi_{n/\ell}: \mathbf{Z}[\Gamma_n] \twoheadrightarrow \mathbf{Z}[\Gamma_{n/\ell}] \hookrightarrow \mathbf{Z}[\Gamma_n]$ corresponds to
the map $A_n \to A_n$ defined by setting $Y_\ell = 0$.
Since $\pi_{n/\ell}(v) = 0$, substituting $Y_\ell = 0$ in $f$ gives a
relation in $J_n+J_n'$, i.e., $f = Y_\ell\cdot g + h$ where $g$ is
homogeneous of degree $r-1$, $h \in J_n+J_n'$, and $Y_\ell$ does not occur in $h$.
But then $Y_\ell\cdot g$ represents $v$, so the minimality assumption on $f$
implies that $h = 0$. Therefore $Y_\ell \mid f$ for every $\ell$ dividing $n_+$,
so $\mathbf{Y}_n\mid f$ and $v \in \I_n$. This proves (iii).
Let $g := \gcd(\{\ell-1 : \ell\mid n_+\})$.
Then $g \mathbf{Y}_n \in J_n$.
It follows from \eqref{star} that the monomial $\mathbf{Y}_n$ only
occurs in elements of $J_n+J_n'$ with coefficients divisible by $g$.
Therefore $\I_n$ is cyclic of order $g$, and so is $\otimes_{\ell\mid n_+}\Gamma_\ell$.
Clearly the map $\otimes_{\ell\mid n_+}\Gamma_\ell \to \I_n$ of (iv) is surjective,
so it must be an isomorphism.
\end{proof}
If $v \in I_n^r/I_n^{r+1}$, let $\projI{n}{v}$ denote the projection of $v$ to
$\I_n$ under the splitting of Proposition \ref{crit}(i).
We will use the following lemma without explicit reference in some of our computations
in \S\ref{prekssect} and \S\ref{rpks}. Its proof is left as an exercise.
\begin{lem}
Suppose $d\mid n$, $v \in \mathcal{I}^{\mathrm{new}}_{n/d}$, and $w \in I_n^{r(d)}/I_n^{r(d)+1}$. Then
$$
\projI{n}{vw} = \projI{n}{v\pi_d(w)} = v \projI{d}{\pi_d(w)}.
$$
\end{lem}
\section{Kolyvagin systems}
\label{kssect}
Fix an odd prime $p$. To prove Theorem \ref{mainthm} we need to
introduce Kolyvagin systems, as defined in \cite{kolysys}. (See in particular
\cite[\S6.1]{kolysys}, and also \cite{babykolysys}, for the case of Kolyvagin systems
associated to even Dirichlet characters that we use here.)
Let $\hat{F}^\times$ denote the $p$-adic completion of $F^\times$.
Similarly, for every rational prime $\ell$
let $F_\ell := F \otimes \Q_ \ell$, $\mathcal{O}_\ell := \mathcal{O}_F \otimes \Z_\ell$,
and define $\hat{F}^\times_\ell$ and $\hat{\mathcal{O}}^\times_\ell$
to be their $p$-adic completions.
We define the ``finite subgroup'' $\hat{F}^\times_{\ell,\mathrm{f}}$
to be the ``unit part'' of $\hat{F}^\times_\ell$
$$
\hat{F}^\times_{\ell,\mathrm{f}} := \hat{\mathcal{O}}_\ell^\times \subset \hat{F}^\times_\ell.
$$
If $\ell = \lambda{\lambda^\tau}$ splits in $F$,
define the ``transverse subgroup''
$\hat{F}^\times_{\ell,\mathrm{tr}} \subset \hat{F}^\times_\ell$ to be the
(closed) subgroup generated by $(\ell,1)$ and $(1,\ell)$, where we identify $F_\ell^\times$
with $F_\lambda^\times \times F_{\lambda^\tau}^\times \cong \Q_ \ell^\times \times \Q_ \ell^\times$.
Then we have a canonical
splitting
$\hat{F}^\times_\ell = \hat{F}^\times_{\ell,\mathrm{f}} \times \hat{F}^\times_{\ell,\mathrm{tr}}$,
and since $p$ is odd
\begin{equation}
\label{ftrs}
(\hat{F}^\times_\ell)^-
= (\hat{F}^\times_{\ell,\mathrm{f}})^- \times (\hat{F}^\times_{\ell,\mathrm{tr}})^-.
\end{equation}
\begin{defn}
\label{phifsdef}
If $\ell \ne p$ splits in $F$, define the {\em finite-singular isomorphism}
$$
\phi^{\mathrm{fs}}_\ell : (\hat{F}^\times_{\ell,\mathrm{f}})^- \xrightarrow{\sim} (\hat{F}^\times_{\ell,\mathrm{tr}})^- \otimes I_\ell/I_\ell^2
$$
by
\begin{align*}
\phi^{\mathrm{fs}}_\ell(x) &= (\ell,1) \otimes ([x_\lambda,F_\lambda(\boldsymbol{\mu}_\ell)/F_\lambda]-1)
+ (1,\ell) \otimes ([x_{\lambda^\tau},F_{\lambda^\tau}(\boldsymbol{\mu}_\ell)/F_{\lambda \tau}]-1) \\
&= (\ell,\ell^{-1}) \otimes ([x_\lambda,F_\lambda(\boldsymbol{\mu}_\ell)/F_\lambda]-1)
\end{align*}
where
$x = (x_\lambda,x_{\lambda^\tau})
\in \hat{F}_\lambda^\times \times \hat{F}_{\lambda^\tau}^\times
= \hat{\mathbf{Q}}_\ell^\times \times \hat{\mathbf{Q}}_\ell^\times$
with $x_{\lambda^\tau} = x_\lambda^{-1} \in \hat{\mathbf{Z}}_\ell^\times$,
and $[\;\cdot\;,F_\lambda(\boldsymbol{\mu}_\ell)/F_\lambda]$ is the local Artin symbol.
(Concretely, note that if $u \in \Z_\ell^\times$ then $[u,F_\lambda(\boldsymbol{\mu}_\ell)/F_\lambda]$
is the automorphism in $\Gamma_\ell$ that sends $\zeta_\ell$ to $\zeta_\ell^{u^{-1}}$.)
Then $\phi^{\mathrm{fs}}_\ell$ is a well-defined isomorphism
(both the domain and range are free of rank one over $\Z_p/(\ell-1)\Z_p$),
independent of the choice of $\lambda$ versus $\lambda^\tau$.
\end{defn}
\begin{defn}
\label{ksdef}
Let $\mathcal{N}_p := \{n \in \mathcal{N} : p \nmid n\}$.
A {\em Kolyvagin system} $\boldsymbol{\kappa}$
(for the Galois representation $\Z_p(1) \otimes \omega_F$) is a collection
$$
\{\kappa_n \in (\hat{F}^\times)^- \otimes \I_n : n \in \mathcal{N}_p\}
$$
satisfying the following properties for every rational prime $\ell$.
Let $(\kappa_n)_\ell$ denote the image of $\kappa_n$ in
$(\hat{F}^\times_\ell)^- \otimes \I_n$.
\begin{enumerate}
\item
If $\ell \nmid n$, then
$(\kappa_n)_\ell \in (\hat{F}^\times_{\ell,\mathrm{f}})^- \otimes \I_n$.
\item
If $\ell \mid n_+$, then
$(\kappa_n)_\ell = (\phi^{\mathrm{fs}}_\ell \otimes 1)(\kappa_{n/\ell,\ell})$.
\item
If $\ell \mid n/n_+$, then $\kappa_n = \kappa_{n/\ell}$.
\end{enumerate}
Let $\mathbf{KS}(F)$ denote the $\Z_p$-module of Kolyvagin systems
for $\Z_p(1) \otimes \omega_F$.
\end{defn}
\begin{rem}
Let $\mathcal{N}_p^+ := \{n \in \mathcal{N}_p : \text{all $\ell \mid n$ split in $F/\mathbf{Q}$}\}$.
In \cite{kolysys}, a Kolyvagin system was defined to be a collection of classes
$\{\kappa_n \in (\hat{F}^\times_\ell)^-\otimes (\otimes_{\ell \mid n} \Gamma_\ell) : n \in \mathcal{N}_p^+\}$,
and $\phi^{\mathrm{fs}}_\ell$ took values in $(\hat{F}^\times_{\ell,\mathrm{tr}})^- \otimes \Gamma_\ell$.
We use Proposition \ref{crit}(iv) to replace $\otimes_{\ell \mid n_+} \Gamma_\ell$ by
$\I_n$ and \eqref{caug} to replace $\Gamma_\ell$ by $I_\ell/I_\ell^2$,
which will be more convenient for our purposes here. Also,
a Kolyvagin system $\{\kappa_n : n \in \mathcal{N}_p^+\}$ as in \cite{kolysys} extends uniquely to
$\{\kappa_n : n \in \mathcal{N}_p\}$ simply by setting $\kappa_n := \kappa_{n_+}$ for $n \in \mathcal{N}_p - \mathcal{N}_p^+$.
\end{rem}
The following theorem is the key to our proof of Theorem \ref{mainthm}.
\begin{thm}
\label{rankone}
Suppose $\boldsymbol{\kappa}, \boldsymbol{\kappa}' \in \mathbf{KS}(F)$.
If $\kappa_1 = \kappa'_1$, then $\kappa_n = \kappa'_n$ for every $n \in \mathcal{N}_p$.
\end{thm}
\begin{proof}
We follow \S6.1 of \cite{kolysys}, with $R = \Z_p$, $\rho = \omega_F$,
$T = \Z_p(1) \otimes \omega_F$,
and with the Selmer structure denoted $\mathcal{F}$ in \cite{kolysys}.
By Lemma 6.1.5 and Proposition 6.1.6 of \cite{kolysys}, the hypotheses needed to
apply the results of \S5.2 of \cite{kolysys} all hold, and the core rank of $T$ is $1$.
By Theorem 5.2.10(ii) of \cite{kolysys}, $\mathbf{KS}(F)$ is a free $\Z_p$-module
of rank one. Therefore (switching $\boldsymbol{\kappa}$ and $\boldsymbol{\kappa}'$ if necessary)
there is an $a \in \Z_p$ such that $\boldsymbol{\kappa}' = a\boldsymbol{\kappa}$, i.e., $\kappa'_n = a\kappa_n$
for every $n \in \mathcal{N}_p$. If $\boldsymbol{\kappa}$ is identically zero, then so is $\boldsymbol{\kappa}'$ and
we are done. If $\boldsymbol{\kappa}$ is not identically zero, then (since the ideal class group
of $F$ is finite) Theorem 5.2.12(v) of \cite{kolysys} shows that $\kappa_1 \ne 0$.
Since $\kappa'_1 = \kappa_1$ in the torsion-free
$\Z_p$-module $(\hat{F}^\times)^-$ (in fact property (i) above shows that
$\kappa_1 \in (\mathcal{O}_F^\times \otimes \Z_p)^-$), we must have $a = 1$.
\end{proof}
\section{Pre-Kolyvagin systems}
\label{prekssect}
Keep the fixed odd prime $p$.
The right-hand and left-hand sides of Conjecture \ref{dconj} are ``almost''
Kolyvagin systems. If they were Kolyvagin systems, then since they agree when
$n=1$ (Proposition \ref{n=1}), they would agree for all $n$ by Theorem \ref{rankone},
and Theorem \ref{mainthm} would be proved.
In this section we define what we call ``pre-Kolyvagin systems'',
and show that a pre-Kolyvagin system can be transformed into a Kolyvagin system.
Using Theorem \ref{rankone}, we deduce (Corollary \ref{prekskscor} below) that if
two pre-Kolyvagin systems agree when $n=1$, then they agree for every $n$.
In \S\ref{cupks} and \S\ref{rpks}, respectively, we will show that the left- and right-hand sides
of Conjecture \ref{dconj} are pre-Kolyvagin systems. Then Theorem \ref{mainthm} will
follow from Corollary \ref{prekskscor} and Proposition \ref{n=1}.
If $x \in (\hat{F}^\times)^- \otimes I_n^r/I_n^{r+1}$, let
$x_\ell$ denote the image of
$x$ in $(\hat{F}^\times_{\ell})^- \otimes I_n^r/I_n^{r+1}$,
and if $\ell \in \mathcal{N}_p$ splits in $F/\mathbf{Q}$, let
$x_{\ell,\mathrm{f}} \in (\hat{F}^\times_{\ell,\mathrm{f}})^- \otimes I_n^r/I_n^{r+1}$ and
$x_{\ell,\mathrm{tr}}\in (\hat{F}^\times_{\ell,\mathrm{tr}})^- \otimes I_n^r/I_n^{r+1}$
denote the projections of $x_\ell$ induced by the splitting \eqref{ftrs}.
Let $\projI{n}{x} \in (\hat{F}^\times)^- \otimes \I_n$ denote the projection
of $x$ induced by the splitting of Proposition \ref{crit}(i),
and similarly for $\projI{n}{x_\ell}$ and $\projI{n}{x_{\ell,\mathrm{f}}}$.
\begin{defn}
\label{Mdef}
If $n \in \mathcal{N}$ and $d = \prod_{i=1}^t \ell_i$ divides $n_+$, let $M_{n,d} = (m_{ij})$
be the $t \times t$ matrix with entries in $I_n/I_n^2$
$$
m_{ij} =
\begin{cases}
\pi_{n/d}(\mathrm{Fr}_{\ell_i}-1) & \text{if $i = j$},\\
\pi_{\ell_j}(\mathrm{Fr}_{\ell_i}-1) & \text{if $i \ne j$}.
\end{cases}
$$
We let $M_d := M_{d,d}$, where $\pi_1(\mathrm{Fr}_\ell-1)$ is understood to be zero,
so that all diagonal entries of $M_d$ are zero. Define
$$
\mathcal{D}_{n,d} := \det(M_{n,d}) \in I_n^t/I_n^{t+1}, \quad
\mathcal{D}_d := \det(M_d) \in \mathcal{I}^{\mathrm{new}}_d \subset I_n^t/I_n^{t+1}.
$$
By convention we let $\mathcal{D}_{n,1} = \mathcal{D}_1 = 1$.
Note that $\mathcal{D}_{n,d}$ and $\mathcal{D}_d$ are
independent of the ordering of the prime factors of $d$.
\end{defn}
\begin{defn}
\label{preksdef}
A {\em pre-Kolyvagin system} $\boldsymbol{\kappa}$
(for $\Z_p(1) \otimes \omega_F$) is a collection
$$
\{\kappa_n \in (\hat{F}^\times)^- \otimes I_n^r/I_n^{r+1} : n \in \mathcal{N}_p\}
$$
where $r=r(n)$,
satisfying the following properties for every rational prime $\ell$:
\begin{enumerate}
\item
If $\ell \nmid n$, then
$(\kappa_{n})_\ell \in (\hat{F}^\times_{\ell,\mathrm{f}})^- \otimes I_n^r/I_n^{r+1}$.
\item
If $\ell \mid n_+$, then
$
(1 \otimes \pi_{n/\ell})\kappa_n = \kappa_{n/\ell}\,\pi_{n/\ell}(1-\mathrm{Fr}_\ell).
$
\item
If $\ell \mid n_+$, then
$\projI{n}{(\kappa_{n})_{\ell,\mathrm{tr}}} = (\phi^{\mathrm{fs}}_\ell \otimes 1)(\projI{n/\ell}{(\kappa_{n/\ell})_\ell})$.
\item
If $\ell \mid n_+$, then $\sum_{d \mid n_+} \projI{n/d}{(\kappa_{n/d})_{\ell,\mathrm{f}}}\,\mathcal{D}_d = 0$.
\item
If $\ell \mid n/n_+$, then $\projI{n}{\kappa_{n}} = \projI{n/\ell}{\kappa_{n/\ell}}$.
\end{enumerate}
Let $\mathbf{PKS}(F)$ denote the $\Z_p$-module of pre-Kolyvagin systems
for $\Z_p(1) \otimes \omega_F$.
\end{defn}
\begin{defn}
\label{pktok}
If $\boldsymbol{\kappa} = \{\kappa_n : n \in \mathcal{N}_p\}$ is a pre-Kolyvagin system,
define $\tilde{\boldsymbol{\kappa}} = \{\tilde\kappa_n : n \in \mathcal{N}_p\}$ by
$$
\tilde\kappa_n := \sum_{d \mid n_+} \kappa_{n/d}\,\mathcal{D}_{n,d}.
$$
\end{defn}
\begin{lem}
\label{detMlem}
Suppose $n \in \mathcal{N}_p$ and $d \mid n$.
\begin{enumerate}
\item
If $\ell \mid d$ then $\pi_{n/\ell}(\mathcal{D}_{n,d}) = \pi_{n/d}(\mathrm{Fr}_\ell-1)\mathcal{D}_{n/\ell,d/\ell}$.
\item
If $\ell \nmid d$ then $\pi_{n/\ell}(\mathcal{D}_{n,d}) = \mathcal{D}_{n/\ell,d}$.
\item
$\pi_{d}(\mathcal{D}_{n,d}) = \mathcal{D}_d \in \mathcal{I}^{\mathrm{new}}_d$.
\end{enumerate}
\end{lem}
\begin{proof}
Suppose $\ell \mid d$. The column of $\pi_{n/\ell}(M_{n,d})$ corresponding to
$\ell$ consists of all zeros except for $\pi_{n/d}(\mathrm{Fr}_\ell-1)$ on the diagonal. The
first assertion follows from this, and (ii) and (iii) follow directly from the definition.
\end{proof}
\begin{prop}
\label{preksks}
The map $\boldsymbol{\kappa} \mapsto \tilde\boldsymbol{\kappa}$ of Definition \ref{pktok}
is a $\Z_p$-module isomorphism $\mathbf{PKS}(F) \xrightarrow{\sim} \mathbf{KS}(F)$ between free $\Z_p$-modules of rank one.
\end{prop}
\begin{proof}
The $\Z_p$-linearity is clear. The injectivity is clear as well, since it follows easily
by induction that if $\tilde\kappa_n = 0$ for all $n$, then $\kappa_n = 0$ for all $n$.
We next show that if $\boldsymbol{\kappa}$ is a pre-Kolyvagin system, then $\tilde\boldsymbol{\kappa}$ is
a Kolyvagin system. In other words, we need to show for every $n \in \mathcal{N}_p$ that
\begin{enumerate}
\renewcommand{\theenumi}{(\alph{enumi})}
\item
$\tilde\kappa_n \in (\hat{F}^\times)^- \otimes \I_n$,
\item
if $\ell \nmid n$ then $(\tilde\kappa_n)_{\ell} \in (\hat{F}^\times_{\ell,\mathrm{f}})^- \otimes \I_n$,
\item
if $\ell \mid n_+$ then $(\tilde\kappa_n)_{\ell,\mathrm{tr}} = (\phi^{\mathrm{fs}}_\ell \otimes 1)((\kappa_{n/\ell})_\ell)$,
\item
if $\ell \mid n_+$ then $(\tilde\kappa_n)_{\ell,\mathrm{f}} = 0$,
\item
if $\ell \mid n/n_+$ then $\tilde\kappa_n = \tilde\kappa_{n/\ell}$.
\end{enumerate}
Fix $n \in \mathcal{N}_p$, and suppose that $\ell \mid n_+$. Then
\begin{align*}
(1 \otimes \pi_{n/\ell})(\tilde\kappa_n)
&= \sum_{d \mid n_+, \ell\nmid d} (1 \otimes \pi_{n/\ell})(\kappa_{n/d} \mathcal{D}_{n,d})
+ \sum_{d \mid n_+,\ell\mid d} (1 \otimes \pi_{n/\ell})(\kappa_{n/d} \mathcal{D}_{n,d}) \\
&= \sum_{d \mid (n_+/\ell)} \kappa_{n/(d\ell)} \,\pi_{n/\ell}(\mathcal{D}_{n,d\ell})
+ (1 \otimes \pi_{n/(d\ell)})(\kappa_{n/d}) \pi_{n/\ell}(\mathcal{D}_{n,d}).
\end{align*}
Fix a divisor $d$ of $n_+/\ell$. By Lemma \ref{detMlem}(i),
$$
\kappa_{n/(d\ell)} \,\pi_{n/\ell}(\mathcal{D}_{n,d\ell}) =
\kappa_{n/(d\ell)}\,\pi_{n/(d\ell)}(\mathrm{Fr}_\ell-1) \mathcal{D}_{n/\ell,d}.
$$
Also,
$(1 \otimes \pi_{n/(d\ell)})(\kappa_{n/d}) = \kappa_{n/(d\ell)}\,\pi_{n/(d\ell)}(1-\mathrm{Fr}_\ell)$
by Definition \ref{preksdef}(ii), so by Lemma \ref{detMlem}(ii)
$$
(1 \otimes \,\pi_{n/(d\ell)})(\kappa_{n/d}) \pi_{n/\ell}(\mathcal{D}_{n,d})
= \kappa_{n/(d\ell)}\,\pi_{n/(d\ell)}(1-\mathrm{Fr}_\ell)\mathcal{D}_{n/\ell,d}.
$$
Thus $(1 \otimes \pi_{n/\ell})(\tilde\kappa_n) = 0$
for every $\ell$ dividing $n$. Since $(\hat{F}^\times)^-$ is a free $\Z_p$-module,
it follows from Proposition \ref{crit}(iii) that
$\tilde\kappa_n \in (\hat{F}^\times)^- \otimes \I_n$. This is property (a) above.
By (a), and using that $\pi_d(\mathcal{D}_{n,d}) \in \mathcal{I}^{\mathrm{new}}_d$, we have
$$
\tilde\kappa_n = \projI{n}{\tilde\kappa_n}
= \sum_{d \mid n_+}\projI{n/d}{\kappa_{n/d}}\,\pi_{d}(\mathcal{D}_{n,d}).
$$
If $\ell \nmid n$, then property (i) of Definition \ref{preksdef} of a pre-Kolyvagin system
shows that $\projI{n/d}{(\kappa_{n,d})_\ell} \in (\hat{F}^\times_{\ell,\mathrm{f}})^- \otimes \mathcal{I}^{\mathrm{new}}_{n/d}$
for every $d$, so
$
(\tilde\kappa_n)_\ell \in (\hat{F}^\times_{\ell,\mathrm{f}})^- \otimes \I_n.
$
This is (b).
Now suppose $\ell \mid n_+$.
For (c), using property (i) of Definition \ref{preksdef} we have
$$
(\tilde\kappa_n)_{\ell,\mathrm{tr}} = \sum_{d \mid n_+} (\kappa_{n/d})_{\ell,\mathrm{tr}}\mathcal{D}_{n,d}
= \sum_{d \mid (n_+/\ell)} (\kappa_{n/d})_{\ell,\mathrm{tr}}\mathcal{D}_{n,d}.
$$
Projecting into $\I_n$, and using (a), (ii) of Definition \ref{preksdef},
and Lemma \ref{detMlem}(ii), we have
\begin{align*}
(\tilde\kappa_n)_{\ell,\mathrm{tr}} &= \projI{n}{(\tilde\kappa_n)_{\ell,\mathrm{tr}}}
= \sum_{d \mid (n_+/\ell)} \projI{n}{(\kappa_{n/d})_{\ell,\mathrm{tr}}\mathcal{D}_{n,d}} \\
&= \sum_{d \mid (n_+/\ell)} \projI{n}{(\phi^{\mathrm{fs}}_\ell \otimes 1)((\kappa_{n/(d\ell)})_\ell)
\,\pi_{n/\ell}(\mathcal{D}_{n,d})} \\
&= \sum_{d \mid (n_+/\ell)}
\projI{n}{(\phi^{\mathrm{fs}}_\ell \otimes 1)((\kappa_{n/(d\ell)})_\ell)\mathcal{D}_{n/\ell,d}} \\
&= \projI{n}{(\phi^{\mathrm{fs}}_\ell \otimes 1)(\tilde\kappa_{n/\ell})}
= (\phi^{\mathrm{fs}}_\ell \otimes 1)(\projI{n/\ell}{\tilde\kappa_{n/\ell}})
= (\phi^{\mathrm{fs}}_\ell \otimes 1)(\tilde\kappa_{n/\ell}).
\end{align*}
This is (c). For (d), using (a), Lemma \ref{detMlem}(iii), and
(iv) of Definition \ref{preksdef} we have
$$
(\tilde\kappa_n)_{\ell,\mathrm{f}} = \projI{n}{(\tilde\kappa_n)_{\ell,\mathrm{f}}}
= \sum_{d \mid n_+} \projI{n/d}{(\kappa_{n/d})_{\ell,\mathrm{f}}}\projI{d}{\pi_{d}(\mathcal{D}_{n,d})}
= \sum_{d \mid n_+} \projI{n/d}{(\kappa_{n/d})_{\ell,\mathrm{f}}}\mathcal{D}_{d} = 0.
$$
Finally, suppose that $\ell \mid n/n_+$. Using Definition \ref{preksdef}(v)
and property (a) above,
$$
\tilde\kappa_n = \projI{n}{\tilde\kappa_n}
= \sum_{d \mid n_+} \projI{n/d}{(\kappa_{n/d})}\mathcal{D}_{d}
= \sum_{d \mid (n/\ell)_+} \projI{n/(d\ell)}{(\kappa_{n/(d\ell)})}\mathcal{D}_{d}
= \projI{n/\ell}{\tilde\kappa_{n/\ell}} = \tilde\kappa_{n/\ell}.
$$
This completes the proof that $\tilde\boldsymbol{\kappa}$ is a Kolyvagin system.
Since $\mathbf{KS}(F)$ is a free $\Z_p$-module of rank one \cite[Theorem 5.2.10(ii)]{kolysys},
to complete the proof it remains only to show that the map
$\mathbf{PKS}(F) \to \mathbf{KS}(F)$ is surjective.
If $\tilde\boldsymbol{\kappa} \in \mathbf{KS}(F)$, then (since $\mathcal{D}_{n,1} = 1$) we can define
inductively a collection
$\boldsymbol{\kappa} := \{\kappa_n \in (\hat{F}^\times)^- \otimes I_n^r/I_n^{r+1} : n \in \mathcal{N}_p\}$ such that
$\sum_{d \mid n_+} \kappa_{n/d}\,\mathcal{D}_{n,d} = \tilde\kappa_n$ for every $n$.
It is straightforward to check that $\boldsymbol{\kappa}$ is a pre-Kolyvagin system; since we
will not make use of this, we omit the proof. By Definition \ref{pktok} the image of
$\boldsymbol{\kappa}$ in $\mathbf{KS}(F)$ is $\tilde\boldsymbol{\kappa}$.
\end{proof}
\begin{cor}
\label{prekskscor}
Suppose $\boldsymbol{\kappa}, \boldsymbol{\kappa}' \in \mathbf{PKS}(F)$.
If $\kappa_1 = \kappa'_1$, then $\kappa_n = \kappa'_n$ for every $n \in \mathcal{N}_p$.
\end{cor}
\begin{proof}
Let $\tilde\boldsymbol{\kappa}$ and $\tilde\boldsymbol{\kappa}'$ be the images of $\boldsymbol{\kappa}$ and $\boldsymbol{\kappa}'$, respectively,
under the map of Definition \ref{pktok}. Then $\tilde\boldsymbol{\kappa}$ and $\tilde\boldsymbol{\kappa}'$ are
Kolyvagin systems, and $\tilde\kappa_1 = \kappa_1 = \kappa'_1 = \tilde\kappa'_1$.
Therefore $\tilde\boldsymbol{\kappa} = \tilde\boldsymbol{\kappa}'$ by Theorem \ref{rankone}, so by the injectivity
assertion of Proposition \ref{preksks} we have $\boldsymbol{\kappa} = \boldsymbol{\kappa}'$, i.e.,
$\kappa_n = \kappa'_n$ for every $n \in \mathcal{N}_p$.
\end{proof}
We will use the following definition and lemma to replace property (iv) in the
definition of a pre-Kolyvagin system by an equivalent property that will be
easier to verify. See Remark \ref{5.9} below.
\begin{defn}
\label{symmdef}
If $n \in \mathcal{N}$, let $\mathfrak{S}(n)$ denote the set of permutations of the primes
dividing $n_+$, and let $\mathfrak{S}_1(n) \subset \mathfrak{S}(n)$ be the subset
$$
\mathfrak{S}_1(n) := \{\sigma \in \mathfrak{S}(n) :
\text{the primes not fixed by $\sigma$ form a single $\sigma$-orbit}\}.
$$
If $\sigma \in \mathfrak{S}(n)$ let $d_\sigma := \prod_{\ell \mid n_+, \sigma(\ell) \ne \ell} \ell$,
the product of the primes not fixed by $\sigma$, and define
$$
\Pi(\sigma) := \prod_{q \mid d_\sigma}\pi_q(\mathrm{Fr}_{\sigma(q)}-1).
$$
\end{defn}
\begin{lem}
\label{5.8}
Suppose that $A$ is an abelian group, $\ell$ is a prime that splits in $F/\mathbf{Q}$,
and $x_n \in A \otimes \I_n$ for every $n \in \mathcal{N}_p$.
Then the following are equivalent:
\begin{enumerate}
\item
For every $n$ divisible by $\ell$, $\sum_{d \mid n_+} x_{n/d}\,\mathcal{D}_d = 0$.
\item
For every $n$ divisible by $\ell$,
$\displaystyle x_n =
-\sum_{\pile{\sigma\in\mathfrak{S}_1(n)}{\sigma(\ell) \ne \ell}}
\mathrm{sign}(\sigma) x_{n/d_\sigma} \Pi(\sigma)$.
\end{enumerate}
\end{lem}
\begin{proof}
We show first that (ii) implies (i) (which is the implication we use later in this paper).
Let $\mathfrak{S}'(d) \subset \mathfrak{S}(d)$ denote the derangements, i.e., the permutations
with no fixed points.
Then we can evaluate the determinant $\mathcal{D}_{d} = \det(M_d)$ as follows.
Let $m_{q,q'}$ be the $(q,q')$-entry in $M_d$. Then
\begin{equation}
\label{detp}
\mathcal{D}_d = \sum_{\sigma \in \mathfrak{S}(d)} \mathrm{sign}(\sigma) \prod_{q \mid d} m_{q,\sigma(q)}
= \sum_{\sigma \in \mathfrak{S}'(d)} \mathrm{sign}(\sigma)\Pi(\sigma),
\end{equation}
where the second equality holds since the diagonal entries of $M_d$ vanish.
Fix an $n$ divisible by $\ell$, and let
$$
S_1 = \sum_{d \mid n_+, \ell \nmid d} x_{n/d}\,\mathcal{D}_d, \quad S_2
= \sum_{d \mid n_+, \ell \mid d} x_{n/d}\,\mathcal{D}_d.
$$
Using property (ii) we have
\begin{equation}
\label{cancel}
S_1 = -\sum_{\pile{d \mid n_+}{\ell\nmid d}}
\sum_{\pile{\sigma\in\mathfrak{S}_1(n/d)}{\sigma(\ell) \ne \ell}}
\mathrm{sign}(\sigma) \projI{n/(dd_\sigma)}{(x_{n/(dd_\sigma)})_{\ell}}
\Pi(\sigma)\mathcal{D}_d.
\end{equation}
Fix a divisor $\delta$ of $n_+$ that is divisible by $\ell$. We will show that the coefficient
of $x_{n/\delta}$ in $S_1$ in \eqref{cancel} is $-\mathcal{D}_\delta$, which exactly cancels
the coefficient of $x_{n/\delta}$ in $S_2$.
Using \eqref{detp}, the coefficient of $x_{n/\delta}$ in $S_1$ in \eqref{cancel} is
$$
-\sum_{d \mid (\delta/\ell)}\sum_{\pile{\sigma\in\mathfrak{S}_1(n/d)}{d_\sigma=\delta/d}}
\biggl(\mathrm{sign}(\sigma)\Pi(\sigma)\sum_{\eta \in \mathfrak{S}'(d)}\mathrm{sign}(\eta)\Pi(\eta)\biggr)
= -\sum_{d \mid (\delta/\ell)}\sum_{\pile{\sigma\in\mathfrak{S}_1(n/d)}{d_\sigma=\delta/d}}
\sum_{\eta \in \mathfrak{S}'(d)}\mathrm{sign}(\sigma\eta)\Pi(\sigma\eta).
$$
For every $\rho \in \mathfrak{S}'(\delta)$ there is a unique triple $(d,\sigma,\eta)$ such that
$$
\text{$d \mid \delta/\ell$, \;$\sigma \in \mathfrak{S}_1(n/d)$, \;$d_\sigma = \delta/d$,
\;$\eta \in\mathfrak{S}'(d)$, \;and \;$\rho = \sigma\eta$}.
$$
To see this,
simply write $\rho$ as a product of disjoint cycles, let $\sigma$ be the cycle containing
$\ell$, and let $d = \delta/d_\sigma$ and $\eta = \sigma^{-1}\rho$. Thus
the coefficient of $x_{n/\delta}$ in $S_1$ in \eqref{cancel} is (using \eqref{detp} again)
$$
-\sum_{\rho \in \mathfrak{S}'(\delta)} \mathrm{sign}(\rho)\Pi(\rho) = -\mathcal{D}_\delta.
$$
Therefore $\sum_{d \mid n_+} x_{n/d}\,\mathcal{D}_d = S_1 + S_2 = 0$, so (i) holds.
Although we will not need it, here is a simple argument to show that (i) implies (ii).
Suppose that $X := \{x_n \in A \otimes \I_n : n \in \mathcal{N}_p\}$
satisfies (i).
If $\ell \mid n$, then (since $\mathcal{D}_1 = 1$) we can use (i) recursively to express $x_n$
as a linear combination of $x_d$ with $\ell \nmid d$. Thus $X$ is uniquely
determined by the subset $X' := \{x_n \in A \otimes \I_n : n \in \mathcal{N}_p, \ell \nmid n\}$.
Clearly $X'$ determines a unique collection $Y := \{y_n \in A \otimes \I_n : n \in \mathcal{N}_p\}$
satisfying (ii), with $y_n = x_n$ if $\ell \nmid n$. We showed above
that (ii) implies (i), so $Y$ satisfies (i). Since (i) and
$X'$ uniquely determine both $X$ and $Y$, we must have $X = Y$, and so $X$ satisfies (ii).
\end{proof}
\begin{rem}
\label{5.9}
We will apply Lemma \ref{5.8} as follows. Let $A := (\hat{F}^\times_{\ell,\mathrm{f}})^-$, and let
$x_n := \projI{n}{(\kappa_n)_{\ell,\mathrm{f}}}$. Then Lemma \ref{5.8} says that we
can replace property (iv) in Definition \ref{preksdef} of a pre-Kolyvagin system
by the equivalent statement:
\begin{enumerate}
\item[(iv)$'$]
if $\ell \mid n_+$, then
$
\displaystyle
\projI{n}{(\kappa_{n})_{\ell,\mathrm{f}}} =
-\sum_{\pile{\sigma\in\mathfrak{S}_1(n)}{\sigma(\ell) \ne \ell}}
\mathrm{sign}(\sigma) \projI{n/d_\sigma}{(\kappa_{n/d_\sigma})_{\ell}}
\Pi(\sigma).
$
\end{enumerate}
\end{rem}
\section{The cyclotomic unit pre-Kolyvagin system}
\label{cupks}
Fix an odd prime $p$.
If $n \in \mathcal{N}$, let $s(n)$ be the number of prime factors of $n/n_+$.
In this section we will show
that the collection $\{2^{-s(n)}\tilde\theta'_n : n \in \mathcal{N}_p\}$ is a
pre-Kolyvagin system. Recall that
$$
\mathcal{N}_p^+ := \{n \in \mathcal{N}_p : \text{all $\ell \mid n$ split in $F/\mathbf{Q}$}\}.
$$
\begin{prop}[(Darmon)]
\label{9.4}
If $n \in \mathcal{N}_p$ then
$$
\sum_{d \mid n_+} \tilde\theta'_{n/d} \prod_{\ell \mid d}\pi_{n/d}(\mathrm{Fr}_\ell-1)
= 2^{s(n)}\beta_{n_+} \quad \text{in $(\hat{F}^\times)^- \otimes \I_n$}
$$
where for $n \in \mathcal{N}_p^+$, $\beta_{n} \in (\hat{F}^\times)^- \otimes \I_n$ is the Kolyvagin
derivative class denoted $\kappa(n)$ in \cite[\S6]{darmon}, or $\kappa_{n}$ in
\cite[Appendix]{babykolysys}.
\end{prop}
\begin{proof}
This is Proposition 9.4 of \cite{darmon}.\footnote{There is a typo in
\cite[Proposition 9.4]{darmon}. The last two $T$'s should be $TQ$, as in
\cite[Lemma 8.1]{darmon}.}
(Note that $\kappa(n)$ in \cite[\S6]{darmon} and $\kappa_{n}$ in
\cite[Appendix]{babykolysys} are defined to lie in
$(\hat{F}^\times)^- \otimes (\mathbf{Z}/\gcd(\ell-1: \ell | n)\mathbf{Z})$, after fixing generators of every
$\Gamma_\ell$. Without fixing such choices, the elements defined in \cite{darmon}
and \cite{babykolysys} live naturally in $(\hat{F}^\times)^- \otimes \I_n$.)
\end{proof}
\begin{thm}
\label{thetapreks}
The collection $\{2^{-s(n)}\tilde\theta'_n : n \in \mathcal{N}_p\}$ is a
pre-Kolyvagin system.
\end{thm}
\begin{proof}
We need to check the five properties of Definition \ref{preksdef}.
For $n \in \mathcal{N}_p^+$, let $\beta_{n}$ be as in Proposition \ref{9.4}.
Since $\beta_{n_+} \in (\hat{F}^\times)^- \otimes \I_n$ for every $n$, it follows
easily by induction from Proposition \ref{9.4} that
$\tilde\theta'_n \in (\hat{F}^\times)^- \otimes I_n^r/I_n^{r+1}$,
where $r$ is the number of prime factors of $n_+$.
This is property (i) of Definition \ref{preksdef}.
Suppose $\ell \mid n_+$. A standard property of cyclotomic units shows that
$$
\mathbf{N}_{F(\boldsymbol{\mu}_n)/F(\boldsymbol{\mu}_{n/\ell})}\alpha_n = \alpha_{n/\ell}/\alpha_{n/\ell}^{\mathrm{Fr}_\ell^{-1}}.
$$
It follows from the definition of $\theta'_n$ that
\begin{multline*}
(1 \otimes \pi_{n/\ell})(\theta'_n)
= \sum_{\gamma \in \Gamma_n} \gamma(\alpha_n) \otimes \pi_{n/\ell}(\gamma)
= \sum_{\gamma \in \Gamma_{n/\ell}} \gamma(\mathbf{N}_{F(\boldsymbol{\mu}_n)/F(\boldsymbol{\mu}_{n/\ell})}\alpha_n) \otimes \gamma\\
= \sum_{\gamma \in \Gamma_{n/\ell}}
\gamma\bigl(\alpha_{n/\ell}/\alpha_{n/\ell}^{\mathrm{Fr}_\ell^{-1}}\bigr) \otimes \gamma
= \sum_{\gamma \in \Gamma_{n/\ell}} \gamma(\alpha_{n/\ell}) \otimes \gamma\,\pi_{n/\ell}(1-\mathrm{Fr}_\ell)
= \theta'_{n/\ell}\,\pi_{n/\ell}(1-\mathrm{Fr}_\ell).
\end{multline*}
Since $\ell \mid n_+$ we have $s(n) = s(n/\ell)$, so
this verifies property (ii) of Definition \ref{preksdef}.
Projecting each of the summands in Proposition \ref{9.4} into $(\hat{F}^\times)^- \otimes \I_n$,
one sees that all terms with $d > 1$ vanish, yielding
$$
\projI{n}{2^{-s(n)}\tilde\theta'_{n}} = \projI{n}{\beta_{n_+}} = \beta_{n_+}.
$$
Properties (iii), (iv), and (v) of Definition \ref{preksdef} follow from the
corresponding properties of the $\beta_{n_+}$.
See \cite[Proposition A.2]{kolysys} or \cite[Theorem 4.5.4]{eulersys}
for (iii), and \cite[Theorem A.4]{kolysys}
or \cite[Proposition A.2]{babykolysys} for property (iv)$'$ of Remark \ref{5.9}.
Property (v) is immediate, since $\beta_{n_+}$ depends only on $n_+$.
\end{proof}
\section{The regulator pre-Kolyvagin system}
\label{rpks}
In this section we study relations among the regulator
elements $R_n$, to show that the collection
$\{h_nR_n : n \in \mathcal{N}_p\}$ is a pre-Kolyvagin system.
\begin{lem}
\label{cnr}
Suppose $n \in \mathcal{N}$, $\ell \mid n_+$, and
$\{\lambda_0-\lambda_0^\tau,\ldots,\lambda_r-\lambda_r^\tau\}$
is a standard basis of $X_n^-$ with $\lambda_r\lambda_r^\tau = \ell$.
Then $\{\lambda_0-\lambda_0^\tau,\ldots,\lambda_{r-1}-\lambda_{r-1}^\tau\}$
is a standard basis of $X_{n/\ell}^-$, and we can choose an oriented basis
$\{\epsilon_0,\ldots,\epsilon_r\}$ of $(1-\tau)\mathcal{E}_n$ such that
$\{\epsilon_0,\ldots,\epsilon_{r-1}\}$ is an oriented basis of $(1-\tau)\mathcal{E}_{n/\ell}$.
With any such bases, $\mathrm{ord}_{\lambda_r}(\epsilon_r) = -h_{n/\ell}/h_n$ and
$$
\grx{\lambda_r}{n/\ell}{\epsilon_r} = \frac{h_{n/\ell}}{h_n}\,\pi_{n/\ell}(1-\mathrm{Fr}_\ell) \in I_{n/\ell}/I_{n/\ell}^2.
$$
\end{lem}
\begin{proof}
Everything except the final sentence is clear.
Comparing the determinants of the logarithmic embeddings
$$
(1-\tau)\mathcal{E}_{n/\ell} \map{\xi_{n/\ell}} X_{n/\ell}^-, \qquad (1-\tau)\mathcal{E}_n \map{\xi_n} X_n^-
$$
with respect to our given bases, we see that
$$
\det(\xi_n) = \log|\epsilon_r|_{\lambda_r}\det(\xi_{n/\ell})
$$
because $\log|\epsilon_i|_{\lambda_r} = 0$ for $0 \le i < r$. Since our bases
are oriented, both determinants are positive. Hence
$$
|\epsilon_r|_{\lambda_r} = \ell^{-\mathrm{ord}_{\lambda_r}(\epsilon_r)} > 1
$$
so $\mathrm{ord}_{\lambda_r}(\epsilon_r) < 0$.
The exact sequence
$$
(1-\tau)\mathcal{E}_n \map{\mathrm{ord}_{\lambda_r}} \mathbf{Z} \map{\cdot\lambda_r} \mathrm{Pic}(\mathcal{O}_F[\ell/n])
\longrightarrow \mathrm{Pic}(\mathcal{O}_F[1/n]) \longrightarrow 0
$$
shows that
$$
[\mathbf{Z} : \mathrm{ord}_{\lambda_r}(\epsilon_r)\mathbf{Z}] = h_{n/\ell}/h_n,
$$
so $\mathrm{ord}_{\lambda_r}(\epsilon_r) = - h_{n/\ell}/h_n$ as claimed.
Since $F(\boldsymbol{\mu}_{n/\ell})/F$ is unramified at $\lambda_r$,
$$
\grx{\lambda_r}{n/\ell}{\epsilon_r}
= (\mathrm{Fr}_{\ell}^{\mathrm{ord}_{\lambda_r}(\epsilon_r)}) - 1
= \mathrm{ord}_{\lambda_r}(\epsilon_r) (\mathrm{Fr}_{\ell} - 1)
= -h_{n/\ell}/h_n (\mathrm{Fr}_{\ell} - 1)
$$
in $I_{n/\ell}/I_{n/\ell}^2$.
\end{proof}
\begin{prop}
\label{regreg1lem}
Suppose $n \in \mathcal{N}$, $\ell \mid n_+$, and $r = r(n)$. Then
$$
(1 \otimes \pi_{n/\ell})(h_n R_n) = h_{n/\ell} R_{n/\ell}\,\pi_{n/\ell}(1-\mathrm{Fr}_{\ell})
\in F^\times \otimes I_n^r/I_n^{r+1}.
$$
\end{prop}
\begin{proof}
To compute $R_n$, fix bases for $X_n^-$ and $\mathcal{E}_n^-$ as in Lemma \ref{cnr}.
By definition
$$
R_n := \left|
\begin{array}{ccccccc}
\epsilon_0 & \epsilon_1 & \cdots & \epsilon_r \\
\grx{\lambda_1}{n}{\epsilon_0} & \grx{\lambda_1}{n}{\epsilon_1} & \cdots & \grx{\lambda_1}{n}{\epsilon_r} \\
\vdots & \vdots && \vdots\\
\grx{\lambda_r}{n}{\epsilon_0} & \grx{\lambda_r}{n}{\epsilon_1} & \cdots & \grx{\lambda_r}{n}{\epsilon_r}
\end{array}
\right|,
$$
and then $(1 \otimes \pi_{n/\ell})(R_n)$ is the determinant of the matrix
obtained by applying $\pi_{n/\ell}$ to rows $2$ through $r+1$ of this matrix.
For $i < r$, $\epsilon_i$ is a unit at $\lambda_r$, so the local Artin symbol
$[\epsilon_i,F(\boldsymbol{\mu}_n)_{\lambda_r}/F_{\lambda_r}]$ lies in the inertia group $\Gamma_\ell$.
Hence $\pi_{n/\ell}(\grx{\lambda_r}{n}{\epsilon_i}) = \grx{\lambda_r}{n/\ell}{\epsilon_i} = 0$
for $i < r$, and so
$$
(1 \otimes\pi_{n/\ell})(R_n) = \left|
\begin{array}{ccccccc}
\epsilon_0 & \cdots & \epsilon_{r-1} & \epsilon_r \\
\grx{\lambda_1}{n/\ell}{\epsilon_0} & \cdots & \grx{\lambda_1}{n/\ell}{\epsilon_{r-1}}
&\grx{\lambda_1}{n/\ell}{\epsilon_r} \\
\vdots && \vdots & \vdots\\
\grx{\lambda_{r-1}}{n/\ell}{\epsilon_0} & \cdots & \grx{\lambda_{r-1}}{n/\ell}{\epsilon_{r-1}}
&\grx{\lambda_{r-1}}{n/\ell}{\epsilon_r} \\
0 & \cdots & 0 & \grx{\lambda_{r}}{n/\ell}{\epsilon_r}
\end{array}
\right|.
$$
The upper left $r \times r$ determinant is the one used to define $R_{n/\ell}$, so
$$
(1 \otimes\pi_{n/\ell})(R_n) = R_{n/\ell} \grx{\lambda_r}{n/\ell}{\epsilon_r}
= \frac{h_{n/\ell}}{h_n } R_{n/\ell} \, \pi_{n/\ell}(1-\mathrm{Fr}_{\ell})
$$
by Lemma \ref{cnr}.
\end{proof}
Fix an odd prime $p$ as in \S\S\ref{kssect} and \ref{prekssect},
and keep the rest of the notation of those sections as well.
\begin{lem}
\label{finlem}
If $n \in \mathcal{N}_p$, $\ell$ is a prime not dividing $n$, and $r = r(n)$, then
$$
(R_n)_{\ell} \in (\hat{F}_{\ell,\mathrm{f}}^\times)^- \otimes I_n^r/I_n^{r+1}.
$$
\end{lem}
\begin{proof}
Since $\ell \nmid n$, if $\epsilon \in \mathcal{E}_n^-$ then
$\epsilon_\ell \in (\hat{\mathcal{O}}_\ell^\times)^-
= (\hat{F}_{\ell,\mathrm{f}}^\times)^- \subset (\hat{F}_{\ell}^\times)^-$.
Now the lemma is clear, since $R_n \in \mathcal{E}_n^- \otimes I_n^r/I_n^{r+1}$.
\end{proof}
\begin{prop}
\label{fsprop}
Suppose $n \in \mathcal{N}_p$ and $\ell \mid n_+$.
Then
$$
\projI{n}{h_n(R_n)_{\ell,\mathrm{tr}}} = (\phi^{\mathrm{fs}}_\ell \otimes 1)(\projI{n/\ell}{h_{n/\ell}(R_{n/\ell})_\ell}).
$$
\end{prop}
\begin{proof}
Note that
$(\phi^{\mathrm{fs}}_\ell \otimes 1)(\projI{n/\ell}{h_{n/\ell}(R_{n/\ell})_\ell})
\in (\hat{F}_{\ell,\mathrm{tr}}^\times)^- \otimes \I_n$
is well-defined, since Lemma \ref{finlem} shows that
$(R_{n/\ell})_{\ell} \in (\hat{F}_{\ell,\mathrm{f}}^\times)^- \otimes I_{n/\ell}^{r-1}/I_{n/\ell}^{r}$.
As in the proof of Proposition \ref{regreg1lem}, fix a basis
$\{\lambda_0-\lambda_0^\tau,\ldots,\lambda_r-\lambda_r^\tau\}$ of $X_n^-$
with $\ell = \lambda_r\lambda_r^\tau$, and
an oriented basis $\{\epsilon_0,\ldots,\epsilon_r\}$ of $(1-\tau)\mathcal{E}_n$ as in Lemma \ref{cnr}. Then
$$
(R_n)_{\ell,\mathrm{tr}} = \left|
\begin{array}{ccccccc}
(\epsilon_0)_{\ell,\mathrm{tr}} & \cdots & (\epsilon_{r-1})_{\ell,\mathrm{tr}} & (\epsilon_r)_{\ell,\mathrm{tr}} \\
\grx{\lambda_1}{n}{\epsilon_0} & \grx{\lambda_1}{n}{\epsilon_1} & \cdots & \grx{\lambda_1}{n}{\epsilon_r} \\
\vdots & \vdots && \vdots\\
\grx{\lambda_r}{n}{\epsilon_0} & \grx{\lambda_r}{n}{\epsilon_1} & \cdots & \grx{\lambda_r}{n}{\epsilon_r}
\end{array}
\right|
= \mathrm{ord}_{\lambda_r}(\epsilon_r)
\left|
\begin{array}{ccccccc}
1 & \cdots & 1 & (\ell,\ell^{-1}) \\
\grx{\lambda_1}{n}{\epsilon_0} & \grx{\lambda_1}{n}{\epsilon_1} & \cdots & \grx{\lambda_1}{n}{\epsilon_r} \\
\vdots & \vdots && \vdots\\
\grx{\lambda_r}{n}{\epsilon_0} & \grx{\lambda_r}{n}{\epsilon_1} & \cdots & \grx{\lambda_r}{n}{\epsilon_r}
\end{array}
\right|
$$
since $(\epsilon_r)_{\ell,\mathrm{tr}} = (\ell,\ell^{-1})^{\mathrm{ord}_{\lambda_r}(\epsilon_r)}$,
and $(\epsilon_i)_{\ell,\mathrm{tr}} = 1$ for $i < r$.
(Recall that when we evaluate these determinants using \eqref{minors}, the multiplicative notation
in $(\hat{F}^\times_\ell)_\mathrm{tr}$ changes to additive notation in the tensor product
$(\hat{F}^\times_\ell)_\mathrm{tr} \otimes I_\ell^r/I_\ell^{r+1}$, so $1$'s in the
top row become $0$'s, and $(\ell,\ell^{-1})^{\mathrm{ord}_{\lambda_r}(\epsilon_r)}$ becomes
${\mathrm{ord}_{\lambda_r}(\epsilon_r)}\cdot(\ell,\ell^{-1})$.)
We have $\mathrm{ord}_{\lambda_r}(\epsilon_r) = -h_{n/\ell}/h_n$ by Lemma \ref{cnr}.
For $i < r$ we have $\mathrm{ord}_{\lambda_r}(\epsilon_i) = 0$, so
$\grx{\lambda_r}{n}{\epsilon_i} = \grx{\lambda_r}{\ell}{\epsilon_i} \in I_\ell/I_\ell^2$ and
$$
\phi^{\mathrm{fs}}_\ell((\epsilon_i)_{\ell}) = (\ell,\ell^{-1}) \otimes \grx{\lambda_r}{n}{\epsilon_i}
\in (\hat{F}^\times_\ell)_\mathrm{tr} \otimes I_\ell/I_\ell^2.
$$
Thus
\begin{align*}
(R_n)_{\ell,\mathrm{tr}} &= -\frac{h_{n/\ell}}{h_n}(-1)^{r}(-1)^{r-1}\left|
\begin{array}{ccccccc}
\phi^{\mathrm{fs}}_\ell((\epsilon_0)_{\ell}) & \cdots & \phi^{\mathrm{fs}}_\ell((\epsilon_{r-1})_{\ell}) \\
\grx{\lambda_1}{n}{\epsilon_0} & \cdots & \grx{\lambda_1}{n}{\epsilon_{r-1}} \\
\vdots & \vdots & \vdots\\
\grx{\lambda_{r-1}}{n}{\epsilon_0} & \cdots & \grx{\lambda_{r-1}}{n}{\epsilon_{r-1}}
\end{array}
\right|\\[7pt]
&= \frac{h_{n/\ell}}{h_n} \left|
\begin{array}{ccccccc}
\phi^{\mathrm{fs}}_\ell((\epsilon_0)_{\ell}) & \cdots & \phi^{\mathrm{fs}}_\ell((\epsilon_{r-1})_{\ell}) \\
\grx{\lambda_1}{n/\ell}{\epsilon_0}+\grx{\lambda_1}{\ell}{\epsilon_0} & \cdots
& \grx{\lambda_1}{n/\ell}{\epsilon_{r-1}}+\grx{\lambda_1}{\ell}{\epsilon_{r-1}} \\
\vdots & \vdots & \vdots\\
\grx{\lambda_{r-1}}{n/\ell}{\epsilon_0}+\grx{\lambda_{r-1}}{\ell}{\epsilon_0} & \cdots
& \grx{\lambda_{r-1}}{n/\ell}{\epsilon_{r-1}}+\grx{\lambda_{r-1}}{\ell}{\epsilon_{r-1}}
\end{array}
\right|
\end{align*}
(the $(-1)^r$ because we moved column $r+1$ to column $1$, and the $(-1)^{r-1}$
because we moved row $r+1$ to row $2$).
When we expand the last determinant (including expanding the sums
$\grx{\lambda_{j}}{n/\ell}{\epsilon_{i}}+\grx{\lambda_{j}}{\ell}{\epsilon_{i}}$),
each term that
includes one of the $\grx{\lambda_j}{\ell}{\epsilon_i}$ lies in $I_\ell^2$
(since the top row also contributes one element of $I_\ell$). Thus all such terms
project to zero in $\I_n$, and so
$$
\projI{n}{(R_n)_{\ell,\mathrm{tr}}} = \frac{h_{n/\ell}}{h_n} ~\projI{n}{\det(A)}
$$
where
$$
A = \left[
\begin{array}{ccccccc}
\phi^{\mathrm{fs}}_\ell((\epsilon_0)_{\ell}) & \cdots & \phi^{\mathrm{fs}}_\ell((\epsilon_{r-1})_{\ell}) \\
\grx{\lambda_1}{n/\ell}{\epsilon_0} & \cdots & \grx{\lambda_1}{n/\ell}{\epsilon_{r-1}} \\
\vdots & \vdots & \vdots\\
\grx{\lambda_{r-1}}{n/\ell}{\epsilon_0} & \cdots & \grx{\lambda_{r-1}}{n/\ell}{\epsilon_{r-1}}
\end{array}
\right].
$$
But then $\det(A) = (\phi^{\mathrm{fs}}_\ell \otimes 1)((R_{n/\ell})_\ell)$, so the proposition follows.
\end{proof}
Suppose $n, n' \in \mathcal{N}$, $n \mid n'$, and $r = r(n)$.
Define
$$
\bR{n}{n'} := \left|
\begin{array}{ccccccc}
\epsilon_0 & \epsilon_1 & \cdots & \epsilon_r \\
\grx{\lambda_1}{n'}{\epsilon_0} & \grx{\lambda_1}{n'}{\epsilon_1} & \cdots & \grx{\lambda_1}{n'}{\epsilon_r} \\
\vdots & \vdots && \vdots\\
\grx{\lambda_r}{n'}{\epsilon_0} & \grx{\lambda_r}{n'}{\epsilon_1} & \cdots & \grx{\lambda_r}{n'}{\epsilon_r}
\end{array}
\right| \in \mathcal{E}_n^- \otimes I_{n'}^r/I_{n'}^{r+1},
$$
using any standard basis of $X_n^-$ and oriented basis of $(1-\tau)\mathcal{E}_n$.
In particular $\bR{n}{n} = R_n$.
\begin{prop}
\label{indstep}
Suppose $n \in \mathcal{N}$ and $\ell \nmid n$.
\begin{enumerate}
\item
If $\ell$ is inert in $F/\mathbf{Q}$, then $h_{n\ell}\projI{n}{R_{n\ell}} = h_n\projI{n}{R_n}$.
\item
If $\ell$ splits in $F/\mathbf{Q}$ and $v \in I_n$, then
$$
h_{n}\projI{n\ell}{\bR{n}{n\ell}\,v} = \projI{n}{R_n}\pi_\ell(v)
- \sum_{\mathrm{primes}~ q \mid n_+}h_{n/q}\projI{n}{\bR{n/q}{n}\,v}\pi_{\ell}(\mathrm{Fr}_q-1)
$$
in $\mathcal{E}_n^- \otimes \mathcal{I}^{\mathrm{new}}_{n\ell}$.
\end{enumerate}
\end{prop}
\begin{proof}
Let $r$ be the number of prime divisors of $n_+$, so $X_{n}^-$ and
$(1-\tau)\mathcal{E}_{n}$ are free $\mathbf{Z}$-modules of rank $r+1$.
Choose a standard basis of $X_{n}^-$ and an oriented basis of $(1-\tau)\mathcal{E}_{n}$.
For $1 \le i \le r = r(n)$, let
$$
a_i = (\grx{\lambda_i}{n}{\epsilon_0}, \grx{\lambda_i}{n}{\epsilon_1}, \ldots,
\grx{\lambda_i}{n}{\epsilon_r}),
\quad
b_i = (\grx{\lambda_i}{\ell}{\epsilon_0}, \grx{\lambda_i}{\ell}{\epsilon_1}, \ldots,
\grx{\lambda_i}{\ell}{\epsilon_r}).
$$
Then
\begin{equation}
\label{5}
\bR{n}{n\ell} = \left|
\begin{array}{ccccccc}
\epsilon_0 & \cdots & \epsilon_r \\
&a_1 + b_1 \\
& \vdots \\
& a_r+ b_r
\end{array}
\right|
=
\sum_{T \subset \{1,\ldots,r\}} \det(A_T)
\end{equation}
where $A_T$ is the matrix whose top row is $(\epsilon_0, \ldots, \epsilon_r)$ and
whose $(i+1)$-st row for $1 \le i \le r$ is $b_i$ if $i \in T$ and $a_i$ if $i \notin T$.
Note that $\det(A_\emptyset) = R_{n}$, and that
the entries of each $b_i$ are in $I_\ell/I_\ell^2$.
Suppose first that $\ell$ is inert in $F/\mathbf{Q}$, so $(n\ell)_+ = n_+$.
Then $\projI{n}{\det(A_T)} = 0$
if $T$ is nonempty (since $\I_n$ has no ``$\ell$ component''), so \eqref{5}
shows that
$$
\projI{n}{\bR{n}{n\ell}} = \projI{n}{\det(A_\emptyset)} = \projI{n}{R_{n}}.
$$
Further, since $\ell$ is inert in $F/\mathbf{Q}$ we have $X_{n\ell}^- = X_{n}^-$,
$(1-\tau)\mathcal{E}_{n\ell} = (1-\tau)\mathcal{E}_{n}$, and $h_{n\ell} = h_{n}$. Thus
$\bR{n}{n\ell} = R_{n\ell}$, and so
$$
h_{n\ell} \projI{(n\ell)}{R_{n\ell}} = h_n \projI{n}{\bR{n}{n\ell}}
= h_{n} \projI{n}{R_{n}}.
$$
This is (i).
Now suppose that $\ell$ splits in $F/\mathbf{Q}$.
Since the entries of each $b_i$ are in $I_\ell$, if $\#(T) \ge 2$ we have
$\projI{n\ell}{\det(A_T)v} = 0$. Thus \eqref{5} gives
\begin{equation}
\label{3}
\projI{n\ell}{\bR{n}{n\ell}\,v} = \projI{n\ell}{\det(A_\emptyset)v}
+ \sum_{i=1}^r \projI{n\ell}{\det(A_{\{i\}})v}.
\end{equation}
By definition of $R_n$,
\begin{equation}
\label{4}
\projI{n\ell}{\det(A_\emptyset)v} = \projI{n\ell}{R_{n}\,v}
= \projI{n}{R_{n}}\pi_\ell(v).
\end{equation}
To compute $\det(A_{\{i\}})$,
let $q = \lambda_i\lambda_i^\tau$, and assume that our oriented basis of
$(1-\tau)\mathcal{E}_{n}$ was chosen so that
$\{\epsilon_0,\ldots,\epsilon_{r-1}\}$ is an oriented basis of $(1-\tau)\mathcal{E}_{n/q}$
with respect to the standard basis of $X_{n/q}$ obtained by removing
$\lambda_i-\lambda_i^\tau$ from
$\{\lambda_1-\lambda_1^\tau,\ldots,\lambda_r-\lambda_r^\tau\}$.
For $1 \le j \le r-1$, $\epsilon_j$ is a unit at $\lambda_i$, so
$\grx{\lambda_i}{\ell}{\epsilon_j} = 0$. Thus
\begin{multline*}
\det(A_{\{i\}}) = \left|
\begin{array}{ccccccc}
\epsilon_0 & \cdots & \epsilon_{r-1} & \epsilon_r \\
\grx{\lambda_1}{n}{\epsilon_0} & \cdots & \grx{\lambda_1}{n}{\epsilon_{r-1}} & \grx{\lambda_1}{n}{\epsilon_r} \\
\vdots && \vdots & \vdots\\
0 & \cdots & 0 & \grx{\lambda_i}{\ell}{\epsilon_r} \\
\vdots && \vdots & \vdots \\
\grx{\lambda_r}{n}{\epsilon_0} & \cdots & \grx{\lambda_r}{n}{\epsilon_{r-1}} & \grx{\lambda_r}{n}{\epsilon_r}
\end{array}
\right|
= (-1)^{r+i} \left|
\begin{array}{ccccccc}
\epsilon_0 & \cdots & \epsilon_{r-1} \\
\grx{\lambda_1}{n}{\epsilon_0} & \cdots & \grx{\lambda_1}{n}{\epsilon_{r-1}} \\
\vdots & \vdots & \vdots\\
\grx{\lambda_{r}}{n}{\epsilon_0} & \cdots & \grx{\lambda_{r}}{n}{\epsilon_{r-1}}
\end{array}
\right| \grx{\lambda_i}{\ell}{\epsilon_r} \\
= (-1)^{r+i} \bR{n/q}{n}\,\grx{\lambda_i}{\ell}{\epsilon_r}
\end{multline*}
(where the second determinant has no $\lambda_i$ row). Further, an argument
identical to that of Lemma \ref{cnr} shows that
$$
\grx{\lambda_i}{\ell}{\epsilon_r} = (-1)^{r+i+1}\frac{h_{n/q}}{h_{n}}\,\pi_\ell(\mathrm{Fr}_q-1)
\in I_\ell/I_\ell^2.
$$
Therefore
$$
\det(A_{\{i\}}) = -\frac{h_{n/q}}{h_{n}} \bR{n/q}{n}\,\pi_\ell(\mathrm{Fr}_q-1).
$$
Multiplying \eqref{3} by $h_{n}$ and using \eqref{4} gives
$$
h_{n}\projI{n\ell}{\bR{n}{n\ell}\,v} = h_n\projI{n}{R_{n}}\,\pi_\ell(v)
- \sum_{q \mid n_+}h_{n/q}\projI{n\ell}{\bR{n/q}{n}\,v\,\pi_\ell(\mathrm{Fr}_q-1)}.
$$
Since $\bR{n/q}{n} \in I_{n}^r/I_{n}^{r+1}$, we have
$$
\projI{n\ell}{\bR{n/q}{n}\,v\,\pi_\ell(\mathrm{Fr}_q-1)} = \projI{n}{\bR{n/q}{n}\,\pi_n(v)}\,\pi_\ell(\mathrm{Fr}_q-1).
$$
This completes the proof of the proposition.
\end{proof}
If $n \in \mathcal{N}$, recall (Definition \ref{symmdef})
that $\mathfrak{S}(n)$ denotes the set of permutations of the primes
dividing $n_+$, $\mathfrak{S}_1(n) \subset \mathfrak{S}(n)$ is the subset
$$
\mathfrak{S}_1(n) := \{\sigma \in \mathfrak{S}(n) :
\text{the primes not fixed by $\sigma$ form a single $\sigma$-orbit}\},
$$
and if $\sigma \in \mathfrak{S}(n)$ then $d_\sigma := \prod_{\sigma(\ell) \ne \ell} \ell$
and $\Pi(\sigma) := \prod_{q \mid d_\sigma} \pi_q(\mathrm{Fr}_{\sigma(q)}-1)$.
\begin{thm}
\label{finitepart}
If $n \in\mathcal{N}_p$ and $\ell \mid n_+$, then
$$
\projI{n}{h_n(R_{n})_{\ell,\mathrm{f}}} =
-\sum_{\pile{\sigma\in\mathfrak{S}_1(n)}{\sigma(\ell) \ne \ell}}
\mathrm{sign}(\sigma) \projI{n/d_\sigma}{h_{n/d_\sigma}(R_{n/d_\sigma})_{\ell}}
\Pi(\sigma).
$$
\end{thm}
\begin{proof}
As usual, fix a basis
$\{\lambda_0-\lambda_0^\tau,\ldots,\lambda_r-\lambda_r^\tau\}$ of $X_n^-$
with $\ell = \lambda_r\lambda_r^\tau$, and
an oriented basis $\{\epsilon_0,\ldots,\epsilon_r\}$ of $(1-\tau)\mathcal{E}_n$ as in Lemma \ref{cnr},
so that $\{\epsilon_0,\ldots,\epsilon_{r-1}\}$ is an oriented basis of $(1-\tau)\mathcal{E}_{n/\ell}$.
Then
$$
(R_n)_{\ell,f} = \left|
\begin{array}{ccccccc}
(\epsilon_0)_{\ell,\mathrm{f}} & \cdots & (\epsilon_{r-1})_{\ell,\mathrm{f}} & (\epsilon_r)_{\ell,\mathrm{f}} \\
\grx{\lambda_1}{n}{\epsilon_0} & \cdots & \grx{\lambda_1}{n}{\epsilon_{r-1}} & \grx{\lambda_1}{n}{\epsilon_r} \\
\vdots && \vdots & \vdots\\
\grx{\lambda_r}{n}{\epsilon_0} & \cdots & \grx{\lambda_r}{n}{\epsilon_{r-1}} & \grx{\lambda_r}{n}{\epsilon_r}
\end{array}
\right|.
$$
For each $i$, we have
$\grx{\lambda_r}{n}{\epsilon_i} = \grx{\lambda_r}{n/\ell}{\epsilon_i} + \grx{\lambda_r}{\ell}{\epsilon_i}$.
If $i < r$, then $\epsilon_i$ is a unit at $\lambda_r$ so
$\grx{\lambda_r}{n/\ell}{\epsilon_i} = 0$. Thus
$$
(R_n)_{\ell,f} = \left|
\begin{array}{ccccccc}
(\epsilon_0)_{\ell,\mathrm{f}} & \cdots & (\epsilon_r)_{\ell,\mathrm{f}} \\
\grx{\lambda_1}{n}{\epsilon_0} & \cdots & \grx{\lambda_1}{n}{\epsilon_r} \\
\vdots && \vdots\\
\grx{\lambda_r}{\ell}{\epsilon_0} & \cdots & \grx{\lambda_r}{\ell}{\epsilon_r}
\end{array}
\right| +
\left|
\begin{array}{ccccccc}
(\epsilon_0)_{\ell,\mathrm{f}} & \cdots & (\epsilon_{r-1})_{\ell,\mathrm{f}} & (\epsilon_r)_{\ell,\mathrm{f}} \\
\grx{\lambda_1}{n}{\epsilon_0} & \cdots & \grx{\lambda_1}{n}{\epsilon_{r-1}} & \grx{\lambda_1}{n}{\epsilon_r} \\
\vdots && \vdots & \vdots\\
0 & \cdots & 0 & \grx{\lambda_r}{n/\ell}{\epsilon_r}
\end{array}
\right|.
$$
The map
$\epsilon \mapsto \grx{\lambda_r}{\ell}{\epsilon} = [\epsilon,F_{\lambda_r}(\boldsymbol{\mu}_\ell)/F_{\lambda_r}]-1$ is an isomorphism from
$(\hat{F}_{\ell,\mathrm{f}}^\times)^- = (\hat{\mathcal{O}}_\ell^\times)^-$ to $(I_\ell/I_\ell^2) \otimes \Z_p$, and is
zero on $(\hat{F}_{\ell,\mathrm{tr}}^\times)^-$ because $\ell$ is a norm in the extension
$F_{\lambda_r}(\boldsymbol{\mu}_\ell)/F_{\lambda_r} = \Q_ \ell(\boldsymbol{\mu}_\ell)/\Q_ \ell$. Hence the first
determinant in the equation above is zero, because the top and bottom rows are
linearly dependent. Also, if $i < r$ then $\epsilon_i$ is a unit at $\lambda_r$,
so $(\epsilon_i)_{\ell,\mathrm{f}} = (\epsilon_i)_{\ell}$ and
$$
(R_n)_{\ell,\mathrm{f}} = \left|
\begin{array}{ccccccc}
(\epsilon_0)_{\ell} & \cdots & (\epsilon_{r-1})_{\ell} \\
\grx{\lambda_1}{n}{\epsilon_0} & \cdots & \grx{\lambda_1}{n}{\epsilon_{r-1}} \\
\vdots && \vdots\\
\grx{\lambda_{r-1}}{n}{\epsilon_0} & \cdots & \grx{\lambda_{r-1}}{n}{\epsilon_{r-1}}
\end{array}
\right|\grx{\lambda_r}{n/\ell}{\epsilon_r}
= (\bR{n/\ell}{n})_\ell\,\grx{\lambda_r}{n/\ell}{\epsilon_r}.
$$
By Lemma \ref{cnr}, $\grx{\lambda_r}{n/\ell}{\epsilon_r} = -(h_{n/\ell}/h_n)\,\pi_{n/\ell}(\mathrm{Fr}_\ell-1)$.
Thus
\begin{equation}
\label{1}
h_n\projI{n}{(R_n)_{\ell,\mathrm{f}}} = -h_{n/\ell}\projI{n}{(\bR{n/\ell}{n})_{\ell}\,\pi_{n/\ell}(\mathrm{Fr}_\ell-1)}.
\end{equation}
We can now ``simplify'' \eqref{1} by inductively expanding the right-hand side
using Proposition \ref{indstep}. Specifically, expand
$\projI{n}{(\bR{n/\ell}{n}\,\pi_{n/\ell}(\mathrm{Fr}_\ell-1)}$
using Proposition \ref{indstep}(ii). Then expand each of the resulting
$\projI{n/\ell}{(\bR{n/(\ell q)}{n/\ell}\,\pi_{n/(q\ell)}(\mathrm{Fr}_q-1)}$
using Proposition \ref{indstep}(ii) again. Continue until no terms $\bR{m/q}{m}$
remain. The resulting sum consists of one term
$$
(-1)^k \projI{n/(q_1\cdots q_k)}{h_{n/(q_1\cdots q_k)}(R_{n/(q_1\cdots q_k)})_\ell} \prod_{i=1}^k \pi_{q_i}(\mathrm{Fr}_{q_{i+1}}-1)
$$
for each sequence $q_1 = \ell,q_2,\ldots,q_k$ of distinct primes dividing $n_+$ (with
$q_{k+1} = \ell$). Identifying this sequence with the $k$-cycle
$\sigma := (\ell,q_2,\ldots, q_k) \in \mathfrak{S}_1(n)$ gives the formula of the theorem,
since $\mathrm{sign}(\sigma) = (-1)^{k-1}$.
\end{proof}
\begin{thm}
\label{Rpreks}
The collection $\{h_nR_n : n \in \mathcal{N}_p\}$ is a pre-Kolyvagin system.
\end{thm}
\begin{proof}
We need to check the five properties of Definition \ref{preksdef}.
Property (i) is Lemma \ref{finlem}, (ii) is Proposition \ref{regreg1lem},
(iii) is Proposition \ref{fsprop}, (iv) is Theorem \ref{finitepart} along with
of Remark \ref{5.9}, and (v) is Proposition \ref{indstep}(i).
\end{proof}
\section{Proof of Theorem \ref{mainthm}}
\label{pfsect}
\begin{proof}[Proof of Theorem \ref{mainthm}]
Fix an odd prime $p$.
By Theorems \ref{thetapreks} and \ref{Rpreks}, we have pre-Kolyvagin systems
$$
\{2^{-s(n)}\tilde\theta_n : n \in \mathcal{N}_p\}, \quad \{-h_nR_n : n \in \mathcal{N}_p\}.
$$
By Proposition \ref{n=1}, $\tilde\theta'_1 = -h_1R_1$ in $\mathcal{O}_F^\times/\{\pm1\}$.
Hence by Corollary \ref{prekskscor},
\begin{equation}
\label{6}
2^{-s(n)}\tilde\theta_n = -h_nR_n
\quad\text{in $(F^\times)^- \otimes \I_n \otimes \Z_p$ for every $n \in \mathcal{N}_p$}.
\end{equation}
If $p \mid n \in \mathcal{N}$, then Proposition \ref{crit}(iv) shows that
$(p-1)\I_n = 0$. Therefore $(F^\times)^- \otimes \I_n \otimes \Z_p = 0$ and
\eqref{6} holds vacuously in this case.
Since \eqref{6} holds for every $n \in \mathcal{N}$ and every odd prime $p$,
this completes the proof of Theorem \ref{mainthm}.
\end{proof}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,245
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This ultra-light, skin-friendly UV ray protection blocks UVA and UVB from harming your skin at SPF 30 and PA+++. TOTAL DAYTIME PROTECTION 30 is enhanced with Grape Seed and Green Tea Leaf Extracts for antioxidant properties.
High in antioxidant properties that help rejuvenate the skin, prevent damage from the sun and promote elasticity. It is also known to reduce skin swelling and help soothe and condition sensitive skin.
An effective and powerful antioxidant which can reduce the damage done by free radicals and ultraviolet radiation. It also works as emollient.
Apply as the last step in your facial skin care regimen and prior to makeup application. Spread evenly over entire face, avoiding immediate eye area.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,760
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{"url":"https:\/\/e-eduanswers.com\/mathematics\/question13567685","text":", 18.10.2019 06:10, COOLIOMARIS\n\n# \u200b\n\n### Other questions on the subject: Mathematics\n\nMathematics, 21.06.2019 14:30, smith3mgy\nSimonne used the following steps to simplify the given expression. 12 - 3(-2x + 4) step 1: 12 + (\u20133)\u00b7(\u20132x) + (\u20133)\u00b7(4) step 2: 12 + 6x + (\u201312) step 3: 12 + (\u201312) + 6x step 4: 0 + 6x step 5: 6x what property of real numbers was used to transition from step 3 to step 4? a. identity property of addition b. inverse property of addition c. associative property of addition d. commutative property of addition\n$$\\frac{3}{5} \\times \\frac{1}{2}$$","date":"2020-09-19 17:38:38","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 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\": 0.7762574553489685, \"perplexity\": 1382.3880652242312}, \"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-2020-40\/segments\/1600400192783.34\/warc\/CC-MAIN-20200919173334-20200919203334-00195.warc.gz\"}"}
| null | null |
Politics 22.11.2016 Bill Wirtz
The former French Minister of Economy, Emmanuel Macron, is going to run for president in the French elections in April 2017. In late August 2016, he had resigned in order to focus on his independent presidential bid. The former banker is not only fairly popular, but also known for his fearless reforms regarding economic liberalism, or at least what is called "economic liberalism" in France. Can he reverse the deadly spiral of deficit spending and tackle entitlement reform?
In August 2014, the government reshuffle of Prime Minister Manuel Valls, desired by president Hollande, was supposed to evict the remaining hardcore socialist members of the government. Minister of the Economy Arnaud Montebourg, known for his radical views, especially those regarding opposing globalisation, was replaced by a young and ambitious high-ranking government advisor: Emmanuel Macron (38), known as the "anti-Montebourg", would become the youngest member of the French executive in over 50 years. Being a former associate of the investment bank Rothschild & Cie, Macron is one of the few ministers with long-term experience in the private sector.
Macron's Law
Major legislative changes in France are appropriated to the initiator of the law. Macron's law (officially: Law for growth, effectiveness and equality of economic opportunities) contained a myriad of changes to legislation regarding Economic law, Labour law and Transport law. Macron opened up the intercity bus market, a measure that created competition on the market, lowered transportation costs and created 13,000 private sector jobs.
Among all the measures that Macron's Law contains, two were most controversial and ended up defining his two years in the French government.
For one, there was the reform of labour regulations regarding work on Sundays. Macron not only extended the exceptions made to allow businesses to open on Sundays, but also increased the total number of permits granted by local authorities. The second measure intended to induce flexibility into the profession of notaries, most importantly through creating 247 zones throughout France, in which notaries don't have to be sworn in by the government and can freely exercise their profession. This basically liberalises the notary market.
Reasons to be optimistic
A young, skilled and fearless independent politician with an appreciation for free markets, what's not to like? The liberalisations of Macron's Law turned out to be beneficial for market competition and offer: since the opening of the intercity bus-market began, the number of passengers using buses has multiplied by 7. With bus tickets only half the price of conventional train tickets, companies such as Flixbus, Ouibus, Isilines or Megabus have flooded the market with interesting offers for consumers. The real and tangible effect of this freeing of the market is pushing conservatives and independents further towards a laissez-faire attitude.
According to the latest polls, the former Minister of Economy would not qualify for the second round of the presidential elections, but would considerably shake things up by drawing from socialists, republicans and the far-right of Marine Le Pen.
Emmanuel Macron's image of fiscal conservatism and social liberalism is appealing to many voters who feel disenfranchised after decades of flat-out failures by both major parties, and in fear of the unknown that is the radical National Front. As a former Minister for the socialist party (who was chosen for his political insight and ambition), Macron lends a voice of reason to the left, all while pushing towards economic liberalism. Macron could become an important asset for free markets in upcoming administrations.
Reasons to be pessimistic
Even though some voters might come around to the advantages of liberalisation, let's not forget that thousands of people protested Macron's Law all throughout, with massive strikes even including certain parts of the judiciary. His law ended up splitting the socialist majority in parliament, so that prime minister Manuel Valls had to trigger article 49 paragraph 3, allowing him to pass a bill without the approval of the house, as long as the National Assembly does not pass a vote of no-confidence. Needless to say, Macron triggered a major uproar in his very own majority, which they prefer not to repeat.
France has lost its way when it comes to idealism: with the end of Charles De Gaulle, the dream of a charismatic leader, guiding the country in the right direction, is Utopian at best. The last three presidents embodied political frustration: the unpopularity and inability to act of the incumbent president is not really his terrible fate, it has become the underlying trademark of the job.
The bottom line is this: no politician can undergo the necessary changes if the people don't change their mind on the role of government. Deficit spending can only be reduced once people consider social services to be their own responsibility, not the one of the taxpayer. If the French public is unwilling to reconsider the size of government, then the upcoming of politicians like Macron will only be sporadic. They will make a step forward, but their successors will make three steps back.
Picture: Creative Commons Official LeWeb Photos
This piece solely expresses the opinion of the author and not necessarily the organization as a whole. Students For Liberty is committed to facilitating a broad dialogue for liberty, representing a variety of opinions. If you're a student interested in presenting your perspective on this blog, you can submit your own piece to submissions@speakfreely.today.
Tags: Elections France Macron Politics
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The Elephant in 775 Rooms: Buckingham Palace Renovations Exhibit Excessive Government Spending
Bill Wirtz is the former Editor-in-Chief of SpeakFreely. He works as policy analyst for the Consumer Choice Center.
Turkey deserves credit for its refugee policy
9.10.2017 Bartu Özden
According to my observations, Turkish politics has recently become a hot topic in the European media. It is surely both possible and necessary to...
History, Politics, Society
Cicero's Divinely Ordained Individualism
6.9.2018 Paul Meany
Cicero was a renowned Roman orator, statesman and writer. He was an enemy of one man rule and a self-described constitutionalist. During the...
5 Things You Should Know About The French Presidential Elections – European Students For Liberty – European Students For Liberty
[…] Emmanuel Macron, former minister of the Economy, quit the socialist government and is now running as an independent candidate. He is widely known for the Macron Law (officially: Law for growth, effectiveness and equality of economic opportunities). This law contained a myriad of changes to legislation regarding Economic law, Labor law and Transport law. Macron opened up the intercity bus market, a measure that created competition on the market, lowered transportation costs and created 13,000 private sector jobs. Furthermore, there was the reform of labor regulations regarding work on Sundays: Macron not only extended the exceptions made to allow businesses to open on Sundays, but also increased the total number of permits granted by local authorities. Another measure intended to induce flexibility into the profession of notaries, most importantly through creating 247 zones so-called "free establishment zones" throughout France, in which notaries don't have to be sworn in by the government and can freely exercise their profession. This basically liberalizes the notary market and brings down costs for consumers. […]
You Clicked on this Article because Marine Le Pen is in the Thumbnail – European Students For Liberty – European Students For Liberty
[…] has been taking on speed in recent weeks: former Minister of the Economy Emmanuel Macron is running as an independent, Thatcher-admiring François Fillon won the Republican primary, former president Nicolas Sarkozy […]
Leave a Reply to 5 Things You Should Know About The French Presidential Elections - European Students For Liberty - European Students For Liberty Cancel reply
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,008
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{"url":"https:\/\/mathematica.stackexchange.com\/questions\/201255\/inverse-matrix-not-computing","text":"# Inverse matrix not computing [closed]\n\nHi just wondering why my Inverse of matrix will not compute? I don't believe its wrapped from \/\/MatrixForm because I copied it by hand into a new document and it still would not work.\n\nAny ides?\n\nm2 = {{1, 1, 1, 1}, {0, Exp[L], Exp[0*L], Exp[0*L]}, {-1, 1, 0, 0}, {0, Exp[L], 0*Exp[-w2*L], 0*Exp[w2*L]}} Y = {0, f, 0, 0} c = Inverse[m2].Y\n\n\u2022 The error says your matrix is singular... this means there is no inverse. \u2013\u00a0bill s Jun 30 '19 at 2:21\n\u2022 And why would you keep in terms such as Exp[0 * L]? \u2013\u00a0David G. Stork Jun 30 '19 at 3:13\n\n$$\\left( \\begin{array}{cccc} 1 & 1 & 1 & 1 \\\\ 0 & e^L & 1 & 1 \\\\ -1 & 1 & 0 & 0 \\\\ 0 & e^L & 0 & 0 \\\\ \\end{array} \\right)$$","date":"2021-03-01 20:11:00","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\": 1, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.17180170118808746, \"perplexity\": 2474.731725432434}, \"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-2021-10\/segments\/1614178362899.14\/warc\/CC-MAIN-20210301182445-20210301212445-00302.warc.gz\"}"}
| null | null |
pip install mbed-cli
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,275
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Home science A massive asteroid shower hit Earth and the moon 800 million a long time back, study claims
A massive asteroid shower hit Earth and the moon 800 million a long time back, study claims
One particular such incident was a large asteroid shower that bombarded both of those the Earth and the moon 800 million many years in the past, according to a new review.
All through this monstrous shower, the asteroids that collided with Earth were being substantially larger than the asteroid liable for the extinction of the dinosaurs 66 million a long time in the past, the scientists imagine.
There is also evidence that an asteroid shower that impacted Earth 470 million many years ago could have brought on a fall in sea amount, icy conditions and contributed to biodiversity.
Speedy-ahead to 66 million several years back, and the Chicxulub effect crater, beneath the Yucatán Peninsula in Mexico and situated offshore around the city of Chicxulub, formed when a substantial meteorite between 6.8 and 50.3 miles in diameter hit the Earth.
This recently found celebration, nevertheless, that occurred 800 million several years back concerned an asteroid shower with a total mass involving 30 to 60 times that of the asteroid that established Chicxulub.
This affect occurred prior to the Cryogenien period of time between 635 million and 720 million decades back, when Earth was lined in icy deserts. This was an period of wonderful environmental and organic changes, the researchers claimed.
The study posted Tuesday in the journal Nature Communications.
This implies "that it is not peculiar that an asteroid shower 800 million years back could possibly have brought on the ice age, simply because a total mass flux 800 million decades in the past is 10 -100 occasions larger than people of Chicxulub impact and/or a meteoroid shower 470 million several years ago," mentioned Kentaro Terada, direct research author and professor at Osaka College in Japan, in an e-mail.
READ SpaceX to set another big milestone with Starling launch on busy day
Lunar craters tell a tale
Because of to erosion and resurfacing that happens on Earth brought on by volcanoes and other geologic processes, it's challenging for experts to study how our earth was impacted in the previous by asteroids and day when they occurred. Any affect craters to Earth prior to 600 million decades ago, scientists imagine, have been erased.
That's why the moon, which is mainly unchanged by erosion and weathering, has delivered a fruitful alternate route for scientists to analyze craters and piece collectively the shared record of Earth and the moon.
In this new examine, researchers utilised data collected by the Japanese House Agency's lunar orbiter Kaguya. Of the 59 craters on the moon that the researchers noticed that experienced diameters larger than 12.4 miles or 20 kilometers in diameter, 8 of the craters appear to have fashioned at the exact time. This consists of the Copernicus crater, which is 57.8 miles in diameter.
Apollo 12 astronauts, the next manned mission to land on the moon, sampled what they thought to be materials ejected from the Copernicus crater when it was designed. The samples, which they gathered soon after landing on November 19, 1969, had been dated to 800 million years outdated, in accordance to NASA.
These 8 craters probably shaped simultaneously when an asteroid 62 miles or 100 kilometers in diameter was disrupted, impacting equally Earth and the moon.
The setup for life on Earth?
All through the asteroid shower, a huge sum of phosphorus was delivered to Earth and "substantial quantities of risky things these as carbon, nitrogen and water ended up equipped on the area of the dry moon," Terada claimed.
READ Thanksgiving 2020: Here's what astronauts will eat in space (video)
Phosphorus could have acted as a dietary component to encourage a lot more algae advancement on Earth, Terada claimed. It really is also achievable that the arrival of factors through asteroids on Earth could have "motivated marine biogeochemical cycles, severe perturbations to Earth's local climate process and the emergence of animals," the authors wrote in the research.
The asteroid Eulalia, a C-variety asteroid in the asteroid belt among Mars and Jupiter, may perhaps have brought about the asteroid shower, the scientists theorized. C-variety asteroids incorporate carbon and are the most common asteroids in our solar procedure.
Eulalia could be the father or mother overall body of Ryugu, an asteroid explored and sampled by the Japanese Hayabusa2 mission. Samples from the asteroid are currently on their way again to Earth. Ryugu is a rubble pile asteroid formed like a spinning best. A rubble pile asteroid is a grouping of rocks held alongside one another by gravity relatively than solitary objects.
There are similarities between Eulalia and Ryugu's surface area that have prompt Ryugu could have once been component of Eulalia.
If anything brought on Eulalia to be disrupted, with fragments breaking off and forming different asteroids, it truly is feasible that this developed an asteroid shower that impacted Earth and the moon, even though also developing near-Earth asteroids.
Terada, a member of first characterization team of Ryugu sample, will help date the samples returned from Ryugu, which could ensure its "parent" and ascertain if it was made for the duration of this asteroid disruption event.
"If we get the age of 800 million many years from the Ryugu sample, I will be so enthusiastic," Terada mentioned.
READ Researchers reach big breakthrough in preserving integrity of audio waves
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How to blur people's faces in photos with signal
Hitman 3 is coming to Nintendo Switch on January 20th
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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The Black Georgians
Arts, culture and events
Black Cultural Archives have unveiled a five month exhibition, that features one of the most noteworthy events in Black history: The Georgian Era.
This new exhibition reveals the everyday lives of Black people in Britain during the Georgian period, 1714-1837. This is a critical historical enquiry that questions what we understand of Englishness and paints a picture of Black people's daily lives fraught with oppression and restriction amongst a degree of social mobility and integration.
This was the first era of large scale settlement of Blacks in Britain. Notable Black activists such as: Oluadah Equiano (pictured above); Ignatius Sancho; and Ottobah Cugoano all demanded freedom from slavery and were the definitive voices of the 18th Century. The Black Cultural Archives exhibition focuses on the lives of these activists and beyond while telling the story in a wider context using the themes: the beginning; life in Georgian Britain; death and departure; and the legacy.
One remarkable character that lived in the momentous 18th Century was Cesar Picton: born a slave in Senegal, but dying a rich man in Kingston upon Thames.
Treated as a commodity by Captain Parr, Cesar was given as a gift to Sir John Phillips, an abolitionist. Using the money he bequeathed from the Philips family, he set up a business and became a coal merchant in Kingston.
As his business grew, he bought more property and eventually lived the life of the gentleman and ultimately achieved financial security. When Cesar died, he left worldly goods to friends, showing even in the most oppressive climate, you can still come out on top.
Black Cultural Archives, 1 Windrush Square, Brixton, SW2 1EF.
For more information, visit www.bcaheritage.org.uk
Celebrating the launch of ELEVATE on the South Bank
Lambeth Council has launched ELEVATE, Lambeth's mission to support diverse talent and open up the creative workplace.
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{
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Кавалье, Ален (род. 1931) — французский кинорежиссёр, сценарист, кинооператор
Кавалье, Жан (1681—1740) — борец за свободу веры, главный предводитель камизаров в Севеннской войне, герцог Севеннский
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8">
<meta http-equiv="X-UA-Compatible" content="IE=edge">
<meta name="description" content="">
<meta name="author" content="">
<!-- Note there is no responsive meta tag here -->
<link rel="shortcut icon" href="../../docs-assets/ico/favicon.png">
<title>Non-responsive Template for Bootstrap</title>
<!-- Bootstrap core CSS -->
<!-- <link href="../../dist/css/bootstrap.css" rel="stylesheet"> -->
<link href="/classpath/fobo/bootstrap.css" rel="stylesheet" type="text/css" />
<!-- Custom styles for this template -->
<link href="non-responsive.css" rel="stylesheet">
<!-- Just for debugging purposes. Don't actually copy this line! -->
<!--[if lt IE 9]><script src="../../docs-assets/js/ie8-responsive-file-warning.js"></script><![endif]-->
<!-- HTML5 shim and Respond.js IE8 support of HTML5 elements and media queries -->
<!--[if lt IE 9]>
<script src="https://oss.maxcdn.com/libs/html5shiv/3.7.0/html5shiv.js"></script>
<script src="https://oss.maxcdn.com/libs/respond.js/1.3.0/respond.min.js"></script>
<![endif]-->
</head>
<body>
<!-- Fixed navbar -->
<div class="navbar navbar-default navbar-fixed-top" role="navigation">
<div class="container">
<div class="navbar-header">
<button type="button" class="navbar-toggle" data-toggle="collapse" data-target=".navbar-collapse">
<span class="icon-bar"></span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
</button>
<a class="navbar-brand" href="#">Project name</a>
</div>
<div class="navbar-collapse collapse">
<ul class="nav navbar-nav">
<li class="active"><a href="#">Home</a></li>
<li><a href="#about">About</a></li>
<li><a href="#contact">Contact</a></li>
<li class="dropdown">
<a href="#" class="dropdown-toggle" data-toggle="dropdown">Dropdown <b class="caret"></b></a>
<ul class="dropdown-menu">
<li><a href="#">Action</a></li>
<li><a href="#">Another action</a></li>
<li><a href="#">Something else here</a></li>
<li class="divider"></li>
<li class="dropdown-header">Nav header</li>
<li><a href="#">Separated link</a></li>
<li><a href="#">One more separated link</a></li>
</ul>
</li>
</ul>
<ul class="nav navbar-nav navbar-right">
<li><a href="#">Link</a></li>
<li><a href="#">Link</a></li>
<li><a href="#">Link</a></li>
</ul>
</div><!--/.nav-collapse -->
</div>
</div>
<div class="container">
<div class="page-header">
<h1>Non-responsive Bootstrap</h1>
<p class="lead">Disable the responsiveness of Bootstrap by fixing the width of the container and using the first grid system tier.</p>
</div>
<h3>What changes</h3>
<p>Note the lack of the <code><meta name="viewport" content="width=device-width, initial-scale=1.0"></code>, which disables the zooming aspect of sites in mobile devices. In addition, we reset our container's width and are basically good to go.</p>
<h3>Regarding navbars</h3>
<p>As a heads up, the navbar component is rather tricky here in that the styles for displaying it are rather specific and detailed. Overrides to ensure desktop styles display are not as performant or sleek as one would like. Just be aware there may be potential gotchas as you build on top of this example when using the navbar.</p>
<h3>Non-responsive grid system</h3>
<div class="row">
<div class="col-xs-4">One third</div>
<div class="col-xs-4">One third</div>
<div class="col-xs-4">One third</div>
</div>
</div> <!-- /container -->
<!-- Bootstrap core JavaScript
================================================== -->
<!-- Placed at the end of the document so the pages load faster -->
<!-- <script src="https://code.jquery.com/jquery-1.10.2.min.js"></script> -->
<!-- <script src="../../dist/js/bootstrap.min.js"></script> -->
<script id="jquery" src="/classpath/fobo/jquery.js" type="text/javascript"></script>
<script src="/classpath/fobo/bootstrap.js" type="text/javascript"></script>
</body>
</html>
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{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,598
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Q: Python random.shuffle() producing same result even after I call random.seed()? # N-Queens Problem
import random
# Define size of board
SIZE_OF_BOARD = 8
POPULATION_SIZE = 50
population = []
startingIndividual = []
# Populate starting individual
for x in range(0, SIZE_OF_BOARD):
startingIndividual.append(x)
for x in range(0, POPULATION_SIZE - 1):
population.append(startingIndividual)
for x in population:
random.seed()
random.shuffle(x)
print(population)
So, when I run this program I get the same permutation 50 times. Even though I reseed shuffle for each list I am shuffling, the result is still the same. Where exactly am I going wrong?
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
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using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
namespace Algorithms.Sorting
{
public class InsertionSorter
{
public int[] performInsertionSort(int[] A, int n) //This function will sort an array using insertion sort.s
{
int sortedIndex = 0;
for(int i=1; i<n; i++)
{
if(A[i]<A[sortedIndex])
{
int updatedIndex = i;
while (sortedIndex < n && sortedIndex >=0)
{
if (A[updatedIndex] < A[sortedIndex])
{
int temp = A[sortedIndex];
A[sortedIndex] = A[updatedIndex];
A[updatedIndex] = temp;
updatedIndex = sortedIndex;
}
sortedIndex -= 1;
}
}
sortedIndex = i;
}
return A;
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
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{"url":"https:\/\/www.semanticscholar.org\/paper\/Annealed-estimates-on-the-Green-function-Marahrens-Otto\/f0a4ac303afe01062b29cd5836821fca22c1d5cd","text":"# Annealed estimates on the Green function\n\n@article{Marahrens2013AnnealedEO,\ntitle={Annealed estimates on the Green function},\nauthor={Daniel Marahrens and Felix Otto},\njournal={Probability Theory and Related Fields},\nyear={2013},\nvolume={163},\npages={527-573}\n}\n\u2022 Published 16 April 2013\n\u2022 Mathematics\n\u2022 Probability Theory and Related Fields\nWe consider a random, uniformly elliptic coefficient field $$a(x)$$a(x) on the $$d$$d-dimensional integer lattice $$\\mathbb {Z}^d$$Zd. We are interested in the spatial decay of the quenched elliptic Green function $$G(a;x,y)$$G(a;x,y). Next to stationarity, we assume that the spatial correlation of the coefficient field decays sufficiently fast to the effect that a logarithmic Sobolev inequality holds for the ensemble $$\\langle \\cdot \\rangle$$\u27e8\u00b7\u27e9. We prove that all stochastic moments of the\u2026\n77 Citations\n\n### On annealed elliptic Green function estimates\n\n\u2022 Mathematics\n\u2022 2014\nWe consider a random, uniformly elliptic coefficient field $a$ on the lattice $\\mathbb{Z}^d$. The distribution $\\langle \\cdot \\rangle$ of the coefficient field is assumed to be stationary. Delmotte\n\n### Green\u2019s function for elliptic systems: existence and Delmotte\u2013Deuschel bounds\n\n\u2022 Mathematics\n\u2022 2016\nThis paper is divided into two parts: In the main deterministic part, we prove that for an open domain $$D \\subset \\mathbb {R}^d$$D\u2282Rd with $$d \\ge 2$$d\u22652, for every (measurable) uniformly elliptic\n\n### Green's function for elliptic systems: Moment bounds\n\n\u2022 Mathematics\nNetworks Heterog. Media\n\u2022 2018\nThe result implies optimal stochastic Gaussian bounds on the Green's function and its derivatives in the realm of homogenization of equations with random coefficient fields with finite range of dependence.\n\n### Berry\u2013Esseen theorem and quantitative homogenization for the random conductance model with degenerate conductances\n\n\u2022 Mathematics\nStochastics and Partial Differential Equations: Analysis and Computations\n\u2022 2018\nWe study the random conductance model on the lattice $${\\mathbb {Z}}^d$$Zd, i.e. we consider a linear, finite-difference, divergence-form operator with random coefficients and the associated random\n\n### Normal approximation for the net flux through a random conductor\n\nWe consider solutions of an elliptic partial differential equation in $${\\mathbb R}^d$$Rd with a stationary, random conductivity coefficient. The boundary condition on a square domain of width L is\n\n### Quantitative estimates in stochastic homogenization for correlated coefficient fields\n\n\u2022 Mathematics\nAnalysis & PDE\n\u2022 2021\nThis paper is about the homogenization of linear elliptic operators in divergence form with stationary random coefficients that have only slowly decaying correlations. It deduces optimal estimates of\n\n### Asymptotics of parabolic Green's functions on lattices\n\nFor parabolic spatially discrete equations, we consider Green's functions, also known as heat kernels on lattices. We obtain their asymptotic expansions with respect to powers of time variable $t$ up\n\n### H\u00f6lder regularity for a non-linear parabolic equation driven by space-time white noise\n\n\u2022 Mathematics\n\u2022 2015\nWe consider the non-linear equation $T^{-1} u+\\partial_tu-\\partial_x^2\\pi(u)=\\xi$ driven by space-time white noise $\\xi$, which is uniformly parabolic because we assume that $\\pi'$ is bounded away\n\n### Corrector Estimates for Elliptic Systems with Random Periodic Coefficients\n\n\u2022 Mathematics\nMultiscale Model. Simul.\n\u2022 2016\nTwo different approaches are used to obtain moment bounds on the gradient of the corrector, independent of the domain size, for the second order elliptic system associated with random coefficients that are assumed to be coercive and stationary.\n\n## References\n\nSHOWING 1-10 OF 32 REFERENCES\n\n### On homogenization of elliptic equations with random coefficients\n\n\u2022 Mathematics, Computer Science\n\u2022 2000\nThe upper bound of $\\varepsilon^\\alpha$ for the rate of convergence is obtained, where $\\alpha$ is a constant which depends on the dimension $d\\ge 2$ and the deviation of $a(x,\\omega)$ from the identity matrix.\n\n### An optimal variance estimate in stochastic homogenization of discrete elliptic equations\n\n\u2022 Mathematics\n\u2022 2011\nWe consider a discrete elliptic equation with random coefficients $A$, which (to fix ideas) are identically distributed and independent from grid point to grid point $x\\in\\mathbb{Z}^d$. On scales\n\n### Normal approximation for a random elliptic equation\n\nWe consider solutions of an elliptic partial differential equation in $$\\mathbb{R }^d$$Rd with a stationary, random conductivity coefficient that is also periodic with period $$L$$L. Boundary\n\n### Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics\n\n\u2022 Mathematics\n\u2022 2013\nWe study quantitatively the effective large-scale behavior of discrete elliptic equations on the lattice $$\\mathbb Z^d$$Zd with random coefficients. The theory of stochastic homogenization relates\n\n### A Central Limit Theorem for the Effective Conductance: Linear Boundary Data and Small Ellipticity Contrasts\n\n\u2022 Mathematics\n\u2022 2014\nGiven a resistor network on $${\\mathbb{Z}^d}$$Zd with nearest-neighbor conductances, the effective conductance in a finite set with a given boundary condition is the minimum of the Dirichlet energy\n\n### An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations\n\n\u2022 Mathematics\n\u2022 2014\nWe establish an optimal, linear rate of convergence for the stochastic homogenization of discrete linear elliptic equations. We consider the model problem of independent and identically distributed\n\n### Disorder, entropy and harmonic functions\n\n\u2022 Mathematics\n\u2022 2011\nWe study harmonic functions on random environments with particular emphasis on the case of the infinite cluster of supercritical percolation on $\\mathbb{Z}^d$. We prove that the vector space of\n\n### A quantitative two-scale expansion in stochastic homogenization of discrete linear elliptic equations\n\n\u2022 Mathematics\n\u2022 2013\nWe establish an optimal, linear rate of convergence for the stochastic homogenization of discrete linear elliptic equations. We consider the model problem of independent and identically distributed\n\n### Estimates On the Variance of Some Homogenization Problems\n\n\u2022 Mathematics\n\u2022 1998\nWe use`p estimates together with Brascamp-Lieb inequalities to obtain bounds on the variance of the solution u\" to the elliptic equation r\" a(x=\"; ')r\"u\" = r\" f . The variance is shown to be O(\"r)\n\n### Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane (Pms-48)\n\n\u2022 Mathematics\n\u2022 2009\nThis book explores the most recent developments in the theory of planar quasiconformal mappings with a particular focus on the interactions with partial differential equations and nonlinear analysis.","date":"2022-09-26 00:24:13","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\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8532209396362305, \"perplexity\": 1020.2628443118164}, \"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-2022-40\/segments\/1664030334620.49\/warc\/CC-MAIN-20220925225000-20220926015000-00347.warc.gz\"}"}
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\section{Introduction}
A connected graph is a {\em cactus} if every edge belongs to at most one cycle.
A {\em cactus forest} is a graph such that every connected component is a cactus.
In this paper, we consider the following problem.
\begin{definition}[{\sc Cactus Vertex Deletion}]
Given a graph $G = (V, E)$ and an integer $k \ge 0$, the problem asks whether $G$ has a vertex set $X \subseteq V$ with $|X| \le k$ whose removal leaves a cactus forest.
\end{definition}
The problem is one of {\em vertex deletion problems for hereditary properties}, which have been both intensively and extensively studied in the field of parameterized algorithms and complexity.
The best known problem in this context is {\sc Vertex Cover}.
The problem asks whether there is a vertex set of size at most $k$ whose removal leaves an edge-less graph.
A naive algorithm solves {\sc Vertex Cover} in $O^*(2^k)$ time\footnote{The notation $O^*$ suppresses a polynomial factor of the input size.}, and after a series of improvements, the fastest known algorithm is due to Chen et al. \cite{ChenKX10:Improved}, which runs in time $O^*(1.2738^k)$.
Another example of this kind of problems is {\sc Feedback Vertex Set}.
The problem asks whether an input graph $G = (V, E)$ has a vertex set of size at most $k$ that hits all the cycles in the graph.
In other words, the goal of this problem is to compute $X \subseteq V$ with $|X| \le k$ such that the graph obtained from $G$ by deleting $X$ is a forest.
This problem is also intensively studied, and several deterministic and randomized algorithms have been proposed so far~\cite{Becker:Randomized:2000,Cygan:Solving:2011,Downey:Fixed:1995,Guo:Compression:2006,IwataK19:Improved,LiN20:Detecting}.
The current best running time is due to Iwata and Kobayashi~\cite{IwataK19:Improved} for deterministic algorithms and Li and Nederlof~\cite{LiN20:Detecting} for randomized algorithms, which run in time $O^*(3.460^k)$ and $O^*(2.7^k)$, respectively.
The gap between the running time of deterministic and randomized algorithms sometimes emerges for vertex deletion problems to ``sparse'' hereditary classes of graphs, such as {\sc Feedback Vertex Set}.
For instance, {\sc Pseudo Forest Vertex Deletion} can be solved deterministically in time $O^*(3^k)$~\cite{BodlaenderOO18:faster} and randomizedly in time $O^*(2.85^k)$~\cite{GowdaLPPS20:Improved} and {\sc Bounded Degree-$2$ Vertex Deletion} can be solved deterministically in time $O^*(3.0645^k)$~\cite{Xiao16:parameterized} and randomizedly in time $O^*(3^k)$~\cite{FengWLC15:Randomized}.
Among others, the known gap on {\sc Cactus Vertex Deletion} is remarkable: Bonnet et al.~\cite{BonnetBKM16:Parameterized} presented a deterministic $O^*(26^k)$-time algorithm, while Kolay et al.~\cite{KolayLPS17:Quick} presented a randomized $O^*(12^k)$-time algorithm.
In this paper, we narrow the gap between the running time of deterministic and randomized algorithms by giving an improved deterministic algorithm for {\sc Cactus Vertex Deletion}.
\begin{theorem}\label{thm:CVD}
{\sc Cactus Vertex Deletion} can be solved deterministically in time $O^*(17.64^k)$.
\end{theorem}
As a variant of {\sc Cactus Vertex Deletion}, we consider {\sc Even Cycle Transversal} defined as follows.
A cactus is called an {\em odd cactus} if every cycle in it has an odd number of vertices.
\begin{definition}[{\sc Even Cycle Transversal}]
Given a graph $G = (V, E)$ and an integer $k \ge 0$, the problem asks whether $G$ has a vertex set $X \subseteq V$ with $|X| \le k$ whose removal leaves a forest of odd cacti.
\end{definition}
Note that a graph has no cycles of even length if and only if it is a forest of odd cacti~\cite{KolayLPS17:Quick}.
Kolay et al.~\cite{KolayLPS17:Quick} gave an $O^*(12^k)$-time randomized algorithm and Misra et al.~\cite{MisraRRS12:Parameterized} gave an $O^*(50^k)$-time deterministic algorithm for {\sc Even Cycle Transversal}.
In this paper, we improve the running time of the deterministic algorithm for {\sc Even Cycle Transversal} as well as {\sc Cactus Vertex Deletion}.
\begin{theorem}\label{thm:d-CVD-ECT}
{\sc Even Cycle Transversal} can be solved deterministically in time $O^*(17.64^k)$.
\end{theorem}
The idea of our algorithms follows that used in \cite{BonnetBKM16:Parameterized}.
We solve the disjoint version of {\sc Cactus Vertex Deletion} with a branching algorithm.
In this version, given a vertex subset $S \subseteq V$ such that $|S| \le k + 1$ and the subgraph induced by $V \setminus S$, denoted $G[V \setminus S]$, is a cactus forest, the problem asks whether there is a vertex subset $X \subseteq V \setminus S$ such that $|X| \le k$ and $G[V \setminus X]$ is a cactus forest.
To solve this problem, Bonnet et al.~\cite{BonnetBKM16:Parameterized} gave a branching algorithm with the measure and conquer analysis~\cite{FominGK09:measure}.
They used measure $k + {\tt cc}(G[S])$, where ${\tt cc}(G[S])$ is the number of connected components in $G[S]$, and proved that each branch of their algorithm strictly decreases this measure.
The main difficulty with using this measure is that when we consider a vertex $v \in V \setminus S$ such that $v$ has at least two neighbors only in a single connected component in $G[S]$, then one of the branch, for which $v$ is determined to be not deleted, does not decrease the measure.
We also use the measure and conquer analysis with a slightly elaborate measure $\alpha k + \beta \cdot {\tt cc}(G[S]) + \gamma\cdot{\tt b}(G[S])$, where $\alpha,\beta,\gamma$ are some constants and ${\tt b}(G[S])$ is the number of bridges in $G[S]$, which allows us to decrease the measure efficiently: When $v$ is determined to be not deleted, the number of bridges in $G[S]$ is decreased in the above situation since otherwise $v$ belongs to a $K_4 - e$ minor.
We believe that although our measure is slightly involved compared to that in \cite{BonnetBKM16:Parameterized}, the algorithm itself and its analysis would be simpler than theirs.
\section{Preliminaries}
\paragraph{Graphs.}
Throughout the paper, graphs have no self-loops but may have multiedges.
Let $G = (V, E)$ be a graph.
We write $V(G)$ and $E(G)$ to denote the sets of vertices and edges of $G$, respectively.
For two distinct vertices $u, v$ in $G$, we denote by $m(u, v)$ the number of edges between $u$ and $v$.
Let $v \in V$.
The {\em degree} of $v$ is the number of edges incident to it.
We denote by $N_G(v)$ the set of neighbors of $v$ in $G$.
Note that as $G$ may have multiedges, $|N_G(v)|$ may not be equal to its degree.
For $X \subseteq V$, the subgraph of $G$ induced by $X$ is denoted as $G[X]$.
We denote by ${\tt cc}(G)$ the number of connected components in $G$.
A vertex $v \in V$ is called a {\em cut vertex} of $G$ if ${\tt cc}(G[V \setminus \{v\}]) > {\tt cc}(G)$ and an edge $e \in E$ is called a {\em bridge} of $G$ if ${\tt cc}(G - e) > {\tt cc}(G)$, where $G - e$ is the graph obtained from $G$ by deleting $e$.
Note that it holds that ${\tt cc}(G - e) = {\tt cc}(G) + 1$ when $e$ is a bridge of $G$.
The number of bridges in $G$ is denoted by ${\tt b}(H)$.
\begin{lemma}\label{lem:cc+b}
Let $H$ be a multigraph with $h$ vertices.
Then, it holds that ${\tt cc}(H) + {\tt b}(H) \le h$.
\end{lemma}
\begin{proof}
Let $H'$ be the graph obtained from $H$ by removing all bridges of $H$.
Then, ${\tt cc}(H) + {\tt b}(H) = {\tt cc}(H') \le h$.
\qed\end{proof}
A {\em block} of a graph $G$ is a maximal vertex set $B$ of $G$ such that $G[B]$ is connected and has no cut vertices.
Note that a graph consisting of two vertices with at least one edge is a block.
It is easy to see that every block in a cactus forest is either a cycle, an edge, or an isolated vertex.
In particular, we call $B$ a {\em leaf block} if it has at most one cut vertex.
We say that vertices $v_1, \ldots, v_t \in V(B)$ are {\em consecutive} in $B$ if for each $1 \le i < t$, $v_i$ is adjacent to $v_{i+1}$ in $B$.
\paragraph{Iterative compression.}
Our algorithm employs the well-known {\em iterative compression} technique invented by Reed, Smith, and Vetta~\cite{ReedSV04:Finding}.
They gave an algorithm for {\sc Odd Cycle Transversal} based on this technique.
The essential idea can be generalized as follows.
Let $\mathcal C$ be a hereditary class of graphs, that is, for $G \in \mathcal C$, every induced subgraph of $G$ also belongs to $\mathcal C$.
The technique is widely used for designing algorithms of vertex deletion problems to hereditary classes of graphs.
The crux of the technique can be described as the following lemma.
\begin{lemma}[\cite{ReedSV04:Finding}]\label{lem:compression}
Let $\mathcal C$ be a hereditary class of graphs.
Given a graph $G = (V, E)$ and an integer $k$, the problem of computing $X \subseteq V$ with $|X| \le k$ such that $G[V \setminus X] \in \mathcal C$ can be solved in time $O^*((c + 1)^k)$ if one can solve the following problem in time $O^*(c^k)$:
Given a subset $S \subseteq V$ of cardinality at most $k + 1$ with $G[V \setminus S] \in \mathcal C$, the problem asks to find $X \subseteq V \setminus S$ with $|X| \le k$ such that $G[V \setminus X] \in \mathcal C$.
\end{lemma}
For {\sc Cactus Vertex Deletion}, the latter problem is defined as follows.
\begin{definition}[{\sc Disjoint Cactus Vertex Deletion}]
Given a graph $G = (V, E)$, an integer $k \ge 0$, and $S \subseteq V$ such that $G[V \setminus S]$ is a cactus forest, the problem asks to find a vertex set $X \subseteq V \setminus S$ with $|X| \le k$ whose removal leaves a cactus forest.
\end{definition}
Let us note that we can assume that $G[S]$ is also a cactus forest as otherwise the problem is trivially infeasible.
\paragraph{Measure and conquer analysis.}
Our algorithm for {\sc Disjoint Cactus Vertex Deletion} is based on a standard branching algorithm with the measure and conquer analysis~\cite{FominGK09:measure}.
Given an instance $I$ of the problem, we define a measure $\mu(I)$ that is non-negative real and design a branching algorithm that generates subinstances $I_1, \ldots, I_t$ with $\mu(I) > \mu(I_i)$ for $1 \le i \le t$.
To measure the running time of the algorithm, we use a branching factor $(b_1, \ldots, b_t)$, where $\mu(I) - \mu(I_i) \ge b_i$ for each $i$.
It is known that the total running time of this branching algorithm is upper bounded by $O^*(c^{\mu(I)})$, where $c$ is the unique positive real root of equation
\begin{align*}
x^{-b_1} + x^{-b_2} + \cdots + x^{-b_t} = 1,
\end{align*}
assuming that from any instance $I$ with $\mu(I) > 0$, its subinstances can be generated in polynomial time and for any instance $I$ with $\mu(I) = 0$, the problem can be solved in polynomial time.
We refer the reader to the book~\cite{FominD10:Exact} for a detailed exposition for the measure and conquer analysis.
\section{An improved algorithm for {\sc Disjoint Cactus Vertex Deletion}}
This section is devoted to developing an algorithm for {\sc Disjoint Cactus Vertex Deletion} that runs in time $O^*(16.64^k)$, proving Theorem~\ref{thm:CVD} by Lemma~\ref{lem:compression}.
\begin{lemma}\label{lem:DCVD}
Suppose that $|S| \le k + 1$.
Then, {\sc Disjoint Cactus Vertex Deletion} can be solved in time $O^*(16.64^k)$.
\end{lemma}
Let $I = (G, S, k)$ be an instance of {\sc Disjoint Cactus Vertex Deletion}, where $G = (V, E)$ is a multigraph, $S \subseteq V$.
Recall that we assume $G[V \setminus S]$ and $G[S]$ are both cactus forests as otherwise the problem is trivially infeasible.
Let $\mu(I) = \alpha\cdot k + \beta\cdot{\tt cc}(G[S]) + \gamma\cdot{\tt b}(G[S])$, where $\alpha,\beta,\gamma$ are chosen later.
In the following, we assume that $\beta \ge \gamma$.
For the sake of simplicity, we write, for $X \subseteq V$, ${\tt cc}(X)$ and ${\tt b}(X)$ to denote ${\tt cc}(G[X])$ and ${\tt b}(G[X])$, respectively.
As $G$ may have multiedges, every cactus forest can be characterized as the following form.
\begin{proposition}[\cite{FioriniJP10:Hitting}]
Let $D$ be the graph of two vertices and three parallel edges between them.
A graph is a cactus forest if and only if it does not contain a subgraph isomorphic to any subdivision of $D$.
\end{proposition}
We call a subdivision of $D$ an {\em obstruction}.
In particular, $D$ itself is also an obstruction.
The algorithm consists of several branching rules and reduction rules.
We say that a reduction rule is {\em safe} if the original instance has a yes-instance if and only if the instance obtained by applying the rule is a yes-instance.
We also say that a branching rule is {\em safe} if the original instance is a yes-instance if and only if at least one of the instances obtained by applying the rule is a yes-instance.
Our algorithm described below {\em determines} whether $G$ has a solution $X$ for {\sc Disjoint Cactus Vertex Deletion}.
However, the algorithm easily turns into one that {\em finds} an actual solution if the answer is affirmative.
We apply these rules in the order of their appearance.
The algorithm terminates if $V(G) = S$ or $k = 0$, and it answers ``YES'' if and only if $k \ge 0$ and $G$ is a cactus forest.
The following reduction and branching rules are trivially safe.
\begin{reduce}\label{rr:isolated-component}
If $G[V \setminus S]$ contains a component $C$ that has no neighbors in $S$, then delete all the vertices in $C$.
\end{reduce}
\begin{reduce}\label{rr:deg-1}
If $G[V \setminus S]$ contains a vertex of degree one in $G$, then delete it.
\end{reduce}
\begin{reduce}\label{rr:must-be-in-solution}
If $G[V \setminus S]$ contains a vertex $v$ such that $G[S \cup \{v\}]$ is not a cactus forest, then delete $v$ and decrease $k$ by one.
\end{reduce}
\begin{branch}\label{br:must-be-in-solution}
If $G[V \setminus S]$ contains vertices $u, v \in V \setminus S$ with $m(u, v) \ge 3$, branch into two cases: (1) delete $u$ and decrease $k$ by one; (2) delete $v$ and decrease $k$ by one.
\end{branch}
The branching factor of Branching rule~\ref{br:must-be-in-solution} is $(\alpha, \alpha)$.
By applying these rules, we make the following assumption on each vertex in $V \setminus S$.
\begin{assume}
Every vertex $v \in V \setminus S$ has degree at least two in $G$ and there are at most two edges between two vertices.
\end{assume}
As $G$ is a multigraph, some vertex may have only one neighbor even if its degree is greater than one.
If $G[V \setminus S]$ contains a vertex $v$ with $|N_G(v)| = 1$, this vertex also can be removed since it is not a part of an obstruction, assuming that $m(u, v) \le 2$ with $u \in N_G(v)$.
This implies the following reduction rule.
\begin{reduce}\label{rr:neighbor-1}
If $G[V \setminus S]$ contains a vertex $v$ with $|N_G(v)| = 1$, then delete it.
\end{reduce}
Thus, we further make the following assumption on each vertex in $V \setminus S$.
\begin{assume}\label{assume:al2}
Every vertex $v \in V \setminus S$ has at least two neighbors in $G$.
\end{assume}
Suppose that there is a vertex $v \in V \setminus S$ that has at least two neighbors in $S$.
By Reduction rule~\ref{rr:must-be-in-solution}, there is no component in $G[S]$ that contains at least three vertices of $N_G(v) \cap S$.
Let $W = N_G(v) \cap S$.
We denote by $t_1$ (resp. by $t_2$) the number of components in $G[S]$ that contain exactly one vertex (resp. two vertices) of $W$.
Let $C$ be a component in $G[S]$ that has at least one vertex of $W$.
If $|W \cap C| = 2$, say $w, w' \in W \cap C$, every edge on the path between $w$ and $w'$ in $G[C]$ is a bridge as otherwise $G[C \cup \{v\}]$ contains an obstruction, which implies that $v$ is removed by Reduction rule~\ref{rr:must-be-in-solution}.
Then, there is at least one bridge on the path between $w$ and $w'$ in $G[C]$.
Thus, ${\tt b}(C \cup \{v\}) \le {\tt b}(C) - 1$.
If $|W \cap C| = 1$, $G[C \cup \{v\}]$ has ${\tt b}(C) + 1$ bridges.
Hence, we have
\begin{align*}
\beta \cdot {\tt cc}(S \cup \{v\}) &\le \beta \cdot {\tt cc}(S) - \beta(t_1 + t_2 - 1), \\
\gamma \cdot {\tt b}(S \cup \{v\}) & \le \gamma \cdot{\tt b}(S) + \gamma(t_1 - t_2).
\end{align*}
Consider the value $t_1(\beta - \gamma) + t_2(\beta + \gamma) - \beta$, that is, a lower bound of $\mu((G, S, k)) - \mu((G, S \cup \{v\}, k))$.
If $t_1 + t_2 \ge 2$, the value is at least $\beta - 2\gamma$.
This follows from the fact that the value is minimized when $t_1 = 2$ and $t_2 = 0$ under $\beta \ge \gamma \ge 0$.
If $t_1 + t_2 = 1$, $t_1$ must be zero since $|W| \ge 2$.
In this case, the value is at least $\gamma$.
This implies the following branching rule, which is clearly safe, has branching factor $(\alpha, \min(\beta - 2\gamma, \gamma))$.
\begin{branch}\label{br:v-has-two-neighbors-in-two-components}
Suppose $G[V \setminus S]$ contains a vertex $v$ that has at least two neighbors in $S$.
Then, branch into two cases: (1) delete $v$ and decrease $k$ by one; (2) put $v$ into $S$.
\end{branch}
Thus, we make the following assumption on each vertex in $V \setminus S$.
\begin{assume}\label{assume:at-most-1-in-S}
Every vertex $v \in V \setminus S$ has at least two neighbors in $G$ and at most one of them belongs to $S$.
\end{assume}
We can remove a vertex having exactly two neighbors by adding an edge between its neighbors.
The following lemma justifies this reduction.
\begin{lemma}\label{lem:eliminate-deg2}
Let $v \in V \setminus S$ be a vertex with exactly two neighbors $u, w$ in $G$.
Suppose that $p = \max(m(u, v), m(v, w)) \le 2$.
Let $G'$ be the graph obtained from $G$ by deleting $v$ and adding $p$ parallel edges between $u$ and $w$.
Then, $G$ has a cactus deletion set of size at most $k$ if and only if $G'$ has a cactus deletion set of size at most $k$.
\end{lemma}
\begin{proof}
Since every obstruction in $G$ containing $v$ also has both $u$ and $w$, there is a smallest cactus deletion set $X$ that does not contain $v$.
Such a set is also a cactus deletion set of $G'$ and vise versa.
\end{proof}
By Lemma~\ref{lem:eliminate-deg2}, the following reduction rule is safe.
\begin{reduce}\label{rr:deg-2}
Suppose that $G[V \setminus S]$ contains a vertex $v$ with $N_G(v) = \{u, w\}$.
Then delete $v$ and add $\max(m(u, v), m(v, w))$ parallel edges between $u$ and $w$.
\end{reduce}
This implies that the following assumption is made.
\begin{assume}\label{assume:deg-3}
Every vertex $v \in V \setminus S$ has at least three neighbors.
Moreover, at most one of them belongs to $S$ and hence at least two of them belong to $V \setminus S$.
\end{assume}
Since $G[V \setminus S]$ is a cactus forest, there is a leaf block $B$.
By Assumption~\ref{assume:deg-3}, $B$ contains at least three vertices.
Suppose that $B$ has exactly three vertices $u, v, w$.
As $B$ is a leaf block, we can assume that both $u$ and $v$ are not cut vertices of $G[V \setminus S]$.
By Assumption~\ref{assume:at-most-1-in-S}, both $u$ and $v$ have exactly one neighbor in $S$, which can be an identical vertex.
If there is a component $C$ in $G[S]$ that contains both a neighbor of $u$ and a neighbor of $v$, then $G[C \cup \{u, v, w\}]$ has an obstruction, which yields the following branching rule with branching factor $(\alpha, \alpha, \alpha)$.
\begin{branch}\label{br:k3-single-comp}
Suppose that there is a leaf block $B$ with $V(B) = \{u, v, w\}$ in $G[V \setminus S]$.
Suppose moreover that each of $u$ and $v$ has exactly one neighbor in $S$ and that these neighbors belong to a single component in $G[S]$.
Then, branch into three cases: (1) delete $u$; (2) delete $v$; (3) delete $w$.
For each case, decrease $k$ by one.
\end{branch}
Otherwise, the neighbors of $u$ and $v$ belong to distinct components in $G[S]$.
Let $C_u$ and $C_v$ be the components of $G[S]$ that have neighbors of $u$ and $v$, respectively.
If $w$ has a neighbor in $C_u$ or $C_v$, then $G[S \cup \{u, v, w\}]$ contains an obstruction.
In this case, we apply Branching rule~\ref{br:k3-single-comp} as well.
Thus, either $w$ has no neighbor in $S$ or $w$ has exactly one neighbor in a component $C_w$ in $G[S]$ with $C_w \neq C_u$ and $C_w \neq C_v$.
In both cases, we apply the following branching rule.
\begin{branch}\label{br:k3-distinct-comp}
Suppose that there is a leaf block $B$ with $V(B) = \{u, v, w\}$ in $G[V \setminus S]$.
Suppose moreover that each of $u$ and $v$ has exactly one neighbor in $S$ and that these neighbors belong to distinct components in $G[S]$.
Then, branch into four cases: (1) delete $u$; (2) delete $v$; (3) delete $w$; (4) put $u$, $v$, and $w$ into $S$.
For (1), (2), and (3), decrease $k$ by one.
\end{branch}
To see the branching factor of this rule, suppose first that $w$ has no neighbor in $S$.
Then, $G[S \cup \{u, v, w\}]$ contains ${\tt cc}(S) - 1$ components and ${\tt b}(S) + 2$ bridges.
Thus, Branching~rule~\ref{br:k3-distinct-comp} has branching factor $(\alpha, \alpha, \alpha, \beta - 2\gamma)$.
Suppose otherwise that $w$ has an exactly one neighbor in a component $C_w$ in $G[S]$ with $C_w \neq C_u$ and $C_w \neq C_v$.
Then, $G[S \cup \{u, v, w\}]$ contains ${\tt cc}(S) - 2$ components and ${\tt b}(S) + 3$ bridges.
Thus, Branching~rule~\ref{br:k3-distinct-comp} has branching factor $(\alpha, \alpha, \alpha, 2\beta - 3\gamma)$.
By Branching rules~\ref{br:k3-single-comp} and \ref{br:k3-distinct-comp}, the following assumption is made.
\begin{assume}\label{assume:leaf-block-size-4}
Every leaf block $B$ in $G[V \setminus S]$ contains three consecutive vertices, each of which is not a cut vertex in $G[V \setminus S]$ and has exactly one neighbor in $S$.
\end{assume}
Let $u, v, w$ be three consecutive vertices in $B$, each of which is not a cut vertex in $G[V \setminus S]$ and has exactly one neighbor in $S$.
Let $u', v', w'$ be the neighbors of $u, v, w$ in $S$, respectively.
There are four cases~(Figure~\ref{fig:c4}).
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{c4.pdf}
\caption{An Illustration of four branching cases under Assumption~\ref{assume:leaf-block-size-4}.}
\label{fig:c4}
\end{figure*}
Suppose that there is a component $C$ in $G[S]$ that contains these neighbors ((a) in Figure~\ref{fig:c4}).
Then, $G[C \cup \{u, v, w\}]$ has an obstruction, yielding the following Branching rule~\ref{br:non-k3-single-comp} that has branching factor $(\alpha, \alpha, \alpha)$.
\begin{branch}\label{br:non-k3-single-comp}
Suppose that there is a leaf block $B$ with $|V(B)| \ge 4$ in $G[V \setminus S]$.
Let $u, v, w$ be three consecutive vertices in $B$, each of which is not a cut vertex in $G[V \setminus S]$ and has exactly one neighbor in $S$.
Suppose that these neighbors belong to a single component in $G[S]$.
Then, branch into three cases: (1) delete $u$; (2) delete $v$; (3) delete $w$.
For each case, decrease $k$ by one.
\end{branch}
Suppose next that exactly two of $u', v', w'$ are contained in a single component $C$ in $G[S]$.
There are essentially two cases: (1) $u' $ and $v'$ are contained in $C$ ((b) in Figure~\ref{fig:c4}) or (2) $u'$ and $w'$ are contained in $C$ ((c) in Figure~\ref{fig:c4}).
In case (1), $G[S \cup \{u, v, w\}]$ contains ${\tt cc}(S) - 1$ components and ${\tt b}(S) + 2$ bridges.
In case (2), $G[S \cup \{u, v, w\}]$ contains ${\tt cc}(S) - 1$ components and ${\tt b}(S) + 1$ bridges.
For these cases, we apply the following Branching~rule~\ref{br:non-k3-distinct-comp}, which has branching factors $(\alpha, \alpha, \alpha, \beta - 2\gamma)$ and $(\alpha, \alpha, \alpha, \beta - \gamma)$ for these cases.
\begin{branch}\label{br:non-k3-distinct-comp}
Suppose that there is a leaf block $B$ with $|V(B)| \ge 4$ in $G[V \setminus S]$.
Let $u, v, w$ be three consecutive vertices in $B$, each of which is not a cut vertex in $G[V \setminus S]$ and has exactly one neighbor in $S$.
Suppose that these neighbors are not contained in a single component in $G[S]$.
Then, branch into four cases: (1) delete $u$; (2) delete $v$; (3) delete $w$; (4) put $u$, $v$, and $w$ into $S$.
For (1), (2), and (3), decrease $k$ by one.
\end{branch}
Finally, suppose any two of $u', v', w'$ are not contained in a single component in $G[S]$ ((d) in Figure~\ref{fig:c4}).
Again, we apply Branching~rule~\ref{br:non-k3-distinct-comp} to this case.
Since $G[S \cup \{u, v, w\}]$ contains ${\tt cc}(S) - 2$ components and ${\tt b}(S) + 5$ bridges, Branching~rule~\ref{br:non-k3-distinct-comp} has branching factor $(\alpha, \alpha, \alpha, 2\beta - 5\gamma)$.
The entire algorithm for {\sc Disjoint Cactus Vertex Deletion} is given in Algorithm~\ref{alg:code}.
The reduction and branching rules cover all cases for the instance $I$ and all the rules are safe.
Thus, the algorithm correctly computes a cactus deletion set $X \subseteq V \setminus S$ with $|X| \le k$ if it exists.
By choosing $\alpha = 1$, $\beta = 0.4052$, $\gamma = 0.0726$, the running time is dominated by the branching factor $(\alpha, \alpha, \alpha, \beta - 2\gamma) = (1, 1, 1, 0.26)$.
By Lemma~\ref{lem:cc+b}, we have $\beta \cdot {\tt cc}(S) + \gamma \cdot {\tt b}(S) \le \beta (k + 1)$.
Therefore, the running time of the algorithm is
\begin{align*}
O^*(c^{\mu(I)}) &\subseteq O^*(c^{\alpha \cdot k + \beta \cdot {\tt cc}(S) + \gamma \cdot {\tt b}(S)}) \\
&\subseteq O^*(c^{1.4052k}),
\end{align*}
where $c < 7.3961$ is the unique positive real root of equation $3x^{-1} + x^{-0.26} = 1$.
This yields the running time bound $O^*(16.64^k)$ for {\sc Disjoint Cactus Vertex Deletion}.
\begin{algorithm}[h]
\caption{A pseudocode of the algorithm for {\sc Disjoint Cactus Vertex Deletion}}\label{alg:code}
\begin{algorithmic}
\Procedure{\tt DCVD}{$G = (V, E), S, k$}
\If{$k \ge 0$ and $V = S$}
\State {\bf return} {\bf true}
\EndIf
\If{$k < 0$}
\State {\bf return} {\bf false}
\EndIf
\If{$G[V \setminus S]$ has a component $C$ that has no neighbors in $S$} \Comment{Reduction~rule~\ref{rr:isolated-component}}
\State {\bf return} {\tt DCVD}$(G[V \setminus C], S, k)$
\EndIf
\If{$G[V \setminus S]$ has a vertex $v$ of degree one in $G$} \Comment{Reduction~rule~\ref{rr:deg-1}}
\State {\bf return} {\tt DCVD}$(G[V \setminus \{v\}], S, k)$
\EndIf
\If{$G[V \setminus S]$ has $v$ such that $G[V \cup \{v\}]$ is not a cactus forest} \Comment{Reduction~rule~\ref{rr:must-be-in-solution}}
\State {\bf return} {\tt DCVD}$(G[V \setminus \{v\}], S, k - 1)$
\EndIf
\If{$G[V \setminus S]$ has vertices $u$ and $v$ with $m(u, v) \ge 3$} \Comment{Branching~rule~\ref{br:must-be-in-solution}}
\State {\bf return} {\tt DCVD}$(G[V \setminus \{u\}], S, k - 1) \lor $ {\tt DCVD}$(G[V \setminus \{v\}], S, k - 1)$
\EndIf
\If{$G[V \setminus S]$ has a vertex $v$ with $|N_G(v)| = 1$} \Comment{Reduction~rule~\ref{rr:neighbor-1}}
\State {\bf return} {\tt DCVD}$(G[V \setminus \{v\}], S, k)$
\EndIf
\If{$G[V \setminus S]$ has a vertex $v$ having at least two neighbors in $S$} \Comment{Branching~rule~\ref{br:v-has-two-neighbors-in-two-components}}
\State {\bf return} {\tt DCVD}$(G[V \setminus \{v\}], S, k - 1) \lor $ {\tt DCVD}$(G[V], S \cup \{v\}, k)$
\EndIf
\If{$G[V \setminus S]$ has a vertex $v$ with $N_G(v) = \{u, w\}$} \Comment{Reduction~rule~\ref{rr:deg-2}}
\State Let $G' = G[V \setminus \{v\}]$.
\State Add $\max\{m(u, v), m(v, w)\}$ parallel edges between $u$ and $w$ to $G'$.
\State {\bf return} {\tt DCVD}$(G', S, k)$
\EndIf
\If{$G[V \setminus S]$ has a leaf block $B$ with $V(B) = \{u, v, w\}$} \Comment{Branching~rules~\ref{br:k3-single-comp} and \ref{br:k3-distinct-comp}}
\For{$x \in V(B)$}
\If{{\tt DCVD}$(G[V \setminus \{x\}], S, k - 1)$}
\State{\bf return true}
\EndIf
\EndFor
\If{$G[S \cup V(B)]$ is a cactus forest}
\State {\bf return} {\tt DCVD}$(G[V], S \cup V(B), k)$
\EndIf
\State {\bf return false}
\EndIf
\If{$G[V \setminus S]$ has a leaf block $B$ with $|V(B)| \ge 4$} \Comment{Branching rules \ref{br:non-k3-single-comp} and \ref{br:non-k3-distinct-comp}}
\State Let $B' = \{u, v, w\}$ be consecutive vertices in $B$ that are not cut vertices in $G[V \setminus S]$.
\For{$x \in V(B')$}
\If{{\tt DCVD}$(G[V \setminus \{x\}], S, k - 1)$}
\State {\bf return true}
\EndIf
\EndFor
\If{$G[S \cup V(B')]$ is a cactus forest}
\State {\bf return} {\tt DCVD}$(G[V], S \cup V(B'), k)$
\EndIf
\State {\bf return false}
\EndIf
\EndProcedure
\end{algorithmic}
\end{algorithm}
\section{An improved algorithm for {\sc Even Cycle Transversal}}
Recall that {\sc Even Cycle Transversal} asks whether, given a graph $G = (V, E)$ and an integer $k$, $G$ has a vertex set $X$ of size at most $k$ such that $G[V \setminus X]$ is a forest of odd cacti.
As in the previous section, we solve the disjoint version of {\sc Even Cycle Transversal} and give an $O(16.64^k)$-time algorithm for it, assuming that $|S| \le k + 1$.
\begin{definition}[{\sc Disjoint Even Cycle Transversal}]
Given a graph $G = (V, E)$, an integer $k \ge 0$, and $S \subseteq V$ such that $G[V \setminus S]$ is a forest of odd cacti, the problem asks to find a vertex set $X \subseteq V \setminus S$ with $|X| \le k$ whose removal leaves a forest of odd cacti.
\end{definition}
A key difference from {\sc Disjoint Cactus Vertex Deletion} is that we need to take the length of cycles into account.
However, in Reduction~rule~\ref{rr:deg-2}, we replace (a chain of) cycles with two multiple edges between two extreme vertices, which does not preserve the length of cycles in the original graph.
Given this, we consider a slightly generalized problem.
In addition to the input of {\sc Disjoint Even Cycle Transversal}, we are given a binary weight function $\omega\colon E \to \mathbb \{0, 1\}$ on edges, and the length of a cycle is defined to be the total weight of edges in it.
Indeed, when $\omega(e) = 1$ for all $e \in E$, the problem corresponds to {\sc Disjoint Even Cycle Transversal}.
Let $S \subseteq V$ such that $G[V \setminus S]$ is a forest of odd cacti.
We first apply Reduction~rules~\ref{rr:isolated-component}, \ref{rr:deg-1}, \ref{rr:must-be-in-solution}, and \ref{rr:neighbor-1} and Branching~rules~\ref{br:must-be-in-solution} and \ref{br:v-has-two-neighbors-in-two-components}, which are trivially safe for {\sc Disjoint Even Cycle Transversal}.
Moreover, we add the following reduction rule, which is also trivially safe.
\begin{reduce}\label{rr:ECT:must-be-in-solution}
If there is a vertex $v \in V \setminus S$ such that $G[S \cup \{v\}]$ has a cycle of even length, then delete it and decrease $k$ by one.
\end{reduce}
We can check in linear time whether an edge-weighted graph is a forest of odd cacti and hence Reduction~rule~\ref{rr:ECT:must-be-in-solution} can be applied in linear time as well.
Up to this point, Assumption~\ref{assume:at-most-1-in-S} is made.
By Reduction~rule~\ref{rr:must-be-in-solution} and Branching~rule~\ref{br:must-be-in-solution}, we also assume that $m(u, v) \le 2$ for every pair of vertices in $G$.
Suppose that $m(u, v) = 2$.
If the parities of two edges between $u$ and $v$ are the same, the length of the cycle consisting of these edges is even.
Thus, we apply the following branching rule in this case.
\begin{branch}\label{br:even-K2}
Suppose that $u, v \in V \setminus S$ and $m(u, v) = 2$ for some $u$.
Let $f, f'$ be the edges between $u$ and $v$.
If $\omega(f) = \omega(f')$, branch into two cases: (1) delete $u$ and decrease $k$ by one; (2) delete $v$ and decrease $k$ by one.
\end{branch}
By Reduction~rule~\ref{rr:ECT:must-be-in-solution} and Branching~rule~\ref{br:even-K2}, the following assumption is made.
\begin{assume}\label{assume:parity}
For every pair of vertices $u, v$ with $m(u, v) = 2$, the parities of the weights of edges between them are opposite.
\end{assume}
Now, let us consider a vertex $v \in V \setminus S$ that has exactly two neighbors in $G$.
Let $u$ and $w$ be the neighbors of $v$.
By Assumption~\ref{assume:at-most-1-in-S}, at least one of $u$ and $w$ belongs to $V \setminus S$.
Similarly to Lemma~\ref{lem:eliminate-deg2}, we define a graph $G'$ by deleting $v$ from $G$ and adding $p$ parallel edges between $u$ and $v$, where $p = \max(m(u, v), m(v, w))$.
We define the weight function $w'$ for $G'$ as follows.
If $p = 1$, we set the weight of the introduced edge $e = \{u, w\}$ as $\omega'(e) = \omega(f) + \omega(f')$, where $f$ (resp.~$f'$) is the edge between $u$ and $v$ (resp.~between $v$ and $w$) and the sum is taken under addition modulo two.
If $p = 2$, at least one of the pairs $\{u, v\}$ or $\{v, w\}$ has multiple edges.
By Assumption~\ref{assume:parity}, these two edges have different parities.
A crucial observation is that if there is a cycle passing through exactly one of these edges, there is another cycle passing through the other edges, which has the different parity.
By setting $\omega'(e) = 0$ and $\omega'(e') = 1$, such cycles are preserved in $G'$.
\begin{lemma}\label{lem:ECT:feasible}
The instance $(G, \omega, S, k)$ is a yes-instance if and only if $(G', \omega', S, k)$ is a yes-instance.
\end{lemma}
\begin{proof}
Consider a cycle $C$ of even length that passes through $v$ in $G$.
By Assumption~\ref{assume:parity}, $C$ must pass through both $u$ and $w$.
Thus, there is a feasible solution $X \subseteq V \setminus S$ for $(G, \omega, S, k)$ with $v \notin X$ and $\{u,w\} \cap X \neq \emptyset$.
By the construction of $G'$, the cycle obtained from $C$ by omitting $v$ is an even cycle of $G'$.
Hence, $X$ is a feasible solution for $(G', \omega', S, k)$.
It is not hard to see that this correspondence is reversible and hence the lemma follows.
\end{proof}
This lemma ensures that the weighted version of Reduction~rule~\ref{rr:deg-2} is safe for {\sc Even Cycle Transversal}, and then Assumption~\ref{assume:deg-3} is made as well.
The rest of branching rules are the same with {\sc Disjoint Cactus Vertex Deletion}, which yields an $O^*(16.64^k)$-time algorithm that solves {\sc Disjoint Even Cycle Transversal} as well.
\begin{lemma}\label{lem:ECT:disjoint}
Suppose $|S| \le k + 1$. Then, {\sc Disjoint Even Cycle Transversal} can be solved in time $O^*(16.64^k)$.
\end{lemma}
By Lemma~\ref{lem:compression}, {\sc Even Cycle Transversal} can be solved in time $O^*(17.64^k)$, completing the proof of Theorem~\ref{thm:d-CVD-ECT}.
\section*{Acknowledgments}
This work is partially supported by JSPS KAKENHI Grant Numbers JP18H04091, JP18H05291, JP18K11168, JP18K11169, JP19K21537, JP20H05793, JP20K11692, and JP20K19742.
The authors thank Kunihiro Wasa for fruitful discussions.
\bibliographystyle{plain}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,522
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Q: Java array initialization within argument list How come the first call to someMethod doesn't compile without being explicit that it's String[]?
It's fine to use an array initializer to create a String[] array but you can't use it to pass an argument. Are the curly braces used in some other fashion for passing arguments that derails how I'd expect this to behave?
public void someMethod(String[] arr){
//do some magic
}
public void makeSomeMagic(){
String[] arr = {"cat", "fish", "cow"};
//Does not compile!
someMethod({"cat", "fish", "cow"});
//This compiles!
someMethod(new String[]{"cat", "fish", "cow"});
//This compiles!
someMethod(arr);
}
The compiler error is the following:
The method someMethod(String[]) in the type Moo is not applicable for the arguments (String, String, String)
A: If you don't want to use explicit String[], use:
public void someMethod(String... arr){
//do some magic
}
…
someMethod("cm", "applicant", "lead");
The three periods after the final parameter's type indicate that the final argument may be passed as an array or as a sequence of arguments.
Read more.
A: You can only use the { "hello", "world" } initialization notation when declaring an array variable or in an array creation expression such as new String[] { ... }.
See Section 10.6 Array Initializers in the Java Language Specification:
An array initializer may be specified in a declaration, or as part of an array creation expression (§15.10), creating an array and providing some initial values
A: Or you can use varargs:
public void someMethod(String... arr){
//do some magic
}
public void makeSomeMagic(){
someMethod("cat", "fish", "cow");
}
It's basically a fancy syntax for an array parameter (vararg must be the last parameter in method signature).
A: You can use the curly braces to initialize an array. In every else case it is used to define blocks of statments.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,114
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Marine veteran amputee gets amphibious prosthetic leg on Long Island
By Stacey Sager
EAST MEADOW, Long Island (WABC) -- He is a model of inspiration and determination, a Marine who lost a leg while serving in Afghanistan in 2004.
In fact, 6% of injured veterans are amputees.
Dan Lasko, 33, is diving into his future with new confidence.
He's got a new, amphibious prosthetic leg from Northwell Health which he calls a "game changer," especially for an athletic Marine Corps Veteran who loves swimming.
"It's like Michael Phelps in there," Lasko said. "I feel really confident and really fast."
Lasko lost his lower left leg after an IED ambush in Afghanistan 13 years ago.
His wife and two young sons had grown accustomed to daddy taking off his leg before he could get into the water, but not anymore.
"There's no more waiting, when I get to the pool or to the beach, 'hold on, I got to change legs,' now I can just go straight into the water, and have a great time with them," Lasko said.
It's Important to understand that Lasko is just the first of many who will want this type of leg. The cost from Northwell will be determined by the exact needs of each patient.
The research and development came from two Long Island firms. At one of them is a former diver who had lost his left leg too years ago.
"Everyone turned to me and said, 'well you're a swimmer, what do you need to make yours better?'" said Matt Flynn, Eschen Prosthetic and Orthotic Laboratories.
"And I tell you, that the team had a real, emotional impact on that," said Joe Garone, Composite Prototyping Center.
And so researchers at Northwell, using a 3-D printer and a nylon-carbon fiber base, worked for months on the prototype that can be worn in water and on shore, and tweaked for every type of patient.
"It can change how much resistance it has, and therefore that equates to how fast they can swim," said Todd Goldstein, Northwell Health 3-D Printer Researcher.
For this veteran who's competed in 30 triathlons and six marathons since he lost his leg, there couldn't be a better fit.
If you are interested in getting an amphibious leg, it is still in trials, but you can email Northwell Health for more information at prosthetic@northwell.edu.
health & fitnesseast meadow central parkmineolaprostheticswimmingveteranamputeenorthwell health
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,261
|
Q: Generic and (early?) binding in Swift 1.2 func f<T>(a:T)->String { return "Generic" }
func f(a:Int)->String { return "Integer" }
func alias<T>(a:T)->String { return f(a) }
f(1) // "Integer"
f("string") // "Generic"
alias(1) // "Generic" (shouldn't be "Integer" ?)
alias("string") // "Generic"
I understand that some form of early (static) binding is happening but I don't understand why. Is it by design or a bug? if by design then alias(a) =/= f(a) but the definition reads exactly alias(a) = f(a) !
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,000
|
package org.elasticsearch.index.mapper;
import com.google.common.collect.ImmutableMap;
import org.elasticsearch.Version;
import org.elasticsearch.action.admin.indices.mapping.get.GetMappingsResponse;
import org.elasticsearch.cluster.metadata.IndexMetaData;
import org.elasticsearch.common.settings.Settings;
import org.elasticsearch.common.xcontent.ToXContent;
import org.elasticsearch.common.xcontent.XContentBuilder;
import org.elasticsearch.common.xcontent.XContentFactory;
import org.elasticsearch.common.xcontent.XContentHelper;
import org.elasticsearch.common.xcontent.XContentParser;
import org.elasticsearch.index.IndexService;
import org.elasticsearch.test.ElasticsearchSingleNodeTest;
import java.io.IOException;
import static org.elasticsearch.common.xcontent.XContentFactory.jsonBuilder;
import static org.hamcrest.Matchers.equalTo;
import static org.hamcrest.Matchers.nullValue;
public class DynamicMappingTests extends ElasticsearchSingleNodeTest {
public void testDynamicTrue() throws IOException {
String mapping = jsonBuilder().startObject().startObject("type")
.field("dynamic", "true")
.startObject("properties")
.startObject("field1").field("type", "string").endObject()
.endObject()
.endObject().endObject().string();
DocumentMapper defaultMapper = createIndex("test").mapperService().documentMapperParser().parse(mapping);
ParsedDocument doc = defaultMapper.parse("type", "1", jsonBuilder()
.startObject()
.field("field1", "value1")
.field("field2", "value2")
.bytes());
assertThat(doc.rootDoc().get("field1"), equalTo("value1"));
assertThat(doc.rootDoc().get("field2"), equalTo("value2"));
}
public void testDynamicFalse() throws IOException {
String mapping = jsonBuilder().startObject().startObject("type")
.field("dynamic", "false")
.startObject("properties")
.startObject("field1").field("type", "string").endObject()
.endObject()
.endObject().endObject().string();
DocumentMapper defaultMapper = createIndex("test").mapperService().documentMapperParser().parse(mapping);
ParsedDocument doc = defaultMapper.parse("type", "1", jsonBuilder()
.startObject()
.field("field1", "value1")
.field("field2", "value2")
.bytes());
assertThat(doc.rootDoc().get("field1"), equalTo("value1"));
assertThat(doc.rootDoc().get("field2"), nullValue());
}
public void testDynamicStrict() throws IOException {
String mapping = jsonBuilder().startObject().startObject("type")
.field("dynamic", "strict")
.startObject("properties")
.startObject("field1").field("type", "string").endObject()
.endObject()
.endObject().endObject().string();
DocumentMapper defaultMapper = createIndex("test").mapperService().documentMapperParser().parse(mapping);
try {
defaultMapper.parse("type", "1", jsonBuilder()
.startObject()
.field("field1", "value1")
.field("field2", "value2")
.bytes());
fail();
} catch (StrictDynamicMappingException e) {
// all is well
}
try {
defaultMapper.parse("type", "1", XContentFactory.jsonBuilder()
.startObject()
.field("field1", "value1")
.field("field2", (String) null)
.bytes());
fail();
} catch (StrictDynamicMappingException e) {
// all is well
}
}
public void testDynamicFalseWithInnerObjectButDynamicSetOnRoot() throws IOException {
String mapping = jsonBuilder().startObject().startObject("type")
.field("dynamic", "false")
.startObject("properties")
.startObject("obj1").startObject("properties")
.startObject("field1").field("type", "string").endObject()
.endObject().endObject()
.endObject()
.endObject().endObject().string();
DocumentMapper defaultMapper = createIndex("test").mapperService().documentMapperParser().parse(mapping);
ParsedDocument doc = defaultMapper.parse("type", "1", jsonBuilder()
.startObject().startObject("obj1")
.field("field1", "value1")
.field("field2", "value2")
.endObject()
.bytes());
assertThat(doc.rootDoc().get("obj1.field1"), equalTo("value1"));
assertThat(doc.rootDoc().get("obj1.field2"), nullValue());
}
public void testDynamicStrictWithInnerObjectButDynamicSetOnRoot() throws IOException {
String mapping = jsonBuilder().startObject().startObject("type")
.field("dynamic", "strict")
.startObject("properties")
.startObject("obj1").startObject("properties")
.startObject("field1").field("type", "string").endObject()
.endObject().endObject()
.endObject()
.endObject().endObject().string();
DocumentMapper defaultMapper = createIndex("test").mapperService().documentMapperParser().parse(mapping);
try {
defaultMapper.parse("type", "1", jsonBuilder()
.startObject().startObject("obj1")
.field("field1", "value1")
.field("field2", "value2")
.endObject()
.bytes());
fail();
} catch (StrictDynamicMappingException e) {
// all is well
}
}
public void testDynamicMappingOnEmptyString() throws Exception {
IndexService service = createIndex("test");
client().prepareIndex("test", "type").setSource("empty_field", "").get();
FieldMapper mapper = service.mapperService().fullName("empty_field");
assertNotNull(mapper);
}
public void testTypeNotCreatedOnIndexFailure() throws IOException, InterruptedException {
XContentBuilder mapping = jsonBuilder().startObject().startObject("_default_")
.field("dynamic", "strict")
.endObject().endObject();
IndexService indexService = createIndex("test", Settings.EMPTY, "_default_", mapping);
try {
client().prepareIndex().setIndex("test").setType("type").setSource(jsonBuilder().startObject().field("test", "test").endObject()).get();
fail();
} catch (StrictDynamicMappingException e) {
}
GetMappingsResponse getMappingsResponse = client().admin().indices().prepareGetMappings("test").get();
assertNull(getMappingsResponse.getMappings().get("test").get("type"));
}
private String serialize(ToXContent mapper) throws Exception {
XContentBuilder builder = XContentFactory.jsonBuilder().startObject();
mapper.toXContent(builder, new ToXContent.MapParams(ImmutableMap.<String, String>of()));
return builder.endObject().string();
}
private Mapper parse(DocumentMapper mapper, DocumentMapperParser parser, XContentBuilder builder) throws Exception {
Settings settings = Settings.builder().put(IndexMetaData.SETTING_VERSION_CREATED, Version.CURRENT).build();
ParseContext.InternalParseContext ctx = new ParseContext.InternalParseContext("test", settings, parser, mapper, new ContentPath(0));
SourceToParse source = SourceToParse.source(builder.bytes());
ctx.reset(XContentHelper.createParser(source.source()), new ParseContext.Document(), source);
assertEquals(XContentParser.Token.START_OBJECT, ctx.parser().nextToken());
ctx.parser().nextToken();
return DocumentParser.parseObject(ctx, mapper.root());
}
public void testDynamicMappingsNotNeeded() throws Exception {
IndexService indexService = createIndex("test");
DocumentMapperParser parser = indexService.mapperService().documentMapperParser();
String mapping = XContentFactory.jsonBuilder().startObject().startObject("type")
.startObject("properties").startObject("foo").field("type", "string").endObject().endObject()
.endObject().string();
DocumentMapper mapper = parser.parse(mapping);
Mapper update = parse(mapper, parser, XContentFactory.jsonBuilder().startObject().field("foo", "bar").endObject());
// foo is already defined in the mappings
assertNull(update);
}
public void testField() throws Exception {
IndexService indexService = createIndex("test");
DocumentMapperParser parser = indexService.mapperService().documentMapperParser();
String mapping = XContentFactory.jsonBuilder().startObject()
.startObject("type").endObject()
.endObject().string();
DocumentMapper mapper = parser.parse(mapping);
assertEquals(mapping, serialize(mapper));
Mapper update = parse(mapper, parser, XContentFactory.jsonBuilder().startObject().field("foo", "bar").endObject());
assertNotNull(update);
// original mapping not modified
assertEquals(mapping, serialize(mapper));
// but we have an update
assertEquals("{\"type\":{\"properties\":{\"foo\":{\"type\":\"string\"}}}}", serialize(update));
}
public void testIncremental() throws Exception {
IndexService indexService = createIndex("test");
DocumentMapperParser parser = indexService.mapperService().documentMapperParser();
// Make sure that mapping updates are incremental, this is important for performance otherwise
// every new field introduction runs in linear time with the total number of fields
String mapping = XContentFactory.jsonBuilder().startObject().startObject("type")
.startObject("properties").startObject("foo").field("type", "string").endObject().endObject()
.endObject().string();
DocumentMapper mapper = parser.parse(mapping);
assertEquals(mapping, serialize(mapper));
Mapper update = parse(mapper, parser, XContentFactory.jsonBuilder().startObject().field("foo", "bar").field("bar", "baz").endObject());
assertNotNull(update);
// original mapping not modified
assertEquals(mapping, serialize(mapper));
// but we have an update
assertEquals(XContentFactory.jsonBuilder().startObject().startObject("type").startObject("properties")
// foo is NOT in the update
.startObject("bar").field("type", "string").endObject()
.endObject().endObject().string(), serialize(update));
}
public void testIntroduceTwoFields() throws Exception {
IndexService indexService = createIndex("test");
DocumentMapperParser parser = indexService.mapperService().documentMapperParser();
String mapping = XContentFactory.jsonBuilder().startObject()
.startObject("type").endObject()
.endObject().string();
DocumentMapper mapper = parser.parse(mapping);
assertEquals(mapping, serialize(mapper));
Mapper update = parse(mapper, parser, XContentFactory.jsonBuilder().startObject().field("foo", "bar").field("bar", "baz").endObject());
assertNotNull(update);
// original mapping not modified
assertEquals(mapping, serialize(mapper));
// but we have an update
assertEquals(XContentFactory.jsonBuilder().startObject().startObject("type").startObject("properties")
.startObject("bar").field("type", "string").endObject()
.startObject("foo").field("type", "string").endObject()
.endObject().endObject().string(), serialize(update));
}
public void testObject() throws Exception {
IndexService indexService = createIndex("test");
DocumentMapperParser parser = indexService.mapperService().documentMapperParser();
String mapping = XContentFactory.jsonBuilder().startObject()
.startObject("type").endObject()
.endObject().string();
DocumentMapper mapper = parser.parse(mapping);
assertEquals(mapping, serialize(mapper));
Mapper update = parse(mapper, parser, XContentFactory.jsonBuilder().startObject().startObject("foo").startObject("bar").field("baz", "foo").endObject().endObject().endObject());
assertNotNull(update);
// original mapping not modified
assertEquals(mapping, serialize(mapper));
// but we have an update
assertEquals(XContentFactory.jsonBuilder().startObject().startObject("type").startObject("properties")
.startObject("foo").startObject("properties").startObject("bar").startObject("properties").startObject("baz").field("type", "string").endObject().endObject().endObject().endObject().endObject()
.endObject().endObject().endObject().string(), serialize(update));
}
public void testArray() throws Exception {
IndexService indexService = createIndex("test");
DocumentMapperParser parser = indexService.mapperService().documentMapperParser();
String mapping = XContentFactory.jsonBuilder().startObject()
.startObject("type").endObject()
.endObject().string();
DocumentMapper mapper = parser.parse(mapping);
assertEquals(mapping, serialize(mapper));
Mapper update = parse(mapper, parser, XContentFactory.jsonBuilder().startObject().startArray("foo").value("bar").value("baz").endArray().endObject());
assertNotNull(update);
// original mapping not modified
assertEquals(mapping, serialize(mapper));
// but we have an update
assertEquals(XContentFactory.jsonBuilder().startObject().startObject("type").startObject("properties")
.startObject("foo").field("type", "string").endObject()
.endObject().endObject().endObject().string(), serialize(update));
}
public void testInnerDynamicMapping() throws Exception {
IndexService indexService = createIndex("test");
DocumentMapperParser parser = indexService.mapperService().documentMapperParser();
String mapping = XContentFactory.jsonBuilder().startObject().startObject("type") .startObject("properties")
.startObject("foo").field("type", "object").endObject()
.endObject().endObject().endObject().string();
DocumentMapper mapper = parser.parse(mapping);
assertEquals(mapping, serialize(mapper));
Mapper update = parse(mapper, parser, XContentFactory.jsonBuilder().startObject().startObject("foo").startObject("bar").field("baz", "foo").endObject().endObject().endObject());
assertNotNull(update);
// original mapping not modified
assertEquals(mapping, serialize(mapper));
// but we have an update
assertEquals(XContentFactory.jsonBuilder().startObject().startObject("type").startObject("properties")
.startObject("foo").startObject("properties").startObject("bar").startObject("properties").startObject("baz").field("type", "string").endObject().endObject().endObject().endObject().endObject()
.endObject().endObject().endObject().string(), serialize(update));
}
public void testComplexArray() throws Exception {
IndexService indexService = createIndex("test");
DocumentMapperParser parser = indexService.mapperService().documentMapperParser();
String mapping = XContentFactory.jsonBuilder().startObject()
.startObject("type").endObject()
.endObject().string();
DocumentMapper mapper = parser.parse(mapping);
assertEquals(mapping, serialize(mapper));
Mapper update = parse(mapper, parser, XContentFactory.jsonBuilder().startObject().startArray("foo")
.startObject().field("bar", "baz").endObject()
.startObject().field("baz", 3).endObject()
.endArray().endObject());
assertEquals(mapping, serialize(mapper));
assertEquals(XContentFactory.jsonBuilder().startObject().startObject("type").startObject("properties")
.startObject("foo").startObject("properties")
.startObject("bar").field("type", "string").endObject()
.startObject("baz").field("type", "long").endObject()
.endObject().endObject()
.endObject().endObject().endObject().string(), serialize(update));
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,226
|
Persona 5 Joker "Coming Soon" to Super Smash Bros. Ultimate, Says Sakurai
Allisa James
April 6, 2019 1:56 PM EST
NewsPS4PS3Switch
Masahiro Sakurai reveals a hint as to when Persona 5 Joker is dropping for Smash Ultimate during his Famitsu Awards speech.
At the recent Famitsu Awards ceremony, Super Smash Bros. director Masahiro Sakurai stated that the Super Smash Bros. Ultimate DLC fight character Joker — from Atlus's Persona 5 — will be "coming soon." The phrase, caught by Twitter user PushDustin, was also used when he spoke about Mewtwo during a NicoNico interview back in 2015:
At the Famitsu Awards, Sakurai stated that Joker is coming soon*
*もうそろそろです。
I believe he used the same phrase in the NicoNico Interview with Mewtwo, which was done around March 22nd. Metwo was released on April 15th for Club Nintendo players.
— PushDustIn (@PushDustIn) April 5, 2019
Interesting to note that the interview was conducted on March 22nd of that year and the fighter was first released on April 15th. Whether that's indicative of how long it will take for Joker to launch remains to be seen.
When Sakurai accepted the well-deserved MVP award, he discussed at length how difficult developing Super Smash Bros. Ultimate has been on him and his staff. He also remarked that he's impressed with the work other studios are doing. He ended his speech with the phrase, "I will forget the troubles I had, but the game remains." A more fitting, if not completely literal, translation of that mantra would be "The problems are temporary, but the game is forever."
While those are certainly wise words, they do carry more weight than normal as Sakurai is rather infamous for overworking himself while developing for the franchise. Hopefully, after the DLC fighters have been released, he and his staff can finally take their well-deserved break.
Lately, there has been a deluge of leaks and hints regarding the release of Joker for Super Smash Bros. Ultimate, such as his fighter render revealed in a Best Buy ad and his release window no longer stating "coming in April." Persona 5 has gotten its own share of news, as Atlus opened a Persona 5 S teaser site with a hidden message in its source code.
Persona 5 Joker Render Revealed for Super Smash Bros. Ultimate
Persona 5's Joker Release Window for Super Smash Bros. Ultimate No Longer Says "April"
Atlus Opens Persona 5 S Teaser Site With Hidden Message in Source Code
Super Smash Bros. Ultimate is available exclusively for Nintendo Switch, and can be purchased through Amazon. Meanwhile, if you haven't picked up your copy of Persona 5 yet you can get the physical game on Amazon or by purchasing it on discount digitally through the PSN store. The game is available exclusively on PS4 and PS3, with more information on Persona 5: The Royal coming later this month.
Persona 5 Sneaks its Way Onto the PlayStation Hits Lineup
Persona 5 The Royal Gets New Details and Screenshots of Female MC
Dragon Ball Z Kakarot Will Also Let Us Play Vegeta, Gohan, and Piccolo
Persona 5, The Legend of Zelda, and Metroid Prime Trilogy Listed For Nintendo Switch
Persona 5, The Legend of Zelda: A Link to the Past, and Metroid Prime Trilogy reported for Switch after found Best Buy listings.
Skyward Sword in Final Production Phase
E3 2010: Sonic Free Riders Announced for Kinect
E3 2010: Microsoft Project Natal Experience Event Disappoints
Cars Toon: Mater's Tall Tales Announced for Wii
Wayne Gretzky Named Cover Athlete of All-New EA Sports NHL SLAPSHOT
Persona 5 Official Artbook Released in English as an E-Book for First Time
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,910
|
Q: jQuery Masonry - re-align tiles when removed I have a script on the page that removes "tiles" using a filter.
Is it possible to re-align everything when tiles are removed?
This doesn't seem to be doing it:
$wrapper.masonry( 'reloadItems' );
I'm using the following to filter:
$('.tags:not(.tags:contains('+filter_text+'))').parent().fadeOut('slow');
Cheers!
A: Use the "remove" method of Masonry for removing elements instead of removing them using jQuery yourself.
So, instead of:
$wrapper.find(<filter criteria>).remove();
Do:
$wrapper.masonry("remove", $wrapper.find(<filter criteria>));
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,427
|
The knockout phase of the 2011–12 UEFA Champions League began on 14 February with the round of 16, and concluded on 19 May 2012 with the final at Allianz Arena in Munich, Germany.
Times are CET/CEST, as listed by UEFA (local times are in parentheses).
Round and draw dates
All draws were held at UEFA headquarters in Nyon, Switzerland.
Format
The knockout phase involved the sixteen teams who finished in the top two in each of their groups in the group stage.
Each tie in the knockout phase, apart from the final, was played over two legs, with each team playing one leg at home. The team that had the higher aggregate score over the two legs progressed to the next round. In the event that aggregate scores finished level, the away goals rule was applied, i.e. the team that scored more goals away from home over the two legs progressed. If away goals were also equal, then thirty minutes of extra time were played, divided into two fifteen-minutes halves. The away goals rule was again applied after extra time, i.e. if there were goals scored during extra time and the aggregate score was still level, the visiting team qualified by virtue of more away goals scored. If no goals were scored during extra time, the tie was decided by a penalty shoot-out. In the final, the tie was played as a single match. If scores were level at the end of normal time in the final, extra time was played, followed by penalties if scores remained tied.
In the draw for the round of 16, the eight group winners were seeded, and the eight group runners-up were unseeded. A seeded team was drawn against an unseeded team, with the seeded team hosting the second leg. Teams from the same group or the same association could not be drawn against each other. In the draws for the quarter-finals onwards, there were no seedings, and teams from the same group or the same association could be drawn with each other.
Qualified teams
Bracket
Round of 16
The first legs were played on 14, 15, 21 and 22 February, and the second legs were played on 6, 7, 13 and 14 March 2012.
|}
Matches
1–1 on aggregate. APOEL won 4–3 on penalties.
Chelsea won 5–4 on aggregate.
Milan won 4–3 on aggregate.
Bayern Munich won 7–1 on aggregate.
Barcelona won 10–2 on aggregate.
Real Madrid won 5–2 on aggregate.
Benfica won 4–3 on aggregate.
2–2 on aggregate. Marseille won on away goals.
Quarter-finals
The first legs were played on 27 and 28 March, and the second legs were played on 3 and 4 April 2012.
|}
Matches
Real Madrid won 8–2 on aggregate.
Bayern Munich won 4–0 on aggregate.
Chelsea won 3–1 on aggregate.
Barcelona won 3–1 on aggregate.
Semi-finals
The first legs were played on 17 and 18 April, and the second legs were played on 24 and 25 April 2012.
|}
Matches
3–3 on aggregate. Bayern Munich won 3–1 on penalties.
Chelsea won 3–2 on aggregate.
Final
Notes
References
External links
2011–12 UEFA Champions League, UEFA.com
Knockout phase
2011-12
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,552
|
Q: YouTube Data API Get Channel by Handle Introducing handles: A new way to identify your YouTube channel
Does the YouTube Data API support querying for a channel by it's @handle? This does not seem to be supported.
ex: https://www.youtube.com/@lionsgatemovies
forUsername param
GET https://www.googleapis.com/youtube/v3/channels?part=id,snippet&forUsername=@lionsgatemovies
{
"kind": "youtube#channelListResponse",
"etag": "RuuXzTIr0OoDqI4S0RU6n4FqKEM",
"pageInfo": {
"totalResults": 0,
"resultsPerPage": 5
}
}
id param
GET https://www.googleapis.com/youtube/v3/channels?part=id,snippet&id=@lionsgatemovies
{
"kind": "youtube#channelListResponse",
"etag": "RuuXzTIr0OoDqI4S0RU6n4FqKEM",
"pageInfo": {
"totalResults": 0,
"resultsPerPage": 5
}
}
None of the supported filter params seem to be appropriate:
{
"error": {
"code": 400,
"message": "No filter selected. Expected one of: mySubscribers, forUsername, mine, managedByMe, categoryId, id",
"errors": [
{
"message": "No filter selected. Expected one of: mySubscribers, forUsername, mine, managedByMe, categoryId, id",
"domain": "youtube.parameter",
"reason": "missingRequiredParameter",
"location": "parameters.",
"locationType": "other"
}
]
}
}
A: You can use Search API with q parameter set to @handle
curl \
'https://youtube.googleapis.com/youtube/v3/search?part=snippet&maxResults=25&q=%40kevinrooke&type=channel&key=[YOUR_API_KEY]'
{
"kind": "youtube#searchListResponse",
"etag": "AYlro9VG2vMtdew4OQiWoQM8Rs0",
"regionCode": "LT",
"pageInfo": {
"totalResults": 1,
"resultsPerPage": 1
},
"items": [
{
"kind": "youtube#searchResult",
"etag": "ls9E_ctoa-RLsqznJwxWlHHIE1s",
"id": {
"kind": "youtube#channel",
"channelId": "UCTdxV_ItCZMayyzGkw7P_qQ"
},
"snippet": {
"publishedAt": "2017-05-27T03:56:38Z",
"channelId": "UCTdxV_ItCZMayyzGkw7P_qQ",
"title": "Kevin Rooke",
"description": "Interviews with the builders bringing the Lightning Network to life. ⚡kerooke@fountain.fm.",
"thumbnails": {
"default": {
"url": "https://yt3.ggpht.com/ytc/AMLnZu-SpmaNjx7KOMqs5Cr7ZthU60BaQApzt89_dHOlcg=s88-c-k-c0xffffffff-no-rj-mo"
},
"medium": {
"url": "https://yt3.ggpht.com/ytc/AMLnZu-SpmaNjx7KOMqs5Cr7ZthU60BaQApzt89_dHOlcg=s240-c-k-c0xffffffff-no-rj-mo"
},
"high": {
"url": "https://yt3.ggpht.com/ytc/AMLnZu-SpmaNjx7KOMqs5Cr7ZthU60BaQApzt89_dHOlcg=s800-c-k-c0xffffffff-no-rj-mo"
}
},
"channelTitle": "Kevin Rooke",
"liveBroadcastContent": "none",
"publishTime": "2017-05-27T03:56:38Z"
}
}
]
}
A: As of this moment (17th Nov 2022), YouTube has yet to update the Data API with @handle support.
A: The channelId was scattered around in the Html. You can easily parse them after fetching the url with the handle.
const html = await(await fetch(url)).text()
const channelId = html.match(/(?<=channelId(":"|"\scontent="))[^"]+/g)[0];
A: Following is the python snippet while we are waiting for YouTube API's official support. This is inspired by goodhyun's wonderful thoughts.
import requests
import re
# return YouTube channel id via handle or False if failed
def scraping_get_channel_id_from_handle(handle:str):
if handle.find('@') == -1:
handle = '@' + handle
url = 'https://www.youtube.com/' + handle
resp = requests.get(url)
if resp.status_code == 200:
found = re.findall('"channelId":"([^"]*)","title"', resp.text)
return found[0]
else:
return False
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 474
|
package org.myrobotlab.arduino;
import java.io.ByteArrayOutputStream;
import java.io.OutputStream;
import java.nio.ByteBuffer;
import java.nio.ByteOrder;
import java.util.Arrays;
import java.util.concurrent.atomic.AtomicInteger;
import org.myrobotlab.logging.Level;
import org.myrobotlab.arduino.virtual.MrlComm;
import org.myrobotlab.string.StringUtil;
/**
* <pre>
*
Welcome to Msg.java
Its created by running ArduinoMsgGenerator
which combines the MrlComm message schema (src/resource/Arduino/arduinoMsg.schema)
with the cpp template (src/resource/Arduino/generate/Msg.java.template)
Schema Type Conversions
Schema ARDUINO Java Range
none byte/unsigned char int (cuz Java byte bites) 1 byte - 0 to 255
boolean boolean boolean 0 1
b16 int int (short) 2 bytes -32,768 to 32,767
b32 long int 4 bytes -2,147,483,648 to 2,147,483, 647
bu32 unsigned long long 0 to 4,294,967,295
str char*, size String variable length
[] byte[], size int[] variable length
All message editing should be done in the arduinoMsg.schema
The binary wire format of an MrlComm is:
MAGIC_NUMBER|MSG_SIZE|METHOD_NUMBER|PARAM0|PARAM1 ...
</pre>
*/
import org.myrobotlab.logging.LoggerFactory;
import org.myrobotlab.logging.LoggingFactory;
import java.io.FileOutputStream;
import org.myrobotlab.service.Runtime;
import org.myrobotlab.service.Servo;
import org.myrobotlab.service.interfaces.SerialDevice;
import org.slf4j.Logger;
/**
* Singlton messaging interface to an MrlComm
*
* @author GroG
*
*/
public class VirtualMsg {
// TODO: pick a more reasonable timeout.. 3 seconds is high.
private static final int ACK_TIMEOUT = 3000;
public transient final static Logger log = LoggerFactory.getLogger(VirtualMsg.class);
public static final int MAX_MSG_SIZE = 64;
public static final int MAGIC_NUMBER = 170; // 10101010
public static final int MRLCOMM_VERSION = 68;
// send buffer
private int sendBufferSize = 0;
private int sendBuffer[] = new int[MAX_MSG_SIZE];
// recv buffer
private int ioCmd[] = new int[MAX_MSG_SIZE];
private AtomicInteger byteCount = new AtomicInteger(0);
private int msgSize = 0;
// ------ device type mapping constants
private int method = -1;
public boolean debug = false;
// when using a real service, invoke should be true, for unit tests, this should be false.
private boolean invoke = false;
private int errorServiceToHardwareRxCnt = 0;
private int errorHardwareToServiceRxCnt = 0;
boolean ackEnabled = false;
private volatile boolean clearToSend = true;
public static class AckLock {
// track if there is a pending message, when sending a message
// this goes to true. when getting an ack it goes to false.
volatile boolean pendingMessage = false;
}
transient AckLock ackRecievedLock = new AckLock();
// recording related
transient OutputStream record = null;
transient StringBuilder rxBuffer = new StringBuilder();
transient StringBuilder txBuffer = new StringBuilder();
public static final int DEVICE_TYPE_UNKNOWN = 0;
public static final int DEVICE_TYPE_ARDUINO = 1;
public static final int DEVICE_TYPE_ULTRASONICSENSOR = 2;
public static final int DEVICE_TYPE_STEPPER = 3;
public static final int DEVICE_TYPE_MOTOR = 4;
public static final int DEVICE_TYPE_SERVO = 5;
public static final int DEVICE_TYPE_SERIAL = 6;
public static final int DEVICE_TYPE_I2C = 7;
public static final int DEVICE_TYPE_NEOPIXEL = 8;
public static final int DEVICE_TYPE_ENCODER = 9;
// < publishMRLCommError/str errorMsg
public final static int PUBLISH_MRLCOMM_ERROR = 1;
// > getBoardInfo
public final static int GET_BOARD_INFO = 2;
// < publishBoardInfo/version/boardType/b16 microsPerLoop/b16 sram/activePins/[] deviceSummary
public final static int PUBLISH_BOARD_INFO = 3;
// > enablePin/address/type/b16 rate
public final static int ENABLE_PIN = 4;
// > setDebug/bool enabled
public final static int SET_DEBUG = 5;
// > setSerialRate/b32 rate
public final static int SET_SERIAL_RATE = 6;
// > softReset
public final static int SOFT_RESET = 7;
// > enableAck/bool enabled
public final static int ENABLE_ACK = 8;
// < publishAck/function
public final static int PUBLISH_ACK = 9;
// > echo/f32 myFloat/myByte/f32 secondFloat
public final static int ECHO = 10;
// < publishEcho/f32 myFloat/myByte/f32 secondFloat
public final static int PUBLISH_ECHO = 11;
// > customMsg/[] msg
public final static int CUSTOM_MSG = 12;
// < publishCustomMsg/[] msg
public final static int PUBLISH_CUSTOM_MSG = 13;
// > deviceDetach/deviceId
public final static int DEVICE_DETACH = 14;
// > i2cBusAttach/deviceId/i2cBus
public final static int I2C_BUS_ATTACH = 15;
// > i2cRead/deviceId/deviceAddress/size
public final static int I2C_READ = 16;
// > i2cWrite/deviceId/deviceAddress/[] data
public final static int I2C_WRITE = 17;
// > i2cWriteRead/deviceId/deviceAddress/readSize/writeValue
public final static int I2C_WRITE_READ = 18;
// < publishI2cData/deviceId/[] data
public final static int PUBLISH_I2C_DATA = 19;
// > neoPixelAttach/deviceId/pin/b16 numPixels/depth
public final static int NEO_PIXEL_ATTACH = 20;
// > neoPixelSetAnimation/deviceId/animation/red/green/blue/white/b32 wait_ms
public final static int NEO_PIXEL_SET_ANIMATION = 21;
// > neoPixelWriteMatrix/deviceId/[] buffer
public final static int NEO_PIXEL_WRITE_MATRIX = 22;
// > neoPixelFill/deviceId/b16 address/b16 count/red/green/blue/white
public final static int NEO_PIXEL_FILL = 23;
// > neoPixelSetBrightness/deviceId/brightness
public final static int NEO_PIXEL_SET_BRIGHTNESS = 24;
// > neoPixelClear/deviceId
public final static int NEO_PIXEL_CLEAR = 25;
// > analogWrite/pin/value
public final static int ANALOG_WRITE = 26;
// > digitalWrite/pin/value
public final static int DIGITAL_WRITE = 27;
// > disablePin/pin
public final static int DISABLE_PIN = 28;
// > disablePins
public final static int DISABLE_PINS = 29;
// > pinMode/pin/mode
public final static int PIN_MODE = 30;
// < publishDebug/str debugMsg
public final static int PUBLISH_DEBUG = 31;
// < publishPinArray/[] data
public final static int PUBLISH_PIN_ARRAY = 32;
// > setTrigger/pin/triggerValue
public final static int SET_TRIGGER = 33;
// > setDebounce/pin/delay
public final static int SET_DEBOUNCE = 34;
// > servoAttach/deviceId/pin/b16 initPos/b16 initVelocity/str name
public final static int SERVO_ATTACH = 35;
// > servoAttachPin/deviceId/pin
public final static int SERVO_ATTACH_PIN = 36;
// > servoDetachPin/deviceId
public final static int SERVO_DETACH_PIN = 37;
// > servoSetVelocity/deviceId/b16 velocity
public final static int SERVO_SET_VELOCITY = 38;
// > servoSweepStart/deviceId/min/max/step
public final static int SERVO_SWEEP_START = 39;
// > servoSweepStop/deviceId
public final static int SERVO_SWEEP_STOP = 40;
// > servoMoveToMicroseconds/deviceId/b16 target
public final static int SERVO_MOVE_TO_MICROSECONDS = 41;
// > servoSetAcceleration/deviceId/b16 acceleration
public final static int SERVO_SET_ACCELERATION = 42;
// < publishServoEvent/deviceId/eventType/b16 currentPos/b16 targetPos
public final static int PUBLISH_SERVO_EVENT = 43;
// > serialAttach/deviceId/relayPin
public final static int SERIAL_ATTACH = 44;
// > serialRelay/deviceId/[] data
public final static int SERIAL_RELAY = 45;
// < publishSerialData/deviceId/[] data
public final static int PUBLISH_SERIAL_DATA = 46;
// > ultrasonicSensorAttach/deviceId/triggerPin/echoPin
public final static int ULTRASONIC_SENSOR_ATTACH = 47;
// > ultrasonicSensorStartRanging/deviceId
public final static int ULTRASONIC_SENSOR_START_RANGING = 48;
// > ultrasonicSensorStopRanging/deviceId
public final static int ULTRASONIC_SENSOR_STOP_RANGING = 49;
// < publishUltrasonicSensorData/deviceId/b16 echoTime
public final static int PUBLISH_ULTRASONIC_SENSOR_DATA = 50;
// > setAref/b16 type
public final static int SET_AREF = 51;
// > motorAttach/deviceId/type/[] pins
public final static int MOTOR_ATTACH = 52;
// > motorMove/deviceId/pwr
public final static int MOTOR_MOVE = 53;
// > motorMoveTo/deviceId/pos
public final static int MOTOR_MOVE_TO = 54;
// > encoderAttach/deviceId/type/pin
public final static int ENCODER_ATTACH = 55;
// > setZeroPoint/deviceId
public final static int SET_ZERO_POINT = 56;
// < publishEncoderData/deviceId/b16 position
public final static int PUBLISH_ENCODER_DATA = 57;
// < publishMrlCommBegin/version
public final static int PUBLISH_MRL_COMM_BEGIN = 58;
// > servoStop/deviceId
public final static int SERVO_STOP = 59;
/**
* These methods will be invoked from the Msg class as callbacks from MrlComm.
*/
// public void getBoardInfo(){}
// public void enablePin(Integer address/*byte*/, Integer type/*byte*/, Integer rate/*b16*/){}
// public void setDebug(Boolean enabled/*bool*/){}
// public void setSerialRate(Integer rate/*b32*/){}
// public void softReset(){}
// public void enableAck(Boolean enabled/*bool*/){}
// public void echo(Float myFloat/*f32*/, Integer myByte/*byte*/, Float secondFloat/*f32*/){}
// public void customMsg(int[] msg/*[]*/){}
// public void deviceDetach(Integer deviceId/*byte*/){}
// public void i2cBusAttach(Integer deviceId/*byte*/, Integer i2cBus/*byte*/){}
// public void i2cRead(Integer deviceId/*byte*/, Integer deviceAddress/*byte*/, Integer size/*byte*/){}
// public void i2cWrite(Integer deviceId/*byte*/, Integer deviceAddress/*byte*/, int[] data/*[]*/){}
// public void i2cWriteRead(Integer deviceId/*byte*/, Integer deviceAddress/*byte*/, Integer readSize/*byte*/, Integer writeValue/*byte*/){}
// public void neoPixelAttach(Integer deviceId/*byte*/, Integer pin/*byte*/, Integer numPixels/*b16*/, Integer depth/*byte*/){}
// public void neoPixelSetAnimation(Integer deviceId/*byte*/, Integer animation/*byte*/, Integer red/*byte*/, Integer green/*byte*/, Integer blue/*byte*/, Integer white/*byte*/, Integer wait_ms/*b32*/){}
// public void neoPixelWriteMatrix(Integer deviceId/*byte*/, int[] buffer/*[]*/){}
// public void neoPixelFill(Integer deviceId/*byte*/, Integer address/*b16*/, Integer count/*b16*/, Integer red/*byte*/, Integer green/*byte*/, Integer blue/*byte*/, Integer white/*byte*/){}
// public void neoPixelSetBrightness(Integer deviceId/*byte*/, Integer brightness/*byte*/){}
// public void neoPixelClear(Integer deviceId/*byte*/){}
// public void analogWrite(Integer pin/*byte*/, Integer value/*byte*/){}
// public void digitalWrite(Integer pin/*byte*/, Integer value/*byte*/){}
// public void disablePin(Integer pin/*byte*/){}
// public void disablePins(){}
// public void pinMode(Integer pin/*byte*/, Integer mode/*byte*/){}
// public void setTrigger(Integer pin/*byte*/, Integer triggerValue/*byte*/){}
// public void setDebounce(Integer pin/*byte*/, Integer delay/*byte*/){}
// public void servoAttach(Integer deviceId/*byte*/, Integer pin/*byte*/, Integer initPos/*b16*/, Integer initVelocity/*b16*/, String name/*str*/){}
// public void servoAttachPin(Integer deviceId/*byte*/, Integer pin/*byte*/){}
// public void servoDetachPin(Integer deviceId/*byte*/){}
// public void servoSetVelocity(Integer deviceId/*byte*/, Integer velocity/*b16*/){}
// public void servoSweepStart(Integer deviceId/*byte*/, Integer min/*byte*/, Integer max/*byte*/, Integer step/*byte*/){}
// public void servoSweepStop(Integer deviceId/*byte*/){}
// public void servoMoveToMicroseconds(Integer deviceId/*byte*/, Integer target/*b16*/){}
// public void servoSetAcceleration(Integer deviceId/*byte*/, Integer acceleration/*b16*/){}
// public void serialAttach(Integer deviceId/*byte*/, Integer relayPin/*byte*/){}
// public void serialRelay(Integer deviceId/*byte*/, int[] data/*[]*/){}
// public void ultrasonicSensorAttach(Integer deviceId/*byte*/, Integer triggerPin/*byte*/, Integer echoPin/*byte*/){}
// public void ultrasonicSensorStartRanging(Integer deviceId/*byte*/){}
// public void ultrasonicSensorStopRanging(Integer deviceId/*byte*/){}
// public void setAref(Integer type/*b16*/){}
// public void motorAttach(Integer deviceId/*byte*/, Integer type/*byte*/, int[] pins/*[]*/){}
// public void motorMove(Integer deviceId/*byte*/, Integer pwr/*byte*/){}
// public void motorMoveTo(Integer deviceId/*byte*/, Integer pos/*byte*/){}
// public void encoderAttach(Integer deviceId/*byte*/, Integer type/*byte*/, Integer pin/*byte*/){}
// public void setZeroPoint(Integer deviceId/*byte*/){}
// public void servoStop(Integer deviceId/*byte*/){}
public VirtualMsg(MrlComm arduino, SerialDevice serial) {
this.arduino = arduino;
this.serial = serial;
}
public void begin(SerialDevice serial){
this.serial = serial;
}
// transient private Msg instance;
// ArduinoSerialCallBacks - TODO - extract interface
transient private MrlComm arduino;
transient private SerialDevice serial;
public void processCommand(int[] ioCmd) {
int startPos = 0;
method = ioCmd[startPos];
// always process mrlbegin..
if (debug) {
log.info("Process Command: {} Method: {}", Msg.methodToString(method), ioCmd);
}
if (method == PUBLISH_ACK) {
// We saw an ack! we ack this internally right away, and down below in the generated code,
// call publishAck on the MrlCommPublisher
Integer function = ioCmd[startPos+1]; // bu8
ackReceived(function);
}
if (method != PUBLISH_MRL_COMM_BEGIN) {
if (!clearToSend) {
log.warn("Not Clear to send yet. Dumping command {}", ioCmd);
System.err.println("\nDumping command not clear to send.\n");
return;
}
} else {
// Process!
log.info("Clear to process!!!!!!!!!!!!!!!!!!");
this.clearToSend = true;
}
switch (method) {
case GET_BOARD_INFO: {
if(invoke){
arduino.invoke("getBoardInfo");
} else {
arduino.getBoardInfo();
}
break;
}
case ENABLE_PIN: {
Integer address = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer type = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer rate = b16(ioCmd, startPos+1);
startPos += 2; //b16
if(invoke){
arduino.invoke("enablePin", address, type, rate);
} else {
arduino.enablePin( address, type, rate);
}
break;
}
case SET_DEBUG: {
Boolean enabled = (ioCmd[startPos+1] == 0)?false:true;
startPos += 1;
if(invoke){
arduino.invoke("setDebug", enabled);
} else {
arduino.setDebug( enabled);
}
break;
}
case SET_SERIAL_RATE: {
Integer rate = b32(ioCmd, startPos+1);
startPos += 4; //b32
if(invoke){
arduino.invoke("setSerialRate", rate);
} else {
arduino.setSerialRate( rate);
}
break;
}
case SOFT_RESET: {
if(invoke){
arduino.invoke("softReset");
} else {
arduino.softReset();
}
break;
}
case ENABLE_ACK: {
Boolean enabled = (ioCmd[startPos+1] == 0)?false:true;
startPos += 1;
if(invoke){
arduino.invoke("enableAck", enabled);
} else {
arduino.enableAck( enabled);
}
break;
}
case ECHO: {
Float myFloat = f32(ioCmd, startPos+1);
startPos += 4; //f32
Integer myByte = ioCmd[startPos+1]; // bu8
startPos += 1;
Float secondFloat = f32(ioCmd, startPos+1);
startPos += 4; //f32
if(invoke){
arduino.invoke("echo", myFloat, myByte, secondFloat);
} else {
arduino.echo( myFloat, myByte, secondFloat);
}
break;
}
case CUSTOM_MSG: {
int[] msg = subArray(ioCmd, startPos+2, ioCmd[startPos+1]);
startPos += 1 + ioCmd[startPos+1];
if(invoke){
arduino.invoke("customMsg", msg);
} else {
arduino.customMsg( msg);
}
break;
}
case DEVICE_DETACH: {
Integer deviceId = ioCmd[startPos+1]; // bu8
startPos += 1;
if(invoke){
arduino.invoke("deviceDetach", deviceId);
} else {
arduino.deviceDetach( deviceId);
}
break;
}
case I2C_BUS_ATTACH: {
Integer deviceId = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer i2cBus = ioCmd[startPos+1]; // bu8
startPos += 1;
if(invoke){
arduino.invoke("i2cBusAttach", deviceId, i2cBus);
} else {
arduino.i2cBusAttach( deviceId, i2cBus);
}
break;
}
case I2C_READ: {
Integer deviceId = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer deviceAddress = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer size = ioCmd[startPos+1]; // bu8
startPos += 1;
if(invoke){
arduino.invoke("i2cRead", deviceId, deviceAddress, size);
} else {
arduino.i2cRead( deviceId, deviceAddress, size);
}
break;
}
case I2C_WRITE: {
Integer deviceId = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer deviceAddress = ioCmd[startPos+1]; // bu8
startPos += 1;
int[] data = subArray(ioCmd, startPos+2, ioCmd[startPos+1]);
startPos += 1 + ioCmd[startPos+1];
if(invoke){
arduino.invoke("i2cWrite", deviceId, deviceAddress, data);
} else {
arduino.i2cWrite( deviceId, deviceAddress, data);
}
break;
}
case I2C_WRITE_READ: {
Integer deviceId = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer deviceAddress = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer readSize = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer writeValue = ioCmd[startPos+1]; // bu8
startPos += 1;
if(invoke){
arduino.invoke("i2cWriteRead", deviceId, deviceAddress, readSize, writeValue);
} else {
arduino.i2cWriteRead( deviceId, deviceAddress, readSize, writeValue);
}
break;
}
case NEO_PIXEL_ATTACH: {
Integer deviceId = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer pin = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer numPixels = b16(ioCmd, startPos+1);
startPos += 2; //b16
Integer depth = ioCmd[startPos+1]; // bu8
startPos += 1;
if(invoke){
arduino.invoke("neoPixelAttach", deviceId, pin, numPixels, depth);
} else {
arduino.neoPixelAttach( deviceId, pin, numPixels, depth);
}
break;
}
case NEO_PIXEL_SET_ANIMATION: {
Integer deviceId = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer animation = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer red = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer green = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer blue = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer white = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer wait_ms = b32(ioCmd, startPos+1);
startPos += 4; //b32
if(invoke){
arduino.invoke("neoPixelSetAnimation", deviceId, animation, red, green, blue, white, wait_ms);
} else {
arduino.neoPixelSetAnimation( deviceId, animation, red, green, blue, white, wait_ms);
}
break;
}
case NEO_PIXEL_WRITE_MATRIX: {
Integer deviceId = ioCmd[startPos+1]; // bu8
startPos += 1;
int[] buffer = subArray(ioCmd, startPos+2, ioCmd[startPos+1]);
startPos += 1 + ioCmd[startPos+1];
if(invoke){
arduino.invoke("neoPixelWriteMatrix", deviceId, buffer);
} else {
arduino.neoPixelWriteMatrix( deviceId, buffer);
}
break;
}
case NEO_PIXEL_FILL: {
Integer deviceId = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer address = b16(ioCmd, startPos+1);
startPos += 2; //b16
Integer count = b16(ioCmd, startPos+1);
startPos += 2; //b16
Integer red = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer green = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer blue = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer white = ioCmd[startPos+1]; // bu8
startPos += 1;
if(invoke){
arduino.invoke("neoPixelFill", deviceId, address, count, red, green, blue, white);
} else {
arduino.neoPixelFill( deviceId, address, count, red, green, blue, white);
}
break;
}
case NEO_PIXEL_SET_BRIGHTNESS: {
Integer deviceId = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer brightness = ioCmd[startPos+1]; // bu8
startPos += 1;
if(invoke){
arduino.invoke("neoPixelSetBrightness", deviceId, brightness);
} else {
arduino.neoPixelSetBrightness( deviceId, brightness);
}
break;
}
case NEO_PIXEL_CLEAR: {
Integer deviceId = ioCmd[startPos+1]; // bu8
startPos += 1;
if(invoke){
arduino.invoke("neoPixelClear", deviceId);
} else {
arduino.neoPixelClear( deviceId);
}
break;
}
case ANALOG_WRITE: {
Integer pin = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer value = ioCmd[startPos+1]; // bu8
startPos += 1;
if(invoke){
arduino.invoke("analogWrite", pin, value);
} else {
arduino.analogWrite( pin, value);
}
break;
}
case DIGITAL_WRITE: {
Integer pin = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer value = ioCmd[startPos+1]; // bu8
startPos += 1;
if(invoke){
arduino.invoke("digitalWrite", pin, value);
} else {
arduino.digitalWrite( pin, value);
}
break;
}
case DISABLE_PIN: {
Integer pin = ioCmd[startPos+1]; // bu8
startPos += 1;
if(invoke){
arduino.invoke("disablePin", pin);
} else {
arduino.disablePin( pin);
}
break;
}
case DISABLE_PINS: {
if(invoke){
arduino.invoke("disablePins");
} else {
arduino.disablePins();
}
break;
}
case PIN_MODE: {
Integer pin = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer mode = ioCmd[startPos+1]; // bu8
startPos += 1;
if(invoke){
arduino.invoke("pinMode", pin, mode);
} else {
arduino.pinMode( pin, mode);
}
break;
}
case SET_TRIGGER: {
Integer pin = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer triggerValue = ioCmd[startPos+1]; // bu8
startPos += 1;
if(invoke){
arduino.invoke("setTrigger", pin, triggerValue);
} else {
arduino.setTrigger( pin, triggerValue);
}
break;
}
case SET_DEBOUNCE: {
Integer pin = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer delay = ioCmd[startPos+1]; // bu8
startPos += 1;
if(invoke){
arduino.invoke("setDebounce", pin, delay);
} else {
arduino.setDebounce( pin, delay);
}
break;
}
case SERVO_ATTACH: {
Integer deviceId = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer pin = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer initPos = b16(ioCmd, startPos+1);
startPos += 2; //b16
Integer initVelocity = b16(ioCmd, startPos+1);
startPos += 2; //b16
String name = str(ioCmd, startPos+2, ioCmd[startPos+1]);
startPos += 1 + ioCmd[startPos+1];
if(invoke){
arduino.invoke("servoAttach", deviceId, pin, initPos, initVelocity, name);
} else {
arduino.servoAttach( deviceId, pin, initPos, initVelocity, name);
}
break;
}
case SERVO_ATTACH_PIN: {
Integer deviceId = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer pin = ioCmd[startPos+1]; // bu8
startPos += 1;
if(invoke){
arduino.invoke("servoAttachPin", deviceId, pin);
} else {
arduino.servoAttachPin( deviceId, pin);
}
break;
}
case SERVO_DETACH_PIN: {
Integer deviceId = ioCmd[startPos+1]; // bu8
startPos += 1;
if(invoke){
arduino.invoke("servoDetachPin", deviceId);
} else {
arduino.servoDetachPin( deviceId);
}
break;
}
case SERVO_SET_VELOCITY: {
Integer deviceId = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer velocity = b16(ioCmd, startPos+1);
startPos += 2; //b16
if(invoke){
arduino.invoke("servoSetVelocity", deviceId, velocity);
} else {
arduino.servoSetVelocity( deviceId, velocity);
}
break;
}
case SERVO_SWEEP_START: {
Integer deviceId = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer min = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer max = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer step = ioCmd[startPos+1]; // bu8
startPos += 1;
if(invoke){
arduino.invoke("servoSweepStart", deviceId, min, max, step);
} else {
arduino.servoSweepStart( deviceId, min, max, step);
}
break;
}
case SERVO_SWEEP_STOP: {
Integer deviceId = ioCmd[startPos+1]; // bu8
startPos += 1;
if(invoke){
arduino.invoke("servoSweepStop", deviceId);
} else {
arduino.servoSweepStop( deviceId);
}
break;
}
case SERVO_MOVE_TO_MICROSECONDS: {
Integer deviceId = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer target = b16(ioCmd, startPos+1);
startPos += 2; //b16
if(invoke){
arduino.invoke("servoMoveToMicroseconds", deviceId, target);
} else {
arduino.servoMoveToMicroseconds( deviceId, target);
}
break;
}
case SERVO_SET_ACCELERATION: {
Integer deviceId = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer acceleration = b16(ioCmd, startPos+1);
startPos += 2; //b16
if(invoke){
arduino.invoke("servoSetAcceleration", deviceId, acceleration);
} else {
arduino.servoSetAcceleration( deviceId, acceleration);
}
break;
}
case SERIAL_ATTACH: {
Integer deviceId = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer relayPin = ioCmd[startPos+1]; // bu8
startPos += 1;
if(invoke){
arduino.invoke("serialAttach", deviceId, relayPin);
} else {
arduino.serialAttach( deviceId, relayPin);
}
break;
}
case SERIAL_RELAY: {
Integer deviceId = ioCmd[startPos+1]; // bu8
startPos += 1;
int[] data = subArray(ioCmd, startPos+2, ioCmd[startPos+1]);
startPos += 1 + ioCmd[startPos+1];
if(invoke){
arduino.invoke("serialRelay", deviceId, data);
} else {
arduino.serialRelay( deviceId, data);
}
break;
}
case ULTRASONIC_SENSOR_ATTACH: {
Integer deviceId = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer triggerPin = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer echoPin = ioCmd[startPos+1]; // bu8
startPos += 1;
if(invoke){
arduino.invoke("ultrasonicSensorAttach", deviceId, triggerPin, echoPin);
} else {
arduino.ultrasonicSensorAttach( deviceId, triggerPin, echoPin);
}
break;
}
case ULTRASONIC_SENSOR_START_RANGING: {
Integer deviceId = ioCmd[startPos+1]; // bu8
startPos += 1;
if(invoke){
arduino.invoke("ultrasonicSensorStartRanging", deviceId);
} else {
arduino.ultrasonicSensorStartRanging( deviceId);
}
break;
}
case ULTRASONIC_SENSOR_STOP_RANGING: {
Integer deviceId = ioCmd[startPos+1]; // bu8
startPos += 1;
if(invoke){
arduino.invoke("ultrasonicSensorStopRanging", deviceId);
} else {
arduino.ultrasonicSensorStopRanging( deviceId);
}
break;
}
case SET_AREF: {
Integer type = b16(ioCmd, startPos+1);
startPos += 2; //b16
if(invoke){
arduino.invoke("setAref", type);
} else {
arduino.setAref( type);
}
break;
}
case MOTOR_ATTACH: {
Integer deviceId = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer type = ioCmd[startPos+1]; // bu8
startPos += 1;
int[] pins = subArray(ioCmd, startPos+2, ioCmd[startPos+1]);
startPos += 1 + ioCmd[startPos+1];
if(invoke){
arduino.invoke("motorAttach", deviceId, type, pins);
} else {
arduino.motorAttach( deviceId, type, pins);
}
break;
}
case MOTOR_MOVE: {
Integer deviceId = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer pwr = ioCmd[startPos+1]; // bu8
startPos += 1;
if(invoke){
arduino.invoke("motorMove", deviceId, pwr);
} else {
arduino.motorMove( deviceId, pwr);
}
break;
}
case MOTOR_MOVE_TO: {
Integer deviceId = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer pos = ioCmd[startPos+1]; // bu8
startPos += 1;
if(invoke){
arduino.invoke("motorMoveTo", deviceId, pos);
} else {
arduino.motorMoveTo( deviceId, pos);
}
break;
}
case ENCODER_ATTACH: {
Integer deviceId = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer type = ioCmd[startPos+1]; // bu8
startPos += 1;
Integer pin = ioCmd[startPos+1]; // bu8
startPos += 1;
if(invoke){
arduino.invoke("encoderAttach", deviceId, type, pin);
} else {
arduino.encoderAttach( deviceId, type, pin);
}
break;
}
case SET_ZERO_POINT: {
Integer deviceId = ioCmd[startPos+1]; // bu8
startPos += 1;
if(invoke){
arduino.invoke("setZeroPoint", deviceId);
} else {
arduino.setZeroPoint( deviceId);
}
break;
}
case SERVO_STOP: {
Integer deviceId = ioCmd[startPos+1]; // bu8
startPos += 1;
if(invoke){
arduino.invoke("servoStop", deviceId);
} else {
arduino.servoStop( deviceId);
}
break;
}
}
}
// Java-land --to--> MrlComm
public synchronized byte[] publishMRLCommError(String errorMsg/*str*/) {
if (debug) {
log.info("Sending Message: publishMRLCommError to {}", serial.getName());
}
ByteArrayOutputStream baos = new ByteArrayOutputStream();
try {
appendMessage(baos, MAGIC_NUMBER);
appendMessage(baos, 1 + (1 + errorMsg.length())); // size
appendMessage(baos, PUBLISH_MRLCOMM_ERROR); // msgType = 1
appendMessage(baos, errorMsg);
byte[] message = sendMessage(baos);
if (ackEnabled){
waitForAck();
}
if(record != null){
txBuffer.append("> publishMRLCommError");
txBuffer.append("/");
txBuffer.append(errorMsg);
txBuffer.append("\n");
record.write(txBuffer.toString().getBytes());
txBuffer.setLength(0);
}
return message;
} catch (Exception e) {
log.error("publishMRLCommError threw",e);
return null;
}
}
public synchronized byte[] publishBoardInfo(Integer version/*byte*/, Integer boardType/*byte*/, Integer microsPerLoop/*b16*/, Integer sram/*b16*/, Integer activePins/*byte*/, int[] deviceSummary/*[]*/) {
if (debug) {
log.info("Sending Message: publishBoardInfo to {}", serial.getName());
}
ByteArrayOutputStream baos = new ByteArrayOutputStream();
try {
appendMessage(baos, MAGIC_NUMBER);
appendMessage(baos, 1 + 1 + 1 + 2 + 2 + 1 + (1 + deviceSummary.length)); // size
appendMessage(baos, PUBLISH_BOARD_INFO); // msgType = 3
appendMessage(baos, version);
appendMessage(baos, boardType);
appendMessageb16(baos, microsPerLoop);
appendMessageb16(baos, sram);
appendMessage(baos, activePins);
appendMessage(baos, deviceSummary);
byte[] message = sendMessage(baos);
if (ackEnabled){
waitForAck();
}
if(record != null){
txBuffer.append("> publishBoardInfo");
txBuffer.append("/");
txBuffer.append(version);
txBuffer.append("/");
txBuffer.append(boardType);
txBuffer.append("/");
txBuffer.append(microsPerLoop);
txBuffer.append("/");
txBuffer.append(sram);
txBuffer.append("/");
txBuffer.append(activePins);
txBuffer.append("/");
txBuffer.append(Arrays.toString(deviceSummary));
txBuffer.append("\n");
record.write(txBuffer.toString().getBytes());
txBuffer.setLength(0);
}
return message;
} catch (Exception e) {
log.error("publishBoardInfo threw",e);
return null;
}
}
public synchronized byte[] publishAck(Integer function/*byte*/) {
if (debug) {
log.info("Sending Message: publishAck to {}", serial.getName());
}
ByteArrayOutputStream baos = new ByteArrayOutputStream();
try {
appendMessage(baos, MAGIC_NUMBER);
appendMessage(baos, 1 + 1); // size
appendMessage(baos, PUBLISH_ACK); // msgType = 9
appendMessage(baos, function);
byte[] message = sendMessage(baos);
if (ackEnabled){
waitForAck();
}
if(record != null){
txBuffer.append("> publishAck");
txBuffer.append("/");
txBuffer.append(function);
txBuffer.append("\n");
record.write(txBuffer.toString().getBytes());
txBuffer.setLength(0);
}
return message;
} catch (Exception e) {
log.error("publishAck threw",e);
return null;
}
}
public synchronized byte[] publishEcho(Float myFloat/*f32*/, Integer myByte/*byte*/, Float secondFloat/*f32*/) {
if (debug) {
log.info("Sending Message: publishEcho to {}", serial.getName());
}
ByteArrayOutputStream baos = new ByteArrayOutputStream();
try {
appendMessage(baos, MAGIC_NUMBER);
appendMessage(baos, 1 + 4 + 1 + 4); // size
appendMessage(baos, PUBLISH_ECHO); // msgType = 11
appendMessagef32(baos, myFloat);
appendMessage(baos, myByte);
appendMessagef32(baos, secondFloat);
byte[] message = sendMessage(baos);
if (ackEnabled){
waitForAck();
}
if(record != null){
txBuffer.append("> publishEcho");
txBuffer.append("/");
txBuffer.append(myFloat);
txBuffer.append("/");
txBuffer.append(myByte);
txBuffer.append("/");
txBuffer.append(secondFloat);
txBuffer.append("\n");
record.write(txBuffer.toString().getBytes());
txBuffer.setLength(0);
}
return message;
} catch (Exception e) {
log.error("publishEcho threw",e);
return null;
}
}
public synchronized byte[] publishCustomMsg(int[] msg/*[]*/) {
if (debug) {
log.info("Sending Message: publishCustomMsg to {}", serial.getName());
}
ByteArrayOutputStream baos = new ByteArrayOutputStream();
try {
appendMessage(baos, MAGIC_NUMBER);
appendMessage(baos, 1 + (1 + msg.length)); // size
appendMessage(baos, PUBLISH_CUSTOM_MSG); // msgType = 13
appendMessage(baos, msg);
byte[] message = sendMessage(baos);
if (ackEnabled){
waitForAck();
}
if(record != null){
txBuffer.append("> publishCustomMsg");
txBuffer.append("/");
txBuffer.append(Arrays.toString(msg));
txBuffer.append("\n");
record.write(txBuffer.toString().getBytes());
txBuffer.setLength(0);
}
return message;
} catch (Exception e) {
log.error("publishCustomMsg threw",e);
return null;
}
}
public synchronized byte[] publishI2cData(Integer deviceId/*byte*/, int[] data/*[]*/) {
if (debug) {
log.info("Sending Message: publishI2cData to {}", serial.getName());
}
ByteArrayOutputStream baos = new ByteArrayOutputStream();
try {
appendMessage(baos, MAGIC_NUMBER);
appendMessage(baos, 1 + 1 + (1 + data.length)); // size
appendMessage(baos, PUBLISH_I2C_DATA); // msgType = 19
appendMessage(baos, deviceId);
appendMessage(baos, data);
byte[] message = sendMessage(baos);
if (ackEnabled){
waitForAck();
}
if(record != null){
txBuffer.append("> publishI2cData");
txBuffer.append("/");
txBuffer.append(deviceId);
txBuffer.append("/");
txBuffer.append(Arrays.toString(data));
txBuffer.append("\n");
record.write(txBuffer.toString().getBytes());
txBuffer.setLength(0);
}
return message;
} catch (Exception e) {
log.error("publishI2cData threw",e);
return null;
}
}
public synchronized byte[] publishDebug(String debugMsg/*str*/) {
if (debug) {
log.info("Sending Message: publishDebug to {}", serial.getName());
}
ByteArrayOutputStream baos = new ByteArrayOutputStream();
try {
appendMessage(baos, MAGIC_NUMBER);
appendMessage(baos, 1 + (1 + debugMsg.length())); // size
appendMessage(baos, PUBLISH_DEBUG); // msgType = 31
appendMessage(baos, debugMsg);
byte[] message = sendMessage(baos);
if (ackEnabled){
waitForAck();
}
if(record != null){
txBuffer.append("> publishDebug");
txBuffer.append("/");
txBuffer.append(debugMsg);
txBuffer.append("\n");
record.write(txBuffer.toString().getBytes());
txBuffer.setLength(0);
}
return message;
} catch (Exception e) {
log.error("publishDebug threw",e);
return null;
}
}
public synchronized byte[] publishPinArray(int[] data/*[]*/) {
if (debug) {
log.info("Sending Message: publishPinArray to {}", serial.getName());
}
ByteArrayOutputStream baos = new ByteArrayOutputStream();
try {
appendMessage(baos, MAGIC_NUMBER);
appendMessage(baos, 1 + (1 + data.length)); // size
appendMessage(baos, PUBLISH_PIN_ARRAY); // msgType = 32
appendMessage(baos, data);
byte[] message = sendMessage(baos);
if (ackEnabled){
waitForAck();
}
if(record != null){
txBuffer.append("> publishPinArray");
txBuffer.append("/");
txBuffer.append(Arrays.toString(data));
txBuffer.append("\n");
record.write(txBuffer.toString().getBytes());
txBuffer.setLength(0);
}
return message;
} catch (Exception e) {
log.error("publishPinArray threw",e);
return null;
}
}
public synchronized byte[] publishServoEvent(Integer deviceId/*byte*/, Integer eventType/*byte*/, Integer currentPos/*b16*/, Integer targetPos/*b16*/) {
if (debug) {
log.info("Sending Message: publishServoEvent to {}", serial.getName());
}
ByteArrayOutputStream baos = new ByteArrayOutputStream();
try {
appendMessage(baos, MAGIC_NUMBER);
appendMessage(baos, 1 + 1 + 1 + 2 + 2); // size
appendMessage(baos, PUBLISH_SERVO_EVENT); // msgType = 43
appendMessage(baos, deviceId);
appendMessage(baos, eventType);
appendMessageb16(baos, currentPos);
appendMessageb16(baos, targetPos);
byte[] message = sendMessage(baos);
if (ackEnabled){
waitForAck();
}
if(record != null){
txBuffer.append("> publishServoEvent");
txBuffer.append("/");
txBuffer.append(deviceId);
txBuffer.append("/");
txBuffer.append(eventType);
txBuffer.append("/");
txBuffer.append(currentPos);
txBuffer.append("/");
txBuffer.append(targetPos);
txBuffer.append("\n");
record.write(txBuffer.toString().getBytes());
txBuffer.setLength(0);
}
return message;
} catch (Exception e) {
log.error("publishServoEvent threw",e);
return null;
}
}
public synchronized byte[] publishSerialData(Integer deviceId/*byte*/, int[] data/*[]*/) {
if (debug) {
log.info("Sending Message: publishSerialData to {}", serial.getName());
}
ByteArrayOutputStream baos = new ByteArrayOutputStream();
try {
appendMessage(baos, MAGIC_NUMBER);
appendMessage(baos, 1 + 1 + (1 + data.length)); // size
appendMessage(baos, PUBLISH_SERIAL_DATA); // msgType = 46
appendMessage(baos, deviceId);
appendMessage(baos, data);
byte[] message = sendMessage(baos);
if (ackEnabled){
waitForAck();
}
if(record != null){
txBuffer.append("> publishSerialData");
txBuffer.append("/");
txBuffer.append(deviceId);
txBuffer.append("/");
txBuffer.append(Arrays.toString(data));
txBuffer.append("\n");
record.write(txBuffer.toString().getBytes());
txBuffer.setLength(0);
}
return message;
} catch (Exception e) {
log.error("publishSerialData threw",e);
return null;
}
}
public synchronized byte[] publishUltrasonicSensorData(Integer deviceId/*byte*/, Integer echoTime/*b16*/) {
if (debug) {
log.info("Sending Message: publishUltrasonicSensorData to {}", serial.getName());
}
ByteArrayOutputStream baos = new ByteArrayOutputStream();
try {
appendMessage(baos, MAGIC_NUMBER);
appendMessage(baos, 1 + 1 + 2); // size
appendMessage(baos, PUBLISH_ULTRASONIC_SENSOR_DATA); // msgType = 50
appendMessage(baos, deviceId);
appendMessageb16(baos, echoTime);
byte[] message = sendMessage(baos);
if (ackEnabled){
waitForAck();
}
if(record != null){
txBuffer.append("> publishUltrasonicSensorData");
txBuffer.append("/");
txBuffer.append(deviceId);
txBuffer.append("/");
txBuffer.append(echoTime);
txBuffer.append("\n");
record.write(txBuffer.toString().getBytes());
txBuffer.setLength(0);
}
return message;
} catch (Exception e) {
log.error("publishUltrasonicSensorData threw",e);
return null;
}
}
public synchronized byte[] publishEncoderData(Integer deviceId/*byte*/, Integer position/*b16*/) {
if (debug) {
log.info("Sending Message: publishEncoderData to {}", serial.getName());
}
ByteArrayOutputStream baos = new ByteArrayOutputStream();
try {
appendMessage(baos, MAGIC_NUMBER);
appendMessage(baos, 1 + 1 + 2); // size
appendMessage(baos, PUBLISH_ENCODER_DATA); // msgType = 57
appendMessage(baos, deviceId);
appendMessageb16(baos, position);
byte[] message = sendMessage(baos);
if (ackEnabled){
waitForAck();
}
if(record != null){
txBuffer.append("> publishEncoderData");
txBuffer.append("/");
txBuffer.append(deviceId);
txBuffer.append("/");
txBuffer.append(position);
txBuffer.append("\n");
record.write(txBuffer.toString().getBytes());
txBuffer.setLength(0);
}
return message;
} catch (Exception e) {
log.error("publishEncoderData threw",e);
return null;
}
}
public synchronized byte[] publishMrlCommBegin(Integer version/*byte*/) {
if (debug) {
log.info("Sending Message: publishMrlCommBegin to {}", serial.getName());
}
ByteArrayOutputStream baos = new ByteArrayOutputStream();
try {
appendMessage(baos, MAGIC_NUMBER);
appendMessage(baos, 1 + 1); // size
appendMessage(baos, PUBLISH_MRL_COMM_BEGIN); // msgType = 58
appendMessage(baos, version);
byte[] message = sendMessage(baos);
if (ackEnabled){
waitForAck();
}
if(record != null){
txBuffer.append("> publishMrlCommBegin");
txBuffer.append("/");
txBuffer.append(version);
txBuffer.append("\n");
record.write(txBuffer.toString().getBytes());
txBuffer.setLength(0);
}
return message;
} catch (Exception e) {
log.error("publishMrlCommBegin threw",e);
return null;
}
}
public static String methodToString(int method) {
switch (method) {
case PUBLISH_MRLCOMM_ERROR:{
return "publishMRLCommError";
}
case GET_BOARD_INFO:{
return "getBoardInfo";
}
case PUBLISH_BOARD_INFO:{
return "publishBoardInfo";
}
case ENABLE_PIN:{
return "enablePin";
}
case SET_DEBUG:{
return "setDebug";
}
case SET_SERIAL_RATE:{
return "setSerialRate";
}
case SOFT_RESET:{
return "softReset";
}
case ENABLE_ACK:{
return "enableAck";
}
case PUBLISH_ACK:{
return "publishAck";
}
case ECHO:{
return "echo";
}
case PUBLISH_ECHO:{
return "publishEcho";
}
case CUSTOM_MSG:{
return "customMsg";
}
case PUBLISH_CUSTOM_MSG:{
return "publishCustomMsg";
}
case DEVICE_DETACH:{
return "deviceDetach";
}
case I2C_BUS_ATTACH:{
return "i2cBusAttach";
}
case I2C_READ:{
return "i2cRead";
}
case I2C_WRITE:{
return "i2cWrite";
}
case I2C_WRITE_READ:{
return "i2cWriteRead";
}
case PUBLISH_I2C_DATA:{
return "publishI2cData";
}
case NEO_PIXEL_ATTACH:{
return "neoPixelAttach";
}
case NEO_PIXEL_SET_ANIMATION:{
return "neoPixelSetAnimation";
}
case NEO_PIXEL_WRITE_MATRIX:{
return "neoPixelWriteMatrix";
}
case NEO_PIXEL_FILL:{
return "neoPixelFill";
}
case NEO_PIXEL_SET_BRIGHTNESS:{
return "neoPixelSetBrightness";
}
case NEO_PIXEL_CLEAR:{
return "neoPixelClear";
}
case ANALOG_WRITE:{
return "analogWrite";
}
case DIGITAL_WRITE:{
return "digitalWrite";
}
case DISABLE_PIN:{
return "disablePin";
}
case DISABLE_PINS:{
return "disablePins";
}
case PIN_MODE:{
return "pinMode";
}
case PUBLISH_DEBUG:{
return "publishDebug";
}
case PUBLISH_PIN_ARRAY:{
return "publishPinArray";
}
case SET_TRIGGER:{
return "setTrigger";
}
case SET_DEBOUNCE:{
return "setDebounce";
}
case SERVO_ATTACH:{
return "servoAttach";
}
case SERVO_ATTACH_PIN:{
return "servoAttachPin";
}
case SERVO_DETACH_PIN:{
return "servoDetachPin";
}
case SERVO_SET_VELOCITY:{
return "servoSetVelocity";
}
case SERVO_SWEEP_START:{
return "servoSweepStart";
}
case SERVO_SWEEP_STOP:{
return "servoSweepStop";
}
case SERVO_MOVE_TO_MICROSECONDS:{
return "servoMoveToMicroseconds";
}
case SERVO_SET_ACCELERATION:{
return "servoSetAcceleration";
}
case PUBLISH_SERVO_EVENT:{
return "publishServoEvent";
}
case SERIAL_ATTACH:{
return "serialAttach";
}
case SERIAL_RELAY:{
return "serialRelay";
}
case PUBLISH_SERIAL_DATA:{
return "publishSerialData";
}
case ULTRASONIC_SENSOR_ATTACH:{
return "ultrasonicSensorAttach";
}
case ULTRASONIC_SENSOR_START_RANGING:{
return "ultrasonicSensorStartRanging";
}
case ULTRASONIC_SENSOR_STOP_RANGING:{
return "ultrasonicSensorStopRanging";
}
case PUBLISH_ULTRASONIC_SENSOR_DATA:{
return "publishUltrasonicSensorData";
}
case SET_AREF:{
return "setAref";
}
case MOTOR_ATTACH:{
return "motorAttach";
}
case MOTOR_MOVE:{
return "motorMove";
}
case MOTOR_MOVE_TO:{
return "motorMoveTo";
}
case ENCODER_ATTACH:{
return "encoderAttach";
}
case SET_ZERO_POINT:{
return "setZeroPoint";
}
case PUBLISH_ENCODER_DATA:{
return "publishEncoderData";
}
case PUBLISH_MRL_COMM_BEGIN:{
return "publishMrlCommBegin";
}
case SERVO_STOP:{
return "servoStop";
}
default: {
return "ERROR UNKNOWN METHOD (" + Integer.toString(method) + ")";
} // default
}
}
public String str(int[] buffer, int start, int size) {
byte[] b = new byte[size];
for (int i = start; i < start + size; ++i){
b[i - start] = (byte)(buffer[i] & 0xFF);
}
return new String(b);
}
public int[] subArray(int[] buffer, int start, int size) {
return Arrays.copyOfRange(buffer, start, start + size);
}
// signed 16 bit bucket
public int b16(int[] buffer, int start/*=0*/) {
return (short)(buffer[start] << 8) + buffer[start + 1];
}
// signed 32 bit bucket
public int b32(int[] buffer, int start/*=0*/) {
return ((buffer[start + 0] << 24) + (buffer[start + 1] << 16)
+ (buffer[start + 2] << 8) + buffer[start + 3]);
}
// unsigned 32 bit bucket
public long bu32(int[] buffer, int start/*=0*/) {
long ret = ((buffer[start + 0] << 24)
+ (buffer[start + 1] << 16)
+ (buffer[start + 2] << 8) + buffer[start + 3]);
if (ret < 0){
return 4294967296L + ret;
}
return ret;
}
// float 32 bit bucket
public float f32(int[] buffer, int start/*=0*/) {
byte[] b = new byte[4];
for (int i = 0; i < 4; ++i){
b[i] = (byte)buffer[start + i];
}
float f = ByteBuffer.wrap(b).order(ByteOrder.BIG_ENDIAN).getFloat();
return f;
}
public void onBytes(byte[] bytes) {
if (debug) {
// debug message.. semi-human readable?
String byteString = StringUtil.byteArrayToIntString(bytes);
log.info("onBytes called byteCount: {} data: >{}<", byteCount, byteString);
}
// this gives us the current full buffer that was read from the seral
for (int i = 0 ; i < bytes.length; i++) {
// For now, let's just call onByte for each byte upcasted as an int.
Integer newByte = bytes[i] & 0xFF;
try {
byteCount.incrementAndGet();
if (byteCount.get() == 1) {
if (newByte != MAGIC_NUMBER) {
byteCount = new AtomicInteger(0);
msgSize = 0;
Arrays.fill(ioCmd, 0); // FIXME - optimize - remove
// warn(String.format("Arduino->MRL error - bad magic number %d - %d rx errors", newByte, ++errorServiceToHardwareRxCnt));
log.warn("Arduino->MRL error - bad magic number {} - {} rx errors", newByte, ++errorServiceToHardwareRxCnt);
}
continue;
} else if (byteCount.get() == 2) {
// get the size of message
if (newByte > 64) {
byteCount = new AtomicInteger(0);
msgSize = 0;
// This is an error scenario.. we should reset our byte count also.
// error(String.format("Arduino->MRL error %d rx sz errors", ++errorServiceToHardwareRxCnt ));
log.error("Arduino->MRL error {} rx sz errors", ++errorServiceToHardwareRxCnt);
continue;
}
msgSize = newByte.intValue();
} else if (byteCount.get() == 3) {
// This is the method..
int method = newByte.intValue();
if (methodToString(method).startsWith("ERROR")) {
// we've got an error scenario here.. reset the parser and try again!
log.error("Arduino->MRL error unknown method error. resetting parser.");
byteCount = new AtomicInteger(0);
msgSize = 0;
continue;
}
// If we're not clear to send, we need to unlock if this is a begin message.
if (!clearToSend && (method == Msg.PUBLISH_MRL_COMM_BEGIN)) {
// Clear to send!!
log.info("Saw the MRL COMM BEGIN!!!!!!!!!!!!! Clear To Send.");
clearToSend = true;
}
if (!clearToSend) {
log.warn("NOT CLEAR TO SEND! resetting parser!");
// We opened the port, and we got some data that isn't a Begin message.
// so, I think we need to reset the parser and continue processing bytes...
// there will be errors until the next magic byte is seen.
byteCount = new AtomicInteger(0);
msgSize = 0;
continue;
}
// we are in a valid parse state.
ioCmd[byteCount.get() - 3] = method;
} else if (byteCount.get() > 3) {
// This is the body of the message copy it to the buffer
ioCmd[byteCount.get() - 3] = newByte.intValue();
} else {
// the case where byteCount is negative?! not got. You should probably never see this.
log.warn("MRL error rx zero/negative size error: {} {}", byteCount, Arrays.copyOf(ioCmd, byteCount.get()));
//error(String.format("Arduino->MRL error %d rx negsz errors", ++errorServiceToHardwareRxCnt));
continue;
}
// we have a complete message here.
if (byteCount.get() == 2 + msgSize) {
// we've received a full message
int[] actualCommand = Arrays.copyOf(ioCmd, byteCount.get()-2);
if (debug) {
log.info("Full message received: {} Data:{}", VirtualMsg.methodToString(ioCmd[0]), actualCommand);
}
// process the command.
processCommand(actualCommand);
publishAck(method);
// re-init parser
Arrays.fill(ioCmd, 0); // optimize remove
msgSize = 0;
byteCount = new AtomicInteger(0);
}
} catch (Exception e) {
++errorHardwareToServiceRxCnt ;
// error("msg structure violation %d", errorHardwareToServiceRxCnt);
log.warn("msg_structure violation byteCount {} buffer {}", byteCount, Arrays.copyOf(ioCmd, byteCount.get()), e);
// TODO: perhaps we could find the first occurance of 170.. and then attempt to re-parse at that point.
// find the first occurance of 170 in the bytes subbytes
// Maybe we can just walk the iterater back to the beginning based on the byte count .. and advance it by 1.. and continue.
i = i - byteCount.get()+1;
log.error("Trying to resume parsing the byte stream at position {} bytecount: {}", i, byteCount);
log.error("Original Byte Array: {}", StringUtil.byteArrayToIntString(bytes));
System.err.println("Try to consume more messages!");
msgSize = 0;
byteCount = new AtomicInteger(0);
// TODO: this is wonky.. what?!
i = 0;
return;
}
}
return;
}
String F(String msg) {
return msg;
}
public void publishError(String error) {
log.error(error);
}
void appendMessage(ByteArrayOutputStream baos, int b8) throws Exception {
if ((b8 < 0) || (b8 > 255)) {
log.error("writeByte overrun - should be 0 <= value <= 255 - value = {}", b8);
}
baos.write(b8 & 0xFF);
}
void appendMessagebool(ByteArrayOutputStream baos, boolean b1) throws Exception {
if (b1) {
appendMessage(baos, 1);
} else {
appendMessage(baos, 0);
}
}
void appendMessageb16(ByteArrayOutputStream baos, int b16) throws Exception {
if ((b16 < -32768) || (b16 > 32767)) {
log.error("writeByte overrun - should be -32,768 <= value <= 32,767 - value = {}", b16);
}
appendMessage(baos, b16 >> 8 & 0xFF);
appendMessage(baos, b16 & 0xFF);
}
void appendMessageb32(ByteArrayOutputStream baos, int b32) throws Exception {
appendMessage(baos, b32 >> 24 & 0xFF);
appendMessage(baos, b32 >> 16 & 0xFF);
appendMessage(baos, b32 >> 8 & 0xFF);
appendMessage(baos, b32 & 0xFF);
}
void appendMessagef32(ByteArrayOutputStream baos, float f32) throws Exception {
// int x = Float.floatToIntBits(f32);
byte[] f = ByteBuffer.allocate(4).order(ByteOrder.BIG_ENDIAN).putFloat(f32).array();
appendMessage(baos, f[3] & 0xFF);
appendMessage(baos, f[2] & 0xFF);
appendMessage(baos, f[1] & 0xFF);
appendMessage(baos, f[0] & 0xFF);
}
void appendMessagebu32(ByteArrayOutputStream baos, long b32) throws Exception {
appendMessage(baos, (int)(b32 >> 24 & 0xFF));
appendMessage(baos, (int)(b32 >> 16 & 0xFF));
appendMessage(baos, (int)(b32 >> 8 & 0xFF));
appendMessage(baos, (int)(b32 & 0xFF));
}
void appendMessage(ByteArrayOutputStream baos, String str) throws Exception {
appendMessage(baos, str.getBytes());
}
void appendMessage(ByteArrayOutputStream baos, int[] array) throws Exception {
// write size
appendMessage(baos, array.length & 0xFF);
// write data
for (int i = 0; i < array.length; ++i) {
appendMessage(baos, array[i] & 0xFF);
}
}
void appendMessage(ByteArrayOutputStream baos, byte[] array) throws Exception {
// write size
appendMessage(baos, array.length);
// write data
for (int i = 0; i < array.length; ++i) {
appendMessage(baos, array[i]);
}
}
synchronized byte[] sendMessage(ByteArrayOutputStream baos) throws Exception {
byte[] message = baos.toByteArray();
if (message.length > MAX_MSG_SIZE) {
log.error("**** message size {} > MAX_MSG_SIZE {} - not sending ****", MAX_MSG_SIZE, message.length);
return message;
}
if (ackEnabled) {
// wait for a pending ack to be received before we process our message.^M
waitForAck();
}
// write data if serial not null.
if (serial != null) {
// mark it pending before we write the data.
if (ackEnabled){
// flip our flag because we're going to send the message now.
// TODO: is this deadlocked because it's synchronized?!
// TODO: should this be set regardless of if the serial is null?
markPending();
}
serial.write(message);
// TODO: if there's an exception, we should clear our pending status?
if (ackEnabled) {
// wait for a pending ack to be received before we process our message.^M
waitForAck();
}
}
return message;
}
public void markPending() {
if (debug) {
log.info("Setting pending flag.");
}
synchronized (ackRecievedLock) {
ackRecievedLock.pendingMessage = true;
ackRecievedLock.notifyAll();
}
}
public boolean isRecording() {
return record != null;
}
public void record() throws Exception {
if (record == null) {
record = new FileOutputStream(String.format("%s.ard", arduino.getName()));
}
}
public void stopRecording() {
if (record != null) {
try {
record.close();
} catch (Exception e) {
log.info("Error closing recording stream. ", e);
}
record = null;
}
}
public static String deviceTypeToString(int typeId) {
switch(typeId){
case 0 : {
return "unknown";
}
case 1 : {
return "Arduino";
}
case 2 : {
return "UltrasonicSensor";
}
case 3 : {
return "Stepper";
}
case 4 : {
return "Motor";
}
case 5 : {
return "Servo";
}
case 6 : {
return "Serial";
}
case 7 : {
return "I2c";
}
case 8 : {
return "NeoPixel";
}
case 9 : {
return "Encoder";
}
default: {
return "unknown";
}
}
}
public void enableAcks(boolean b){
ackEnabled = b;
// if (!localOnly){
// shutdown MrlComm from sending acks
// below is a method only in Msg.java not in VirtualMsg.java
// it depends on the definition of enableAck in arduinoMsg.schema
// enableAck(b);
// }
}
public void waitForAck(){
if (!ackEnabled) {
return;
}
// if there's a pending message, we need to wait for the ack to be received.
if (ackRecievedLock.pendingMessage) {
synchronized (ackRecievedLock) {
try {
ackRecievedLock.wait(ACK_TIMEOUT);
} catch (InterruptedException e) {
}
if (ackRecievedLock.pendingMessage) {
log.error("Ack not received, ack timeout!");
// TODO: should we just reset and hope for the best? maybe trigger a sync?
// ackRecievedLock.pendingMessage = false;
arduino.ackTimeout();
}
}
}
}
public void ackReceived(int function) {
synchronized (ackRecievedLock) {
ackRecievedLock.pendingMessage = false;
ackRecievedLock.notifyAll();
}
}
public int getMethod(){
return method;
}
public void add(int value) {
// this explodes - sendBufferSize increases forever ... duh index not valid
// is this suppose to be round robin buffer ?
// sendBuffer[sendBufferSize] = (value & 0xFF);
// sendBufferSize += 1;
}
public int[] getBuffer() {
return sendBuffer;
}
public static void main(String[] args) {
try {
// FIXME - Test service started or reference retrieved
// FIXME - subscribe to publishError
// FIXME - check for any error
// FIXME - basic design - expected state is connected and ready -
// between classes it
// should connect - also dumping serial comm at different levels so
// virtual arduino in
// Python can model "real" serial comm
String port = "COM10";
LoggingFactory.init(Level.INFO);
/*
Runtime.start("gui","SwingGui");
VirtualArduino virtual = (VirtualArduino)Runtime.start("varduino","VirtualArduino");
virtual.connectVirtualUart(port, port + "UART");
*/
MrlComm arduino = (MrlComm)Runtime.start("arduino","MrlComm");
Servo servo01 = (Servo)Runtime.start("servo01","Servo");
/*
arduino.connect(port);
// test pins
arduino.enablePin(5);
arduino.disablePin(5);
// test status list enabled
arduino.enableBoardStatus(true);
servo01.attach(arduino, 8);
servo01.moveTo(30);
servo01.moveTo(130);
arduino.enableBoardStatus(false);
*/
// test ack
// test heartbeat
} catch (Exception e) {
log.error("main threw", e);
}
}
public void onConnect(String portName) {
if (debug) {
log.info("On Connect Called in Msg.");
}
// reset the parser...
this.byteCount = new AtomicInteger(0);
this.msgSize = 0;
ackReceived(-1);
}
public void onDisconnect(String portName) {
if (debug) {
log.info("On Disconnect Called in Msg.");
}
// reset the parser... this might not be necessary.
this.byteCount = new AtomicInteger(0);
this.msgSize = 0;
ackReceived(-1);
}
public static boolean isFullMessage(byte[] bytes) {
// Criteria that a sequence of bytes could be parsed as a complete message.
// can't be null
if (bytes == null)
return false;
// it's got to be at least 3 bytes long. magic + method + size
if (bytes.length <= 2)
return false;
// first byte has to be magic
if ((bytes[0] & 0xFF) != Msg.MAGIC_NUMBER)
return false;
int method = bytes[1] & 0xFF;
String strMethod = Msg.methodToString(method);
// only known methods.
// TODO: make the methodToString return null for an unknown lookup.
if (strMethod.startsWith("ERROR"))
return false;
// now it's got to be the proper length
int length = bytes[1] & 0xFF;
// max message size is 64 bytes
if (length > 64)
return false;
// it's a exactly a full message or a message and more.
if (bytes.length >= length+2)
return true;
return false;
}
public boolean isClearToSend() {
return clearToSend;
}
public void setInvoke(boolean b){
invoke = b;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,997
|
{"url":"http:\/\/tex.stackexchange.com\/questions\/110089\/microtype-has-issues-with-mathdesign-mathsf-font","text":"# microtype has issues with mathdesign \\mathsf font [closed]\n\nI am using\n\n\\documentclass{amsart}\n\\usepackage{microtype}\n\\usepackage[bitstream-charter,cal=cmcal]{mathdesign}\n\n\nSometimes \\mathsf{...} produces\n\n pdfTeX error (font expansion): auto expansion is only possible with scalable fonts.\n\n\nAppears when the \\mathsf is typeset in the end of a line and only with certain characters (e.g. P,..).\n\nSo, which fonts does mathdesign use for \\mathsf? Should I replace it with other font? Suggestions? Solutions?\n\nminimal 'working' example:\n\n\n\\documentclass[12pt]{article}\n\\usepackage[T1]{fontenc}\n\\usepackage{microtype}\n\\usepackage[bitstream-charter]{mathdesign}\n\n\\begin{document}\nLorem ipsum dolor sit amet, consectetur adipiscing elit. Morbi sdf$\\mathsf P$ elementum, elit in varius viverra, mauris augue cursus purus, vel ullamcorper nisi orci pretium nunc.\n\\end{document}\n\n\nThis example is minimal, i.e. dropping \\usepackage[T1]{fontenc} avoids the error.\n\n-\nWelcome to TeX.sx! Could you please extend your code snippet into a minimal working example (MWE) that reproduces the error? \u2013\u00a0 cgnieder Apr 22 at 17:33\nYour MWE runs without errors on my machine (microtype 2013\/03\/13 v2.5 and mathdesign 2006\/01\/29 v1.55) \u2013\u00a0 cgnieder Apr 22 at 18:30\nVersion 3.1415926-2.4-1.40.13 (TeX Live 2012\/Debian) (format=pdflatex 2012.12.23), fontenc 2005\/09\/27 v1.99g, microtype 2010\/01\/10 v2.4, mathdesign 2006\/01\/29 v1.55 \u2013\u00a0 DCh Apr 22 at 18:50\nthis looks like you should update microtype. \u2013\u00a0 cgnieder Apr 22 at 19:25\nfontenc 2005\/09\/27 v1.99g, microtype 2013\/03\/13 v2.5, mathdesign 2006\/01\/29 v1.55 - the same error massage \u2013\u00a0 DCh Apr 22 at 21:07\n\n## closed as too localized by lockstep, Andrew Swann, Martin Schr\u00f6der, Claudio Fiandrino, zerothApr 24 at 14:05\n\nThis question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.\n\nThis has nothing to do with microtype.\n\\usepackage[bitstream-charter]{mathdesign} does not provide sans serif font and the default is used. Type1 version of default font may not be installed (this seems to be the case in standard TeX setup), it is in cm-super\nI don't think that moving around the call of fontenc does anything. The fact is that you don't have the Type1 version of the default sans serif font, which is available with the CM-Super fonts (to be installed). \u2013\u00a0 egreg Apr 24 at 8:33\nI can not confirm that. \\usepackage[T1]{fontenc} \\usepackage[bitstream-charter]{mathdesign} ... S\\textsf{S}$\\mathsf{S}$ gives this \\usepackage[bitstream-charter]{mathdesign} \\usepackage[T1]{fontenc} ... this Installing cm-super solves the issue though. (But the sans serif font is still in wrong scale.) Seems like cm-super font is not included in 'standard' TeX setup.. Could someone explain what is going on and how to properly use sans serif fonts with mathdesign? \u2013\u00a0 DCh Apr 24 at 13:51","date":"2013-12-20 11:33:08","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\": 0.88241046667099, \"perplexity\": 9086.244877354977}, \"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-2013-48\/segments\/1387345771702\/warc\/CC-MAIN-20131218054931-00019-ip-10-33-133-15.ec2.internal.warc.gz\"}"}
| null | null |
\section{Introduction}
Thermal convection at a rough surface is a paradigm for a great variety of heat transfer processes in natural and engineering environment. As well as in the case of a smooth surface, the temperature field in the close vicinity of the wall is the key to understand and to predict the magnitude of the convective heat flux from a solid body to a surrounding fluid. Although convection at rough surfaces has been studied quite frequently in the past, experimental data being highly resolved in space and time are rare and frequently obtained at very specific or badly reproducible applications. Our work provides a set of temperature data obtained in confined natural convection that meets such requirements. The particular aim of this work is to contribute to a more universal understanding of such heat transfer processes and to improve their general predictability.
The heat transfer in many geophysical and engineering applications occurs at surfaces that are not smooth. For instance, the heat exchange between an urbanized or tree covered landscape to the atmosphere represent such a process. Nowadays, there is not yet a simplified and universal model predicting the heat flux precisely. Since this quantity is one of the main contributions to the total energy balance of the Earth, it directly affects the mean temperature in the atmosphere. Hence, uncertainties in the prediction of the convective heat transfer rate bias the prognosis of the global warming. Heat exchangers, in which rough surfaces increase the efficiency of the heat transfer represent another typical example with no less economical and ecological importance. For instance, they are widely used in huge facilities to generate electrical energy, but they are also installed in billions of households to produce heat using gas and oil burners. The literature on this subject is as diverse as the specific applications of thermal convection at rough surfaces. It is not our intension here, and probably it is even impossible, to give a comprehensive review on the literature in this field. Instead, we will refer to a few typical examples from Geophysics and Engineering as well as to a few fundamental studies that show the great diversity of studies done in the past.
For instance, there are many publications in geophysical journals that address the convective heat transfer between the Earth's ground and the atmosphere. One of the first studies of this subject has been carried out by Thom {\it et al.} \cite{Thom1975}, who investigated the fluxes of sensible and latent heat over a level forested region. The authors applied the so-called aerodynamic method of flux determination (a profile-gradient based model), but they had to conclude ``that the complexity and uncertainty of obtaining reliable flux estimates over tall vegetation by the aerodynamic model preclude its use ...''. About one and a half decades later, Schumann proposed a simple and more general model to describe a convective boundary layer above a homogeneously rough ground at zero mean wind \cite{Schumann1988}. His model was derived from momentum and heat balances in the atmospheric surface layer and incorporates closures based on Monin-Obukhov theory. It provides explicit relations of the turbulent temperature as well as the velocity fluctuations with respect to the vertical distance from the ground. Moreover, it proposes scaling laws $Nu\sim Ra^{\gamma}$ with $\gamma$ being $1/3$ for a smooth and $1/2$ for a rough surface. In their paper, the authors also suggest a limit $Ra_t$ to discriminate between smooth and rough surfaces:
\begin{equation}
Ra_t>560\left(\frac{H}{z_0}\right)^{8/3}, \hspace{1cm} \textrm {for~ Pr}=0.7.
\label{limitsmallrough}
\end{equation}
The Rayleigh, the Nusselt and the Prandtl number are defined as:
\begin{equation}
Ra=\frac{\beta g\Delta \vartheta H^3}{\nu\kappa}, \hspace{1.0cm} Nu=\frac{\dot{q}_{k}}{\dot{q}_{d}}, \hspace{1.0cm} Pr=\frac{\nu}{\kappa} .
\label{Definition_Ra_Nu}
\end{equation}
In these definitions, the variable $H$ is the interfacial layer of the atmosphere enclosing the layer with the temperature inversion (or in a more general view the total thickness of the fluid layer), $z_0$ is the height of the surface roughness, $\beta$ is the thermal expansion coefficient, $g$ is the gravitational acceleration, $\Delta \vartheta$ is the temperature drop across the fluid layer and $\nu$ and $\kappa$ are the kinematic viscosity and the thermal diffusivity of the fluid, respectively. The Nusselt number is defined as the ratio between the convective and the diffusive heat fluxes $\dot{q}_{k}$ and $\dot{q}_{d}$, the Prandtl number is the ratio between the kinematic viscosity and the thermal diffusivity. In a more recent work, a group of researchers calculated numerical simulations to investigate turbulent convection at periodic as well as randomly structured rough surfaces \cite{Toppaladoddi2014}. The primary aim of this work was to understand the effect of various kinds of roughness on the magnitude of the turbulent fluxes of heat and momentum and how the shape of roughness may change the melting or the growth of the Arctic see ice in the oceans. The authors developed a two-dimensional MPI code that is based on the Lattice Boltzmann method, and they applied this code to channel flow and turbulent Rayleigh-B\'enard convection. For the latter problem they found a variation of both, the scaling exponent $\gamma$ as well as the pre-factor $C$ in the $Nu=C~Ra^{\gamma}$ scaling law and they identified an enhanced plume production from the tips of the roughness elements to be the reason for this variation.
In the engineering community, the diversity of convective heat transfer problems at rough surfaces is even broader. It covers convection at the outer side of buildings, the heat transfer in heat exchangers or the cooling of turbines and power electronic circuits, just to mention a few examples. In a recent work published in Applied Thermal Engineering, Palyvos investigated the thermal losses from building surface or roof mounted solar convector to the ambient air due to wind induced convection \cite{Palyvos2008}. In this reviewing paper, the author critically discusses various existing correlations to estimate the convective heat transfer coefficient of buildings, and he proposed an own, alternative model. He also recognized that ``almost all these correlations are based on empirical data and exhibit a significant lack of physical generality''. Consequently, Palyvos states that ``More realism is needed, i.e. field rather than laboratory measurements, as well as some sort of standardization in the choice of such things as the wind velocity sensors or the measurement topology, e.g. height above ground and/or distance from the fa\c{c}ade wall or roof, etc. In this way, a much smaller set of well proven and generally accepted correlations may emerge, which will greatly help the designer/modeler.'' Following the request towards more universality, Ghodake investigated the enhancement of the convective heat transfer coefficient in heat exchangers with various ribs numerically \cite{Ghodake2016}. Using ANSYS CFD, the authors analysed different shapes of ribs such as V-shaped, triangular and rectangular as well as semicircular ones. They also varied the angle of attack of the flow. The most general conclusion of this, rather empirical case study, is the hypothesis that the enhancement of the heat transfer is associated with an increase of separations and reattachments over the ribbed wall. This boosts the fluid mixing, creates flow unsteadiness and interrupts the development of the thermal boundary layer. Another study of the convective heat transfer at rough surfaces which was dedicated to flat and finned heat sinks for the cooling of semiconductors has been reported by a group of Italian researchers \cite{Ventola2014}. Unlike in the references above, the authors investigated a grainy surface as to be formed from a direct metal laser sintering process. Using a self-made test facility, they measured the convective heat transfer coefficient at a great variety of surfaces with different grain. In particular, they deserve credit for developing a theoretical model that is based on a surface fully covered by sand of uniform grain (as used in \cite{Schlichting2000}. The key idea of this general model is ``to estimate the size of the eddies that dominate the heat transfer close to the wall by a combination of the size of the roughness elements, i.~e. $k_p$ and $k_a$ (the peak surface roughness and the grain size diameter),
and the viscous length scale $\eta$. In the conclusion of their work, the authors state that the observed enhancement of the heat transfer at grainy surfaces could not be explained by the increase of the effective surface area but must be caused by a variation of the fluid-mechanical properties of the boundary layer.
While a great number of papers deals with specific applications of thermal convection at rough surfaces, only a few publications focus on this problem from a rather fundamental perspective. In the late 1960s, Townes \& Sabersky and Gowen \& Smith conducted laboratory experiments, in which they investigated canonical model flows over rough surfaces \cite{Towns1966,Gowen1968}. They measured the temperature and the velocity field in the very thin convective boundary layer. But, the experiments were insufficient to discriminate, e.~g. between diffusive and turbulent transport because of the lack of an adequate spatial and temporal resolution. In the last two decades it became also very popular to study natural convection with rough surfaces in the so-called Rayleigh-B\'enard(RB) set-up. This system consists of a fluid layer of the thickness $H$ that is heated from below and cooled from the top. Convection sets in due to the density difference between the hot fluid at the bottom and the cold fluid at the top plate. Only two control parameters exist to describe the model flow completely, the Rayleigh number $Ra$ and the Prandtl number $Pr$ (for the definitions see Eq.~\ref{Definition_Ra_Nu}). In many applications as well as in all laboratory experiments, the fluid layer is laterally confined by sidewalls. In this case, the aspect ratio $\Gam$, being the ratio between the lateral extent of the fluid layer $D$ and its thickness $H$ completes the set of control parameters. The first researchers who introduced roughness to the Rayleigh-B\'enard set-up were groups in Hongkong \cite{Shen1995,Du2000} and Lyon \cite{Ciliberto1999}. Shen et al. used a pair of rough plates structured by a lattice of square pyramids with a fixed height $h_s$ and a fixed spacing $d_s=2h_s$ at their surface and build up RB experiments of aspect ratios $\Gam=1$ and $\Gam=0.5$. They measured the near-wall temperature and velocity field in these two test cells applying a special particle tracking technology that is based on thermo-sensitive particles. In particular, they measured the temperature of the fluid as a function of the vertical distance $z$ from the horizontal plates using a small movable thermistor (Thermometrics, AB6E3-B10KA202J). The essential result of their work is the discovery that the emission of plumes completely changes compared against a smooth surface. Furthermore, they found that the horizontal shear flow along the rough surface creates eddies in between the pyramids, which enhances the detachment of thermal plumes and increase the total heat transport with respect to the smooth surface. Furthermore, they found that this process does not change the exponent $\gamma=2/7$ in the $Nu\sim Ra^{\gamma}$ scaling law. Ciliberto and Laroche investigated spherical roughness elements randomly or periodically arranged at the surface of the heated bottom plate. Unlike Shen et al., they reported an increase of the exponent to $\gamma>2/7$ for randomly distributed obstacles. We wish to mention here three more references on turbulent RB convection with roughness showing the full discrepancy that actually exists on this field. Qiu {\it et al.} and Wei {\it et al.} reported an increase of both, the exponent $\gamma$ as well as the pre-factor $C$ of the $Nu=C Ra^{\gamma}$ relation \cite{Qiu2005,Wei2014}. Roche {\it et al.} even observed an exponent $\gamma=0.5$ in a RB cell with 110~µm deep V-shaped grooves at the top and bottom plates \cite{Roche2001}. The cell was operated with gaseous Helium at a temperature of about 5~K and the transition in the exponent starts at a Ra number of $Ra\approx10^{12}$. The authors explain the change in the scaling exponent with a laminar--turbulent transition of the boundary layer. However, they could not verify this hypothesis, since direct measurements of the near-wall flow field and, hence, the proof of the laminar-turbulent transition were not feasible in this experiment. Another kind of roughness elements that are also used in our own experiment, has been introduced by Tisserand {\it et al.} \cite{Tisserand2011}. The researchers operated a water-filled RB cell at room temperature and they prepared only the surface of the heated bottom plate with cubic-shaped elements of a height corresponding to the thickness of the thermal boundary layer. The cuboids were arranged in a square array of square plots with a period of twice the length (width) of the elements. The particular idea of this, asymmetric cell was to compare the thermal impedances of the lower half of the cell (rough) and the upper half of the cell (smooth), when they interact with the same bulk flow. In summary of their measurements, the authors conclude that the convective heat transport between a solid surface and a fluid is largely a local process and does virtually not depend on the global structure of the flow. They also could confirm the observation from all previous experiments that the enhancement of the Nusselt number exceeds the increase of the surface area due to the roughness. Salort {\it et al.} and Liot {\it et al.} continued the work on this specific roughness structure \cite{Salort2014,Liot2016}. In a collaborative work between the \'Ecole Normale Sup\'erieure de Lyon and the Technische Universitaet Ilmenau, the researchers measured the global heat transport as well as the local temperature and the velocity fields in the vicinity of the rough surface. The measurements of the global heat flux and a first, however poorly resolved measurement of the near-wall temperature field has been undertaken in the Lyon water cell. The local velocity field has been measured in the large-scale RB experiment called the ``Barrel of Ilmenau''. Because of its large size, the diameter amounts to 7.15~m and the height amounts up to 6.30~m, it actually provides the best spatial resolution of any kind of boundary layer measurement in thermal convection at very high Ra numbers. The aim of this joint work was to understand the specific mechanism, how does the convective heat transfer at the rough surface get enhanced by the destabilization of the boundary layer. In fact, the authors could validate that because of the roughness elements the critical Ra number $Ra_c$ decreases, and the boundary layer becomes turbulent. They also found a significant heat transfer enhancement beyond this bound, although the scaling exponent $\gamma$ does not increase to the asymptotic limit $\gamma_{asymp}=0.5$ as predicted by Goluskin \& Doering in their upper bound analysis \cite{Goluskin2016}. Another systematic investigation of the effects of roughness geometry on turbulent RB convection over rough plates has been reported by Xie \& Xia very recently \cite{Xie2017}. The authors applied pyramid-shaped roughness elements periodically distributed at both plates. Their measurements cover Ra numbers between $7.5 \times 10^7<Ra<1.3 \times 10^{11}$ and Prandtl numbers between $3.57<Pr<23.34$. The authors of this paper classified the heat transport scaling into three regimes. Regime~1 is considered to be the dynamically smooth regime with a $Nu\sim Ra^{\gamma}$ scaling equally to the smooth case. In the Regimes~2 and 3, the scaling of the heat transport is enhanced and the width-to-height ratio of the roughness elements controls the variation of the scaling exponent. The authors also found that with increasing width-to-height ratio the clustering and the lifetime of plumes grow at least in high Pr number fluids, like e.~g. water.
Among this experimental work, there are also a few numerical studies in which thermal convection problems at rough surfaces are addressed. The major challenge of any kind of Direct Numerical Simulation (DNS) is to resolve the boundary layer flow field and, in particular, the fine structure of the flow around the obstacles along with the turbulence in the strongly mixed bulk region. In order to fulfill this requirement, a huge computational effort is needed. Many numerical works are limited to two dimensions. Three of the very recent 2d DNS studies on convection with rough surfaces have been published by Toppaladoddi {\it et al.} \cite{Toppaladoddi2015,Toppaladoddi2017} and Zhu {\it et al.} \cite{Zhu2017}. However, it seems to be not trivial to transfer the results of these simulations to a real 3d geometry, since all kinds of roughness elements generate a strongly three-dimensional flow field near the surface which significantly changes the convective heat flux between the surface and the fluid. One of the rare examples of a 3d DNS has been reported by Stringano {\it et al.} \cite{Stringano2006}. He run a set of DNS for thermal convection at a grooved structure, at whereby the grooves are arranged at the plate in concentric rings. The simulations were performed in a cylindrical RB cell of aspect ratio $\Gam=1/2$ at $Pr=0.7$ and cover a domain in Ra number between $2\times 10^6<Ra<2\times 10^{11}$. The results show an increase of the scaling exponent $\gamma$ if the thickness of the thermal boundary layer becomes smaller than the groove height. Another, very recent numerical work on thermal convection at a non-smooth solid-liquid interface has been published by Wagner \& Shishkina \cite{Wagner2015}. They investigated a cubic RB set-up with four parallelepiped shaped, large obstacles equidistantly attached at the bottom and top plates. The height of the obstacles significantly exceeds the thickness of the thermal boundary layer, which is in strong contrast to the work discussed above. Unlike in all other work, the authors of this study found the increase of the heat flux at their particular surface structure to be lower than the increase of the area caused by the obstacles. However, the total heat flux is still higher compared with a smooth case, and it depends on the height and the distance of the obstacles. The authors evaluated their hypothesis by means of the numerical data up to a maximum Ra number $Ra=10^8$, but they could not validate their data experimentally.
Without raising any claim of completeness of the references above, it can be stated that a great variety of experimental, numerical and theoretical work on thermal convection at rough surfaces has been done. The results of all this work almost coincide in the fact that the enhancement of the heat transfer due to the surface roughness is larger than the pure increase of the surface area (except the work of Wagner \& Shishkina \cite{Wagner2015}). It also seems to be a consensus that the roughness elements induce thermal plumes or even turbulence that increase the mixing in the near-wall flow field. However, there is an obvious contradiction wether or not the scaling exponent in the relation $Nu\sim Ra^{\gamma}$ changes to $\gamma=1/2$, the asymptotic bound predicted for ultra high Ra numbers. Furthermore, there is not yet a universal model meeting all the specific cases of roughness that predicts the convective heat transfer coefficient with an adequate accuracy. An undoubtedly very valuable contribution to derive such a model would be a better knowledge of the local temperature field in the vicinity of the rough surface. This is the main approach of our work. In this spirit, we provide a set of precise and highly resolved temperature data in the parameter domain, where the roughness elements significantly modify the near-wall flow field and we will compare this data with its counterpart from a smooth surface.
\section{Experimental Set-up}
\subsection{Rayleigh-B\'enard cell}
In order to obtain such data in a laboratory experiment, this experiment has to meet two requirements: i) the Rayleigh number has to be sufficiently high to trigger roughness effects in the near wall flow field, ii) the temperature sensor must be much smaller than the characteristic length scales in the boundary layer flow field. These are typically the thickness of the boundary layer or the size of the roughness elements. In the large-scale Rayleigh-B\'enard experiment ``Barrel of Ilmenau'' both requirements are fulfilled very well. A maximum Rayleigh number of $Ra_{max}=10^{12}$ can be set. This is more than one order of magnitude higher than the transitional Ra number at which roughness effects have been observed in the Lyon experiment \cite{Tisserand2011}. At this Ra number, the thickness of the boundary layer amounts to a few millimeters, but the temperature sensor that we used for our measurements has a diameter of only 130~µm, and a length of 330~µm in length. Hence, it is about 50 times smaller than the thickness of the boundary layer and permits a spatial resolution that cannot be reached in any other existing RB cell. The ``Barrel of Ilmenau'' consists of a well insulated container of cylindrical shape with an inner diameter of $D=7.15$~m. It is filled with fresh air. The Prandtl number $Pr=0.7$ remains virtually constant over the entire range of temperatures set during the measurements. A heating plate at the lower side release the heat to the air layer and a cooling plate at the upper side remove it. Both plates are carefully levelled perpendicular to the vector of gravity with an uncertainty of less than 0.15~degrees. The thickness of the air layer $H$ can be varied continuously between $6.30~{\rm m}>H>0.15~{\rm m}$ by moving the cooling plate up and down. The temperature of both plates can be set to values between $20^\circ {\rm C}<T_h<80^\circ {\rm C}$ (heating plate) and $10^\circ {\rm C}<T_c<30^\circ {\rm C}$ (cooling plate). Due to the specific design of both plates (for details see \cite{duPuits2013}), the temperature at their surfaces is very uniform and the deviation does not exceed 1.5~\% of the total temperature drop $\Delta T=T_h-T_c$ across the air layer. The variation of the surface temperature over the time is even smaller and remains below $\pm 0.02$~K. The sidewall of the model room is shielded by an active compensation heating system to inhibit any heat exchange with the environment. Electrical heating elements are arranged between an inner and an outer insulation of 16 and 12 cm thickness, respectively. The temperature of the elements is controlled to be equal to the temperature at the inner surface of the wall. In order to test the efficiency of the system, we set the same temperature of $30.0^\circ \rm{C}$ at the heating and the cooling plates. In case that a heat flux throughout the sidewall exists, the temperature of the interior of the cell would deviate, for outgoing heat towards a lower and for ingoing heat towards a higher temperature. We have measured $29.9^\circ \rm{C}$ indicating that the heat exchange with the environment is very small and can be neglected.
\begin{figure}
\centerline{\includegraphics[width=10cm]{Fig01_Rectangular_Cell_Setup_V1.png}}
\caption{Rectangular test section to measure the near-wall temperature field in turbulent Rayleigh-B\'enard convection with one rough surface at the heating plate. The test section is build in into the large-scale Rayleigh-B\'enard experiment ``Barrel of Ilmenau''.}
\label{BOI_Setup}
\end{figure}
Our study of convection reported here was undertaken in a smaller volume of rectangular base area that has been separated from the large test section (see Fig.~\ref{BOI_Setup}). The base area of the inset (defined as x,y-plane) is 2.50~m by 0.625~m, its height (z-coordinate) is 2.50~m. As shown in Fig.~\ref{BOI_Setup}, the original heating and cooling plates of the large cylindrical test cell keep the inset sealed and maintain the stable and well-defined boundary conditions at the bottom and the top of the air layer. We covered the bottom plate with artificial roughness elements, so that our present inset gets actually the same as reported in a recent publication where we investigated the near wall velocity field \cite{Liot2016}. The particular idea behind the slender, cuboid geometry with aspect ratios $\Gam_x=1$ and $\Gam_y=0.25$ is to create a quasi two-dimensional space, in which the orientation of the global flow pattern is fixed with respect to the alignment of the roughness elements. One more important benefit of this configuration with the smaller cuboid RB cell surrounded by the larger cylindrical one is the almost perfect adiabatic boundary condition at the sidewall of the inner test section. This is due to the fact that the vertical temperature profiles $T(z)$ in the large container and in the inner test section (see Fig.~\ref{BOI_Setup} are virtually the same. A heat flux throughout the sidewall of the inner cell is, thus, impossible.
We created the roughness structure at the heating plate by sticking more than 400~little Aluminum blocks at its surface. In order to maintain a uniform temperature of the surface, we took particular care of a good thermal contact between the Aluminum blocks and the surface of the plate. The obstacles with a size of $d=30\times 30~\rm mm^2$ (base area) and $h=12$~mm (height) were arranged in a periodical distance of $l=2d$ as well in x as in y-direction (see Fig.~\ref{Microthermistor Setup}). We specified the height of the obstacles with respect to the typical thickness of the boundary layer in the way that it is smaller at the lowest and larger at the highest Ra number adjustable in our RB cell. The latter quantity has been estimated from the global relation $\delta_{th}=H/2Nu$. The size and the arrangement of the obstacles in the present work is scaled to fit the geometry and the fluid-mechanical similarity with the Lyon experiment \cite{Tisserand2011,Liot2016}. They use a cylindrical cell with a diameter of $D=0.50~\rm m$ and a height of $H=1.00~\rm m$. The working fluid is water with a Prandtl number $Pr=2.5...6.2$. Due to the smaller size, the Lyon experiment is more flexible with respect to a variation of the boundary conditions (no roughness, various shapes of roughness, single-sided or double-sided roughness). Moreover, it is much better suited to measure the global heat flux precisely and many interesting results has been obtained there. However, it is very difficult to measure the local temperature field in the vicinity of the rough surface with high spatial and temporal resolution. Particularly, this is related to the fact that the boundary layer is typically about 2~mm thick in the Lyon RB cell and even the smallest, water-resistent temperature sensor having a size of about 400~µm, is too large to fully eliminate effects of spatial integration of the temperature field.
\subsection{Temperature measurement technique}
\begin{figure}
\centerline{\includegraphics[width=12cm]{Fig02_Microthermistor_Setup_V2.png}}
\caption{Arrangement of the artificial roughness elements at the surface of the heating plate with the temperature sensor placed above the Groove area (online: blue). The micro-thermistor is the little dark pearl mounted between the lower tips of the two needles.}
\label{Microthermistor Setup}
\end{figure}
The measurements reported here will close this gap. As shown in Fig.~\ref{Microthermistor Setup} (true-to-scale), the micro-thermistor that we used to measure the local temperature is much smaller than both, the height of the obstacles and the thickness of the thermal boundary layer, respectively. The temperature sensitive element is the little dark pearl of ellipsoidal shape mounted between the lower tips of the two needles. The typical Reynolds number of the local flow field around the sensor is of the order of $Re_l\approx 1$. This is an important quantity to estimate the effect of the sensor to the near-wall flow field. The thermistor is connected via two 18~µm wires to the conducting support left and right of the sensor. In order to minimize a potential measurement error caused by the strong wall-normal temperature gradients within the thermal boundary layer, the sensor and the connecting wires are aligned parallel to the iso-surfaces of the mean temperature running parallel to the plate surface \cite{Kaiser2012}. The sensor support, a 4~mm rod of brass with the two needles at one side, is mounted at a precise linear motion system with a position accuracy of 0.002~mm. The position at which the sensor touches the surface of the plate, is defined as one half of its diameter or in absolute numbers $z=0.065$~mm. It can be found using a microscope camera.
Another specific requirement on temperature measurements in turbulent flows is the capability of the sensor to track even the fastest temperature fluctuations in the flow. This does not only depend on the properties of the sensor, but also on the characteristics of the flow around it. More precisely, this is mainly related to the size and the advection velocity of the smallest vortices in the turbulent flow. A good estimation for the required resolution in space and time are the Kolmogorov microscales $\eta=(\nu^3/\varepsilon)^{1/4}$ and $\tau_{\eta}=(\nu/\varepsilon)^{1/2}$ with $\nu$ and $\varepsilon$ being the kinematic viscosity and the rate of dissipation of turbulent kinetic energy, respectively. For the lower and the upper end of the Ra number domain of our rectangular test section, $Ra_1=4.6\times 10^9$ and $Ra_2=4.7\times 10^{10}$ - the Kolmogorov length scales amount to $\eta_1=6.4$~mm and $\eta_2=3.0$~mm while the Kolmogorov time scales are $\tau_{\eta,1}=2.7$~s and $\tau_{\eta,2}=0.6$~s, respectively. In order to resolve these scales, the sensor has to be smaller and faster than these numbers. With respect to the size of our thermistor, this demand is fulfilled very well. We also evaluated the response time of the thermistor in a simple laboratory set-up shown in the left sub-figure of Fig.~3. The sensor was placed in a well-defined flow, whose velocity can be set between 0.0~m/s and 1.0~m/s. This is the typical domain of flow velocities close to the rough plate in our convection experiment. Using a Laser beam, we heated the sensor up. Having achieved a steady state, we switched the Laser off, and we measured the decay time. It is quite common to quantify this curve in a single value, at which the sensor has achieved 70~\% of the total jump between the high and the low temperature - the response time $\tau_{70}$. We follow this and plot the response time of our micro-thermistor in the right sub-figure of Fig.~3. For all flow velocities, even for the case that the surrounding air is in rest, the response time is shorter than the Kolmogorov time scale, and thus, the sensor is capable to resolve even the fastest temperature fluctuations in the boundary layer flow field. In addition to the measurement of the response time, we calibrated the sensor against a primary standard thermometer of PT~100 type with an uncertainty of 0.02~K.
\begin{figure}
\vspace{0.5cm}
\makebox[7cm]{\includegraphics[width=5 cm]{Fig03a_Response_time_setup.png}}
\makebox[7cm]{\includegraphics[width=7 cm]{Fig03b_Response_time.png}}
\label{response_time}
\caption{Laboratory set-up to measure the response time of the temperature sensor under well-defined flow conditions (left sub-figure). Measured response time $\tau_{70}$ with respect to the flow velocity (right sub-figure).}
\end{figure}
We also take care of a potential self-heating of the thermistor due to the injected current. We operate the sensor in an active bridge configuration applying a very low sensor current of $I=5\times 10^{-6}$~A. At this current, the measurement error due to the effect of self-heating is estimated to be less than 0.01~K and, hence, below the calibration uncertainty. The output signal of the bridge was acquired by a computer based measurement system with a sampling interval of 5~ms and a dynamic range of 18 bits giving a resolution of the temperature of 5 1/2 digits.
\subsection{Measurement procedure}
Our temperature measurements were undertaken in the same rectangular test section, as recently used to study the velocity field at a rough surface. We chose the same Rayleigh numbers $Ra_1=4.6 \times 10^9$ and $Ra_2=4.7 \times 10^{10}$ corresponding to temperature drops of $\Delta T=3$~K and $\Delta T=40$~K, respectively. While $Ra_1$ is below transitional effects may start to appear, $Ra_2$ is beyond the transition observed in the global heat flux in the Lyon experiment \cite{Tisserand2011} as well as justified in the near-wall velocity field in the Ilmenau cell \cite{Liot2016}. We have undertaken our present temperature measurements at the centre of the heated bottom plate. With respect to the installed roughness pattern and the orientation of the mean flow, we define three distinct surface regions in our set-up (cf. Fig.~\ref{Microthermistor Setup}) that we refer to as: i) ``Grooves''~--~these are the valleys in which the air flow is virtually not disturbed by the obstacles, ii) ``Notches''~--~these are the surface areas in between the obstacles where the obstacles block the flow, and iii) ``Top''--these are the upper surfaces of the obstacles. We measured the temperature profiles $T(z)$ at each of these surface regions, considering the strong variations that have been reported with respect to the measured velocity field \cite{Liot2016}. Each of the profiles was measured point-by-point, moving the sensor in a vertical line away from the surface of the plate. It is assembled from 35 different z-positions starting at $z=0.065$~mm ($z/H=2.6 \times 10^{-5}$) and ending up at $z=147.66$~mm ($z/H=0.059$). Time series of 540,000 samples have been acquired over a period of 2700~s at every single position providing sufficient data for an adequate statistical convergence. In the subsequent sections, we will discuss the data and we will derive a few more general conclusions that describe the local heat transport in turbulent convection with rough surfaces.
\section{Results}
\subsection{Statistical analysis of the temperature field}
One of the unquestionable results of all experimental work on convection at rough surfaces is the fact that the increase of the heat flux is larger than the increase of the surface due to the roughness elements (see e.~g. Du \& Tong \cite{Du2000} or Ciliberto \& Laroche \cite{Ciliberto1999}). Our highly resolved temperature measurements that start very close to the surface of the heating plate enable us, to investigate the local temperature field and to quantify the local heat transport at distinct surface areas. To this aim, we measured temperature profiles at all three surface regions ``Groove'', ``Notch'' and ``Top'' (cf. Fig.~\ref{Microthermistor Setup}), and we compare the results with earlier measurements at a smooth surface. The latter measurements have been undertaken in two different set-ups: i) in the full-size ``Barrel of Ilmenau'' with a diameter of $D=7.15~{\rm m}$ and a height of $H=6.30~{\rm m}$ for the higher Ra number $Ra=5.2\times 10^{10}$, and ii) in a smaller inset of cylindrical shape with a diameter of $D=2.50~{\rm m}$ and a height of $H=2.50~{\rm m}$ for the lower Ra number $Ra=3.4\times 10^{9}$. The working fluid is also air with a Pr number of $Pr=0.7$ in these two experiments. The sensor and the measurement technique used for the measurements at the smooth surface were the same as applied at the rough one. We wish to refer here to two references \cite{Li2012,duPuits2013}, where the reader will find more details on the facility as well as the original data from both, the lower and the higher Ra number measurements.
\begin{figure}
\centerline{\includegraphics[width=15cm]{Fig04_Mean_lowRa_V1.png}}
\caption{Low Rayleigh number: Profiles of the mean temperature $\Theta(z/h)$ measured at the rough surface at $Ra_1=4.6 \times 10^9$, Top: black circles, Notch: black squares, Groove: black triangles, and at the smooth surface ($Ra=3.4 \times 10^9$): open circles. The dashed vertical line at $z/H=4.8 \times 10^{-3}$ indicates the height of the obstacles.}
\label{Tmean_lowRa}
\end{figure}
\begin{figure}
\centerline{\includegraphics[width=15cm]{Fig05_Mean_highRa_V1.png}}
\caption{High Rayleigh number: Profiles of the mean temperature $\Theta(z/h)$ measured at the rough surface at $Ra_2=4.7 \times 10^{10}$, Top: black circles, Notch: black squares, Groove: black triangles, and at the smooth surface ($Ra=5.2 \times 10^{10}$): open circles. The dashed vertical line at $z/H=4.8 \times 10^{-3}$ indicates the height of the obstacles.}
\label{Tmean_highRa}
\end{figure}
We start our discussion with the profiles of the normalized mean temperature $\langle \Theta\rangle$:
\begin{equation}
\langle \Theta\rangle=(\langle T(t)\rangle -T_B)/(T_H-T_B)
\label{mean_temperature}
\end{equation}
versus the normalized distance $(z/H)$ from the plate surface that are plotted in Fig.~\ref{Tmean_lowRa} for $Ra_1=4.6 \times 10^9$ and in Fig.~\ref{Tmean_highRa} for $Ra_2=4.7 \times 10^{10}$. Here, the temperature $T(t)$ represents the temperature time series measured using the micro-thermistor at the various distances $z$ from the surface of the plate. $T_B$ is the so-called bulk temperature measured using a separate Resistance Temperature Device (RTD) of PT~100 type in the center of the cell which is well mixed and of virtually uniform temperature. $T_H$ is the temperature of the heating plate also measured using a RTD within the plate. In order to plot all profiles in the correct geometric relation, we indicate the height of the obstacles as a dashed vertical line at $z/H=4.8 \times 10^{-3}$, and we shift the profile of the mean temperature at the top of the obstacles by this height. At both Ra numbers, the temperature gradient at the top of the obstacles, which is a direct measure of the local heat flux density according to Fourier's law
\begin{equation}
\dot{q}=-\lambda ~\partial \langle T \rangle / \partial z|_{z=0},
\label{fouriers_law}
\end{equation}
significantly exceeds as well the gradients in the Groove and the Notch as that gradient measured at the smooth surface. This is a first conclusion of generality, the heat flux enhancement observed at rough surfaces is mainly accounted for by an increase of the local heat transfer coefficient at the top of the obstacles. And, this is true below and beyond the critical Ra number, at which a transition in the scaling exponent $\gamma$ was observed. For completeness, we have listed the numbers in Table~\ref{Gradients}.
The transition of the scaling exponent $\gamma$ beyond a critical Ra number $Ra_c$ \cite{Ciliberto1999,Roche2001, Qiu2005} represents as well an interesting and not well understood feature of thermal convection at rough surfaces. This is quite similar to RB convection with smooth plates, however, the transition starts at a much lower $Ra_c$. Schumann provided a first approximation to predict its onset \cite{Schumann1988}. Applying Eq.~\ref{limitsmallrough} to our measurements, we obtain a critical Ra number $Ra_c=560~(2500~{\rm mm}/12~{\rm mm})^{8/3}=8.5 \times 10^8$ as the bound for this transition. In an alternative manner, Du \& Tong predicted the onset of roughness effects at all, if the thickness of the boundary layer $\delta_T$ falls below the height of the obstacles $h$ \cite{Du2000}. Applying this prediction to our experiment, it means that $\delta_T$ must fall below $h=12$~mm. This appears at a critical Nu number $Nu_c=H/2h=2500~{\rm mm}/24~{\rm mm}=104$. Since comprehensive data of the scaling $Nu\sim Ra^{\gamma}$ at $Pr=0.7$ is not available for a rectangular box of two different aspect ratios and rough top and bottom plates, we estimate the corresponding Ra number using the Grossmann-Lohse model that is developed for smooth boundaries \cite{Grossmann2001}. We obtain $Ra_c=5.6\times 10^9$, which is only about half a decade higher than Schumann's prediction. From the work by \cite{Liot2016}, we also have a prediction, when roughness effects start to modify the heat transport in our specific test section of rectangular shape. The authors of this work investigated the velocity field in the Notch region and they found the fluid being confined within the Notch below a critical Ra number, while there is a fluid exchange with the bulk flow beyond it. They also analysed the scaling of the the local wall heat flux that they measured directly using heat flux sensors. From both measurements, they found a consistent limit for the onset of roughness effects which is as high as $Ra_c=1.5\times10^{10}$. This is in a fair agreement with the two predictions by Schumann and Du\&Tong, and the little differences with respect to those models might be a consequence of the finite aspect ratio as well as specific geometry of the rectangular test section with aspect ratios $\Gam_x=1$ and $\Gam_y=0.25$.
In the work presented here, we wish to find out whether or not the transition observed in the scaling of the local heat transport as well as in the local flow field in the Notch can be attributed to a variation in the local temperature field and can we learn something more from our highly resolved temperature measurements. To this end we plot the profiles of the mean temperature at the Top of the obstacles in a semi-logarithmic manner (see Figure~\ref{Mean_top_comparison}). The profiles are scaled by the various thicknesses of the boundary layer at $Ra_1$ and $Ra_2$ (see Table~\ref{Gradients}) and thus, they are directly comparable. We used the displacement thickness for this normalization that is defined as \begin{equation}
\delta_{T}=\int_0^{z_{max}} \left\{1-\overline{\Theta}(z)\right\}dz.
\label{eq:displacement_thickness}
\end{equation}
It is quite obvious that the profile at the higher Ra number significantly differs from those two at the lower Ra number and at the smooth plate, respectively. The curve is steeper and in the flow region $1<z/\delta_T<10$, it exhibits a logarithmic behavior. In particular, the latter fact is a typical attribute of a turbulent boundary layer. In fact, the roughness elements shift the laminar-turbulent transition of the boundary layer and, hence, the transition of the scaling exponent towards lower Ra numbers with respect to the prediction in RB convection with smooth plates \cite{Grossmann2001}. However, it should be noted here as well that the variation of the scaling exponent $\gamma$ is expected to be smaller with respect to the transition in RB convection with smooth surfaces since the enhancement of the local heat flux concerns only the Top region and, thus only one forth of the total surface area.
\begin{table}
\begin{center}
\begin{tabular}{lcccc}
& ~~ $\partial \langle \Theta \rangle / \partial (z/H)|_{z/H=0}$ &~~ $\partial \langle T \rangle /\partial z ~[\rm{Km^{-1}]}$ &~~ $\delta_T/H~ [\times 10^{-3}]$ & ~~$\delta_T ~\rm{mm}$ \\[3pt]
$Ra_1=4.6 \times 10^9$ \\
Groove & 562.1 & 255 & 4.394 & 10.99 \\
Notch & 514.9 & 238 & 4.691 & 11.72 \\
Top & 784.9 & 340 & 2.326 & 5.82 \\
smooth & 480.0 & 225 & 4.473 & 11.18\\[3pt]
$Ra_2=4.7 \times 10^{10}$ \\
Groove & 848.9 & 4406 & 3.461 & 8.65 \\
Notch & 494.6 & 2790 & 4.466 & 11.17 \\
Top & 1048.2 & 5500 & 1.888 & 4.72 \\
smooth & 824.7 & 170 & 2.445 & 15.40 \\
\end{tabular}
\end{center}
\caption{Gradients of the mean temperature and (displacement) thickness of the thermal boundary layer at the distinct areas at rough and smooth surfaces for two Rayleigh numbers. (Please note that the dimensional temperature gradient as well as the dimensional thickness of the thermal boundary layer at the smooth plate ar $Ra_2=4.7 \times 10^{10}$ are not fully comparable with the other numbers since the total height of this experiment was $H=6300~\rm{mm}$ instead of $H=2500~\rm{mm}$) for all other measurements.}
\label{Gradients}
\end{table}
\begin{figure}
\centerline{\includegraphics[width=15cm]{Fig06_Mean_top_comparison_V1.png}}
\caption{Profiles of the mean temperature above the top of the obstacles at $Ra_1=4.6 \times 10^9$ (gray circles) and $Ra_2=4.7 \times 10^{10}$ (black circles) compared with the profile at the smooth plate at $Ra=3.4 \times 10^9$ (open circles). The dashed line fits the profile at the higher Ra with a logarithm according to $\Theta=0.102~{\rm log}(z/\delta_T)+0.21$.}
\label{Mean_top_comparison}
\end{figure}
The interpretation of the measured mean temperature profiles in the Groove and the Notch regions is not that easy and it seems to be hard to predict a quantitative effect of the observed modification in the flow field to the scaling of the global heat transport. For the lower Ra number $Ra_1=4.6 \times 10^9$, all profiles including the temperature gradients at the wall collapse fairly well with each other and with the profile at the smooth plate (except of course that profile measured above the Top). For the higher Ra number $Ra_2=4.7 \times 10^{10}$, the profiles of the mean temperature exhibit clear differences. In particular, in the region $0.001<z/H<0.048$, the latter number corresponds with the height of the roughness elements, those profiles are more flat compared with the reference at the smooth plate. This indicates an enhancement of the convective heat transport $\langle w'T'\rangle$, but not necessarily a transition to a turbulent state of the boundary layer in the Groove and in the Notch regions. In order to explain these variations of the temperature profiles, it might be useful to have again a look back into the recent study of the velocity field undertaken in the same RB cell with one rough surface \cite{Liot2016}. The authors of this paper report a transition of the flow field in the Notch, when the Ra number exceeds about $Ra_c=1.5\times10^{10}$. For more clearness of the reader, we re-plot Figs.~4 and 5 of that paper here. The figures show that the fluid remains confined in the Notch below $Ra_c$, and it forms a little RB cell with a local Ra number of $Ra_{l1}\approx 100$. This number is based on the height of the obstacles and the vertical temperature drop across the Notch. It considerably remains below the stability limit $Ra_s=1707$, at which Lord Rayleigh and Jeffreys predicted the onset of convection in an laterally infinite fluid layer \cite{Rayleigh1916,Jeffreys1926}. However, we found little temperature fluctuations in the Notch (to be discussed along with the profiles of the standard deviation below), which indicates that heat is not only transported by diffusion. A little fraction of convective transport also takes place at the lower Ra number. Beyond $Ra_c$ the situation in and around the Notch completely changes. The fluid is no longer trapped inside the Notch and starts to leave it. This induces a much stronger convective transport process between the fluid in the Notch and the fluid above the obstacles and significantly enhances the efficiency of the heat transport. This becomes also visible in the gradient of the mean temperature, which is much smaller than below the transition. The process, that which enables the fluid to leave the Notch at the higher Ra number, is mainly triggered by local shear forces due to the mean wind, but it is also supported by local buoyancy forces within the notch. The latter ones can be estimated again computing the local Ra number which amounts to $Ra_{l2}\approx 1100$. This value is still below the stability limit. However, the velocity measurements show that both contributions together are strong enough to trigger the exchange of fluid between the Notch and the mean flow. The temperature profile in the Groove at the higher Ra number is also more flat than that at a smooth surface. For this observation, we do not have an explicit explanation. However, it seems to be very likely that the transition of the flow field in the Notch also affects the flow field in the Groove.
\begin{figure}
\centerline{\includegraphics[width=13cm]{Fig07_Flow_field.pdf}}
\caption{Mean velocity fields inside the Notch at $Ra=4.7 \times 10^9$ (left subfigure) and $Ra=4.0 \times 10^{10}$ (right subfigure). The scale of the arrows is arbitrary and differs from one plot to another to allow better visualization of the flow \cite{Liot2016}}
\label{Mean_velocity_field}
\end{figure}
\begin{figure}
\vspace{0.5cm}
\centerline{\includegraphics[width=12cm]{Fig08_Potential_flow_structures.pdf}}
\caption{Sketch of possible flow structure inside a Notch. (a) Thermally stratified, no convection, (b) internal convection, no fluid exchange, (c) external convection with fluid exchange (original Fig.~5 from \cite{Liot2016}) }
\label{Potential_flow_structures}
\end{figure}
In a next step, we are going to discuss the fluctuations, respectively the standard deviation of the near-wall temperature field. The fluctuations are a specific signature of thermal plumes are believed to contribute significantly to the heat transport process. We compute the standard deviation of the temperature time series, and we normalize the result by the temperature drop between the surface temperature of the heating plate and the temperature in the bulk:
\begin{equation}
std (\Theta) = \frac{1}{T_H-T_B}\sqrt{\frac{1}{N-1}\sum_{i=1}^N (T_i-\langle T\rangle)^2},
\end{equation}
We plot the profiles of the normalized standard deviation for both Ra numbers in the diagrams of Fig.~\ref{Tstd_lowRa} and Fig.~\ref{Tstd_highRa}, respectively. Again, the profiles at the top of the obstacles are shifted by $\Delta(z/H)=4.8 \times 10^{-3}$. Since all profiles are scaled by the total temperature drop between the heating plate and the bulk, they can be directly compared. In order to analyse some potential variation of the flow with respect to the smooth case, we have added profiles of the temperature fluctuations over a smooth surface.
We start discussing the lower Ra number case at $Ra_1=4.6 \times 10^9$ (see Fig.~\ref{Tstd_lowRa}). The profile of the temperature fluctuations in the Groove collapse very well with that measured at the smooth reference. This is not surprising and requires no further discussion, since the flow situation in the Groove is quite comparable with the flow along a smooth plate. In contrast to this, the fluctuations in the Notch are generally smaller, which can be explained by the confinement of the fluid in between the obstacles and the decoupling of this fluid volume from the fully turbulent mean wind (see Figs.~\ref{Mean_velocity_field} and \ref{Potential_flow_structures}). Beyond the height of the obstacles at $(z/H)>4.8 \times 10^{-3}$ the temperature fluctuations at all specific regions of the rough surface are generally smaller than those at the smooth surface. This sounds a bit mysterious, since due to the obstacles with all their corners and edges, we rather expected an opposite behavior. One potential explanation might be the different geometry of the rough and the smooth cells. While the flow is locked in a single direction in the rough (rectangular) cell, it has the freedom to change its orientation in the smooth (cylindrical) one. The flow in the latter case is, thus, more complex and larger velocity fluctuations are very likely. This must be reflected in the temperature field as well since both fields are strictly coupled. Insofar, the observation discussed above, might rather be associated with the shape of the cell than with the existence of roughness at the plate surface.
For the higher Ra number case, meaning beyond the transition in the global $Nu(Ra)$ relation, the profiles of the temperature fluctuations clearly differ from those measured at the lower Ra number case case. The maximum of the fluctuations in the Groove is shifted from a position above the height of the obstacles to a position in between them, indicating that the thickness of the boundary layer really becomes smaller than the height of the obstacles. In this content, it is also interesting to point out that the fluctuations in the Notch exhibit a buckle at $z/H=1.8 \times 10^{-3}$. This kink as well as the steeper increase of the profile very close to the wall are consequences of the onset of the flow exchange within the Notch and the turbulent mean wind above the roughness elements (cf. Fig.~\ref{Mean_velocity_field}). With respect to the convective term $\langle w'T'\rangle$, these are also indicators of a generally more efficient heat transport than in the case of the lower Ra number.
\begin{figure}
\centerline{\includegraphics[width=15cm]{Fig09_Std_lowRa_V1.png}}
\caption{Low Rayleigh number: Profiles of the normalized standard deviation $\rm{std}(\Theta(z/H))$ at various locations of the rough surface ($Ra_1=4.6 \times 10^9$), Top: black circles, Notch: black squares, Groove: black triangles, and at the smooth surface ($Ra=3.4 \times 10^9$): open circles. The dashed vertical line at $z/H=4.8 \times 10^{-3}$ indicates the height of the obstacles.}
\label{Tstd_lowRa}
\end{figure}
\begin{figure}
\centerline{\includegraphics[width=15cm]{Fig10_Std_highRa_V1.png}}
\caption{High Rayleigh number: Profiles of the normalized standard deviation $\rm{std}(\Theta(z/H))$ at various locations of the rough surface ($Ra_2=4.7 \times 10^{10}$), Top: black circles, Notch: black squares, Groove: black triangles, and at the smooth surface ($Ra=5.2 \times 10^{10}$): open circles. The dashed vertical line at $z/H=4.8 \times 10^{-3}$ indicates the height of the obstacles.}
\label{Tstd_highRa}
\end{figure}
A final interesting aspect that we are able to discuss in this section thanks to our highly resolved temperature measurements, is the question, at which distance from the plate the flow ``forgets'' the specific geometry of the surface. With respect to the profiles of the mean temperature and the temperature fluctuations we identify this as the z-position, at which the ``rough'' profiles collapse with the ``smooth'' case. Considering all the curves in Figs.~\ref{Tmean_lowRa},~\ref{Tmean_highRa},~\ref{Tstd_lowRa}, and \ref{Tstd_highRa}, this point can be roughly identified with a specific distance of twice the height of the obstacles. Insofar, it is undoubtedly verified that the the enhancement of the convective heat flux due to an artificial roughness of the solid surface is, at least in the domain of Ra numbers investigated here, associated with a modification of the near-wall temperature field.
\subsection{Temperature fluctuations and plume dynamics}
\begin{figure}
\centerline{\includegraphics[width=14cm]{Fig11a_hist_groove_V1.png}}
\centerline{\includegraphics[width=14cm]{Fig11b_hist_notch_V1.png}}
\centerline{\includegraphics[width=14cm]{Fig11c_hist_top_V1.png}}
\vspace{0.1cm}
\caption{Histograms of the samples of the normalized temperature $\Theta(t)$ at various locations of the rough surface. In all plots, the full and the dashed lines represent the rough case at $Ra_2=4.7 \times 10^{10}$ and $Ra_1=4.6 \times 10^9$, respectively. For comparison, we have added the smooth case at $Ra=3.4 \times 10^9$ (dotted line). Upper row: Groove, intermediate row: Notch, lower row: Top. From left to right $z/H=0.00234$, $z/H=0.0048$, and $z/H=0.0099$ ($z/H=0.0048$, $z/H=0.0071$, and $z/H=0.0096$ for the lower row).}
\label{Histograms}
\end{figure}
In the preceding two sections, we have discussed some statistical quantities of turbulent RB convection with rough surfaces and we have compared the results with earlier measurements at smooth surfaces. In this section, we wish to investigate the particular properties of the temperature fluctuations, that are a signature of the evolution of thermal plumes at the roughness elements. In particular close to the wall, where a strong, wall-normal temperature gradient appears, the fluctuations directly reflect the underlying flow field. In so far, their analysis contributes to understand the specific dynamics of the evolution of thermal plumes and the turbulent heat transport in the near-wall flow region. Based on the discussion above and taking into account some observations from previous work we wish to check a few particular quantities that potentially indicate a transition of the near wall flow field from a fluctuating, but still rather laminar to a turbulent flow state. These aspects are designated in the following list.
\begin{list}{•}{
\setlength{\leftmargin}{0.45cm}
\setlength{\itemindent}{-0.45cm}}
\item ~~There is a transition of the flow field in the Notch if the the Rayleigh number is increased from $Ra_1=4.6 \times 10^9$ to $Ra_2=4.7 \times 10^{10}$ (see Fig.~\ref{Mean_velocity_field}).
\item ~~A logarithmic region in the mean temperature profile above the top of the obstacles emerges at the higher Rayleigh number $Ra_2=4.7 \times 10^{10}$ (see Fig.~\ref{Mean_top_comparison}).
\item ~~In the Notch, the profile of the temperature fluctuations exhibits a ``buckle'' at $Ra_2=4.7 \times 10^{10}$ (cf. Fig.~\ref{Tstd_highRa}).
\item ~~There is a hypothesis that the frequency and the dynamics of the plume emission at a rough surface changes with respect to a smooth one (see \cite{Du2000}).
\end{list}
First, we analyse the probability density function of the temperature fluctuations. To this aim, we plot selected histograms of the normalized temperature $\Theta(t)$ in Fig.~\ref{Histograms}. In all plots, the dashed lines represent the lower Ra number case below the transitional effect sets in, the full lines represent the higher Ra number case beyond the transition, and the dotted lines represent the distribution of the samples at a smooth surface at $Ra=3.4 \times 10^9$. For reasons of clarity, we only show the lower Ra number case for the smooth surface, since it equals the higher Ra number case. All histograms are partitioned into 50 bins and they are normalized by the total number of samples $N$ within a single time series. For a more clear representation, we fitted the discrete data points of the histograms by a spline function. Due to the uniform normalization of all distributions, the histograms are directly comparable. The upper row shows the distribution of the normalized temperature fluctuations within and above a Groove. We have chosen specific distances of $z/H=0.00234$ (left), $z/H=0.0048$ (middle), and $z/H=0.0099$ (right) to describe the typical dynamics of the temperature field at the half, the full and the double height of an obstacle ($h=12~\rm mm$). With respect to the typical flow situation in the Groove, it can be expected that the temperature fluctuations in this flow region behave very similar at all three distances $z/H$ compared with the smooth plate. Solely, the distribution very close to the plate at $z/H=0.00234$ and at the lower Ra number $Ra_1=4.6 \times 10^9$ (left upper sub-figure is slightly shifted towards the temperature of the heating plate $\Theta=1$. This might be an effect of a heat transport from the sidewalls of the obstacles to the fluid flow in the Groove that distort the mean temperature field close to the roughness elements. The probability distributions of the temperature in the Notch (middle row) significantly differ from that of the smooth case. Independently, on whether or not the critical Ra number for the transition of the local flow field is exceeded, the distributions at half and full height of the obstacles are higher, narrower and clearly skewed towards the temperature of the heating plate. There is also a difference between the low and the high Ra number distribution at $z/H=0.00234$ reflecting as well the transition of the flow field within the Notch as the observed ``buckle'' in the profile of the temperature fluctuations. In the lower row, the histograms of the temperature above the top of the obstacles is shown at $z/H=0.0048$, $z/H=0.0071$, and $z/H=0.0096$. This corresponds to positions next to the top surface of an obstacle, as well as 6~mm and 12~mm above. In particular, these plots are of interest, since they show the typical distribution of the temperature fluctuations in the region, where the mean temperature profile at the higher Ra number exhibits a logarithmic trend and where they significantly deviate from the lower Ra number and from the smooth case.
Eventually, we wish to turn back to the hypothesis that roughness modifies the number and the frequency of emitted plumes at the top of the obstacles. This modification is commonly reported along with an increase of the global heat transport, however, the scaling exponent $\gamma$ remains unchanged. On the other hand, there are various experiments, in which the scaling exponent $\gamma$ changes, when a certain critical limit in Ra is exceeded. Among other experiments, such a variation in $\gamma$ has been also observed in the two equivalent RB cells in Ilmenau and Lyon \cite{Liot2016}. It seems to be clear that a laminar-turbulent transition of the boundary layer, even it appears only locally at the Top of the obstacles, will definitely affect the scaling exponent (see Fig.~\ref{Mean_top_comparison}). However, it is not confirmed that a change in the dynamics of the plume evolution does the same. In the subsequent discussion, we will focus on this we will particularly compare the plume dynamics at smooth and rough surfaces. Although, there is no final definition what a thermal plume is, people commonly consider fluid parcels that dissolve from the boundary layer at the heating (or cooling) plate and which feature a higher (or lower) temperature with respect to the surrounding fluid as such an event. Plumes arise as well at the smooth as at the rough surface and they significantly affect the heat transport in thermal convection. If they arise more frequently, and/or their typical characteristics, like their size or their energy content, change due to the roughness elements, they modify the typical bahavior of the temperature fluctuations in the fluid layer close to the plate. More precisely, one would expect an increase of the temperature fluctuations and/or a shift of their probability distribution towards the temperature at the plate surface. In this content, it might be useful to remind the profiles of the normalized standard deviation $\rm{std}(\Theta(z/H))$, in particular those measured beyond the height of the obstacles ($z/H>0.0048$). These profiles are plotted in Fig.~\ref{Tstd_lowRa} (low Ra number) and Fig.~\ref{Tstd_highRa} (high Ra number) and as we already discussed above, the temperature fluctuations are unexpectedly lower at the rough than at the smooth surface. The effect is not very large and it is more pronounced for the lower Ra number case. For the higher Ra number it almost vanishes and, in so far, there is no indication of significant higher plume activity from analyzing the standard deviation of the temperature time series. The same holds for the probability distribution of the temperature fluctuations that we plot for various distances from the plate surface in Fig.~\ref{Histograms}. The most relevant curves are the measurements beyond the height of the obstacles at $z/H=0.0099$ shown in the right sub-figures. In all three diagrams representing the distribution of the temperature fluctuations above the Groove, the Notch, and the Top region, a significant difference between the two cases with the rough plate and the case with the smooth plate appear. In summary, we can state that neither the profiles of the temperature fluctuations nor the probability distributions indicate any signature of an enhanced plume activity or any variation in their typical properties, at least for our specific roughness profile.
\begin{figure}
\centerline{\includegraphics[width=14cm]{Fig12a_FFT_near_wall_V3}}
\centerline{\includegraphics[width=14cm]{Fig12b_FFT_log_region_V3.png}}
\vspace{0.1cm}
\caption{Power spectra of the temperature fluctuations at $z/\delta_T=0.6$ (upper diagramm) and $z/\delta_T=3.6$ (lower diagramm). The gray (online: blue) and the black lines represent the rough case at $Ra_1=4.6 \times 10^9$ and $Ra_2=4.7 \times 10^{10}$, respectively. For comparison, the spectra of the temperature fluctuations above a smooth surface at $Ra=3.4 \times 10^9$ has been added as light gray line (online: orange). The insets show the respective measurement positions in the mean temperature profiles as detailed shown in Fig.~\ref{Mean_top_comparison}.}
\label{FFT}
\end{figure}
An even more sophisticated tool to study the dynamics of the temperature fluctuations in the time domain is the transformation of the signal in a Fourier space. Such an analysis provides the spectrum of the amplitude of the temperature fluctuations over the various frequencies. From this, we may associate specific properties of the spectra obtained, like its slope, its cut off or distinct peaks, with specific characteristics of the plumes and the underlying turbulent background. Since plumes are believed to preferably arise at the Top of the obstacles, we run a Fast Fourier Transformation (FFT) of selected temperature-time series above the roughness elements (TOP) for our analysis. We focus on two specific distances above the top of the obstacles. With reference to Fig.~\ref{Mean_top_comparison}, we compute a first set of FFTs at $z/\delta\approx0.6$, the normalized distance at which the profiles of the mean temperature at $Ra_1=4.6 \times 10^9$ and $Ra_2=4.7 \times 10^{10}$ obey their maximum difference. A second set of FFTs has been computed for the distance $z/\delta\approx3.6$, the position, at which the mean temperature profile at $Ra_2$ exhibits the logarithmic trend. In order to make the frequency spectra at various Rayleigh numbers comparable, we scale the frequency $f$ by the advection velocity $\langle v\rangle$ of the mean flow along the heated bottom plate and plot all spectra versus the wave number $k=2\pi f/\langle v\rangle$. The specific velocities for this normalization amounts to $\langle v_{Top,Ra1}\rangle=0.12~{\rm ms^{-1}}$, $\langle v_{Top,Ra2}\rangle=0.30~{\rm ms^{-1}}$ for the rough surface \cite{Liot2016} and $\langle v_{Smooth,Ra=3.5e11}\rangle=0.10~{\rm ms^{-1}}$ for the smooth case \cite{Li2012}. We plot the spectra at the smaller distance $z/\delta\approx0.6$ above the top of the obstacles in the upper diagram of Fig.~\ref{FFT} and the spectra at the larger distance $z/\delta\approx3.6$ in the lower diagram. For comparison, we add the spectra above a smooth surface at a slightly lower Rayleigh number $Ra=3.4 \times 10^9$. As a reminder, the insets of both diagrams show the profiles of the mean temperature for the two rough cases indicating the measurement positions that have been Fourier transformed. In both diagrams the ``rough'' and the ``smooth'' spectra at the lower Rayleigh number $Ra_1$ almost collapse. This indicates that neither the evolution and the dynamics of the plumes nor the spatial structure of the turbulent background flow is modified by the pure existence of the obstacles. Only beyond the transition, at the higher Ra number $Ra_2$, the spectra at the rough surface exhibit differ from that at the smooth one. It exhibits a more flat slope and the cut-off wave number - this is the point, at which the spectrum collapses with the noise level - is about twice as high compared with the spectra at the lower Ra number. The flatter profile and the higher cut-off wave number indicate faster and more intensive fluctuations of the temperature (and the velocity) field. The factor of ``two'' in the kick-off wave number can be interpreted as halving the typical size of the smallest turbulent structures that appear close to the rough surface. This is much more than the decrease of the boundary layer thickness between those two Ra numbers, which amounts to $\delta_T(Ra_1)=5.82~\rm{mm}$ and $\delta_T(Ra_1)=4.72~\rm{mm}$ (see Table~\ref{Gradients}), respectively, and this confirms that a real transition of the boundary layer takes place. On the other hand, we did not find any distinct bump over the entire spectra neither for the two rough nor for the smooth case. The emission of plumes happen, thus, always randomly, and this emission scheme is neither affected by introducing roughness elements nor by exceeding the critical Ra number for the boundary layer transition at the rough surface.
\section{Conclusions}
We report highly resolved temperature measurements in turbulent Rayleigh-B\'enard (RB) convection with a rough surfaces. The measurements have been undertaken in a rectangular test section at two Rayleigh numbers $Ra_1=4.6 \times 10^9$ and $Ra_2=4.7 \times 10^{10}$. The particular idea of this work is, to study the near-wall temperature field below and beyond a critical Rayleigh number $Ra_c$, at which a transition in the scaling of the global heat transfer relation $Nu\sim Ra^{\gamma}$ has been observed as well in our experiment as in an equivalent one with water that is operated by a group at the \'Ecole Normale Sup\'erieure de Lyon \cite{Liot2016,Tisserand2011}. We used a very tiny micro-thermistor of only 130~µm in diameter and 330~µm in length for our measurements, which is about forty times smaller than the smallest thickness of the boundary layer at the top of the obstacles. The high spatial resolution of our measurements as well as the very short response time of the sensor, which are below the typical Kolmogorov microscales that occur in our experiment, enable us to reveal a potential variation of the local temperature field between the smooth and the rough cases as well as between the low and the high Ra number rough cases.
In our work, we analysed profiles of the mean temperature and the temperature fluctuations as well as the probability distribution of the temperature fluctuations at various locations with respect to the roughness elements. Our measurements demonstrate that the heat flux enhancement generally observed at rough surfaces results from an increase of the local heat transfer coefficient at the top of the obstacles. We also show that the transition of the scaling exponent $\gamma$ in the global heat flux relation $Nu\sim Ra^{\gamma}$ can be attributed to a modification of the temperature field at the top of the obstacles as well as in the flow regions in between them. Below a critical Ra number $Ra_c \approx 10^{10}$, the profile of the mean temperature $T(z)$ at the top of the obstacles basically equals with that measured at a smooth surface. Beyond this point the profile tends to exhibit a logarithmic trend, a typical signature of a turbulent boundary layer. The transition at the rough surface appears at a critical Ra number of $Ra_c \approx 10^{10}$, which is about three to four orders of magnitude lower than the limit that has been predicted for RB convection with smooth plates \cite{Grossmann2001}. The profiles of the mean temperature in the Notch and in the Groove changes as well beyond the critical Ra number. In both flow regions, the gradients are smaller and we attribute this to the onset of local convection that increases the local (and the global) heat transport. We also observed that the variation of the temperature field due to the surface roughness only covers a fluid layer with a thickness of about twice the height of the obstacles. Beyond this distance from the wall, the temperature field above the rough surface is virtually unchanged if one compare it with a smooth surface. In this far-wall region, neither the profiles of the mean temperature nor the probability distribution of the temperature fluctuations show any significant differences.
We also check the hypothesis of Du \& Tong , who discovered that the dynamics of the plume emission at a rough surface change with respect to the smooth case \cite{Du2000}. While the mean temperature field and the probability distribution of the fluctuations do not change, when roughness elements have been installed the typical size of the plumes and the frequency of their emission increases, however, even exceeding the critical Ra number for the transition of the boundary layer. For our specific roughness structure of cubic obstacles in a $l=2d$ periodical distance, we do not find any signature of a variation of the plume dynamics with respect to the introduction of roughness elements as long as one remains below the transition. This is certainly in contrast to the work, done by Du \& Tong, but might be related to the different types of roughness elements.
\bigskip
\noindent \textbf{ACKNOWLEDGMENTS}
\medskip
The authors wish to acknowledge the German Research Foundation under the grant number PU436/1-2, the German Academic Exchange Service under the project ID 57128323 and the European Union under the Grant Agreement number 312778, who gratefully supported the work reported here. We also thank the Veritas Sensors LLC providing the ultra-small micro-thermistors used in the work reported here. Moreover, we wish to thank Vigimantas Mitschunas, Sabine Abawi and Olivier Liot for their assistance to operate the experimental facility and to run the measurements.
\bigskip
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{"url":"http:\/\/math.stackexchange.com\/questions\/293593\/joint-pdf-for-2-hypergeometric-random-variables","text":"# Joint pdf for 2 hypergeometric random variables\n\nI have 2 random hypergeometric variables $X \\sim \\mathcal{H}(N, K, n)$ and $Y(X) \\sim \\mathcal{H}(N-n, L-n+X, n)$ with $N = K + L$. How can I write the joint pdf of $X$ and $Y$?\n\nI guess that $P(X=a \\wedge Y = b) = P(X=a)P(Y=b)$ but I'm not really sure that $X$ and $Y$ are independent.\n\nI know that $P(X=a)$ follows the first hypergeometric distribution and $P(Y=b \\mid X=a)$ follows the second. Then, I can state the following:\n\n$$P(Y=b \\mid X=a) = \\frac{P(Y=b \\wedge X=a)}{p(X=a)} \\Rightarrow P(Y=b \\wedge X=a) = P(Y=b \\mid X=a)P(X=a)$$\n\nIs this the right way to evaluate the joint pdf?\n\n-\nThe Bayes rule that you wrote is correct. For independency one needs to check if $P(Y=b|X=a)=P(Y=b)$ is correct or not. \u2013\u00a0 Seyhmus G\u00fcng\u00f6ren Feb 3 '13 at 15:05\nInstead of $Y(X) \\sim \\mathcal{H}(N-n, L-n+X, n)$, I'd normally write $Y\\mid X \\sim \\mathcal{H}(N-n, L-n+X, n)$. That way it's clear that it's not $Y$ that is determined by $X$, but rather it's the conditional probability distribution of $Y$ that is determined by $X$. But don't make the mistake of construing \"$Y\\mid X$\" as a noun. There's no object called $Y\\mid X$. \u2013\u00a0 Michael Hardy Feb 3 '13 at 16:51\nI've retagged: please do not use an ambiguous abbreviation (\"pdf\") when the expanded tag is already available (\"probability-distributions\"). \u2013\u00a0 Willie Wong Feb 4 '13 at 8:58","date":"2015-08-03 01:22:22","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9130864143371582, \"perplexity\": 275.75879856479213}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-32\/segments\/1438042989331.34\/warc\/CC-MAIN-20150728002309-00243-ip-10-236-191-2.ec2.internal.warc.gz\"}"}
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\section{Introduction}\label{sc:intro}
Bose--Einstein condensation in alkali-metal gases was observed experimentally in 1995~\cite{And1995.Sci269.198,Bra1995.PRL75.1687,Dav1995.PRL75.3969,Bra1997.PRL79.1170}. A few years later, these pioneering experiments were followed by the creation of singly quantized vortices~\cite{Mat1999.PRL83.2498,Mad2000.PRL84.806} and vortex lattices~\cite{Mad2000.JMO47.2715,Abo2001.Sci292.476,Ram2001.PRL87.210402} in such systems. Since then, the study of vortices in Bose--Einstein condensates (BECs) has flourished both theoretically and experimentally~\cite{Fet2009.RMP81.647,And2010.JLTP161.574} due to their close connection with phase coherence and superfluidity. In particular, their stability has been the subject of extensive research~\cite{Dod1997.PRA56.587,Rok1997.PRL79.2164,Pu1999.PRA59.1533,Iso1999.PRA60.3313,Svi2000.PRL84.5919,Vir2001.PRL86.2704,Sim2002.PRA65.033614,Kaw2004.PRA70.043610,Jac2005.PRA72.053617,Huh2006.PRA74.063619,
Lun2006.PRA74.063620,Cap2009.JPB42.145301,Kuo2010.PRA81.023603}.
In a loop encircling a quantized vortex, the phase of the condensate order parameter undergoes an integer multiple $\kappa$ of $2\pi$ windings. In principle, a vortex in a BEC can have any winding number $\kappa$. However, it is well known that a vortex with $|\kappa|>1$ typically has a higher energy than the corresponding number of separated single-quantum vortices. Consequently, vortices with large winding numbers are prone to splitting~\cite{Shi2004.PRL93.160406,Mot2003.PRA68.023611,Gaw2006.JPhysB39.L225,Huh2006.PRL97.110406,Mat2006.PRL97.180409,Iso2007.PRL99.200403,Kar2009.JPhysB42.095301,Kuo2010.PRA81.033627}, which renders them challenging to create with dynamical methods, such as using a focused laser beam to stir~\cite{Mad2000.PRL84.806} or slice through~\cite{Ino2001.PRL87.080402} the BEC, rotating it with an asymmetric trap potential~\cite{Hod2001.PRL88.010405}, or colliding condensates separated by tailored optical potentials~\cite{Sch2007.PRL98.110402}. Being able to produce vortices with large winding numbers would provide access to novel vortex splitting patterns beyond the typical linear chain that prevails for $|\kappa|\leq 4$~\cite{Shi2004.PRL93.160406,Iso2007.PRL99.200403}. Due to the distinct nature of the different splitting patterns predicted for large values of $\kappa$~\cite{Kuo2010.PRA81.033627}, observing the decay of such vortices would allow for a lucid comparison between theory and experiment. Moreover, it has been speculated that giant-vortex splitting may create necessary conditions for the initialization of superfluid turbulence~\cite{Abr1995.PRB52.7018,Ara1996.PRB53.75}.
In addition to the above-mentioned dynamical methods, vortices can be created with the so-called topological phase engineering technique~\cite{Nak2000.PhysicaB284.17,Iso2000.PRA61.063610,Oga2002.PRA66.013617,Mot2002.JPCM14.13481} (see Ref.~\cite{Pie2008.book.vortices} for review), in which the spin degree of freedom of the BEC is controlled adiabatically by a time-dependent nonuniform magnetic field. Since the method does not rely on the relaxation of condensate dynamics, it is especially well suited for producing multiquantum vortices. Indeed, the first two-quantum and four-quantum vortices in dilute BECs were created by applying the technique to spin-1 and spin-2 BECs confined in a magnetic Ioffe--Pritchard (IP) trap~\cite{Lea2002.PRL89.190403}.
Subsequent theoretical studies have demonstrated that the topological phase engineering technique can also be used to implement a so-called vortex pump~\cite{Mot2007.PRL99.250406,Xu2008.PRA78.043606,Xu2008.NJP11.055019,Xu2010.PRA81.053619,Kuo2010.JLTP161.561}. In this device, a fixed amount of vorticity is added to the BEC in each control cycle, and thus its repeated application would---stability issues notwithstanding---enable the creation of vortices with arbitrarily large winding numbers. The original proposal~\cite{Mot2007.PRL99.250406} involved creating $2F$ quanta of vorticity per cycle in a spin-$F$ BEC with a magnetic-field configuration consisting of the standard IP trap and an additional hexapole magnetic field. Backed by numerical simulations, the pump was shown to be operable both fully adiabatically and partly nonadiabatically. Later, Xu \emph{et al.}~\cite{Xu2008.PRA78.043606} presented a different pumping cycle for the IP trap in which the hexapole field was replaced with a uniform transverse field. Unfortunately, both of these control cycles suffer from the fact that the magnetic fields provide radial confinement only during part of the cycle, and thus, the fully adiabatic operation of the pump necessitates an optical trap to confine the BEC radially. Since the purpose of the IP trap has been to confine the atomic cloud in the first place, there has been little incentive to supplement it with an optical trap or a hexapole field. Therefore, pumping schemes not requiring such extra ingredients would be desirable from a practical standpoint.
Recently, Xu~\emph{et al.} showed theoretically that vortex pumping can be applied to quantum superpositions to generate countercirculation states~\cite{Xu2010.PRA81.053619}. The authors used a novel control cycle which is particularly suitable for the time-averaged orbiting potential (TOP) trap~\cite{Pet1995.PRL74.3352} and has the advantage that the radially confining quadrupole field can be kept on throughout the entire cycle. However, since only one of the components in the superposition state could be trapped magnetically, a three-dimensional optical trap, as well as a strong optical plug potential piercing the vortex core, had to be employed~\cite{Xu2010.PRA81.053619}.
The aim of this article is to bring the vortex pump closer to experimental realization by showing that it can be implemented with mature, existing technologies alternative to the ones considered in Refs.~\cite{Mot2007.PRL99.250406,Xu2008.PRA78.043606,Xu2008.NJP11.055019,Xu2010.PRA81.053619,Kuo2010.JLTP161.561}. To this end, we demonstrate that vortices can be efficiently pumped in the TOP trap without using, in contrast to Ref.~\cite{Xu2010.PRA81.053619}, additional optical potentials to confine the BEC in the radial direction or to pin the vortex core. Instead, the radial confinement is provided solely by the magnetic field throughout the pumping process, and optical trapping is required only in the axial direction. We present simulations based on the Gross--Pitaevskii equation which indicate that several pumping cycles can be carried out before a majority of the particles escape from the trap or before the generated multiquantum vortex splits into singly quantized vortices due to dynamical instabilities~\cite{Kuo2010.PRA81.033627,Pie2007.PRA76.023610}. On the other hand, we also show that even a relatively weak optical plug potential is efficient in preventing the splitting and in reducing the loss of particles, thereby enabling the controlled creation of isolated vortices with large winding numbers.
The remainder of this article is organized as follows. In Sec.~\ref{sc:theory}, we present the zero-temperature mean-field theory of the spin-1 BEC, describe the control cycle of the vortex pump, and discuss in detail the confinement of the condensate during the cycle. Section~\ref{sc:results} presents our numerical results, which we relate to realistic experimental setups in Sec.~\ref{sc:experimental}. Finally, Sec.~\ref{sc:conclusion} concludes the article with a discussion.
\section{Theory and methods}\label{sc:theory}
\subsection{Mean-field model}
We consider a dilute spin-1 BEC in the zero-temperature limit, thereby neglecting the possible effects due to noncondensed atoms. In the standard mean-field treatment, the spin-1 condensate is described by a three-component order-parameter field that we write in the eigenbasis of the spin-1 matrix $F_z$ as $\vec{\Psi}=\left(\Psi_{+1},\Psi_{0},\Psi_{-1}\right)$. Its time dependence is given by the spin-1 Gross--Pitaevskii (GP) equation~\cite{Ohm1998.JPSJ67.1822,Ho1998.PRL81.742}
\begin{eqnarray}
\label{eq:GPE}
i\hbar\partial_t\vec{\Psi}({\mathbf{r}},t) &=& \Big( {\mathcal H} + g_\mathrm{n} \vec{\Psi}^\dagger\vec{\Psi} \nonumber \\
&& + g_\mathrm{s} \vec{\Psi}^\dagger \mathbf{F} \vec{\Psi} \cdot \mathbf{F} \Big) \vec{\Psi}({\mathbf{r}},t).
\end{eqnarray}
The single-particle Hamiltonian operator ${\mathcal H}$ is given by
\begin{equation}\label{eq:ham}
{\cal H} = -\frac{\hbar^2}{2m}\nabla^2 + V_\mathrm{opt}({\mathbf{r}}) + \mu_\mathrm{B} g_F \vect{B}\left({\mathbf{r}},t\right)\cdot \mathbf{F},
\end{equation}
where $m$ denotes the atomic mass, $g_F$ is the Land\'{e} factor, $\mu_\mathrm{B}$ is the Bohr magneton, $\vect{B}({\mathbf{r}},t)$ denotes the external magnetic field, and $\vect{F}=\left(F_x,F_y,F_z\right)$ is a vector of the standard spin-1 matrices~\cite{Ued2010.book.Bose}. Optical potential terms are contained in $V_\mathrm{opt}({\mathbf{r}})= V_\mathrm{tr}(z) + V_\mathrm{plug}\left( r \right)$, where $ V_\mathrm{tr}(z)=m\omega_z^2 z^2/2$ is a strong axial harmonic trap and $V_\mathrm{plug}\left(r\right)=A \exp\left(-r^2/d^2\right)$ describes a possibly present Gaussian-shaped repulsive plug potential of amplitude $A\geq 0$ and width $d$. Here, $r=\sqrt{x^2+y^2}$ is the radial coordinate. The coupling constants $g_\mathrm{n}$ and $g_\mathrm{s}$ appearing in Eq.~\eqref{eq:GPE} measure the strengths of the local density--density and spin--spin interactions, respectively. They are related to the $s$-wave scattering lengths $a^{(0)}$ and $a^{(2)}$ into spin channels with total spin 0 and $2\hbar$ by the expressions $g_\mathrm{n}=4\pi\hbar^2\left[a^{(0)}+2a^{(2)}\right]/3m$ and $g_\mathrm{s}=4\pi\hbar^2\left[a^{(2)}-a^{(0)}\right]/3m$. The order parameter is normalized such that $\int d^3 r \vec{\Psi}^\dagger \vec{\Psi} = N_0$, where $N_0$ is the number of particles in the BEC.
\subsection{Magnetic fields and the pumping cycle}
The operation principle of the vortex pump is to control the spin degree of freedom of the condensate locally by slowly tuning the magnetic field $\vect{B}({\mathbf{r}},t)$ in a cyclic manner such that the system acquires a fixed amount of vorticity per cycle~\cite{Mot2007.PRL99.250406,Xu2008.PRA78.043606,Xu2008.NJP11.055019,Xu2010.PRA81.053619,Kuo2010.JLTP161.561}. In the pumping scheme considered here, the spin-1 atoms are assumed to be magnetically confined in the standard TOP trap~\cite{Pet1995.PRL74.3352}. It consists of a quadrupole field $\vect{B}_\mathrm{q}$, which has axial symmetry about the $z$ direction, and a rapidly rotating, spatially uniform magnetic field $\vect{B}_\mathrm{rot}$ oriented along the $xy$ plane. In addition, we assume that the TOP trap is accompanied by a uniform axial bias field $B^\mathrm{b}_z(t) \hat{\vect{z}}$ that can be controlled independently of the other fields. The total magnetic field can be written as
\begin{equation}\label{eq:B}
\vect{B}({\mathbf{r}},t) = \vect{B}_\mathrm{q}({\mathbf{r}}) + \vect{B}_\mathrm{rot}(t)+B^\mathrm{b}_z(t)\hat{\vect{z}},
\end{equation}
where $\vect{B}_\mathrm{q}({\mathbf{r}}) = B'\left(x\hat{\vect{x}}+y\hat{\vect{y}}-2 z \hat{\vect{z}}\right)$ is the quadrupole field with the radial gradient $B'$ and the rotating transverse bias field is given by
\begin{equation}\label{eq:Brot}
\vect{B}_\mathrm{rot}(t) = B_\mathrm{rot}(t) \left[\cos\left(\omega_\mathrm{rot} t\right) \hat{\vect{x}}+\sin\left(\omega_\mathrm{rot} t\right)\hat{\vect{y}} \right],
\end{equation}
where $\omega_\mathrm{rot}$ denotes its angular frequency of rotation about the $z$ axis. The bias field strengths are assumed to be bound by $B_0$ such that $B^\mathrm{b}_z(t)\in\left[-B_0, B_0\right]$ and $B_\mathrm{rot}(t)\in \left[0, B_0 \right]$. We point out that the field configuration of Eq.~\eqref{eq:B} has already been employed in BEC experiments~\cite{Hod2000.JPhysB33.4087,Hod2001.PRL86.2196}.
In order to facilitate vortex pumping in the TOP trap, we use the control cycle presented in Fig.~\ref{fig:cycle}. It is carried out by tuning two magnetic-field parameters, $B^\mathrm{b}_z$ and $B_\mathrm{rot}$, and can be divided into part A ($0\leq t \leq T_\mathrm{A}$) and part B ($T_\mathrm{A} \leq t \leq T_\mathrm{A}+T_\mathrm{B} =: T$). Part A is similar to the original proposals of topological phase engineering~\cite{Nak2000.PhysicaB284.17,Iso2000.PRA61.063610,Oga2002.PRA66.013617,Mot2002.JPCM14.13481,Mot2007.PRL99.250406} and the experiments~\cite{Shi2004.PRL93.160406,Lea2002.PRL89.190403,Lea2003.PRL90.140403,Kum2006.PRA73.063605,Oka2007.JLTP148.447,Shi2011.JPB44.075302}, and it is responsible for increasing the circulation in the spin-1 BEC by two quanta. It is executed by reversing the axial bias field with the rotating field switched off,
\begin{equation}\label{eq:part_A}
\left. \begin{array}{ll} B_\mathrm{rot}(t) = 0 \\ B^\mathrm{b}_z(t)= B' \rho_0 \tan\left[\frac{2t-T_\mathrm{A}}{T_\mathrm{A}}\arctan\left(\frac{B_0}{B'\rho_0} \right) \right] \end{array}\right\}\ 0\leq t \leq T_\mathrm{A},
\end{equation}
where $B_0$ should be large enough to render the BEC essentially spin polarized along the $z$ axis at $t=0$ and $t=T_\mathrm{A}$. To improve adiabaticity, the time dependence for $B^\mathrm{b}_z$ has been chosen such that spins at a distance of $\rho_0$ from the $z$ axis are turned with constant speed, but part A can also be performed by reversing $B^\mathrm{b}_z(t)$ linearly in time~\cite{Lea2002.PRL89.190403}. In part B, the axial bias field is returned to its initial value while ramping up and down the rotating field,
\begin{equation}\label{eq:part_B}
\left. \begin{array}{ll} B_\mathrm{rot}(t) = B_0\sin\beta(t) \\ B^\mathrm{b}_z(t)=B_0 \cos\beta(t) \end{array}\right\}\ T_\mathrm{A} \leq t \leq T,
\end{equation}
where $\beta(t)=\pi\left(t-T_\mathrm{A}\right)/T_\mathrm{B}$. Part B was originally proposed by Xu~\emph{et al.}~\cite{Xu2010.PRA81.053619}, and it is designed to preserve the accumulated vorticity. The cyclic repetition of parts A and B will therefore increase the vortex winding number of the spin-1 BEC by two per cycle.
\begin{figure}
\begin{center}
\includegraphics[
width=210pt,
keepaspectratio]{FIG1}
\end{center}
\caption{\label{fig:cycle} (a) Control cycle of the vortex pump in the $(B_\mathrm{rot},B^\mathrm{b}_z)$ plane,
where $B_\mathrm{rot}$ and $B^\mathrm{b}_z$ are the strengths of the transverse and axial magnetic bias fields, respectively.
The cycle starts at $(B_\mathrm{rot},B^\mathrm{b}_z)=(0,-B_0)$ and proceeds clockwise. (b) Time dependence of $B_\mathrm{rot}$ (solid line) and $B^\mathrm{b}_z$ (dashed line) [Eqs.~(\ref{eq:part_A}) and (\ref{eq:part_B})] during the cycle.
The cycle is divided into parts A and B as indicated.
In part A, the axial bias field $B^\mathrm{b}_z$ can also be reversed linearly. }
\end{figure}
To efficiently steer the condensate spin by the magnetic field $\vect{B}\left({\mathbf{r}},t \right)$ requires that the Zeeman energy dominates over the kinetic energy at each point in space. Hence, to guarantee adiabaticity, $|\vect{B}\left({\mathbf{r}},t \right)|$ should be sufficiently large in the region occupied by the BEC. This condition is not fulfilled at the origin when $B^\mathrm{b}_z$ crosses zero at $t=T_\mathrm{A}/2$, and thus it is desirable to prevent particles from entering this area. This can be accomplished by introducing the repulsive plug potential $V_\mathrm{plug}(r)$ along the $z$ axis. The plug not only improves adiabaticity but also serves to stabilize the created multiquantum vortex against splitting~\cite{Kuo2010.PRA81.033627,Kuo2010.JLTP161.561}. The plug can be realized for pancake-shaped BECs by a focused blue-detuned laser beam as has been done in various experiments~\cite{Dav1995.PRL75.3969,Abo2001.Sci292.476,Ram2001.PRL87.210402,Sim2005.PRL94.080404,Nee2010.PRL104.160401}.
In this article, we present results for vortex pumping both with and without the plug potential.
\subsection{Confinement during pumping}\label{subsc:confinement}
An essential difference between this article and earlier work concerning the vortex pump~\cite{Mot2007.PRL99.250406,Xu2008.PRA78.043606,Xu2008.NJP11.055019,Xu2010.PRA81.053619,Kuo2010.JLTP161.561} is that here we never employ an optical trapping potential in the radial direction. Instead, radial confinement is provided by the magnetic field throughout the entire pumping cycle. In the case of spin-1 BECs, the magnetically trapped weak-field seeking state (WFSS) corresponds locally to the highest-energy eigenstate of the Zeeman Hamiltonian $g_F \mu_\mathrm{B}\left({\mathbf{r}},t\right) \vect{B} \cdot \mathbf{F}$, with the effective trap potential given by the local eigenvalue $|g_F\mu_\mathrm{B}\vect{B}({\mathbf{r}},t)|$. Even though this potential is solely responsible for the radial confinement, a strong optical trap is still needed in the axial direction to keep the atomic cloud centered around $z=0$ throughout the pumping cycle. Changes in $B^\mathrm{b}_z$ shift the $z$ coordinate of the zero-value point of the total magnetic field, and without the optical $z$ confinement, carrying out the cycle would merely move the whole BEC along the $z$ axis. Hence, the Hamiltonian in Eq.~\eqref{eq:ham} includes the harmonic axial trapping potential $V_\mathrm{tr}(z) = m\omega_z^2 z ^2 / 2$ with a trap frequency $\omega_z$ that is assumed to be large enough to render the condensate pancake shaped, which means that the order parameter can be taken to have the form $\vec{\Psi}\left({\mathbf{r}},t\right)=\vec{\Psi}_\mathrm{2D}\left(x,y,t\right)\zeta(z)$, where $\zeta(z)=\exp\left(-z^2/2 a_z^2\right)/\sqrt[4]{\pi a_z^2}$ and $a_z = \sqrt{\hbar/m\omega_z}$ is the axial oscillator length. This enables us to integrate out the $z$ variable in Eq.~\eqref{eq:GPE} and obtain an effectively two-dimensional GP equation with the magnetic field determined at $z=0$.
Let us consider the shape of the magnetic potential in the vicinity of the origin. During part A of the cycle ($0\leq t \leq T_\mathrm{A}$), the strength of the magnetic field is given by
\begin{eqnarray}
|\vect{B}| &=&\sqrt{(B'x)^2+(B'y)^2+\left[B^\mathrm{b}_z-2B'z\right]^2}\nonumber \\ &\approx& |B^\mathrm{b}_z|- \frac{2B'|B^\mathrm{b}_z|}{B^\mathrm{b}_z}z +\frac{B'^2}{2|B^\mathrm{b}_z|} r^2,
\label{eq:BstrengthA}\end{eqnarray}
where in the expansion we have neglected third- and higher-order terms in $B'r/|B^\mathrm{b}_z|$ and $B'|z|/|B^\mathrm{b}_z|$. Therefore, the magnetic field at $t=0$ gives rise to an approximately harmonic potential in the radial direction with the trap frequency
\begin{equation}\label{eq:omega0}
\omega_0 = B'(0) \sqrt{\left|\frac{g_F \mu_\mathrm{B}}{m B^\mathrm{b}_z(0)}\right|}.
\end{equation}
It is convenient to measure all quantities in terms of $\omega_0$, and thus we express lengths in units of the corresponding oscillator length $a_0=\sqrt{\hbar/m \omega_0} \gg a_z$, energies in units of $\hbar\omega_0$, time in units of $1/\omega_0$, and the magnetic field in units of $\hbar\omega_0/|g_F|\mu_\mathrm{B}$. Variables expressed in these units are henceforth denoted with a tilde.
According to Eq.~\eqref{eq:BstrengthA}, the profile of the radial confinement will change during part A. Initially, the potential is harmonic, with the effective trap frequency $\propto |B^\mathrm{b}_z(t)|^{-1/2}$ increasing in time. At $t=T_\mathrm{A}/2$, the trap becomes purely linear in $r$ with the gradient $|g_F|\mu_\mathrm{B} B'$. The axisymmetric modulations of the trap profile will cause shrinking of the BEC and excitation of its breathing mode. Although these effects do not critically hinder the operation of the pump, they can nevertheless be reduced by introducing time dependence into $B'$ such that it is decreased when $|B^\mathrm{b}_z|$ is ramped down during part A. Here, we use the dependence
\begin{equation}\label{eq:B'}
B'(t)=\left\{\begin{array}{ll} B_0'\left[\frac{B_\mathrm{min}'}{B_0'}+\left(\frac{2t-T_\mathrm{A}}{T_\mathrm{A}}\right)^2\left(1-\frac{B'_\mathrm{min}}{B_0'}\right)\right], & 0 \leq t \leq T_\mathrm{A}, \\ B'_0, & T_\mathrm{A} < t \leq T. \end{array} \right.
\end{equation}
with $B_0'=B'(0)$ and $B_\mathrm{min}' \approx 0.4 B'_0$. Varying $B'$ is not necessary but it improves the accuracy of the pump and reduces the loss of particles.
In part B, the frequency $\omega_\mathrm{rot}$ of the rotating bias field is chosen to be low compared with the frequencies of transitions between different magnetic substates but large compared with the effective radial trap frequency. Typically, magnetic trap frequencies are of order $10^2\,\mathrm{Hz}$ while the transition frequencies are of order $10^6\,\mathrm{Hz}$. Thus, a reasonable choice would be, e.g., $\omega_\mathrm{rot} \sim 10^4\,\mathrm{Hz}$. These conditions ensure that the atoms will not undergo transitions to other substates and be lost from the magnetic trap but instead move in an effective potential given by the time average of the instantaneous magnetic potential over one rotation period of the field $\vect{B}_\mathrm{rot}$. For $T_\mathrm{A} \leq t \leq T$, the fast-time-averaged field strength can be expanded in a power series in $B'r/B_0$ and $B'|z|/B_0$ as
\begin{eqnarray}\label{eq:BstrengthB}
\frac{\omega_{\mathrm{rot}}}{2\pi}\int_{I_t} dt'|\vect{B}(t')| & \approx & B_0 - 2B'z \cos\beta + \frac{B'^2}{4B_0}\left(1+\cos^2\beta\right)r^2 \nonumber \\ &+&
\frac{2 B'^2\sin^2\beta}{B_0} z^2,
\end{eqnarray}
where the third- and higher-order terms have been discarded and the interval of integration is $I_t=\left[t-\pi/\omega_\mathrm{rot}, t+\pi/\omega_\mathrm{rot}\right]$. Equation~\eqref{eq:BstrengthB} implies that the effective radial trap frequency will decrease by $\sim 29\%$ during part B, but this should not significantly disturb the pumping process.
\section{Results}\label{sc:results}
We study the temporal evolution of a spin-1 BEC during vortex pumping by numerically solving the GP equation, Eq.~\eqref{eq:GPE}, with the $T$-periodic time dependence of the magnetic field $\vect{B}({\mathbf{r}},t)$ given by Eqs.~\eqref{eq:part_A}, \eqref{eq:part_B}, and \eqref{eq:B'}. After factoring out the $z$ dependence of the order parameter as $\vec{\Psi}\left({\mathbf{r}},t\right)=\vec{\Psi}_\mathrm{2D}\left(x,y,t\right)\zeta(z)$, Eq.~\eqref{eq:GPE} is discretized on a uniform grid with a finite-difference method and integrated in time for several consecutive pumping cycles using a split-operator approach. The dimensionless coupling constants are chosen to have the values $\tilde{g}_\mathrm{n}= N_0 m g_\mathrm{n}/\sqrt{2\pi \hbar^4 a_z^2 } =250$ and $\tilde{g}_\mathrm{s}= N_0 m g_\mathrm{s}/\sqrt{2\pi \hbar^4 a_z^2 } = -0.01\,\tilde{g}_\mathrm{n}$, the latter corresponding to spin-1 condensates of ${}^{87}$Rb~\cite{Kla2001.PRA64.053602,Kem2002.PRL88.093201,Wid2006.NJP8.152,Note1}. The durations for parts A and B of the control cycle are given by $\tilde{T}_A = \omega_0 T_\mathrm{A} = 3$ and $\tilde{T}_\mathrm{B}= \omega_0 T_\mathrm{B} = 2$, respectively, and the parameters in Eqs.~\eqref{eq:part_A}, \eqref{eq:part_B}, and \eqref{eq:B'} have the values $\tilde{B}_0= |g_F|\mu_\mathrm{B}B_0/\hbar\omega_0=200$, $\tilde{\rho}_0=\rho_0/a_0=5$, $\tilde{B}_0'=|g_F|\mu_\mathrm{B} a_0 B'_0/\hbar\omega_0 = \sqrt{200}$, and $\tilde{B}_\mathrm{min}'=|g_F|\mu_\mathrm{B} a_0 B'_\mathrm{min}/\hbar\omega_0 =6$. The Land\'{e} factor $g_F$ is taken to be negative as in the case of spin-1 ${}^{87}$Rb. The frequency of the rotating field is set to $\tilde{\omega}_\mathrm{rot} =\omega_\mathrm{rot}/\omega_0 =85$. We present results both with and without an optical plug potential of amplitude $\tilde{A}=A/\hbar\omega_0 = 10$ and width $\tilde{d}=d/a_0=2$. Before the pumping is started, a relaxation method is used to bring the BEC to the lowest-energy WFSS with the magnetic field in its $t=0$ configuration.
\begin{figure*}
\begin{center}
\includegraphics[
width=0.9\textwidth,
keepaspectratio]{FIG2}
\end{center}
\caption{\label{fig:T5-profiles} (Color online) Areal particle density and complex phase of the order-parameter components $\Psi_{+1}$ (at $t=l T$, $l\in\mathbb{N}$) and $\Psi_{-1}$ (at $t=l T + T_\mathrm{A}$) in the $xy$ plane during pumping (a) without and (b) with an optical plug potential of amplitude $A=10\hbar \omega_0$ and width $d=2 a_0$. The field of view in the panels is (a) $12a_0 \times 12a_0$ and (b) $14a_0\times 14 a_0$. The two parts of the control cycle have the durations $T_\mathrm{A}=3/\omega_0$ and $T_\mathrm{B}=2/\omega_0$, and the dimensionless coupling constants are set to $\tilde{g}_\mathrm{n}=250$ and $\tilde{g}_\mathrm{s}=-0.01 \tilde{g}_\mathrm{n}$.}
\end{figure*}
Figure~\ref{fig:T5-profiles} shows the squared moduli and the complex phases of the most relevant order-parameter components $\Psi_{+1}$ (at $t=l T$, $l\in\mathbb{N}$) and $\Psi_{-1}$ (at $t=l T + T_\mathrm{A}$) during the pumping process.
The accumulation of two quanta of vorticity during part A of each cycle is clearly visible in the phase fields at $t=l T+T_\mathrm{A}$. Part B of the cycle, during which the rotating bias field is on, is observed to leave the vorticity unaffected. The pumping also causes breathing of the BEC, as indicated by its oscillating spatial extent and by the nonzero radial derivatives of the phase fields. The excitation of the breathing mode is attributed to the changing magnetic confinement during the pumping cycle [see Eqs.~\eqref{eq:BstrengthA} and \eqref{eq:BstrengthB}].
Axisymmetric vortex states with large winding numbers $\kappa$ have been found to be dynamically unstable against splitting in pancake-shaped, harmonically trapped single-component BECs for most values of the interatomic interaction strength, with the degree of instability generally increasing with increasing $\kappa$~\cite{Kuo2010.PRA81.023603,Kuo2010.PRA81.033627}. Therefore, when the stabilizing plug potential is not employed in the pumping, the created multiquantum vortex is expected to split after it has accumulated a sufficiently large winding number. In Fig.~\ref{fig:T5-profiles}(a), the onset of splitting is visible around $t=T+T_\mathrm{A}$, when $\kappa=4$. As shown for $t=5T+T_\mathrm{A}$, the process eventually results in a line of singly quantized vortices and is thus ascribed to a dynamically unstable excitation mode with orbital angular momentum of $\pm 2\hbar$ per particle with respect to the condensate~\cite{Kuo2010.PRA81.033627}. On the other hand, when the relatively weak plug is employed [Fig.~\ref{fig:T5-L-N}(b)], the vortex does not split despite its significant breathing, and a nearly symmetric 12-quantum-vortex state is observed at $t=5T+T_\mathrm{A}$. We have confirmed numerically that the plug amplitude $A$ can be subsequently ramped down without destroying the state.
Due to the finite pumping period $T$ and magnetic field strength $|\vect{B}|$, the pumping process is not perfectly adiabatic, and there are spins that do not follow the local magnetic field. Since these spin components are no longer trapped, they escape the condensate region. Consequently, the number of particles in the trap decreases during the process. The loss rate depends on the degree of adiabaticity of the pump, i.e., on the pumping speed and on the local field strength $|\vect{B}({\mathbf{r}})|$.
Figure~\ref{fig:T5-L-N} presents the number of particles in the trap, $N=\int_{r\leq R} d^3 r \vec{\Psi}^\dagger\vec{\Psi} \leq N_0 $, and their average orbital angular momentum $\ev{\hat{L}_z}/N=-i\hbar\int_{r\leq R} d^3 r \vec{\Psi}^\dagger \left[ \hat{\vect{z}} \cdot \left( \vect{r} \times \nabla \right) \right] \vec{\Psi}/N$ as functions of time during the pumping process. Here, the particles are considered lost after their distance from the $z$ axis exceeds $R=13 a_0$. Therefore, the curves in Fig.~\ref{fig:T5-L-N} also include contributions from unconfined atoms that have not drifted away from the trap region. Even after taking this into account, we find that a substantial portion of the atoms still remain in the WFSS at $t=5T$. As expected, the optical plug [Fig.~\ref{fig:T5-L-N}(b)] is observed to significantly reduce the loss of particles.
Whereas the number of vortices in a scalar condensate is quantized, the orbital angular momentum $\ev{\hat{L}_z}$ is a continuous quantity. The angular-momentum curves in Fig.~\ref{fig:T5-L-N} indicate the increment of vorticity by two during part A of each cycle and show that $\ev{\hat{L}_z}/N$ increases monotonously during the first few cycles. The considerable deviation of $\ev{\hat{L}_z}$ from the ideal value $2l\hbar N$ after $l$ cycles is mainly due to the contribution of the untrapped atoms that remain in the region $r\leq R$. Moreover, the slight increase in $\ev{\hat{L}_z}/N$ during each part B is attributed to the small center-of-mass motion induced by the rotating transverse bias field.
\begin{figure*}
\begin{center}
\includegraphics[
width=1.0\textwidth,
keepaspectratio]{FIG3}
\end{center}
\caption{\label{fig:T5-L-N} (Color online) Particle number $N$ and average orbital angular momentum $\langle \hat{L}_z \rangle/N$ of the BEC as functions of time during the vortex pump simulations presented in Fig.~\ref{fig:T5-profiles}: (a) no optical plug potential; (b) Gaussian-shaped plug of amplitude $A=10\hbar\omega_0$ and width $d=2 a_r$. }
\end{figure*}
\section{Experimental feasibility}\label{sc:experimental}
Let us briefly relate the proposed topological vortex pump to a realistic experimental setup. As an example of previously realized experimental parameters, the experiment of Ref.~\cite{Nee2010.PRL104.160401} studied condensates of $2\times 10^6$ $^{87}$Rb atoms in the $5\,^2S_{1/2}\;|F=1,\,m_F=-1\rangle$ state. The atoms were confined in a TOP trap with a bias field of $B_{\mathrm{rot}}=5$~G and a quadrupole field with the radial gradient $B'=27$~G/cm. An additional red-detuned Gaussian laser beam propagating in the $xy$ plane provided strong confinement along the $z$~axis and negligible confinement in the radial plane; the TOP trap provided the significant portion of radial confinement. This laser beam had a wavelength of 1090~nm, a power of $\sim$\,0.5~W, and radii of $\sim$\,20~$\mu$m along $z$ and $\sim$\,2~mm along $r$. The combined optical and magnetic trap gave trapping frequencies of $\left(\omega_0,\omega_z \right) = 2\pi\times \left(8,90\right)$~Hz, low compared with the TOP trap rotation frequency of $\omega_{\mathrm{rot}}=2\pi\times 4$~kHz.
As a proposed implementation of the vortex pump, we consider the spin-1 condensates of ${}^{87}$Rb and the following field parameters. First, in place of a red-detuned trapping laser, we assume the use of a blue-detuned beam that has a Gaussian profile in the $xy$ plane and a first-order Hermite--Gauss profile along $z$. Since the atoms would be trapped in the dark region between the two halves of the beam, the laser would not provide any radial confinement. A 1-W, 532-nm beam, with Gaussian radii of $\sim$\,9~$\mu$m along $z$ and $\sim$\,500~$\mu$m along $r$, will give an axial trapping frequency of $\omega_z \approx 2\pi \times 900$~Hz. This field remains constant throughout the pumping cycle and is large enough to support the atoms against gravity and the magnetic-field forces along $z$.
Secondly, for the magnetic-field parameters, we assume the field values given above that correspond to Ref.~\cite{Nee2010.PRL104.160401}. Thus, we take the maximum strength of the uniform fields to be $B_0=5$~G [Eqs.~\eqref{eq:part_A} and \eqref{eq:part_B}], which is reached for the axial bias field $B_z^\mathrm{b}$ at the beginning and end of part A of the cycle, i.e., at $t=0$ and $t=T_\mathrm{A}$. Between these times, the rotating component of the TOP trap is off, the quadrupole field with the gradient $B'_0=27$~G/cm is on, and the bias field pushes the zero-value point of the total magnetic field $\vect{B}$ from nearly 1~mm above the pancake-shaped BEC to nearly 1~mm below the BEC. For this trap, the harmonic approximation of Eqs.~\eqref{eq:BstrengthA} and~\eqref{eq:BstrengthB} is well justified. Equation~\eqref{eq:omega0} yields $\omega_0 \approx 2\pi \times 11$~Hz, and hence the simulations in Sec.~\ref{sc:results} correspond to $T_\mathrm{A} = 3/\omega_0=44$~ms. As assumed in Eq.~\eqref{eq:B'}, $B'\left(t\right)$ can optionally be ramped to a minimum value of $B'_\mathrm{min} \approx 11$ G/cm during part~A. In part B, the rotating bias field $\vect{B}_\mathrm{rot}$ is ramped on while the bias field $B_z^\mathrm{b}$ reverses direction, bringing the zero-value point of $\vect{B}$ in a spiraling trajectory around the BEC prior to $\vect{B}_\mathrm{rot}$ being ramped back off. Our simulations fix the duration of this stage at $T_\mathrm{B} = 2/\omega_0=29$~ms.
Additionally, for the parameters assumed above, the implementation of an optical plug would be straightforward. The assumed value of the plug radius used in our simulations, $d=2 a_0$, corresponds to a Gaussian $1/e^2$ beam radius of $2\sqrt{2}a_0 \approx 9$~$\mu$m, similar to the beam used in Ref.~\cite{Nee2010.PRL104.160401}.
For numerical convenience, our simulations have assumed smaller numbers of atoms than would be ideally used in an experiment, as well as lower values of magnetic fields than those typically found in TOP traps~\cite{Note2}. Nevertheless, based on the validity of the harmonic approximation and the readily achievable time and length scales, the primary features seen in the simulations should be preserved and observable with experimentally feasible parameters.
\section{Conclusion}\label{sc:conclusion}
In summary, we have discussed how to implement a vortex pump for a BEC in a TOP trap, resorting only to standard experimental techniques and magnetic-field configurations that are already available in BEC laboratories. We showed that the pumping can be carried out without using an additional optical potential to trap the atoms in the radial direction. Instead, the radial confinement is provided solely by the magnetic field, and a harmonic optical potential is employed only in the axial direction. Our simulations demonstrated that even if the pumped multiquantum vortices are not stabilized by a Gaussian-shaped plug potential piercing their core, several pumping cycles can still be carried out before the vortex splits clearly. On the other hand, already a relatively weak plug potential was found to prevent the splitting and to reduce the loss of atoms from the trap.
Our results are expected to facilitate the experimental realization of the vortex pump. This achievement would represent an important milestone in vortex physics, since it would provide a controlled method to produce almost any desired amount of vorticity. From a theoretical point of view, the vortex pump is a fascinating example of adiabatic quantum dynamics for which the control parameters of the system are varied cyclically but the system does not return to its initial eigenspace. In fact, the appearance of vortices can be interpreted as the accumulation of a position-dependent geometric Berry phase~\cite{Ber1984.ProcRSocA392.45} for individual spins of the condensate atoms~\cite{Nak2000.PhysicaB284.17,Iso2000.PRA61.063610,Oga2002.PRA66.013617,Mot2002.JPCM14.13481,Lea2002.PRL89.190403,Mot2007.PRL99.250406,Pie2008.book.vortices}.
\begin{acknowledgments}
The authors thank J. A. M. Huhtam\"aki, P.~J. Jones, V.~Pietil\"a, and E.~Ruokokoski for insightful comments and discussions. CSC - IT Center for Science Ltd. is acknowledged for computational resources. P.K. and M.M. have been supported by the Emil Aaltonen Foundation and by the Academy of Finland under Grants No.~135794, 138903, 141015, and through its Centres of Excellence Program under Grant No.~251748 (COMP). B.P.A. acknowledges the support of the US National Science Foundation Grant PHY-1205713. P.K. thanks the Finnish Cultural Foundation, the KAUTE Foundation, and the Magnus Ehrnrooth Foundation for financial support.
\end{acknowledgments}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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{"url":"https:\/\/katten-oppasser.de\/undefined-symbol.html","text":"undefined symbol. Undefined Symbol Theater is very proud to announce that the Playwrights Center of San Francisco has partnered with us to produce Singulariteen at the San Francisco Fringe 2013! This is a very exciting development and it will enable us to produce the best show possible. Symbols can also remain undefined when a symbol reference in a relocatable object is bound to a symbol definition in an implicitly defined shared object. Hello, I tried to build a program based on the Catalyst, the paraview and my program is ok to compile, but I got this error when I run the program:. Uncaught (in promise) TypeError: Cannot read properties of undefined (reading 'Symbol(immer-state)') heroku redux. We have removed some processor-independent functionality from processor-specific libraries. Some extensions, though, define the following conventions of addition and multiplication:. TensorRT import - Undefined symbol. For instance, suppose we have the definition of convergence of sequence. This is a frequent mistake, already discussed several times in this forum. While researching further it appears the issue might be specific to PHP version series 5. Undefined first referenced symbol in file vector::insert_aux(DPEEDFU*, const DPEEDFU&). This error means there is a mismatch between the byte size of Unicode characters used by the environment\u00a0. hello guys i'm trying to build this unity game to Xcode but I'm facing this problem. The undefined symbols for architecture x86_64: can be easily fixed by defining a value inside the missing declared statements. This problem can have any of the following causes: Cause 1. Generating an Executable Output File When generating an executable output file, the link-editor's default behavior is to terminate with an appropriate error message should any symbols remain undefined. x86_64 Additional info: I checked version of this library for Fedora31 (krb5. error #10234-D: unresolved symbols remain. That means the c++ std::string abi doesn\u2019t match between building pytorch source and building cpp extensions. trying test code which uses app_easy_timer and im getting the following compile error Error: L6218E: Undefined symbol app_easy_timer (referred from\u00a0. Here is a brief overview of what I'm. You need to be using the dllexport keyword to declare a function to be exported. 3: undefined symbol: EVP_KDF_ctrl, version OPENSSL_1_1_1b\" Version-Release number of selected component (if applicable): krb5-libs-1. Answer: There is no symbol that literally means \u201cundefined\u201d as far as I know. This can be resolved by forcing liblua. Also, put the code in php tag next time _____. And here is the output from running the command that Zirias provided. client import FirewallClient, FirewallClientIPSetSettings, \\\\ File. Its caused by _GLIBCXX_USE_CXX11_ABI=1 when compile pytorch from source. For example, you would say: __declspec (dllexport) int __cdecl sum (int a, int b); Refer to the Microsoft Exporting from a DLL Using __declspec (dllexport) document. Thanks Axel, i tried a simple case to load one libhello. PROBLEM: _main is an undefined symbol. The symbol has no well-defined meaning by itself, but an expression like {} \u2192 is shorthand for a divergent sequence, which at some point is eventually larger than any given real number. \uff081\uff09 \u4f7f\u7528file \u547d\u4ee4\u67e5\u770b so\u5e93\u7684\u67b6\u6784\uff0c\u770b\u770b\u662f\u5426\u4e0e\u5e73\u53f0\u4e00\u81f4. What is meant by undefined symbol error in C++? It means you haven't written a function, or you haven't created a variable, or you haven't\u00a0. You want to use try\/catch exception handling so you must use c++ not c. ) add using namespace std; before the definition of main. Hi goldwake, I have been having the same issue as you I think. Assuming a number is positive, dividing it by a very small positive number that approaches zero. For what it's worth, I also am unaware of any such symbol, at least anything which is generally accepted by the math community. ld: 0711-317 ERROR: Undefined symbol:. I'm trying to install TensorRT 5 from the tar file on Ubuntu 16. How does this relate to How to upload two. error #10010: errors encountered during linking; \"testing-VecN-library. I created a basic spec file for my NavigationStack component. Then he just extracted TBB into \/usr\/local\/ which was where even I had extracted my TBB library and the shared library was able to detect TBB's shared libraries there and it worked!. thanhkien84 opened this issue Nov 26, 2018 \u00b7 3 comments Labels. I've finally managed to compile my V3 project in V4,\u00a0. 1 Successfully installed keras-2. Once I pip installed the wheels file, I tried import tensorrt in a python shell, but get Traceback (most recent call la\u2026. const char *foo = InvalidateImage (bar); Because it is a library, you would not notice this until you attempt to run the program which uses this symbol. Syntax --undefined=symbol Usage Causes the linker to: Create a symbol reference to the specified symbol name. 104-2 Perl wrappers for cairo local\/cairo-ubuntu 1. Undefined symbol _file_set first referenced in file \/usr\/ucblib\/libucb. What is the cause of the \"undefined symbol. [Bug ld\/29086] -Wl,--wrap=foo with LTO leads to undefined symbol, cvs-commit at gcc dot gnu. This default behavior allows the shared object to import symbols from either relocatable objects or from other shared objects when the object is used to create a dynamic executable. EDIT (PARTIAL) SOLUTION!!!!: After following these steps of the link below I managed to fix the scipy error, I uninstalled numpy, scipy and scikit-learn. It was added in a newer version of GAUSS than you have. Because DrawCar is not a part of the Main function! Nor can it be Move your public variables out of the main method and into the main body of the class. It's a program processing messages from an external system and seemed to fail only when the processing of the message triggered some given operations. 0 update 3, these symbols are *Undefined* in the libraries libmkl_def. Look at the library dependencies using the ldd tool. Undefined symbol error, but it's in the static library : r\/C_Programming. 10 | grep -i CRYPTO_set_locking_callback. In my situation (Debian 9) the order was correct: the php-mysqlnd. \u2022 You use the report field in a calculated field, and then you remove the report field. The symbols of infinity In analysis, measure theory and other mathematical disciplines, the symbol is frequently used to denote an infinite pseudo-number, along with its negative,. When the link-editor is generating a shared object output file, it allows undefined symbols to remain at the end of the link-edit. error: Undefined symbol '_DisplayError' referenced in \"c:\\Users\\i87278\\Desktop\\OCR exp\\cvibuild. Moschops (7244) It means you haven't written a function, or you haven't created a variable, or you haven't linked against the library or object code that contains the missing function or variable. I have already created a static library that works, but when I test my dynamic library, I get an unresolved symbol. 1 on CentOS 8 and was having immense stability problems. For c++ you need to change these things. Accept Solution Reject Solution. After all input files have been read and all symbol resolution is complete, the link-editor searches the internal symbol table for any\u00a0. For example, if you try to calculate the value of pi using an infinite series, you will eventually reach a point where the value is undefined. LI72305: UNDEFINED SYMBOLS FOR TEMPLATE FUNCTIONS. so | grep pthread_setname_np 000000000000e0f0 w DF. o I am migrating the code to Sun 5. [ 93%] Linking C executable Test_BitIO Undefined symbols\u00a0. undefined symbol hello guys, i have just installed mplab. I don't know whether you may run into a problem when you append all those libraries to LD_LIBRARY_PATH rather than putting the needed additional ones ahead of the default ones. By default putch () is defined as an empty function. conf says: LoadModule php5_module modules\/libphp5. Undefined, a variable lacking initialization. By disabling cookies, some features of the site will not work. Hello, I am currently working on a program that does real-time facial recognition using a raspberry pi v2 camera module. I am experimenting an issue with my Keil project. If it's different, please provide the info that Wile_E asked for. Dec 19, 2007 3:46PM edited Dec 20, 2007 12:48PM in Developer Studio C\/C++\/Fortran Compilers. so 807578 Member Posts: 13,959 Green Ribbon Jan 8, 2009 5:14AM edited Jan 13, 2009 1:36PM in Developer Studio C\/C++\/Fortran Compilers. If you have a project file loaded, make sure your source code file (. cpp Syntax to Your Document - Code Example 3 - Adding the Similarity. When the library should be loaded the system raised an UnsatisfiedLinkException withe the message: \"undefined symbol: __dso_handle\". h Syntax to Your Document - Code Example 4. lgi\/Makefile has to be edited like below. Undefined symbols and where to find them. Add a comment | Sorted by: Reset to default. mexa64': Undefined symbol: mxErrMsgTxt\" It is my understanding that mxErrMsgTxt is legacy code and shouldn't be used. Understanding \"Undefined Symbol\" Error Messages. What is the maths symbol for undefined?. Hi I'm trying to use prof and so have to use the static version of our libraries. error: undefined symbol: stdout. 0018212: cmake: undefined symbol: archive_write_add_filter_zstd. It means the linker has looked through all the compiled code you told it to, and it still can't find what it's looking for. The official dedicated python forum. Inside the jail, I get no return, but from host, I get the following: Code: [email\u00a0protected]:~ # objdump -TC \/usr\/lib\/libpthread. so: undefined symbol: _ZN12ninebot_algo10AprAlgoLog9instance_E. How to remove common errors in TurboC compiler || C++ graphics || Linker error || undefined symbol. qt creator symbols not found for architecture x86 64 Undefined symbols for architecture x86 64 | Compiling CPP c++ programs in MAC or\u00a0. 3 to manage virtual environment and python versions. I'm developing a DLL in LabWindows\/CVI 2012. undefined symbol: __ZN2at19UndefinedTensorImpl10. 45] ifcfg-rh: dbus: couldn't initialize system bus: Could not connect: Connection refused dbus is showing symbol lookup errors Traceback (most recent call last): File \"\/usr\/bin\/firewall-cmd\", line 31, in from firewall. 0 configuration and merge the differences between 8. what is an undefined symbol error? Last edited on Dec 18, 2011 at 1:58pm. LIB + MATH (S,T,C,M,L - for model) + C (S,T,C,M,L - for model) Undefined Symbol Linking \"C. org, 2022\/05\/04; Prev by Date:. Change your LinkWith attribute to: [assembly: LinkWith (, ForceLoad = false, SmartLink = true)] This will make Xamarin. I hardly know what Qt_5 is for except that it is an IDE. gss, line 12] In this example, we can tell that arimaFit is a GAUSS procedure or function. \u2022 You change the field type to a type other than data. It means you haven't written a function, or you haven't created a variable, or you haven't linked against the library or object code that contains the missing function or variable. The program is actually a library that is loaded by another program. Resolving Undefined Symbol linker messages. Written by Embarcadero USA on Thursday, 2 July 1998\u00a0. Copy link thanhkien84 commented Nov 26, 2018. a, just reverse the sequence: lib2. For c++ you need to change these things 1. gareth July 29, 2020, 9:22pm #1. Undefined Behaviour in C and C++; What are common programming errors or 'gotchas' in Python? What is the Symbol Table? What is a reference variable in C++? What are all the common undefined behaviours that a C++ programmer should know about? Difference between Compile Time Errors and Runtime Errors in C Program. The following list provides solutions to some of the more common causes of \"undefined symbol\" errors: Undefined Symbol When TLINKing from DOS Command Line ===== The TLINK command line must have the libraries in the following order ( GRAPHICS. OS : Windows 7 64 bit Laz: Lazarus 1. This is usually caused by a misspelled identifier name, or missing declaration of the identifier used. \" In math, this term refers to a value that is not assigned to any specific number. Undefined symbols can affect the link-edit process according to the type of symbol, together with the type of output file being generated. cpp file) which has main in it is listed in the. According to Math - Symbol for Undefined, dividing a number by 0 may be represented by UNDEF, but the staff are unaware of any specific symbol meaning \"undefined\". This is why you are getting the undefined symbol errors since CVI can't find the exported functions. do you have any idea of this problem. The program was working but I was having latency issues and read that I. yum doesn't work: yum search pycurl This problem occurred: \/usr\/lib64\/python2. This site uses cookies to store information on your computer. UNDEFINED SYMBOL AT COMPILE TIME An undefined symbol at compile time indicates that the named identifier was used in the named source file, but had no definition in the source file. This is the first time I try to upgrade to 2. Deep Learning (Training & Inference) TensorRT. The symbols i_malloc, i_free, are defined in libmkl_core. iOS ask the native linker to remove unused code from the native library, and if you're lucky, the reference to the inexistent method will be in unused code, so it will end up removed. undefined symbol: g_unicode_script_get_type. Re: :0: error: (499) undefined symbol: Monday, August 24, 2015 6:17 PM ( permalink ) +4 (4) Your code cannot compile without a \"main\" function, so we are guessing here with half of your project. What causes the undefined symbol. The error Undefined symbols for architecture arm64: \"_OBJC_CLASS_$_SKAdImpression\" during the iOS build . what is an undefined symbol error? It means you haven't written a function, or you haven't created a variable, or you haven't linked against . so: undefined symbol: CRYPTO_num_locks . There are two way to solve this problem: build cpp extensions with -D_GLIBCXX_USE_CXX11_ABI=1. Good judgement is the result of experience \u2026 Experience is the result of bad judgement. After all of the input files have been read and all symbol resolution is complete, the link-editor searches the internal symbol table for any symbol references . Dividing by zero is not considered infinity (\u221e), it is UNDEF. I'm trying to set up a HelloWorld subscriber on a VxWorks 7 system but I am getting a number of undefined symbols. Bug 714140 - undefined symbol: Perl_Gthr_key_ptr. I want to post a simple solution to a problem I had with a dynamic linked library which was coded in c++. The symbol has no well-defined meaning by itself, but an expression like. 'Undefined symbols for architecture' on iOS I've just started out adding Fabric support to my Navigation router. Import Error: undefined symbol: png_riffle_palette_neon. Last edited on Dec 18, 2011 at 12:22pm. Error string: Could not load library (Poco exception = \/home\/marco\/catkin_ws\/devel\/lib\/\/librqt_template_plugin. Undefined Symbol F01343D00789 Sequence Num. Free Pre-Algebra, Algebra, Trigonometry, Calculus, Geometry, Statistics and Chemistry calculators step-by-step. These seem typically to include -lm -lc -lgcc -lgcc_s -ldl -lc (libc linked twice). these errors are inside a custom library. 1 According to Math - Symbol for Undefined, dividing a number by 0 may be represented by UNDEF, but the staff are unaware of any specific symbol meaning \"undefined\". The symbols of infinity [ edit ] In analysis , measure theory and other mathematical disciplines, the symbol \u221e {\\displaystyle \\infty } is frequently used to denote an infinite pseudo-number, along with its negative, \u2212 \u221e {\\displaystyle -\\infty }. Undefined, a function or variable lacking a declaration. If a sequence$(a_n)_{n=m}^\\infty$is not converging to any real number, we say that the sequence$(a_n)_{n=m}^\\infty$is divergent and we leave$\\lim_{n\\to\\infty}a_n$undefined. And i want to try a simple program: #include #include #pragma config WDT = OFF void main (void) {printf(\"hello world\"); while(1);} if i want to build the program i get the following error: Build C:\\Documents and Settings\\ruud\\My Documents\\probeer for device 16F690. Turned out that RHEL had TBB installed by default and my colleague had to delete the TBB files in \/usr\/include and \/usr\/lib64\/. After all of the input files have been read and all symbol resolution is complete, the link-editor searches the internal symbol table for any symbol references that have not been bound to symbol definitions. Undefined Symbols (Linker and Libraries Guide). Resolving \"undefined symbol __Vectors' referenced in expression\" error. Tags: Posted by JohanEkdahl: Tue. Undefined behavior, computer code whose behavior is not specified under certain conditions. M A T H E M A T I C S \u00b7 Symbols used in Set Theory Math Classroom, Math Teacher, Teaching Math, Math Vocabulary. 3 When trying to execute previously well-functioning keras\/tens. Marking as WontFix since this is an issue that should be fixed in Chromium, and it does not impact the CEF automated builders (config here). G0025 : Undefined symbol: 'arimaFit' [arimafit-example. 0 for quite some time and decided to update to 1. I am getting following errors when compiling source code in Keil-5. channelFoam: symbol lookup error:\/OpenFOAM\/OpenFOAM-2. 4-4 A rendering library for Kate streams using Pango and Cairo local\/pixman 0. undefined symbol: _ZN12ninebot_algo10AprAlgoLog9instance_E. This problem occurs if the following conditions are true: \u2022 You put a field in a footer or in the report body. Description of problem: After upgrade to Fedora 32, Matlab 2020a complain about: \"symbol lookup error: \/lib64\/libk5crypto. I'm working on my Cmake build scripts and in the process I've found a weird error. Undefined, an unavailable linker symbol. I\u2019m trying to install TensorRT 5 from the tar file on Ubuntu 16. To do this, I am using the BFD library. \"error: undefined symbol: _ZdlPvm, version Qt_5\". 0\/platforms\/linux64GccDPOpt\/lib\/libincompressibleLESModels. a, it is too late -- lib1 has already been searched. Undefined symbol in dynamic library. Undefined symbol: _sampleTextMethod. Dividing a number by zero is usually considered undefined. means that you failed to include all the libraries you need in your cc. CodeProject, 20 Bay Street, 11th Floor Toronto, Ontario, Canada M5J 2N8 +1 (416) 849-8900. txt the symbol shows up in the file, but has no address associated with it. h Code of the Complete Syntax - Code Example 2 - Adding the Similarity. Stack Exchange network consists of 180 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. or \"undefined symbol: PyUnicodeUCS2\\_\" errors. SOLUTION: main is the entry point for every C\/C++ program. ini file was at the top of the list of modules. Repeat this for every Undefined symbol in the linker error message, and you'll be . Resolving \"undefined symbol __Vectors' referenced in expression\" error But this did not solve the problem! In the explorer project accidentally found that in the folder: CMSIS, in this folder another folder EFM32GG and it has two files (startup_gcc_efm32gg. Assuming a number is positive, dividing it by a very small positive number that approaches zero would yield +\u221e. We are grateful to PCSF for their support of new works. If is not in the domain of , then this is written as () \u2191, and is read as \"() is undefined\". This is an issue reported earlier and it remains for the following versions of tensorflow and keras: Successfully installed tensorflow-2. dbus is timing out and processes are failing to connect Jan 1 01:00:00 hostname NetworkManager[1534]: [546. To change the modules order just change the trailing number so you should have \"10-mysqlnd. For example, continuing with the files main. i have 2 separate small projects i tried to compile, one just be trying to learn how to use the xml library, and the other is a multi file c program just for fun, both compile fine with only some small warnings, the usually stuff like \"inter to pointer. Iam attempting a script to return the current cursor position using the getyc macro I have #included the curses. so which are depended on each other?If it's the same: please don't open two topics on the same - well, topic, at the same time. h however on compilation (with gcc) it errors with Undefined symbol. If the command line has the sequence lib1. c) but they are empty (zero length)!. unable to load undefined symbol _z15InvalidateImageSs I am trying to determine why I am getting this error. Undefined symbol error after compiling a new LES model. c(4): Undefined symbol _output_word in module ccode Warn : :Program has no entry point both files are compiled and assembled under one project and under. Functions are *not* defined in header files, but only declared, . The issue must be from some incompatibility between Qt_5 in my computer and the one that runs with Coppeliasim, but i don't know in what way as this is all new to me. 1: \/bin\/sh: Undefined symbol \"[email protected]_1. Answer (1 of 5): There is no generally accepted symbol for \"undefined. APAR is sysrouted FROM one or more of the following: APAR is sysrouted TO one or more of the following: Fix information. mustaqimM reacted with thumbs up emoji. Normally it's preferrable to start with a 9. 0-3 C++ bindings to Cairo vector graphics library local\/libtiger 0. This is a common problem that many web developers come across while working with several web utilities for designing programs. - John Omielan Dec 23, 2018 at 3:03. On some analysis using nm , I found that the symbols which are flagged as UNDEFINED are in fact getting defined in the temporary objects created in the SunWS. If that undefined name is not referenced in your source, then it usually. so\" fails: undefined symbol: sqlite3_libversion The only solution I am aware of at the present time is to either disable the PDO SQLite3 extension, by editing the system PHP configuration file \"php. Summary: undefined symbol: Perl_Gthr_key_ptr Keywords: Status: CLOSED RAWHIDE Alias: None Product: Fedora Classification: Fedora Component: perl-Compress-Raw-Zlib Sub Component: Version: rawhide Hardware: Unspecified OS: Unspecified. Static libraries are searched in order for symbols that have been referenced but not yet defined. o is compiled into an object file everybit BUT there is no reference in this line to everybit hence the undefined symbols gcc -o everybit_harvey main. 5 configuration file that has not been fully migrated. o: undefined reference to symbol '[email protected]@GLIBC_2. Perhaps you have such a long path that it doesn't find all the necessary ones. After creating a Dynamic Link Library project in Debug configuration, I set the Build\u00bbTarget Type to Static Library. Error: L6218E - undefined symbol. so LIBFLAG = -shared CCSHARED = -fPIC LIBS +=$ (shell $(PKG_CONFIG) --libs lua) endif endif. error: undefined symbol: stderr. There is no symbol that literally means \u201cundefined\u201d as far as I know. When running the SAP HANA hardware and cloud measurement tools as adm user a similar error is thrown: \". Learn more about pango, matlabwindow, linux, symbol lookup error MATLAB. This showed up in a nightly CI job whose configuration has not changed.$ file file: symbol lookup error: file: undefined symbol: magic_setparam $which file \/usr\/local\/bin\/file$ ldd $(which file) linux-vdso. whenever i try to compile a c program, the linker gives that \"undefined symbol\" stuff. Eventually got to the logs and found a lot of messages like this: Service 'indexer' exited with status 127. undefined symbol \"cout\", \"cin\", and \"endl\". 807578 Member Posts: 13,959 Green Ribbon. ld: 0711-345 Use the -bloadmap or -bnoquiet option to obtain more. user24036 October 31, 2021, 9:08am #1. You have to remove local library and local Git, then reinstall Git with dependencies by sudo rm -v$(which git) sudo rm -v\u00a0. Warning: module 0xffff8000006ffaf0 holds reference to undefined symbol _ZNSt6localeC1Ev. ) replace #include with #include. I am creating a dynamic library of object files that contain functions to emulate the behavior of objdump and nm. a does not need any symbols from lib2. \"undefined symbol\" when trying to compile c?. Get a virtual cloud desktop with the Linux distro that you want in less than five minutes with Shells! With over 10 pre-installed distros to choose from, the worry-free installation life is here! Whether you are a digital nomad or just looking for flexibility, Shells can put your Linux machine on the device that you want to use. Using apt-get install I installed the following things: apt-get install libopenblas-base apt-get install libopenblas-dev apt-get install python-dev apt-get install gcc apt-get install gfortran. Answers (1) This happens when you use an 8. What is the cause of the \"undefined symbol: amdgpu_bo_list_create_raw\" error on RHEL8 VMware Guest?. build pytorch with -D_GLIBCXX_USE_CXX11_ABI=0. Example: if the library calls '_nmalloc' that function does not exist in our libraries (we supply new header files in Borland C++. SOLUTION: undefined symbol: __dso_handle \u2014 oracle. There is no symbol that literally means \"undefined\" as far as I know. \u201cUndefined symbol: protocol descriptor for Swift. The first 3 minutes are automatically charged: \\$2. I am using the STM32CubeF7 HAL to programa a module I need. By continuing to use our site, you consent to our cookies. Error is coming because you are trying to access class member variable x, y, z from outside in the main() where x, y, z is not declared. Usually one states \"we leave the statement undefined\". Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have. undefined symbol: X means always that X should be exported from one of loaded libraries, but it's not. ExpressibleByFloatLiteral\u201d Code Answer \/\/ You need to exclude architecture arm64 if you are\u00a0. 16-2 Cairo vector graphics library, with Ubuntu's LCD rendering patches local\/cairomm 1. In most cases output is directed to the serial port. Issue an implicit --keep=symbol to prevent any\u00a0. Any one suggest me what i am missing to define symbol?. The meaning of the message is that the name you are using as an operand on the RCALL instruction cant be found by the assembler. As this is more or less obvious then here's some advice\/tips to work on: Double check the spelling. undefined symbol: __ZN2at19UndefinedTensorImpl10_singletonE #370. 3 as per the following PHP bug report: PHP :: Bug #48614 :: Loading \"pdo_sqlite. Warning: module 0xffff8000006ffaf0 holds reference to undefined symbol _ZNSt6_MutexD1Ev. Use Math Input Mode to directly enter textbook math notation. These symbol references are referred to as undefined symbols. function is running before my redux state is available, TypeError: Cannot read properties of undefined (reading 'find') 0 Cannot read properties of undefined (reading 'map') in react js while using redux. 5\" here I cannot find path, because every thing get error, I cannot use ls or find file and folders or any other command in my \/rescue\/sh area. Closed rmrao opened this issue Jun 21, 2019 \u00b7 3 comments Closed undefined symbol: __ZN2at19UndefinedTensorImpl10_singletonE #370. CentOS 8: undefined symbol: EVP_KDF_ctrl in libcrypto. This is default behaviour on linux systems by Makefile. Performing standard arithmetic operations with the symbols is undefined. If you are not happy with the use of these cookies, please review our Cookie Policy to learn how they can be disabled. However, in the code I downloaded, it's already commented out and replaced by mexErrMsgIdAndTxt. ephore October 23, 2018, 8:37am #1. When I try to build my project after changing the Target Type, LabWindows\/CVI throws several link errors: Undefined symbol'__CompiledDebuggingLevel' referenced in \"C:\\Program Files\\National Instruments\\CVI2012\\bin\\msvc\\cvistart. my problem is \"undefined symbol try\" how can I handle this Jan 7 '09 #8. so is loaded, while i can\u2019t call the function defined in the libhello. Make sure you write a function called main (all lowercase) in your program. Make a new virtual environment with pyenv called XXXX. Once I pip installed the wheels file, I tried. Trying to execute CMake gives the error: cmake: symbol lookup error: cmake: undefined symbol: archive_write_add_filter_zstd. \u5373 symbol lookup error: libpathplan. o -arch x86_64 -framework CoreServices Undefined symbols: \"_bitarray_reverse\", referenced. undefined symbol: _ZTIN10tensorflow8OpKernelE_ #32. I am trying to compile a LabWindows\/CVI project, but I am getting errors that are referenced as Link Error or Undefined Symbol errors. Dividing by a very small negative number that approaches zero would yield -\u221e. By other hand, in Computer science there are some symbols: undefined, null and NaN. Pressed CTRL+H to find and replaced PM_Payment_WORK to pmRemittanceTemp. Undefined value, a condition where an expression does not have a correct value. Regenerated the report and found 2 missing fields, replaced the missing fields with an alternative from the. ImportError: \/home\/pybind11_example. Thank you, but I can't find where is the problem, I deleted everything from proj\\lb\\i386-win32 and at lazarus I Run \"run \/ clean - build -> right corner, delete\" and then build. gercurx Any ideas where I can find a solution or what I've missed (7 Replies). 6-1 The pixel-manipulation library for X and cairo local\/python-cairo 1. At a customer site, we had a C++ program (renamed for the purpose of this blog to myprogram) which was failing after running for some time. Ultimately some of the linker warnings can be ignored (for any observer sets that are not being used by your code), but, if the sections and section references aren't being linked correctly, then ignoring them will be ignoring a real problem. 2 & using -compat=4 for compilation. You should find out in which library requested symbol\u00a0. Those are generated freshly by the build process but haven't been used for years. However there is always a workaround! Below the steps I followed: I exported the report to a package file. That said, array identifiers ARE pointers, but the compiler is nice about it so doing ptrPrintVal = &printVal; or doing ptrPrintVal. It's the \u201cUndefined symbols for architecture x\u201d error. Your problem is not being sure which language you are coding in. There are four main reasons why a procedure or function might be undefined in your GAUSS code: 1. I have problem with error 017: undefined symbol. Undefined Symbol is in a Microsoft Library ===== 1) If the libraries call functions that exist in the MSC RTL (Runtime Library) but do not exist in our library, you will get 'undefined symbol' errors during link. The printf () function calls another function putch () to output the character after formatting. ImportError: \/home\/pybind11_example. It is often represented with the word UNDEF. 0 with Service Pack 4 to Microsoft Dynamics Version 2010 with Service Pack 1, when the customer sent me the package,. Undefined symbols for architecture arm64: \"_AVPPluginSetDebugLogFunction\" , referenced from : _OSXMediaPlayer_AVPPluginSetDebugLogFunction_m851849289 in Bulk_Assembly - CSharp_1. extern InvalidateImage (const char *); and later using it. \"Invalid MEX-file '{path-to}\/CCGHeart. To Solve undefined symbol: _PyUnicode_DecodeUnicodeEscape Error By using typed-ast latest version Issue was solved in My case. Last week I have upgraded this customer from Microsoft Dynamics GP Version 10. This last one provides the missing symbol: Raw. To allow printf () to write to a specific output the putch () function needs to be redefined. - Solution 2: Fixing This Error for a Different Program Recreating This Problem in Your Program - Code Example 1 - Adding the Main. Finally, comparing the size of the library files on this system and at another customer site with the exact same version of the software gave us the answer: somehow a different version of one of our library landed there and didn\u2019t contain the missing symbol. The solution is to add following line right in the start of your c++ code: extern void *__dso_handle. Re: Undefined symbol: atexit \u00ab Reply #4 on: January 23, 2022, 04:44:54 pm \u00bb Yes, openssl deprecated a lot of cyphers in recent versions, but they are still in the Pascal bindings\/header file. GitHub Gist: instantly share code, notes, and snippets. I have written a program to calculate the price of a phone call. Using the centos:8 Docker image as of this morning. rmrao opened this issue Jun 21, 2019 \u00b7 3 comments Comments. It means you haven't written a function, or you haven't created a variable, or you haven't linked against\u00a0. Solved: error Undefined symbol. I have no idea why Turbo C isn't complaining about the user of the \"public\" keyword on them, but it's probably because it's a very old compiler. Solve a simple undefined symbols error These error messages tell us that line 6 of our code is using two symbols that have not yet been\u00a0.","date":"2022-05-25 20:40:57","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.26064425706863403, \"perplexity\": 3460.4693357759056}, \"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-2022-21\/segments\/1652662593428.63\/warc\/CC-MAIN-20220525182604-20220525212604-00039.warc.gz\"}"}
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Mc Elligott Cup Final : Austin Stacks 2-18; Rathmore 2-7.
Hail our super heroes – winners of the McElligott Cup (County JF Championship) – played in a damp Castleisland on this Friday evening (7th Sept) – the first time we took part in this competition. This was as entertaining a game of football as we have witnessed all season. The first half was 'nip and tuck' and even though we opened the scoring in the second minute with a well-taken point from play by Alan Duggan it was the Rathmore boys who dominated play in the opening ten minutes after which they led by 1-3 to 0-2 – with our second point also coming from Alan Duggan from play.
But we were now coming more into the game and in the space of the next six minutes we had drawn level with four unanswered points from four different players. Alan Duggan was on target again, this time from a free after William Kirby was fouled going through, and the other three points came from play by Theo Diggin, Fergal Ryan and Donagh McKivergan.
The East Kerry boys had a slight edge over us for the rest of the half – putting over four more points to three for us. Fergal Ryan scored two and Alan Duggan the other after being put through by Donagh McKivergan with a quickly taken free. This left us trailing by one point at the break (1-7 to 0-9) after playing into a strong enough wind in that first half accompanied by continuous rain.
On the resumption Gearóid McCann won the throw-in and Donagh McKivergan raced forward to put over after just 12 seconds to level the game for the second time. Then four minutes later William Kirby put us back in the lead for the first time since our opening score of the game and this was followed in the 37th minute by a pointed free by Donagh McKivergan after Brandon Patterson was fouled. But the score that really put us on the road to victory came just over a minute later. It started with a brilliant run by Darragh Scanlon and when the ball broke in the square it was that brilliant veteran, William Kirby who coolly steered the ball to the net – what a super hero this man is – playing in his 24th season with the Club. Four more points followed – all from play – from four different players – Theo Diggin, Dale Counihan, Fergal Ryan and Alan Duggan – before another super hero really put the game to bed with our second goal of the game in the 48th minute – none other than Dale Counihan. So good was our performance in the second half that we just conceded one score in those closing 30 minutes – a goal in the 52nd minute. Our closing two points of the game came from Fergal Ryan to cap a fine performance by the wing forward who ended up chief scorer with six points in total – all from play. In fact all but two of our 2 goals and 18 points came from play.
So a near perfect performance by all twenty one who lined out on this Friday. Seamus Bastible couldn't be faulted for Rathmore's two well-taken goals in the 8th and 52nd minutes. Apart from his kick-outs being near perfect he was well protected by his three front men – the two Shanes (Kelliher and Walsh) and Paul Galvin and Barry Shanahan who moved into the number 3 spot in the second half. How many times did those four men make perfect turnovers throughout the hour and set up attack after attack. Our two wing backs, team captain, Andrew Morrissey and Eoghan Galvin didn't put a foot wrong. What an inspirational captain Andrew is – both on and off the field. His pre-game team talks, for example, would rouse any team to perform at its best. Gearóid McCann and Darragh Scanlon dominated the centre all through and the five starting forwards were outstanding as can be seen from the stats below. Michael Linnane didn't get on the score-sheet on this occasion but it was Michael's hat-trick of goals in the semi final that contributed significantly to us getting to the final. All our six subs could easily have made the starting 15 with Dale Counihan on fire when introduced early in the second half – scoring 1-1 in the process. And all of the extended panel who were unlucky not to have performed in the final on this occasion played their part during the season in helping this weekend's victory.
And very few teams will win anything without the help of our back room staff. Paddy Barry, Eamon O'Reilly, Kieran Kelliher and Brian Morgan took on the management of our Junior team this year and this victory was the culmination of what these men have achieved.
Team: Seamus Bastible, Shane Kelliher, Shane Walsh, Paul Galvin, Andrew Morrissey (Capt), Barry Shanahan, Eoghan Galvin, Gearóid McCann, Darragh Scanlon, Theo Diggin (0-2), Alan Duggan (0-5 – one free), Fergal Ryan (0-6), Donagh McKivergan (0-3 – one free), William Kirby (1-1, Michael Linnane.
Subs used: John Dennis (ht), Brandon Patterson (ht), Dale Counihan (35m) (1-1), Paddy Fitzgibbon (55m), James Quigley (55m), John Murphy (56m).
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Welcome To Chariklo, A Ringed World In The Solar System Just Examined By The Webb Telescope
No, Earth's Inner Core Is Not Reversing The Direction Of Its Spin
For A Better Relationship, Start By 'Unlearning' These 3 Toxic Thought Patterns
These Venture Veterans Have Launched A $350 Million Fund Aimed At Digitizing Life Sciences
See The 'Green Comet' Close To The Big Dipper After Sunset This Weekend
Satellite Sees Iceberg Twice The Size Of New York City Breaking Away From Antarctica
7 Reasons To Ignore The Hype About The 'Green Comet' (And Why You Need Binoculars)
Commercial Asteroid Mining Now Has A 2023 Launch Date To Scout Its First Target
ForbesInnovationScience
A 'Potentially Hazardous' Asteroid Just Soared Past Earth. It's First Of Five That Will Get Close To Us This Week
I inspire people to go stargazing, watch the Moon, enjoy the night sky
Jun 2, 2021, 04:01am EDT |
The reason why 2021 KT1 is news is that NASA estimates that it's between 492 feet/150 meters and ... [+] 1,082 feet/330 meters in diameter and it wasn't observed until late in May 2021 just a week before its closest pass.
An asteroid called 2021 KT1 yesterday safely passed Earth, but it was seen very late—and it's not the only space rock that will this week come close to your planet.
The reason why 2021 KT1 is news is that NASA estimates that it's between 492 feet/150 meters and 1,082 feet/330 meters in diameter. It wasn't observed until late in May 2021 just a week before its closest pass.
NASA says that anything 50 feet/150 meters or larger—so about twice the size of the Statue of Liberty—to be "potentially hazardous" if they get within 4.6 million miles/7.5 million kilometers.
2021 KT1 passed just a little closer than that, which technically makes it "potentially hazardous," but it's important to get this in context and grasp the size of our Earth-Moon system. The Moon is, on average, 238,855 miles/384,400 kilometers from Earth, so 2021 KT1 passed well beyond the orbit of our satellite. About 19 lunar distances from Earth, in fact.
It's moving at over 11.2 miles/18.1 kilometers per second, or 37,282 miles/64,000 kilometers per hour and won't be back until May 19, 2058.
What is an 'Apollo' asteroid?
Meet The Unknown Immigrant Billionaire Betting Her Fortune To Take On Musk In Space
A Covid Test Targeting The Fast-Spreading XBB.1.5 'Kraken' Variant Is Being Developed By Scientists
Electric Vehicle Charging Will Soon Require 12 Times More Energy Than Used Today, Warns Motorway Services Boss
2021 KT1 is classed as an "Apollo" asteroid. It's one of a group of near-Earth asteroids named after the first one to be discovered, 1862 Apollo, in 1932. Apollo asteroids cross Earth's orbit so they are of particular interest.
More close asteroids have been spotted
However, during May 2021 a smaller asteroid called 2021 GT3 got to within just 159,000 miles/256,000 kilometers of Earth. NASA's Jet Propulsion Laboratory (JPL) says that no fewer than four more asteroids will get close to Earth this week:
June 2, 2021: the 63 feet/19 meters 2021 JW6 will get to within 1.9 million miles/3.1 million kilometers
June 2, 2021: the 53 feet/16 meters 2021 KE1 will get to within 3.5 million miles/5.7 million kilometers
June 3, 2021: the 140 feet/43 meters 2021 KN1 will get to within 1.4 million miles/2.3 million kilometers
June 5, 2021: the 290 feet/88 meters 2021 KF2 will get to within 3.1 million miles/5 million kilometers
What is an asteroid and where do they come from?
Relatively small rocky bodies orbiting the Sun, asteroids come from the Asteroid Belt between the orbits of Mars and Jupiter. They're essentially the leftovers—the debris—from the formation of our Solar System.
How to see an asteroid
Asteroids are incredibly difficult to see with the naked eye, but a pair of binoculars can help you glimpse the very largest asteroids we know of. One example currently visible is Vesta.
Named after Roman mythology's virgin goddess of home and hearth, Vesta is a whopper, measuring about 326 miles/525 kilometers in diameter.
It's currently in the constellation of Leo, which hangs in the western sky after dark in the northern hemisphere. You can see it by following these charts on TheSkyLive.com, though you will need binoculars or a small telescope.
How dangerous is the asteroid belt?
Vesta is in the largest celestial body in the Solar System's main asteroid belt. Does that sound like a dangerous place? It's not. Despite C3PO claiming that "the chances of successfully navigating an asteroid field is approximately 3,720 to 1" in the Star Wars movie, these asteroids are very spread-out. The chances of hitting one is more like one in a billion.
When will a large asteroid strike Earth?
The odds of an asteroid hitting Earth are much larger, depending on the size; a 0.62 miles/1 kilometer asteroid is reckoned to hit Earth every 500,000 years, on average, while a massive 3 miles/5 kilomteter objects in a one-in-20 million years event.
Wishing you clear skies and wide eyes.
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When you're in the industry world your major aim should be to surge in front of your competition. It is crucial that your products is of top quality but it's important too to market it in order that it reaches all households. You are able to consider for example cell phone producers. Normally two products have the standard features however the one with greater sales is one which established fact than the other. What's being emphasized here's you need to speak aloud concerning the product you're offering and that you can do with the proper implementation of advertising ideas. Advertising is paramount for your success in the industry.
Generally our conception concerning the advertising is restricted towards the big banners we have seen around the roadsides, within the malls and also the massive advertisements within the electronic in addition to print media. Using the rising recognition from the internet, the pops ups and promotions for those sites will also be now area of the advertising industry. For that big business houses they are standard mode of advertising as well as the smaller sized businessmen it's a no-no factor due to the greater budgets. There's you don't need to get disheartened since these aren't the only advertising ideas or methods to market your product. Disbursing pamphlets, utilization of sales rep are the cheap and efficient methods for advertising. One should realize that something that will make certain the customer will get to understand about your products is a means of advertising.
To mention a couple of from the cheap and advertising ideas you are able to rely on disbursing the pamphlets and flyers within the areas which are inside your achieve. With these flyers you may also distribute the discounts. These coupons can get the folks discussing your products and also the contemporary knowledge states the person to person continues to be the easiest method to start the advertising.
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Be a normal person in the company sittings inside your locality. Talk about your products with others. They may provide you with some nice advertising ideas. The greater you retain your ears and eyes open the greater you will likely gain. Project yourself because the key player within the niche of the product and make certain that the product as if it's the only viable choice available for sale. These advertising ideas are certain to help make your project a success on the market.
NEXT POST Next post: Home Based Internet Business: It Can Be Done!
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4 haunted dorms at Ohio University
SyFy's Scariest Places on Earth deemed Ohio University as one of it's spookiest locations in an episode titled "Satan's Dormitory" in 2009, but the tales told in this 30 minute show continue to haunt the students of the University in multiple dorms today.
Wilson Hall
Wilson Hall, on West Green, is said to be the most haunted of all the residential halls because of its very precise location in the exact middle of five local cemeteries as well as a mysterious death that took place during the 70s. Since then, students have told multiple stories of room 428 and anyone who has lived there can tell stories of doors slamming and items moving around the room. One student died violently after practicing occult in that very room and now the room is locked off, deemed "unlivable" by the University.
Lauren Murphy, a current resident of Wilson, was studying at her desk one day with the door completely closed when it opened on its own. She got up to check if someone was there or if there was possibly a draft, but discovered nothing. Around the same time, a floor mate of Murphy's was sitting in his room when his mirror fell, without being touched, and shattered everywhere.
Like Wilson, other dorms on campus have scary stories.
Washington Hall
On the other side of campus, Washington Hall residents have quite a few chilling stories involving paranormal activity.
Haley Stultz, a senior, lived in Washington her freshman year during the 2012-13 school year. During her second semester her and her roommate would continuously wake up in the middle of the night to the sound of someone typing on a laptop. Both regularly checked to see if the other was up late working on homework, but every time they heard the noise, their laptops were securely shut and both girls were in bed asleep. The noise continued about 3 times a week for a few weeks when Stultz decided to make a move.
"I just figured if I said something nice maybe he would stop," said Stultz. "So, I named the ghost Harold and told him we didn't mind if he was there as long as he was nice and stopped waking us up!"
A few days later, Stultz was telling a few friends the story when another Washington resident overheard. She asked what was going on and Stultz explained the story.
"It was the weirdest thing. I told her we named the ghost and she asked what we named him. I told her Harold and it ends up she named him the exact same thing. I don't know. It's just really weird," said Stultz.
Michelle Reinold, resident from 2012-2013: "I lived in Washington freshman year and this girl went to shower and locked her room after her and her roommate was out of town and when she came back her room was stilled locked but all her drawers and closets were open"
Rachel Caddey, resident from 2014-2015: "My microwave would go off in the middle of the night and there would be knocking at our door and no one was there. Our beds would shake and wake us up. Our dorm room door, cabinets, and closet doors would open on their own"
Crawford Hall
About 22 years ago in 1993, a young student named Laura fell from her fourth floor Crawford window on to the pavement and died. Since then, every year residents have told stories tied to her death.
Maria Doll, a resident of Crawford during her freshman year (2013-2014), said she didn't know anything about the hauntings of her new home until one night when her roommate went out on a week night but Doll stayed in. She set an alarm on her phone for 3 A.M. just to make sure her roommate returned safely. At three, she woke up, looked over and saw her roommates feet, noticing an ankle bracelet and a few minutes later fell back asleep.
Around 7 A.M., her roommate returned. Confused, Doll asked why she had come home and left again. But her roommate said she that this was her first time home since she had left the night before. Doll explained that she had seen her there at three when she woke up to check on her and noticed her ankle bracelet and everything.
"Then she lifted her pant legs and told me she didn't wear an ankle bracelet," said Doll. "I was so freaked out. I know someone was there. I was completely awake. That was our first experience with Laura."
Doll explained that almost every floor had experiences with "Laura". Eventually, Doll and her roommate accepted her as their third roommate when the TV would turn off and the doors would slam closed without help from a human.
Jefferson Hall
"I lived fourth floor Jeff my freshman year," said Maggie Etherington. "We always heard what seemed like marbles dropping and rolling across the floor above us, but we lived on the top floor so no one was above us."
Etherington explained that it was nothing incredibly scary just strange things that would happen every once and awhile and made her and her floor mates question if the dorm had a few supernatural residents.
The historic tale of Jefferson Hall involves a group of students who decided to explore their dorm on a lazy night in 1996, when they discovered a schoolteacher hovering above her chair behind a desk, dressed in 1950s style clothing. The crew ran back to their Residential Assistant but by the time they arrived back at the empty room, the door was locked.
To learn more visit these sites: Forgotten Ohio, HerCampus
Author: Sophia Ciancone
I am a senior at Ohio University studying journalism, planning to graduate in May of 2016. As my days at Ohio University come to an end, I hope to soak up the beauty and culture that thrive in Athens. I have a wide variety of interests from writing to a passion for social media. I spend my time getting involved. If I have a minute of free time I find myself starting up a new project, making sure every bit of my time is well spent. View all posts by Sophia Ciancone
Author Sophia CianconePosted on November 13, 2015 December 1, 2015 Categories Court Street, HistoryTags Athens Ohio, Crawford Hall, haunted, Jefferson Hall, ohio university, Satan's Dormitoroy, Washington Hall, Wilson Hall
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Q: How to access iframe from main document javascript I am using an iframe provided by a third party company (due to what our business is, it's either pay for an iframe, or pay for them to build a website, and the iframe was better suited for what we need) However, there are certain things that we can't do with the iframe, which would be useful. However, as it's things which are temporary, and cost a few hundred £ each time we want to change anything, it would be better if we could access it via the parent page, which contains the iframe.
For example, we have a form in the iframe which defaults to a certain product (depending on the page) you are on, however you can search for others. However, for some products we don't currently deal with them, so it would be nice to be able to have an alert box which pops up on the main page, which tells the user we don't deal with this.
For example, if in someone types "Swimming Party" we could have a message pop up that says "Sorry, we currently can't offer any swimming parties, however if you call NUMBER we will book one for you as soon as possible"
And also, if someone types in we would be able to capitalize the first letter if someone put the name as "steve" and not "Steve" (for example)
Sorry if this isn't very descriptive of what we need, but we've never really had to deal with an issue like this, however, the limitations imposed mean we have to look for a simple sounding solution to a much more complex problem.
A: Try contentDocument or contentWindow.document. The mockup below works for me. Some notes:
1) You have to wait for the whole document to be loaded, i had a tiny bit of trouble with selecting the iframe before the content was loaded. If you sue jquery, this won't be a problem. ;)
2) If the contents of the iframe are cross domain, you'll have to inject some script into the iframe to extract the html nodes you're interested in.
Main Page
<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8" />
<title>Main</title>
</head>
<body id="mainbody">
<div>MAIN</div>
<iframe id="myFrame" src="test_3.html"></iframe>
<script src="test.js"></script>
</body>
</html>
Iframe Page
<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8" />
<title>Inner</title>
</head>
<body id="iframebody"><div id="nameDiv">MY NAME</div></body>
</html>
Script
document.body.onload = function() {
document.querySelector('#myFrame').contentDocument.querySelector('#nameDiv').textContent = 'NEW NAME';
};
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"redpajama_set_name": "RedPajamaStackExchange"
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Horní Teplice (německy Ober Wekelsdorf) je část města Teplice nad Metují v okrese Náchod. Nachází se na severu Teplic nad Metují. V roce 2009 zde bylo evidováno 90 adres. V roce 2001 zde trvale žilo 119 obyvatel.
Horní Teplice je také název katastrálního území o rozloze 6,1 km2.
Galerie
Reference
Externí odkazy
Části Teplic nad Metují
Sídla v Broumovské vrchovině
Sídla v CHKO Broumovsko
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## Apparition & Late Fictions
## ALSO BY THOMAS LYNCH
POETRY
Skating with Heather Grace
Grimalkin and Other Poems
Still Life in Milford
NONFICTION
The Undertaking: Life Studies from the Dismal Trade
Bodies in Motion and at Rest: On Metaphor and Mortality
Booking Passage: We Irish and Americans
## Apparition & Late Fictions
A NOVELLA AND STORIES
## Thomas Lynch
W. W. NORTON & COMPANY
NEW YORK LONDON
Copyright © 2010 by Thomas Lynch
All rights reserved
"Love, We Must Part Now" from Early Poems and Juvenilia by Philip Larkin. Used with permission from Faber and Faber Ltd. "Love, We Must Part Now" from Collected Poems by Phillip Larkin. Copyright © 1988, 2003 by the Estate of Philip Larkin. Used by permission of Farrar, Straus and Giroux, LLC. "I Would Like My Love to Die" from Collected Poems by Samuel Beckett. Used with permission from Faber and Faber Ltd. "I Would Like My Love to Die" from Collected Poems in English and French by Samuel Beckett, © 1977 by Samuel Beckett. Used by permission of Grove/Atlantic Inc. "Let It Be" © 1970 Sony/ATV Tunes LLC. All right administered by Sony/ATV Music Publishing, 8 Music Square West, Nashville, TN 37203. All rights reserved. Used by permission. "A Deep Sworn Vow" and "When You Are Old" by William Butler Yeats. Used with permission from A.P. Watt Ltd.
For information about permission to reproduce selections
from this book, write to Permissions,
W. W. Norton & Company, Inc.,
500 Fifth Avenue, New York, NY 10110
Library of Congress Cataloging-in-Publication Data
Lynch, Thomas, 1948–
Apparition & late fictions: a novella and stories /
Thomas Lynch.—1st ed.
p. cm.
ISBN 978-0-393-04207-8
I. Title. II. Title: Apparition and late fictions.
PS3562.Y437A87 2010
813'.54—dc22
2009038981
W. W. Norton & Company, Inc.
500 Fifth Avenue, New York, N.Y. 10110
www.wwnorton.com
W. W. Norton & Company Ltd.
Castle House, 75/76 Wells Street, London W1T 3QT
This book is for
Mary Tata
## Contents
ACKNOWLEDGMENTS
Catch and Release
Bloodsport
Hunter's Moon
Matinée de Septembre
Apparition
## Acknowledgments
THESE STORIES have been improved by the scrutiny of early readers including Mary Tata, George Martin, Tommy Lynch, Heather Grace Lynch, Nick Delbanco, Bret Lott, A. L. Kennedy, Keith Taylor, Richard McDonough, Sejal Sutaria, Pat Lynch, Dan Lynch, Michael Heffernan, George Bornstein, and Margaret Lazarus Dean, and by the guidance and good counsel of my editors, Jill Bialosky and Robin Robertson. For these and for the assistance of Mike Lynch, Ken Kutzli, and Sean Lynch, I am permanently grateful, as I am to the editors of the following journals where these stories first appeared, sometimes in different versions:
"Catch and Release" and "Bloodsport" first appeared in Witness.
"Bloodsport" also appeared in The Best American Mystery Stories 2001, edited by Lawrence Block (Houghton Mifflin, 2001).
"Hunter's Moon" first appeared in Granta.
"Apparition" first appeared in a shorter version as "Block Island" in The Southern Review and in the anthology Not Safe, but Good: Stories Sharpened by Faith, vol. 1, edited by Bret Lott (Thomas Nelson Books, 2007).
## Apparition & Late Fictions
## Catch and Release
THE THERMOS BOTTLE with his father's ashes in it rested on the front seat of the drift boat. He was glad to have the morning's busywork behind him and to be in the river. The green thermos with the silver cap looked inconspicuous enough.
Neither the waitress at the All Seasons Diner nor the other guides meeting their clients over biscuits and sausage gravy had noticed it. Nor had the woman from the tackle shop with whom he had arranged a car spot for his truck and trailer. He told her he'd be floating Walhalla to Custer and left her a set of keys. He took some twenty-pound shooting line, some ten-pound leader and eight-pound tippet, some split shot and a Snickers bar, some feathers and yarn. He'd been tying his own flies for years. "On account," he told her, putting the gear on the counter.
"You'll be a long way downstream from the other guides, Danny," she told him. "Most of 'em are doing Green Cottage to Gleason's Landing. Salmon all over the gravel, they say. Or Gleason's to Bowman's or Rainbow to Sulac. No one's below Upper Branch but you."
"That rain'll push some fresh one's in," he said. "Some steelhead and lakeruns, maybe. First of October. It's time."
"Well, you'll have some peace and quiet at least. It's a zoo up here with guides and canoes and walk-ins. Mind the bow hunters. Season opens today."
"Peace and quiet, yes."
He gathered his supplies and left.
Only Enid, the woman with whom he slept some nights, who managed his website and kept track of his bookings and packed his shore lunches, had been curious.
"What's with the thermos?" she'd asked when he stood in the door in the dark with his waders and vest. She knew he only drank Mountain Dew.
"A client's," he'd said, and bent to kiss her.
"Good luck," she whispered, and rolled over and returned to sleep. He pulled the quilt up over her bare shoulder. For a moment he wondered if he should stay.
AND PUSHING off from the Walhalla landing, in the first light of the first morning of the first October since his father died, with his lame dog Chinook curled in the boat's bow, his father's ashes in a thermos on the front seat, himself easing the oars into the downstream current—the three of them adrift in the Pere Marquette, the forest on either side of which was ablaze with the changes of Michigan's autumn—he thought it was nearly like taking his father fishing again and that the thermos bottle was a perfect camouflage and that he didn't know if such things ought to feel like weeping or like laughter. He loved the damp rotting smell of autumn, the breeze that bore it through the tunnel of the river, the pockets of fog, the marsh and mudbanks, the litter of fallen and falling trees, the unseen traffic in the woods, the distance his drift put between himself and all the other details and duties of ordinary time. He loved the snug hold of the river on his boat, the determination of its current, the certain direction, the quiet.
And though this time of year he could put sixty days of guiding together—from late August to late October—and though his arms and shoulders would burn from the rowing and his hands blister from the oars and ache from the tying of knots and his fingers would sting with line cuts and fish bites and embedded hooks, it never really seemed like work. But it was his work. Three hundred dollars a day, less 5 percent for the lodge that booked the trip, less the cost of lunches and tackle and car spots, gas and gear, plus tips—these were the heydays of the year when yuppies from the suburbs all over the Midwest would drive their SUVs to Baldwin to dress up in their designer fly-fisherman costumes and catch the biggest fish of their lives on the lightest tackle. Danny took their pictures, took their money, filled them full of lore and stories, and sent them back to what he imagined were their trophy wives and dreary day jobs, glad that he had passed on all of that to become a trout bum and live the life he figured God intended him to live. This time of year, the only days of rest came from cancellations. The sales rep from Akron, booked for today, had called to say he couldn't get away.
THE THERMOS had been his stepmother's idea. Though his father had been a minister all his life, and had officiated at hundreds of funerals, he had steadfastly refused to say what he wanted for his own arrangements. "You'll know what to do," is all he'd ever say when questioned on the matter. "Bury me or burn me or blow me out of a cannon—I don't care. I'll have my heaven. It's yours to do. That's one funeral I won't have to preach."
So when it happened, it was Margaret, his stepmother, who had decided that, after the wake and funeral, the body would be cremated. When Danny's sister and brothers had questioned her, suggesting that he'd rather be buried at the family lot next to their mother, Margaret said it would make their father portable and divisible and, after all, he loved to travel and never got to do enough of it, and he always had tried to be fair with them all.
Thus, to his youngest son, the artist, she gave a portion of the ashes, which he mixed with acrylics and oils and began a series of portraits of his father. To his middle son, who was the associate pastor at the Presbyterian church, she gave a portion to be buried in a bronze urn under a tastefully diminutive stone in the local cemetery with his name and dates on it, next to his first wife. The graveside service, like the wake and funeral, was widely attended by fellow clergy, local Presbyterians, and townspeople. To his daughter she gave a portion sealed in a golden locket with an emerald on it which could be worn as a pendant or ankle bracelet in Ann Arbor where she practiced law. And to Danny she'd given the thermos—an old green Stanley one that looked like something carried by construction workers—and said, "Why don't you take your father fishing?"
DANNY REMEMBERED his father taking him fishing, that first time in the river, when he was a boy, how the water tightened around his body, the thick rubber of the Red Ball waders constricting in the current. It was late March. It was cold and clear and he wondered how his father ever found this place, hours from home, driving in the dark to get to the river at first light. How they stood overlooking the river from the top of the hill, its multiple interwoven channels his father called "the Braids" because it was in this area the river split and turned and coiled around itself before returning to its orderly flow between two banks below Indian Bridge. He remembered his father saying they were going to a secret spot, back in the swamp, called Gus's Hole, named for the man who had discovered it and told someone who told someone who told Danny's father. Then hiking down the oak ridge with their rods and net and gear. His father held his hand as they stepped into the river and he could feel, for the first time, the weight of the water moving, something urgent and alive; and how his father held his hand high in the air, like dancers doing twirls through the deeper water, the boy he was then bobbing in the water, his father holding him aloft to keep the river from rushing over the top of his waders.
They fished hardware then—Little Cleos and Wobblers and other spoons that looked in the water like wounded baitfish. Or clattering plugs that nose-dived in the current and jiggled just above the bottom. Later they would graduate to long poles and lighter tackle and various spinners. His father always swore by Mepps #4s with little egg-colored beads and copper blades. For steelhead they'd roll spawn bags through holes behind logjams and fallen trees and finally they gave up their spinning gear for fly rods and reels and the slow perfections of their own patterns—caddis and stoneflies and hex nymph variations—of preparation, presentation, catch and release.
But that first time Danny remembered nothing so much as the slow tug of the lure in the current, the fresh sensation of the river's bottom and the current's ways as he worked it through snags, over gravel, around stumps, and into the dark pools where his father told him fish were waiting. And he remembered the first fierce hit of the brown trout, how it rose in a fury and leapt and ran upstream, then downstream, taking line from the reel, and his father's calm counsel behind him, "That's it, Danny, that's it," and the beauty of it in the net and the mystery of it and the sense that he had of having been chosen by the fish because he had made the long trip in the dark, endured the cold, and the long walk through the swamp to stand in the right spot and cast to the right place, and listened to his father's instructions and let the lure sink and work its way until it brought the eight-pound fish and the eight-year-old boy together.
Every winter, spring, and fall since then he'd fished the Pere Marquette—from mid-August to mid-October for the salmon's spawning run, until late December for the steelhead that would follow them to feast upon the eggs, then March and April for the spawning brown trout and steelhead. And long before Danny knew that the river had been named for a dead Jesuit missionary who'd died a couple of centuries before where the river emptied into Lake Michigan, long before he'd learned the names of birds and trees, waterfowl and wildflowers, he'd learned the habits of the river, its eddies and sluices, backwaters and rapids, its clay banks and sand and gravel bottoms, and the habits of the fish who spawned and lived and died in it. He knew the names of every hole and pocket and pool and gravel that was known for fish, from the Birch Hole to the Trestle Hole to Alligator Alley and the Aquarium, Hole 9, the Sand Hole, Wadels and the Highbanks, Beaver Den, the Improvements, the Maples, Stumps Lodge, and a hundred more, a fair few of which, unknown to anyone before himself, he had named. After a lackluster college career and half-hearted efforts in the working for General Motors as a driver at the Proving Grounds, he had taken his father's free advice to do something he was "really truly passionate about." He moved from the small house he shared with some high school friends to Baldwin, in western Michigan, near the river's headwaters, and resolved to be a trout bum. He guided in the fall and spring and went to Alaska or Montana in the summers and guided there. All winter he tied flies which he sold to the local outfitters and private buyers. He taught workshops for visiting executives, field-tested new gear for the manufacturers, worked his website and fished alone, studying the river, whenever he didn't have a client. It was enough to keep body and soul together. He was not yet thirty, never going to be rich, but was, nonetheless, among the most respected guides on the river, well known for what he knew that no one else knew better.
And though his father had given up winter fishing over the years, and would find excuses about the spring runs, they had shared the salmon and the autumn for twenty years. When his father loaned Danny the down payment on the cabin and ten acres of scrub pine west of town, he took in trade what he called a "Life Lease" on the third week of September—the height of the season by his calculations.
This was the first year he never showed.
This was the September he had died instead. Something "acute" and "myocardial," the doctor had called it when Danny's father woke in the early morning, walked into the bathroom, and lay down on the green tile floor and died.
"He never said a word," Danny's stepmother said, "not a word. Never called out. Never made a sound." Here she would pause and take a breath. "It was as if he didn't want to disturb anyone, as if he knew it was his end." She'd catch her breath again. "A glass of water on the counter, a bottle of aspirin open, him on the floor, he never made a sound."
THE DRIFT boat in the river never made a sound. Even the oars in the oarlocks were quiet. Danny could hear the fog lifting, the breeze and the leaf-fall, the wing flap of a heron lifting off. He could hear the water over the downed limbs of trees, under the riverbanks, the curl of a muskrat into the current. He could hear the winter oak and sumac and tag-elder and the air and his dog breathing and the perfect silence in the thermos. He wanted to scream.
He wanted to fill the elements with his heartache and anger—to shake a fist in the face of creation and ask God what exactly he had in mind that made his father die too soon. Why, when he was just at the point in his life when he could begin to enjoy it, after all the years of working for the church, out in the middle of the night when someone died, up in the morning to visit the sick or counsel the heartsore or balance the books, evenings with the deacons and the session, all those Saturday nights huddled in the office over his homily, lunching with locals, glad-handing the politicos. Why, after all those years of mortgage and tuition and car payments, always behind the eight ball on a churchman's stipend, always worried about the salvation of souls, the brick and mortar; why, when he could finally begin to relax a little? Why now, why his father, why, goddammit. Why?
But Danny shouted nothing into the woods. He only pushed the oars and worked the water to the first bend in the river, where he eased to the south bank, dropped the anchor, and stepped out of the boat. Chinook jumped from the bow into the water, up to the bank, and disappeared into the woods. The badly twisted left leg—he'd been hit by a car as a puppy—never slowed him down anymore. He'd learned to run with it and run without it when he turned up deer bedding in the forest and chased them through the woods or rooted for voles in the undergrowth.
Danny pulled out his father's rod case, assembled the ten-foot eight-weight rod, attached his father's ancient reel, ran the line through the guides, and began to tie his terminal tackle.
"Your father would want you to have it," is what his stepmother had said, taking Danny to the basement back at home where the gear was kept. "And he'd want your son to have it, if you ever have one."
Danny couldn't imagine being a father. But for the first time it occurred to him that if he ever became a father, his children would never know their grandfather. This seemed wrong.
Danny had known his father's father. And had loved and admired the man who occupied his youth like a force of nature. He wondered if getting to heaven meant you got to ask these reasonable questions. Why? How? When? Why not?
He opened his father's fly box.
He felt his face tighten around his eyes. He began to weep.
It was not the bone-wracking sob he'd been holding back for months now, ever since the morning he'd gotten word. It was not the shoulder-shaking, wind-robbing breakdown with his name on it he knew he'd have to have eventually. It was only the catch in his breath and the tightening behind his eyes and the improbable twitching of his eyebrows at the sight of his father's little collection of caddis and stonefly patterns, some streamers, eggs in different colors. But it was not so much the flies as the knots, the bits of line like little hangman's knots in the eye of each hook where his father had bitten them off after they hadn't produced or after a day's fishing or after they'd produced enough. His father had never learned more than two knots. The nail knot for leader and tippet and the cinch knot for flies. Danny had brought home wonderful variants from the Katmai Lodge in Alaska where the salmon ran to fifty pounds and from the cutthroat waters in Colorado and Montana. But his father never wanted to learn them. He trusted his tying and never lost a fish, he was fond of affirming, on account of knots. His held, he'd insisted, however old-fangled.
Danny wiped his eyes, glad that he was fishing the low water, glad for the little release, glad for the feel of his father's knots as he bit one off a yellow egg and tied it freshly to his line.
The water had risen with last night's rain and darkened some and the sun was not yet high enough to see into the hole but he knew this bend by heart and reckoned the fish would be suspended in it and tied one split shot on and cast it backhand slightly up stream toward the opposite bank. He could feel the split shot in the gravel at the top of the hole, and could feel it fall into the deeper run, and watched the loop in his line straighten in the current and held the rod out in front of him as the line moved through the water. At the bottom of the run he began to strip back line and looped it out upstream again, the familiar balance between fisherman and rod tip and terminal tackle establishing a cadence to the casting, the fly's drift, and the retrieval so that he was covering the hole every thirty seconds. He avoided, in his cast, the bronze leaves of an overhanging oak tree, and in his drift, the submerged stump at the bottom of the run. He could hear Chinook deep in the forest rolling in something dead and rotting. Part wolf, part malamute, he could not help himself, Danny thought.
On his eighth or tenth cast, the fly stopped early in the run and Danny set the hook and a huge buck king salmon rose shaking its head to the surface. Yes! he could hear himself saying to himself as he felt the first surge of the fish against him. This was always new, always the first time, never routine, this wonder that two separate natures could intersect like this. The fish roiled on the surface, dove deep into the hole, and held there, the line pulsing, rod pulsing with the weight of it, then it ran upstream toward the logjam above the gravel, taking line off the reel as it worked its way. The look Danny got of it in the shallows made him guess, eighteen, twenty, maybe twenty-one pounds. And it was fresh—silver and blue—only out of the lake a couple of days, still weeks away from spawning and fatigue and death. Danny loved the ferocity of fish in the lower water. He could see the tiny egg in the bony upper jaw. Right in the snot-locker, he could hear himself saying, and turned to look, because he'd never in all these years gotten out of the habit of turning to look for his father's amazement and excitement and approval. But there was no one grabbing for the camera or the video cam or the net or anything. There was only the boat snug to the riverbank and the thermos bottle on the front seat and Danny, momentarily disconnected from the fish's struggle, failed to turn the fish back downstream, and in the lapse the salmon got under the logjam and, free of the reel's drag and the rod's wisdom, it snapped the eight-pound tippet easily. Danny felt the line go limp, he could see the fish jump behind the logjam, his father's egg pattern still in its jaw. He turned and looked back to the boat. There was nothing. He was alone. For the first time since his father's dying he felt his father's death, the dull, undeniable force of it, and it dropped him to his knees in the river, the water nearly running over the top of his chest waders, its cold embrace a kind of comfort as his body shook in spasms. He held his father's fishing rod aloft as the weight of the river beat against his chest. The fog was lifting now; the morning warming as the sun rose. He whistled for Chinook, stood, and walked back to the boat.
Standing beside the boat in the water, he reached the thermos on the front seat, unscrewed the stainless steel top that served as a coffee cup, set it on the bow. Then he loosened the cap in the mouth of the thing and poured a handful of his father's ashes into his right hand. They were dull gray and sharp-edged and utterly dry. They had about them the feel of vermiculite or kitty litter or something freeze-dried that could, with the addition of water, be reconstituted. He grinned at the prospect that "just adding water" might bring his father back. He let his hand drop into the river, where he opened his fist and released the ashes into the current. The little white puff, like a cloud of milt when the buck salmon spawn, disappeared downstream.
Danny waited for something like an overwhelming feeling, and when none was forthcoming, he recapped the thermos, climbed back in the boat, weighed his anchor, and eased into the downstream current. He found himself sighing, breathing deeply, grinning inexplicably.
Chinook kept up through the woods, chasing along the riverbank, sometimes leaping in the river to swim beside the boat or behind it or crossing in front of it like a furry dolphin. He'd climb the bank on the other side and rummage among the overgrowth for muskrat and squirrels and possum. Long ago he'd learned to run with the twisted left leg, and when he needed the most speed, when he'd kick deer out of their bedding, to simply tuck it up and run on three legs. Danny reckoned it never slowed him down but kept him from going too far astray. Never out of earshot, a whistle from Danny would bring the dog to the riverbank to check in; then it would disappear again into the deep green of the forest. The dog stayed tuned to the sound of the boat and to its master's noises. Danny was glad he hadn't killed the thing.
He'd only just gotten the dog a few years before. He'd chosen it from a litter in Alaska, his first season there, and brought it home with him to Michigan. Unaccustomed to traffic that would not stop, the dog had run into a street in Mount Pleasant, where Danny had been partying with old college friends, and was hit by a car taking an old couple to the casino. They felt horrible. They offered to pay for any damages to the dog. Danny had taken the dog to the vet, who suggested the veterinary school at Michigan State University. The dog was sedated and Danny raced to East Lansing, where student vets set the leg, put it in a cast, and thanked Danny for his confidence in them. When the cast came off, Chinook's leg was badly twisted. Another vet suggested a surgeon in Chicago. Danny drove the dog there and the surgeon, taking a look at the damages, told Danny he'd really be better off to put the dog down. The leg, he said, would never be right. The older Chinook got the more it would be in the way. He could, of course, remove the leg and the dog might even get on without it. Other dogs did. But a three-legged dog? The surgeon said he could euthanize Chinook.
He'd called his father. His father said that he trusted Danny's instincts. He said he knew that Danny would know the right thing to do. He said he admired him for taking on the care of the dog and these difficult decisions. His father had told him to say his prayers, to listen to God, and the right thing would come to him, whatever it was. He told Danny to take a day or two to think it over. He told him he'd be willing to help, whatever the decision was.
Danny rowed past the first confluence. The sun was working up the sides of the trees. The day was going to be blue. He found himself talking to his father's ashes.
Let go, let God, he heard himself saying.
He was thinking of that first brown trout. He remembered wanting to take it home to show his mother. His father said they could either take it home and it would die and they would eat it or they could let it go. His father was working the lure out of its jaw. Danny was holding the handle of the net the fish was flopping in. He couldn't believe the options. Kill it, eat it, show his mother. Let it go. Never see it again. It was so beautiful. How could he let it go. How could he kill it. He remembered how it made him feel cold, for the first time, having to make such an awful choice.
At the second confluence, the river turned westward and the topography flattened and the current slowed into an area all of the guides called "the Serengeti," because it was a huge flat expanse of cat marsh a mile wide on either side of the main flow. Surrounded by a high ridge of ancient white pines, winter oaks and sugar maples, the area seemed like a huge bowl of natural wonders. Whenever his father fished this float with him, he'd always call it a National Geographic moment when they'd make their way into the Serengeti. Once they'd seen a pair of fully matured bald eagles and a young eagle with them in a tree. They'd seen osprey and owls and plenty of heron, and in one shallow backwater slough, outside the current, they'd seen salmon by the thousands milling about, as if resting from the furious pursuit of reproduction and mortality that had triggered their return to the river. The irony of this discovery was that these were literally fish in a barrel, but because there was no current, there was no way to present a fly to them in a way that would induce them to strike it. Danny and his father could only row in and out among the hordes of fish remarking on nature's strange abundance.
After the Serengeti the boat bearing Danny and his dog and his father's ashes drifted into the Braids. This was a maze of river and stream and rivulet that from overhead might look like a bunch of pretzels. It was rarely fished because guides could get lost in it, following the wrong shoot to its dead end, at which point the boat would have to be walked out against the current, or circling some loop outside of the mainstream. Local meat fishermen used to hike in with burlap bags and clear out a hole with weighted snagging hooks on thick line and stiff rods. But the DNR had impounded a pickup and a bunch of gear some years before, so even the snaggers left the swamp alone now. No fish was worth the loss of a truck.
Danny remembered, as a boy, fishing Gus's Hole with his father, when a pair of snaggers appeared in the hole just above the one he and his dad were fishing. He could hear the boozy laughter and the splash of the treble hook and the line ripping through the water as they yanked and yanked until the hook lodged in the belly or the side or the tail of a salmon. He could hear the leap and splash of the foul-hooked fish and the shouting of the snaggers, how they'd hoot and howl and they'd crank the huge reels and how the short stiff rods seemed like baseball bats and how they'd take ten fish out of a hole, then move on through the swamp.
At the Sand Hole, a twenty-yard expanse between fallen cedars, Danny tied a single egg on. He stood above the hole, cast quarter down, sweeping the red, above the hen salmon he could see fanning the gravels with her tail, the males lined up like a squadron behind her. Behind the red, a pocket of dark water the width of a dining room table seemed to suck the egg into its deeper habitats. As the egg worked its way into the sweet spot where the current and seam and the foam line converged, he let out line so the fly would sink. Here is where he figured the steelhead held—where the O2 was highest, the food flow was fastest, and the deeper water gave them cover, downstream from the spawning salmon and browns. All the fanning of the gravel would send a constant buffet of hex nymph and stoneflies and caddis along with fresh spawn from the maturing salmon. He imagined the steelhead, mouth agape, holding in the current, gulping anything that looked like protein floating by. On his second cast, the fish slammed the egg with such ferocity Danny lurched forward, set his foot, then set the hook. The water broiled and he could see the pale white mouth of the huge rainbow which dove deep, then rocketed out of the water, then dove and ran downstream, jumping twice, then turning back so fast the line slackened and Danny reeled as fast as he could. From wherever he was in the woods, Chinook could hear the line peeling against the drag, the fat smacking of the fish on the water, Danny shouting "Yes! Yes!" and ran to witness the commotion. It was fifteen minutes before the huge silver fish, fifteen pounds of hunger and outrage, was backed into the shallows where Danny held his line tight, the rod aloft, and, genuflecting, grabbed the steelhead by its tail. The tiny hook was deeply embedded in its jaw and with a pair of forceps Danny worked the hook free, held the gasping, wriggling fish to his face, admiring its teeth, its steely eyes, then knelt in the water and worked it back and forth in the current, letting the water wash through its gills, restoring the spent thing's equilibrium. After thirty or forty seconds of this, he could feel the fish's muscles flexing, Danny loosened his hold of the huge tail just enough for the fish to sense freedom. In a flash it was gone.
It was his favorite thing—to hunt the stealthy, transparent, invisible fish, to know enough about its habits to isolate it in all that dark water, to present a fly or an egg of his own making, the right size and color, at the right angle at the right depth at the right speed to trigger the thing to animal desire, then to fight the thing in its own environs, counting on his knots, his timing, and the proper setting of his drag, then to catch the thing, to hold it, and then to let it go.
"Love 'em and leave 'em," is what his father used to say, and it was true. No dinner of salmon or steelhead, and they'd had plenty, ever made him feel as full as the utter mastery involved with returning the captive to its freedom, the genuine pleasure of letting it go.
Among his clients it was well known now that Danny would put you on fish, teach you to catch fish, work hard to net them, and take your picture with them. And you could keep one male chinook or coho to take home to barbecue for the neighbors, with too much beer and blathermentia, but whatever plenty the season provided, whatever the limit the State of Michigan allowed, one salmon was the limit on Danny's boat and none was better and no browns or steelhead were ever kept.
"But they're all just going to die," one of his clients once protested when Danny released a thirty-pound hen.
"We're all just going to die," Danny told him, "sooner or later, a hundred percent."
Danny washed the fish slime off his hands, poured another handful of his father's ashes out, kissed the fist he held them in, and let them go into the dark water.
He sat with his dog on the bank for a few minutes, waiting for no particular reason, watching the salmon working the reds. Everything around him seemed a metaphor for his father—the leaf-fall, the clear water, the fish in their futile quest. He tried to remember when he first became aware that his father approved of him, his life, and whether it was a gift outright or whether he had earned it.
Early on they'd argued over church and Sundays. He'd quit after his mother died. He was ten when she had died of "complications" involving "medications" and "depression." And though he would stand in the first pew on the left side of the church with his stepmother and siblings—for his father had married Margaret within the year—with a hymnal in his hand for a few years after that, he was never really "there" and didn't really sing the songs or believe or understand the light that came through stained glass or how it had happened—how or why his mother had been removed from their lives. At thirteen he simply refused to go and gave his father and Margaret to believe that if forced he'd make a scene that would embarrass them more than his absence. He took to fishing Sundays at the dam below the park in town for bluegill and crappie and sucker and carp. He'd learned to like the quiet and the privacy and the feel of fish on the other end of his line. And when he told his father years later that the river, the Pere Marquette, was his church and chapel and Bible and choir, that he felt closer to God there and closer to himself there and closer to his mother there and closer to life, his father had nodded, if not approval, then at least acceptance. "The Lord," the churchman told his son, "has a fondness for fishermen." This was near enough to forgiveness.
And there'd been issues over education. Danny had been a lackluster student. After high school he'd managed to get accepted to Central Michigan University. It was the nearest school to Baldwin and the Pere Marquette. He enrolled in a course of General Studies, but between the draw of the Indian casino on the edge of town and the Pere Marquette less than two hours away, nothing in the curriculum could keep his interest long. Whenever he could he'd drive to the river and float a stretch researching the pocket water, the structure and the habits of resident and migrant fish. Home for Easter that first year, Danny's father told him he could not succeed by doing things halfway.
"Do something you're really passionate about. What's the worst that can happen? You're young. You can afford to fail but you can't afford not to try."
This was permission enough for Danny. He quit school. Worked on a landscape crew for three months, and with the cash bought a tent, three new Orvis rods and Billy Pate reels, and moved to Baldwin in early August, with his drift boat outfitted, pitching his tent on the river, where he slept and tied flies when he wasn't fishing. He'd begun hanging out at the PM Lodge and the local bars and tackle shops, picking up guide trips where he could. It wasn't long before word got around about his talents for putting clients on fish, for finding the right drift through difficult holes, for working the river in difficult conditions. Some fishing guides were taxi drivers. Rowing to some famous meat hole and sitting tight all day, then rowing out. Danny fished the slots and shoeboxes, the lesser-known pools where fish held on deep gravel or between difficult snags and stumps. He'd taught himself what shade did and rain did and the moon did to the habits of fish and the conduct of water.
When he climbed back in the boat, Chinook bolted downstream, then into the forest. Danny pulled the anchor up and felt the current's slow embrace circle the boat and take it in. He dipped his oars. The thermos, now nearly empty, rolled slightly on the front seat.
As he approached Gus's Hole Danny whistled for his dog. He anchored in the current and listened close. Sometimes his heart filled with the beauty and the silence of the place. Deep in the swamp, beyond the energies of other guides and walk-ins, the sense of isolation and privacy was comforting. And Gus's Hole itself held few fish anymore. The current had changed enough over the years to widen and flatten out the confluence of water. Still the area was fishy and the air was full of the smell of rotting salmon and the sickly-sweet smell of some larger putrefaction in the woods. Danny reckoned it was a deer shot out of season by a bow hunter who couldn't trail it through the swamp. In the distance he could hear Chinook howling. He whistled again and waited. After five minutes he could hear the dog approaching.
Danny wondered if the dog knew how close it had come to being killed in this place. It was to Gus's Hole Danny had brought the dog two years before, after all the experts had weighed in with their advice. They'd fished all day and Danny climbed the high banks to the oak ridge from which he and his father had first looked down on the curling braids of river that formed the swamp that winter years ago when he was eight. He had a field shovel and a pistol he had borrowed from a fellow guide. At the top, he began digging the dog's grave, the work quickening with anger and slowing with sadness, the variable speeds of the labor like the division of his heart. He had the grave dug, the pistol loaded, the blanket he intended to wrap the corpse in ready. He whistled for the dog and called "Chinook!" He heard the dog bark and saw, from the high ridge, the dog emerge from a marshy island a quarter of a mile downstream. "C'mon, Chinook," Danny called, and the dog ran the bank upstream fifty yards, then dove into the current, crossing to a sandbar in the middle, shook himself dry, then dove again, swimming upstream fifty more yards before coming to the base of the high banks. He shook himself dry again and bounded up the steep hill without slowing. At the top of the high banks he leapt into Danny's embrace. This prodigious bit of cross-country swimming and sprinting and climbing, regardless of the crookedness of his leg, proved the dog well able for his habitat. It convinced Danny that he should not kill him, not now, not ever, unless he was truly pained or endangered by his handicap.
And when his father heard, he had approved. "Surely," he had said, "we must tolerate some imperfections in the ones we love."
Two years later the dog's "imperfection" hadn't held him back a bit.
Danny sat on the bank and shook the thermos. There was maybe a cup of his father's ashes left in it. He wondered what to do. The river had its share of him, working its way now, Danny reckoned, downstream to Scottville and Ludington and out into the big lake and maybe to the mouth of the Chicago River and into the Mississippi and through the middle ground of America into the Gulf of Mexico, to the larger amalgam of oceans that held the continents afloat on the globe.
He poured the remnant into the silver cap of the thermos. He dipped the thermos into the current and poured the cold water into the capful of ashes. With his knife he mixed the water and the ashes into a kind of gray paste, like thick oatmeal, and then he ate it with a spoon.
## Bloodsport
MOST TIMES the remembrance was triggered by color—that primary red of valentines or Coca-Cola ads—the color of her toenails, girlish and perfectly polished. He remembered her body, tiny and lifeless and sickeningly still as she lay opened and autopsied on the prep-room table. He could still bring to mind, these many years since, the curl of the knot in the viscera bag the pathologist had tied, with all of her examined organs inside, and the raw edge of the exit wound in her right leg and the horrible precision of the hole in her breast where the man who murdered her put the muzzle of the gun.
And he remembered the dull inventory of detail, the hollow in her mother's voice the morning she called him at the funeral home.
"Elena's been shot, Martin. Up in Baldwin. She's at the Lake County Morgue. Go and get her, Martin. Bring her home."
ELENA HAD been only fifteen when her father died—the darkly beautiful daughter of a darkly beautiful mother and a man who'd had cancer. He was laid out in an 18-gauge metal casket. The funeral was huge. Martin could remember standing between them, Elena and her widowed mother, when they'd come to see the dead man's body. He figured he was ten years older than the daughter, ten years younger than the mother. He had asked, as he'd been trained to ask, if everything was "satisfactory." It was the failure of words that always amazed him.
"He got so thin."
"Yes."
"At least he's not suffering anymore."
"No."
"Thank you, Martin."
"Yes."
And he remembered how Elena, after trying to be brave for her mother, after standing and staring at the lid of the casket as if she could tough it out, as if she could look but not see, had let her gaze fall on the face of her dead father and cried, in one great expiration of pain, "Oh Daddy! Please, no," and nearly doubled over at the middle, holding her tummy, and how her knees buckled and how he grabbed her before she fell to the floor. And how she had pressed her sobs into his shirt and how he'd hugged her close and felt her holding on and could smell her hair and feel the form and perfect sadness in her shaking body and how he'd said that everything would be all right because he really didn't know what to say. It made him feel necessary and needed and he wanted to hold her and protect her and make everything better, because she was beautiful and sad and though he could not fix it he would not let her go until she could stand on her own two feet again. And he thought that being the only embalmer in town was no bad thing when you stood among the widowed and orphaned and they would thank you for the unhappy work you'd done on their people.
FIVE YEARS after that and it was Elena, killed by her husband with a gun.
Martin could not get his mind off how mannish the violence was, how hunter-gatherly, how very do-it-yourself, for the son-of-a-bitch, according to the coroner, to stand on the front deck of their double-wide out in the woods while she loaded the last of her belongings in the car—her boom box and a last armful of hanging things—how he must have carefully leveled the rifle, his eyes narrowing to sight her in. He put the first bullet through her thigh. An easy shot from fifteen yards.
He must have wanted to keep her from running.
"The way you would with any wild thing," the fat pathologist, smelling of stale beer, had told Martin in the morgue, taking the cigar out of his mouth to hold forth like an expert. "You hobble it first, then you don't have to chase a blood trail through the woods all night." He warmed to his subject. "Bow hunters go for the heart or lungs most times. They don't mind chasing through swamps and marshes after a wounded buck. It's part of the sport to them. But shooters go for the head shot or the legs."
And as she lay in the thick leaf-fall beside the car, bleeding from the severed femoral artery, he'd walked over, put the barrel to her left breast, and squeezed off another round.
"She'd have bled to death either way," the pathologist said. The sight of that fat hand with the cigar touching the spot on Elena's thigh where the bullet tore its exit out sickened Martin. And when the same hand pulled the sheet back to show the terrible carnage to her torso—the postmortem incisions very loosely stitched up and the black and blue and red little wound where her killer must have reckoned her heart would be, Martin quickly moved his stretcher beside the morgue tray, covered her body, and took charge before the pathologist carried his feckless lecture any further. He signed the logbook beside Elena's name and case number, got the death certificate marked Gunshot wounds to leg and chest in the section that asked for the cause of death and Homicide where it asked for manner and had her name and date of death, all of it scrawled in the sloppy hand of the pathologist, and got her out of there.
All the way home he tried to imagine how it must have happened—if anyone could have heard it, the small-caliber outrage of it, as if she'd been a doe feeding among the acorns or come to the salt lick, her large brown eyes full of panic and stillness. He wondered if she knew he was dangerous. He wondered if she realized, after the first shot, that he was going to kill her. He wondered if she died with fear or resolve. He wondered if bleeding from the first wound, she might have passed out, and never saw the face of her killer or the barrel or the gun or felt it on her body or saw his eyes as he pulled the trigger.
Taken as a thing itself, considered within the broad range of human conduct, undistracted by his professional duties, Martin regarded the aberration of the dead girl's body riding behind him as utterly incomprehensible. How could someone kill someone so coldly, someone with whom you had made plans, had sex, watched television, promised love? It left him with a functional ambiguity. Martin tried to assemble a reasonable sentence in which the last phrase went like and then he shot her, twice, because...but he was always unsuccessful.
He looked in the rearview mirror at the length of the stretcher in the back of the hearse with its tidy blue cover under which Elena's body was buckled in, her head on the pillow, a small bag with her bloodied clothes, her jewelry and personal effects beside her. He tried to connect this horror with his remembrance of a sad, beautiful girl sobbing at the graveside of her dead father a few years before, waiting for the priest to finish with his prayers. The morning was blue and sunlit, the buds of maples just busting loose, the men who'd been pallbearers lined up on one side of the grave, Elena and her mother and grandmother on the other. And all around, a couple hundred who'd come to pay their respects—women who worked with Elena's mother at the real estate office, men who worked with her father at the shop, parishioners from Our Lady of Mercy and kids from the freshman class of the high school. And after the priest had finished, Martin had nodded to the pallbearers to remove their gloves and solemnly place them on the casket—a little gesture of letting go. And then, from the pile of dirt next to the grave, under the green grass matting, he'd given a small handful of dirt, first to the dead man's mother, then to the dead man's wife, and then to Elena; and at his direction, each stepped up to the casket and traced a cross on the top with the dirt that Martin had given them. He put a hand on their elbows as they stepped on the boards in a gesture of readiness and ever-vigilant assistance. And after that, Martin made the announcement he had practiced saying out loud the night before.
"This concludes the services for Mr. Delano."
He reminded himself to speak slowly, to enunciate, to articulate, to project.
"The family wishes to thank each of you for your many kindnesses—for the floral tributes and Mass cards and most especially for your presence with them this morning."
He took a breath, tried to remember what part came next.
"You are all invited to return now to Our Lady of Mercy Parish Hall where a luncheon has been prepared in Mr. Delano's memory. You may step now directly to your cars."
At this direction, people began to move away, relieved at the end of the solemnities, talking freely, trading news and sympathies. Martin had been pleased with the performance. Everything had gone off just as he'd planned—a fitting tribute, a good funeral. The pallbearers walked away as a group, looking official. Someone assisted the grandmother from the grave. Elena's mother, her eyes tired and red, took Martin's arm as they walked to the limousine, holding the rose Martin had given her, the crowd of people parting as they made their way. And Martin was thinking this is no bad thing for people to see what a dependable man their new funeral director was—a reliably upright, lean-on-me kind of man—less than a year out of mortuary school, mortgaged to the eyes for the business he'd bought from the widow of the man who'd been here before, but clearly a responsible, dependable citizen, someone to be called on, night or day, if there was trouble.
At the door to the car Mrs. Delano stopped, turned toward Martin with a brave smile, tilted her head slightly, opened her arms, and Martin, sensing that she wanted him to, without hesitation bent to embrace her. She was saying "Thank you, Martin" and "I could never have made it through this without you" loud enough for bystanders to hear and he was patting her back professionally, all caring and kindness as you would with any hurt or wounded fellow human, saying to her, "You did good, he'd be proud of you," and she was patting his shoulders, and then, once the hug was over, holding the hankie to her eyes, she quickly disappeared into the backseat of the car in a rush of grief and relief and gratitude, and Martin straightened up and held the door.
Elena, who'd been following Martin and her mother to the limousine, holding two roses she'd picked from her father's casket spray, paused at the car door and, perhaps because she was following her mother's lead, perhaps thinking it was the proper thing to do, looked Martin in the eyes and said, "Thank you, thank you for everything," and reached up to lock her hands around Martin's neck, and just as Martin was starting to say, in a voice all caring and kindness, "You're very welcome, Elena," she rose on her tiptoes, pressed her body firmly against his, and kissed him squarely on the mouth. Martin could feel her chest on his chest, her small hands holding the sides of his face, and her soft mouth opening slightly and the wetness of it on his lips. He let go of the door handle and held her at the waist, first pulling her toward him, then, opening his eyes, gently pushing her away, and when she stopped kissing him, he could feel his face reddening and he was wondering if the priest and the pallbearers and the townspeople could see his blush and the flash of desire he could feel in himself and the wish beginning to form in his mind that everyone would disappear so that he could hold her and touch her and comfort her and have her and then, but before he could pat her on the back professionally, before he could say, "There, there, everything is going to be all right," before he had a chance to restore the air of solemnity and order, Elena proffered, with a brave smile, one of the roses she was holding. He took it from her and, as her mother had, Elena disappeared headfirst into the back of the long black Cadillac.
For weeks after that Martin had tried to figure that kiss, its meaning. Surely she could not have known how much he might have imagined, after holding her at her father's casket, her sweetness and innocence and beauty and vulnerability and how much a man like Martin felt like protecting and consoling and holding and touching her. Surely he'd been discreet, during the long hours of visitation at the funeral home, standing at the back of the chapel watching Elena and her mother greeting the neighbors and family and friends who came. She could hardly have any idea of her own beauty, the perfect form, her small arms, her lithe body, her dark eyes, her breasts, the softness of her walking. Martin had been determined to look professional, caring, concerned. He had made an effort not to stare. Surely it was only that she was overwhelmed with the graveside duties, the deep emotions, or having seen her mother embrace Martin, and awkward about the currencies of thanks and the conventions of familiarity, she had overstated her gratitude, overpaid her sense of indebtedness. Or maybe in some way that Martin could not sort out entirely, the attachment to her father, torn apart by his untimely dying, was looking for another "male" attachment? Martin remembered a movie he'd seen in which a young widow whose husband gets killed in World War II takes a teenage boy into her bed, in her grief. Yes, love and grief, maybe something complex like that.
THERE WAS a safety in dealing with only the parts—the arteries and chemistries, the closure of eyes and lip lines, the refitting of cranium and sternum, the treatment of cavities and viscera, the placement of hands, the suturing of wounds and incisions, the rouge and lipstick and nail polish, the dressing and hairdo and casketing. Duty had a way of separating Martin from what it was he was doing. Stuffing the opened cranium with cotton, fitting the skullcap back in place and easing the scalp back over the skull, thereby restoring the facial contours, and minding the tiny stitches from behind one ear to behind another was only part of the process of embalming; and embalming was only part of the process of laying out the dead, which was only part of the process of the funeral, and the funeral was only a part of the larger concept of a death in the family, and a death in the family was a more manageable prospect, more generic, somehow, than the horror—round and witless and recognizable and well beyond his professional abilities—of a lovely girl, grown lovelier as a woman, who leaned on him and counted on him and had kissed him once as if she meant it and who moved away and then got shot like an animal in the woods by a man about whom Martin knew next to nothing.
FOR MONTHS after her father's funeral, Martin kept an eye out for Elena. Her mother came to pay the bill, and pick up more holy cards and thank-you notes. And then she came to order a stone. Beloved Husband and Father, is what it said. Martin had advised her against a double marker. She was young and would surely remarry, he thought.
And Martin would always ask, "How is Elena doing?" in his most professional, caring voice.
"She's having some trouble with her schoolwork. She doesn't sleep well. I'm a little worried."
Martin gave Elena's mother a list of grief support groups, run by the local hospice and area churches. He reminded her that there used to be "a year of mourning" and said that Elena's feelings were probably "very normal" and that "time heals all wounds."
"Yes," said Elena's mother. "It's just so hard."
She thanked Martin again for everything and said she hoped he'd understand if she said she hoped she wouldn't be seeing him again.
Martin smiled and nodded and said he understood completely.
THE NEXT June, Martin read in the local paper how Elena had been captain of the debate team that went to the regional finals in Ann Arbor, and the year after that she had gone to Italy on a Rotary Exchange Scholarship, and in her senior year she was pictured on the front page smiling in her prom dress beside the son of the man who owned the Lincoln Mercury dealership in town, over a caption that read A Night to Remember, and Martin remembered how very happy she looked, how very pretty. After that he pretty much lost track of her.
"AFTER HER father died," Elena's mother told Martin when she came in to pick out a casket and arrange the funeral, "she seemed a little lost."
Martin listened and nodded as Elena's mother, looking so much older now, outlined the details of her dead daughter's life. She'd finished school, applied to college, spent the summer after graduation waitressing in a bar-restaurant in western Michigan, to get out on her own and earn a little money.
"She met him there. At the Northwoods Inn."
He worked for the county road commission and came in weeknights after work and weekends after fishing or hunting. He was handsome and chatty. He had a trailer in the woods. He gave her compliments and brought her flowers and bought her beer and cheeseburgers. And when it came time to go to university, to get the education her father had saved for, she called her mother and told her that she was moving in with this man.
"I didn't approve but what could I do, Martin? Her father would never have allowed it. But what could I do?"
Martin shook his head and nodded.
"I told her she was throwing her life away on a summer fling, but she said she loved him. She loved him and he killed her, shot her like a damn dog, Martin."
Elena's mother's sobs grew heavy. Martin poured her a glass of water, moved the box of Kleenex nearer to her.
"Thank you, Martin," she said. "I'm sorry."
"Not at all," he said. "It's okay."
"In no time she was pregnant and he said he wanted to 'do right' by her. I told her he would always feel trapped, or always feel like he had done her a big favor, always feel like he was such a big man and she was just nothing without him, but she said she loved him and maybe it was all meant to happen like this and what could I do, Martin? What could I do? Her father would have gone up there and brought her home, but I had no one, no one."
They were married in the county offices in a civil ceremony, Elena wearing her prom dress and her new husband wearing a cowboy hat and a blue jean jacket and a string tie.
Elena's mother took the wedding snapshot out of her purse and told Martin to "cut him off of there and use that picture for the paper and the holy cards. She was so happy then."
Elena miscarried in her third month and took a job working dispatch for the Sheriff's Office.
By the following midsummer things were getting bad. Her husband's appetite for Budweiser and bloodsport hadn't abated.
"She'd call home crying, Martin. He still went to the bar weeknights. He'd come home boozy and, well, unpredictable. And he spent the weekends tramping through the woods shooting small game, which he'd bring home for her to clean and cook."
He'd go out at night and snag spawning salmon and bring them back to freeze and smoke and put up in jars.
"Her letters home got so sad, Martin—'He doesn't bathe enough,' she wrote me once, 'he seems so angry.'"
She had taken from her purse a packet of pink envelopes and was holding them and rocking a little in the chair across the desk from Martin.
"She had such beautiful handwriting."
Martin nodded, smiled, understood.
"She called me crying horribly once and I asked her if he'd hit her but she said no, no. He had killed a fawn, right outside their trailer. It had come with its mother to feed at the pile of carrots he baited them with. They were in bed. Sunday morning. He sat up, walked to the window, went to the door where he kept his rifle. It was months before the legal season. He shot it right from the door. The fawn, Martin. The little fawn."
She was shaking now again, sobbing and rocking in the chair.
"Do you know what he told her, when she yelled at him for shooting the fawn?"
Martin shook his head.
"He told her it couldn't live without its mother anyway."
Now she was sobbing and shaking fitfully and Martin reached across the desk to take hold of her hands, in which she held the packet of her daughter's letters.
"We don't have to do this now," Martin told her.
But she wanted to go on, to get it out, to get this part behind her.
AFTER HE killed the baby deer, Elena applied to the state university in Mount Pleasant using the return address at the Sheriff's Office. When the letter came from the admissions department, beginning Dear Ms. Delano: Congratulations! she made a copy and mailed it home with a note asking her mother if there was still money left for her education.
"'Of course,' is what I told her," Elena's mother told Martin. "I wanted her to get her education before she settled down. After she lost the baby, she had no reason to stay with him. And he was drinking and depressed. He worked and drank and grew more distant. She could see she had made a big mistake. I could tell she wasn't happy."
Elena told her mother how she gave her husband back his leather coat and the tiny diamond ring and said she would always care about him but that she had been too young and she felt she owed it to her father to return to school and get her life on track and she would always treasure their time together but she really had to go. She thought it would be the best thing for them both. She was sure he wasn't happy either.
The night before she had planned to leave, she did her hair and polished her nails and cooked him pheasant and they ate by candlelight—"for old times' sake," she had told her mother when she called to say she'd be home tomorrow. She really wanted no hard feelings. It had been her mistake and she was sorry to have involved him in it. Surely they would always be friends.
"He's okay with it. He doesn't like it but he's okay with it," is what she told her mother when her mother asked her how he was taking it.
And, near as the coroner and the sheriff could piece it together, it was after everything she owned had been loaded in the car, the trunk full of books and photo albums, the backseat packed with her stereo and a rack of hanging clothes and the front passenger seat with the one suitcase full of toiletries and socks and underwear; maybe she was turning to wave goodbye before going, or maybe he'd been drinking Budweiser all night, or maybe he'd helped her and then went berserk, but whatever happened, whether it was passion or calculation, before she sat into the driver's seat, he got the rifle from wherever he kept it, near as they could figure by the angle of the wound, he stood on the front porch, aimed, and fired, then walked over to where she lay in the leaf-fall beside the car and shot her again, in the breast.
This was the part that Martin could never imagine—the calculation of shooting her in the leg, then slowly, deliberately walking over and pressing the barrel against her left breast and pulling the trigger. Wouldn't such madness in a man give signs before? Wouldn't the first gunshot wake him from the dream?
Elena's mother was rocking in the chair across from Martin, sobbing quietly, clutching the letters, staring at the snapshot of her daughter on the desk standing next to the man who had just killed her.
"You pick out the casket, Martin. I can't do it. Something like her father's. Please, Martin. You do it."
He used the cherry casket with the moss pink velvet interior, and though it was considerably more costly than what Elena's father was buried in, he charged the same and thought it was the least he could do.
AND NOW, twenty years since, nearing fifty, he could still not shake the sense of shame, that the men in her life had let her down badly. The father who died too young, the husband who murdered her, even the embalmer who could only treat her viscera with cavity fluid, inject her arms and legs and head, stitch the horrible incisions of the postmortem—from left shoulder to breastbone, breastbone to right shoulder, then breastbone to pubic bone—the little bulge in her tummy where the bag full of organs made her look almost expectant, then cover the stitches with cotton and adhesive. And then put a little blush on her cheeks, brush her lipstick on, curl and comb her hair. He had dressed her in the sweater and jeans her mother brought in and lifted her into the casket, put her First Communion rosary in her hand, a crucifix in the head panel of the casket, and put an arm around her mother when she came to look.
"Oh no, no, no," she sobbed, her shoulders rising and falling, her head shaking, her body buckling at the sight of her daughter's dead body. Martin held her at the elbows, whispering, "Let it go, I'm so sorry," because he never could think of the right thing to say.
Over time Martin learned to live with the helplessness and the sadness and the shame. He quit trying to figure the right thing to say. He listened. He stayed.
Still, all these years since, whenever the right shade of red turned up, he could see the fat old pathologist and his cigar and stupid tutorial manner there in the morgue with its cold smell of disaster and formalin, and the hearse that he drove up to get her that October. And the way they lay in coolers in the corner of the room, the two bodies in trays beside one another—Elena and the son-of-a-bitch that shot her.
He had shot himself, after killing her. He walked back in the house, sat on the edge of the bed, and taking the muzzle of the rifle in his teeth, pulled the trigger with his thumb, dividing his face at the septum in the process.
"Isn't that always the way?" the old pathologist had said, yanking the tray out with Elena's body on it. "It's lovesickness. A man kills his wife, then kills himself. A woman kills her man, then does her nails."
Martin hated those sentences and couldn't forget them. That they rang true sometimes and false at others had never been a comfort.
EVENTUALLY, AFTER the wake and Mass, her body was buried beside her father's, leaving another grave on the other side for her mother. It was all Martin could do—to get her where she was supposed to be. Her mother had a stone cut that read Beloved Daughter with a rose between her dates and another with her own name on it and her year of birth and a dash and had it placed over the open grave beside her husband. She moved away some few years after that. Martin never heard from her again.
## Hunter's Moon
SOME DAYS on his walk Harold Keehn thought about his wives. Some days it was caskets. Others it was the heartbreaking beauty of the natural world such as he had come to know it. Often as not the consolidation of these topics was seamless and the names and particulars would race through his brain like a litany in code that only he could decipher. Elizabeth, goldfinch, Primrose Maple, hemlock, Helen, Mandarin Bronze, osprey, glacier, 18-gauge Perma-Seal, Autumn Oak, chickadee, trillium, Joan. The list always ended with Joan, his third wife, whom he'd buried last April in a Clarksville Princess Mahogany with a tufted dusty rose velvet interior, in Mullett Lake Cemetery between two blue spruce saplings he'd planted there. The naming gave him a sense of mastery, as if he'd had some say in all of it.
When she had died in early January her body was kept in the cemetery's stone winter vault, waiting for the frost's hold on the ground to give way in the spring and the grave to be opened.
"We don't dig much after the deer season opens," is what Harley Flick, the local sexton, told him, when the graves were arranged for last November, when Harold knew the end was near.
When the racing of names got out of hand Harold would stare intently at the path in front of him, count the cadence of his footfall or breathing and pray for his mind to go blank and hush. Then he could hear the air in the leaves, the lapping of water, the brisk movement of wildlife in the undergrowth. He could imagine the larval stages of next year's hatch of dragonflies and hexagenia, caddis and stoneflies, the imperceptible growth of antlers and turtle shells, the long pilgrimage of hatchling and fingerling, the return of the grayling and wolverine. He would try and sense his body's oneness with the pace and nature of the world around him. Better not to think too much, he often thought.
He thought it unlikely he'd ever marry again.
HE DID three or four miles a day, along the abandoned railroad bed through the woods, between his place on the southwest corner of the lake and the village to the north; or south along the west edge of the river mouth, circling the wetlands, where carp spawned in late May and early June, under the interstate and up to the highway, then back again. Some days he'd do more if the weather was fine and his knees didn't ache, or the sciatica hadn't hobbled him, and he was glad for the time out of the house where he found the days, though shortening now, impossibly long. On the best of days he could imagine himself walking all the way to Cheboygan, on out the Straits Highway at the north end of town, along the edge of the big water to Mackinaw City, over the bridge to the Upper Peninsula and into whatever oblivion God had in mind for him. Maybe to Munising or Seney or Manistique—he loved the sound of northern names. And the names of tribes that had named those places: Algonquin and Huron and Chippewa. Or walking south all the way down the mitten of Michigan along the old railway lines through Gaylord and Grayling, Saginaw and Bay City, all the way to Rochester where the tracks passed alongside the house he'd lived in years ago with his first wife before the names of things made much difference to him.
Time occupied for him a kind of geography, the north of which he thought of as the future and the south of which he thought of as the past and where he was at any given moment was the immediate present tense of his personal history, the known point on the map of what he'd call his life and times. It kept him from feeling entirely lost. Some days the future was west and the past east and the moment was shooting craps out in Vegas or some other fantasy, but it always suited him best to think of the whole miserable business as linear. The prospect of time bringing him back around to the point he set forth from was a crueler joke than he could imagine, though the faces of clocks, the evidence of the sun and moon, the repetition of themes in his own life were, of course, disquieting. Today is a gift, the sign outside the Topinabee Church read this morning when he'd gone for his oats at the Noka Café. That's why they call it the present! Better than last week's bromide, Harold thought: Fresh spirits have no expiration date!
Harold stood on his porch, stretching both arms to reach the ceiling, then he dropped his sweatpants and pissed in the general direction of the neighbors' place. Everyone was gone this time of year, back to their jobs and schools and schedules. There were some color tourists and weekenders, but mostly he had the place to himself. He spread his legs, bent at the waist, touched his palms to the ground, feeling the back of his thighs stretch, having to bend his knees ever so slightly now. Then he stood up straight and stretched his arms up over his head again, easing the standing pain in his lower back and right buttock. Then he hitched up his pants, did a couple of slow squats to loosen his knees and side-to-sides to ease the tightness in his groin, and stepped off the porch, pursing his lips to suck in the air. The decadent smell of leaf-fall, the crunch of his footfall in the road's top gravel, the sparkling light of the advancing afternoon, the sweet crispness in the cooling air, the sore pads of his feet, the ringing in his ears—these were all familiar.
If he was hungry after his walk, he told himself, he'd drive into Topinabee for a slab of whitefish or a burger at the Noka. That might kill the time left until nightfall. Once it was dark he could fall asleep watching some cable news or old reruns. Always good to have a plan.
He could hear the dog barking in the distance—Larry Ordway's bat-faced mongrel bitch—frenzied and barking at God knows what. Harold looked along the roadside for a proper stick.
IF HE'D remained married to Elizabeth, today would be their anniversary. Was it forty years yet? He'd lost count. October 29—the day the stock market crashed and the Great Depression got under way. The day, he'd heard on the radio this morning, the National Organization for Women was founded, the day he was married for the first time, that late October during Vietnam. That figures, he thought, thinking how Elizabeth had left him broke, depressed, vanquished, and confused about women, suffering a kind of post-traumatic stress.
She had left him for a woman.
It all seemed a bit of a blur to him now, and feeling the nerve ends in his right leg warm to the pace he was keeping up Grace Beach Road, he was glad for nature's forgetfulness, how the pain in his ass could be dulled some and the numbness in his legs could be walked out. The afternoon light angling through the woods, the blue sky, the bird noise in the trees, the air rushing in and out of him: life as he knew it, here in the moment, in the gift of the present such as it was, was nothing but a walk in the woods of northern lower Michigan, in mid-autumn.
They'd had a ranch house on three acres in one of the best suburbs north of Detroit. He was a sales rep for Clarksville Casket. All of Michigan—over four hundred funeral homes—was his territory. They had a daughter, Angela, a dog, Maggie, a rosy future. And even if he'd married Elizabeth because he thought marriage was sensible and inevitable, and because he figured as well her as any other; even if he found her, while very attractive, not entirely admirable, even if she had married him to get out of her crazy mother's house, even if they both woke some mornings wondering if they each might have done better for themselves, they had assembled a life. If he had not loved her completely, utterly, irretrievably, he thought then and he thought now, coming to the intersection of Grace Beach and Grandview Beach Roads, he had loved her sufficiently.
Larry Ordway's dog was in full fury now, the sharp blasts of its barking amplified by the general silence in the rest of the world through which Harold's approaching footfall in the gravel was all the more discernible. Harold's grip on the stick tightened in anticipation of the dog's charge down the driveway in real or feigned attack. One never knew what to expect of the bitch. He wanted to be ready for all contingencies. From half a mile off, the dog's distemper sounded menacing. Maybe a raccoon trapped, or skunk or deer, or some late-season cottager going by on a bike or on foot. The dog was a pest—another in a line of disagreeable mixed breeds that had guarded Ordway's empire over the years. It was an empire of sheds and outbuildings, scrap vehicles and rusting implements surrounding his double-wide in the woods at the side of the abandoned railroad easement. The current mutt kept sentry at the top of a long drive that gave on to the road where it curved to cross the tracks. It would come snarling and barking down the drive, chasing off everything that came into its view.
ANGELA, THEIR daughter, was lovely and bright; their lives seemed full of possibilities. They had a manageable mortgage, good credit, good friends, made love twice a week, belonged to the local Congregational church where Elizabeth sang in the choir and was known for a casserole she brought to funeral luncheons and potlucks. Harold ushered for Sunday services. They were the happy young couple with the pretty child.
When Elizabeth turned thirty she went back to school to finish the degree she'd abandoned when she got pregnant. Angela went to day care. Elizabeth commuted to the university and took classes in English and Women's Studies.
Harold was gone a part of most weeks working his way up and around the state, calling on northern and western accounts. Other times he worked Detroit and the suburbs. He'd go as far west as Lansing, as far north as Saginaw, and still be home in time for dinner. He'd check the death notices in the Sunday papers to try to get a sense of who'd be in their offices and when. He'd try to see his best accounts every other month, others once a season, others twice a year, some just at convention and some he'd call or send a card to now and then. Some bought better over lunch, others after a few drinks, some over coffee in their offices. Harold had learned to cultivate his relationships with the primary buyers—most often the owner or the owner's son. He'd listen to whatever he had to listen to—their theories on why one unit sold and another didn't, their bad-mouthing of the competition, worries over the cremation trend, stories of the latest strange cases: double suicides, remarkably obese cases, multiple fatalities at industrial sites or on the interstates, anything. One week he'd work the city among the ethnic firms—Poles and Romanians, blacks and Jews—then the cushy suburbs of Grosse Point and Bloomfield Hills, up through Pontiac to Flint. Another week he'd work the firms in tri-cities and all the small farm towns in between, spending as much time with the Woolevers in Midland and Cases in Saginaw and Penziens and Stapishes in Bay City with their multiple rooftops and hundreds of calls as the Struthers firm in Reese who did forty funerals a year, but all of them copper or bronze or premium hardwood, paid for in cash by old German farmers. Then he'd take a run out through Ypsilanti and Ann Arbor and Jackson along I-94 to Kalamazoo and Muskegon, then up to Grand Rapids and up the west side of the state through the rich resort towns, Traverse City and Charlevoix, Petoskey and Harbor Springs. Once in the spring and once in the fall he'd try to make it through the UP. He'd buy drinks for his accounts at their district meetings and their yearly golf outings and pop for lunches and dinners with his best accounts. He loved the long hours alone in the car and the vacant landscape and the open roads. He'd been through the CB radio craze—his handle was "Boxman"—and car phones and cell phones, all the gadgets. The drive along Route 2 to the west, then north to Seney, then up to Munising, then west to Marquette was a favorite drive. His accounts up there ordered caskets by the truckload and were accustomed to infrequent deliveries. They'd back up their best units in basements and garages and keep six months' worth of inventory on hand and borrow from their colleagues in the next county if they ran short of a particular unit. And the drive along the east side up the Lake Huron shore from Pinconning and Standish where he'd always lunch at Wheelers for the way it hadn't changed over the years, still serving malted milkshakes in big silver tins and burgers with fried onions and real French fries. Then through Au Gres and Oscoda, Greenbush and Harrisville, all the way up to Alpena and Rogers City along the long blue edge of the state. He'd listen to radio preachers or farm stations that gave the price of sugar beets and alfalfa. Or Paul Harvey or Rush Limbaugh or public radio—it hardly mattered. He called on every firm in every town, promising each to keep their "line" of Clarksvilles "exclusive" to prevent comparison shopping between competing firms. If one bought a Tuscany 20-gauge, with lilac crepe for little old ladies, he'd sell the other firm a Silver Rose with pink velvet. If one took the Pietà or Last Supper, the other was pitched the Praying Hands or Old Rugged Cross. He kept sales charts on them all and pushed them to beat last year's averages, convincing them that the more they spent on caskets, the greater return they'd eventually realize on their "investments." He left stacks of notepads, pens, and packets of breath mints, each with Clarksville's logo and his contact particulars imprinted. He gave his best accounts custom-made coffee mugs and playing cards with their firm's name embossed next to Clarksville & Keehn—A Winning Team!
Elizabeth hadn't exactly left him. She'd put him out. She kept the house, their daughter, the newer of their two cars, and showed him the door.
"You're welcome to stay if you want to," she told him, "but I'm sleeping from now on with Eleanor."
It happened so fast it was a blur to him now. He'd gone from the more or less happy paterfamilias to a man living at loose ends. They'd been married twelve years and it was over in months. Or maybe he was only the last one to know. Either way, he found himself paying the mortgage on a house he no longer lived in, payments on a car he no longer drove, and support for a daughter, now ten years old, he saw all too rarely. That he was paying alimony to a woman with whom he no longer slept vexed him especially at the time. His consortium had been replaced by Dr. Eleanor Dillingham, who taught a course in American Women Poets at the community college.
"Poetry," she had been quoted as saying, "like suicide, is something more women attempt and more men accomplish. It is time we changed all that!" The required reading for the course was a book about a madwoman who lived in someone's attic and began, "Is a pen a metaphorical penis?"
The thought of Dr. Dillingham making love to Elizabeth both excited and disturbed Harold. He could understand their attraction to each other and wondered at the time why they couldn't include him. He found Eleanor oddly fetching himself. He moved out the first weekend in December that year.
One of his accounts gave him the use of rooms over the funeral home garage where he could put a phone and sleep when he wasn't on the road. He took night calls and helped with the removals in trade. A grieving family gathered around the deathbed of a loved and lamented head of the household sent twinges of regret and anger through Harold, but the face he showed to mourners was commiseration. With no particular home to return to, his trips got longer and longer. His sales began to increase as he spent more and more time on the road. Money began to roll in again. He opened an account for his daughter's college fund. He wanted her to go to Alma College, a Presbyterian school in the middle of the state, where he called on the Dewey Funeral Home. The male students there all looked like Young Republicans. Or Albion, where there were mostly Methodists and they didn't teach Women's Studies. Or maybe Calvin College, where modest Dutch Reform girls and boys were monitored for proper conduct.
Come Fridays he'd stay on wherever he was. He'd take a room in Grand Rapids or Ishpeming or Port Huron. He'd coax his last account out for a boozy dinner and offer to help out with the Saturday funerals and sit in the motel watching TV or in the lounge listening to the terrible music and hoping to get lucky with one of the local women.
He bought the house on the lake the year after the divorce. He'd heard about it from the account in Indian River who had buried the widow who had lived in it for years. It was old and a little ramshackle but it had a good foundation and two hundred feet of frontage and could be bought right from the old woman's children, who lived in Ohio.
Harold saw in it a refuge from the rootlessness he felt—a place to bring his daughter for those summer vacations and winter breaks his visitation rights entitled him to. He envisioned her long into the rosy future, learning to fish and ski and name the species of things with him, then perhaps bringing a boyfriend up for the weekend. He saw her wedding on the lawn under a great tent on a blue day in July or August, and his eventual grandchildren gathered round the barbecue for family reunions, playing badminton or horseshoes, or paddling canoes out on the sparkling water, all housed in the dormer he would in due course build over the big garage—everyone together, all distance and divisions healed. He saw, however dimly, all of them circling his last illness, or bearing him to some proper disposition, then returning to the family home on the lake to toast his life and times and memory—a man, they would say, who had played the cards life dealt him, and played them well, and would be sorely missed.
Quality time, he told himself; less is more, if better spent.
It had a stone fireplace in the main room and a good well and a wide front porch and a screened porch off the front bedroom upstairs from which he could watch the lake and the stars and the northern lights. The sunrise poured into the windows; he slept to the lapping of lake water. He compensated for fewer visits downstate by inviting his accounts up for a weekend's fishing or snowmobiling or hunting or cards. He'd take them to the Breakers Bar in Topinabee, get them a little liquored up, buy them a steak, pitch the latest additions to the line, and take their orders. He'd write off the expenses on his tax returns. His market share kept getting better and better. His up-line in Indiana were all full of praise. There was talk of a job at headquarters, maybe something salaried in marketing.
The business was all about "protection" then. Just like diamonds and deodorants, tampons and defense budgets, caskets were sold on the Cold War notion that they could be "sealed" and "safe" and would "protect" the body more or less "forever" from leakage or embarrassment or unforeseen dangers. It was the marketing theme. Harold would spend hours extolling the virtues of Clarksville's gum-rubber gaskets, "cathodic protection"—a magnesium bar than ran down the bottom of their heavier-gauge steels which was said to retard rust—and the 48-ounce Solid Copper Omega which "grew stronger through age by oxidation." "Precious metals," he called the bronze and the copper units. "Permanent protection." The more product knowledge his accounts had, the more product they'd buy and the more they'd sell and the better he'd do.
The second summer after he and Elizabeth divorced, Angela came up to the lake and stayed with him for a couple of weeks. She was just on the brink of becoming a teen. Harold could sense the changes coming. He resolved to have her bedroom redecorated when she'd come up next year "for the whole summer." But after he married Helen, she first demurred—a conflict with summer volleyball—and then refused. Angela said that her mother said that her father was an incurable heterosexual, bound to dominate women, a member of the patriarchy, a serial rapist, and a hopeless case. Angela told him all of this in a letter Harold figured her mother typed. She was turning thirteen. Elizabeth's anger at his remarriage was something he never understood. That she passed it along to Angela was even more inexplicable. Harold called often but Angela was always "out." He sent flowers for her birthday. He sent her cards and letters she never responded to. It hurt him, but he blamed it on her mother. He tried to stay in touch but figured he was best off leaving well enough alone. She had enough to deal with with all the changes in her little life. He didn't want to pull her into a conflict. He thought the day would come when she was older, that she would understand her father better and they could reestablish a relationship. It never happened.
When she was sixteen, Angela was hit by a train. She'd been walking the tracks that ran between Main Street and the Mill Pond early on a Sunday morning. The autopsy showed that she'd been drinking and was pregnant. And though the death was ruled accidental, Harold wondered if she might have thrown herself in front of the train, and if so which of her parents should be blamed the most. It also occurred to him that maybe the man who had impregnated her pushed her in front of the train. He wondered if it was a man or a boy. Had she been looking for another father figure in her life or just playing around with one of the pimply jerk-offs she'd grown up with. Knowing the cause of death while not knowing the cause of the cause of death sent Harold into a spiral of such steep descent that he resolved to put it all out of his mind, void as that darkness was of any place names he could recognize; he feared the point of no return.
Elizabeth arranged to have Angela's body cremated, to which Harold agreed, but he insisted on putting her in a Melrose Cherry instead of the cardboard casket they always used when there wasn't going to be a public viewing. "A shame to burn such a beautiful piece of wood," Elizabeth had said, when they went to the funeral home to identify their daughter's body. She rubbed her hand along the deeply polished finish, averting her eyes from the body in the box. He wanted to hit her. He wanted to put his fist through her face and shut her mouth forever. But he only nodded and thought, Yes, a shame.
Angela's face was unmarked, all the injuries apparently "internal." He and Elizabeth divided the ashes. Each got a little cherub-shaped urn with half their daughter's remains inside. He never knew what his former spouse had done with her half of their daughter. He never asked. He hadn't spoken to Elizabeth since.
HAROLD DIDN'T know Larry Ordway, or even if Larry was his name. It might have been Lenny or Louie or Lester. All the sign said was L. Ordway Private, scrawled on a white board in blue paint and nailed to a tree stump at the end of the drive. Harold couldn't remember when he settled on Larry or why exactly, or how he'd come to several conclusions about a man he had never met. There was the cross that went up in the woods one year, painted white and gold and ten feet tall and wired for lights, big bulbs, that shone through the dark eerily—a sign of the born-again Christian fundamentalist, Michigan militia type of head case he reckoned Larry Ordway must be. And there were the mean-spirited dogs always challenging him at the bend in the road, barking, baring their teeth, the current one now five or six years into its miserable life, looking like the hound of hell with its frothing muzzle and pointed ears. Harold had feared it and picked up his pace to get past its purview and kept an eye out even when he was out of range. He had even taken long detours through the woods to keep out of the dog's way, his fear getting the best of him until once, the week after Joan had been diagnosed, Harold had waited for the dog, growled back at it as it hunched and snarled, and taunted it with waving arms and timed it perfectly so that when the cur lunged within kicking range, he caught it squarely in its yapping mouth, a perfect punt, flipping it on its backside and sending it yelping back up the drive. It hadn't really challenged Harold since. Its barking was vicious but still it kept its cowering distance. Harold kept a stick handy, just in case, half hoping it would give him another go. Sometimes now, after he'd gotten past Ordway's, he'd hear movement in the dense woods on either side of the railway easement, aside and behind him, and wondered if the dog was following along, waiting to pounce or wanting to make friendly. Harold didn't know but didn't trust the thing. He'd turn sometimes and look behind him. Once or twice he thought he caught a glimpse of the bitch crossing the tracks in a blur, maybe stalking him, waiting for its chance to settle the score.
At Hobo Beach, where the trail ran nearest the lake, Harold stopped to sit and watch the water for one of the eagles that nested nearby or the osprey that nested on a platform placed in the river mouth by the DNR, or anything else that might happen. The rapturous descent of fishing birds, the haunted call of loons, the hovering of kingfishers, the uncommon beauty of common mergansers—these incarnations now remembered, late in the day, late in the year, made him feel a fortunate pilgrim indeed. Above the treeline on the far side of the lake the fat face of the full moon was emerging. Harvest moon, he thought. No, hunter's moon, then beaver moon, then cold. He ran through the names of moons such as he knew them. There would be moon shadows tonight and light on the water.
All the docks and boat hoists were stacked on the beach, waiting for the coming winter's freeze and deep snows and next spring's thaw—worm moon, he thought in March, pink moon April, May, the flower moon—before being hauled out and reassembled in the water for the high season. The boaters and sunbathers all gone south and the snowmobiles not yet thinking about coming north, the off-season vacancy of late October struck Harold as the best of all times of year.
This golden harvest aspect of it all, a feast for all souls, the sense of finished work and jobs done well—"Autumn Oak" is what he named the clear-finished, hand-polished number with the fallen-leaf appliqué stitched into the cap panel and the tailored beige linen interior when he first beheld it, and his bosses at Clarksville had agreed with him. They even designed an urn to go with it—same wood, same finish, same falling leaves machine-etched into it—a package deal for the cremation crowd. "The Autumn Oak Ensemble," they called it in the catalogue, and sold them by the truckload all over the place—the most popular of Clarksville's hardwood line.
HE SHOULD have been better to Helen. She had never been anything but good to him. Maybe if he'd been more trusting, less damaged goods, not so angry. Maybe if Angela had lived. He didn't know. He often wished he could do that over again, make it up to her.
They'd met at the convention in Grand Rapids. Her booth for Barber Music Systems was next to Clarksville's on the exhibit floor. By the last day of the convention he worked up the nerve to ask her to dinner. She was younger, plain-faced, smarter and more pleasant than anyone he'd been out with in years. Her father had started the business, which sold background music systems for mortuaries—hours of hymns or new age music to "Break the Terrible Silence of Grief." Under Helen's guidance the company was marketing video memorial tributes.
They dated for six months, then got engaged, then got married. They honeymooned at Casa de Campo in the Dominican Republic. He golfed, she sunbathed. It was good but brief. They divorced soon after Angela's death. Helen had the marriage annulled and was soon enough remarried to a man from Forethought Preneed who didn't drink as Harold had begun to and didn't sleep around, the way Harold did in the months following his daughter's death. Helen wasn't bitter. She just wanted out. She wished him well, but wouldn't hang around for his "self-destruction."
He saw her at conventions after that. She'd always smile sweetly and keep her distance. He'd readied the little speech to make amends and ask forgiveness, to say it had all been his fault and bad timing and the drink, of course, but when he approached her in her booth the year before he retired, her eyes looked panicky, she put one hand over her mouth and the other out straight as if to warn him off. "I'm sorry," he said, "I just wanted to say..." but she looked frightened and her eyes were filling with tears and she was backing away as if from some peril or contagion, so he stopped and turned away and said nothing. He thought maybe he'd write it all in a letter, to let her know he knew how bad he'd been to her. But he never got around to it and thought now, sitting by the lake, watching the moon rise, that he likely never would.
He stood and sighed heavily and stretched and, dropping his pants, pissed in the sand and, after hitching up again, walked back from the water's edge to the trail along the railway bed he'd been on. It was here he'd seen an indigo bunting once, perched on the head of a cattail by the water, a blueness he had never seen before or since. He felt a chill in the evening air. Looking down the long tunnel between the trees, Harold watched for deer crossing, or porcupine, or beaver. He looked back the way he'd come for signs of Larry Ordway's dog. Nothing moved. Once years back he had seen what he swore was a black bear pausing in the clearing with its nose in the air. He knew that the woods held red fox and gray wolf but he had never seen them. Signs of life in the deep interior, seen and unseen, quickened in him the kind of gladness he remembered having as a boy. The emergency flight of pheasant and wild turkey, the passing shadows of osprey and vulture and gull, the trees fed on by nuthatch and pileated woodpecker, felled by beaver, scraped by whitetail bucks, mauled he imagined by bear and raccoon—these apparitions of the world's natural order were a comfort to him for reasons he could not articulate.
It was Harold Keehn who had convinced Clarksville to market wooden caskets, the rich grains, the homey cabinetry warmth of it all. And Harold who came up with the idea of planting trees, saplings only, pennies only by way of expense, in cahoots with the forestry department, for every Clarksville casket sold. The Memory Tree Program had been a huge success. It assuaged the baby boomers' natural concerns about ecology and conservation and renewable resources. He'd pitched the whole market shift at a sales conference in a presentation he called "Don't Let Your Business Go Up in Smoke" in which he noted the growing popularity of cremation and the natural consumer preference for boxes that would burn. "Permanence" and "Protection" had given way to "Natural Beauty" and "Sensible Choices." A woman's right to choose, thought Harold, applied to the recently widowed as well as the recently impregnated, noting the coincident rise in the abortion and cremation rates. "If you can't beat 'em, join 'em," he told his bosses at Clarksville headquarters. "Give them plenty of choices." They called their cremation catalogue Options by Clarksville and filled it with urns and cremation caskets.
Had it not been for the drinking and carousing that everyone knew about but no one mentioned, he might have been given a vice presidency, stock options, an office at headquarters in Indiana. As it was they thanked him for his insights and cut his territory. He didn't care. Inflation kept his commissions high. He had nothing to save for. His dead daughter's college fund went into a couple of mutual funds and swelled during the 1990s. He had more than he'd ever need. His place was paid off. After Angela had died, after Helen left, very little mattered until he met Joan at an AA meeting in a church basement in Topinabee.
She had looked to be about the age Angela would have been if she had lived. She always listened to Harold carefully when he talked at the weekly meetings. She nodded and smiled when it came his turn to tell how he was powerless over alcohol and his life had become unmanageable or how he'd come to believe that a power greater than himself could restore him to sanity or the particulars of his searching and fearless moral inventory. She would nod and smile and at the end she would squeeze his hand after the Our Father or the recitation of the Serenity Prayer and give him a hug and tell him, "Easy does it, Harold." She was so happy, it seemed, so very happy. And whatever calamity or sadness brought her to AA, details of which she shared frankly on occasion, she seemed to inhabit a permanent present tense, free of the past and future, afloat on the moment she occupied. She was pretty and had a graceful body and eyes like the blue of the indigo bunting he had only seen once. When Harold asked her if he could ask her out to dinner, she said she'd rather bring him dinner at home. It was a chicken and rice casserole, and pecan pie for dessert. She stayed that night and the night after that and on the weekend they moved her out of the rooming house in Cheboygan, her entire estate fitting easily in the trunks of their two cars. There had never been any talk of marriage. They were companions and occasional lovers, generous with each other in ways that were new to Harold. He took her walleye fishing and built fires in the fall and winter. She read to him in bed and cooked him breakfast. He took her on sunset cruises along the lakeshore in an old wooden inboard he bought for such occasions, savoring the changing light and night skies and the silence that would sometimes settle between them. She quit her job at the marina where she did payables and receivables. He did most of his business by phone and fax. They went to dinner in Petoskey and Mackinaw City and Indian River, movies in Gaylord and Cheboygan. She abided his long walks, his long silences, his darker moods. Whenever she touched him, or talked to him, or looked at him, Harold felt alive. And though he never could figure out why she came and stayed—she was twenty-three years younger and might have had a more exciting life—his gratitude was manifest and he treated her accordingly.
He bought a small RV and they would leave just before Memorial Day, driving around the country on no particular schedule, returning after Labor Day when all the summer neighbors had returned to their downstate lives. He kept a list of the place names they had been to. She kept albums full of photos, each of them posing for the other, in front of some diner or park entrance or stop in the road. They had lived together there on the lake for over ten years. The best years of his life, he would always say. The best of hers, she would say in return. Only when her death seemed certain had they agreed to marry so that Harold could be her next of kin. Her family in Lansing was long estranged. None came during her sickness or after she died, though Harold made the requisite phone calls. Whatever happened between her and her family happened years ago, the detachment having achieved a point, apparently, of no return.
Joan's cancer took a year and a half from first diagnosis, to the surgery to remove a lung, through the radiation and chemo and eventual reoccurrence—the "irregularity," they called it when it showed back up—to the morning last winter when, after an awful seizure, because it had grown into her brain, she died with Harold sitting helplessly by.
After the burial he'd ordered a stone with her name and dates on it—Joan Winters Keehn—but he'd never seen it, though he passed the cemetery often enough: he never thought of her as there. But some nights over the past months, he'd go out in the motor home and sit at the table where they'd play gin rummy nights on the road; or he'd crawl into the bed where they had slept in their summer travels, pressed to one another, his right hand cupping her small left breast in their genial embrace. Some nights alone out there in the RV in the driveway, he'd wonder if it was time to take up drink again. So far he hadn't on the advice that Joan herself had always given out, that there was no sadness that couldn't be made more miserable by the addition of a Class A depressant. Still, the brand names of whiskeys were beginning to make their way onto the lists of names he kept—Jameson's, Bushmills, Powers, and Paddy—with the names of birds and the names of caskets, the names of moons and towns and tribes and names of his lost wives.
Harold Keehn could imagine Adam in the garden, that first index finger working overtime, assigning to every new thing he saw, fresh, orderly syllables—aardvark, apple, elephant, waterfall—as if to name it was to know it or own it or anyway to have, if not dominion over it, some consortium with it. He wondered how it must have been when that first man first whispered "Eve" and the woman turned to look into his eyes.
When Harold found himself at the south edge of Topinabee, the hum of the highway coming through the woods on his left and the moon on the water on his right, he knew he'd walked too far. How had he lost track of it all? And turning back to go the way he'd come, he wondered if there would be enough light left in the day for the way home. Even at his best pace it would take him nearly an hour. Suddenly he was aware of his body and its pains and aches. His knee was grinding, his feet aflame, the small of his back full of crippling twinges. He was fatigued. The air was getting colder now and the wind off the lake increasing. He resolved to keep, in spite of everything, a steady pace.
He'd quit the casket trade at the right time, Harold thought. It was no longer the permanence and protection of the metal ones, or the warmth and natural attractions of the woods. Now it was all gimmicks and knickknacks. Interchangeable corner hardware—tackle boxes for fishermen, plastic mum plants for gardeners, little faux carrots and kitchen utensils for women who cooked, all molded in plastic—How silly, he thought. And "memory drawers"—the little box-within-the-box to put farewell notes and mementos in—smarmy malarkey thought up by "focus groups" and test markets. Back in his day it was the salesman on his rounds that came back with the best ideas. What the public wanted in a casket, Harold had told the honchos at home office, was a way to "get a handle" on it all—a death in the family—the once-in-a-lifetime aspect of it all. Trouble was it was the ultimate one-to-a-customer deal. And hard enough to get folks enthused about even the one.
Now he was aware of the angling lights that lit the way before him. The golden rays of evening washing through the trees on his right and the silver of the moonrise over the lake on his left illumined the track of railway bed before him. It took his breath away, the beauty of it. His chest was heavy. He sighed.
When Joan's Princess Mahogany was moved last April, from the stone winter vault to the freshly opened grave, the seams of the boards in the casket lid were splitting where the epoxy had dried in the cold interior of the holding tomb. Condensation, desiccation, extremes in freeze and thaw: Even the best of boxes will eventually rot, he thought. Everything in nature disappears. Harley Flick let Harold bury his daughter's ashes, the half that he had kept in the house these many years, in the room she never came to stay in, in the same new grave as Joan was buried in. He poured his daughter's ashes over his third wife's casket where they filled in the open seams in the lid. Then he borrowed Harley's shovel and filled half the muddy grave himself before Harley finished the job with his John Deere backhoe.
When Harold turned off the rail easement by Larry Ordway's cross, the dog lay dozing in the road at the bottom of the drive and hardly budged when Harold walked by. He'd let his stick go miles back. Turning down Grace Beach Road, on the last leg of the journey, he looked back and saw the dark shape of the dog behind him. It was not barking or bothered or giving chase, just following him at Harold's own pace, silhouetted by the last light of the sun behind it. He was aware of his heart racing and his breath laboring and the general ache of his body sharpening and the fatigue of the long walk overtaking him. If the dog attacked he could not fend him off. But the dog did not attack, only followed Harold home, footsore, winded, aching, spent.
Harold slumped on the bottom step of his front porch, watching the last light pour out of the day and the moonlight widening over the flat surface of water and the darkness tightening all around him. He avoided the impulse to name some stars that appeared in the firmament, or to name some fish swimming unseen in the dark waters, or whatever living things moved in the woods. He wouldn't be going to Topinabee. He didn't want a drink. He wouldn't build a fire tonight. In his flesh he felt entirely quenched. It was enough to let his vision blur watching the water and the moon and to find Larry Ordway's dog, if Larry was Ordway's name at all, curled beside him free of menace, watching nothing happen, thinking nothing of substance, void of memory or purpose or expectation. Neither the names of breeds nor the names of dogs nor the names of their owners troubled him anymore. The dog kept watch all night and did not howl at the rising or the falling moon.
## Matinée de Septembre
AISLING PREFERRED FRENCH press coffee, pinhead oat meal, cymbidium orchids, and Mahler adagiettos. She loved the tiny courtesies—the door held open, handwritten thank-you notes, engraved invitations, a man who rose when she left the table—a thing Nigel had always done. She found an attention to detail assuring: the pilots' crisp epaulets, the fashionable scarves of the stewardesses. The way Delibes' Flower Duet was played during the boarding process, to calm the possibly anxious passengers, the little packet with the toothpaste and toothbrush and blindfold and booties, the blanket wrapped in plastic, the headphones, the manifest efficiency of the cabin crew dispensing preflight orange juices, their starched white cotton shirts and blouses. British Air seemed, like all things British, more civilized somehow than the American carriers. Aisling wondered if her sense of it was defensible.
She settled into seat 51H of the nethermost World Traveler section of the plane with the certain knowledge that even here, in steerage, she would be treated with the same dignity as the balding men and their trophy wives, already on their second cocktails in first class with their personal TVs and cushy quarters.
Bigger seats for bigger asses, is what she thought, a little startled by her sudden hostility but pleased to think that they had to pay so much more for the same time in the air, the same travel, more or less the same amenities. Oh, they might get better movies, real silverware, more legroom, and, of course, first on, first off boarding and deplaning privileges. But at the end of the day they would still be asses—big, fat, balding asses whose wives only traveled with them for the shopping ops, the change of scenery, and the chance of meeting someone really interesting. She rubbed her right temple where a headache was forming. Aisling could identify in herself more sudden shifts in temper lately and a low-grade, ever-present contempt for white males past a certain age—older men with money and position, confidence and self possession. Was it age or the early onset of something? Perhaps a product of her education—postfeminist, postmodern. That she had been briefly married to a man of this description; that she was the daughter of just such a man—these were among the exceptions that proved the rule. No need for the perfect to upbraid the good, she told herself. Even if asses, men could be of use.
Aisling was privately pleased with the size of her own ass, its shape and contour, tiny really for a woman at forty and still very firm. An occasionally vegetarian youth, a whole-foods adulthood, the eschewing of red meats in favor of fish, the odd bit of free-range chicken, no sugars or breads or potatoes, plenty of greens and roughage, brown rice, and regular exercise—jogging as a girl, long walks as a woman—neither the sedentary habits of an academic life, the shape-shifting perils of pregnancy, nor the occasional binges of chocolate and cheeses or some finer wine; and yes, she would have to confess, every now and then a Big Mac and fries, just for the decadence of it. No conspiracy of age or maternity or indulgence had added more than a dozen pounds in two dozen years to the body she had as a girl of sixteen. "A small package," is what Nigel had called her during their courtship, when he couldn't get enough of her. "A small package in a large world," he would whisper in the voice he had wooed her in. There had been a lovely imbalance to their lovemaking. He was always so grateful, so full of praise, naming her specific parts and his reverence for them. It was his age. They'd married when she wasn't yet thirty and he was just gone sixty. She'd been his student, then his assistant, then his lover, then his wife, then his widow. Now she was his bibliographer and minder of his reputation. She had his letters and notebooks, his unpublished poems and variorums. And, though her thick black hair was graying, though her dark brown eyes appeared too often tired, and she read with bifocals now, she remained a woman with good skin, a girlish figure in which she took, seeing so many women of her age gone bulky, a secret pride and quiet pleasure. Age was irreversible. But if she was not voluptuous as a girl, she neither sagged as a woman. If her breasts were small—"fried eggs," her father called them, inappropriately, in her teens—they defied now the pendulous gravity of larger, fleshier, fatter bodies. She had, despite the baby and her age, the bosom of a woman half her age. She looked good in no bra or a Wonderbra, pantsuits or little black dresses, vintage lingerie or plaid pajamas. And she knew men noticed her, and that her figure quickened in them, if not desire, then admiration, even appreciation.
She sat up straight in 51H, tucked the pillow and the blanket on one side of her, the canvas bag with her American publisher's logo on it, full of magazines, hand cream, bottled water, and travel essentials on the other side of her. She still had plenty of room for comfort—a small package, flying steerage from London to Detroit on British Airways Flight 202 on the last leg of six weeks of literary duties—a professor and poet of note, a person of substance and discernment, trusted and tenured, who though seated in the back of the plane was nonetheless flying on someone else's dime. She prayed for a vacant spot between her and the red-haired woman, twenty years her senior, applying her makeup in the window seat.
She could have flown as easily in the front as in the back. Money was never the issue with her. She was—though few of her colleagues or students knew it—the daughter of one of Detroit's first families. Her father had made a fortune in auto parts and owned a share of one of the professional sports teams. He'd established a trust fund in Aisling's name before she'd started grade school. She'd had access to it since she was twenty-one. She could have drawn on those resources, or on the not-inconsiderable accounts of her late husband. She'd recently sold some of his papers to the university archivist for enough to remodel their bungalow in Burns Park—one of Ann Arbor's respectable neighborhoods. It had more advanced degrees per capita than even the wealthier exurbs. And she was as well paid as anyone on the English faculty. Still, both the front of the bus and the back had their special privileges and the idea that she traveled for free was more appealing than the idea that she traveled in style.
She watched as other passengers boarded the plane and for a moment wondered if these might be the people she'd be dying with. Might some malfunction or shoe bomber or flock of seabirds bring the plane down over the Atlantic? Or some small outpost in Scotland—everyone blown to bits in the carnage? The whiff of disaster and mortality was agitating and she searched in her purse for her medication and something a colleague had given her for airsickness. She pressed on the point of the pain in her temple, rubbed her eyes. She wondered if she could get a drink.
She smiled at the cabin steward, a young Indian or Pakistani, passing out pamphlets. He was so pretty. Olive-skinned, dark-eyed, twenty-something, slightly built but muscular. His crisp white short-sleeved shirt with the epaulets, the blue tie. He must be gay, she thought. "The best ones always are. Then she scolded herself for thinking such thoughts—right out of a sitcom or chick flick. But they are! she thought, and scolded herself again. She could see that he paid attention to his body hair and the press of his trousers. But if he weren't, she wondered, which of her parts would his hands first go to—breasts or cheekbone, buttocks or genitalia? Or might he, as Nigel had, trace the slow curve of her eyebrow with his index finger? She did not scold herself for such thoughts, but regretted nonetheless the absence of answers. She sighed, resigned to the prospect that that part of her life was over.
Aisling read the Menu and Destination Guide he'd given her. How very helpful, such a nice touch. Like flying in a café instead of a bus. Dinner would be a choice of "Southwestern-Style Chicken Breast with Sweet Corn Salsa and Rice" or "Prosciutto and Ricotta Cheese Lasagna on a Bed of Spinach." There would be wines and cheeses and a sweet. Detroit's average July/August temperature was 24 degrees centigrade, the booklet helpfully informed. There was a quick conversion guide: C × 2 \+ 30 = F. Hotter than either Ireland or England had been and nearly twice as much summer rainfall as the British Isles. The packet included a map of Michigan and the city of Detroit and its environs. Familiarity indeed breeds contempt, she thought. She remembered the packet on the flight coming over, six weeks before. How she had savored the shape of the names: Marylebone, Bloomsbury, Holborn, Strand, Soho, The City, and Mayfair—she loved the contorted streetscape and the wide slice of the Thames through the middle and she had squinted to look at the names of the bridges: Lambeth, Westminster, Waterloo. She'd looked at the space between Theobalds Road and Guilford Street, bordered on the west by Southampton Row and, on the east, by Gray's Inn Road, in Holborn, where her friend Vanessa lived in Orde Hall Street and where she'd stayed while in London before she and Vanessa took the train from Paddington to Exeter for their duties in Devon. It was Vanessa who had put her name in for the Arvon Foundation course, as someone who would make an excellent co-tutor.
Vanessa had been to Ann Arbor the year before to do a reading and to lecture on Modern British Poets. Aisling had liked her immediately, her poems, her no-nonsense style in workshops and lectures, and her rich midlife sexuality. "Why is it," she had asked Aisling at the reception following her triumphant reading in the Rackham Amphitheater, "visiting male poets get offers of sex from the graduate students, and visiting women get unsolicited manuscripts?" There'd been a little scandalous buzz about Vanessa and one of the first-year M.F.A. fictionists with whom she had left the reception, arm in arm.
Aisling had invited her to lunch the next day. They had traded books, email addresses, gossip about other poets. They had stayed in touch.
Once the Arvon Foundation had invited Aisling to England, it was easy enough to find other programs willing to host her. She'd read and presented a paper at the Yeats Summer School, lectured at Queens and Cheltenham.
Her colleagues at the university, stuck with their lackluster summer stints at literary summer camps and writers' colonies, could not fail to be impressed by Aisling's summer duties in the British Isles. She had sent them postcards of Totleigh Barton—a two-story manse from the twelfth century covered with thatch, surrounded by outbuildings and the green Devon countryside or of Yeats's grave at Drumcliff, or of Pre-Raphaelites from the Tate Gallery.
In Aisling's department at the university, most of the men angled for Bread Loaf in Vermont, famous for its flings, or maybe a stint at Yaddo or MacDowell Colony or Sewanee, or a summer term at a nicely situated private college. Only the elders in her department expected to cross the Atlantic on the strength of their work. Some, of course, had been to European conferences in their specialties—Renaissance Writers or Women's Studies or Translations of Modern Russian Writers. But they were scholars on scholarly business—may as well, thought Aisling, have been bankers or ophthalmologists. Aisling traveled as artist and academic, poet and scholar, maker of beauty and witness to it.
Whatever claims to fame her colleagues had, they would have to admit that she had developed an international standing, however modest. Two slim volumes from a respectable university press in America and now the promise of a book on the shelves in Britain and Ireland. She would add it all to the personal page on the faculty website the university maintained.
She had been flown and housed and fed and paid and publicly feted for her time and talents. She had slipped the dull gravity of the everyday schedule and ordinary geography and been borne aloft, by the power of words she had written in private, published in fetching if fairly limited editions, and she was now being returned home from the ancient outposts of the English-speaking world.
"Please fasten your seat belt, madam."
The pretty Indian or Pakistani was leaning across her, bringing her seat back and tray table to their full upright position. He had one hand on the button on the armrest at her side, the other behind her, moving the headrest forward. She could smell his soap and talcum powder. Something from Harrods or Jermyn Street.
"May I get a drink?"
"We're about to take off."
"All the more reason," she smiled at him.
"Once we're airborne, I'll be back around, madam."
Eye candy, Aisling thought to herself, and scolded herself for thinking it. He must be gay. She scolded herself again.
The plane was speeding down the runway. Aisling closed her eyes and wondered if the pills were working. Though she wasn't sure of God anymore, she was praying for safety and deliverance. She did not want to die with these people. The whining children two rows ahead of her, the school soccer team at the head of her section, the redheaded woman applying awful lipstick beside her—what if these were her neighbors in death? She had much more living to do.
Suddenly the thought of returning to university in a couple of weeks, the return to the dull routines of the classroom and committee work, the needs of students and colleagues, the pressures of performance—it was all more than disturbing to her. She was sure it was the source of what was now a splitting headache, the stiffness in her shoulders, a panic taking shape. Whether it was this or the sudden press of mortality that air travel always stirred in her—as the jet raced down the runway toward its takeoff—she resolved to extend her summer travels. She was not yet ready to go home to Burns Park to wait out the remainder of August sweltering in town, preparing syllabuses and lecture notes. No, she would make for someplace without duties or details, social or literary obligations; someplace where she could let herself be pampered and excessive, waited on and catered to, where she could read for pleasure, sleep at all hours, bask in the absence of obligations; where she might restore herself after long travel and hard labor in distant places, before returning to the daily grind of the fall semester. A fortnight of utter self-indulgence, ease, and tranquillity, she thought, just what she needed. This craving for freedom, release, forgetfulness, she would indulge it. She could not only afford it, she thought, she could not afford not to do it—the better for her students in the long run, to restore herself before pouring herself out in service of their needs. Better to return to her office in Angell Hall, to the eventual committee meetings and faculty teas, looking as well rested as she was well traveled. The headaches and insomnia that seemed now her ever-present scourges might be mended by two weeks of ease.
So ran her thoughts as the plane climbed upward and outward from its earthbound gravity and the signal sounded that allowed them to unlock their seat belts and move about the cabin. Aisling decided to study the maps in the back of her in-flight magazines and to use the hours of her return flight to decide on a further destination. She was determined to land in Detroit with a location in mind to escape it all again, to spend the last two weeks of August in pursuit of bliss and rejuvenation.
AISLING BLACK was born in Birmingham, an upmarket northern suburb of Detroit, in the year Martin Luther King and Bobby Kennedy were killed and Nixon was elected president. She was the only daughter of a man who made a respectable fortune selling safety glass to Detroit's automakers. Her mother taught elementary school, read Plath and Sexton, and died young of subarachnoid hemorrhage. Her father never remarried, rather threw himself more deeply into work, finding ways to embed antennae and defrosting elements in the glass, amassing thereby a quite substantial fortune. A dead mother and often absent father produced a brainy, sometimes brooding girl who edited her grade school newspaper, won all the academic prizes in high school, tested well enough to get scholarships to all the better Eastern universities. After an undergraduate degree in Vermont she got her M.F.A. and Ph.D. in California, where she had studied with the famous poet. If she hadn't yet created much in the way of beauty, she nonetheless knew beauty when she saw it. And she had seen in Nigel's long, brutish poems, full of the blood and bone of his father's butcher shop in Seattle, something raw and sensual. He was one of Roethke's last students, had taught with Berryman at Minnesota, quoted Lowell and Bishop and Robert Frost, and was fairly manic, famously bingey, and, Aisling thought, a brilliant teacher. And he was handsome in the way men are who know how to wear the proper jacket and tie, a hat in winter, a cashmere coat; men who could be counted on for good directions and clean handkerchiefs, even if they sometimes seemed to look too long in the wrong direction or drank too much or said outrageous things. He could speak in metaphors, assigning depth to the everyday, connecting elements of the tedious and mundane to the stuff of art and literature. He had praised her poems in his workshop and attended her reading in the student union. When he invited her to travel with him, the year between her master's and doctoral studies, on a cross-country tour to mark the publication of his selected poems, she accepted. It was out of character for her to do so. But he made it sound more like a job than an assignation and an important part of her education. And it was. He was twice her age and worldly in ways she wanted to learn from. She started as his companion and personal assistant, someone whose discipline about travel arrangements and scheduling allowed him freedom from such drudgeries. He would show up, the crazy genius, wow the audience with his performance, sign the books, and say outlandish things that would guarantee coverage in the local press. She would get him to the next stop in one piece, make all the arrangements for lodging and meals and local transport. She worked with his publisher to do press releases, managed the book signings, the radio and print interviews. Eventually they became lovers. Sex with a man past his prime was more tuned to the timing of her own body than the more urgent or furtive lovers she'd been with before. If his performance was flagging, his desire was intense and it aroused her. Everything took longer with Nigel. Talk was an essential part of the seduction. So was food. His patience was consummate, his gratitude endearing, his amazement at her little body, its parts and zones and regions and responses, a constant blushing. "Classic beauty on a peasant's form," he said, lathering her body in the shower. He genuflected and kissed her. She came to love him. They moved in together. She proposed marriage. He agreed.
Theirs had been the perfectly bargained marriage. Each got something in the deal. He wanted her youth and beauty and trust-fund security, no less her scholarly instincts and disciplines—the better to burnish his postmortems. She wanted his age and experience and manic freedom. She wanted access to his generation of writers and poets. He loved her young friends always asking him questions about poets now dead whom he remembered. Both assumed it would not last forever—"Such habits are not suited to the long haul," he'd said—which made any awkwardness the more bearable.
When she finished her doctorate—her dissertation on the Maud Gonne poems of W. B. Yeats was published by a reputable university press—he gladly agreed to follow her back to Michigan, where she could be nearer her widowed father. The department had made them a "package" deal to get them to move from California. He was given an endowed chair with the graduate writing program. He had minimal teaching duties, charge over the visiting writers program, and funds allotted for an annual conference. She was given a fellowship and the promise of a tenure-track position. They bought a house on Granger Avenue in Burns Park with a big backyard and front porch and good kitchen. Their soirees for visiting writers were sources of great gossip among faculty and graduate students and were sometimes reviewed in the Ann Arbor News. Nigel knew everyone still alive and assisted the university in attracting the best writers to Ann Arbor. He knew the requirements of visiting poets, the foibles of writers on the road; he was great at introductions and literary intrigues. And he made Aisling his ever-present "partner in crime," a woman half his own age whose manifest intelligence made her choice to be with him all the more a prize—he loved being seen with her and made no secret of his dependence on her.
His tenderness when her only pregnancy ended after eight months in the stillbirth of a tiny son made her love him unequivocally. He had a stone cut with the infant's name, his own name, and the only date of record on it, and would take Aisling to the little grave in Maplewood, near the Arboretum, Sunday mornings in those first months of their bereavement. It was those Sunday visits to his dead son's grave that became the basis for the folio of poems, provisionally titled Nativity, which formed the core of a memoir she wrote the year following Nigel's death that was shortlisted for the Pulitzer and fast-tracked Aisling's tenure with the university. In the few years since, she had published another book of her own poems and a critical study of Nigel's work; she had read and lectured all over the country, from the Library of Congress to Huntington Library, and been featured in profiles on National Public Radio and PBS. Much as she tried to avoid the role of poetry's heartsore young widow, there was no doubt that her years with Nigel, her well-documented bereavements, and her relative youth and beauty made her something of an item on the literary circuit. She drew a crowd where others mostly drew a few. Her classes filled early at the university, with more young men than young women in them. Her poems, reviews, scholarly papers were all solicited by the better journals and magazines. Her fees for lectures and readings kept rising and rising. The invitations required the retention of a speaker's agent. She was even considering hiring a personal assistant—someone to help with the correspondence and calendar and travel details—but so far the right person hadn't materialized. Besides, she often thought, the daughter of a chief executive officer has sufficient management and discipline bred in the bone to handle a literary and academic career. Her celebrity, such as it was, had provided a secure income, the choice about how much work she would do, plenty of exciting opportunities, and all within her range of expertise. She had exceeded her own expectations of herself, becoming not only a witness to artists but an artist herself. She was at forty both teacher and the subject taught, poet and critic and woman of substance.
She kept herself fit with the usual regimens. She gathered her long black and graying hair into a variety of buns and braids, updos and ponytails. She dressed in vintage ensembles mostly bought secondhand at the Salvation Army and she spent with abandon on designer shoes. She kept her distance from doctors, gardened in three seasons, took pills for insomnia and mild depression, smoked in private on occasion, and had a tattoo on her right buttock that matched the one on her late husband's left. They'd had them done in Spokane the April before moving to Ann Arbor with money they made from reading together. In the years since Nigel's death she had not had sex and seldom missed it. Sometimes she would take a long bath with scented candles and Chinese soaps and bring herself to orgasm. The knowledge that there were men who would still be eager and willing to have sex with her was, in some ways to Aisling, better than sex with any of them might eventually be, requiring as it would a degree of intimacy and theater she was not so certain she was even capable of. She had abandoned all prospects of parenthood, spoke to her aging father infrequently, and lived now at some little distance from her body and soul. If not entirely happy, she was content.
IT WAS the ad in the in-flight magazine for the Grand Hotel that guided Aisling Black to Mackinac—the tiny island in Lake Huron and the Straits of Mackinaw between Michigan's upper and lower peninsulas. She'd been there with her father as a child and remembered her enchantment with the place, the enforced absence of automobiles, the busy docks and harbor, the horses and buggies and Victoriana, the easy opulence of the hotel, the dress code for dinner, the afternoon teas and string quartets. Timeless...read the banner over the photo of the hotel. There was a website and phone number. She could see herself doing nothing for a long time on the broad porch of the great hotel, watching the inland sea sparkling in the distance, restoring herself, renewing her stamina, readying for the school term that would begin in September. After she cleared customs in Detroit, she phoned the Grand, reserved a lake-view room until the end of the month, drew cash from an ATM in the terminal, got a seat on the five o'clock flight to Pellston Airport, a cab twelve miles north to Mackinaw City, where she was deposited along with her much-traveled bags at the Arnold Line Docks, and after the brief ferryboat ride across the Straits, she found herself on the island.
"Professor Black?" said a young man in footman's dress, top-hatted and gloved, and when she nodded, he took her bag, took her gently by the elbow, and led her to the shiny red-wine-colored coach with The Grand Hotel stenciled in gilt italics on the door. He looked like one of her undergrads, dressed for the prom or student musical—Great Expectations, she thought, Mr. Darby himself! He helped her into the coach, climbed up to take the reins, and brought her by turns up through the town to a wide thoroughfare which led uphill to the huge white clapboard edifice on the southwestern-facing bluff which was truly, now that she saw it aglow in the night, a grand hotel. The creak of the wooden coach, the strain of the draft horses up the hill past a stone church and wooden townhouses, the absence of anything automotive in the air—the racket of horns or emissions of motors—all of it conspired to slow her approach, after the furious day of taxis and airlines and ferries. It was, she knew, a kind of Disney, a theme park of sorts, a pretense of a place beyond the reach of real time. All the same, she could feel it almost metabolically—the pace of things slowing to the footfall of horses, of couples strolling the gaslit sidewalks or stretched out on the greensward listening to distant music—everything could wait, nothing was urgent, time could be taken, the moment held up for examination. She could feel herself getting sleepy. He took her immediately to her room, where a basket of fruits and cheeses and a bottle of wine had been left with a welcome note from the hotel's manager. He turned the light on in the bathroom, drew the curtains shut, placed her bags in the closet, and gave her the key to the room and honor bar. He said she could call the front desk in the morning about her registration. She could tell he could tell she was very tired.
"What time is it?" she asked him.
"Going ten. Where have you come to us from, Professor?"
"London, Detroit, well, Ann Arbor, by way of London today."
"A long journey. You'll want a good night's sleep."
"Yes, sleep's the thing." She proffered a twenty-dollar bill.
"Oh no, Professor, I really couldn't. There's no tipping expected or permitted at the Grand. If there's anything at all, anything, just call...Enjoy your stay."
He was out the door and she was left standing in the middle of a large corner room that looked out over the grounds of the hotel, the lakeshore and the Straits of Mackinaw, and in the distance, the lights of the Mackinaw Bridge, which connected the two peninsulas of her native state. She stripped and showered briefly, wrapped herself in the hotel robe, hugging herself in the plush terrycloth, savored a berry, left the wine untouched, then crawled under the down comforter, assembled the pillows according to custom, and considered the distance she'd come that day. Then Aisling Black fell into a sleep the details of which she would not remember. Whether she dreamed of her dead husband or baby, or new phantasms of love and desire, or random remnants from her travels; the pulsing in her temple, the usual insomnia, the cares of the day, all of it gave way to the blank oblivion of comfort and keen fatigue.
She woke briefly before dawn and did not know if she was fully awake or only between deeper slumbers. She rolled from one shoulder to the other, sat up in bed, and with gathering consciousness wondered if she oughtn't to make some notes toward an opening lecture for her senior seminar—a reading and writing course she taught to undergraduates in which she tried to get them to use the work of masters as predicates for their own creative efforts. The Sincerest Form was the title of a book one of her colleagues had written on the uses of imitation for young writers. She took it as the title of her course in the catalogue. Part of her lecture, indeed part of the paper she had recently delivered at the Yeats International Summer School, focused on the formulation, attributed to T. S. Eliot, that "all poets borrow, great poets steal." As evidence of this dictum she had produced W. B. Yeats's late nineteenth-century transcription of a late sixteenth-century sonnet by Pierre de Ronsard, "pour Hélène."
Quand vous serez bien vieille, au soir, à la chandelle,
Assise auprès du feu, dévidant et filant,
Direz, chantant mes vers, en vous émerveillant:
Ronsard me célébrait du temps que j'étais belle.
The poem, a cautionary tale to reticent lovers whose beauty will not outlive the lines of poems, is borrowed by Yeats to woo and warn his life's great and unattainable love, Maud Gonne.
When you are old and grey and full of sleep,
And nodding by the fire, take down this book,
And slowly read, and dream of the soft look
Your eyes had once, and of their shadows deep...
Aisling made her case that poetic license was, in fact, license to steal, in service of the art itself, from all the poems that have come before. All art, she would further argue, communes with all of the art that went before it. "It's what we do," she'd say, quoting Auden, "to break bread with the dead." And she called on her students to become priests and priestesses of the holy forms of language and literature. She would regale them with the romance between Yeats and Maud Gonne, how he had pursued her and she had refused him, his many proposals, but kept up their "intercourse" all of their lives. She would read them what he wrote in his journal on the day that he first saw her—the thirtieth of January, 1889: I was twenty-three years old when the troubling of my life began. And she would read them his breathless first description of her: I had never thought to see in a living woman so great beauty. It belonged to famous pictures, to poetry, to some legendary past.
Aisling knew that her students were similarly vexed, being likewise in their early twenties, by the beauty and heartbreak they were finding in one another, and that their aptitudes for poetry and language might be, if she could attach them to their sharpening romantic and sexual appetites, forever fixed in their temperaments. Sex and death, she would always instruct them, are the only subjects worth thinking about. And then she would tell them she stole that from Yeats, who had written it in a letter to Olivia Shakespeare. Love and grief, she would further instruct them, share the one body. And she knew that they knew that this was true. She encouraged them to imitate the poems they most admired, to match them, as Yeats had with Ronsard, line for line, theme by theme, rhyme scheme by rhyme scheme, until they found themselves sounding more like themselves than like their dead or elder mentors. In fulfillment of which assignments she had received more of the ridiculous than the sublime—Let us go then, you and I, / Out for some brewskies and pizza pie—but still she had them thinking about poetry as something they could not only read, but do. What matter if most of them ended up as marketing assistants and investment bankers, sales reps and attorneys—hers was to move them as best she could to become readers and writers, if not for a living, at least as accessories to their lives.
It was going toward daylight now and Aisling had papers from her briefcase spread on the bed. She scolded herself for abandoning her resolve to find deeper rest and so she turned off the nightlight, closed her eyes, and lay flat beneath the coverlet, touching herself, trying to recall what it was to be touched.
When she opened the curtains just before noon the light in her windows all but blinded her. She looked out eastward over the town and harbor to Round Island and Bois Blanc and the ferryboats plying the Straits from Mackinaw City and St. Ignace. The small abandoned lighthouse at the harbor's mouth was one she had seen on postcards for years as an icon of Michigan's Great Lakes. To the southwest was the five-mile bridge between Michigan's peninsulas and beyond it the edge of Lake Michigan, its freshwater blue unlike any sea or ocean, sparkling in the mid-August light. She unlocked the bank of tall windows and opened them for the fresh blast of summer air and the rattle of horse-drawn traffic below. She looked around the large bedchamber with its nineteenth-century sofa and Queen Anne wingbacks and inlaid cherry writing table and straight-back chair. There were cushioned window seats along the side windows, a queen-size four-poster bed, and tiled bathroom with Jacuzzi and glass shower stall. There were richly framed prints of seaside tableaux—Daubigny's The Beach at Villerville, Seurat's pointillist masterpiece of La Grande Jatte, and one she recognized as September Morn, of a nude girl standing ankle deep in the water, though she couldn't remember the artist's name. A bit too vivid and pastel for her tastes, she thought, all signature Carleton Varney. Nigel would never have slept in such a room; she winced at the little needle of guilt that such excess should be wasted on herself alone. Still, the views were spectacular, the feather bed had given her the best night's sleep she'd had in months, and the light, as she stood in it, was an ointment and balm to all that ailed her. She would make the best of her time here, whatever the cost was, it hardly mattered. She ordered breakfast and newspapers to the room, asked that the florist send up a centerpiece for the coffee table, set about putting away her things, and set up her laptop on the writing desk, determined to remain in the rented robe for most of the day. Much of the clothing she'd traveled to Europe with was all a little stale from the long journeys, overly wrinkled, too dull for the midsummer ambience of her new surroundings. She examined the catalogue of hotel services and on-site shops and found a serviceable outlet for women's fashions in one of the first-floor boutiques. She'd pay a visit after her brunch. She studied her feet and scheduled a pedicure with the salon. She avoided the temptation to check her email or call her father or make known to her colleagues in Ann Arbor her whereabouts. She wanted to remain outside the loop and range of the known and ordinary world and to bask in the bright late-summer light, free of any obligation. She studied the local directory for newsagents, bookstores, points of interest. She resolved to rent a bike and see the island, named by the Ojibwas for its turtle shape and regarded as sacred to the spirit world. All in her own good time, she thought. But for now she promised to do nothing of substance beyond growing accustomed to the space she inhabited. She breakfasted, dressed casually in khaki slacks and a white blouse, wore sunglasses and a ball cap, as if going incognito, and made her way out to inspect her surroundings. By midafternoon she had registered with the front desk, looked in the dining room and café and terrace bar; she'd walked the length of the front porch—six hundred and sixty feet: longest porch in the world, the signage said—walked down to the pool and formal gardens, then back to the hotel for the spa and some shopping. She had, in the same time, acquired a new wardrobe, a new sense of wellness, and a new appetite for whatever she might find on the menus. She napped, and then surveyed the ensemble she'd bought in the hotel's boutique. She'd splurged on a pair of strappy black patent leather cork-soled wedges to go with her pedicure. She found a simple black linen dress with a deep V neckline that gathered softly just below her knees that she thought would look perfect with the turquoise pashmina shawl she'd found in London. Near the checkout she found a pair of seed bead dangling earrings, also in turquoise, and a blue moiré water silk headband to match. For Aisling, each element of an ensemble was vested with meaning or metaphor or symbol so that wearing it, she became an accessory, somehow, to a larger narrative the denouement of which remained unknown. Seeing the new garments laid out on the bed filled her with curiosity about her place in it all. She dressed in quiet, consulting the bureau mirror after every element of her outfit was added.
When she appeared in the lobby just after high tea, half an hour before dinner would be served, she looked rested and reborn. The hotel manager, a man in his thirties trying to look older than he was, standing sentinel in his blue suit and club tie and button-down shirt, hastened across the lobby to offer any assistance.
"Good evening, Mrs....?"
"Black," she told him, "Aisling Black."
He shook her hand and smiled earnestly.
"Is this your first visit to Grand Hotel?"
"My first in years," she said, "I used to come here with my father for conventions. I remember him meeting the governor here. And I was here the year they were making the movie."
"Yes, Jane Seymour and Christopher Reeve...how very sad...to think of him so damaged by falling from a horse...They made that movie in '79."
"Somewhere in Time?" She had never seen it.
"That's it, Somewhere in Time, indeed."
"I remember them filming in this lobby."
"Well, you're welcome back after all these years. What is it you do now?"
"I write, I'm at the University of Michigan."
"Yes, yes! Professor Black. I saw that you'd arrived last night. Mark Twain was here, you know...lectured on the front porch, July of 1895. Perhaps we could impose upon you to give a reading?"
"How very kind of you, but really, no, no, I'm here to be healed from all kinds of work."
"Of course, Professor, I understand."
"I'm Aisling, just Aisling, but thank you anyway."
"I'm Michael Musser, my family has owned the place for years. Please let me know if there's anything, anything we can do to make your stay—"
"Thank you, Michael, I will of course."
They had arrived at the entrance to the dining room.
"Would you like a cocktail before dinner, Profess—Aisling? It's early, what would you like?"
"Maybe a scotch?"
"And soda? Water?"
"No, only scotch. Then a table, the fresh air...I'm famished suddenly."
"And will you be dining alone?"
She nodded and smiled.
He arranged with the maître d' of the dining room to seat her at a window table so that she could have a view of the porch, the grounds, and the lake. He brought her a tumbler of single malt, asked if he could get her anything else, promised to be at her beck and call, then left her to enjoy her dinner.
Aisling was used to dining alone. The time formerly spent conversing with Nigel between ordering and courses she now passed people-watching, assigning fictions to the assorted configurations at different tables. The elegant and amply mirrored dining room was a perfect venue for this, as people unaccustomed to dressing for dinner rose to the occasion of their surroundings. Most of the families with school-age children were back to their lives now, leaving the hotel and the island to seniors and young couples and what seemed like a convention of visiting engineers or architects. The whiskey warmed her. A black man in a white dinner jacket brought her a menu. She was happy to have something to hold and read as she watched the tables fill. The mirrored columns that ran the length of the long dining room allowed her to consider people at various angles without appearing to stare. A woman with a goiter and a fat husband, the pretty couple on their honeymoon, from Pinconning, she imagined, where his family had made a little fortune in cheese and hers farmed sugar beets. This was his first suit and she'd been a virgin. Or maybe not, Aisling thought; maybe he was home on leave from Afghanistan and she was an escort from Traverse City, the long luxury weekend and the hooker paid for by a kindly uncle who couldn't abide the thought of the bright-faced boy going back to the war before sleeping with a woman. Or maybe it was she who was home and he the escort and a rich lesbian aunt who paid. The possibilities were inexhaustible. She loved to bask in the liberties of fiction and thought she might like to write a novel. The four blue-haired women, maybe sisters or widows from the same Presbyterian church, here for a hospice conference or bridge tournament. The handsome brothers and their fashionable wives, men in their fifties, here to celebrate a mother's birthday maybe? Anything was possible, or was not.
It was in the mirror, at an angle, that Aisling first saw the young woman decanting water into glasses at a table slightly behind her and slightly left. Whether it was the grace with which she moved among them or the startling beauty of her face, unlike any Aisling had ever seen, or the poise and shapeliness of her figure, half hidden in a mannish dinner jacket, black trousers, white shirt and bow tie, it was hard to know; but Aisling heard herself catch her breath and her right hand moved to cover her mouth, which had opened to make the sound of vowels as she let the caught breath go. She was dressed to match the rest of the waitstaff except for the maître d' and headwaiters, who were dressed in tails. Hers was an apparently junior or apprenticed position as she never served food or wine and only cleared tables after they'd been vacated by diners. Her skin was mocha-colored and her eyes still blacker and her bearing was regal though she was not tall. And while she could have been a native Detroiter or Tanzanian, Aisling guessed she was Jamaican. Crews from that island nation had been coming to Mackinac for years, spending April until late October working the Grand and then returning home for the winter season at the Wyndham Resort in Kingston. When she came to fill Aisling's water goblet the older woman blushed and feigned composure whilst the younger, whose name badge read Bintalou, smiled shyly and backed away. Though Aisling tried to appear engaged with the menu, she was in fact following her every move around the room, using mirrors for the wider scrutiny they provided. Nothing on the menu, from the "Braised Pork and Vegetable Spring Roll in Ginger-Mushroom Soy Sauce" appetizer to the "Almond-Macadamia Crusted Trout" entrée could take her attention from the beauty. When the waiter asked Aisling what she had decided on, she ordered another scotch and told him to choose something for her; everything on the menu looked sumptuous and she'd trust his instincts. Bintalou was sixteen or seventeen, Aisling guessed, and near enough Aisling's own size—just over five feet and a hundred pounds. Their eyes were near enough the same color, though the girl's were deeper, darker, entirely benign. Her black hair was braided in tight cornrows and her bosom and her bottom were rounder, higher, more ample and sinewy than Aisling's, and her body was bolder, more powerful. She had tiny wrists, long, delicate fingers, thin arms, perfect ankles. Her blackness was richer, more articulate, than Aisling's whiteness. The newness of it, its purity and youth, its incarnate beauty were utterly compelling. Aisling could feel the heat of a blush rising just watching the girl move about the room among the clatter of dinnerware and conversations, the industry of the waitstaff, the piano music in the corner, the chandeliers and candlelight—everything was silent and incidental to the aura and cocoon of luminary beauty her vision moved in.
The girl's connection to the maître d' was a curiosity. He paid her a great deal of attention, seeming to instruct her at every opportunity. The obvious deference paid by the other servers was, Aisling reckoned, connected to this.
Aisling nursed the second tumbler of scotch slowly, and picked indifferently at the choices the waiter had made, a spinach salad with pecans and goat cheese and dried cherries and an entrée of wild mushroom ravioli. She kept hoping the girl would come to fill her water glass or clear the plates from her table, but as the dining room filled, it was clear the Jamaicans worked in teams, working separate sections of the dining room, and that Bintalou was assigned to one nearer the main entrance. Aisling ordered a dessert, the better to extend her silent audience. And after dessert, she chose a brandy, Calvados Coquerel, to tarry all the longer.
She had watched the light go out of the day, replaced by a full moon that lit the surface of the Straits below. And because the evening was fine and warm, she decided to walk the long porch after dinner while other guests lazed in the wicker rockers that lined the porch on one side, with the planters of red geraniums on the other. Even from the outside, with the lights of the bridge and the village below and the moonlight on the water all around, Aisling was focused on the interior as she walked, looking through windows for the girl moving among the last of the evening's dinner guests.
Aisling's sleep that night was vexed by dreams of her dead husband and son. She had not taken a pill to sleep because she'd been drinking. The night was fitful and she woke full of longing—though she could not say exactly for what. She wondered if she'd be able to make two weeks. She took a long shower and dressed for a day of walking. Downstairs she made her way into the dining room, not so much for breakfast as to read the papers and to see if the girl was working the breakfast shift. She recognized the waitstaff from the night before but there was no sign of Bintalou. Better to take her beauty sleep, Aisling thought, and ordered orange juice and toast and coffee.
The papers were full of the coming elections, worries over the local and national economy, the crisis in home mortgages and foreclosures, the falling values in the automotive stocks. A large portion of her own trust fund was held in General Motors stock since they were the largest purchaser of her father's safety glass. In the first nine months of the year the stock had lost half its value. The prospect loomed that the company would be bankrupt before the year was out and that her little fortune would be seriously diminished. The stock of her father's company, the other large portion of her trust fund, had also lost considerable value.
Aisling had always adopted a style between the struggling artist and the sensibly situated academic, but she was comforted by the knowledge that wealth, inherited from her father, but wealth all the same, spared her the common worries over livelihood and future and freed her from daily interest in the marketplace. She was raised in privilege and was accustomed to ignoring the politics and economics of the larger world in deference to her intellectual and artistic pursuits. She wondered if the "one-state recession" that the papers claimed Michigan was in, because of the "failing domestic automotive industry," would actually have an effect on her. She had seen it in Europe—the comparative devaluation of the dollar, the higher costs of everything, the fears among retailers and hoteliers. The ongoing wars, the distrust of American leadership, the global markets in disarray, the price of oil—there had even been a surcharge on the ferryboat over to the island, blamed on the cost per gallon of gasoline. And now that she looked about the room, she could see that here in high season, the middle of August, in the finest hotel in Michigan, many of the tables were empty at breakfast, much as the seats on the ferry over had been. Even the hotel's "Modern Baroque" décor, reconsidered in the morning light, with its striped wallpapers and floral print chintz, by Dorothy Draper's celebrated protégé, seemed now a little dated and overwrought—the old century's effort to update the century before. The table settings of Syracuse china, "since 1871–USA," now that she looked closely, were faded and dull, and the Irish linen tablecloths frayed and somewhat overlaundered. As she looked around the room half filled with elderly visitors wearing the name badges of conventioneers and senior tours, eating their omelets and oatmeal and fruits, the grandeur of the Grand Hotel seemed to Aisling perhaps a little past its prime. Perhaps the economy really was in serious peril, Aisling thought, and marveled that she should live to be forty before such a thought ever occurred to her. The first needle of a headache stabbed her right temple. She sipped her water, closed her eyes, inhaled deeply.
The merchants she gossiped with in town that morning all ranged between commiseration and denial. Oh sure, the season's traffic was a little off, nothing to panic over, the crowds would return once the price of gasoline went down. The local fudge shops still made big slabs of the local specialty on marble altars in the shopwindows. The liveries renting riding horses and carriages or offering guided tours of the island still kept the downtown sounding like an old western with hoofbeats on the pavement. The auto industry had been failing for years—laboring under its pension and health care costs, executive salaries, poor decisions about consumer preferences. Mackinac relied on its six hundred horses—hackneys and drafts and even-tempered riding stock—for taxi and tours and haulage. The rich smell of fried food and horse manure was a welcome change from the diesel smell of London and Belfast, and the soft breeze off the big water made it all bearable under the August sun. Aisling wore a smart panama and sunglasses, walking shorts and a blue denim shirt, and browsed among the souvenir shops with nothing in mind. She listened to a trio—guitar and fiddle and hammered dulcimer—playing bluegrass music in front of one of the Main Street hotels. She walked the docks of the marina in the harbor where sailing boats tethered to their buoys and power yachts snug in their berths shone in the sparkling water. She bought sunscreen and applied it and walked up the town among the day-trippers.
At the end of the Main Street was a large village green overseen by a life-size bronze of Jacques Marquette, in soutane and cape, atop a tall granite base. He was among the first Europeans to see the Straits and wintered on the island in 1670, disabusing the native Huron and Ojibway of their heathenry. Aisling sat on a bench among the families and sunbathers sprawled on the green and read from a pocket history she bought at the bookstore:
Father Jacques Marquette, the great Jesuit missionary and explorer, died and was buried by two French companions somewhere along the Lake Michigan shore on May 18, 1675. He had been returning to his mission at St. Ignace, which he had left in 1673 to go exploring in the Mississippi country. The exact location of his death has long been a subject of controversy. A spot close to the southeast slope of a hill, near the ancient outlet of the Pere Marquette River, corresponds with the death site as located by early French accounts and maps and a constant tradition of the past. Marquette's remains were reburied at St. Ignace in 1677, when Hurons disinterred his remains and bore his bones back in a cedar box.
She read about the ancient fishing tribes, remnants of which, dating to the tenth century, had been found, who thought of the island as the home of "Gitche Manitou"—the Great Spirit. And about the fur trade and John Jacob Astor's export of beaver pelts from the island. She read about how it became the second national park, in 1875, after Yellowstone, and later was given to Michigan as its first state park.
Above the green on a bluff overlooking the harbor was Fort Mackinac, a limestone encampment built by the British in 1780 to control the Straits during the Revolutionary War. I'll tour that another day, she told herself, avoiding her natural temptation to scholarship instead of rest and lazing. She repeated her resolve to do nothing of substance and blamed the sudden return of her headache on reading through sunglasses or too much sun. She decided to walk back to the Grand, maybe nap for an hour, and then maybe a swim.
It was coming out of a three-story green clapboard house—one of the dormitories for Grand Hotel staff—on Cadotte Avenue that Aisling saw Bintalou, dressed in traditional housekeeping attire and making her way with a young man, arm in arm towards the Grand. As lovely as she'd been the night before, in the late-morning light she was more beautiful still. Her blackness shone. Her stride was athletic and elegant and her pace lengthened the distance between her and Aisling as they walked uphill. Try as she might, Aisling could not catch up to the girl and her companion. She became aware of her windedness and fatigue, which combined with her now throbbing headache to become a little panic, plodding up the east side of the boulevard while the object of her deepest interest strode up the west. The young man she walked with wore the top hat and red frock coat of the Grand's carriage drivers. In every other way unprepossessing, he was not tall, nor handsome, nor did he seem especially gifted, except that he was walking next to her. Aisling couldn't say for certain if they were simply companions or actually lovers. They seemed impossibly young for the mysteries of it all. Still, Aisling had read enough undergraduate papers to know that few of her students came to university as virgins. Their compositions often included cheerful narratives of "hook-ups" and "fuck buddies," "friends with benefits" and variations on the theme—sex-for-the-sake-of-sex encounters, unencumbered by deeper meanings, detached from feelings beyond a general fondness. This was at odds with Aisling's upbringing and experience, which assigned to every touch or caress, kiss or entanglement a meaning and purpose and direction. That she could no longer make out the manifest cues of one body's relations with another disturbed her greatly. Perhaps it owed to her own sexual circumstance, whereby, with the exception of the years with Nigel, she labored between spasms of ineptitude and decrepitude—conditions that rhymed too perfectly to ever include in one of her poems.
By the time she reached the lobby of the Grand, Bintalou already had assumed her station in the Salle à Manger, tending to the endless luncheon buffet, and the young man, true to his uniform, was sitting atop one of the hotel coaches under the main entrance portico, much like the one that had transported her hither from the docks the night of her arrival. But for the piercing pain behind her eyes, she would have gone in for a long lunch and the chance to watch the beauty move among the fruits and soups, coldcuts and pastries. Instead she went up to her room, drew the drapes, and took a nap.
In what remained of her two weeks at the Grand, Aisling's infatuation with the Jamaican girl only grew more fervid and consuming. She woke every morning with a plan to pursue chance encounters. She hid her quizzing of Michael Musser, ever the helpful host, about his hotel and waitstaff, in sufficiently innocuous banter so as to avoid concerns. She came early to the dinner service at night and almost always stayed until the end, lingering over desserts and coffees, then taking up her watch on the east end of the long porch, hoping to see Bintalou exit by the service door below, whereupon she would hasten to follow her to what she knew was her room in the front right corner of the third floor of the green house at the bottom of Cadotte Avenue. By turns she had come to know the girl's routines and haunts, her variable schedule and her days off. She'd found out that Bintalou was, indeed, the daughter of the dining room's majordomo. The distinguished-looking man in tails and vest—Henry Goodison—was in charge of seating and oversaw the gloved waiters and the lesser servers. He had been working summers in the Salle à Manger for forty years and was, to the Mussers, central to the smooth dispensation of service. Henry Goodison had been the ideal head man and factotum. Honest, loyal, proud in his bearing, and a natural leader. Not only had he a memory for the names of returning guests, but he would ask about their children and children's children, all the while displaying the crisp efficiency and stewardship of his position. He'd become the natural liaison between the Mussers and the Jamaicans, settling disputes over wages and scheduling and monitoring the conduct of his countrymen in their off hours to prevent any scandal or disruptions. His only lapse in perfect conduct was a generation ago, when he had impregnated one of the housekeeping staff, the daughter of his neighbor in Kingston. Bintalou had been raised by her mother, alone, inasmuch as Mr. Goodison already had a wife and family. He had been supportive financially, but otherwise kept his distance. And only when Bintalou's mother died had she come to stay with the Goodisons. Her father's long history of coming and going between the Caribbean and Mackinac Island had made it possible for Bintalou to get a visa for the summer—a far more difficult thing, according to Michael Musser, since the attacks of September 11. Indeed, much of the summer help on Mackinac now was Polish or Mexican or Ukrainian. Only the dining room of the Grand remained the province of the West Indians. The Mussers had promised Henry Goodison a room and board for his daughter and part-time work and training—her age, fifteen, making a full-time position quite impossible. Besides, it gave her time to be with other young people from around the world and from Michigan, and provided her father the chance to reconnect with her and gave her plenty of time off to enjoy the new environs.
Aisling's fascination with the girl was outside the range of all former experience and, she believed, at odds with her orientation and discipline. The two or three same-sex romances she had had as an undergraduate were, for the most part, exploratory, as much to satisfy curiosities for her partners as for herself. She was, if not recently, nonetheless a heterosexual. Her dreams conformed, her memories. She missed the sound of Nigel's strong piss in the toilet in the morning, the scratch of his razor over his face, the smell of him—whiskey and tobacco. She missed his mannish foibles and self-certainties. She missed his love. His desire.
Her growing fantasies in which Bintalou figured so prominently owed, she reasoned, to a confluence of longings—for a child, for a lover, for a life of passion and meaning beyond the meager rehearsals of her own. The stunning young Jamaican, Aisling further reckoned, was the embodiment of all that she'd pursued as an artist and a scholar—a beauty beyond the idea of beauty, rather the living, breathing incarnation of beauty itself. This prospect frightened and excited her.
When she finally and with some regret asked Mr. Musser to prepare her final bill, on the evening before her scheduled departure, he said he'd be sad to see her go and hoped that she'd be a regular visitor in summers to come. He said he would organize a car to meet her at the dock in Mackinaw City and drive her to the airport in Pellston in plenty of time to make her flight to Detroit.
Aisling Black left the Grand in the early morning, a fog with the first chill hint of autumn in it hovering over the harbor. She got an early ferry, watched as the boat turned out of the harbor and the great white edifice of the Grand came into view. "Bintalou," she whispered to herself, then turned her sights on the distant mainland.
The man holding the sign on the docks with her name on it threw her bags in the back of his rusting brown minivan, and said that there'd been a slight change of plans. He was sorry to report that her flight from Pellston was canceled.
"Mustn't have been enough seats bought up. They always blame it on the weather or some 'mechanicals.' But it's dollars and cents, simple as that, ma'am. No sense spending all that jet fuel on less than a full flight."
"But I have a ticket, I have to get home. I have classes in Ann Arbor tomorrow."
"Not to worry, ma'am, I've thought of that. I can have you back south in no time. Hell, Ann Arbor's four hours. Door to door, faster than the jet, what with all the security and searches and waiting around and crap. And I'll charge you less." He was holding the door of the minivan, and beckoning with the other arm for her to sit in for the drive south.
"No...no, that's not what I planned on. You don't understand. I've a splitting headache; I can't take four hours on the road. I have a ticket and reservation. I'm sorry, just take me to the airport."
"No sense in going to the airport, ma'am, the flight has been canceled, you can check for yourself." He held out his cell phone. "I have 'em on speed dial. I'm trying to be helpful here. I'll charge you less than the airlines, ma'am. A hundred and fifty, no need to tip. Hell, the gas will cost me nearly that."
Now Aisling's head was pounding and she was confused. She didn't want to get in his car. The other passengers were making their way to the car park. They had their own vehicles and places to go. She backed up and sat down on the bench by the boat. She was feeling pushed to a decision she didn't want to make. How could she get in a car with a perfect stranger? Why would the airlines cancel a flight? The day was clearing. Conditions were fine. Why should she have to change her plans?
"I'm sorry, I can't go now. It's quite impossible. Kindly leave me my bags." She took a twenty-dollar bill from her purse. "For your trouble, please...I really can't go. I won't. Impossible."
"Whatever you say, ma'am. Customer's always right." And he muttered something under his breath. He refused the gratuity and left her bags beside her and got in his minivan and drove away.
After a few moments to gather herself, Aisling moved to the shade of the Arnold Line ticket office. She sat in the pavilion wondering what it was she should feel. Outrage? Relief? A secret thrill? When questioned by the ticket agent about her plans, she bought a one-way ticket for the ten o'clock boat back to the island and retraced her morning's journey back to the hotel.
When she presented herself in Michael Musser's office, he congratulated her on the decision to stay. He was sorry for the trouble in traveling but pleased that it had returned her to the Grand. The room she had vacated was being tidied for another incoming guest but he would find her a room as fine or nicer for the same rate as her old room. She was welcome at the Grand for as long as she'd like. He took her up on the elevator to inspect the suite.
"A stroke of luck," he said when she approved the new quarters. "We had a cancellation this very morning. It's the economy. Like a plague. No confidence. Everyone is cutting back. Of course, they always claim a death in the family, or a sickness. I understand." Then, catching himself as the purveyor of doom, he made hasty apologies and backed out of the room.
Aisling settled herself in the tall wingback chair by the bank of windows that looked out to the Straits, amazed at the morning's happenstance, and a little peeved at herself for her vacillations, irked at her ignorance of her own desires. She would have to let the department know about her delay in returning. She composed the email in her head, not wanting to talk to the department in person. She would amplify the matter of a canceled flight and hope that no one thought it strange that she had stranded herself within two hundred and fifty miles of the university. How, after all, could she explain herself to them. She could hardly make sense of it all herself. She could have rented a car herself, driven down the interstate at a leisurely pace and been home before evening. Both her graduate workshop and undergraduate seminar met once a week for a three-hour session. So after missing the first week's duties tomorrow, she'd have a full week to get herself reconfigured and make her way home. She sat trying to clear her mind of contingencies and nodded off in the wingback into a restless sleep.
NOW THE poet gave herself over to the chance unwinding of the days before her and to the pursuit of beauty outside of the forms her training and temperament had always used. Rather than put the hours to work over sonnets or sestinas or the further cataloguing of her husband's work, rather than plan further lectures, assignments, classroom exercises or review her students' writing samples, rather than send new work out to the better journals or plan a conference for the coming year, she sat for hours gazing at the water and the way the daylight worked upon it. She succumbed to her preoccupation with the young Jamaican, working her schedule around the girl's schedule. When she was working the afternoon teas, Aisling partook of the afternoon teas. When she was working the dining room, Aisling lingered there. When she was out for the day with the young coachman, Aisling just followed on foot as the young couple shopped among the stalls in town, or toured the massive summer "cottages"—Queen Annes and Greek Revivals and Carpenter Gothics—on bluffs overlooking both ends of town. These were summer homes with names like "Far View" and "Tootle Cottage," "Ingleneuk" and "Windemere," built by the moguls of another age, Midwest capitalists who made their fortunes in textiles or railroads, lumber or banking; doctors and soldiers and clergymen. The cupolas and turrets, widow's walks and towers rose over the broad porches to look out on Lake Huron and the Straits and west of the great bridge in the distance, all down the long blue edge of summer in Michigan. Aisling, of course, tried to keep her distance, but it wasn't long before she could tell that Bintalou and her paramour—though of that she was not yet certain—had become aware of her pursuit. There was no good way to explain how it was she was always happening to be in their vicinity. What would she eventually tell whoever it was who might eventually ask? That she was transfixed by the vision of the girl; that the girl seemed in every one of her aspects, utterly sublime, and that since gazing upon her beauty, all other ruminations on the theme of beauty seemed poor imitations and wastes of time. She had made studies in her notebook, hoping to draw out some verse on this theme, but it was useless. On the subject of beauty Aisling could wax eloquent, but in the presence of it she was smitten to silence. All she could do was follow on, solitary in her pursuit, hoping to get some succor for the hunger it whetted in her for more of the same. She had grown more sleepless, more mindless of her resolve to rest, more mindful of the madness the girl had ignited in her. But she was helpless to do anything about it. She asked the hotel physician for a palliative for her constant headache, which now only sharpened when the girl was nowhere to be found. Aisling had confined her time downstairs in the hotel to the times Bintalou was working. On the girl's days off, Aisling searched out her whereabouts, checking the hotel environs first, often asking one of the coachman's fellow drivers where might Blake Shields be that day—for she had researched his name and other particulars—knowing that they would likely be together.
The daily papers were full of concerns over the mortgage crisis and home foreclosures. While waiting for a sighting of Bintalou one late morning, Aisling Black sat out on the hotel porch reading about the steep decline in home values over the recent months—some by as much as 25 percent—and the "negative equity" many homeowners now had in their properties. There were gathering worries over banking and insurance, the stock market losses near 40 percent, and the rising rate of unemployment over 7 percent across the country, nearing 10 percent in Michigan. Aisling could only wonder what it all meant, having never considered the threat of ruin. She couldn't help noticing the declining occupancy at the Grand. A rear section of the dining room was now closed off. The hotel's usual population of guests and staff seemed in decline. There were fewer golfers on the golf course, swimmers in the pool, revelers in the bars, old-timers at tea. The traffic in bikes and horse buggies was noticeably thinner. The fudge shops were closing earlier, the ferryboat schedules condensing, the souvenir shops discounting their trinkets, first by 10 percent, then by 25. The days were growing shorter irretrievably.
One night after the dining room had closed Aisling followed Bintalou and Henry Goodison and Blake Shields to the ferry docks, where they took the boat over to St. Ignace, the nearest port on the Upper Peninsula. Aisling could not help but follow, so much it had become her custom now, seating herself several rows behind the trio on the lower deck of the ferry and following them from the dockside to the casino that was their destination. It was owned by the Chippewa Indians and offered banks of slot machines, keno, roulette and poker, blackjack and craps. She watched as they settled into their games. She took a seat at the blackjack table next to the one where the headwaiter was playing and Bintalou was sipping a soda behind him. Blake Shields was trying his luck with nickel slots. Aisling was perfectly positioned to watch the girl overlooking her father's cards, her face changing with his changing fortunes. Henry Goodison was well known and evidently well regarded in St. Ignace. Suited officials from the casino's management stopped to greet him on their rounds. Cocktail waitresses brought him drinks. Several of his fellow gamers stopped to pay respects. He introduced them all to his daughter, who seemed more than delighted to be out with her dad, hugging him around his shoulders, kissing his large ears, and cheering when he won, offering consolations when he lost. Aisling envied their easy endearments, having long since grown distant from her own father. She found it hard to concentrate on the cards she was being dealt and kept losing more money than she ever intended, but was enthralled, nonetheless, with the proximity of the beautiful child and the games of chance. Aisling was amazed at the crowd of gamblers packed into the igloo-shaped casino. They were elderly, many were disabled, some of them looked near the edge of ruin. She imagined herself among them, in a venue out of Dante as people wagered not from their plenty but from their want. And in among the blinking lights and flashing machineries, Bintalou and her distinguished father hugged and giggled and smiled as he won.
It was after the last ferryboat back to the island by the time Henry Goodison was ready to call it a night. Aisling followed the trio to the docks, where they invited her to share a ride back on a private boat and to split the cost of the passage with them. It was as near as Aisling had ever been to the girl, who smiled shyly but kept up her talk with her father in their own patois, which was, like all local brogues and dialects, indecipherable to their fellow travelers.
"Greek to me, too," said Blake Shields, seeing Aisling straining to make some sense of the Jamaicans' talk. "I get bits and pieces of it now and then." He was trying to be courteous, leaning in closely to be heard in the open boat taking them back to the island. "Are you enjoying your stay on the island?"
Aisling praised the peace and tranquillity, confessed that she'd stayed past her time, had missed another week of classes downstate but couldn't seem to work up the will to leave. He told her he'd be leaving in a week, that the community college he went to in Petoskey always started later because all the students worked the season in northern Michigan.
"What about your girlfriend?" Aisling asked, as if she were only making conversation.
"Binta?" he said, as if it wasn't certain, the nature of their relationship. "She's missed a week already too. She attends an all-girls college—really it's a high school—in Jamaica. But her father's letting her stay until I leave next week. My folks will get her to the airport."
The young man kept talking about how he had a bad case of island fever, couldn't wait to get back to the mainland, was disappointed with the tips this summer—"Everyone's down twenty, thirty percent"—but Aisling was elsewhere in her private thoughts. She was looking out into the dark water as the hum of inboard motor dulled their voices. She was thinking of what she would do without her, trying to imagine them both back to their separate lives. She closed her eyes and could see young schoolgirls in the Lesser Antilles where Nigel had taken her over Christmas and New Year's not long after the pregnancy. He'd rented a villa overlooking Cruz Bay on the smallest of the U.S. Virgin Islands. She could see Bintalou now, as she remembered them, black schoolgirls in their baggy uniforms—yellow cotton blouses over pleated blue skirts, loose neckties, and white anklets—moving like pretty schools of fish in unison through the village on their way to and from school, mornings, afternoons, how he had been so perfectly solicitous of her, so tender, bringing her orchids and poems and bottled water, trying to hasten her heart's slow healing.
IN THE fourth week of her stay on Mackinac, Professor Aisling Black made certain observations about the world around her. The panic in the press, local and national, was growing more fraught with every day. There were rumors of banks on the brink of collapsing, an insurance carrier and a well-known investment house had bellied up, the candidates for the coming elections were full of dire predictions and promises. "Toxic assets" and "derivatives" had entered the talk among townspeople and merchants. The manicurist in the hotel salon told Aisling that last week a family of guests had been asked to leave when the fees for their stay far exceeded the limit on the credit card they'd provided at check-in. A yacht in the harbor had been repossessed: everywhere were signs of what the woman called "a contagion" among the moneyed set. And while Aisling's own interests had never included money or concerns over solvency, her instincts as a scholar triggered an appetite for researchable data. She spent a day on the Internet examining the financial press and their websites, emailing her own accountant, checking her online bank accounts. She would have called her father but they'd grown distant over the years since her mother's death and she could not imagine a conversation that would allow either one of them to reengage.
What was worse was the knowledge of Bintalou's coming departure. Aisling tried to imagine her life after the girl was gone. She had come to see herself almost entirely as an acolyte to the incarnate beauty.
By now she had figured out the predictable elements of Bintalou's routines. She knew, for example, that Sunday mornings would find her at worship in the Little Stone Congregational Church on the edge of the golf course between her dormitory lodge and the hotel. And though Aisling had given up any practice of religion, she attended the services, sang the old hymns along with the faithful, was generous when the basket was passed, and watched closely as the light through the stained glass illumined the angelic visage of her beloved. The pastor of the church was a man with a trained preaching voice, a toupee, and a precision about his clerical dress which, like the rest of the island, labored to seem from ages past. He preached about the gifts of the body and the spirit—good health and contentedness, fitness and fidelity—"things beyond the purchase of any treasure!" Then he led the congregation in song, their voices rising to "Softly and Tenderly Jesus Is Calling," which Aisling sang with real conviction and during which Bintalou, singing along with the rest of the church, turned and looked Aisling deep in her eyes, which had gone blurry with tears at the penultimate verse of the hymn:
Time is now fleeting, the moments are passing,
Passing from you and from me;
Shadows are gathering, deathbeds are coming,
Coming for you and for me.
That is when she knew that the long weeks of her pursuit of beauty might soon be coming to an end.
Thereafter, like any lover preparing to please her lover, Aisling took more time with her selection of costume, accessories, jewelry, and fragrance. She spent abundantly in the hotel shops and spa, in the latter of which she gave herself over to a full day of ministrations, from the Age Defying Purification Facial to the Herbal Stimulating Body Wrap and Hot Stone Massage. Considering herself in the full-length mirror, wrapped in a terrycloth and satin towel, the sight of her graying hair made her grimace.
When she returned to her room, coiffure and cosmetics perfectly done, she seemed a much-restored version of herself, from years before, in the weeks and months of her courtship with Nigel, readied and willing and eager to love. A version of the future that continued to take shape in Aisling's imagination was the one that involved an approach to Henry Goodison and the offer to take his daughter on as her personal assistant. The girl could come to live with her in Ann Arbor. The house in Burns Park was surely sufficient. Aisling would enroll the girl in Greenhills School, one of the best preparatory academies in the country, and eventually, of course, at the university. She would further agree to winter holidays in Jamaica and summers on Mackinac, but Bintalou would get the best of educations, eventual citizenship, a generous stipendium, and better prospects for the future. She could assist Aisling with the household duties, cooking and correspondence, shopping and social obligations. She would travel with Aisling on the widening circuit of literary duties, seeing the country and the world. It would be a remarkable tuition for a young woman and increase her own chances measurably; surely Henry would see the wisdom in all of this. Surely he'd want what was best for his daughter. They could share mealtimes, intellectual pursuits, even clothing—Aisling was sure all her things would fit—daily hopes and little heartbreaks. She could become the sister, daughter, mother, and partner, now that she thought of it, she never had. It was hereabouts that the revelation of this plan would begin to obscure itself in Aisling's contemplations.
It was mid-September—long past the time she should have returned to school—when the professor rented one of the hotel's bicycles and followed at a distance while Bintalou and her young coachman rode ponies out to the north side of the island. All the August crowds were gone. The senior citizen tours remained in town, taking in the blacksmith's shop and Beaumont Museum, browsing for bargains in the stores, rummaging among the kitsch and knockoffs, the replicas of former treasures, the knickknacks and copied curios.
By the time they turned northwestward around Mission Point, out past the Arch Rock and Voyageur's Bay, there was no one on the road but the pair on horseback and Aisling on her bike in pursuit. The late morning was warm and windless and bright blue, the lake water glistening and smooth. Aisling felt like a girl again on the Schwinn coaster pedaling along Lake Shore Road, further and further from the town. It was four miles to Point aux Pins—the tail of the turtle the island was shaped like—and as Aisling turned south she found the ponies tied to a picnic table off the road. This end of the island was heavily forested with white cedars and silver birches and the land pushed out overlooking the lake. Aisling settled her bike in the woods and walked through the trees toward the water past the remnants of an abandoned log cottage, coming to a high promontory out of which trees grew at angles and the land beneath gave way to a high bank going down to the lake. She could hear their voices rising up from below. There was no sign of the footpath they must have taken and Aisling was about to double back to the old cottage to see if she could find a way when she considered the base of a cedar tree growing out at an angle of forty-five degrees and another one next to it, their dense branches interlacing forming something of a natural observatory. Crawling out astraddle the trunk of the one tree, first balancing off the other tree, then bending forward to embrace the first, she found herself hidden in the leaning trees fifty feet overhead a patch of sandy beach the couple below her had found on an otherwise rocky shoreline. The outgrowth of arborvitae, its flat filigree spray of leaves, provided perfect cover. She could see them and could not be seen.
Bintalou had brought a little lunch of fruit and cheeses from the Grand's buffet and from the same backpack produced some bottled water and a pair of towels. After eating they both stood, as if on signal, removed their shorts and T-shirts, their white undergarments, folded them carefully in the towels, and together, quite naked, ran into the lake wearing only their shoes because the bottom was rocky. Their shouts rose upward from the shock of water.
"It's cold!" cried Bintalou.
"Dive in," he called back, "you'll get used to it!"
Aisling's limbs tightened her embrace of the tree and edged a little forward for a better view, the bark of the white cedar scratching her knees.
They were so gorgeous in the water, her perfect blackness, his perfect white, his blondness, her dark brunette, their buttocks and genitalia, those lovely, unspeakably perfect breasts. She could hardly breathe now that she saw the girl standing ankle deep in the shallows, bent ever so slightly forward, her left hand on her right elbow, hugging herself, her right hand in between her knees trying to warm itself, or cover herself—she could not say—looking off in the distance at the Les Cheneaux. Aisling's eyes were tearing up, her vision blurring; she was so grateful now for everything. Eventually the girl dove headlong into the deeper water and swam out to meet her companion, the two of them playing like children at the lake, trying to dunk one another, splashing each other, laughing, their bodies shining in the light and water.
Aisling considered climbing back to the land, finding her way down to the beach to join them, organizing excuses for how she had just "happened upon this place," all very innocent and serendipitous. She would remove her own clothes as she approached them so as not to make them uncomfortable, so as not to make them cover themselves. She wanted to be naked with them, to tell them everything, to hear their histories and to tell them hers, how the heart bears its unspeakable cargo to lay it down at the feet of beauty.
But she could not move, she could not speak, her eyes were closing with her own arousal. She hugged the tree more passionately, pressing herself against its thickened bark. She felt her pulse rising in every vessel, behind her eyes, in her temples, in her limbs.
Whether she saw them or dreamed them or prayed them to step from the water and lie on the beach beside one another in the midday sun, their heads on the pillows they had made of their clothes, first touching one another with such tenderness, then taking one another unto themselves; whether it happened as she saw it or did not see it, she could not know. Whether it was the lapping of lake water or the caught breath between them, their own entanglements or hers alone, she could not be certain. It hardly mattered. She lay in the soft embrace of the cedars, her arms and legs gone limp, her cheek turned softly against the tree. Whether ecstasy or aneurysm, apocalypse or broken heart, seizure, stroke, or coup de chance; whether every beauty echoes another—one's innocence, all art—whether to live for it or die for it, no longer mattered. The vision before her seemed to beckon, though her own vision was doubled, blurred by the leafy shade and light, her tears, the body or bodies, or souls—she could not know. She could feel her embrace of the cedar loosening. Her body and her being letting go.
## Apparition
I
IT WAS Good Riddance that put Adrian Littlefield on the lecture circuit. Before Riddance he'd self-published two pastel-covered self-help books with fashionably gerundive titles—Learning to Love in the Present Tense and Making the Best of the Worst-Case Scenarios. They were widely ignored and only sold to his family and folks at the church where he was the assistant pastor in those early years of marriage and parenthood. He'd done a workbook on dividing family duties between spouses and written a couple of articles for the ministerial press but otherwise was going nowhere and barely making ends meet until, after Clare left him, he wrote Good Riddance—Divorcing for Keeps and it changed his life.
It held that some divorces, like some marriages, are made in heaven. And we ought to be thankful for them. The key to living in concert with God's Will or the Natural Order or what the Fates had in mind for you was to learn to accept the direction your love life was taking you, even when it meant the end of love. Divorce was neither the result of too much of one thing nor too little of another, too many heartbreaks or too few. It was, like tsunamis and famines, hurricanes and genocides, God's way of culling humanity's herd of lovers, for reasons that were unknowable to mortals, but part, nonetheless, of a larger plan. Shit happens—Adrian Littlefield told his readers unambiguously—we must go with the flow. That's life, get over it, get on with it.
It took him six months to write it, a month to get an agent, and another month to get a contract and a fat advance from a publisher in New York. It was an immediate hit. The first printing sold out in a week. The paperback rights went for half a million. There was talk of a documentary for one of the cable networks.
The invitations followed. He worked up a little forty-minute shtick for the keynotes and workshops. He could bend his presentation, peppered with tastefully suggestive humor, around the occupational curiosities of any professional association. He had an infomercial in postproduction—one of those hour-long specials where he wore baggy clothes and did a little chalk talk with a really attractive and earnest-looking studio audience. He traveled three months out of every four from his home office in Findlay, Ohio, where he housed, behind his sprawling redbrick Queen Anne mansion on the corner of South Main and Second Streets, in a carriage house he'd refurbished with money from his books and lectures, the Center for Post-Marital Studies—an elaborate tax shelter, along with the foundation that raised funds to advance the work of the CPMS, which was primarily to pay its principal apostle—himself—to spread the word such as it was revealed to him. He'd kept pace with the rapidly expanding technologies of communications from bulk mail to blast fax, to email to website and blogosphere. Some nights, posting the latest news to the website, or linking a recent interview so his followers could listen in, he felt not a little like Paul writing epistles to the various churches. If, as that great circuit-rider wrote, it is better to marry than to burn, was it not much better still to divorce than to smolder? "Exes," Adrian's favorite slogan held, "you can't live with 'em and you can't shoot 'em. A little Good Riddance goes a long way."
Walking in the moonlight across the gardened yard between his office in the coach house and his residence, he would often consider how far he'd come in Findlay, Ohio, after all, from the little clapboard manse on Cory Street behind the church to this three-story palace with its towers and turrets, bay windows and balconies, its dozen cut-brick chimneys, limestone lintels and sills, its stained-glass transoms and fish-scale slates and fluted copper downspouts, its twelve-foot ceilings, tile baths, and mammoth basement with caverns and wine cellar, coal bins and cisterns, its dumbwaiter and dentil molds and third-floor ballroom, its nine species of hardwood floors and cabinets and crown moldings. And he would meditate on the apocryphal book of Sirach, its wisdoms on comfort and pleasure in 14:14, chapter and verse, to wit: "Defraud not thyself of the good day, and let not the part of a good desire overpass thee." He had come to admire about the Hebrews and Greeks their sense that this life might be as good as it gets, that whatever might be coming after this—Hades or Shades or heavenly mansions—mightn't be something to bank much on. Maybe it's much like the devil, he thought: the heaven you know is better than the one you don't. Adrian would look up into the moon's face, whatever there was of it, and offer thanks for the progress he had made since he'd been left by his wife and quietly exiled from St. Mark's Methodist, across the street and three blocks north, all those years ago. That local Methodists could not help but pass his place and might pause to wonder in their daily rounds if the mansions being prepared for them in heaven would ever be as sumptuous as the one their former associate pastor now occupied in Findlay was part of the good day of which Adrian would not defraud himself. Such sentiments he knew did not ennoble him but there they were and he could not deny them. The retinue of Mexican landscapers, housekeepers, and handymen required in every season, the steady traffic of visitors and conferees who would come for weeklong intensive residencies over which Adrian presided like a maharishi to the formerly married, the occasional news or documentary crew, the place's reputation for haute cuisine, wine tastings, harp and flute recitals, a growing collection of private art, book launches—these were all the stuff of local rumor and gossip. He had become in Findlay a man of parts, none exactly known entirely. To Methodists and their united brethren, no less to the nonbelievers hereabouts, Adrian had become a local notable and celebrity. After pissing off the back porch into the thick beds of variegated hostas that bordered the place, he would enter the back door with a benediction, "God bless all here and bless this house," pour himself a glass of something, and make his way upstairs to his private rooms.
In the years since Riddance was released, he'd written The Good Riddance Workbook and Questions & Answers About Good Riddance, thereby giving the trilogy his publisher said really would saturate the market. A coffee table book, provisionally titled Second Chances and featuring the personal stories and posed photographs of successfully divorced people from all over the country, was in production. It showed them in their new habitats and fashions, smiling knowingly, or looking healthy and well and newly "centered," preoccupied with lives of evident merit and higher purpose. One of Adrian's devotees who did architectural photography was doing the images while Adrian was preparing some introductory text. The more that offers came into the Center for PMS, the more he raised his fees for speaking, which had the effect of making him seem worth that much more, which brought, of course, more offers. He counted it an irony, and a pleasant one, that he'd become, de facto, an itinerant preacher—the calling he'd felt early and often in his youth—albeit preaching a secularized gospel that was a hybrid of pop psyche, warm fuzzies, personal witness, and cultural study. That he homilized not from the pulpits of great cathedrals but from the lecterns of convention hotels struck him as part of the Creator's plan for him. He'd stopped saying "Gawd" in the deeply reverential tone of his Methodist training and taken rather to the user-friendly, guilt-free parlance of nature and creation. It was, he told himself, more "inclusive." That he traveled like a Wesleyan but was paid like a free-market capitalist filled him with a sense of this life's mysteries. Likewise, he'd given up "the Reverend" for simply "Dr." Adrian Littlefield for the scholarly, vaguely medicinal ring that it added to his brief. He had a D.D. mail order from one of the agencies that advertised in clergy magazines and an honorary doctorate from his alma mater, Ohio Wesleyan, where he'd given the commencement speech the year after his book came out. He had done postgraduate work at the Methodist Theological Seminary in Delaware—a course of study he let lapse after the divorce. He was certainly not the first to observe that the high priests of the current culture had secular rather than sacred credentials. Not lost on him either was the happy fact that divorcing, conventionally regarded as a failure, had been the essential key to his success. Though the scandal had cost him his job as the long-suffering and underpaid associate pastor at St. Mark's, as soon as his wife left him, he became the tragically abandoned and heroically single father of two youngsters. Once the book came out, with excerpts published in Redbook and Esquire, interviews on public radio, profiles on prime-time network shows, he'd become a kind of local hero. He'd even gone to Chicago for a sit-down with Oprah, to chat and answer the audience questions, which appearance alone had accounted for a massive third printing of Riddance and an audio book.
When he found himself, as he often did, disembarking from some posh hotel, with the hefty stipend cooling in his briefcase, the livery sedan idling at curbside waiting to take him to the plane, the appreciation of conventioneers ringing in his ears, and, as luck would occasionally have it, a woman waking in the bed upstairs to find him gone but not, he could convince himself, entirely forgotten, Adrian Littlefield's heart filled with thanksgiving for the failure of his marriage. All things, he told himself, work together toward some good. And the new life, such as he had come to know it, had restored to him some holy order, a sense of real purpose and calling and voice. If God was a practical joker, well then he would grin and bear it. He offered, in such moments, silent and abundant thanks and praise to Whoever Was In Charge Here, as he had come to call his Lord and Savior, the one he believed might still be out there, wherever, listening to his heart of hearts.
IT WAS the National Association of Family Law Attorneys (NAFLA) annual meeting in Connecticut that brought him to the Foxwoods Casino—a high-stakes bingo parlor parlayed by the Mashantucket Pequot Tribal Nation into one of the most profitable gaming emporiums in the country. It rose out of the hilly forests between Norwich and the coast, like something out of Kubla Khan, its lights blazing in the darkness, a pleasure dome on the "rez."
"More than a hundred thousand an hour, twenty-four hours a day, three hundred and sixty-five days, et cetera, et cetera," the limo driver who met his flight in Hartford said, drawing out the syllables of the et ceteras for emphasis; "that's net-net in profit. Every day. Pretty good revenge, eh? We gave them firewater and reservations, they give us Keno and the dollar slots. They don't call it wampum for nothing, eh!"
The shops and restaurants and conference center and hotel all played into the tribe's master plan—to disabuse as many of their former oppressors of as much money as possible in the most mindless way. It seemed a suitable locale for the divorce attorneys to hold their annual meeting. Dr. Littlefield was the keynote speaker. His fee was fifteen thousand, first-class travel, ground transport, premium lodging and meals. His books and videos sold briskly after his lectures. He signed them Best Wishes—A.L., and if the purchaser was a woman of a certain age and style he'd add, every available benevolence. He thought this sounded a literary chord, and while what he wrote was far from literary, he thought the pretence would do no harm. Also, it gave him time to ask some chatty questions.
"And what do you do?" he'd ask her flatly.
"Whatever it takes," or "Well, that depends," some would flirt, whereas others would offer only "mediation and depositions."
In most cases he would fly into a venue the night before, check in, order a chicken salad from room service, watch a movie, and go to bed. He would be up early the next morning to walk, then ready himself for his standard forty-five-minute speech, fifteen minutes of Q&A and whatever it took to sign all the books. Then he'd collect the balance of his fee and make for the airport and the next venue.
But the NAFLA conference was in late July, in a resort casino near the sea in New England. He'd been going nonstop for most of the year and the pictures in the preregistration packet looked inviting. Along with the predictable workshops on "Custody Issues" and "Grandparents' Rights" and "Pre-and-Post-Marital Agreements," there was a "Traditional Yankee Clambake" scheduled at Mystic Seaport, a golf outing, and "A Day on Beautiful Block Island." He'd never been to Block Island but he had kept a picture of the place in his imagination for years. The brochure photo of handsome couples assembled around fruity drinks, smiling from the wide porch of an elegant nineteenth-century hotel, caught his eye. He told the organizers he'd be staying for the entire conference, and while he wouldn't golf with the attendees, he'd be happy to eat with them and wanted to take the trip to the island. They were happy to book him two more nights in the suite, knowing that their members would be pleased at the chance to visit with their keynote speaker.
THE TOUR bus from Foxwoods to Point Judith took an hour. The boat from Point Judith to Block Island took another—thirteen miles across Block Island Sound from Galilee, Rhode Island, to the town of New Shoreham at the Old Harbor. Stepping aboard the car ferry Anna C., Dr. Adrian Littlefield tried to imagine what crossing water must have added to the romance his former spouse must have felt en route to the first of her infidelities. He took a seat on the middle deck of the ferry among the conferees from NAFLA, who sat in a block in the first few rows of benches, careful to get an aisle seat for the escape he planned once the boat was on its way. They waited for cars and day-trippers to board, admiring the small fleet of fishing vessels in Point Judith. The Enterprise and Lady Helen and Stormy Elizabeth—it was not so much the names of vessels as the black riggings and booms and spools of netting that put him in mind of life's entanglements. He thought he might begin another series of books entitled Life's Entanglements. He took a notebook from his shirt pocket and wrote in it, Life's Entanglements? and underlined the entry.
"Good Riddance was a miracle for me! I have to tell you, Doctor..."
"Oh, thank you," he smiled at the red-haired woman sitting next to him, "that's very flattering."
"No really, Doctor. It gave me permission to...I hope I'm not interrupting your—"
"No, no, not at all. I'm glad you liked it."
"I mean before your book I never would have, you know, felt empowered." She emphasized the middle syllable of "empowered" like she had "permission."
"Yes, yes," Adrian said, and looked deeply into her blue eyes, trying to imagine what idiom from the lexicon of daytime talk shows she would give out with next, a scrutiny which the red-haired woman mistook for interest.
"Of course, there's so much baggage we had to let go of first, my ex and me, before we could, you know, grow up and grow apart, you know, together."
Adrian nodded and smiled and stood and took her hand and held it meaningfully before looking about as a man does looking for the nearest toilet.
"Too much java," he said, to explain himself, and the man next to the red-haired woman pointed toward the rear of the ferryboat where they all had boarded.
"Head's at the back, Dr. Littlefield, we'll save your spot."
Adrian smiled and made his move to the back of the boat just as it was disembarking and climbed the stairs to the open upper deck.
Block Island was the site of his former wife's first infidelity. He'd only ever seen it in pictures—photos that she'd brought home from her trip to New York that April, ostensibly to visit her friend Christina. "I just need to get away from Findlay, and family and kids," she told him, "just for a week, a little perker-upper. Christina has taken the time off of work, we're going to do girl things." Clare was thirty-three at the time. They had a son and a daughter, ages eight and four. They'd been married almost twelve years. Her discontent was palpable.
"Oh, Ben invited us out to the island for the weekend," she told him when she returned from the week away, leaving him with the children and the house and his own work to manage. "You remember Ben, don't you? Uncle Harold's friend? The artist, you know, Harold and Olive's neighbor. Christina and I ran into him in town, at his studio in SoHo, and he invited us out to the island for the weekend. We couldn't get theater tickets so we decided to go. Harold and Olive were going to come too but canceled at the last minute."
Adrian remembered the determinedly plural references and the way it was supposed to work against his suspicions. And the supposed serendipity of it all, all very last-minute and carpe diem, nothing planned. Clare labored to make it all sound like happenstance, which of course made it all the more suspect. He rummaged through the photographs for any that included Christina. But there were none. There was Ben on the sailboat, manfully at the wheel, Ben in the kitchen, turning from the stove, Ben and Clare smiling from their places at a table on the porch of what looked like an old hotel. The table they were sitting at was set for two. Where, he wondered, was Christina? Where were Harold and Olive? Where were the photos of the "girl things" they had done? Afraid of the answers, he never asked.
He'd met Ben once, the year they all drove East to visit Harold and Olive in Westchester. Harold was wealthy and worked in the city making investments for an insurance company. He had an office in Rockefeller Center and spent his lunchtimes all winter skating on the rink there and in the summer jogging through Central Park. All of Harold and Olive's friends were, like them, fiftyish, well-off, fit, and always grinning. None of them smoked. They all took vitamins. Everyone practiced some New Age regimen to guarantee a particular wellness. Ben was the next-door neighbor, "heroically still married to his disabled wife"—an artist who lived on his earnings as an illustrator but was "really just waiting to sell his marvelous oils." He was tall and smooth-skinned and deeply tanned and his white hair and full beard made him look almost biblical. He'd done some covers for Life magazine and the Saturday Evening Post. Clare had confessed to Adrian the crush she'd had on Ben as a girl when visiting at her uncle's after her parents' divorce; Ben had made a fuss over her in some way she never elaborated, something to do with Rockefeller Center. He would have been in his thirties then and she'd always really "felt really very special, you know, that he'd make such a fuss over a fifteen-year-old girl."
Clare's own father had left when she was twelve, for reasons that were never made clear to her. One day he came in and said that he would always love her and then he left. Her mother always looked a little wounded after that but never said a word about the divorce except "I'm a one-man woman," which is why, Clare reasoned, her mother never remarried. After that they spent most summers and most Christmases with Uncle Harold and Aunt Olive.
All these years since, it all looked simple and predictable to Adrian now—the girl abandoned by her father looking for an older man's approval, attention, etc. etc. etc.—it was all embarrassingly usual, unremarkable in every way. A textbook case: she was driven, consciously or subconsciously, to replicate her father's rejection by sexual misconduct that would assure her husband's rejection. They had arranged their little off-season tryst while Adrian, the earnest ignoramus, stayed home with the kids and the church work, the reliable garden-variety cuckold and bumpkin, a brute if he raised objections, a wimp if he didn't, finished either way.
The Anna C. sounded its horn once and made its way out past the harbor's bars and seafood restaurants, out past the sunbathers waving from shore, out past the rock pilings covered with cormorants, into the open water followed by maybe two dozen gulls which soared alongside the ferry for food tossed from passengers. Adrian Littlefield, seated between strangers on the long bench on the top deck, considered his fellow passengers from the isolation he had learned to wrap around himself in public transit. There were couples with children, college students in packs of various sizes, Asian tourists, and pairs of lovers, some obviously married, some obviously not. He watched as they grew more affectionate the further out from port the boat traveled. The touching and hugging and holding, and even kissing, grew more manifest as the mainland grew more distant. This he assigned to the crossing of water, the sense of privacy that passage to an island must add to the sense that all lovers share of being alone against the world and its elements.
One young couple, kissing like newlyweds, looked to Adrian like his son, Damien, and his girlfriend. They were handsome and serious, maybe bound for their honeymoon. The young woman snuggled into his half embrace, safe and sound under his protection. The young man had Damien's wary eyes, keeping a reliable watch over everything, against the ever-present danger of the worst thing happening.
II
"LET'S GET Mom flowers," Damien would say, on the way home from grade school or St. Mark's day care. "That'll surprise her."
The things Adrian remembered after twenty years.
Adrian knew that Damien knew that all was not well between his mother and his father. And he knew the boy would keep trying to fix it.
It still stung him to think of the way the boy, on those increasingly rare occasions when the family dined together, would rise to clear the table and offer to read his sister a bedtime story so "you and Mom can go for a walk together around the neighborhood."
The sweetness in the boy's eyes—his mother Clare's eyes, huge and blue—his sense that it could all be fixed with good behavior and the proper gesture, his hope in the face of a hopeless situation, still made Adrian Littlefield wince, these many years since, at the pain he knew had been inflicted on his son and daughter by the undoing of the marriage and the family. How his son had labored to keep them all together, to keep the little family unit intact, to keep the "happily ever after" fiction going, still stung Adrian in his heart of hearts.
Damien was grown and out on his own now. He lectured in the Religious Studies program at a small private college in Michigan, happily distanced by nearly two decades from all that sadness, happily preoccupied with his own life's course, which included, apparently, a colleague who taught Old Testament Literature and who Damien had told his father "might be the one!" That the failure of his parents' marriage had not entirely disabused Damien of a sense of providence and kindly fates was a comfort to Adrian, who wondered at the resilience of the young, but remained vigilant for signs of permanent damage.
Damien called frequently and emailed often and drove down twice a year—for Father's Day and Christmas. He'd brought Jocelyn with him the last time he came, a pretty and pleasant woman, a few years older than Damien, not beautiful, but fetching, Adrian thought, diminutive and bookish, a perfect pastor's wife or professor's.
Adrian made his visits, too; whenever his lecture circuit took him anywhere near his daughter in Chicago or his son in Michigan, he'd add on a day to pay his call. They'd have dinner, see a movie or some local attraction. They had devised, if not any sense of a family "home"—safe harbor, big extended family dinners, year-in, year-out traditions—nonetheless a kind of family life à la carte, keeping up contacts and appearances, juggling the reconstituted relations admirably.
Still, Adrian's remorse, his guilt for his part in it—the poor choice of spouse he had made, his failure to retain their mother's love, the remarkable sense of liberty once she'd gone, and the turn for the better his own life took once she had vacated their lives and premises—it all still netted out as sharp regret whenever he thought of his daughter, Sarah, named for the patriarch's wife, or his son, Damien, named for the saintly priest who'd served the lepers on Molokai; Damien, a beautiful, sad boy just gone eight, that awful summer all those years ago.
AT THE time, of course, it seemed so sudden; but in hindsight Adrian could see the turns in the journey that ended in their divorce. Clare had given up her household routines and enrolled in classes at the University of Findlay. It was the mid 1980s and she wanted to go into video production and shoot films that would get played on MTV, like Michael Jackson's Thriller, which she watched over and over and worked out to at her aerobics class in the morning; or documentaries about wage discrimination and spousal abuse, the urban poor and world hunger she would eventually sell to public TV. She was waging a war against the tummy and cellulite that had appeared on her body and which she blamed on her pregnancies. She went off in a sweatband and leg warmers and a leotard in the morning, then came home and showered and went off to school for the afternoons. Adrian took Damien to Lincoln Elementary and brought Sarah to the day care at St. Mark's Methodist, then picked Damien up after school and brought them home and made them supper. Clare let it be known that she'd been more or less duped by the culture into marriage and having babies. Adrian was part of the conspiracy—not willfully perhaps, not purposefully, but part of the conspiracy nonetheless—that sought to shackle Clare's life and prospects to these other lives. Damien and Sarah were part of the conspiracy too, needy as they were of so much attention. Like his father, Damien constantly strove to make Clare happy—he was an excellent third grader, he read avidly, drew colorful pictures of happy vistas. Sarah was a tidy preschooler, too young, Adrian convinced himself, to sense as Damien sensed the coming disaster.
In the end it was not Ben Walters, the aging artist back East Clare left them for, but a fellow student from her video production class whose family had money and Clare could imagine financing her own filmmaking. Ben Walters had only been a test—a springtime fling, to see if it could be done, if other men wanted her, if she could do it. The summer she eventually moved out, the summer after the late-spring tryst with Ben, she spent shifting her friendships from hers and Adrian's circle of young parents and church families to the younger, entirely hipper, vastly more imaginative group of undiscovered artists and photographers David Eason hung out with at the university. David dressed in black denim pants and black silk shirts and wore a wide-brimmed black hat and looked artistic. David had long fingers and long curly black hair and dreams of enrolling in NYU's photography school; and Clare thought they'd go off to New York together, live in SoHo, and be young forever, and interesting.
When it dawned on Adrian that his wife might be having an affair, after David Eason's name kept turning up in her monologues on life and school and her widening prospects, after the same sick knot began to tighten in his stomach that he remembered from the months before when she'd returned from her adventures on Block Island, when the phone bill kept turning up lengthy calls to a number he didn't recognize, always placed during hours when he was at the church; when she'd stopped coming to bed with him, and making excuses for her changing schedule, when he could no longer account for the changes in her, he asked if she'd "see someone" with him. A therapist or counselor, "someone to talk to about saving the marriage" because, as he tried to explain to her, "there are two little lives depending on us, to get this right and keep it together." When she refused his suggestion about marriage counseling, he found someone he could talk to himself, to sort out his feelings of anger and fear. Eventually it was this therapist who diagnosed Clare in absentia as "possibly a narcissist" and Adrian as "possibly an enabler, a co-narcissist," a man who would never satisfy a narcissist's appetite for grandiosity, and the more he tried the more feckless he would seem to her and that for her part Clare was probably "already psychologically divorced" from him. It was then that Adrian hired a private investigator to follow Clare around and take pictures and document her adulteries. Adrian believed it would help with the eventual proceedings and with the custody issues. And he thought he should have certainty, beyond reasonable doubt, before dropping a hammer he knew would do irreparable damage. He could imagine a life without Clare but could simply not abide a life distant from his children's lives.
He got photos of Clare and David coming and going at hotels along I-75, tastefully north and south of Findlay, where only transients and other adulterers might be. There were detailed logs of their arrivals, departures, room numbers, dates and times. Adrian calculated that the average stay was 2.75 hours, which he tried to divide into foreplay, intercourse, and afterglow, and was bothered to find himself slightly aroused by these contemplations. After twelve years he felt he knew Clare's sexual repertoire—what she liked to do and have done to her—and the thought of her doing those things with someone else first enraged, then excited, then sickened, then saddened him. There were photos of them having dessert at a Cracker Barrel restaurant near the Lima exit. They looked like teenagers, sharing a piece of Boston cream pie, surrounded by booths full of old married couples, their forks hovering over the pie on the plate, nearly touching, their eyes fixed deeply on each other's eyes. There was a background workup on David Eason, who, except for the fact that he still lived with his parents at age twenty-nine, in the carriage house of one of the great mansions on Sandusky Street, and had been hospitalized once for an unspecified nervous disorder, seemed unremarkable in every way. A month or so into their affair, midway through the spring semester, Clare and David planned a romantic weekend at a boutique hotel in Cincinnati. Clare had told Adrian she was going to film a short movie about the Anderson Ferry that brought cars back and forth across the Ohio River, to fulfill requirements for her video arts class.
"It's the last of the old ferryboats on the Ohio—since 1817! There's a whole crew of us going, production and sound, camera and lighting, we're going to submit it to short film competitions. We'll be staying in dorm rooms at Mount St. Joseph. There are no phones. I'm riding with the other camera operator." Clare had her story perfectly constructed. Adrian had smiled agreeably, and nodded and said nothing except that he'd watch the kids. While she packed a small bag, he called his detective.
That evening, after Clare went off waving from David Eason's Toyota van, Adrian took Sarah and Damien out to St. Michael the Archangel's for the annual parish fair and let them ride the rides and eat cotton candy and sit under the big tent eating spaghetti and meatballs with the Catholics. He walked with them among the booths of crafts and white elephants and children's games and bought them both chances at ringtoss and beanbags. He could not resist the urge to spoil them with diversions and easy pleasures. He peeked into the casino tent with blackjack tables and roulette wheels and tables of bingo games and wondered at the way these otherwise devout people would wallow in sin for a worthy cause. Whereas the Baptists had bake sales and the Methodists did Christmas bazaars, the Episcopalians favored crafts and antiques sales and the Presbyterians were forever doing dinners and teas to raise money for their various causes, there was something to envy in the way these Poles and Germans, Italian and Irish Catholics would indulge their nearly pagan appetites to put some money in the priests' pension fund or a new roof on the rectory or construct a bell tower to sound the Angelus all over town. Food and drink and dance and games of chance—St. Michael the Archangel's parish fair had become the biggest and best of the churchy entertainments in Findlay, the opening of the summer season, and folks from all over Hamilton County came.
Adrian sat at a long table littered with half-empty plastic cups of beer and soda, under the main tent where a band was playing. He gave Damien a roll of tickets for rides and games and told him to hold his sister's hand. He sipped a Coca-Cola and looked at his watch to figure how far south Clare was by now. An accordion and clarinet wheezed between drumbeats from the bandstand at the end of the tent and Adrian found himself fixed on the long-married couples—bald men with plump bellies and women half again their marriage weight, still holding one another, after thirty or forty years, through waltzes and polkas, a little tipsy with the beer, the press of coreligionists and the humid evening air, like figures out of a Flemish painting, all knees and elbows and red faces sweltering under the canvas and party lights, circling in this rollicking dance.
"How you keeping, Adrian? Good of you to come."
The priest startled Adrian from his contemplations.
"Oh, yes, yes, Francis, fine. And how are you? This extravaganza gets bigger every year!"
"First full convergence since the sixteenth century! You wanna raffle ticket?"
Adrian was still catching up to the priest's conversation.
"Convergence? Raffle?"
"Yep, first time since the Council of Trent that Friday the thirteenth and the full moon in June fall on the same day—should be lucky. First prize is a trip to Cancún for two. Clare would like that, Adrian. What say ye?"
The priest could sell anything, Adrian thought, especially in his slightly manic, larger-than-life incarnation as Father Francis Assisi Concannon, priest of God—half huckster, half holy man—the six-foot-four-inch lumbering frame clad in a Hawaiian shirt and Bermuda shorts, one fist full of dollars and another full of raffle tickets. Adrian fished five dollars from his pocket and handed it over.
"How many will that buy?"
"Only one, but it's all you'll need, bucko!"
Adrian and the priest had been friends for ten years since they'd both come to Findlay, fresh from their separate educations and appointments. The one and only meeting of the Findlay Ministerial Association, now defunct, had been held at St. Mark's Methodist. They played racquetball together, took in a movie from time to time, commiserated over parish politics and their bishops, and traded titles of books to read. Father Concannon favored Irish poets while Reverend Littlefield preferred spiritual guides and homiletics. The Collected Poems of William Butler Yeats got traded for The Best of Robert Ingersoll. Beckett got traded for Frederick Buechner. Every so often they'd go to dinner in the next town over, where no one knew them and they could be free of the scrutiny of parishioner and congregant.
"And where's the beautiful Mrs. Littlefield on such a fine moon and June and spooning evening?" The priest loved the sound of the pursed vowels in his mouth second only to the sound of his own voice. He pressed a ticket and pencil on the table for Adrian to fill out.
"She's gone to Cincinnati, to Mount St. Joseph."
"Jaysus, Adrian, she's gone and joined the convent on ye—the nuns will rob you, man."
Mount St. Joseph was one of the many holdings of the Sisters of Charity in Cincinnati—a four-year school for good Catholic girls, a fraction of whom would get a calling and join the order after graduation.
Adrian scribbled his name and phone number on the ticket stub, tore off his portion, and gave the book of tickets back to the priest.
"She's gone past the point where the nuns would take her, Frank."
Adrian wondered if the priest could hear more than small talk in what he'd said, the way he'd said it, and there was this sudden panic that he'd have to explain, to everyone—his parents and children, his senior pastor and his congregation, his neighbors and friends, old and new, the lawyers and taxman, God in heaven, everyone, everyone would know—that he had failed as a husband and father and head of the household; he had failed to keep his marriage intact, he had failed to keep his wife happy and satisfied and at home with her family where she belonged. Because he knew at the moment she was riding southbound with her new lover, David Eason, on I-75, maybe indulging in a little highway sex, his hand in her panties, her face in his lap, the reckless pleasure of it. He figured they were well south of Dayton now. Adrian knew he'd have to account to everyone but he just didn't want to do it now and so he smiled into the priest's inquisitive gaze.
"A film project, Francis, something for school."
ADRIAN'S WIFE and her new lover had dinner on Mount Adams, in a small Italian restaurant, Guido's on the Hill, then strolled among the boutiques and galleries, then went to their room at the Cincinnatian and reappeared the following morning, when, hand in hand, they were photographed leaving the hotel together, each with their cameras and equipment bags. They drove out River Road, along the north shore of the Ohio to the southwest, and were photographed photographing Anderson's Ferry, no doubt to supply some bit of alibi. They even rode it across the Ohio to the Kentucky shore, then back again. The photo of the ferry disappearing into the sunlit fog with the hills barely visible and half a dozen cars lined up on deck was almost artistic. Then they drove up Anderson Ferry Road to Delhi Road, where Adrian's detective took a photo of David taking a photo of Clare, posing among the summer semester students, at the door to one of the dorms, Clare trying her best to be "one of the girls" though she was more than a decade older than them. Then they returned to the hotel and window-shopped downtown. She bought him a straw boater at Batsakes hat shop. He bought her flowers. It was all in the report. They returned to their room for their second night of bliss, new lovers in early June, carefree in a not-too-distant city.
Adrian was not proud of the fact that he'd hired DiBardino to spy on Clare, but he figured he really had to have proof, not only for the eventual proceedings, but for himself, before he could call an attorney from outside the congregation, have him file for divorce and an "ex parte" order granting custody of "the minor children" to him and possession of the marital home pending the outcome of the proceedings. He really had to know. He was not proud of the fact that he'd talked to an attorney who'd given him the number for the private eye and told him how to get the goods on Clare.
"What you want is chapter and verse, open-and-shut, a slam dunk. Play for keeps, Reverend, play to win it," the attorney told him. He said he'd need three thousand up front and would bill him for the balance, if there was any more. He promised discretion and anonymity.
It was early Sunday afternoon, after the eleven o'clock service to which he always took his children and at which he preached, every other week, when DiBardino brought him the file and the photos. Reading it was like working a rotten tooth loose, the dull ache sharpening, then subsiding, then sharpening again, the nerve exposed, then numbing inexplicably.
It was the sentence in the report that read, "Subjects were observed embracing in front of a church on Mount Adams," that finished it for him. He knew that she had taken her new lover there because it was where Adrian had taken her the night he had proposed to her almost a dozen years before. They'd had dinner at the Rookwood Pottery and walked up to Immaculata Church on the highest point of Cincinnati's seven hills. Adrian had planned this part. They'd walked up Guido Street to where it dead-ended in the small front courtyard of the church, surrounded by cast-iron fencing and overlooking the city. They had looked down on the wide turn of the Ohio River, and the city with its bridges, and the southwestern expanse of Kentucky and America, and pledged their love and planned their future there. They'd driven down from Delaware, where he was finishing his studies at Methodist Theological and she was a student at Ohio Wesleyan. They'd spent the day walking around Eden Park among the gardens and observatories. It was late summer and Clare was the golden girl of his dreams with whom he'd had sex maybe a half a dozen times since the night she came to him, to the flat he'd rented off campus, and kissed him and let her clothes be taken off of her. Though neither of them were virgins, neither was really experienced either. They were, in his memory of it, innocents. It was in front of the church atop Mount Adams he had taken from his pocket the quarter-karat diamond ring his mother had helped him buy the weekend before and he pressed it into her palm and said he wanted to be married to her and to live with her forever and to build a future with her and would she be his wife? She said nothing at first, only slipped the ring on her finger, kissed him deeply and pressed her head against his chest, sighed and said of course she would. "My darling," she called him, "of course." In his memory of it, they seemed in love.
It was, in Adrian's heart, the place they were truly pledged, truly promised to one another, truly wed. In the early years of their marriage, when he wondered whether they were going to make it, it was that place, that moment, and its nearly cinematic replay in his memory that always convinced him they were meant for one another. And it was the wash of moonlight through the window of the Holiday Inn in downtown Cincinnati, which shone on Clare's bare shoulders as she knelt over his outstretched body that night, that still illumined his recollection of their sex, in slow motion, like a silent film—how she slowly bent to kiss him, letting her mouth with its warmth and quickened breathing work its way up and down his body, her hands so smooth, her arms outstretched, touching at once his right temple and his eyelids and his inner thighs, then taking him slowly into her mouth, hushing with the fingers of her left hand the catch of his breath, then in her own time, when he could not imagine any greater ecstasy, straddling him, taking him into herself—this was the consummation of their love, silver in every former remembrance, transcendent and sacramental, anointed, bathed in light, and now, now gone terribly, irretrievably dark.
That she would share their places with someone else seemed a more intimate betrayal than even sex. He called the attorney; he wept giving their particulars; full names, birth dates, date of marriage; the papers would be filed in the morning. She was served as she left her class with David Monday night. Adrian had left on Sunday afternoon and taken the children to visit his parents in Grand Rapids. He left a manila envelope on their bed, with photos of her and David coming and going from their assorted rendezvous and a note that told her he had "chapter and verse" on her "film project" and would use them against her if she contested anything. It would be a "slam dunk." She was to move out. "Now." He underlined the word. He included a check for five thousand dollars which he'd borrowed from the same church elder who'd loaned him the retainer for the attorney. She could use that to get set up in her "new life." She could keep the diamond ring, take anything she regarded as hers, and get out. He would contact the Western Ohio Conference of the UMC to find out whatever paltry amount had accrued to his pension fund and insurance account. If it was more than the five thousand, he'd pay her the difference. If it was less, she could keep the change. He wanted no fiscal entanglements between them. He would, he assured her, never keep the children from her, neither would he allow her to take them with her wherever it was she saw herself going. On this point Adrian was fairly certain that she would not put up much of a fight because what she really wanted was to travel light, free of encumbrances, into a new life with a new man off to New York where her truly artistic self would surface once the dull weight of husband and household and maternity had been lifted from her. He offered her, in this manila envelope, a package deal—freedom, some finances, and a permanent if à la carte relationship with her children, in trade for her getting out and letting everyone settle into the lives they would lead without her. Adrian, of course, had had the counsel of his attorney, who told him it would all go much better for him and for his children if there were truly no hope of reconciliation, if he could get Clare to move out of the "marital home." The photos, the check, the promises of freedom and a future of "quality time" with her children, and the not-entirely-articulated but none-too-hidden threat of embarrassment, Adrian figured would be enough.
"Ohio is a no-fault state," his attorney had told him, "but when it comes to the division of property and the custody of minor children, who is at fault still matters to judges and jury panels." Adrian had said as much in his note, which he paper-clipped to the photos and the check.
Over and over he had read the portions from the Gospel of Matthew—in the fifth chapter and the nineteenth—which held, "Whosoever shall put away his wife, except it be for fornication, committeth adultery." He read the same text in a different translation which replaced "fornication" with "marital infidelity." He had, he assured himself, the perfect right to put Clare away, to save his children from her craziness, to free himself from this overwhelming pain.
In the end, of course, she moved out, not because of his threats but because of himself, and the dull life with him she no longer wanted and could no longer endure. She moved out because that's exactly what she wanted to do and Clare could be counted on to do what she wanted.
When Adrian and Damien and Sarah came home that Tuesday in the middle of June, to the house they had all shared on the corner of South Cory Street and Lima Avenue, some blocks south of the river that flowed through Findlay, Clare and her things were unmistakably gone. She had moved out, into an apartment on the other side of Findlay. She told the children she would always love them and would be coming back for them as soon as she got herself established.
"Everything is going to be really terrific! You're going to love New York! There's so much to see and do there. Not like 'Finally, Ohio'—wait and see!" She'd come back for the last bits and pieces of her things. The white Toyota van was idling at the curb with the back doors open for Clare to pitch garbage bags full of her clothes and linens.
They kept nodding and smiling and weeping and hugging her, their little hands and faces holding and searching and wondering why this was all happening and why couldn't she stay with them and be their mother and she would always be their mother no matter what and someday she was sure they'd understand and Adrian went into the house and vomited, because he felt so helpless, so totally lost in the tears of things, so angry and heart-rent and utterly helpless, and their mother's voice trailing off as she left them on the porch waving and sobbing and jumped in the van with her lover and drove off into the future. Adrian stood looking out the screen door at the small figures of his children on the porch—all of this happening in slow motion now—as the van disembarked, and they raised their little hands and waved, and waved, and waved.
No remembrance of these events was free of the guilt Adrian still felt for the damage done his son and daughter, his complicity in its infliction.
Now what he could remember was the creak of the spring on the screen door as he pushed it open, and held it open and said, Come in now, and how the two of them turned, limp from the waving and weeping, and how Damien took his sister Sarah's hand and brought her back inside their suddenly and terribly broken home.
This was the moment, these many years since, that Adrian Littlefield could never forgive himself for—for failing his darling son and precious daughter so profoundly, for doing them such unspeakable damage by failing to keep their mother home, the marriage together, the household intact, life as they had come to know it safe and warm and blessed with abundant love. He couldn't protect his children from this hurt. His wife, their mother, had just driven off with her new lover, in a white van, leaving them all fixed and wriggling in the here and now without a clue as to what the future would hold. And as much as his heart hurt for his children's damage, standing at the kitchen counter buttering bread to make them grilled cheese sandwiches, and pouring out three matching glasses of milk, and cutting up a green apple into six little wedges, and placing this meal before them, and holding their tiny hands saying "Let us pray," and hearing their little voices give out with the accustomed grace, to wit: "God is great, God is good..." some corner of his broken, brooding heart quickened with the hope that his own life and times might just have gotten better, easier, simpler, saner somehow. Good riddance, he did not say out loud, while they ate their sandwiches wordlessly, but he said it nonetheless, good riddance, indeed.
That night he tucked them into bed and said their prayers with them, including the part about God blessing "Mommy and Daddy and Gramma and Grandpa and all the children in the world who don't have homes" and promised that everything was going to be all right and they'd go tomorrow for Vacation Bible Camp. Then he sat out on the porch watching the darkness tighten around the neighborhood and the bats circling out of the trees up and down the street and sipped from a tumbler of whiskey he'd taken to pouring himself, at the suggestion of the same church elder who'd loaned him money. "It'll help you sleep," is what he'd said.
And as often happened then and now, something out of Scriptures came to mind. May your fountain be blessed, and may you rejoice in the wife of your youth. A loving doe, a graceful deer—may her breasts satisfy you always, may you ever be captivated by her love. Why be captivated, my son, by an adulteress? Adrian remembered this reading from their wedding. He had chosen it from Proverbs, Chapter 5, as a reminder of the gifts of fidelity. Of course, he chose it as a caution against mannish misconduct—the biblical and still-conventional wisdom which assigned to men brute passions and moral weakness and more or less assumed that he would stray while she, the "loving doe" and "graceful deer" would remain true and unblemished by temptation. Now he read it as a cruel twist of his cuckolding, that even the Scriptures seemed to mock him. He had been happy enough, satisfied, captivated by her love, such as he'd known it, captivated by an adulteress, after all.
His fountain seemed blighted, the wife of his youth banished. The loving doe and graceful deer she'd seemed for years, now seemed a snarling bitch; her breasts gone dry, now satisfying David Eason, the silly fuck, captivated by an adulteress with stretch marks, hemorrhoids, more wear and tear than he might've imagined. Adrian gulped the bourbon. It made his eyes water. The taste of it, deliciously sinful for a Methodist, burned his tongue and the back of his throat and made him feel suddenly wonderful, capable, incredibly released, as if the whole of this disaster might be managed. But at night he wept.
WHEN HE woke in the morning, dry-mouthed, slightly hungover, but the children alive and sleeping soundly and himself alive and the sky remarkably not fallen, it seemed to Adrian they might all survive, the wife of his youth's departure notwithstanding. He pulled the photos DiBardino had given him from out of the large manila envelope marked confidential and carefully tore them all to shreds, all but the one of Anderson's Ferry, its paddle wheels flanking its barge of cars, its fog lights beaming from above the pilot's cabin—Boone No. 7 Port of Cincinnati—easing out from shore into the crosscurrents of the river, its masts and antennae turning and tuned in, the dense fog lifting off the water. That one Adrian Littlefield kept.
III
WHEN BLOCK Island came into view, he could see the tall sand cliffs, the green headlands, and the litter of sailing boats. From the dock in Old Harbor, Block Island seemed to Adrian like a postcard of the upmarket Yankee resort—huge painted Victorian hotels overlooking the harbor with red, white, and blue buntings hanging from their broad verandas, sloops and schooners and power yachts scattered around the seafront, brightly painted shop fronts done up for the season, and an abundance of cedar-shingled housing, graying but not particularly aged. Everywhere there were tanned and happy people in shorts and sandals and designer eyewear going about no particular business. There were bicycles and mopeds and cars for hire. The dockside was busy with day-trippers and courtesy vans from the various hotels meeting their guests. There were grandparents there to welcome their visiting families and the predictable vignettes of arrival and departure that are all the business of ports of call. Adrian Littlefield waited while the other passengers disembarked. The organizer from the National Association of Family Law Attorneys, holding her clipboard and smiling widely, was reminding the attorneys to "be back for the four o'clock ferry! We have the installation of officers ball tonight at Foxwoods!" This gave the group five hours to tour the island, maybe take a swim, maybe browse the shops for souvenirs. Adrian waited for the rest to leave. He wanted to do his tour alone. He walked up the town, looking over the offerings in store windows, admiring the lithe bodies of women in beachwear, looking into the faces of men. At the top of the Main Street where the road turned sharply left, he came to the National Hotel. It looked familiar to him. He climbed the front steps and took a seat on the long porch where lunch was being served. He ordered iced tea and, from the list of appetizers, steamers in drawn butter, a cup of seafood chowder, and bruschetta. A little taste of everything, he thought. He had a good view of the harbor and the foot traffic coming and going along the Main Street front.
There were fathers with cell phones, their teenagers on holidays with their noncustodial parents—subversive daughters being courted by their new stepmothers, young boys bristling at the new men in their mother's lives. There were young couples traveling en famille, with toddlers and infants and bored preteens.
He could see in the faces of the young husbands the fear he had felt in himself at that age, that he'd be overwhelmed at any minute by the duties and expenses and decisions.
He could see in the faces of their wives the worry and regret and second-guessing. How, they seemed to be asking themselves, had they gotten themselves into this predicament? They had been young and footloose and passionate and now they were homebound and bored and fatigued by motherhood and family life. They had been creative and well-read and interesting. Now they were dull, bored, vexed by the daylong needs of their off spring.
Adrian tried to reckon the ones who would make it and the ones who wouldn't. He tried to guess, by something in the way they walked or interacted, which of the children for whom this would be the last real family vacation. In the future there would be other configurations of adults and siblings in their lives. Partners, companions, significant others, spousal equivalents, stepparents, stepsisters, half brothers. But for many this would be the last vacation where mother and father shared the same time of their lives.
He was aware of a kind of psychic wince that always registered wrongly as a smile on his face whenever he looked at children and thought of his own children's pain, courageously borne in the years after their mother left. Of course, there was nothing he could do. Still he suffered a kind of survivor's guilt that what had been the best change in his life and the lightning rod of his success had hobbled his son and daughter somehow, in ways he sensed but could not measure. He had been a good parent and a good provider but he had not loved their mother. And now, in their young adulthood, he could see in the lives of his son and daughter that essential mistrust of their own hearts, a wariness about the love of others that made it difficult for them to form intimate attachments. He looked for early signals of such things in the manner and conduct of the children passing by.
He could see as well the older men eyeing the younger women and felt a quiet kinship with them. Adrian had counted it among the blessings of age that the abundance of women he found attractive was ever broadening even as his sexual prowess began to falter. The older he got the more and younger women there were to look at. Their beauty, at every age, took more of his breath away than it had when he was a much younger man.
He'd quit begrudging Clare her infidelities. The sense of sexual betrayal had been replaced by an understanding of it as a failure of honesty. Not that she'd had sex with Ben or David or whomever else, but that she'd not been forthright with Adrian. He could forgive her giving into an urgency of desire, but did not forgive her hiding it from him. Nor could he much blame Ben for taking advantage of the situation. They were the necessary precipitators or necessary events—an evolution, a natural elaboration of an order whereby the universe of love and attachment purges itself of anomalies. A man of fifty-something—as Adrian was now, as Ben had been then—could not easily resist the proffered affections of a woman twenty years his junior. Nor could a woman unhappy in her home life, tired of small towns and small children, bored by her husband's regular and routine affections, worried over the pressing and passage of time, be expected to travel in the off-season with a handsome artist to a distant island, to sail and walk deserted beaches and talk over dinner, idling away the remains of the day, and not offer her body to him. Especially when he had made a fuss over her as a girl. It was only natural for a woman at loose ends and a man in his fifties to fall into a fitful consortium should the occasion arise.
Adrian Littlefield had himself made a habit of confirming this in the years since, every chance he got, which was mostly at conventions such as the one he was currently hired to hold forth to. Attended as they always were by more than a few of the recently divorced, or recently traded in for a younger, fresher model, or recently disappointed in love or perennially discontented with life, these professional conferences provided cover for those occasions when the sexually rejected might reconfirm their sexiness. Chief among the obligations—Adrian knew this from his own experience—of every newly divorced man and woman was to demonstrate that it was not a sexual dysfunction that occasioned the breakup. As the keynoter and visiting expert at these confabs and conventions, Adrian was often the focus of much of the free-floating, unattached, ready, willing, and able sexual energy of the registrants, a number of whom, it never failed, would make known in the usual ways to Adrian, their availability for more intimate conversations on related themes. He was possessed, after all, of a certain celebrity in these circles. He was famously single, well-spoken, well-dressed, well-paid, and the center of an hour or two hours' attention during which he would motivate, inspire, entertain, inform, and uplift his listeners. To be gracious, charming, self-effacing was easier after a standing ovation, handsome payment, and a line of supplicants waiting for a signature on a book that bore his face and name on the cover. He had no less an appetite for a stranger's affections than any man or woman did. And while he sometimes missed the predictable lovemaking of the married life, he found it hard to count as anything but good fortune that the years since the dissolution of his marriage had been characterized by sexual encounters more abundant if more distant, more passionate if less precise, hungrier if less often sated, more memorable if often nameless. If each partner in these arrangements felt a little "used," it was, to him and to no few of the women he had had sex with, still pleasant enough. That bodies could pleasure and could be pleasured, free of social, emotional, or intellectual encumbrances, seemed to Adrian a good and wholesome thing. And he made it his mission to attend to his partner in ways that would overwhelm whatever residual regret she might otherwise attach to the "one-night stand." With several of these women he had maintained an ongoing correspondence, some of which had ended sooner, some later, and some of which remained pleasant and unpredictable addendums to his professional life. Sometimes he would invite one of them to join him on an extended speaking tour. They would spend a week, or maybe two weeks, together. They'd begin to behave like real companions. He'd remember how she drank her coffee and order room service accordingly. She'd pack and unpack his things between hotels. Each would pretend an intimacy they both knew did not, and likely would never, really exist. They would tell each other secrets over dinner. It warmed something in Adrian he could never quite identify. Getting to know someone after having sex with them was a reversal of the usual arrangement by which the business of intimacy was in the main conducted, but for a variety of reasons, it appealed to him. The flesh, Adrian sometimes pointed out in his workshops, is far less particular than the heart or the mind when it came to finding "suitable" partners. Sex between people who might not otherwise find anything to admire about one another could be quite, well, satisfactory, especially on a time-fixed basis. Whereas, he would likewise observe, there were people who could be attracted in every possible way, intimate in all ways in the conduct of their lives together, but sexually uninspired. These were but a few among the many mysteries his programs dealt with. And Adrian had seen in the faces of the registrants at Foxwoods, in the small talk of the NAFLA conferees, in the body language of their pairings and couplings and comminglings at the welcome reception the night before, chitchatting with wines and finger foods, the men in their best business-casual attire, the women wanting to look professional but sexy—he had seen it all—the whole register of human want and willingness and desire.
Adrian could see it now, watching from the long porch of the National Hotel, the parade of suffering humankind, bearing their various histories and fears of missed chances and discontents along the esplanade, the mercilessly sunlit day unfolding around them; it was inevitable. "The story of love," as he often told his audience, "to quote Professor Bowlby, is told in three volumes: Attachment, Separation, and Loss." Or if the time allowed only a thumbnail version, "Love," he would say, quoting Roy Orbison, "hurts."
"No pair of words ever added up to more truth than that!" Adrian would often close his keynotes with the observation.
"Boudleaux Bryant wrote it. Roy Orbison sang it. Everyone in the room here knows it—Love hurts! We ante up, we go all in, we play the cards we're dealt as best we can, and still it comes down to a simple two-word arithmetic, this fact of life: Shit happens, Jesus wept, Life sucks, Love hurts. And yet we keep on playing for keeps because Love heals, love sings, love haunts, love holds, love gives, love takes, love warms, love knows, love waits, love weeps, love laughs, love lives, love lasts, God is love and love never ends." This litany of love, for which Adrian had become well-known, always signaled the end of his speech. He would let the last words settle in the air, careful to keep his gaze fixed above their heads, off in the distance beyond the back of the room, then step back from the lectern, let his hands fall to his side and his head bow slightly, which never failed to bring on the first round of applause. "Thank you," he would say, holding his hands to his heart as the applause grew louder. "Thank you, thank you...you are so very kind..." It would bring them to their feet. He would bow again.
ADRIAN LITTLEFIELD looked down the long porch of the National Hotel and tried to envision his ex-wife Clare in her thirties seated at one of these tables with her old artist Ben, wizened and hirsute, each of them pleased with what they had just accomplished by getting sufficiently free of life's entanglements to arrive here on the island together, off-season, unencumbered. Like Ben's wife, Adrian and the children had been jettisoned, thrown overboard to the fish or gulls, somewhere en route between the mainland and island. How knowingly they must have smiled at one another, how free of any moral vexation, how entitled they surely must have felt to their mutual lapses of faith. Or maybe not. Maybe there was some whiff of regret. How could he ever know? How, of course, could Findlay, Ohio, flat and landlocked, compete with an island in the ocean? How could an associate pastor compete with a true artist or a sickly, sexless, disabled wife back in Westchester—how could she compete with a young and eager and interested girl?
ADRIAN PICKED at the elements of his meal. He wasn't as hungry as he'd thought. He asked for the bill, left a large gratuity, and asked the waitress for another cup of coffee and a local phone book.
IV
FATHER FRANCIS Assisi Concannon phoned the Reverend Adrian Littlefield and told him to be ready at five o'clock.
"I can't make it, Francis. I've got kids to watch."
"I'm bringing a sitter. We need a night out."
"No really, Francis, I really can't."
"Never cross a priest, Adrian. It's bad karma. See you at five."
Word about the marital woes of the associate pastor at Findlay's St. Mark's Methodist Church had gotten round all over that part of Ohio. If bad news travels fast, Adrian observed, news involving the private lives of the reverend clergy moved like wildfire. And the tongue is a fire, he recalled from the Letter of James. The tongue is set among our members as a world of iniquity; it stains the whole body, sets on fire the cycle of nature, as is itself set on fire by hell.
A tongue among my members would be just the thing, he thought, then tried to turn his thoughts to godly themes.
"The people who know you know you," his father had told him when Adrian had shared his worry over the gossip that was circulating about his wife leaving him. "And those who don't, don't care." But Adrian knew that people were talking. Clare had spent a few weeks in an apartment on the north end of Findlay, then moved to Bowling Green, then eastward on to Cleveland.
She kept giving Damien new phone numbers which the boy would sit repeating to himself until he had them memorized. To watch his son, just gone eight years old, holding the slip of paper, quizzing himself and his little sister Sarah on their mother's new numbers hurt Adrian and angered him.
From the same mouth come blessing and cursing. Does a spring pour forth from the same opening both fresh and brackish water? Whenever he heard some strand of a rumor about his predicament, he renewed his vow to keep silent rather than add his own fiery tongue—a restless evil full of deadly poison—to the general conflagration.
Oh, to put my tongue to better use, he thought, at the altar of some good woman's pleasure...then caught himself again and tried to budge his brain from the parts of women.
Whether it was coincidence, correlation, or cause and effect, soon after Clare Littlefield moved out of the manse, her cuckolded husband's sermons began to draw more listeners to church. Adrian thought it was mostly the spectacle of a churchman writhing in such worldly pain that packed the pews. Not many of you should become teachers, James the brother of Jesus advised, for you know that we who teach will be judged with greater strictness.
Folks from as far off as Fostoria and Columbus Grove, Bowling Green and Bucyrus would show up for the eleven o'clock services on the every other Sundays Adrian preached. Even his Wednesday night Bible Study, formerly a lackluster assembly of elders and widows, suddenly had new faces, younger faces, and more females than ever before. His former custom of preparing typed sermons, two minutes a page, eight pages per service of carefully constructed remarks on some biblical principal, gave way to a much looser, catch-as-catch-can delivery. This owed to the necessities of his newly single life as the custodial parent of two young children. He hadn't the time proper preparation takes, to study the readings from the Lectionary, and find some way to connect those dots.
In the years of his marriage he'd spend every other Saturday night in his office at St. Mark's, preparing his sermon for the following morning. He liked to think of members of the congregation, driving up and down South Main at all hours, returning from boozy dinners, late movies, the shift change at Cooper Tires, God knows what assignations—how they would see the lights blazing from his basement office and know he was hard at the Lord's work late at night.
"You were burning the midnight oil again," someone would always say, shaking his hand after Sunday services. Adrian thought it was a good thing to be known for. The Reverend Hinkston had a fairly comprehensive library, which he had generously opened to Adrian. It had volumes of old sermons, church histories, homiletic guides, and toastmaster's resources. A formula for his homilies had emerged, by which he sought to speak to his entire flock, young and old, devout and backsliding, male and female, country and townie, educated and simple. He would pick out a couple of homey anecdotes with an evident lesson, maybe a verse from Helen Steiner Rice for the blue hairs, some folksy jokes, and something from the current music, "a little bit country, a little bit rock and roll," to keep the younger crowd alert. To these he would add something from the day's Scripture readings and tie them all together seamlessly, like the papers he wrote in seminary—five paragraphs, compare and contrast, beginning, middle, and end—methodical. Thus, Jesus would be our "bridge over troubled waters" and Moses was an example of "knowing when to fold and knowing when to hold them."
He'd type them up with double spacing and wide margins for notes he might add in longhand afterward, shifting those points he most wanted to stress into capital letters and underlining things he might want to repeat. The old Underwood his father gave him had given way to an Olympia Electronic and then to an IBM Selectric. He would go into the darkened sanctuary and test his delivery in the vaulted acoustics of the worship space. "My brothers and sisters in Christ Jesus Arisen," he would always open. He would listen for the echo of his own voice off the stone walls and open the articulation of the words to allow for this amplification. "Amen and Amen," he would almost always close, in a style borrowed from a TV preacher he'd seen as a boy who would heal people by the laying on of hands. And though his sermons were, in their printed versions, carefully wrought and often very readable, and though he kept copies for his pastoral archives, the homilies he labored over were as enthusiastically ignored and as politely disregarded by his congregants as the Reverend Hinkston's famous "three points and a poem" snorers. Regardless of what it was he was saying, he could effect no manifest change in the congregants' response. The smilers kept smiling, the nodders kept nodding, the sighers kept sighing, the dozers kept getting elbowed by their wives. "Uplifting message this morning," Mrs. Melzer would always say, shaking his hand and pressing the handkerchief to her nostrils. "Covered all the bases today, Reverend," Clark Waters, the head of the DPW would say without fail. "A blessing, a blessing, words from the heart," the breathless Donna Montgomery would always own, holding her right hand over her bosom and offering her left hand for Adrian to take in the kind of straight-armed feint he always associated with characters from a Tennessee Williams play. He would assume his place at the back doors of the church while half of the congregants made for the parking lot and the other half made for the fellowship hall where coffee and donuts and cookies were served.
Whatever the response, Adrian counted himself blessed to have anyone listening at all. So many churches had lost members to the growing hoard of radio and TV preachers—Swaggart and the Bakkers, Jerry Falwell and Pat Robertson—and the local crazies from the Ohio River valley who flooded the local radio airways with variously "old-time," or "feel-good," or "prosperity" gospels and perpetual appeals for "seed offerings" and "love gifts."
Many late Saturday nights or early Sunday mornings, crawling into bed beside Clare, bits and pieces of his sermon still tumbling through his brain, Adrian would press himself to the warm bend of his wife's buttocks, and reaching beneath her night shirt, cup one of her breasts in his hand and bury his face in her long hair, which always smelled to him like Eden. Unfailingly that verse that held how "the word became flesh and dwelt among us" would come to him, along with the prayer that she might wake sufficiently to allow him to dwell among her flesh entirely.
In their early years, before Damien and Sarah, before she returned to school, before he and his work and their dull routines in what she'd begun to call "Finally," Ohio, had become the focus for her discontent, Clare would often give in to this ritual seduction, signaling by a tiny sigh, a catch of her breath, or a little moan, or by pushing her rump more firmly against him, or by rolling on her back and moving his hand with her hand between her legs while still seeming to be deep in sleep. Afterward, as she curled back into her private slumber, the wordless discourse of their lovemaking done, Adrian would count his blessings, giving thanks for the gift of his wife while replaying the tape of his freshly typed sermon in his mind, tapping out the phrases such as he could remember them, word by word, syllable by syllable, with his fingers tapping on Clare's beautiful bottom, a private code the cadence of which would put him eventually to sleep.
In the weeks since she had left him, his homiletics, even Dr. Hinkston, the senior pastor, commented, had become suddenly "more moving, more engaged, more relevant somehow, more meaningful."
"A gift of the Holy Spirit, the fiery tongue!" Dr. Hinkston called it, but Adrian figured it might be the drink that Francis Concannon and he had gotten in the habit of overdoing on Saturday nights, and the low-grade, ever-present, and flickering rage, the mysteries of human suffering and passion, the cross and flame his life had become.
FATHER FRANCIS Concannon couldn't care less about that. His anger was no mystery at all.
"For fuck sake, Adrian," the priest shouted over the phone, "next time I get word about a friend in trouble from a horse's ass instead of the horse's mouth, I'm gonna really be pissed."
"I'm sorry, Francis, I should have called."
Word of his friend's trouble had got to him at St. Michael's as gossip from one of his church ladies.
"At least we don't have to worry about your missus running off with an artist, Father!" said Mrs. Bokuniewicz one Tuesday in late June after morning Mass.
The priest had been a true friend ever since, coming over to Adrian's without invitation, bringing pizza and beer, chicken chop suey, a bag of burgers and fries, and a bottle of Irish, sitting up with Adrian those summer nights in the first weeks of his single parenthood. And when Adrian would take the kids up to bathe them and tuck them into bed and say their prayers, the priest would tidy up downstairs, picking up the toys, folding the laundry, cleaning up the kitchen, feeding the kitten and the dog. Then he'd fix two tumblers of what he called "Dunphy's damage"—generous measures of bronze-colored liquor poured from a green bottle over ice cubes, and insist that Adrian sit out on the screened porch and tell him everything.
"That's the stuff, boyo," the priest would say in a stage-Irish brogue, taking a long sip of the whiskey, "St. Patrick's holy water itself. Any fucking wonder they call it 'spirits'!"
Adrian would sip from his own glass, wince with the burn of it, exhale deeply, and settle into the chair with one ear tuned for the children upstairs, glad to have the priest sharing the watch with him. On such nights the general panic that he'd felt since Clare had left him would subside, if only for the couple of hours that the two men sat out on the screened porch, in the dark, watching the evening traffic on Lima Avenue go by and the streetlamps attracting bugs and the bats circling in the night air.
He'd been so frightened that something would happen to Damien or Sarah. It was an ever-present fear, a general dread, a constant wariness that something bad would happen to them if he let down his guard. He'd always had it, but it was doubled now that his wife had left him and he realized only half the eyes were keeping watch, half the ears were tuned toward their protection, only one body to place between them and peril. He was not sleeping very well. He was always tired. The giant frame of Francis Concannon, his thick hair wild on his head, his great jaw silhouetted by the yellow glow of the streetlamp, sitting on the other side of the small table on the screened porch was a comfort—as if, at least on such evenings, there was another protector in the house and Adrian's vigilance could be dialed down. In his Ohio State sweatshirt, a pipe smoldering in one hand, a glass of whiskey in the other, the priest made an unlikely sentry. But for Adrian his friendship had been a gift. All the area's Protestant clergy had sent polite epistles, promised prayers, cited Scriptures, encouraged him to get in touch if they "could help in any way," but otherwise had kept their distance, as if spousal distemper and divorce were contagions and Adrian the local carrier. Father Concannon had broken the quarantine and entered the pesthouse. He had been unambiguously the friend in need. And Adrian was ever grateful. That he had become Adrian's de facto confessor had been another gift. The deepest secrets of his ruined heart had been dumped on the priest, his black anger and dark rages, his vengeful impulses toward Clare and her new lover—"the motherfucker," Adrian seethed, because she was a mother and she left her children and "he'll never love her like I loved her, he's only fucking her." Sometimes he wept and the priest would reach across the table, take hold of Adrian's shoulder in his massive hand, and steady him till the worst was over.
"Lacrimae fucking rerum, pal," he'd commiserate, "the fucking tears of things."
Adrian couldn't confide much to his senior pastor, the Reverend Hinkston, or to the Northwest Plains district superintendent, the Reverend Carlton Paul Ritter, or to the West Ohio Conference bishop, Bishop Wesley A. Maghee. Nor could he imagine really having a drink, really letting his tongue get loosened with any of them. While they were his connectional and episcopal up-line, he knew that the failure of his marriage presented some difficulties for them. The scandal of a minister's wife running off with another man, leaving children behind, was a cup they'd rather God had let pass. A committee had already been formed at the church to monitor Adrian's "changed status" and report to the district superintendent, who would report to the bishop.
To his credit, the Reverend Hinkston had at least made an effort, however clumsy. He'd told Adrian he'd be happy to counsel him and Clare and let it be known that he'd saved many marriages over his "nearly forty years of servant ministry."
"Cleta and I have met with many a young couple in distress, and prayed with them and set them right—it's all there in the book, you know, that's the guide. Sometimes all it takes is someone with a little more experience, you know, to remind us of the times we were in love. Cleta and I have gotten through many a crisis, and still been married forty-three years. Even sexual dilemmas can be gotten through."
Adrian could not bring himself to tell the Reverend Hinkston that he did not want Clare back. Her having sex with other men was not a "sexual dilemma." It was betrayal. Her lack of remorse, her willingness to leave her children were all, for him, signs of moral or mental disorder, perhaps forgivable, even, he prayed, forgettable, but ultimately—he knew this in his heart of hearts—irreparable. Their deal was done. She had breached the central contract of their marriage, broken their household, reneged on their intimate agreements. That damage was done and could not be undone. They would all have to live with it, for better or worse. Whether madness or passion or unfortunate choices, whether she was just spreading her own wings, realizing her full potential, or just reaching her sexual prime and hungry for a little unfamiliar sex, he neither knew nor any longer wanted to know. He could live with the brokenness, the worry over his children's well-being, the fears he had for the future. He could live with all of that. But he could no longer live with her. Whether this was unchristian, unholy, sinful, or immoral, whether work of the devil or Will of God, he couldn't say. And didn't care; it was what it was.
What Francis Concannon seemed to offer, since he first called Adrian and came barging into the wreckage of his family, was moral immunity, spiritual oasis, a kind of ecclesiastical safe harbor neither too shallow for the hard truth nor too fathomless to obscure the sadness of it. He never quoted the Letters of Paul or the Gospel of Matthew or the Acts of the Apostles to Adrian. Rather, he had brought him food and drink and poems. He brought him time and the moment's safety. He brought his brother cleric the gift of his presence and manifest willingness to bear his portion of the grief and rage and fear. And sitting out on the screened porch, listening to crickets and watching lightning bugs, sipping their "damage," Adrian found himself somehow comforted by the priest's recitations, his tipsy party-pieces, his bits of Yeats:
Others because you did not keep
That deep-sworn vow have been friends of mine;
Yet always when I look death in the face,
When I clamber to the heights of sleep,
Or when I grow excited with wine,
Suddenly I meet your face.
or Beckett:
I would like my love to die
and the rain to be raining on the graveyard
and on me walking the streets
mourning her who thought she loved me.
Even an anonymous, possibly medieval bard who claimed:
The way to get on with a girl
Is to drift like a man in the mist,
Happy enough to be caught,
Happy to be dismissed.
Glad to be out of her way,
Glad to rejoin her in bed,
Equally grieved or gay
To learn that she's living or dead.
The lilt and faux brogue the priest achieved in the recitation of his poems unfailingly made Adrian laugh or weep. The drink assisted.
Best of all of them, Adrian thought, was Philip Larkin. "That awful man," Francis Concannon called him, and gave out with line after line by heart:
There is regret. Always, there is regret.
But it is better that our lives unloose,
As two tall ships, wind-mastered, wet with light,
Break from an estuary with their courses set,
And waving part, and waving drop from sight.
Adrian so wanted to get past it all. He knew it would kill him if he did not let it go. He wanted to be free of it. To be restored to some kind of wholeness, beyond regret, beyond fear, beyond it all—yes, two tall ships, wind-mastered, wet with light—he wouldn't begrudge Clare any happiness, once he had been restored to his; once he had his new course set he'd happily wish her a good riddance: and waving part, and waving drop from sight. In the meantime, though, the dark wound still festered. She had harmed him and harmed his household and he did not wish her well.
NIGHTS AFTER the priest left Adrian slept better, not least because of the bottle of whiskey they would feel duty-bound to empty before they'd say good night. The priest would slump into his Buick and drive off to St. Michael's rectory and Adrian would quench the lights in the manse, promising again to slow down the drinking, thanking God, as he climbed the stairs, for friends like Francis, for the safety of his children on whose slumber he would always peek, bending to kiss them and to hear them breathe.
In his own bedroom he would strip, piss, brush his teeth, and crawl onto the mattress on the floor. Clare had taken the brass bed frame, bought at the Salvation Army store soon after they'd married, when she left. And the large Persian rug that had covered the pine floorboards—it went off rolled up in David Eason's Toyota van, and the oak dresser he'd refinished for her for one of their anniversaries and the matching beveled-glass mirror he'd found for Christmas one year—all of it gone—the fixtures and furnishings of their married life. He lay in the sheets ogling the breasts of the willing, come-hithering beauties in the pages of a girlie magazine he'd taken from an embarrassed teen at church when it fell from her school bag inadvertently. He would study their ankles and the curl of their feet, their tattoos and perfectly tonsured genitalia. He would try to imagine his hand on their breasts, then his mouth, then his ear placed over their sternum where he'd hear their heart beating life and longing. After masturbating, he'd pray to be spared the fate of Onan, who had spilled his seed in the Book of Genesis. He'd resolve to quit the drink, to be a better father, to accept his circumstances, and to do God's will and God's work in accordance with his calling. Whether it was drink or fatigue that made him sleep, he was glad for it whenever it came. If he dreamed, he did not remember it.
A NIGHT of excess and a resolve to quit drinking had kept him abstemious for a couple of weeks. But the priest's phone call had left him little wiggle room for an excuse. Adrian did not want to go out that night. He'd never left the kids with a sitter since Clare had left. He had to prepare a sermon for the morning. Things were getting shaky at the church. Still, he knew he never could refuse the priest, and so resigned himself to be ready at five. He showered and shaved and put on a fresh pair of khakis, his brown penny loafers, a blue button-down shirt, and his dark blue sport coat.
WHETHER IT was the bag of books and crayons and paper, the fact that she was nearer their own size, or that she was the first adult female to stand at the kitchen table since their mother had left three months before, that drew his kids to Mary De Dona, Adrian Littlefield had no idea. Either way, halfway through his list of child care guidelines—what they should eat, when they should go to bed, their family doctor's emergency numbers, his beeper number, the numbers for the police and fire and ambulance—Sarah was seated with her thumb in her mouth, picking crayons from the cigar box and drawing stick figures on a blank page that Mary De Dona had given her. Damien was explaining the decals on his skateboard to Mary De Dona, who sat listening intently to the boy.
"C'mon, Reverend Littlefield," Francis Concannon pleaded, holding the back screen door open, "the car's running, can't you see they'll be fine."
"But I don't know anything about her," Adrian protested as the priest's car backed out of the driveway and made its way up Cory Street, past East Sandusky Street, to West Main Cross.
"For fuck sake, Adrian, what's to know? She teaches fourth grade at our parish school. She's a woman. She's Italian. She knows more about kids than either of us will ever know. They'll be fine. We're taking the night off. We're on a mission."
Still, when the priest stopped for gas before getting on the freeway, Adrian went into the station and called home to see if everything was all right.
"They're fine," Mary De Dona assured him. "We're having grilled cheese sandwiches and then we're walking to Riverbend Park to feed the carp and do some people-watching."
By the time they'd passed Toledo and crossed from Ohio into Michigan, Adrian knew there'd be no turning back.
"Where are we going, Francis?"
"Tijuana...relax. We'll be there before you know it."
Just north of Monroe, the priest rolled up his car window, lit up his pipe, and the car filled with the acrid smell of marijuana.
"I'm your designated driver tonight, Adrian, no boozola for me. Wanna toke?"
Adrian declined the pipe, but it hardly mattered. The car filled with a cloud of smoke and he could feel the sudden lightheadedness he hadn't felt since his pre-seminary days at Wesleyan. The same idiot grin, the same stupid nodding as if he could agree with anything; the same hollow in his stomach—the priest was looking over and smiling inexplicably. Outside of Detroit, they opened the windows again and from under the driver's seat the priest pulled a bottle of after-shave and splashed the car and Adrian.
Crossing over the Detroit River on the Ambassador Bridge, Adrian was counting boats in the water below them when he was seized by a sudden panic. He was going into a foreign country in the company of an apparently crazy priest. He was a single parent from Findlay, Ohio, who'd left his minor children, two hours south, in the custody of an unknown Italian woman who might sell them to the circus or run off with them, for all he knew, or testify against him in some future trial brought by his ex-wife for negligence.
"Where in God's name are we going, Francis?" Adrian pleaded with the priest.
"For dinner in Windsor—and then the ballet. Aren't you hungry? I am starving, man."
Adrian was hungry, and though utterly helpless, he could not work up any anger at the priest or make sufficient protest.
The light of late September was fading from the evening sky and the cities on either side of the river were bathed in the most inextinguishable light, and he decided to give it all to God—his children's safety, his shaky situation at St. Mark's, the sermon he had not prepared, the hapless prospects for his future, his damaged household, his pummeled heart, his hunger, his desire, his unfathomable want, the sores and boils of rejection he felt—all of it he was giving to God, as Job had in that god-awful comfortless book which had only ever raised in Adrian reasonable doubts. It didn't matter. And as they passed through Canadian customs, assuring the border agent that they were U.S. citizens, here for the evening only, and only for pleasure, Adrian turned to his guide and guardian, his pilot and protector, the dry but pleasantly stoned parish priest, and quoted, "Blessed be the name of the Lord."
"Back at you, brother," the priest smiled widely. "Amen!"
AT MARIO'S on Ouellette Street, the priest ordered steaks and baked potatoes and a bottle of Bardolino.
"I was six months at the University of Padua," he told Adrian. "I love their wines—all from the hill country around Verona. Romeo and Juliet, Lago di Garda, the Dolomites...sei la più bella del mondo...Goddammit Adrian, where goes the time?"
All Adrian could do was nod and sigh.
After dinner, the men walked up Ouellette Street to a place on the corner called Studio 4. The priest held the door for Adrian, who eyed him narrowly.
"Have no fear, Reverend Littlefield. Would I lead you astray?"
Inside it was dark and sparsely crowded. There were booths and tables along the brick walls with candles on them. The middle space was one broad serpentine bar winding its way around through adjoining rooms with black-shirted bar men and women serving drinks on one side and on the other, men of all ages, some seated, some standing, all gazing upward at an odd parade of beautiful, awkwardly dancing girls, working their way at intervals along the bar in various stages of exotic undress, from partial to total nakedness.
"Oh God, Francis..." said Adrian. But he could not avert his eyes from their bodies, in general and in particular, and their smiles and their eyes which, the closer he got, seemed to bear him no apparent malice, rather seemed to understand how he was entirely enthralled, quite hopelessly fixed upon their beauty, quite beyond the province of reason or words. He had not seen a naked woman in months except in movies and magazines—the little stash of pornography he kept hidden in his bedroom closet. But these were live and moving beauties. He could smell their powders and perfumes. That he could not touch them was no bother. He only wanted to admire them and bless them for how they did not refuse his gaze.
It was late when they left Windsor, and Adrian was fairly drunk and the lights of the Detroit skyline shining off the river were, like everything, a blur to him. The U.S. Customs guard asked if they had anything to declare.
"Nothing," Father Concannon told him.
"Bringing any guns into Detroit?"
"No, nothing," the priest repeated.
"Maybe I should loan you one of mine!" the fat guard joked, and waved them on.
Adrian slept through Toledo and never woke when the priest stopped for coffee in Bowling Green and had to be helped from the car at home.
"What a night, Francis," Adrian tried to say, as the priest helped him into the house and up the stairs, out of his clothes and into bed. Downstairs the priest found Mary De Dona curled on the couch in her own slumber, the TV flickering out some infomercial, the children's drawings on the coffee table. He shook her gently by the shoulder.
"I'm all right, Father, let it be."
"Will I take you home?"
"Go on, Father, let it be."
The priest stood straight and still, listening for the sounds of anything wrong in the house. No noise issued from upstairs where Adrian and his children were safely asleep. Only the hum of ceiling fans and appliances, the low and orderly breathing of Mary De Dona on the couch, and low din of the flickering TV—all was well, he reasoned, all was well.
Out on the porch Father Concannon pronounced a general absolution: "Ego te absolvo a peccatis tuis in nomine Patris et Filii et Spiritus Sancti. Amen," then drove home and slept the sleep of a child of God.
ADRIAN LITTLEFIELD dreamed he was whole again, his consortium restored, the body of a woman opening to him. In the dream she was a stranger to him, her habits a mystery, their desires new. The feel of her flesh under his fingers tingled with what were entirely new sensations; still, he knew that she knew that he knew that she knew. And her body was smaller than he might have remembered if he had remembered anything, which in his dream it was well known that he didn't. He had no past or future, only now. And her hair was darker and her sweet mouth lovelier and her parts in their own way accustomed to his. Oh God, Oh God, is all he could hear himself trying to say, because now that he thought of it God is love, and those who abide in love abide in God, and God...Oh Love, Oh Yes...Oh...
Wherever he was between awake and the dream, it was Mary De Dona astraddle him, pressing her little body onto his body, with one hand on his chest and with the other pressing her fingers to his open lips, whispering, "Hush, Reverend Littlefield, let it be," still moving on him, though he had come already, still shoving herself against him, him holding her buttocks in his hands, pulling her forward and pressing her back, pressing his heels into the mattress, his head thrown back until she bent to kiss him on his mouth and on his head and then sitting back upright, because now he was hard again and turning her over and pressing himself deeper and deeper into her and her eyes closing and her mouth wide open and the room awash in new morning light and her two hands pulling him into her and she was finished and he was finished again and it was done. Adrian Littlefield rolled on his back, breathless and wide awake.
"Have you a cigarette?" she asked him.
"Cigarette?...Why, no..."
"Porca miseria! Are you good for anything?"
She was pulling her camisole over her head, and stepping into her panties, and buttoning her blue jeans and throwing the green sweatshirt over her with St. Michael the Archangel printed in gold and pulling her curly black hair back and knotting it in a bun. He had not noticed the night before how very comely she was—her rose skin, her brown eyes, the little figure of her standing right in front of him. He wondered at her age.
"Was your ex-wife beautiful?"
Adrian hadn't a clue what to say.
"I don't know...I suppose she was...I don't know...Why...?"
"Your children are beautiful. I've got to go. They shouldn't see me here."
"Wait," he called after her, but she was gone.
RUMMAGING THROUGH his sermon archives for something to say that Sunday morning, the Reverend Adrian Littlefield could not stop grinning. Nor could he focus on the titles of his collection: "If It's Good It's God's," "The Attitude is Gratitude," "The Present's a Gift," "The Corinthians Write Back." He hustled Damien and Sarah into their Sunday clothes, brushed his teeth and gargled with mouthwash, put on his brown suit, and made for St. Mark's, wondering why he couldn't wipe the grin off his face. Nor could he make sense of the morning's readings—from Ezekiel and Psalms, Philippians and Matthew—all he could think of was the bend of her body, the taste of her shoulder, the smell of her hair, her inexplicable kindnesses. Had it been a dream? And rising to his homiletic duties, gazing into the faces of his people, he knew he had nothing to tell them. So he sang.
As for Father Francis Assisi Concannon, holding forth to the faithful at the ten o'clock Mass, he observed it was the Feast of Michael the Archangel and another Sunday in Ordinary Time.
V
IN THE Block Island phone book, Dr. Adrian Littlefield looked among the W's for Ben Walters. There was only one, on Pilot Hill Road. He pulled a map of the island from his back pocket. Pilot Hill Road ran a little ways southwest of town. He walked across the street to the taxi ranks near the boat docks and climbed in an old station wagon with Island Hack & Taxi Tours on it.
"Tour of the island?" the old woman at the wheel asked him.
"How long does it take?" he asked.
"How long do you have?"
"I have to be back for the four o'clock ferry."
"No problem," she told him. "That's acres of time. It really is a tiny island."
Adrian got in and introduced himself.
"Adrian Littlefield," he extended his hand.
"Gloria, Gloria Dodge," she said. "You're welcome to the island. First time here?"
"Yes, well...yes, my first time."
She drove out of the parking lot and turned northward going out past the hotels, the local bars, the long beaches packed with sunbathers. Adrian kept a map and watched the sights and signposts go by. Gloria kept up the travelogue.
"The island is only seven miles long by three miles wide, shaped like a pork chop, less than eleven square miles." Gloria had this "tour" memorized. Adrian looked at his foldout map of the island that he'd been given by the tour organizers. Block Island did, indeed, look like a pork chop, with the narrow bony end to the north and the squat round meaty end to the south. There was a profusion of dune and seascape as they drove out of town.
"That's Scotch Beach there, a little rougher water. It was named for the people who didn't want to pay the dime a week to help with the upkeep of the beaches. They could swim there free."
Adrian smiled and nodded and feigned interest. The old station wagon bumped along out what he read was Corn Neck Road past Bush Lot Hill toward Chaqum Pond at the north end of the island.
"Many people live here year-round?" he asked.
"I guess they figure around nine hundred now. Most of these homes are seasonals. People from Providence and Hartford and Boston and New York. Many of the same families are coming for generations. I've been here my entire life. We raised seven children here. Seventeen grandchildren and nine great-grandchildren. They'll all be coming home next week. My husband's birthday."
"Wow," said Adrian, and looked out at the sea.
At the end of the road was a rock beach, a small parking lot, and off to the west an old lighthouse.
"1867 it was built. They're making it into a museum. My son is on the volunteer committee. If you want to get out and have a look, I'll wait. There's plenty of time."
Adrian shook his head and she backed the car around to head back the road they'd come.
"How old will your husband be?"
"Well, he'd have been eighty but he died last year. Still, we figured we'd get together anyway, and celebrate, you know...it's all just family."
"That must be very hard," said Adrian, a little worried that he'd gotten more information about Gloria than he ever wanted but figuring now there was no turning back.
"Well, of course, we all miss him terribly. He was the dearest man and the grandchildren were so sad, so beautiful...they wanted to have a cake and get out all the pictures. I can't wait to see them. They're coming from as far away as Denver. He was the dearest man, everybody's favorite. He always loved it when they came to visit us here on the island."
"Won't that be nice," Adrian said. "Look at those beautiful yachts!"
They'd come to the marina at the Great Salt Pond and the New Harbor area. Sailboats tied to their moorings rocked in the wide basin. Fashionably clad boaters walked up and down the docks. A restaurant called Dead Eye Dick's was doing a brisk luncheon trade. The old station wagon passed a small graveyard on the left.
"He's buried in there," said Gloria.
Adrian said nothing, hoping the conversation might take another turn.
"Fifty-eight years we were married. But we'd been 'together' for years before that. I met him when I was thirteen. It was February. I was ice-skating with friends and all they could do was laugh at me because I kept falling and I couldn't stand up. The legs would go right out from under me. They were all laughing. It was horrible. And all of a sudden, I look up and this boy is holding out a hockey stick to me. 'Grab on,' he says to me, 'I'll pull you to shore.' And after he had rescued me, he sat me on a bench on the shore and knelt and untied my skates and helped me on with my boots and I thought, What a beautiful boy, what a beautiful boy."
"You were very young," Adrian said. He was studying the map as the road curved south, working its way down the west side of the island. He plotted the winding route that would get them eventually to Pilot Hill Road. Like it or not, Gloria was giving him the full-course tour.
"Well, it was three years later. I was sixteen. I was walking into town and he stopped his truck and asked if I wanted a lift. And at first I said 'No thanks' because I didn't want to be too easy, but he just leaned over and said, 'Are you sure?' and I thought, Well...why not. I climbed in and we've been together ever since. The day they bombed Pearl Harbor he asked me to marry him. He knelt down just the way he had when he rescued me that time, you know, to untie my skates, only this time he asked me to marry him. And I thought what a beautiful man he is. We had the wedding on New Year's Eve. It was 1941. He left for the Navy three days later. He was a UDT man. Well, you know, growing up on the island, he knew how to swim. I was nineteen. He was twenty."
Adrian looked at the old woman beside him and tried to imagine her at nineteen. He tried to imagine what it took to marry a man and sleep with a man who was leaving for war and might never return. He wondered if she had been faithful to him. He wondered if he had been faithful to her. He wondered if they discussed such things in those days.
"What was his name?" Adrian asked her.
"Bob. Well, Robert. Well, Bob...Bob Dodge." Her eyes were red and watering now. Saying his name aloud must have triggered the tears. There was no sadness in her voice. Only resignation. But her eyes were brimming with real tears. She was looking for something to wipe them with.
"I'm sorry, I don't know..."
"Oh no, no worry," Adrian said.
They'd nearly made it to the southeast corner of the island.
Gloria pulled into a parking area below another lighthouse.
"Take a walk out there and you'll be able to see the end of Long Island," she said, "and Mohegan Bluffs and the Southeast Lighthouse. They had to move the thing back a few years ago. It would have fallen into the sea. And you can say you stood in Rhode Island and saw New York."
Adrian had no interest in the lighthouse or Long Island or a walk in the sand but he figured Gloria might want a moment to recompose herself. So he got out and took the path out among the scrub trees and Rosa rugosa shrubs, to where a wooden deck overlooked the high bluffs and the beach maybe two hundred feet below. It was a breathtaking view and he opened his arms wide and closed his eyes and tilted his head back so his face basked fully in the brackish air and the bright sunlight and the slight breeze and the beauty of it. He stood and looked at the seascape. He looked back the way he'd come but everything had disappeared behind the dunes he'd walked through. He wondered how long it might take for Gloria to get herself together.
Below on the long beach he could see fishermen casting lines into the sea and reeling in and casting out again and further up the beach were figures of men and women—maybe a dozen or two dozen—outstretched on the strand taking the sun, and others running into the waves and back or diving into the pounding surf. The noise of the ocean and their voices seemed even more distant than he'd gauged at this height.
Out on the sea he could make out pleasure boats in their various odysseys, seabirds diving, what looked like schools of fish feeding on the surface. He wondered how long he'd have to watch and if the season was the right one to see a whale or a dolphin, or great sea turtles coming ashore to lay their eggs, or any of the countless other creatures that would never appear in Findlay, Ohio, no matter how long one looked out into the light or dark. There were cloud banks in the distance and the line between the sea and the sky, and what he reckoned might be the edge of Long Island far in the distance, grew less and less articulate and for a moment he wondered if he might be entirely lost.
VI
THE MEETING with the bishop had gone fairly badly. In retrospect, Adrian was not surprised. This was Ohio and these were the 1980s, and they were Methodists and what did he expect the bishop to do?
"It's not," Bishop Maghee was eager to assure him, "that Clare left you, or the divorce, or that your children might be scandalized," though he felt duty-bound to cite, in the prayer he began the audience with, that caution from the Gospel Matthew, to wit: If any of you put a stumbling block before one of these little ones who believe in me, it would be better for you if a great millstone were fastened around your neck and you were drowned in the sea...Woe to the one by whom the stumbling block comes.
Nor was it the "very reasonable" concern about a suddenly single man with, "let us say, adult desires," ministering to churchwomen made vulnerable by their "religious dilemmas." The issue of "clerical continence" was not the thing. Though the bishop was pained to remind him, it was not unheard of—a clergyman preying on a parishioner or vice versa.
"Romance and religious fervor are so often confused."
It was simply the concern the bishop had for Reverend Littlefield's emotional well-being which seemed, if reports were even partially true, something he ought to be tending to. And all he was recommending, after all, all he was actually insisting upon, all he had actually conferred with the district superintendent and Reverend Hinkston about was a "little respite" from pastoral duties—neither millstone nor drowning—only a temporary "leave of absence" until after the holidays; three months of personal reflection, "with pay of course," during which time both Adrian and the good people at St. Mark's could "reassess their relationship."
The bishop gave Adrian a list of "Christian counselors," approved by the district, none with offices nearer to Findlay than Toledo and Cleveland, who would be "helpful and discreet" and who would bill the UMC directly for "up to six months of therapy and an evaluation."
"We must take care of the caretakers, Adrian, minister to the minister—that's what we are called to do!" said Bishop Maghee, and looking at his wristwatch, extended a hand for Adrian to shake.
"FOR FUCK sake, Adrian, what'd you say?" Father Francis Concannon was no stranger to the ways of bishops.
He and Adrian were sitting in the leather wingback chairs in the library at St. Michael's Rectory the day after the Reverend Littlefield had been called to see the bishop of the Western Ohio Conference of the United Methodist Church.
"What could I say? 'He who sings prays twice'?"
"Of course you're right. It's like farting at skunks—trying to gainsay a bishop. The whores." Father Concannon said "whores" to rhyme with "lures," and added, "The fucking wankers...my own's a dodgy client just like yours."
The priest had made tea and was pouring it.
"I gave him all the ammo he'd need, what with the singing. And of all things, Beatles tunes."
"The Pentecostals would call it talking in tongues. And brought the snakes in for you to fucking dazzle."
THE REVEREND Adrian Littlefield, recently quit by his adulteress wife, recently angry and lonely and bone tired of the duties of single parenting, recently despairing, recently at wit's end over his prospects, recently drunk, recently stoned, recently a patron of a topless-bottomless bar in Windsor, and only a matter of hours after having sex with his children's babysitter, whose first name he could remember but whose surname had escaped him, had stood in the sanctuary under the massive stained-glass likeness of Christ his Lord and before the faithful congregants of St. Mark's United Methodist Church on South Main Street in Findlay, Ohio, and, uncharacteristically lost for words, cleared his throat, opened his arms as Moses before the Red Sea, and raising his inexplicably grinning face heavenward, instead of commencing a sermon, sang out, off-key but enthusiastically, in evident praise and thanksgiving for all of his recent iniquities:
When I find myself in times of trouble
Mother Mary comes to me
Speaking words of wisdom, let it be.
And in my hour of darkness
She is standing right in front of me
Speaking words of wisdom, let it be.
Let it be, let it be.
He stepped from behind the oaken pulpit, stationed himself in front of the altar, and smiling widely, raised his voice again.
And when the broken hearted people
Living in the world agree
There will be an answer, let it be.
For though they may be parted there is
Still a chance that they will see.
There will be an answer, let it be.
He was into the next verse though he wasn't entirely sure he knew it, And when the night is cloudy, there is still a light that shines on me... and might have even made some sense of all of it, but the Reverend Dr. Hinkston, his senior pastor, sensing something in his associate's manner was terribly amiss, and worried that the eleven o'clock service was about to go seriously off-track, rose from his seat behind the pulpit and led the people in polite applause, then wordlessly, by nod and glaring, signaled the ushers at the back of church to pass the plates for the offering. A couple of spiky-haired teens dressed in black and emblazoned with tattoos and seated with their perpetually embarrassed parents, arose during Adrian's brief solo and, thrusting their fists into the air, took up the chorus: Let it be, let it be, let it be, let it be, whisper words of wisdom, let it be, while one air-keyboarded the piano chords and the other air-guitared the heavy bass notes of the refrain, all the while nodding their spiky heads furiously in time. Marilyn Rubritus, the unmarried and, it was rumored, heavily medicated daughter of the long-widowed Geraldine Rubritus, stood in the middle of the pew she and her mother had occupied quietly for nearly half a century, unknotted her long silver hair from the bun it had been in all her life, till it fell in quick luxuriant waves over the red cashmere sweater she always wore to church, and began to sway with the singing rhythmically, her arms outreaching and her palms upturned, her bony shoulders and skinny hips achieving a kind of tidal sway, and the look on her face one of ecstasy, the way you'd imagine young hippie girls at Woodstock years before. Cleta Locey, the organist, never one to be caught off guard, had taken up the tune, which of course she knew, while the Reverend Hinkston tried, from his side of the sanctuary, to eyeball her to quit and the ushers moved with the gilded plates up the aisles and everyone, reaching for their tithes and offerings, looked back and forth at one another wondering what it was they had just suddenly witnessed. It had lasted only a couple of minutes, but it was sufficiently outside the pale of their erstwhile church experiences that they knew something unforeseen and unplanned had happened.
"Did you see them," whispered Adrian, still grinning like a simpleton, "the heavenly hosts?" as Reverend Hinkston led him to the vestry, while the bell choir, after a series of dagger looks from the senior pastor, and accustomed to playing for time, took up their rendition of "How Great Thou Art."
After removing his stole and cincture and robe, and following the Reverend Hinkston's directive, Adrian collected his children from their Sunday school classes, loaded them in the family Honda, and drove them home. He sang all the way home. And because he could not keep from grinning, Damien and Sarah grinned back at him. For the first time in months they all looked joyous.
"I TELL YOU, Francis, it was the closest thing to inspiration I have ever felt—the breath of God—as if I'd been suddenly loosed from the bonds of gravity and routine. Always before, I'd be trying to say something that'd touch their souls while they'd all be trying not to fall sleep. But this was different. I tell you, Francis, I felt alive and they looked alive to me. For the first time I looked out over that little sea of faces and saw them all as fellow pilgrims. Not fellow United Methodists, or fellow Christians, or fellow sinners. Just fellow humans in search of the way home. And I could see that what they didn't need was another sermon. It's as if I could see myself in them: hapless and lonely, holy and free, all of us somehow in it together, Francis, just trying to find our way, wondering if we're ever going to make it."
The priest nodded and smiled and sipped his tea while Adrian carried on.
"It was an apparition. They looked like innocents, Francis. Angels—every one of them. I tell you, I could see their wings. I could see them readying to take their flight, Cora Perkins, fat Bill Wappner, the grievous Fielding couple with their punk-rock twins. They just all looked so lovely to me—these people whose people I've been burying and marrying and baptizing. Poor Marilyn Rubritus, old Henry Richardson in his wingtips and banker's suit, Art Geyer with his homely wife, the Morris sisters, just turned fourteen and fifteen, proud of their new figures, the way boys are suddenly watching them; Freda Chambers with her goiter and bug eyes, they all looked like cherubim and seraphim and archangels. And all I wanted was to tell them that everything was going to be all right. Everything would turn out fine—I could see it all, Francis—they'd all be just fine and flying again."
"A beatific vision!" Father Francis said. "Was that some dope or what!"
"It wasn't the dope, Francis," Adrian said. "I think it started after Mary...what's her name?"
The priest sat up in the wingback, set his mug on the side table, and leaned into the conversation.
"Mary De Dona? What about her?"
"Well, she came to me in the night. There was this dream and I just woke up and she was...there."
"There?"
"Well, she was on top of me, and she was naked, and, well..."
"The beatific vision!" Father Concannon sat back, pressed his head into the high back of the chair, smiling broadly.
Adrian looked puzzled.
"A 'pastoral' visit!" The priest was evidently not surprised. "God bless her. Now there's a woman with really priestly instincts."
FATHER FRANCIS Assisi Concannon explained to the Reverend Adrian Littlefield how, ever since she'd come to St. Michael the Archangel's parish school, Mary De Dona had made known to him the depth of her devotions. She had expressed in a variety of ways her willingness to do whatever she could to assist him in his priestly mission. She had arrived in Findlay the year before to replace one of their retiring teachers. Little was known about her. She had answered their ad in the Ohio Catholic. She had her teaching certificate and was Catholic, or Italian at least, and parochial schools couldn't be that particular since they paid a good deal less than the public schools. And she was wonderful with the children. That much was obvious from the start. They all loved her and all of the parents loved her and the other teachers all approved as well. Even the nun who served as principal of the school and was famously cranky and stingy with praise, spoke glowingly about Mary De Dona.
"We rented her one of the condos we'd made, in the former convent, off the parking lot. Just behind here. Can't get nuns to live in them anymore. Well, couldn't get nuns anymore, period. But Mary was delighted with the space, the stained-glass windows and tiny rooms, the smell of Murphy's Oil Soap on the woodwork. She moved in with her dog and easels and books and that was that."
The priest recounted to Adrian how one night the winter previous, she had come to the rectory greatly agitated, with her huge black dog in tow and begging for a blessing on the beast.
"It was late and I was alone here, sitting up watching Hill Street Blues, and she said the dog wasn't well, couldn't sleep, and would I bless it. Of course I thought she's some kind of a head case, out there in her bathrobe with red boots on and a stocking cap and the wind lashing and the snow piled everywhere. But I wanted to get back to the TV so I blessed the thing, a quickie, but blessed it nonetheless, and said I hoped that would do it, but she insisted I use some holy water. So before I know it, I've got the two of them—this huge dog and this tiny woman in her nightdress, standing in the vestibule of the rectory, dripping with the weather, and I'm off looking for holy water, with which I eventually drench the two of them—Dante, the dog, that's what she calls it, and Mary De Dona—and bless them both in Latin for fuck sake and she's shivering with the cold and I'm holding open the door for them, and she reaches up behind my head, stands on her tiptoes, and kisses me. On my own mouth! Then off she goes with goddamn Dante, shouting 'Thank you Father, thank you Father' all over the neighborhood."
"I never knew you anointed dogs," said Adrian.
"Well, we usually don't, except on the feast of my own namesake. It's coming up next week, in point of fact. We do cats and canaries, dogs and goldfish. We'll bless, as the fellow said, 'all creatures great and small.' But she was in a panic, what could I do? And she told me it was a 'Catholic' dog. Couldn't I see that by the white cross on its chest and its obvious piety—a fucking head case—still, harmless enough, is what I thought."
"The next night she came over about the same time. She had a plate of brownies. She looked, well, done up, you know. She had some perfume on. She said that Dante had crawled into bed beside her and slept like a baby and she was forever in my debt and would do anything at all to repay the kindness. 'Anything, Father,' she kept saying with those eyes of hers looking up at me, 'anywhere, anytime, anything at all, Father.' I thought she was going to break into that James Taylor tune. I won't say she made a pass at me, Adrian, but there is about Mary De Dona a generosity of spirit I've not encountered in our species before."
"Did you have sex with her, Francis?"
"No, no I didn't...really...couldn't."
"You're a better man than I am then."
"Not better, Adrian, just different."
That his friend could be so tolerant of sin in general and yet so scrupulous in the observance of his priestly vows was at once both perplexing and impressive.
"So what did you tell her about me, Francis? That I was abandoned and desperate and horny?"
"I told her you needed a babysitter. I said that we needed a night out. And yes, I think I mentioned that you were on your own...and might have said something about loneliness. I'm not sure I ever used the word divorce. Nor did I say that Clare had left you, I'm sure I didn't. Only that you were a good man, dealt a bad hand, and we needed a night off."
"Well, I'll have plenty of time off now, it seems."
"The blighters," Father Concannon seethed, "the fucking wankers."
THE FLOWERS that Adrian Littlefield took to call on Mary De Dona were sunflowers. The vase that she put them in was fluted and blue. She stood at the kitchen sink, in her condo in the former convent, cutting the ends of stems, setting them into the vase, waiting for Adrian to think of something to say.
"You shouldn't have. They're beautiful."
The black dog that had barked fiercely when Adrian walked up the porch steps and sniffed his groin and buttocks when he entered the house now lay on the floor at the woman's feet eyeing Adrian warily.
"I wanted to say how very grateful I was—"
"Grateful?"
"For the other night."
He wanted to tell her it had been like grace to him, the way she'd given herself, the way she'd come to him. It was free, abundant, unearned, a gift. Amazing grace, he wanted to say. He wanted to tell her that she had saved his life and restored to him a sense of worthiness. He wanted her to know how damaged he had felt, after Clare had left him, after his marriage failed, how he'd been despairing and depressed and beside himself and how ever since the other night he'd been filled with an inextinguishable sense of general benignity, and that even though he'd been suspended by his bishop from all pastoral duties and hadn't a clue what he was going to do next, he was certain he'd been changed for the better and immeasurably improved by what had happened between them the other night. He had rehearsed words to this effect but could not think of how to say them.
"I'm grateful too," she said. "It was lovely. Do you want to do it again?"
"What?"
"Do you want to have sex with me again?"
There was such decorum about her speech, a daintiness, at odds, it occurred to Adrian, with the boldness of her question.
"Well, yes, but...Yes, of course I do."
"Who's watching your children?"
"Francis...well, Father Concannon. He's taking them for burgers and a movie—The NeverEnding Story."
"How very good of him."
Mary De Dona set the sunflowers in the blue vase on the kitchen table, banished her giant dog to the back porch and the fenced yard, and took Adrian's hand and led him upstairs to the tiny room that was her bedchamber. She closed the door behind them, lit a small candle on the bedside table, pulled the curtains by, and turned to face him. Then she slowly, wordlessly removed his clothing and just as slowly, wordlessly removed her own. Whereupon the two of them fell into intimate if predictable embraces, kissing and licking, touching and sucking, holding and beholding and savoring each other for all of an hour, then another. Then they bathed together in her tub. She dried his back and rubbed some scented oil on him, and let his hands rub some of it on her, and though they agreed he really should be going, that surely The NeverEnding Story would be over now, before he could bring himself to put his clothes back on, he fell to his knees and pressed his face against her, whereupon they took up their embracing in earnest again.
Stretched out on Mary De Dona's narrow bed watching the ceiling fan slowly circling above them, Adrian could not keep from thinking about the latter days of Job, blessed by the Lord with fourteen thousand sheep, six thousand camels, a thousand yoke of oxen, and a thousand donkeys. He thought he might see his children and his children's children and die old and full of days.
"SEX WITH a generous stranger," wrote the Reverend Adrian Littlefield, in the first paragraph of what would become Good Riddance, "is balm to the wounds of the broken hearted." He searched that sentence for something wrong, at odds as it seemed with his religious training, but in his own ears it rang entirely true.
The visitations of Mary De Dona had been a balm to him.
"Divorce is not, it turns out, the worst that can happen. The sky does not fall. The clocks do not stop. The buses run on schedule. Life goes on. The world is full of possibilities."
It was consorting with Mary De Dona—the illicit sound of which exited him—that made him certain of these things and emboldened him to write them down. Their copulations—from the Latin for "fastened together"—had restored his faith in divine providence. Fastened together with Mary De Dona was, he was certain, a state of grace. It always left him grinning and grateful, dreaming in all tenses and feeling infused with what he took to be gifts of a holy spirit. If he did not speak in tongues, the two of them no less spoke with tongues and fingers and mouths and hands, with the caught breath and perfect hush of touching, and all of the interlocking, interweaving, intersecting limbs and parts that had become the parties to their intercourse, such as it was. They hardly spoke. What more was there to say?
She came over on Tuesdays and Thursdays in the evening. She'd play with Sarah and Damien, help with their dinner and bedtimes, sometimes throw in a couple of loads of laundry and tidy the house, then make love to Adrian Littlefield, like any other household duty or routine chore. This too excited him, her matter-of-factness with him. Then she'd go home. On Saturdays, he'd arrange a babysitter, and go to the former convent where she lived, bringing fresh flowers and massage oil, eager to repay her lovely body for all of the kindnesses it had shown his during the week.
"I want to worship at the altar of your every pleasure," he would whisper.
"Hush, churchman! On your knees! Come into the holy of holies," she replied, with a come-hither smile, feigning sacrilege and shyness, taking him by the hand to her tiny room. "Light the candles, like a good altar boy."
"GOD IS good," he wrote with newfound conviction, "and has given us each other to magnify that goodness."
"Paul was wrong," he wrote in a line that would later be quoted in and out of context. "It is good for a man to touch a woman, and good for a woman to touch a man."
Paul's confusion of sex and sin seemed to Adrian at odds with the essence of human nature. How, he asked himself, could the goodwill he bore toward Mary De Dona, the gift he saw her as, the grace he felt awash in when with her, the thanks that was overflowing in him since she came to him, the sign he reckoned that she was of God's love—how could it all be anything but good? What sin could leave one so manifestly at one with creation, at peace with one's being and another's? In Mary De Dona all the dull notions he'd studied in seminary were happily incarnate—resurrection, reconciliation, communion, and rebirth. She was Easter, Christmas, epiphany and apocalypse, a blessing and beatitude, a feast for his soul. Christ might have gone to the cross for him, but Mary De Dona had come to bed with him. Jesus might have raised himself from the dead, but Mary De Dona had restored Adrian to life itself—the life of the body and the mind and spirit that had been killed by the failure of his marriage. Paul was wrong. Adrian knew it now. And if Paul could be wrong, why not James and John and Job and the rest? Not entirely wrong, just occasionally. Just enough for reasonable doubt, a little wiggle room for questioning—that's all he was trying to establish. What if the Bible was only a book, the authors of it merely men who felt, for reasons he could now more fully comprehend, inspired by the loving breath of God?
How had it taken him so long to arrive at this intelligence? It made him want to read it more closely. It made him love the words and fear them less. As human text, a record of mortals searching for glimpses of God, it was engaging and inspiring. As holy writ, inerrant transmissions through prophets and apostles, it seemed as silly now as all pronouncements of infallibility. Still, Adrian had to admit to himself that until very recently, he had accepted the King James Bible as God's Word dictated to and carefully transcribed by such scriveners as God has chosen for reasons best known to God and God alone.
And it was Paul, that great poser and epistler, that first circuit rider in the cause of Christ, the model for John Wesley's bold remark that "the world is my parish," whom Adrian now read with the grain of salt that put the wonder into everything, everything. For never was a man more wrong about women, and therefore wrong about the men who coupled with them, than Paul was when he wrote to the Corinthians. That Paul regarded men as brutes and women as temptresses, fit only to keep each other from "incontinency," and marriage as a better station than passionate, erotic sex, but not as good as celibacy, struck Adrian as unfair to women like Mary De Dona, who seemed to him quite proper vessels and dispensers of God's mercy and grace. To be welcomed into another's bed, into another body, not for the promises you might make, or the shelter you might provide, or the babies you might bear or sire or for all the future possibilities, but rather for the gifts you might bring to the here and now, the moment at hand, not the past or future or pluperfect tense, but for the moment at hand—now there's the thing, thought Adrian, the gift like grace.
God is love, he quoted the beloved apostle, and he who abides in love abides in God and God in him. Whenever Adrian replaced the word love in this dictum with the name of Mary De Dona, apart from the idolatry, the sentence rang entirely true. He felt reborn, re-created, and alive in her.
By Halloween he had over a hundred pages. Part rant, part methodical reasoning—it had become his manifesto, a statement of faith in flux.
"We are God's gifts to one another. We come to one another like grace—out of the blue."
By Thanksgiving he sent what turned out to be the first three chapters to an agent in New York who sent back an agreement for Adrian to sign which gave 15 percent of any sale to the agency.
When a check came in the first week of January for over forty thousand dollars, he could not help but see in it the saving hand of God, coming as it did in the same mail as a letter from the district supervisor detailing how, "after prayerful reflection and full consideration of the needs of the faithful in the Western Ohio Conference, we feel some changes have to be made."
"The second half of the advance will be paid upon delivery of the completed manuscript," wrote Adrian's agent from New York. "Get to work."
"We'd like to offer you a chance to truly expand your ministry through our Pastoral Exchange Program," wrote the district supervisor. There was a pamphlet on the Worldwide Ministerial Exchange of the United Methodist Church and a "call" to trade places for three months with "the Reverend Gilson Miller and his family" from somewhere unpronounceable in England.
"The 'geographic cure,'" said Francis Concannon. "Bishops are mad for it. Outta sight, outta mind. First they move ya, then they lose ya!"
"Might be good for the children, a change of venue, a chance to see a bit of the world." Mary De Dona said travel was good education. "Three months will go by in a blink."
"Would you come with us?" Adrian asked her.
"An international scandal?" she laughed.
She thanked him for asking but said it would be better for him to travel light. She had her work to do and he had his.
HE HAD never heard the voice of God or seen anything that made him certain of God's direction. His "calling" to the ministry of the Methodist Church had never been a road to Damascus experience. It had happened to him like the rest of life had—the slow accumulation of events, some memorable or remarkable, most others not, which taken together had become his life.
Things happen as they are supposed to happen, Adrian told himself. If God did not speak to him, he thought, God was nonetheless the one in charge. This was largely a default position, faith arrived at by the servants' door. If Adrian was not in charge of all that happened—and he clearly wasn't—and yet unable to abide the prospect of no one in charge, then God was whoever was left in the room. This was the only article of faith he still clung to: there was a God and he was not it. So life happened the way it was supposed to happen. If God wasn't directing things, God was at the least watching. Everything might not work out for the best, but everything would be over soon. It was enough on most days to keep him going forward, this knowledge that whoever was in charge here was carrying on.
He put thirty thousand of his windfall down on the big vacant mansion down the street from the church on the better-than-even chance there'd be no job waiting at St. Mark's when he came back from England. At least, he figured, they'd have a home. He left ten thousand with Mary De Dona with instructions to "do it up the way you'd like it," and in his heart of hearts imagined them maybe getting married or moving in together when he returned, absence making their hearts grow fonder, he hoped.
After clearing it with Clare's attorney, he got passports, took the kids out of school, and in the first week of March, Mary De Dona drove Adrian and Damien and Sarah to Detroit, where they boarded an overnight flight for London. Thence on a train to Leeds in West Yorkshire, changing trains to the village of Hebdon Bridge, where one of the congregants from Heptonstall met them at the station and drove the jet-lagged pilgrims further up into the Pennines to the stone manse that would be their home until the first of June. Looking out over what he'd read was the Caldervale, across the valley to hill fields rising up from the river below, Adrian Littlefield gave thanks for safe journeys and prayed for his children's well-being and his own.
VII
"GREAT VIEW, isn't it!"
"Yes, yes it is, spectacular."
Gloria was leaning on the station wagon, finishing a smoke, recomposed.
"Could you see Long Island?"
"I don't know. Things kept appearing, then disappearing. Hard to know where the sea gives way to the land or sky. Everything kept blurring into everything else. But yes, yes, beautiful."
They got back into the car; Gloria started the engine and backed out the drive. Adrian looked at his map to figure the route back to New Shoreham and the Old Harbor. There were still two hours before the return ferry.
"Do you know Pilot Hill Road?" Adrian asked. "Do you know someone by the name of Ben Walters?"
"There was a Walters up that way all right. Just above Tug Hole. His wife was sick in some way. I think she died. Summer people from New York. He painted. You know, pictures. I don't know if he still comes or not. Do you want to go by there? How do you know them?"
"Friend of the family," Adrian said. "If you've got the time, I'd like to have a look."
He didn't know exactly what he wanted to find. He'd never really sorted out his thoughts on the matter. He didn't know what he was supposed to feel or think about it. Ben Walters had only been the first of the infidelities he was sure of. There might have been others before him. It hardly mattered now. He remembered a time when he hated the name and the idea of another man touching his wife in that way. He'd tried to outgrow those primitive feelings. He could remember, as a much younger man, wishing for the kind of marriage Gloria had had—those long years, those children and grandchildren and great-grandchildren, that love and grief and routine. Adrian could remember the times as a young family man driving through small towns in Ohio with Clare on a Sunday afternoon, looking into the tall windows of turn-of-the-century homes with their gingerbread and clapboard and backyard gardens, trying to imagine the orderly good-old-days lives of the inhabitants of such places, where everyone had a huge front porch on which they sat in the evening drinking lemonade and telling stories and waving at the neighbors who'd be walking by. He could remember how he awoke one morning to find he had the very thing, a settled life in Findlay, Ohio, in an old house with a wide porch and wooden floors and knickknacks and radiators and a wife whose unhappiness seemed to grow in direct proportion to his happiness. He'd wanted that life, the settled, Sunday dinner with the family, Rockwell print of an existence where he'd eventually be the senior pastor of a thriving church where everyone knew everyone and everyone's business and kept an eye out for each other's children and were determined to live happily ever after.
At about the same time Clare was getting tired of all of that. She wanted to know if he'd consider moving to New York. He could maybe manage one of her Uncle Harold's companies. They could live in Westchester. There would be more money, she was sure. He could commute to the city by train like Uncle Harold did, to his office in midtown. She could do photography or videos or something artistic and the kids could get a nanny or go to a fashionable day care center and then to a Montessori school. She could come into the city on Friday nights for the theater; they could ice-skate at the rink at Rockefeller Center as she had done as a girl visiting Uncle Harold after her mother and father divorced. They would have interesting friends, an interesting life. She was tired of Ohio and Findlay and the First Methodist Church. She didn't want to be a senior pastor's wife. She didn't want a summer place on Lake Erie. She didn't want to grow old in an old house in the Midwest with a man who was content to be going nowhere.
He told her he thought there were no geographic cures. "Unhappiness," he told her, in the way she hated that ministers had of speaking in slogans, "was portable. Discontent travels light."
Gloria had turned up High Street and it turned to a dirt road at Pilot Hill and she broke suddenly at the entrance to a small two-track drive on the right.
"I think that's the Walters place up there. Go ahead, I'll wait. Take your time."
Adrian hadn't a clue what he was supposed to do, or what it was he wanted to find. What if Ben Walters was there and knew who he was? Maybe he'd read one of his books, or seen him on Oprah or heard him on the radio and knew the connections. What if he didn't know Adrian's connections to the woman he had seduced here almost twenty years ago and how it changed all their lives and left his children motherless and him with his hands full of duty and detail? What if he was only a withered old man walking around in tennis shorts and sandals with leathery skin and a bald head? What would they have to say to each other?
Adrian could feel his heart racing as he worked his way up the little gravel drive to the clearing in the woods where a little cedar-shingled cottage appeared surrounded by a little lawn with a few old Adirondack chairs in a semicircle and badly in need of paint. The place was tiny, a story and a half with a screened porch, inside of which was what looked like the main door. There was no sign of life anywhere around the place. No car, no open windows, the grass a little overgrown. He tried the screen door, walked in, and knocked at the main door and listened hard for the sound of any movement inside. There was none. He tried the doorknob but it was locked.
There was nothing. Only the sound of catbirds in the thick woods around the place, and the smell of the sea, and the movement of the breeze in the greeny things around the place. There was as well the small noise of a wind chime hanging from a hawthorn tree in the yard, and at a distance, as he listened, the noise of children down the hill near what he guessed was the fresh water pond he'd seen on the map at the base of the hill. Adrian looked in the window. The interior was small and dark. A table at one end of the kitchen. A fieldstone fireplace at one end of the main room, and a small hallway leading to what must be the bedroom, or bedrooms—maybe two. There were no signs of recent life inside the house.
Adrian walked around to the bedroom window. Through the sheer curtains he could see an old metal bed, a bureau, and a chair. Off to one side was a sink and a mirror. The bed was made. There was no disorder to the room.
He looked out across the yard. There was a small shed with windows and a small porch. It was, he reckoned, the artist's studio. He looked inside and saw an easel and a table and jars full of brushes and rags hung from hooks on the wall. No canvases or works in progress were anywhere to be seen.
Adrian walked back across the yard and sat in one of the Adirondack chairs and propped his elbows on the wide arms and rested his chin on his folded hands and wondered what to make of the place. Surely, he thought, Ben Walters would be here in midsummer if he was going to be here at all. The place's vacancy had about it a permanence that was, to Adrian, palpable. He figured that Ben Walters, now nearing seventy-five, widowed and alone, his little artistic career having come to nothing, must be summering out in assisted living or a nursing home, probably after the second or third stroke had left him paralyzed or dumbstruck. Anyway, Adrian was sure, Ben was never going to be skippering a sailboat off the coast or walking the beach or sweeping anyone else's wife off her feet, not in this life, and likely not in another. Ben Walters was no longer a man he need contend with. It was good that he'd come here to make this clear.
Adrian tried to imagine how it must have been for Clare. Getting her friend Christine to cover for her, getting here, getting it on with the old fart, the romance of it all, the distance from Findlay, Ohio, the hopes for a new, more exciting life.
She'd gotten as far as Cleveland in the years since. She'd married again and divorced again, and again. She seemed, these long years since, every bit as discontented, only older. The children had each spent hours with counselors learning to love their mother without having to approve of her "inappropriate choices," and to maintain the proper "emotional borders" between themselves and their mother's insuperably chaotic life.
It occurred to Adrian that if she outlived him, much of his hard-earned estate would work its way to her, through the generosity of his children, who would surely use a portion of their inheritance to support their mother, who could be counted on to be, as ever, in need.
Oh well, is what he told himself, when such things came to him.
There was nothing about this place, his coming here that had the sense of portent he had imagined when he first made arrangements to come here. He had simply married the wrong woman. He had chosen wrongly. He had mistaken passion and good sex, easy to muster at twenty-something, for true affection. He knew it early in the marriage. He remembered the mornings he would awake beside her, wishing there were more about her that he really treasured, really admired, really needed. She was not, he knew, a great mother, an exceptional human, or a particularly good woman. They had made good babies, not brought out the worst in each other. But neither had they brought out the best. And he had been, stupidly, willing to live with the consequences of his poor decisions, to tolerate waking next to a woman whom he did not truly admire in trade for lovely children and an ordinary life free of the larger vexations. He had not, in his marriage to her, abused her physically or verbally. He had tried his very best to make her happy. He hadn't drank or gambled or cavorted with other women. He'd done his share of diapers and dishes. He had tried to support her efforts to find more interesting things to "do" than "wife and mothering." He had been, in all ways he thought, the reliable husband, the agreeable helpmate, the hedged bet against hunger and loneliness, the other body in the bed. But he had not, he knew it now, ever loved her entirely. Surely that was a critical fault. Sitting in the Adirondack chair on Ben Walters's deserted front lawn, Adrian wondered if she'd ever known how much he really didn't love her. Maybe something in her sensed that emptiness and railed against his willingness to live in a lackluster if otherwise functional marriage. Maybe it was this that drove her to do what she had done. He had been willing to settle for too little. For him it had always seemed sufficient. For her, enough would never be enough. Which aberration of desire, he now wondered, offended God or Nature or the Fates the most? His willingness to settle or her refusal to? His contentment or her discontent?
Here in a clearing in the woods, in the small yard of the small house where the wife of his youth had long ago betrayed him, it all seemed to him like a mystery now, fathomless and unknowable—the ways of the humans and their hearts.
So it was with Mary De Dona. Much as she'd appeared, she disappeared. She simply wasn't there when Adrian returned from his banishment in England. "You lovely man," is how she addressed the letter she'd left for him with Father Concannon, which thanked him for "his tenderness and generosities" and gave no further details about her plans or whereabouts. She left brief farewell notes for his children too, in the rooms she'd done up for him in their new residence.
"I couldn't tell you, Adrian, I haven't a clue," the priest insisted under questioning. "She turned in her keys and resignation and drove off smiling and waving goodbye, that miserable hound in the backseat of the car."
Adrian was not surprised. He'd come home from West Yorkshire with a finished manuscript, a determination to leave the church, a sense of new message and purpose, and a new place to live in Findlay, Ohio.
He thought he caught a glimpse of Mary De Dona or someone who looked strikingly like her once while he was watching a movie in a hotel room in Minneapolis. Adrian watched and waited for the credits, which listed her name as Sasha Black. Then he lay back on the pillow, fell asleep, and dreamed cinematic dreams. He would search on the Internet but never find her.
He was suddenly worried about missing the boat. He'd lost all track of time somehow.
He jogged back to the car where Gloria was just finishing a cigarette. She pressed it out in the ashtray and started the engine.
"Anyone home?"
"No one, nothing there."
"Too bad," she said. "You've come all this way."
"Oh well," Adrian said. "Another time maybe."
She backed out to Pilot Hill Road and drove Adrian back to the ferry docks. He gave her a hundred-dollar bill and thanked her for the tour.
"But it's only thirty dollars."
"Buy a little cheer for the birthday party."
"That's really good of you," Gloria said, folding the crisp note into her shirt pocket.
"May I ask you," Adrian said, leaning back with one foot out of the car, "may I ask you, Gloria, did you ever find yourself, like ever, in those fifty-eight years, you know, married to Bob, did you ever wonder, What the hell am I doing here? you know with the kids and the work and the routine?"
"No," she said without hesitation, "never once. I was just so glad to have him home, safe, after the war, I'd missed him so. And I thought, I always thought, what a beautiful man, what a good man he was. So I can tell you, we had our hard days, sure. But no, I never wondered about being with him. I wish I were with him now. I can still feel him."
There was a catch in her breath. Her hands dropped from the steering wheel of the old wagon. Adrian said nothing.
"The young these days are so unhappy, so impatient, so full of expectations. All we wanted was to survive it. To be together. To get through, Bob and me, you know, and for the children...Nowadays they just want too much. Whatever they have, they think there must be more. They want so much they don't know what they want."
She was staring at a point in the middle of the steering wheel. She caught her breath again.
"Yes, yes, I suppose that's it."
Adrian wondered what it was he wanted. He had long since lost hope of a woman who could love him like a wife would and love his children like a mother. That mix of passion and sacrifice seemed quite impossible to him now. Not because such women did not exist, but because he lacked what it was they wanted. Though he'd had housekeepers and nannies and tutors and teachers and therapists for his children; though he'd had no shortage of memorable sexual partners; there had not been nor would there be, he now knew with certainty, anyone like Gloria in his life and times—a woman who would mourn and remember the boy he had been, the man he was, the old man he hoped to be, who would love him and outlive him and keep him alive in the daily lives and times of his people, his children and his children's children. He could feel the wave of sadness rising in him that he knew, if he did not move, would overtake him.
He closed the car door and made for the ferry.
"Safe home," she shouted after him.
Boarding the boat, Adrian blew kisses.
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J. R. Uretsky is an artist, performer, musician and art curator living in Providence, Rhode Island.
Education
Uretsky earned her bachelor's degree in studio art at Biola University, her master's degree in fine arts in sculpture at the University of Connecticut, and her graduate certificate in Museum Studies at Harvard University.
Work
Uretsky uses puppetry, textile art, sound, video and performance to create "expressive confessions," evoking emotions in the audience. Her work has been exhibited nationally and internationally. Her work was included in the 2013 DeCordova Biennial at The DeCordova Sculpture Park and Museum. She has also performed and exhibited at Art Basel in Miami, Florida, the Carpenter Center for the Visual Arts at Harvard University, the Rhode Island School of Design Museum as well as the Museum of Art and Design in New York. Uretsky's work has been published by print, online and video journals such as Headmaster Magazine, Gaga Stigmata, Big Red & Shiny and ASPECT: The Chronicle of New Media Art.
Uretsky performs in queer punk band, Bed Death and J.R. and the Worship Band. Her musical influences include Bad Brains, Pedro the Lion and David Bazan. Her other collaborations include an art band called Feminist Conference with drummer and visual artist Rachel Blumberg and cellist Emily Dix Thomas.
In addition to being the curator at the New Bedford Art Museum, Uretsky has curated exhibits at Artspace in New Haven, Connecticut, AS220 and the Wedding Cake House (Dirt Palace) in Providence, Rhode Island, and the Distillery Gallery in Boston, Massachusetts.
An active member of the Providence creative community, Uretsky sits on the Dirt Palace Public Projects Board of Trustees.
Awards
2018 YWCA Woman of Distinction Award for her dedication to uplifting the creative expressions of people of color, women, and LGBTQ individuals.
2013, deCordova Biennial
2012–2013 Artist-in-residence, Dirt Palace
2015 On Our Radar, CREATIVE CAPITAL
2009–2011 Graduate Fellowship, University of Connecticut
Bibliography
Eli, Phillip. "Katy Perry, tighty-whities, and fine art from Ocean State Job Lot." Providence Phoenix.Feb 26, 2014.
Glaser, Brian Christopher. "Inside Out: JR Uretsky." Big Red & Shiny. December 3, 2012.
McQuaid, Cate. "A Woman's Arms a Refreshing Show of Force," The Boston Globe. April 15, 2014.
McQuaid, Cate. "Some standouts in Locally Made," The Boston Globe. August 26, 2013.
McQuaid, Cate. "Cartoony geometrics, 3D 'paintings' standout at the deCordova Biennial," The Boston Globe. October 12, 2013.
Moeller, Robert. "Made in New England: The 2013 deCordova Biennial. Hyperallergic. December 12, 2013
Pyper, John. "The artists of the deCordova Biennial," Big Red & Shiny. October 30, 2013.
Rysz, Tori. "Turning Barriers into Bridges." ArtNewEngland. July–Aug. 2016.
Volmer, Suzanne. "2013 DeCordova Biennial, DeCordova Sculpture Park and Museum." Sculpture 33.7 (2014):72. Print. Photo.
References
Artists from Providence, Rhode Island
Biola University alumni
University of Connecticut alumni
Harvard Extension School alumni
Queer musicians
1985 births
Living people
21st-century American artists
21st-century American women artists
21st-century American musicians
21st-century American women musicians
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 1,360
|
{"url":"https:\/\/www.researcher-app.com\/paper\/133158","text":"3 years ago\n\n# The Radio Sky at Meter Wavelengths: m-Mode Analysis Imaging with the Owens Valley Long Wavelength Array.\n\nRyan M. Monroe, Hugh Garsden, Jonathon Kocz, Jayce Dowell, Gregg Hallinan, Steven W. Ellingson, Benjamin R. Barsdell, Marin M. Anderson, David P. Woody, Frank K. Schinzel, T. Joseph W. Lazio, Jacob M. Hartman, Yuankun Wang, Gregory B. Taylor, Stephen A. Bourke, Michael W. Eastwood, Danny C. Price, Harish K. Vedantham, Lincoln J. Greenhill, M. A. Clark\n\nA host of new low-frequency radio telescopes seek to measure the 21-cm transition of neutral hydrogen from the early universe. These telescopes have the potential to directly probe star and galaxy formation at redshifts $20 \\gtrsim z \\gtrsim 7$, but are limited by the dynamic range they can achieve against foreground sources of low-frequency radio emission. Consequently, there is a growing demand for modern, high-fidelity maps of the sky at frequencies below 200 MHz for use in foreground modeling and removal. We describe a new widefield imaging technique for drift-scanning interferometers, Tikhonov-regularized $m$-mode analysis imaging. This technique constructs images of the entire sky in a single synthesis imaging step with exact treatment of widefield effects. We describe how the CLEAN algorithm can be adapted to deconvolve maps generated by $m$-mode analysis imaging. We demonstrate Tikhonov-regularized $m$-mode analysis imaging using the Owens Valley Long Wavelength Array (OVRO-LWA) by generating 8 new maps of the sky north of $\\delta=-30^\\circ$ with 15 arcmin angular resolution, at frequencies evenly spaced between 36.528 MHz and 73.152 MHz, and $\\sim$800 mJy\/beam thermal noise. These maps are a 10-fold improvement in angular resolution over existing full-sky maps at comparable frequencies, which have angular resolutions $\\ge 2^\\circ$. Each map is constructed exclusively from interferometric observations and does not represent the globally averaged sky brightness. Future improvements will incorporate total power radiometry, improved thermal noise, and improved angular resolution -- due to the planned expansion of the OVRO-LWA to 2.6 km baselines. These maps serve as a first step on the path to the use of more sophisticated foreground filters in 21-cm cosmology incorporating the measured angular and frequency structure of all foreground contaminants.\n\nPublisher URL: http:\/\/arxiv.org\/abs\/1711.00466\n\nDOI: arXiv:1711.00466v1\n\nYou might also like\nDiscover & Discuss Important Research\n\nKeeping up-to-date with research can feel impossible, with papers being published faster than you'll ever be able to read them. That's where Researcher comes in: we're simplifying discovery and making important discussions happen. With over 19,000 sources, including peer-reviewed journals, preprints, blogs, universities, podcasts and Live events across 10 research areas, you'll never miss what's important to you. It's like social media, but better. Oh, and we should mention - it's free.\n\nResearcher displays publicly available abstracts and doesn\u2019t host any full article content. If the content is open access, we will direct clicks from the abstracts to the publisher website and display the PDF copy on our platform. Clicks to view the full text will be directed to the publisher website, where only users with subscriptions or access through their institution are able to view the full article.","date":"2022-07-03 10:48:26","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\": 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\": 0.5609503984451294, \"perplexity\": 5376.469049135709}, \"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-2022-27\/segments\/1656104240553.67\/warc\/CC-MAIN-20220703104037-20220703134037-00223.warc.gz\"}"}
| null | null |
{"url":"https:\/\/pureprimesol.com\/finite-element-analysis-results-static-theories-of-failure-part-%E2%85%B3\/","text":"# Finite Element Analysis Results: Static Theories of Failure: Part \u2163\n\nApplied Engineering Ductile Materials Failure Analysis Theory and Mechanical Stress Terminology Part \u2163\n\nIn the previous section of this series, we discussed the following topics:\n\nAbove are a few foundational pieces for finite element analysis (FEA), stress analysis, and structural analysis. You can get an analytical solution and FEA analysis results without having a design allowable, material properties, or knowing the material classification. However, you won\u2019t know what the result means without having the stress and strain curve information. Or if you take the time to set up each material with its Stress Allowable.\n\n# Static Failure Theories (Ductile Materials)\n\nAll failure theories are material, load, and temperature-dependent. We need to be aware of the ductile-brittle transition temperature if we expect to encounter extreme temperatures. This post will cover the failure theories of Maximum Shear Stress and Distortion Energy. Other material failures, brittle material, plastic deformation, and buckling will be covered in the future.\n\n## Maximum Shear Stress Theory (MSS):\n\nMaximum Shear Stress Theory predicts that failure will occur when the maximum shear stress in a uniaxial tension test exceeds the material\u2019s shear strength. Yielding begins when the maximum shear stress exceeds the allowable stress in an element, exceeding the maximum shear strain values obtained from tensile tests of materials of the same material type.\n\n\\tau_{max} = \\frac{\\sigma_{1} + \\sigma_{3}}{2} = \\frac{S_{y}}{2}\n\nAny stress theory can be plotted and will develop an envelope. The inside area of that envelope is where failure will not occur, leading to a safe design.\n\nMSS is commonly used during the design process and is well-accepted, but we find it could be more accurate. For example, the figure below shows that the failure occurs after MSS predicts it will, leading to an increase in the safety factor, which we might view as a good outcome early in the design process when there are many unknowns.\n\nHowever, during the validation stage of any project, it is not desirable to use MSS as there is a chance that MSS will increase cost needlessly.\n\nAdditionally, the Maximum Shear Stress Theory for Ductile Materials is also known as the Tresca Theory. A factor of safety (n) can be set or determined using the following formula:\n\nn=\\frac{S_{y}}{\\sigma_{1} - \\sigma_{3}}\n\n## Distortion Energy Theory (von Mises yield criterion)\n\nThe distortion-energy theory is the most widely used theory for ductile materials. The Distortion Energy Theory predicts that material will yield when the strain energy per unit volume reaches or exceeds the yield strain energy per unit volume for simple tension or compression.\n\nThe theory was developed by observing that ductile materials stressed hydrostatically under deep water exhibited greater yield strengths than previously obtained at atmospheric pressure.\n\nThe Distortion Energy Theory argues that if you divide strain energy into hydrostatic volume changing energy and angular distortion energy, yielding is primarily affected by the angular distortion energy, causing your model to yield. Below is some point of interest:\n\n\u2022 It is a scalar, unsigned stress quantity resulting in no direction or indication of tension or compression, resulting in an always positive value with no direction.\n\u2022 Allows Multiaxial Stress States To Be Compared To Failure Criteria Obtained From A Uniaxial Tension Test (this is very important)\n\u2022 Only Considers Strain That Produces Distortion (Shape Change)\n\u2022 Only the stresses which act to distort the shape of the part will matter when checking for yield.\n\nWhen plotted, The Distortion Energy Theory develops an envelope, and the interior area of that envelope is considered an area where failure will not occur. Leads to a safe design\n\nThe distortion-energy theory has many names:\n\n\u2022 von Mises theory\n\u2022 von Mises\u2013Hencky\u2019s theory\n\u2022 The shear-energy theory\n\u2022 The octahedral-shear-stress theory\n\nYield begins when the maximum shear stress in a stressed material exceeds the maximum shear stress in a specimen of the same material when that specimen starts to yield.\n\n\\sigma_{e} = \\frac{\\sqrt{2}}{2}\\sqrt{(\\sigma_2-\\sigma_1)^2+(\\sigma_3-\\sigma_1)^2+(\\sigma_3-\\sigma_2)^2}\n\nDistortion Energy Theory is commonly used for analysis situations, but when employed with sound safety factors, Distortion Energy Theory is completely appropriate for design applications. And, for design and analysis purposes, a factor of safety (n) can be set or determined using the following equation:\n\n\\sigma_e=\\frac{S_y}{n}\n\n## The Maximum Normal Stress Theory:\n\nMaximum Normal Stress Theory is the oldest of the failure theories, and most text states that maximum normal stress is not suitable for predicting ductile fracture and failures. The theory shows that failure will always occur whenever the greatest tensile or compressive stress exceeds the uniaxial tensile or compressive strength. And, testing data shows that it correlates best with brittle material.\n\nAs we have shown in an early post, There are three mutually perpendicular planes passing through any given point in a body such that the stresses on each are purely normal, either tension or compression.\n\n\\sigma_{min,max} = \\frac{\\sigma_1+\\sigma_3}{2} \\pm \\sqrt{\\frac{\\sigma_1 - \\sigma_3}{2}+\\tau_{1,3}^2}\n\nHowever, a brittle material will fail when the maximum principal stress exceeds some value, independent of whether other stress tensor components are present. Experiments in uniaxial tension and torsion have corroborated this assumption. Still, the failure envelope clearly shows that the maximum normal stress theory is only valid in the first and third quadrants. It does not account for the interdependence of normal and shear stresses, which affects the second and fourth quadrants.\n\n## Final Thoughts\n\nFEA simulation is a great tool for engineers to investigate structural designs, but you have to know what the results mean. There are two theories on what initiates ductile material to fracture: the distortion energy theory and the maximum shear stress theory.\n\nUltimately, these theories appear to predict failure in structures when the shear stress exceeds a certain limit. However, the distortion energy theory appears to be more accurate with the initial conditions (material properties and geometric properties) than we used in FEA structural analysis.\n\nIt seems as though the Von-Mises stress criterion is most closely approximated by material constants and geometric properties(cross-sections). The Max-stress criterion tends to over-predict failure, especially near yield.\n\nAlthough the Finite Element Method (FEM) is a great tool for engineers to investigate structural designs, you must know what the results mean to be effective.\n\nThis site uses Akismet to reduce spam. Learn how your comment data is processed.","date":"2023-02-06 10:28:55","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.45515722036361694, \"perplexity\": 1299.1054358556198}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764500334.35\/warc\/CC-MAIN-20230206082428-20230206112428-00289.warc.gz\"}"}
| null | null |
<?php
/**
*
*/
namespace Mvc5\Route\Match;
use Mvc5\Arg;
use Mvc5\Http\Request;
use Mvc5\Route\Route;
use function preg_match;
use function strlen;
class Path
{
/**
*
*/
use Plugin\Params;
/**
* @param Route $route
* @param Request $request
* @param string $path
* @param int $offset
* @param callable $next
* @return Request|mixed
*/
protected function match(Route $route, Request $request, string $path, int $offset, callable $next)
{
if (!preg_match('(\G' . $route->regex() . ')', $path, $match, 0, $offset)) {
return null;
}
$offset += strlen($match[0]);
$matched = !isset($path[$offset]);
$request = $request->with([
Arg::CONTROLLER => $route->controller(),
Arg::MATCHED => $matched ?: $offset,
Arg::NAME => $route->name(),
Arg::PARAMS => $this->params($match, $route->defaults() + (array) $request[Arg::PARAMS]),
Arg::PARENT => $request,
Arg::ROUTE => $route
]);
return $matched ? $next($route, $request) : ($route->children() ? $request : null);
}
/**
* @param Route $route
* @param Request $request
* @param callable $next
* @return Request|mixed
*/
function __invoke(Route $route, Request $request, callable $next)
{
return $this->match($route, $request, $request[Arg::URI][Arg::PATH], (int) $request[Arg::MATCHED], $next);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,493
|
Ayub Kheyl () (Ab Khail) is a small village 11 km (7 miles) outside Khost, Afghanistan. It was the site on July 27, 2002, of a firefight by US soldiers, assisted by Afghan militia, against militants in the village. After the fight, the Americans captured the wounded 15-year-old Omar Khadr, who they learned was a Canadian citizen. He was among the youngest persons detained at Guantanamo Bay detention camp and the last Western citizen held there.
The village was known among the Afghanis for its Pashtun residents, who were conservative and considered closely tied to the Taliban.
See also
Khost Province
References
Populated places in Khost Province
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,472
|
{"url":"https:\/\/cre8math.com\/tag\/day099\/","text":"## On Coding X: Computer Graphics\u00a0II\n\nNow that I\u2019ve talked about what I use computer graphics for in my last installment of On Coding, I\u2019d like to talk about the most useful graphics package I\u2019ve come across for writing mathematics: \u00a0TikZ.\n\nFirst a word about the name, TikZ. \u00a0It\u2019s an example of a\u00a0recursive acronym, like GOD used in Douglas Hofstadter\u2019s\u00a0G\u00f6del, Escher, Bach, which stands for \u201cGOD Over Djinn.\u201d \u00a0(By the way, read\u00a0G\u00f6del, Escher, Bach if you haven\u2019t already \u2014 it\u2019s an amazing book about G\u00f6del\u2019s Incompleteness Theorem.)\n\nSo TikZ stands for \u201cTikZ\u00a0ist\u00a0kein\u00a0Zeichenprogramm,\u201d which is German for \u201cTikZ is\u00a0not a drawing program.\u201d \u00a0According to its creator, Till Tantau, this indicates that TikZ is not a way to draw pictures with your mouse or tablet, but is more like a computer graphics language. \u00a0It is designed for use in LaTeX, and is built with a lower-level language called\u00a0PGF, which stands for \u201cportable graphics format.\u201d\n\nSo enough history \u2014 you can read more in the\u00a0TikZ manual.\u00a0 But a word of caution \u2014 it\u2019s over 1000 pages long, so there\u2019s a lot there! \u00a0I\u2019ll only be able to scratch the surface today.\n\nThe two most important features of TikZ, in my opinion, are (1) you can draw very precise graphics because you\u2019re writing in a high-level langauge, not using a WYSIWYG environment, and (2) you can use it in LaTeX, including putting any LaTeX symbols in your graphics. \u00a0This second point is extremely important, since it means when you\u2019re labelling a diagram, the labels (usually involving mathematical symbols) look\u00a0exactly the same way as the do in your text.\n\nI use TikZ almost exclusively for two-dimensional mathematical graphics. \u00a0In a recent paper, for example, I created the following image.\n\nNote how precisely all the elements are rendered \u2014 this is because you give precise coordinates for all the individual graphics elements.\n\nLet\u2019s take a look at another, somewhat simpler example from this paper.\n\nI chose this example for a few reasons. \u00a0First, it requires using a lot of the basic TikZ commands. \u00a0Second, it illustrates the\u00a0practical side of drawing computer graphics. \u00a0No, I\u2019m not going to submit this image to a Bridges art exhibition any time soon \u2014 but if I need to make an image for a paper, I want it to look\u00a0good.\n\nSo I thought I\u2019d take some time to explain various aspects of the code which creates this image. \u00a0Maybe not the most glamorous graphic \u2014 but you\u2019re welcome to look at the 1000+ page manual for hundreds of neat examples. \u00a0Here is a diagram for a finite-state automaton from page 179, for example.\n\nHere\u2019s what you\u2019d need to type in LaTeX\u00a0to get the simple graph I mentioned a moment ago.\n\nLet\u2019s look at several different excerpts from this example. \u00a0The first line opens your TikZ environment, and scales the image. \u00a0The default unit is 1 cm, so you need to set the scale to get your image the right size.\n\nThe next line is just a comment \u2014 I don\u2019t use comments much for simple images. \u00a0But for more complex images, they\u2019re important. \u00a0Just as important as commenting computer code.\n\nThe next line draws a set of axes. \u00a0What\u2019s important here is that I defined my own \u201c\\axes\u201d command \u2014 so you can include user-defined functions as well. \u00a0The advantages of being able to do this are huge. \u00a0You can use\u00a0any command you define in LaTeX in a TikZ image.\n\nAnother great feature is the looping construct, \u201c\\foreach.\u201d \u00a0The red segments above are one unit long, and so the only difference is the starting points. \u00a0Also important here is that you can use mathematical expressions inside TikZ, like the \u201c{(\\x^2+9)\/6}\u201d expression in the curly brackets.\n\nNote the directives \u201c[thick, red]\u201d after the \u201cdraw\u201d command. \u00a0You have control over\u00a0every aspect of the drawing routines in TikZ \u2014 and there are\u00a0lots of them. \u00a0For example, there is the \u201cdashed\u201d directive in line 10. \u00a0But not only can you draw a dashed line, but you can also specify the lengths of the dash marks and the spaces between them, too! \u00a0Further, you can create user-defined styles if you use the same set of directives over and over again. \u00a0This is helpful if you are creating several related images, and you want to tweak the color, for example. \u00a0You only have to change the color specification in the style, and every image which uses that style will be rendered using the new color.\n\nNotice the \u201c\\filldraw\u201d command in line 6. \u00a0I wanted open circles, so I drew the circles in red, and filled them with white. \u00a0If I wanted to, I could have also specified precisely how thick I wanted the circles to be drawn, but the default thickness looked just fine to me.\n\nNotice the use of \u201cnode\u201d and \u201c\\node\u201d throughout. \u00a0In line 9, for example, the \u201c3\u201d is centered at coordinates (3, -0.5). \u00a0Also note the placement of nodes at the end of line segments, as shown in lines 7 and 8. \u00a0Again, it is significant that you can include\u00a0any text you can create in LaTeX in your diagram.\n\nI hope this is enough to give you the feel of using TikZ. \u00a0As I mentioned in the last installment, I also use TikZ to create slideshow presentations since there is so much you can do. \u00a0Remember, there are over 1000 pages of examples of things you can do in TikZ in the manual.\n\nI would like to mention one caveat, however. \u00a0I do sometimes use Mathematica to create images with mathematically intensive equations and calculations. \u00a0TikZ is closer to a markup language in its usage, as I see it \u2014 so any image that requires significant coding to produce is a bit too cumbersome syntactically. \u00a0Here is an example of an image I used Mathematica to create.\n\nBut another nice TikZ feature is that you can import images into your\u00a0environment \u2014 so I can draw complex images in Mathematica, include them in my TikZ picture, and then overlay LaTeX symbols on top of the image. \u00a0With this capability, the possibilities are quite literally endless.\n\nA final comment: \u00a0TikZ and LaTeX are\u00a0both open source, so if you\u2019re interested, you can just download them and start experimenting! \u00a0The learning curve is a little steep, admittedly, but once you\u2019ve climbed up, you\u2019ll be able to create astonishingly beautiful graphical images for any purpose you have in mind. \u00a0Try it!","date":"2017-11-21 15:39:24","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\": 0.8079975843429565, \"perplexity\": 1382.7692607215774}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-47\/segments\/1510934806419.21\/warc\/CC-MAIN-20171121151133-20171121171133-00164.warc.gz\"}"}
| null | null |
#ifndef _MAPGEN_H
#define _MAPGEN_H
#ifdef LIBNOISE
#include <libnoise/noise.h>
#else
#include <noise/noise.h>
#endif
#include "cavegen.h"
#include "map.h"
class MapGen
{
public:
MapGen();
virtual ~MapGen() { }
virtual void init(int seed);
virtual void re_init(int seed); // Used when generating multiple maps
virtual void generateChunk(int x, int z, int map);
private:
std::vector<uint8_t> blocks;
std::vector<uint8_t> addblocks;
std::vector<uint8_t> blockdata;
std::vector<uint8_t> skylight;
std::vector<uint8_t> blocklight;
std::vector<int32_t> heightmap;
int seaLevel;
bool addTrees;
bool expandBeaches;
int beachExtent;
int beachHeight;
bool addOre;
bool addCaves;
bool winterEnabled;
virtual void generateFlatgrass(int x, int z, int map);
virtual void generateWithNoise(int x, int z, int map);
virtual void ExpandBeaches(int x, int z, int map);
virtual void AddTrees(int x, int z, int map);
virtual void AddOre(int x, int z, int map, uint8_t type);
virtual void AddDeposit(int x, int y, int z, int map, uint8_t block, int minDepoSize, int maxDepoSize, sChunk* chunk);
CaveGen cave;
// Heightmap composition
noise::module::RidgedMulti ridgedMultiNoise;
// ##### TREE GEN #####
noise::module::Billow treenoise;
// ##### END TREE GEN ####
/*noise::module::ScaleBias perlinBiased;
noise::module::Perlin baseFlatTerrain;
noise::module::ScaleBias flatTerrain;
noise::module::Perlin seaFloor;
noise::module::ScaleBias seaBias;
noise::module::Perlin terrainType;
noise::module::Perlin seaControl;
noise::module::Select seaTerrain;
noise::module::Select finalTerrain;*/
};
#endif
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,168
|
Q: When I change the activation function of the output layer the model doesn't learn I'm trying to make a binary classifier for sentiment analysis and I'm using the model from the Basic Text Classification Tutorial in TensorFlow, which is the next:
model = tf.keras.Sequential([
layers.Embedding(max_features + 1, embedding_dim),
layers.Dropout(0.2),
layers.GlobalAveragePooling1D(),
layers.Dropout(0.2),
layers.Dense(1)])
This model has a binary accuracy of 80%. However, when I change the activation function of the last layer (the dense layer) to a sigmoid function, the model doesn't learn, the binary accuracy is less than 40%.
model = tf.keras.Sequential([
layers.Embedding(max_features + 1, embedding_dim),
layers.Dropout(0.2),
layers.GlobalAveragePooling1D(),
layers.Dropout(0.2),
layers.Dense(1, activation='sigmoid')])
Why does this happen?
A: This is because when you don't specify an activation function the layers.Dense uses the default activation function which is linear activation function according to here, You can read more about the difference between different activation functions and how they work here.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
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Q: Handling Incoming Parameters - Payment and Transaction with Rails I have an Order and an OrderTransactions model in my Rails4 application. They have a basic has_one and belongs_to relationship between them.
I'm posting requests from /orders/new page to the bank's URL as you can see below:
<%= simple_form_for(@order, :url => "https://testsanalpos.est.com.tr/servlet/est3Dgate", :method => :post) do |f| %>
<% @hashing.each do |k, v| %>
<%= f.input k, input_html: {name: k, value: v}, as: :hidden %>
<% end %>
<%= f.input :participation_id, ... %>
<%= f.button :submit, "Ödeme Yap" %>
<% end %>
The @hashing, hash is coming from my controller =>
class OrdersController < ApplicationController
before_filter :authenticate_user!
before_action :set_order, only: [:show, :edit, :update, :destroy]
skip_before_action :verify_authenticity_token
def new
@order = Order.new
@hashing = {
clientid: POS['clientid'],
oid: Time.now.to_i.to_s,
amount: POS['amount'],
okUrl: POS['okUrl'],
failUrl: POS['failUrl'],
rnd: Time.now.to_i.to_s,
}
end
def create
@order = Order.new(order_params)
respond_to do |format|
@order.purchase
end
end
def success
end
def fail
end
private
def set_order
@order = Order.find(params[:id])
end
def order_params
params.require(:order).permit(:id, :ip_address, :first_name, :last_name, :card_brand, :card_number, :card_verification, :card_expires_on, :user_id, :participation_id)
end
end
Order.rb =>
class Order < ActiveRecord::Base
belongs_to :user
belongs_to :participation
has_one :transaction, :class_name => "OrderTransaction"
def purchase
participation.update_attribute(:payment_status, true)
create_transaction(:action => "purchase", :response => response)
end
end
The bank's page is getting all necessary information from the user like credit card number, card expiration date etc. My application is not doing anything about purchase process, all of them are happening on the bank's side.
Then the bank is returning me a bunch of parameters about payment process. If the payment is success full, bank is posting the parameters to my /orders/success.html.erb and if it fails it is posting to /order/fail.html.erb.
I have 2 problems =>
1) I want the Order model instance is created whatever the response is successful or failed. It seems like that should be happening by create method in controller but it not working :/
2) How can I get the parameters that the bank send to my /fail or /success URL? I need to get them into my application and save them as a OrderTransaction instance in my database. I can see the parameters in my logs like this =>
Started POST "/orders/fail" for 127.0.0.1 at 2014-06-01 13:40:28 +0300
Processing by OrdersController#fail as HTML
Parameters:
{
"TRANID"=>"",
"Faturafirma"=>"OMÜ Uzaktan Eğitim Merkezi",
"islemtipi"=>"Auth",
"refreshtime"=>"5",
"lang"=>"tr",
"amount"=>"30",
"ACQBIN"=>"490740",
"clientIp"=>"193.140.28.145",
"name"=>"AKBANK",
"cardHolderName"=>"dsadas dasdsa",
"okUrl"=>"http://localhost:3000/orders/success",
"storetype"=>"3d_pay_hosting",
"Response"=>"Declined"
....
}
DB Schema =>
create_table "orders", force: true do |t|
t.integer "participation_id"
t.datetime "created_at"
t.datetime "updated_at"
end
create_table "order_transactions", force: true do |t|
t.integer "order_id"
t.string "clientip"
t.string "cardholdername"
t.string "response"
t.string "errmsg"
...
t.datetime "created_at"
t.datetime "updated_at"
end
Routes.rb =>
...
post 'orders/success' => 'orders#success'
post 'orders/fail' => 'orders#fail'
resources :orders, only: [:index, :new, :create]
...
A: Suggest:
Move your creation success failure into a model and pass it such.
def success
Order.create_from_params(order_params)
end
def fail
Order.create_from_params(order_params)
end
Then handle success failure from params using response Decline etc..
class Order<ActivewRecord::Base
def self.create_from_params(order_params)
self.create(inflatewithfields) && self.purchase if params[response'] == 'success'
self.create(inflatewithfields) if params[response'] == 'Decline'
end
end
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,931
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RIDERS OF THE
APOCALYPSE
GERMAN CAVALRY AND
MODERN WARFARE, 1870–1945
DAVID R. DORONDO
NAVAL INSTITUTE PRESS _Annapolis, Maryland_
Disclaimer: The statements of fact, opinion, or analysis expressed in this book are those of the author and do not reflect the official policy or position of the Department of Defense, the U.S. Government, or the U.S. Army.
_This book has been brought to publication with the generous assistance of Marguerite and Gerry Lenfest_.
Naval Institute Press
291 Wood Road
Annapolis, MD 21402
© 2012 by D. R. Dorondo
All rights reserved. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage and retrieval system, without permission in writing from the publisher.
_Library of Congress Cataloging-in-Publication Data_
Dorondo, D. R., 1957–
Riders of the apocalypse : German cavalry and modern warfare, 1870–1945 / D. R. Dorondo.
p. cm.
Includes bibliographical references and index.
ISBN 978–1-61251–086-6 (hbk. : alk. paper) 1. Germany. Heer—Cavalry—History—20th century. 2. World War, 1939–1945—Cavalry operations. I. Title.
UA714.D67 2012
357'.1094309041—dc23
2011051956
This paper meets the requirements of ANSI/NISO z39.48-1992 (Permanence of Paper).
Printed in the United States of America.
20 19 18 17 16 15 14 13 12 9 8 7 6 5 4 3 2 1
First printing
TO THE MEMORY OF
LIEUTENANT COLONEL JOHN F. DORONDO, USAF
(1915–1961)
CHINA MARINE
COMBAT VETERAN
HORSEMAN
And their horsemen shall come from afar: they shall fly as the eagle that hasteth to devour.
They shall come all for violence.
_The Book of Habakkuk_ , I, 8–9
From all the villages, along all the roads, they come together: wagons, horses, refugees with handcarts. Hundreds. Thousands. Endlessly they stream from north and south to the great east-west road and crawl slowly away, day after day, as though the hoof beat of the horse were the metronome of the hour, the measure of the ages.... Behind us crash the waves of war. Before us stretches the infinite succession of wagons. But here there is only the rhythm of the horse's gait, just as it has always gone, imperturbably, for all time.
Marion Gräfin Dönhoff, "Ritt gen Westen"
_Die Zeit_ , 21 March 1946
CONTENTS
_List of Maps_
_Acknowledgments_
1 The Day of the Horseman
2 The Legacy of 1870
3 Not Quite Sunset: The Cavalry in World War I
4 False Dawn: The Interwar Period, 1918–1933
5 The Field of Mars: Cavalry Equipment, Horses, and Doctrine in the 1930s
6 Bucking the Trend: The Cavalry Rides to War, 1939–1940
7 Barbarossa: The 1st Cavalry Division in Russia, 1941–1942
8 Hell's Outriders: Cavalry of the _Waffen_ -SS
9 Pale Horsemen: The 8th _Waffen_ -SS Cavalry Division _Florian Geyer_ , 1942–1943
10 Last Recall: The 1st Cavalry Corps, 1943–1945
_Epilogue:_ Whither the Horses?
_Appendix A_
_Appendix B_
_Appendix C_
_Appendix D_
_Notes_
_Bibliography_
_Index_
MAPS
Map 1. The Franco-Prussian War, August-September 1870
Map 2. The Invasion of Poland, September 1939
Map 3. The First Cavalry Division in the Netherlands, May 1940
Map 4. The First Cavalry Division in France, June 1940
Map 5. Eastern Front, General Situation, Late September 1941
ACKNOWLEDGMENTS
I do hope readers will take a few moments with what follows, for while whatever deficiencies there are in this work remain mine alone, a number of individuals and organizations provided important support to me in the writing of this book. My colleagues in the Department of History at Western Carolina University, notably my Head of Department, Richard Starnes, sustained me throughout with research assistance, kindness, and good humor. CuChullaine and Basha O'Reilly of The Long Riders' Guild gave unfailing encouragement in many ways and opened doors that I did not even know existed. Jeremy James, in _The Byerley Turk_ , wove a tale that has, for reasons of my own, touched me more deeply than he knows. Serena Herter, Tenzin Frisby, and Don Wood placed great trust in me over the years in many ways. They exhibited unfailing generosity to me and my family, and I shall always be grateful. Adam Kane of the Naval Institute Press saw something worthy in this undertaking. To him I owe a particular debt of gratitude, as I do to both Chuck Grear for his fine cartographic skill and Wendy Bolton for her sharp eye and keen instinct for the best words. Seven Meadows Archery of Tacoma, Washington, and Cold Steel, Inc., of Ventura, California, allowed me access to extraordinary mounted weaponry both for this book and for my university course, "The Horse in European History." Ashley Evans and Mark Haskett at WCU's Office of Public Information supplied valuable photographic assistance. Much closer to home, Cheri and India endured my equine enthusiasms with more patience, grace, and equanimity than I deserve. I love them dearly and thank them with all my heart. Finally, there are the horses: Grey Action (Frosty), a gentle soul of a Quarter Horse; PMX Tuxedo Junction (Buster), the quintessential Morgan; Midnight's Roxanne (Roxy), a delightfully headstrong Anglo-Arab; and—standing above them all—My Victory (Buddy), a magnificent Irish Thorougbred. As only a horse can, Buddy sees clearly what remains hidden from my sight. He already knows, as of old, what lies beyond those horizons I have yet to cross. His voice comes to me from that distant place, gently urging me on. When the time comes, I pray that I may follow after him and that he will remember me.
CHAPTER 1
THE DAY OF THE HORSEMAN
Until its final passing in the 1940s, European mounted warfare constituted a reflection, however distant, of the inheritance of the steppe. Though occurring on battlefields usually very different from the endless grassland stretching eastward from the Black Sea to the Tian Shan and Altai Mountains of China and Mongolia, this warfare nevertheless retained characteristics derived from the horse-cultures of the East: aggression typically paid off; riders could usually outflank fixed defenses; speed and maneuverability remained critical to victory. Well into the twentieth century, the presence of the mounted warrior exerted a profound effect not only on Western military thought but also at deeper levels such as politics, religion, literature, and the heraldic arts. In the seventh-century _Lex Baiuvariorum_ , for example, the tribal duke of the Bavarians would not be required to name a co-regent so long as he could mount his own horse unaided and effectively wield his weapons. In the _Heliand_ , an inculturated Saxon version of the Gospel written in the first half of the ninth century, there appears to be a recurrent apprehension about the ever-present threat of mounted attack. Even in the original Christian scriptures, in this case the Book of Revelation, there would also seem to be reference to the much earlier terror evoked in the eastern Roman world by the Parthian horsemen of the first century AD, a terror that the Romans frequently attempted to counter by recruiting non-Roman horsemen of their own. As is too well known to require elaboration here, the whole of medieval chivalry centered upon the mounted warrior. Later, in the early modern period, the age-old specter of the Ottoman Turkish horseman inspired European emulation not only in the formation of units of hussars and uhlans in Hungary, Poland, and Prussia, but also in the very terminology to describe them and their uniforms. Indeed, after 1870 "uhlan" became a synonym across Europe for the much-feared German cavalry, regardless of an individual unit's actual designation, while the Polish lancer's _czapka_ , with its flat top and decorative cordage, became perhaps the most distinctive helmet of the modern age. Not even the world wars eliminated the iconic status of the mounted warrior, for horse-cavalry vocabulary carried on in armored formations' names in Great Britain and France. In today's Germany the official heraldic shields ( _Länderwappen_ ) of two federal states reflect these ancient equestrian traditions. There one sees very prominently displayed the "Saxon Horse" and the "Westphalian Horse" in the armorial bearings of Lower Saxony and North Rhine-Westphalia, respectively.
To be sure, the military and social importance of the horse was also recognized far beyond Europe's bounds. In China, to note but one very distant example, horses, their chariots, and—unfortunately—their drivers were buried in royal Shang tombs, and in post-Soviet Turkmenistan the horse has been conspicuously adopted as reflecting the very essence of Turkmen identity. Closer to Europe's home, in Istanbul's Topkapi Palace, twentieth-century Turkish sultans still sat in tents that nomads would have recognized, tents at whose entrances stood horsetail standards. Interestingly enough, this same Turkish adornment was adopted by Prussian and other German armies and police forces in the form of the standard called the _Schellenbaum_ or "jingling Johnnie," along with Turkish kettle drums and cymbals. The _Schellenbaum_ is still used in the Federal German Armed Forces.
Of course, today's most famous remnant examples of mounted soldiers are also to be found in Europe, namely the Household Cavalry of the British Army and the Republican Guard in France. There are also less-well-known European examples to be found in the very successful Irish Defence Forces' Equitation School and the riding curriculum at the Theresianische Militärakademie, Austria's West Point. However, one should also certainly note non-European units such as the Indian Army's President's Bodyguard. The Indian Army also enjoyed, at least as of 2008, the distinction of fielding the only remaining horse-mounted, un-mechanized cavalry regiment in the world: the 61st Cavalry, not surprisingly home to some of the world's best polo players. In the United States, too, the (armored) 1st Cavalry Division still maintains an official Horse Cavalry Detachment for ceremonial purposes, and in the wake of the fighting in Afghanistan in 2001, the U.S. Army's Command and General Staff College at Ft. Leavenworth temporarily instituted a military horsemanship program for Special Operations personnel.
In addition to certain armies' retention of horses, a number of paramilitary law-enforcement agencies around the world also still employ them. Among others, these forces include the South African Police Service, the Royal Canadian Mounted Police, and the U.S. Border Patrol. Then, too, the metropolitan police forces of many major cities throughout the world still include mounted units and not merely for ceremonial purposes. In the United States the most famous such unit by far is New York's, which was substantially expanded as recently as 2006, but other cities as varied as Honolulu, Las Vegas, and Washington, D.C., were also enjoying what the _New York Times_ then called "a resurgence in horseback policing."
For more than three thousand years, whether in enforcing the law or, more importantly, waging war, the horse was the means by which the warrior gained true mobility, range, and striking power. Either by riding the horse or by having it pull his chariots, supply wagons, or, later, machine guns and artillery, he harnessed the strength, endurance, and above all the intelligence of the horse for his own military purposes; and, despite the horse's flight-response to danger, he even managed make use of its occasional aggressiveness. This latter trait became particularly useful in the massed attack when the horse's herd-instincts could effectively mask threats from which even cavalry mounts trained to the sound of gunfire might otherwise flee. Equine aggressiveness among stallions and late-gelded males could also be brought to bear in the melee of individual combats wherein the horse would see another mount, and not its rider, as the enemy. Such innovations "brought to warmaking the electric concept of campaigning over long distances and, when campaigning resolved itself into battle, of manoeuvering on the battlefield at speed—at least five times the speed of men on foot." It also brought fear: the fear of an enemy able to appear at times and in places of his own choosing, often completely unexpectedly. Indeed, given the context of the present work, it seems worth noting that this pervasive fear rooted itself very deeply in the European psyche. As late as the 1940s, the German government's propaganda consciously attempted to evoke the terror still latent in the folk memory of the menace of the steppe horsemen. The purpose was to encourage resistance to the advancing, and by then largely mechanized, Red Army. Emphasis on the specific ideological threat posed by Communism only overlaid and reinforced the much more ancient dread. The essential element was the fear itself, the perception of Germany's being overrun by "Asiatic hordes." Ironically, the fact that the Red Army still employed large units of horse-mounted Cossacks only further reinforced the propaganda.
The centrality, the emotional pre-eminence, of the mounted warrior even down to the twentieth century's beginnings was not quite as old as the Western way of war itself. From approximately the fifth century BC to the fifth century AD, that way of warfare consisted primarily in the face-to-face fight to the death of the infantry phalanx with cavalry acting (occasionally brilliantly, as under Alexander's command, or as with the Thebans at Leuctra in 371 BC) as an adjunct to the main battle. Nevertheless, the transmission of the mounted warrior's role from the Eurasian steppe and Persia via Greece and Rome to Europe proper created a powerful impetus for the future. This force found its first true expression in the post-Roman, Germanic successor-kingdoms of early medieval Europe. In them the Greco-Roman tradition combined itself with the power of the horse to create an essentially new style of mounted warfare, one aiming at the immediate destruction of the enemy rather than hit-and-run harassment and graduated attrition typical of the steppe warrior.
To the extent that Western cavalry now gave primacy to rapidly closing with and destroying the enemy, the widespread use of the stirrup constituted an important contributory technology. As with the idea of the cavalry itself, the stirrup's use also gradually migrated from East to West. By the tenth century AD it was commonly used by Western European horsemen and, among cavalrymen, helped provide a more stable platform from which to drive home attacks with a couched lance or to strike heavier, downward blows with handheld weapons such as the mace, sword, or axe. It is also suggested that the increased striking power made possible through the combination of effective bits, stirrups, and deeper-seated saddles encouraged the breeding of heavier horses in Western Europe from the Carolingian period forward, ones capable of withstanding the greatly increased collision-impact of lance-wielding heavy cavalry. It should, for example, be remembered that two armored knights' chargers, each weighing at least one thousand pounds (not including the weight of the rider, his weapons, and armor) and moving at speeds approaching 15–20 mph (25 km/h), would generate a tremendous shock. As such face-to-face combat became the idealized norm in Western mounted warfare, breeds possessing a heavier, though not necessarily "cobby" or "carty," conformation followed. Indeed, they helped drive the cycle: the heavier the horse, the greater the weight of man and armor it could carry and impact it could withstand. This capability, in turn, necessitated still heavier breeds to absorb ever-greater collisions, and so on.
Insofar as German cavalry is concerned (but not only there), the longterm consequence of this equid-and-technology evolutionary spiral in the early-modern period was the increasingly rigorous and statesupported breeding program of the sort that produced such fine military horses as the Hanoverian and the Trakehner. And, though lance-onshield combat eventually disappeared, the essential physical dynamics of Western European cavalry combat from AD 870 to 1870 did not. As a result, into the twentieth century Western European cavalry horses, German horses among them, would remain generally taller and of heavier conformation than, for example, their Cossack counterparts. Nevertheless, one never really sees a clear break between a Western way of war involving nothing but infantry and a Western way of war wholly dominated by mounted knights by circa AD 750–800. Rather, there appears to have been a steady, and steadily growing, influence of the concept of mounted warfare permeating Western Europe from the East, undergoing modification and culminating in the full flowering of the chivalric ideal in the High Middle Ages and then continuing through the early-modern period with its introduction of gunpowder weaponry.
In the latter age, however, one of the primary conundrums facing cavalrymen in the Western world was what their role would become given the advent of firearms. The employment of long-range missileweapons by horsemen was, of course, already of ancient lineage. The tactically successful use of such weapons against riders was a crucial military evolution even before the common use of gunpowder. One need only mention Crecy (1346) and Agincourt (1415). The introduction and rapid refinement of firearms merely compounded the range at which common foot soldiers might visit destruction upon their chivalric social betters. To a certain extent, firearms also added to the perceived insidiousness of the foot soldier's shooting the rider from the saddle before the latter could even strike a blow or, more likely, simply killing his horse. The horse did, after all, present a much larger target than the man riding it, and killing the horse automatically stopped the cavalry charge. For example, the nobly born Gaston de Foix, commander of the French army at Ravenna in 1512, and some twenty of his courtly fellows were unceremoniously "gunned down to a man" when they sportingly attempted to pursue already defeated Spanish arquebusiers. Furthermore, if such factors were not already sufficient to make socially refined horsemen unsure about facing mere mechanics on a gunpowder dominated battlefield, there was the occasional accusation that gunmen used an early equivalent of dum-dum bullets in the form of rounds dipped in poisonous substances such as green vitriol (likely an oily metallic sulfate), a charge specifically made during the siege of the English city of Colchester in 1648.
In the late fifteenth and throughout the sixteenth century the principal firearm was, in one version or another, the arquebus. For the time being this weapon, though dangerous, did not truly threaten to displace cavalry from the battlefield. The arquebus did not, for example, possess a high rate of fire. Estimates range from one shot every thirty to forty seconds under ideal conditions (unlikely in a battle) to one shot every "several minutes." Perhaps the best estimate is one shot every two minutes, though this, too, may be generous. Under such circumstances, cavalry could very likely close with opposing infantry before the latter's arquebuses inflicted unacceptable losses of men and horses. Moving at the trot, cavalry on sound horses could cover perhaps 270 yards (approximately 250 meters) per minute, while at the gallop the distance covered would approximately double. Consequently, infantry armed with arquebuses faced unpalatable options. They could fire a volley at the weapon's maximum effective range of about one hundred yards and hope to reload and fire again before the horsemen were upon them. Conversely, they could wait until the range had decreased to a much more lethal fifty yards or so, fire, and then see whether the horsemen survived in sufficient numbers and with sufficient impetus to ride them down. Precisely for this reason, pikemen remained an integral feature of infantry formations throughout the period. The pike—as much as eighteen feet in length—provided close-in protection for arquebusiers who were otherwise doomed, as were artillerymen, if the cavalry got in amongst them. The pikemen were therefore essential to the infantry's survival until the invention of the socket bayonet. That device transformed the shoulder-fired weapon into a means by which infantrymen could defend themselves from cavalry attack while reloading, provided they had the nerve. Large numbers of horses moving at the gallop not only present a tremendous visual spectacle. For anyone standing nearby, they also literally shake the earth. A wall of such creatures, hundreds or even thousands strong, ridden by shouting cavalrymen and running full tilt directly into one's face, would certainly seem to be unstoppable. The adverse psychological effect upon infantrymen, even when they formed the vaunted infantry square, could be enormous.
Of course, the cavalry forces of European armies also attempted to adapt themselves to the use of firearms, most notably in the form of the pistoleers of the mid-sixteenth century. The _caracole_ , through a complicated tactical evolution, resulted precisely from the effort by cavalrymen to make use of firearms themselves to break up opposing infantry formations and thus render possible a battle-winning charge with cold steel. "Cavalry," writes historian Jeremy Black, thus continued to provide "mobility, and that was crucial for strategic, logistical, and tactical reasons. It enabled forces to overcome the constraints of distance, to create equations of numbers, supplies and rate of movement that were very different to those of infantry, and also to force the pace of battle in a very different fashion to that of infantry." The arquebus, the wheel-lock horse-pistol, and, eventually, the eighteenth-century flintlock musket did not render that utility nugatory. Indeed, if the carbine musket and cannon could be fully incorporated into the cavalry, as in fact they were, then the cavalry would continue to have, as in fact it did, a viable combat role on the battlefields of Europe.
According to Michael Roberts, the Swedish king and royal innovator Gustavus Adolphus nevertheless forbade the _caracole_ and instead insisted that the cavalry always charge home with swords drawn, relying on the combined weight of man and horse for tactical success. In place of his horsemen's firearms, or at least supplementing them, he also "was able to arm his units with a light and transportable field piece designed to supply close artillery support for infantry and cavalry alike." Herein one may see the beginnings not only of the vaunted horse-artillery of the Napoleonic era but also what late-twentieth-century military writers would have termed an organic artillery capability for the cavalry. Unsurprisingly, however, not all historians agree that this incorporation of artillery with the cavalry constituted a solution to the cavalry's problem of how to break up firearms-carrying infantry formations. David A. Parrott, for one, maintains that Gustavus Adolphus' effort created no real solution. On the contrary, he writes that the Swedish king's artillery was not "capable of the same degree of mobility as cavalry." While the Swedes had developed cannon firing 3-pound shot over an effective range of some three hundred yards, these "were not mobile as a matter of course" owing to a lack of good-quality horses and easily portable stocks of ammunition. Therefore, "the vaunted reforms of Gustavus Adolphus produced nothing capable of approaching this [mobility] requirement." Parrott therefore concludes that, absent the aforementioned and truly revolutionary innovation that highly mobile horse-artillery would have provided in smashing prepared infantry formations, the cavalry's tactical importance became increasingly that of turning the flanks of opposing armies so that the latter could be taken from the side or rear while pressure was maintained front and center by one's own infantry. Of course, the opponent's own cavalry would be tasked with preventing just such a turning movement, thus setting up the continued face-to-face clash of horsemen employing not only handheld firearms but cold steel, both at pointblank range. It would, therefore, not be "new" tactics deciding the issue in a given cavalry battle but the resolution of the combatants, and at least implicitly the quality of the winners' mounts. In precisely this respect, the importance of secure, high-quality breeding stock for supplying large numbers of remounts assumed a strategic significance.
In Germany the eighteenth century witnessed the establishment or expansion of a number of State studs whose mission was to develop and maintain breed-stock suitable for military and agricultural employment. Two of the most famous of these would eventually play major roles in the breeding of the modern German military horse. These were the East Prussian State Stud at Trakehnen and the Hanoverian State Stud at Celle. These studs and others contributed greatly to the establishment of a solid breed-stock of horses that, if not quite as finely athletic as the English Thoroughbred of the day, were nonetheless very well suited for employment as cavalry mounts. To that extent, they helped revamp the capabilities of a Prussian and, later, German cavalry that no less an observer than Frederick the Great dismissed upon his accession as being "not even worth the devil coming to fetch it away." Nevertheless, the development of more stringent breeding standards, combined with more effective training in individual and close-formation galloping and other exercises by Frederick's cavalry commanders such as Johann Joachim "Papa" von Ziethen and Friedrich Wilhelm von Seydlitz, soon honed the Prussian cavalry into a force of European renown. No longer would the Prussian cavalry be good only for parading, a fact demonstrated for all to see in battles such as Seydlitz' crushing victory over the French at Rossbach in 1757.
Firearms of all kinds had indeed made the battlefield more lethal. That was true enough. Cavalrymen, such as the Prussians at Rossbach, recognized "that a well-timed musket volley could destroy an entire regiment." They also knew, however, that musketry, rifle-fire, or artillery could create opportunities for the cavalry's decisive engagement, provided that cavalry commanders appreciated "the complexity of the [lateeighteenth-century] battlefield." More than ever before, "precise maneuvers, speed, boldness, and timing" would determine the mounted arm's success on battlefields where "the margin of error separating cavalry success and failure" grew ever narrower. This lesson applied not only to Prussian or other German cavalry, but to all the military horsemen of Europe.
These issues became acute between 1800 and 1815, for European cavalry "reached its apotheosis" during the reign of Napoleon I. In his campaigns, cavalry performed those functions—often with consummate skill—that still remained to it on battlefields now coming to be dominated by the emperor's beloved artillery, if not quite yet by truly accurate, long-range volleys from rifled firearms. These roles consisted of screening the French armies' movements and strength from spies and opposing forces. The cavalry also carried out reconnaissance and prepared the conditions for the concentration of divergent French columns at the point of contact with the enemy. Finally, the French horsemen became the ultimate pursuers of broken enemy formations, though the latter were almost never broken by the cavalry itself. Despite the awful psychological effect of a massed cavalry attack made at the gallop, Napoleonic-era infantry squares, bristling with bayonet-tipped muskets and often supported by guns, could only rarely be smashed by direct mounted assault. Nevertheless, Napoleon may be said to have resurrected the cavalry's operational role from its relative diminution in the eighteenth century as reflected in the declining ratio of cavalry to infantry, despite the cavalry's contributions in such famous early eighteenthcentury battles as Blenheim (1704) and, later, Rossbach. Napoleon added skirmishing to the cavalry's remit and, in the 1790s, was one of the first French commanders to employ effective horse-artillery. The latter innovation gave genuine speed and mobility to the "king of battle," greatly increasing the striking power of mounted formations. Furthermore, by disrupting the enemy infantry's formations, a properly coordinated artillery barrage, whether from field guns or horse batteries, could still make possible the European cavalry's ultimate self-expression, namely the pressing home of attacks with the _arme blanche_. At the very least, it was assumed that dragoons and carabiniers could close sufficiently to employ their own shoulder-fired weapons or pistols. Nevertheless, even Napoleon's superb cavalry could not overcome the iron logic of gunpowder weaponry, as demonstrated with such terrible magnificence in the futile attack by fully 10,000 French horsemen against the allied squares at Waterloo. Not even such a grand failure, however, served to dislodge the cavalry from the armies of Europe, if for no other reason than that no substitute for it existed in the missions noted above. Only the cavalry could rapidly execute the vital tasks of long-range reconnaissance, screening, flanking, liaison, and pursuit. Nothing less than the advent of reliable wireless communications and internal-combustion propulsion would truly change that calculus; and even then, the cavalry's departure from the scene "was slow, uneven, and reluctant."
Thus, throughout the second half of the nineteenth and well into the twentieth century, the cavalry—indeed military horsepower generally—could still claim a place on the battlefields of Europe. In the last great cavalry war of Western European history, the Franco-Prussian War, both France and the German States routinely employed light and heavy cavalry at both the tactical and the operational level, though not, as shown below, with equal effectiveness. Later, in World War I, all of the major European armies still marched with huge numbers of cavalry fully integrated into their combat formations, though as the reader will see, nascent motorization (particularly armored cars)—not to mention more effective, long-range artillery and machine guns—vastly restricted what the cavalry might still accomplish, at least on the Western Front. By contrast, on the Eastern Front from Courland and East Prussia to Rumania, horsemen still enjoyed a considerable prestige and found themselves usefully employed both tactically and operationally.
Nevertheless, not even the events of 1914–1918 completely removed cavalry and horse-powered transport from European armies. We are particularly concerned with the fact that this remained so in Germany. Throughout the 1920s and 1930s, the _Reichsheer_ of the Weimar years and, later, the _Heer_ still conceived of important tactical and operational roles for the horse, both in combat and in logistics. Both organizations would plan accordingly, notwithstanding a great deal of propaganda to the contrary. Consequently, when Hitler's government willfully plunged Europe into the greatest war in its history, the German Army still possessed hundreds of thousands of horses in its establishment and not just for pulling supply-wagons, field-kitchens, artillery, and ambulances. German cavalry also went to war in 1939, not as a mere horse-mounted anachronism but as a matter of some necessity. As will be shown, that necessity would only grow before 1945.
One might well argue that that reliance on horses by the _Reichsheer_ and the Nazi-era _Heer_ was misplaced. Germany's military leaders, so the argument would run, ought to have done otherwise. Such an objection is fair enough in the abstract. In this matter, however, as in all historical inquiry, the primary question—as formulated by a noted authority in German military history—should not necessarily address what the German army _ought_ to have done regarding the cavalry's employment. Rather, the question should account for _why_ the German army did what it did. Why still use horse-mounted troops after 1918? Why after 1925, when motorization was becoming a reality? Why after 1935, when the first panzer divisions were being raised? Why, ultimately, even in 1945, when literally thousands of horse-soldiers still found themselves in action? This work constitutes the beginning of an attempt to answer these questions.
Of course, cavalrymen were only as good as their horses, and this treatment of the German cavalry therefore also touches upon one of the great and enduring bonds in the human experience: that between the horseman and his mount. Having moved steadily away from regular, close contact with large animals since the middle of the twentieth century—except among a continuously dwindling number of farmers or perhaps from the safe side of a zoo's enclosures—Western society has become largely ignorant of the profound interaction between horses and humans. Notwithstanding the undoubted commercial successes of recent occasional books, plays, and feature films (the British National Theatre's 2009 triumph _War Horse_ and the U.S. films _Seabiscuit_ and _Secretariat_ come most immediately to mind), horses since 1945 have become the perceived preserve of a "horsey set" of racing owners and/or breeders, huntsmen, or the simply rich. This perception remains current despite the fact that in the United States alone the equine population stood at well over five million at the beginning of the twenty-first century. In the United Kingdom the figure totaled perhaps one million at the same date, though it remains somewhat unclear whether that number resulted from recent natural accretion or severe undercounting in earlier surveys. Given such numbers, particularly in the United States, and based upon the author's own experience, it seems clear that very substantial numbers of horses certainly do not live a life of luxury in racing stables and hunt clubs, nor do they live quite so far apart from their human companions as one might think. Nevertheless, actual contact between those huge numbers of horses and the larger human population in whose midst they live remains minimal for human society as a whole.
Of all the ties binding humans and horses, surely the most poignant and nearly the oldest is the one existing between the military horse and the mounted warrior. If not quite as ancient as warfare itself, this bond is nearly so. But war remains, and has always been, a hard business. Physical destruction abounds. Men, women, children—and animals—die. Of course, no moral equivalence between the death of a horse and that of a man, woman, or child is intended. The assertion of any such equivalence would be grotesque. Nevertheless, the deaths of horses can be piteous. They know real fear. They feel real pain. They seem to suffer real loss. Their size and their very nearness to their riders make their suffering all too palpable, all too visceral, when they are seriously or mortally injured. That nonquantifiable but vivid characteristic called "heart," the inner quality possessed by so many horses that drives them on even at the risk of injury or death, can show itself most heroically when they die. Horses worn out by their lives' exertions can be utterly composed and evidently ready when they go to their graves. The author has seen this firsthand. Those not yet ready to die can fight for life and very often do. The author has seen this as well. Cavalry horses' training could itself sometimes be brutal, but so was the task to which they were set by their human masters. The numerous instances of those same horses' noble behavior in combat (other words simply do not fit) nevertheless attest to a quality far beyond simple, enforced obedience. Just as many of their riders did, just as many soldiers have always done, such horses often showed their most profound dignity when their own lives hung in the balance. Is this mere cavalry romanticism, mere horseman's anthropomorphism? Perhaps it is. Certainly many cavalrymen viewed their mounts merely as equipment to be discarded without further ado when injured or to be replaced without a second thought when killed. Others evidently felt differently. If not, why have war horses, so far as we can reckon, always had individual names from the earliest times down to the vast mounted forces of the late nineteenth and early twentieth century? Beginning in the Napoleonic period, most cavalrymen were literate. Consequently, it was "the first period where the personal relationship between the military horse and the soldier was recorded" in substantial numbers of accounts. The relationship could prove, and was shown to be, as intense as any between humans. Those accounts also provide the first substantive indication of a tale quite likely as old as the military horse itself, a tale of a very special bond forged in the crucible of war, a tale of fierce joy in life and unbearable heartache in death.
CHAPTER 2
THE LEGACY OF 1870
Prior to the dawn of mechanized warfare in the early twentieth century, and indeed for several decades thereafter, no element of the Western world's armies so evoked the exotic and romantic aspects of war as the cavalry. For centuries the cavalryman's kettle-drums and bugles were the sine qua non of martial music. For pageantry, nothing could surpass the panoply of the cavalryman: the sheer mass of his horse, his flowing regimental standards, snapping guidons, jingling tack, polished leather, and flashing steel. But it was not all mere show. Cavalry still evoked real fear. The shock value—and therefore the fear—of a massed cavalry attack was as old as the weapon itself and still persisted in the late nineteenth century. As he had for centuries, the mounted warrior still appeared to be forever "uncatchable, inescapable, unapproachable." Long before the defeat of the foot-slogging Anglo-Saxons by the Norman horsemen at Hastings in AD 1066 and the great flowering of the Age of Chivalry, so fearsome were the mounted charge and its practitioners that they transformed not only European warfare but even European culture itself, as seen as early as the ninth-century Saxon Gospel, _The Heliand_. Indeed, historian H. R. Trevor-Roper, among others, placed the horseman at the epicenter of a fundamental societal change in the chivalric ideal; and no less a military historian than John Keegan speaks of a "cavalry revolution," one in which massed horsemen literally reinvented warfare as a "thing in itself," a means not merely to dominate one's enemy but to annihilate him. War could now become, though it was not always in fact, a product of "militarism."
Perhaps the last great hurrah for this view of the cavalry was the Franco-Prussian War. Though all of the major European armies would still possess huge cavalry forces in World War I, and though the German army, for one, was still fielding new cavalry forces as late as 1943–1944, the last significant and sustained cavalry-versus-cavalry operations occurred in 1870–1871. The romance of the cavalry had yet to be blown away by the full mechanization of European warfare. Feats of the nineteenth-century mounted arm—indeed all arms—could still be celebrated in verse, prose, and song: Tennyson and, later, Kipling come first to mind for English-speakers. More germane, however, was the fact in the aftermath of 1870, German lights such as Theodore Fontane, Richard Wagner, and Johannes Brahms celebrated the Reich's victory over France in moving words and music. The "gigantic historical canvases" of painter Anton von Werner depicting German commanders on the field at Sedan or the proclamation of the German Empire at Versailles could still effectively disguise the battlefield's carnage at Spicheren and Wörth, Metz and Mars-la-Tour. Socially, sartorially, psychologically, European cavalry remained wedded to this military romanticism in spite of the rapidly changing technological world surrounding it.
Curiously, even earlier manifestations of the cavalry's attempted adaptation to technology in the early-modern period, whether in the form of so-called horse-pistols, carbines, or even horse-artillery and the resultant designations of light cavalrymen as hussars, dragoons, uhlans, or chausseurs, did not succeed in permanently or completely divorcing the cavalry from the idea that cold steel remained the ultimate weapon. Very frequently, light-cavalry formations, such as those mentioned above, evolved into versions of their heavy-cavalry rivals—the cuirassiers in France and the _Reiter_ regiments in Prussia—and became possessed of the same dictum, namely that the "consummation of the cavalryman's purpose in life [remained] the charge _en masse."_ Notwithstanding the hussar's braid-encrusted pelisse and rakish busby—a uniform that gave Prince Friedrich Karl von Hohenzollern (commander of the Prussian Second Army in 1870) the nickname "The Red Prince" because he wore it all the time—light cavalry also tended to aspire to the social status and panache of the heavy cavalry regiments, especially that of the armorplated cuirassiers, a status that remained attractive to even the uppermost crust of European society, particularly on the Continent. Even Otto von Bismarck, Prussian and, later, imperial chancellor, held a major's commission in the 1st Heavy Reserve _Reiter_ Regiment and often wore its uniform, much to the serious annoyance of many professional officers around him, one of whom commented "acidly" that wearing a cuirassier's greatcoat was no particular aid to military understanding. And perhaps no mounted regiment in Europe surpassed the splendor of French emperor Napoleon III's "Hundred Guards" cuirassiers, though their flamboyant uniform was not atypical with its mirror-finish steel cuirass and helmet, the latter with gilded crest; two helmet-plumes (white horsehair and red feathers); a sky-blue tunic trimmed with red collar, cuffs, and lapels; gold epaulettes; white trousers; black top-boots; and white gloves.
Fancy or not, the cavalry faced an uncertain future at mid-century. In Prussia and elsewhere after 1850, the cavalry's role in modern armies was being re-examined. Following the victorious war against Austria in 1866, Prussia's leading commander, Helmuth von Moltke the Elder, did something rather unusual for victorious commanders: he analyzed what he and the Prussian army had done wrong. Insofar as the cavalry was concerned, several items were of note. On 27 June 1866 at Langensalza on the River Unstrut in Thuringia, the cavalry of the Hanoverian army (allied with Austria) had just managed to break Prussian infantry squares, suffering severe casualties in the process. This outcome seemed to confirm the cavalry's traditional role as battle-winning shock troops. But in the very next month, on 3 July at Königgrätz, the Prussian cavalry found itself incapable not only of providing effective reconnaissance in the days before the battle but also of effective pursuit of the defeated Austrians afterwards. When Moltke subsequently critiqued his and his armies' performance in a "sensitive memorandum" to the Prussian king in 1868, he gave vent to his views of what the Prussian (and eventually the German) cavalry's future role should be. He stressed that the cavalry could and should still work in tactical concert with artillery and infantry as had the Hanoverians at Langensalza and the Prussians at Königgrätz. Nevertheless, the cavalry should no longer be held back primarily in order to deliver a massed charge at a decisive moment that might never come. While not entirely discounting the latter possibility, he wrote that cavalry should instead be used more extensively for screening, reconnaissance, and security. All these were missions for which horsemen remained uniquely suited. Precisely two years later, in July 1870, Moltke's conclusions were tested in the Franco-Prussian War.
Despite Moltke's admonitions, one roughly contemporaneous observer of the events of 1870–1871 wrote that German cavalry didn't develop effective reconnaissance and screening capabilities until well after the war against France had begun; thus it did not emulate examples such as that set by the U.S. Army's General John Buford during the Gettysburg campaign in the Civil War. The same author criticized the "stubbornness" and the "ill-informed" attitudes of the Europeans in their refusal to learn what he considered the proper lessons from the Civil War. Unlike their European counterparts for whom the cavalry's specialization by type was still at least nominally in effect in 1870, American cavalrymen had long ceased to be functionally divided into "heavy cavalry" (for battle-winning massed attacks delivered with the _arme blanche_ ), "light cavalry" (for screening, reconnaissance, and messengerservice), and "dragoons" (essentially well-mounted infantry). Instead, "the traditional [American] cavalryman has ever been the light dragoon—a soldier trained and equipped to fight mounted or dismounted, to perform screening and reconnaissance, and to act as a scout or messenger. True heavy and true light horse have been rare." Thus the cavalry of the American Civil War, whether Union or Confederate, did the bulk of its fighting on its feet. It broke no fundamental tradition in adapting to increasingly effective firepower. Though saber swinging melees did occur, as at Brandy Station, Virginia, in June 1863, most cavalry action during the Civil War was on foot, the horse serving as much as a means of transport as of attack. Evidently the American cavalryman did not feel morally obligated, as one author put it, to die on horseback, whereas his European counterpart still did in 1870.
Whatever difficulties they had in executing Moltke's vision, the German cavalry of 1870 tended to exhibit much better understanding of their newly important role than did the French. At the beginning of the war, for example, the French cavalry was still guided by the regulations of 1829, the arm having "learnt nothing" in the meantime regarding more modern operations and tactics, according to one contemporaneous observer. Implicitly, this would mean that nothing was learned from the Crimea, the American Civil War, or even the much more recent Austro-Prussian War. Still, says this same observer, the French cavalry was conscious of its "past bravery and patriotism." The absence of effective lessons learned was exacerbated by the fact that when the war began, the French cavalry "had no reserves of horses" and an "[unspecified but evidently large] portion of the effective strength were four-year old remounts."
By contrast, Prussian and other German cavalry—almost always referred to by the French as uhlans whether the cavalry in question were actually lancers or not—consistently demonstrated an ability to reconnoiter more effectively than their French counterparts, even while stubbornly insisting on the ideal of the massed attack. As early as the frontier battle at Wissembourg on the borders of the Palatinate on 4 August 1870 and the roughly coincidental battle at Spicheren near Saarbrücken some forty miles to the northwest on 6 August, the French cavalry utterly failed to determine the scope of the threat facing Napoleon III's armies. In part this was owing to the extraordinary directive of the French marshal Achille Bazaine dated 20 July wherein he stated that "our reconnaissance should not be aggressive." Unfortunately for Bazaine, cavalry still constituted the sole reliable means of gathering information about an enemy's dispositions beyond the line of sight. His directive, therefore, amounted to gouging out his own eyes during the critical phase of the armies' concentration for battle. As it was, the French cavalry remained almost "completely inactive" throughout the period up to and including the Battle of Sedan as regards operational reconnaissance, even if at a tactical level French mounted forces were sometimes capable of effective action. Further, since French cavalry when it did patrol was "not accustomed to patrol far to the front," French commanders typically assumed that German cavalry patrols were followed by much larger forces immediately to the rear even when this was not the case. This misapprehension helps explain French timidity when confronted with the constant presence of far-ranging German mounted units. And while perhaps the case could be made that cavalry proved to be of little practical value in the steep defiles around Spicheren, the same could not be said of the fighting at Wissembourg and the follow-on battles at Froeschwiller, Wörth, and Morsbronn. There the French desperately tried to retrieve their infantry's fortunes through a sacrificial massed attack by General Michel's and General Bonnemain's reserve cavalry, including a full division of cuirassiers.
At Froeschwiller and Wörth, the French 2nd Cavalry Division's 1st and 4th Cuirassiers of the _Brigade Girard_ charged Badenese and Württemberger infantry over ground broken up by palisaded hop-fields and vineyards. As the horsemen were funneled by these obstructions into the intervals between the fields, the 4th Cuirassiers had to ride over twothirds of a mile under sustained rifle-fire. Both regiments suffered heavy losses "without having effected anything." The division's 2nd and 3rd Cuirassiers of the _Brigade Brauer_ attacked over similar terrain made even worse by an "absolutely insurmountable" barricaded ditch. The 2nd Cuirassiers alone lost their colonel and 5 officers killed; more than 130 officers and men wounded; and some 250 horses killed outright or dying subsequently of their wounds. Throughout the attacks, the German infantry was "always out of reach and often out of sight" of the French horsemen.
In the view of recent scholarship of the Franco-Prussian War, the German infantry's standing up to charging cavalry was still a radically new way for infantrymen to fight horsemen, dating back perhaps to Waterloo. Traditionally, infantrymen not formed in squares would tend to throw themselves to the ground to avoid blows from sabers and to make the horses shy away, presuming that the foot soldiers weren't already running for their lives. Now, however, they "simply stood in lines and blazed away." The results of such tactics for the French horsemen repeated themselves elsewhere that day. At the other end of the French line on the far right, for example, the 8th and 9th Cuirassiers of the 1st Cavalry Division's _Brigade Michel_ attacked German infantry in the village of Morsbronn. As earlier on the left, French troopers again charged through the intervals between hop-fields and vineyards and took heavy rifle-fire as they passed. The 8th Cuirassiers lost two-thirds of their horses before the cavalrymen even reached the village. Of the 9th Cuirassiers—and the supporting 6th Lancers of the division's _Brigade Nansouty—_ almost all troopers not killed before they gained the village were subsequently shot down and killed or captured along the village's main street as the horsemen rode headlong into a blockaded dead-end. Afterward, dead horses and men lay so thickly in the street that passage along it was literally impossible. Witnesses and subsequent observers reported that the German bullets had "rattled like hail" against the cuirassiers' steel breastplates and created "a strange music" in the process. The preponderance of unarmored lancers among the French dead at Morsbronn, compared to steel-plated cuirassiers, led at least one historian of the battle to conclude, erroneously, that the breast plate would therefore always be a part of the cavalryman's equipment. Be that as it may, German riflemen had emptied hundreds of saddles and killed and wounded hundreds of men and horses. The French horsemen, for their part, had merely bought a bit of time for their infantry's retreat.
As disastrous as these attacks had been, the French cavalry's failure in reconnaissance had been equally faulty. As at Spicheren, so too at Froeschwiller the French suffered "a disastrous failure...to appreciate the strength and intentions of the Germans." Indeed the day before the Bavarians attacked at Wissembourg (3 August), the local French commander, General Ducrot, reported that the Bavarians' threat was a "simple bluff." Only effective employment of the French cavalry in reconnaissance could have provided timely intelligence of unimpeachable character. By dramatic contrast, orders issuing from the Prussian Royal Headquarters, as well as from those of Prince Frederick Charles' Second Army, often directed the cavalry specifically to "be pushed forward _as far as possible."_ Of course, not all orders were executed as given, and war's inevitable friction affected the reliability of the information passed back up the chain of command. Nevertheless, in the war's crucial opening phase, German cavalry operated consistently more effectively and widely than the French in the critical job of providing intelligence and fixing the enemy in place so that German infantry could be brought to bear.
In the aftermath of the fighting at Spicheren and Froeschwiller/ Wörth, and with the French armies in retreat across the board, the Germany cavalry—despite occasionally losing contact with the enemy—nevertheless showed itself willing and able to act boldly and range widely. In these instances, its behavior sometimes appears reminiscent of the "rides" of American Civil War generals Jeb Stuart, John Hunt Morgan, Nathan Bedford Forrest, Alfred Pleasanton, and Benjamin Grierson. Perhaps the most striking example, though still somewhat paltry when compared to the distances and consequences involved in that earlier conflict, was the German advance to the Moselle between 6 and 14 August 1870. German horsemen thrust in behind the French Army of the Rhine as it fell back on the fortress of Metz, cutting the telegraph connecting Paris and the depot at Nancy. The German riders thereby made cooperation with French forces still at Belfort all the more difficult. In some cases, German cavalry patrols forged as far as forty miles ahead of advancing main columns. On 12 August German cavalry reached the Moselle below Metz at Pont-a-Mousson and, farther south, at Frouard. In both places they crossed the river and again not only cut the telegraph but also the rail lines linking Metz with Nancy and, by extension, Chalons-sur-Marne where the French Government had ordered the formation of a reserve army. In point of fact, most of the German cavalrymen at Pont-a-Mousson were actually captured before they could complete their work of destruction. Nevertheless, they scored psychological victories as dramatic as in the war's opening days when, on 26 July, the young Count Zeppelin and his mounted patrol had been captured while having lunch at the Shirlenhof Inn eight miles behind French lines at Niederbronn, or when Prussian uhlans blew up a French railroad viaduct near Saargemünd on 23–24 July. These examples were now being replicated up and down the line not only at Frouard and Pont-a-Mousson but also by the German cavalrymen who rode brazenly to the very walls of the fortress of Thionville, the gates being shut virtually in their faces, or who openly scouted within one-half mile of the main French camp at Metz. For their part, the French commanders in the latter city appeared to have failed utterly to use their available cavalry for anything like effective reconnaissance. On the contrary, they limited their efforts to placing staff officers as observers in the cathedral's belfry. At a so-called council of war on 10 October, at least one corps commander recognized that the cavalry remaining in the city was "incapable of service," evidently through prior mismanagement and the consequent collapse of morale. Presaging 1914, or even 1940, relatively small numbers of wide-ranging German uhlans and hussars created an effect "out of all proportion to their strength and achievements." It was enough to create that terrifying picture of "'the Uhlans' [ _sic_ ], ruthless, swift, and ubiquitous, which was to frighten the children of France and Europe for forty years to come." Such operational success for the German cavalry most dramatically manifested itself soon thereafter with the stopping of the French withdrawal westward from Metz.
Map 1. The Franco-Prussian War, August-September 1870
In this case the 5th and 6th Cavalry Divisions received orders to scout ahead to the Metz-Verdun road to try to determine the French army's line of retreat. On 14–15 August German mounted units encountered French cavalry and other forces headed westward along the road in the vicinity of Mars-la-Tour and Vionville. The German cavalrymen took the French under fire with horse-artillery and stopped the column in its tracks. Other German formations advanced to the sound of the guns. For their part, the French failed to push their way through what still amounted to a cavalry screen in order to keep open their line of retreat. The result was the halting of the entire French movement along the line of Mars-la-Tour–Vionville–Rezonville–Gravelotte–Metz. Here the German cavalry, materially assisted by French hesitation, played the critical function of finding and fixing the enemy while the German infantry came up to try to cut off the French withdrawal. The German horsemen thus played precisely the roles assigned them by Moltke in his report to the Prussian king in 1868.
Of all the fighting along the road linking Metz and Verdun, certainly the emotional high point for German mounted troops was the so-called Death Ride at Mars-la-Tour of the 12th Cavalry Brigade under General Friedrich Wilhelm von Bredow. In this attack the 1st, 2nd, and 4th Squadrons of the 7th Cuirassiers and the 2nd, 3rd, and 4th Squadrons of the 16th Uhlans charged en masse against prepared French infantry and artillery in order to gain time for faltering German troops and guns to regroup. Taking advantage of swales to approach within several hundred yards of the French positions, the German cavalry burst from the gun-smoke obscuring the battlefield and "flashed by" endangered Hanoverian artillery batteries at the critical moment. Somewhat atypically, the attack was launched straight from the gallop with no preliminary trot to the canter. As the charge got under way, four attached Prussian horse-artillery batteries fired obliquely across the right front of the horsemen. This gunfire, "right before their [the horses'] feet," according to one student of the event, helped pave the way for a successful attack and fit perfectly with Prussian artillery doctrine in 1870 by covering the cavalry's deployment and preparing its attack by direct fire upon the enemy. Charging over a distance of some 1,500 yards (1,300 m), the Prussian cuirassiers and uhlans crashed headlong into the French gun-line, cut down at least two French artillery batteries' gunners, destroyed a mitrailleuse battery, and smashed two squares of French infantry. Unfortunately, the Germans' formations broke up as they went forward, a perennial problem for any massed cavalry attack at that or any other time. They then found themselves counterattacked in turn by French horsemen outnumbering them by a factor of about five. In the fighting that followed, described as "frenzied" and a "tornado" of violence in which all arms of both sides became completely intermingled and heedless of trumpeted commands, the Germans nevertheless managed to extricate themselves and retreat to the safety of their own infantry and covering artillery. In a similar fashion later that same day, but in an event much less well known, the Prussian 1st Guard Dragoons attacked French infantry advancing on and threatening the Prussian left flank's 38th Infantry Brigade on the heights northeast of Mars-la-Tour. Once again, the charge went in under rifle- and mitrailleuse-fire so as to allow the German infantry to disengage. The dragoons rode headlong into the advancing French infantry and accomplished the mission, but with 5 officers, an ensign, 42 men, and 204 horses dead. Six officers, 2 ensigns, 76 men, and 42 horses were wounded. Five troopers went missing. This constituted about 30 percent of the regiment's effective strength. In the case of Bredow's brigade, the losses were more than 50 percent (420 killed and wounded of 800 engaged). They would presumably have been higher still had not the badly rattled French infantry shot down more than 150 of their own counterattacking cuirassiers in the space of a few minutes' confusion. Though described as not merely a "rarity" but as perhaps the "last successful cavalry charge in Western European warfare," Bredow's attack had allowed the German infantry time and space to rally. That, in turn, kept the French from continuing their retreat to the west. The same could be said of the Guard Dragoons. Notwithstanding these terrible losses, losses soon to be far surpassed by French horsemen at Sedan, the German troopers' success buttressed arguments favoring the cavalry's continued utility for the next forty years.
Despite the German cavalry's accomplishments following the war's outbreak and their frightful success at Mars-la-Tour, lessons were being learned regarding cavalry's future role. One of the most important of these lessons appeared to be that "the rifle bullet and the spade [had] made the defensive the stronger form of warfare," at least temporarily. Consequently, and as witnessed by Moltke's earlier memorandum of 1868, the classic cavalry charge against infantry was fast becoming a thing of the past. In the war of 1870, for example, the French _chassepot_ rifle had a maximum range of about 1,300 yards (1,200 m), while the German Dreyse "needle gun's" maximum range was about 650 yards (600 m). And while in both cases the maximum effective range would be much less, they remained a deadly threat to mounted troops. But even certain cavalry units such as dragoons now carried rifled weapons of their own. The Prussian light cavalry, for example, carried a shortened carbine-variant of Dreyse's rifle. The rapid and increasingly widespread issuing of rifled weapons to both foot soldiers and cavalrymen since about 1850, when combined with the means to deliver unprecedentedly large numbers of men to the front via railroads, constituted an important change in European military affairs. What had not yet happened was a real opportunity to test the effects of this change on European battlefields. True, it may be argued that the elder Moltke's initial deployment of Prussia's armies by rail in the invasion of Saxony and Bohemia in 1866 served to show the European importance of at least one of these new technologies and on an almost American scale of distance. Further, insofar as cavalry still formed an integral portion of Prussia's armies, Moltke made provision that rail cars have tether rings and removable partitions built into them so that horses and artillery of all types could be more easily transported. To the extent, however, that the Prussian campaigns of 1866 and 1870 depended at least in their initial stages on deployment by rail with a view to long-distance maneuvering for a decisive _Kesselschlacht_ , one would have thought that the cavalry's importance would have increased and not decreased. That is, while armies deployed to their frontiers by rail, they typically marched thereafter. Only later, as the enemy's railroads were commandeered, would they be expected to bring up reserves and supplies using the iron horse.
As late as 1866 the need for more effective cavalry employment was exacerbated by the fact that Prussian mounted formations were still often placed at the end of marching columns instead of being allowed to range far ahead. Indeed at Königgrätz, the Prussian cavalry still followed behind the infantry. The horsemen did not truly bring their great numbers to bear in the fighting and did not effectively pursue the broken Austrian Army at the end of the day (in part because of late charges by the latter's heavy cavalry as they attempted to buy time for an Austrian withdrawal). Once again, Moltke's report of 1868 noted such deficiencies. The war of 1870 changed all that and witnessed the combination of rail-deployment and massive cavalry operations, even though the latter sometimes had only disastrous tactical results.
Consequently, German and European cavalry in 1870 was not typically used in one of its most potentially important operational spheres, namely the regular, long-range interdiction of the enemy's railways as had so often been the case during the American Civil War. In retrospect, employing cavalry for this purpose should have been self-evident given the railroads' own significance. "If railway lines were intact, the trains smoothly organized [this itself being an important prerequisite], and supply from the railhead unhampered, armies could keep the field so long as there was blood and treasure in the nation to support them." Interdiction of such lines of communication and supply could have played a critical role in making the eventual German victory even more devastating to France than it turned out to be. Using cavalry for this purpose provided "the chance of disorganizing by invasion or _deep raids_ [emphasis added] the mobilization of" the enemy, thus "reducing his plans to chaos, and leaving him defenceless." At least one prominent American military observer in 1870, General Philip Sheridan, saw the German cavalry in action and noted the absence of such efforts. In his view, the German cavalry performed well the traditional roles of covering the front and flanks of advancing armies; and he did not fault the bravery of either the German or French troopers in the massed attack. Nevertheless, he observed, German horsemen never had the far-ranging effect their numbers should have allowed. Had the cavalry "been massed and maneuvered independently of the infantry, it could easily have broken up the French communications, and done much other work of weighty influence in the prosecution of the war."
Whatever shortcomings the German cavalry may have had in Sheridan's estimation, it was nevertheless coming to grips with a salient feature of military operations in the second half of the century. Rapid technological change associated with breech-loading rifles, nascent automatic weapons, rifled artillery, and railways necessitated more effective combined-arms thinking. Defensive positions, otherwise strong and massing the defenders' long-range rifle-fire, might still be overcome by determined opponents using the combined-arms assault of infantry, cavalry, and artillery. Conversely, anything less than attack by combined arms ran the very real risk by 1870, if not by 1860, of decimation by the same massed rifle-fire. Interestingly enough, at Mars-la-Tour Bredow's troopers closed successfully with the French gunners and infantry, in part, precisely because the Prussian horse-artillery fired diagonally across the front of the charging horsemen. This particular tactical doctrine still prevailed in 1914, even though an eventually stalemated Western Front had not yet been foreseen.
The German cavalry of 1870 also continued a tactical employment of horsemen and horse-artillery dating back to Napoleon I. The French emperor had pioneered the combination of artillery (to weaken an enemy's infantry formations) with massed cavalry and infantry assault (to shatter them). Given the technology of the Napoleonic era, trotting horsemen covering some six hundred paces every two minutes (approximately 250 yards/228 meters per minute) could close with the typical artillery piece of the day (firing to a range of eight hundred to nine hundred paces) before the gun could fire more than one or two rounds. Of course, at the canter or gallop the distance closed much more quickly, and many charges covered the final 150 yards or so (137 m) at the latter gait provided that horses were fresh. Therefore, charging cavalry "did not suffer over-much from enemy cannon fire," an observation excepting those unfortunate men and horses who were actually blown apart or eviscerated by canister or round shot. The employment of massed cavalry in corps formation at the decisive moment to defend one's own position or to attack the enemy's also dates to Napoleon. He'd established "the corps...as the largest organizational form for cavalry units." But given the substantially increased range, hitting-power, and rate-of-fire of rifles and artillery by 1870, horsemen charging a prepared infantry formation became much more vulnerable. Indeed, cavalrymen began to experience this painful realization as early as Waterloo, despite the estimated maximum of only 5-percent accuracy for unrifled musketry fire beyond ten yards' range. Unfortunately, the deadlier weapons of 1870 greatly increased the cavalryman's exposure. Assuming the height of a heavy cavalry horse to be sixteen hands or nearly five-and-a-half feet ("hands" being four-inch increments measured from the forefeet to the point of withers with the horse standing square on a flat surface), the rider's head rose to a height of not quite three yards (2.75 m) above the ground. Notwithstanding his helmet and/or cuirass, he was now extremely vulnerable at unprecedentedly long ranges; and this does not even take into account the horse itself. As a target for riflemen or artillerists, the horse possessed the terribly unfortunate combination of a thin skin and a high silhouette even when galloping for brief moments at perhaps thirty miles per hour (48 km/h).
Despite these critical vulnerabilities, cavalrymen—at least at a campaign's beginning when their horses were not yet debilitated—could cover up to 50 miles (80 km) per day when riding hard. Even 80 to 100 miles (up to 160 km) in a twenty-four-hour period were not unheard-of for well-mounted light cavalry. All the while, the horse bore an average load approaching 250 pounds (113 kg). Furthermore, given its ability to swim, not even the tactical obstacles of streams and middling rivers necessarily stood in the cavalry's way, even though rivers such as the Moselle above and below Metz demanded ferries or bridges in order for the cavalry to cross. Therefore, in a premotorized age, and indeed even later, a realistic alternative to horse-mounted units on the European battlefield simply did not exist. Scouting, patrolling, covering the flanks and rear, protecting the withdrawal, raiding—all of these missions remained the tasks of both pure cavalry formations and the mounted units attached to Prussian infantry divisions. By 1866 even the latter included four squadrons of approximately seven hundred horsemen.
Greatly aiding the German cavalry in 1870 was the detailed information they possessed on the French transportation infrastructure as the campaign began. German commanders were said to have had better maps of France than the French armies' own staffs. German longrange cavalry reconnaissance and pursuit displayed persistence after the initial battles on the frontiers, even if it was not always completely effective. The French cavalry, on the other hand, were criticized by a contemporary not only for continued massing of formations when such mass was unnecessary but also for "never send[ing] out a single scout or vedette" in the long retreat westward from the Franco-German frontier. Such tactical ineffectiveness only worsened the logistical nightmares often accompanying French troops during their mobilization and initial deployments. At Metz on 1 August, for example, some two thousand wagons loaded with hay, straw, and oats clogged the city's streets with no other apparent destination in mind. Similarly, French cavalry at Metz had to be employed "day and night as laborers," using their mounts' saddlebags to transport matériel from stalled supply-trains to the city's depots. Not until 23 July did Napoleon III demand the attention of his Minister of War, General Edmond Leboeuf, to the matter of the "establishment of a [national] requisition and remount service" in order to supplement or replace the French cavalry's extant system of regimental depot squadrons. It seems incredible that such a matter wasn't undertaken _before_ the French declaration of war, especially in light of the fact that such a service, among others, would normally "require months if not years of preparation." By that date, the destruction of a goodly portion of the French cavalry at Wissembourg and Froeschwiller was barely two weeks off.
After all, it was not as though the French had no experience in longrange cavalry operations and the remount services necessary to support them. After Jena in 1806, for example, Napoleon I "unleashed his cavalry in a pursuit designed to complete the destruction of the enemy and the enemy state; a deep penetration to spread panic among the enemy population and destroy all hope of recovery." Even so, he had seen in his cavalry not only "an exploitation force or reconnaissance asset" but also a "true shock force that could have effects disproportional to its numerical size" as at Eylau in 1807. If the latter were true, if the massed attack were still to be the French cavalry's main reason for being, then massing them in the rear and holding them in place until the critical moment, though frequently condemned, would be a logical tactical disposition. In fact, the French cavalry had done as much even earlier, as before the revolutionary wars of the 1790s, and one could argue that the idea in fact came from the example of the armies of Frederick the Great at Rossbach in 1757 and Zorndorf in 1758. Unfortunately, between 1807 and 1870, French commanders had apparently forgotten the former examples and remembered only the latter ones. As a matter of common sense, for French commanders—and implicitly for German ones—holding the cavalry in reserve until the decisive moment always brought with it the danger of having the mounted forces sitting useless altogether or being committed too late to make a difference. And despite the greatly increased firepower on the part of the infantry, dismounted combat for the European cavalry was still considered the exception. In any case it could only be undertaken by horsemen armed with the cavalry carbine such as dragoons and hussars in Prussia or _chevaulegers_ in Bavaria. In the event, French dragoons in 1870 often dismounted to volley-fire their carbines on advancing German cavalry. Evidently, however, these defensive tactics were insufficiently tenacious and the dragoons' marksmanship was insufficiently accurate. Consequently, except for this sort of occurrence, only the German cavalry in 1870 managed to be not only consistently wide-ranging in reconnaissance and screening but also able to deliver massed attacks when called upon to do so.
The cavalry's role as envisioned by Moltke in 1868 was certainly not limited to him alone. Cavalry's employment had been studied with renewed interest by Prussian cavalry officers and theorists from about 1863 onward. That does not mean, however, that there existed uniformity of view among them. Colonel Albrecht von Stosch, an officer of the Prussian General Staff who fought in 1866 and 1870 and eventually (and somewhat curiously) became Chief of the Admiralty, wrote that American cavalry in the Civil War had been essentially mounted infantry. Their reliance more on firepower than cold steel for battlefield effectiveness ran counter, he said, to the cavalry's putatively true value as a shock force, a "typically conventional" European view. Other Prussian officers, however, noted in their work that the American use of cavalry as long-range interdiction forces against strategic lines of telegraphic and railroad communications constituted what later generations would call a wave of the future. Nevertheless, and "almost without exception," Prussian students of the cavalry still maintained in 1866 and 1870 that the mounted arm's first duty was to stay mounted, avoid dismounted combat unless absolutely necessary, and attack with cold steel. The prevailing view remained that dismounted cavalry's role in the American Civil War arose from the uneven and overgrown nature of North American battlefields, not from significant changes in firearms' evolution. The dismounted role, it was felt, did not apply in Europe. Nor was the strategic raid viewed as of great military value. As late as 1900, therefore, the German cavalry—like other mounted forces in Europe—would still count the sword and the lance among its principal weapons, and apart from the reconnaissance and screening missions so much emphasized by Moltke, German horsemen would generally be held in reserve for the breakthrough battle that, at least on World War I's Western Front, never came. Therefore, despite Moltke's admonitions and their own successes up to the Battle of Sedan, German cavalry officers preferred to "trust to their own experience" and a recollection of the smashing successes of Frederick the Great. Fundamentally altering the role of the cavalry to follow any other model, particularly an American one, was still alien to German and the larger European traditions in 1870. Both German and French cavalry officers remained "fatally fascinated" by the shock-effect of massed formations of horsemen.
Of the two nations' mounted arms, it is ironic that the French did not more readily adopt another cavalry doctrine, particularly one emphasizing more long-range patrolling. After all, French cavalrymen had been active throughout the 1830s and 1840s in Algeria, where they had responded to the guerrilla war against French colonial rule with the creation of light, wide-ranging mounted units. These included the Ottoman-inspired light cavalry known by their Turkish designation as _sipahis_ and the so-called _Chassuers d'Afrique_. Eventually, three regiments of the latter were also posted to Mexico in the 1860s to bolster the shortlived regime of the French-supported Habsburg emperor Maximilian. Among the noteworthy features of these particular units was the adoption of the Iberian-influenced Barb as the mount of choice, incomparable in its ability to thrive in the arid environments of both North Africa and the high plains and mountains of central and northern Mexico. These were the "little grey Arab horses" whose dead bodies, along with those of their riders, would soon carpet the hillsides above Sedan.
It was toward that city that the German armies marched in the wake of the French defeat at Mars-la-Tour and the following battle at Gravelotte-St.-Privat. In advancing generally west-northwest, the Germans aimed to disrupt the French Government's attempt to raise a relief force for Marshal Bazaine's army now trapped at Metz. This period witnessed the French relief armies' movement and their pursuit by the German from Chalons to Rheims to Sedan from 20 to 28 August. During these days, the German cavalry once again ranged far ahead of the advancing infantry, often by as much as forty or fifty miles (up to 80 km). As they had after the battles on the frontier at the war's beginning, the German horsemen hounded the French and provided vital intelligence. Even so, the riders sometimes lost contact through no fault of their own; the French armies were subjected to what historian Michael Howard called "lunatic change[s] in direction" in their line of march as they tried to maintain contact with faulty supply lines. Once the German cavalry found their quarry, however, they helped delay and harass French forces sufficiently to deflect them ever farther northward toward the borders of Belgium and the fortress of Sedan. All the while the German infantry came up remorselessly from the east and southeast.
At Sedan one sees perhaps the most pointless waste of cavalry in the whole of the war. This occurred in the attempt by the French horsemen, under the command of General Margueritte, to pierce the German lines above the village of Floing to allow for a French breakout to the west. Shot through the face while reconnoitering the German lines, Margueritte could not ride with his troopers. They nevertheless went in gallantly according to observers, including King William of Prussia who witnessed the charge from across the Meuse. As had happened several times since the war's beginning, the result was "a useless and terrible sacrifice...a fearful loss of life with no result whatever." The two brigades of the cavalry reserve making the repeated charges not only didn't effect a breakout; "they did not delay the German infantry five minutes." With the exception of a number of German skirmishers cut down in the initial French charge, the German infantry simply waited and "mowed [the French horsemen] down with volleys." As at Morsbronn near Froeschwiller in the war's opening days, the French cavalry "were shot down before they could get within fifty yards. It was a useless, purposeless slaughter." The five regiments involved suffered some 350 men killed, not counting the wounded and those taken prisoner. One unit of two squadrons had only 58 survivors from the 216 who made the charges. The entire time that the French had been under fire was said to have been perhaps one-quarter of an hour. Rallying twice, the French horsemen came on three times in total. By the third attempt, the cavalry horses were not so much charging as picking their way gingerly over the corpses of the fallen.
Even for those managing to survive the destruction of Margueritte's cavalry, the losses suffered by French mounted and horse-drawn units at Sedan were terrible. At least ten thousand horses were captured in the French surrender. Of those, the Germans killed huge numbers deemed too broken down to keep. One Bavarian battalion alone killed three thousand after being ordered to destroy "any that looked sickly." At distant Metz, too, horses of the French cavalry, artillery, and transport units found themselves not only hated for eating up scarce supplies of grain intended for the nearly starving garrison but slaughtered for food themselves. These units were ordered to cull forty horses each for slaughter, and by 20 September fifty percent of the garrison's cavalry mounts had been butchered. Similar fates also befell large numbers of military horses in the French capital. Once the city was invested, the Parisian diet deteriorated largely to "scraps of bread, red wine, and horse meat."
With the strangulating encirclement of Paris and the subsequent occupation of most of northern France after Sedan, the German cavalry's role became one very familiar to German horsemen in Russia seventy years later: anti-partisan duty. In late 1870 and early 1871, the partisans were the _francs-tireurs_. Sometimes actual guerrillas, sometimes remnants of former French army units, sometimes newly raised formations, the _francs-tireurs_ often provided more effective intelligence to French commanders than had the French cavalry whose traditional role it was. The _francs-tireurs_ also harassed German patrols and attempted to sabotage the Germans' supply lines still stretching back to the Rhine. In this second phase of the war, German cavalry routinely undertook far-ranging patrols to the south and west of Paris in order to alert Moltke to the possibility of a French attempt to relieve the capital. Those same cavalry units carried out missions to extend the system of requisitions ever deeper into the French countryside to supplement their own armies' logistics. Ultimately, they were ordered to "sweep the country clean of _francs-tireurs."_
In the process, the war assumed ever-deeper levels of brutality as a heavy winter arrived. The siege of Paris dragged on, and the French continued stubbornly to resist (even while eventually fighting among themselves during the Commune). Prussian chancellor Otto von Bismarck raged that all _francs-tireurs_ should be summarily shot or hanged. Villages sheltering them, he said, should be burned to the ground. Indeed, reprisals against real or suspected partisans were savage, what one historian of the war called "a wholesale Americanization" of the conflict reminiscent of William T. Sherman's intention to make his Southern enemies in Georgia "howl" during the Civil War. Fortunately for France, the German cavalrymen and their commanders couldn't or wouldn't fulfill all Bismarck's wishes.
In that winter of 1870, the German cavalry's own difficulties made punitive expeditions questionable if not actually impossible. Supplies and remounts became relatively scarce and roads often so badly covered in ice and snow that troopers had to lead their horses instead of riding them. The horsemen were nevertheless forced to keep to the roads because the countryside was sometimes impassable with deep snow. To add insult to injury, German cavalry now also frequently had to be accompanied by infantry. Precisely because of the threat posed by the _francs-tireurs_ in ambushes of slow-moving, road-bound mounted columns, German commanders had to ensure they had infantry support. Of course, tying the cavalry to the speed of the infantry deprived the horsemen of their principal advantage. The long-range capability of the cavalry disappeared "the moment it had to march under the protection of the infantry." The German cavalry's war of movement became a sort of snail-paced war of attrition until the spring thaw arrived. And when the spring did come, so too did France's surrender. The Treaty of Frankfurt of May 1871 recognized not only the humbling of France but the arising of a new Great Power in Europe, a once and future German _Reich_.
At Froeschwiller, Wörth, Mars-la-Tour, and Sedan the massed cavalry charges of both the Germans and the French were not typically intended to shatter fixed infantry formations, though that could sometimes be a fortunate result, as in Bredow's "Death Ride." Rather, in all cases, massed cavalry attacks were launched to retrieve situations in which one's own infantry had been driven from the field or were threatened with that fate, as had also been the case with the Austrian cavalry charge late in the day at Königgrätz in 1866. The objective was to give the infantry sufficient time to retreat and/or re-form. The massed charge therefore became the means not so much to crown the victory as to stave off a defeat. Occasionally, of course, cavalry were ordered to attack under the false impression that the enemy was actually broken and could be pursued. The most egregious example of such a mistake shows in Prussian general Karl Friedrich von Steinmetz's ordering of a mounted attack against the French lines at Gravelotte through a ravine on a raised causeway already choked with the bodies and debris of earlier, failed Prussian infantry assaults. The predictable result was the "slaughter by the hundreds" of the units in question. A "dreadful" French rifle-, automatic-weapons-, and artillery fire hit the cavalry full in the face without the horsemen's "having the least chance of returning it." Naturally, the fault in this case lay not with the cavalry itself but in Steinmetz's gross misjudgment of the tactical situation.
At the same time the cavalry's real worth re-emerged in missions that only horsemen could execute in the nineteenth century: long-range reconnaissance, flanking movements, and the interdiction of the enemy's rail lines and communications. German cavalry proved consistently more adept at these tasks than did the French. After Sedan, however, the German cavalry's operations against the _francs-tireurs;_ the guarding of lines of supply and communication stretching back to the German States; and foraging for the occupation forces assumed precedence. And while these important missions could still be effectively executed by the Germans' mounted troops, these nevertheless found themselves increasingly tied to the infantry for protection against roving columns of French partisans. Thus the German cavalry ran the risk of losing their most significant operational assets—speed and mobility.
As effective as the German horsemen tended to be, one question remains: why did they not emulate the American example of the strategic "ride" so much in evidence in the Civil War? It turns out they did, after a fashion, and somewhat unintentionally. To the extent that German horsemen routinely rode far in advance of marching infantry columns, one sees a long-range, mounted reconnaissance capability similar to that seen in the Civil War. This capability is most evident in the form of wide-ranging German patrols, though not very large ones. They often occurred only in squadron-strength or less. One of the most striking examples of their success showed in their cutting the rail lines at Pont-a-Mousson south of Metz in the follow-up phase after the battles at Spicheren and Froeschwiller. At times in this particular pursuit, the German troopers rode as much as forty miles ahead of their infantry, a figure corresponding closely to the distances covered daily by John Hunt Morgan's cavalry in Kentucky in 1862. German cavalry played an even more important role in helping find and fix the French army in its attempted retreat from Metz to Verdun. The mounted units thus significantly contributed to setting the stage for—and, of course, fighting in—the resulting battles at Mars-la-Tour, Vionville, and Gravelotte-St.-Privat, and, ultimately, the bottling up of the French back in Metz where they'd started. German cavalry also materially helped extend the invaders' reach in the encirclement of Paris after Sedan and in long-distance foraging during the subsequent siege of the French capital. Perhaps most important, throughout the war German cavalry enjoyed what earlier generations called moral superiority over their French opponents. That confidence, despite occasionally very heavy losses, contributed in turn to their ultimate tactical and operational superiority.
One does not, however, see German cavalry engaged in the longrange strategic raiding as conducted by both Confederate and Union horsemen between 1862 and 1865. As often as not, those earlier forays aimed at capturing entire towns, operational theaters' supply dumps, or thoroughly wrecking vast stretches of railroad. The absence of this kind of raiding in 1870–1871 is all the more interesting given the evident Prussian attention paid to the technical aspects of Civil War–era use of railroads for theater-wide deployment of forces, not to mention the importance of railroads in Prussia's victory in 1866 as well as in keeping German armies supplied in 1870. German interest in the Union's and Confederacy's use of railroads did not appear to translate into a changed attitude toward the cavalry's tactics or strategy based upon the American example, at any rate certainly not before 1870. Many German students of the Civil War dismissed both Union and Confederate cavalry as merely mounted infantry, a new type of dragoon, who (somewhat ironically) relied too much on firearms for their effectiveness, rather than on "the 'vehemence and force' of shock tactics," as was evidently still preferred in Continental Europe. This attitude persisted despite the particular admiration for the Confederate cavalry in Prussia by as prominent and successful a Prussian cavalry officer as Prince Friedrich Karl von Hohenzollern.
On the other side, why did the French cavalry not emulate the American example set during the Civil War? Several possible explanations suggest themselves. In the first instance, no prominent French soldiers wrote about the Civil War before 1870, a period in which French armies were often already at war in North Africa or Mexico. Their own lessons learned in mounted operations would presumably have sufficed. Secondly, the American Civil War had occurred "at a distance [greatly removed from France] and in the midst of special circumstances."Not the least of these circumstances was the perceived amateurishness of American armies, Union and Confederate. Consequently their experiences' applicability to the French army was judged to be of limited value at best, though surely the French cavalry school at Saumur recognized that the distance from France to Mexico was not less than that from France to the borders of the Union or the Confederacy. Finally, it was maintained that the heavily "populated, cultivated, and civilized" nature of Western Europe made a French replication of strategic raiding as undertaken by Grierson or Morgan unlikely, if not impossible, despite the fact that more obscure French observers noted the strategic-raiding role that cavalry might still play. Indeed, one might argue that precisely the thickly woven nature of Western Europe's transportation infrastructure would have made strategic raiding even more valuable in offering many more targets than had been the case earlier in the still relatively sparsely settled reaches of Kentucky or Mississippi. As noted at the outset in reference to the French cavalry's lackadaisical reconnaissance and interdiction in the war's opening days, there existed in Paris an "imperturbable complacency" until 1866; and despite rousing itself after Königgrätz to adopt the _chassepot_ and new siege artillery and enact, in 1868, a plan for a thoroughgoing reorganization, the French army in 1870 was frequently simply outfought. And when not outfought, it suffered catastrophically bad leadership. In the forty-three years following the Treaty of Frankfurt, as the new German Reich and the French Republic girded themselves for the next round in their centuries-old rivalry, the cavalry of both countries remained integral to their respective armed forces, as did horsemen in all other European armies. For the victorious Germans of 1871, the question was not so much would there be cavalry in the next war, but rather to what great victories would they ride?
CHAPTER 3
NOT QUITE SUNSET
THE CAVALRY IN WORLD WAR I
By the end of the nineteenth century cavalry operations in European armies had become a matter of some doctrinal uncertainty. More powerful weapons, firing more accurately and at longer ranges, raised the question of the suitability, indeed the survivability, of the cavalry. This was no less the case in Germany than elsewhere. While the Franco-Prussian War had seemed to show that cavalry could still win a battle by means of the massed charge with cold steel, the true value of the German cavalry during that conflict had demonstrated itself in armed reconnaissance with a view to finding and fixing the enemy; screening and securing German forces; interdicting the enemy's communications; and, at war's end, foraging and anti-partisan operations. None of these missions, particularly neither of the first two, had changed by 1900, though some soon would as a result of the widespread application of internal-combustion technology.
At the dawn of the twentieth century, cavalry still possessed the unique ability to move almost at will, though not always rapidly, over the most varied terrain and in nearly all types of weather. Cavalrymen could leave the largely road- and rail-bound infantry literally in the dust. In Western Europe, however, mounted forces faced an interesting potential problem, one that had been noted as early as the late 1860s, namely the congested physical nature of the landscape over which armies might move in future. That portion of the North European Plain stretching from Normandy through France, Belgium, and the Netherlands and into northwestern Germany had a very high population-density by the last quarter of the nineteenth century. With it came a significant degree of industrial urbanization and attendant infrastructure. This infrastructure constituted a set of major obstacles to the free movement of mounted troops: intensively cultivated, and therefore very soft and wet, footing; numerous canals and railway lines; mine-pits and slag heaps; and innumerable fences and garden walls, the latter a delight to fox hunters but a real hindrance for heavily laden cavalrymen and their horses. Making these obstacles even more troublesome were the increasingly vast and complex fortifications strewn right across northwestern Europe from Liège and Namur past Luxemburg to Verdun. It was the latter's job specifically to complicate the movement of armies and thereby hinder invasions or block them altogether.
As the Franco-Prussian War had so amply demonstrated, modern war had become terribly consumptive not only of cavalrymen but also of horseflesh. Despite advances in breeding and veterinary services, lossrates rose still further as the twentieth century dawned. Nevertheless, German and other cavalrymen assumed that horsed regiments would continue to have their place in the order of battle, even in the congested regions of northwestern Europe. The Germans' experience in 1870–1871 had done little to convince them otherwise. On the contrary, German observers felt that the cavalry should be strengthened and modernized, not reduced or—worse—eliminated. For example, one of Germany's most noted military authors of the era, General Friedrich von Bernhardi, called the early-twentieth-century strength of the German cavalry lamentably weak when compared to the mounted forces of France or Great Britain. The Boer War, he wrote, had shown what highly mobile and hard-hitting cavalry columns could still do, even in an age of high-powered infantry weapons. The key, he insisted, lay in ensuring that the German cavalry possessed its own accompanying bicycle-mounted infantry and more effective artillery, as well as training cavalrymen better as marksmen. Such additions would ensure that the horsemen could, if necessary, operate independently and with sufficient firepower to cause the enemy real damage. All the while they would retain their vaunted mobility, even though he never really explained what bicyclists would do once they ran out of road. He also cautioned, however, that every new war would create new conditions and totally unforeseen circumstances to which the cavalry, as all arms, would have to be ready to adapt.
Across the English Channel, Sir John French, who had "established his military reputation by his performance as a cavalryman in the Boer War" and who would later become the first commander of the British Expeditionary Force in France, shared this view. Though speaking for the British, his comments were ones that would have been widely shared in Germany. French wrote that cavalry circa 1900 were being taught to shirk exposure on the battlefield as a result of what he considered undue respect for infantry fire. "We ought," he wrote to the contrary, "to be on our guard against false teachings of this nature...[and the] consequences of placing the weapon above the man" and, implicitly, above the horse. Of course, his own experiences in the Boer War might have taught him otherwise. Between 1899 and 1902, the British cavalry in South Africa "lost 347,000 of the 518,000 [horses] that took part, though the country abounded in good grazing" and possessed a "benign climate." Of those lost, "no more than two per cent were lost in battle. The rest died of overwork, disease, or malnutrition, at a rate of 336 for each day of the campaign."
In the absence of motorized vehicles, however, horses remained critical for mobility in that conflict. This fact represented the only real hope for the cavalry's survival in European armies. Reinforcing the mobile importance of horse-mounted and horse-drawn forces, another feature of the Boer War stood out: among the Trek Boers, "every man was a mounted shot." Like their earlier American counterparts in the Civil War and in the wars with the Plains Indians from 1850 to 1890, Boer horsemen were the quintessential mounted infantry. Though some of them might yet be armed with sabers, their primary weapon remained the rifle, and the horse served principally as a means of effective crosscountry transport. If there were to be a place for mounted formations at the dawn of the twentieth century, would it not have to be that of mounted infantry who would nevertheless fight dismounted? British cavalrymen increasingly thought so after 1902. Accordingly they were equipped and trained with rifles rather than carbines and achieved a level of firepower and accuracy approaching that of the British infantry. They were becoming essentially what in British and British imperial terminology were designated mounted rifles: skilled "horsemen trained to fight on foot, men who are mounted and intend to perform all the duties of cavalry, except that which may best be described as 'the shock.' It is expected of them that they should perform all the outpost [ _sic_ ], reconnoitering, and patrolling of an army in a manner similar to cavalry; the only difference being that they must rely solely upon their fire power for defensive and offensive action."
Commenting on the lessons to be drawn from another war of the period, the Russo-Japanese War of 1904–1905, German and Austrian officers came to a rather different set of conclusions. Under the pseudonym "Asiaticus," a German officer wrote that Russian cavalrymen were too ready to go to ground with their firearms. In doing so, he said, they repeatedly sacrificed the cavalry's greatest asset, namely its mobility. Similarly, Austrian count Gustav Wrangel observed that the Russian horsemen's experience demonstrated that troopers could not serve both firearms and the saber equally well and be skilled riders at the same time. In any case, Wrangel noted, too great a reliance on firearms robbed the cavalryman of his desire to charge the enemy and implicitly deprived him of his real weapons, the sword and the lance.
Such arguments continued unabated, even as rapid technological change continued to force the cavalry to adapt. Combining horse-soldiers with the technology that did exist culminated in the following calculation: railways would be used for initial operational deployment, as they had been for German armies ever since 1866. Increasingly heavy artillery would be the primary offensive preparation against field positions. The latter would then be taken by infantry assault. For its part, the cavalry would still be used for reconnaissance, screening, security, encirclement, and pursuit, if no longer for the battle-winning charge with swords drawn. One may argue, however, there's not much terribly novel in this approach. Cavalry had often been used for precisely these tasks in the Western military tradition ever since Hannibal's charging horsemen cut off the legions' retreat at Cannae and rode down the survivors (a favor the Romans returned at Zama). The mounted warrior's ethos and the tradition of the cavalry's shock value nevertheless lingered up to the outbreak of war in 1914. Even then, however, missions such as long-range screening and reconnaissance or interdiction of the enemy's lines of supply had not fully displaced the assumption that at least some future battles might still be decided by the massed cavalry attack. In a view no doubt shared among more than a few German cavalrymen, the British Army's _Cavalry Training Manual_ of 1907 still pronounced as a matter of principle that rifle fire, however effective it might be, "cannot replace the effect produced by the speed of the horse, the magnetism of the charge, and the terror of cold steel."
One prominent British officer, Colonel G. F. R. Henderson, deduced from the campaigns of the Boer War that the sentiment as expressed in the _Cavalry Manual_ meant that the cavalry at the turn of the century was "as obsolete as the crusaders." If, however, the matter of the infantry's use of the bayonet is considered, then the cavalry's retention of the sword, and even the lance, may not seem so far-fetched, whether in Germany or Great Britain. The same officer had earlier been pleased that the British _Infantry Regulations_ of 1880 had reiterated the psychological and tactical importance of the bayonet at close quarters, despite the by-then-widespread use of smokeless powder, magazine-fed rifles, and rapid-firing field artillery. Admittedly, Henderson modified his opinion about the bayonet's efficacy as a result of the Boer War, just as he did for the sword-armed cavalry. As the events of 1914–1918 repeatedly showed, however, German, British, and other infantry routinely went over the top with bayonets fixed long after the cavalry on the Western Front was deemed utterly useless. Indeed, the success of the Japanese infantry in their costly assaults against prepared Russian positions at Mukden and Port Arthur during the Russo-Japanese War seemed to show that the foot soldier's cold steel could still be employed with decisive effect provided that the attacking infantry had sufficient preparatory artillery support and a sufficient reserve of raw courage while covering the fire-swept zone between the opposing trench lines. Bernhardi, as well as another influential German military writer of the period, Colonel Wilhelm Balck, shared this assessment. Both stressed the "moral factor" (i.e., morale) as much as they stressed the material factor as a determinant of victory. They also applied it equally to the individual soldier and the nation in whose army he served. If, therefore, prominent military thinkers still posited a useful role for the bayonet, and if cold steel really could still frighten an enemy soldier—he need only imagine a foot or more of it being plunged into his gut—then the cavalry's retention of edged weapons and even lances does not seem so odd.
Ironically, however, the massed cavalry attack was in part made more unlikely by the very masses of infantry that some cavalrymen still confidently intended to drive from the field. Railways proved to be just as efficient as prewar planners had hoped in delivering unprecedented numbers of men and equipment for battle. For example, in approximately one month's time after the outbreak of hostilities in 1914, some 312 divisions of French, German, Austrian, and Russian troops had been brought by rail to the battlefronts, a number excluding hundreds of thousands of cavalry mounts and draft horses. Having reached the enemy's territory, however, those same masses of troops in a certain sense became a liability. From the railhead onward, those surging tides of men continued to have to move largely on foot even while officers, cavalry troopers, and artillerymen rode. Furthermore, such huge numbers of troops, regardless of branch of service, had to be supplied by logistics trains still relying primarily on the power of horseflesh. Therefore, horses (and mules) remained a critical element of all the European armies at war's outbreak and not merely in the putatively outmoded cavalry regiments. An indication of horses' continued necessity reveals itself in the following statistic: the single largest category of cargo unloaded in the French ports for the British army throughout the entire period of 1914–1918 was horse fodder. Similarly, the Director of Military Operations in the War Office from 1910 to 1914, Major-General Henry Wilson, ensured that the BEF's mobilization plan included such apparently minor, but nonetheless crucial, details as "the provision of horse-stall fittings and gangways at the French ports" for the hundreds of thousands of horses (and mules) that the British armies in France would need from the start. From the war's earliest days, similar numbers of horses were being mobilized all across Europe for the cavalry, artillery, and transport services: 165,000 in Britain; 600,000 in Austria; more than a million in Russia. The European-wide ratio of horses to men generally was estimated to be 1:3.
In Germany, as in all other combatant nations in 1914, horses were called up in unprecedented numbers from their civil tasks on farms, in businesses, and field sports. On 31 July the upper house of Germany's parliament, the Imperial Federal Council ( _Bundesrat_ ), issued decrees prohibiting the exportation of fodder, provisions, and livestock. Making Germany's equine mobilization even more efficient was the fact that German horses, like their human counterparts, had to be registered in peacetime; thus the military authorities knew where the horses were "at all times." Inaugurated in 1900, this system "involved a regular census and inspection of all horses in the country. Beasts were graded and a picture was built up of the nation's horse stock. A horse muster commission was established in each corps [area] to draw up detailed orders for the impressment of animals. These orders would be carried out prior to the full implementation of Germany's mobilization plan." Augmenting civilian registration and subsequent mobilization were the various State studs. The Hanoverian State Stud based at Celle, for example, alone provided annual deliveries of some 2,500 remounts to the German army by 1914, while the East Prussian State Stud at Trakehnen shipped out fully 7,000 per annum. In addition to these private and State-sponsored resources at home, the German government also continued to look abroad for horseflesh. The U.S. Consul General in Berlin, Robert P. Skinner, was cited in the _New York Times Magazine_ of 3 May 1914 as reporting that the German government was advertising "in certain American newspapers for 500 American thoroughbreds, 1,000 more or less pedigreed horses, and 1,000 draught horses for artillery use." German purchasing agents were also reportedly active in Ireland, having evidently "contracted for every horse on the Irish landscape...up to 1916." Other agents had even "invaded France and bought up 18,000 first-class cavalry and artillery mounts."
Though the war would necessarily nullify such prewar contracts, the Germans' need for such numbers of horses remained clear. A combat-ready corps of the regular German army in 1914 required no fewer than 280 trains comprising more than 12,000 railway cars in order to move from its depot to its deployment area. Those cars included 2,960 specially outfitted to transport only horses. Similarly, the need for provender was enormous, as already indicated. Given 1914's standard daily horse ration of approximately twenty-two pounds (10 kg) of feed and fodder, the German First Army alone required approximately 840 tons of feed and fodder each day for its establishment strength of 84,000 horses of all types. That requirement compares with approximately 555 tons of daily rations for the same army's 260,000 men. "To put it another way, the First Army needed 50% more food for horses than for men, though it had over three times as many men as horses." In all, the German army of 1914 intended to move not only three million men but also fully 600,000 horses merely for the initial campaign in northwestern Europe. These staggering totals required an equally breathtaking commitment of rolling stock to get the troops, their horses, and their equipment to the frontiers. No fewer than 11,000 trains were scheduled in the mobilization plan.
Of course, from the war's opening days on the Western Front, the German army also attempted to requisition horses in occupied territory, precisely because anticipated losses of horseflesh demanded it. In southern Alsace around Belfort, not far from the battlefields of 1870, enforced requisitioning began as early as 2 August, according to reports in the British press. Naturally, such attempts did not go uncontested. One German officer, apparently acting alone, reportedly entered one locale only to be "forced hurriedly to retreat" by enraged civilians. Similarly, in other villages German troops on the same mission were said to have been driven off by pitchfork-wielding Frenchmen. Such searches could quickly escalate to skirmishing. German dragoons attempting to enter Villers-la-Montagne, for example, found themselves forced to retreat by French _chasseurs_ , while a full German mounted regiment's attack at Montfortane failed in the face of French infantry fire.
Nevertheless, at least on the Western Front, the cavalry's tactical and operational importance diminished rapidly as the lines stabilized after the First Battle of the Marne. Well aimed rifle-fire, particularly of the British "Old Contemptibles" of 1914, machine guns, and artillery quickly showed themselves capable of bringing effective gridlock to battlefields eventually made completely inert by the construction of the trenches. Furthermore, one of the cavalry's by-now-classic functions—turning the enemy's flank—proved itself increasingly difficult given the soon-to-be static nature of the lines in France. The force-to-space ratio was so high that infantrymen were finding maneuver ever more problematic absent effective and widespread motorization. It therefore became impossible for cavalry, whether German, French, or British, to envelop flanks that were never sufficiently "in the air" after September 1914. Such a conundrum was particularly troublesome for the German army. As already noted, the army's doctrine since the late 1860s had envisioned the cavalry's playing precisely that flanking role, as it had done in 1870. Once the enemy's flanks had been overlapped and his frontal defenses smashed by fire, but only then, would his positions be taken by assault. If, however, flanks could not be turned, then the cavalry on the Western Front would either have to wait for the increasingly improbable great breakthrough or—the worst of fates for the cavalryman's ethos—fight permanently dismounted.
As the German cavalry rode to war in 1914, their expectations were matched by their counterparts on the Allied side. Since at least the summer of 1911, the French army's General Staff expected that in a war against Germany, the British would dispatch a force of some 150,000 men and 67,000 horses, the latter including mounts for a full cavalry division and two separate mounted brigades. The French and British staffs also fully expected to meet German cavalry in force. Indeed, as early as 1908 the French Superior Council of War had received an analysis of likely German wartime action. That analysis predicted a German drive through at least eastern Belgium around the northern flank of France's frontier defenses. This prediction recognized the Germans' "tradition of enveloping their opponent's flanks," a mission almost impossible in 1914 without the employment of strong mounted forces capable of rapid, wide-ranging movement. As a consequence, no fewer than nine of the German army's eleven cavalry divisions found themselves on the Western Front in 1914. That fact alone indicated the expected importance devolving upon mounted forces in a campaign designed to defeat France before Russia could mobilize effectively.
Each of these German cavalry divisions had an establishment-strength between 4,500 and 5,200 men and some 5,600 horses, including remounts. A typical mounted division included between 3,500 and 3,600 troopers armed with carbines, sabers, and, in many cases lances, to fight specifically as cavalry. In addition, each division had an organic infantry battalion ( _Jäger zu Pferde_ , literally "hunters on horseback") of as many as 1,000 men. Presaging future motorization, the _Jäger_ were typically bicycle- or truck-mounted, though as their name actually indicates, they often rode as well. Frequently, however, they slogged along on the boot-leather express, just as the infantry has done since time immemorial. Interestingly enough, though no one could know it in 1914, this relationship between mounted and dismounted German troops would be exactly reversed in 1939. In World War II it would be the infantry divisions that would have a cavalry squadron in their organic reconnaissance battalions.
In 1914 the standard primary weapon for both the _Jäger_ and mounted troopers alike was the Mauser M1898, 7.92-mm carbine. Nevertheless, some cavalrymen also still carried straight-edged swords. In the heavy cavalry regiments the sword was a thrusting, rather than a cutting, weapon having a 36-inch (91-cm) blade and known as a _Pallasch_. The _Pallasch_ weighed just a few ounces less than three pounds (1.36 kg). Uhlans and other light cavalry carried a similar weapon but one slightly lighter in weight and implicitly intended more for slashing. Still others were armed with sabers as such. As for offensive heavy weapons, German cavalry divisions also possessed—in a manner similar to other European cavalry forces—their own horse-artillery detachment of twelve guns in three batteries per division. According to the German timetable at the start of the campaign in the west, the entirety of such a division was supposed to cover between twelve and twenty miles (up to 32 km) per day. The cavalry expected to cover these miles in one of four recognized gaits: the walk, trot, gallop, and "extended gallop." The canter appears to have been used only for march-pasts. Using these gaits, German horsemen could cover between 125 (walk) and 700 ("extended gallop") paces per minute, a "pace" being about 31 inches (78 cm). In other words, cavalry at the walk—the most frequent marching gait—would cover just over one hundred yards (91 m) per minute or about three-and-a-half miles per hour. While that is about the speed of fast-moving infantry, it must be remembered that horses could keep going for longer periods of time, particularly if the infantryman was carrying his full load of equipment. At the trot, a good ground-covering gait that spares the horse if the trooper posts, the cavalry could cover 275 paces per minute or approximately 8 miles per hour (not quite 13 km/h). In point of fact, the distance covered daily by the cavalry would almost certainly be greater. Given the constant need for reconnaissance forays and screening operations, the horsemen would of necessity have to ride many more miles than that. Thus equipped and ready to move, the German cavalry in 1914, particularly the eight divisions eventually grouped in two corps during the so-called Race to the Sea after the First Battle of the Marne, constituted "the largest body of horsemen ever to be collected in Western Europe before or since." In light of their numbers and their theoretical mobility, they had at least the potential to be everywhere.
Riding West
In the opening campaigns following Germany's declaration of war, the German army's cavalry units on the Western Front played the centuries-old role envisaged for European horsemen, a role reemphasized in the Prussian army in the late 1860s by Helmuth von Moltke the Elder. No one could know it with certainty at the time, but by the end of the First Battle of Ypres in November 1914 the German cavalry would find themselves essentially out of a job as mounted warriors. The entrenchment of the front from the sea to Switzerland, a distance of some 450 miles (724 km), would make traditional cavalry operations impossible. Nevertheless, at the campaign's beginning, the German horsemen were almost always in the van and often well in advance of it. For example, General Otto von Emmich's task group, sometimes referred to as the Army of the Meuse, included General Johannes Georg von der Marwitz's II Cavalry Corps (2nd, 4th, and 9th Cavalry Divisions including the Prussian Guard Cavalry). These forces crossed the Belgian frontier on 4 August 1914 and set out straight for Liège some twenty miles to the west, with mounted "outriders distributing leaflets disclaiming aggressive intent." On 5 August 1914, French general Joseph Joffre responded by authorizing French cavalry to move into Belgium, that country's neutrality now having been violated by advancing German columns. The German cavalry's movements, however, helped alert French commanders to the latter's intention of driving much farther west into Belgium than Paris had earlier anticipated. On 14 August, just before the opening of the planned French march into Belgium, Joffre learned that German mounted patrols had attempted to seize bridges over the Meuse south of Namur. For their own part, French horsemen helped discover that the German armies were in fact moving deeper into Belgium than Joffre had likely expected. When the French Third and Fourth Armies advanced into Belgium, they included a full cavalry corps and a separate cavalry division, along with nine infantry corps. Unfortunately, the German Fourth and Fifth Armies marching to meet them comprised not only ten infantry corps but also two cavalry divisions, as well as six reserve brigades and the garrison forces at Metz. Equally unfortunately, the French cavalry of General J. F. A. Sordet's corps failed, as the French cavalry had often failed in 1870, to find and fix the invaders. The Ardennes country into which the French horsemen advanced admittedly did not lend itself to mounted operations. This southeastern corner of Belgium is fairly rough terrain, covered as it still was at that time by heavily forested hills. Furthermore, the River Semois and its tributaries cut across the French axis of advance. Coincidentally, these same conditions would confront invading German armored forces coming from the other direction thirty-six years later. As it was, despite nine days' riding from 6 to 15 August, the French horsemen never did locate the German columns and only succeeded in breaking down their mounts.
Somewhat to the surprise of the German invaders, the Belgian army fought more tenaciously at Liège than expected. In the fighting around that important fortress-city, Belgian commanders credited the German cavalry with helping prevent a successful Belgian counterattack from Ft. Embourg, one of twelve fortresses enclosing the city, in this case on the River Ourthe southeast of that stream's confluence with the Meuse. Despite having shot down large numbers of German infantry unwisely advancing across open ground around the fort, the Belgian defenders could not follow up their success. Apparently they feared operating in the open in the face of large numbers of German horsemen hovering in the vicinity. The Germans turned this hesitation to their advantage when a detachment of hussars subsequently launched a small-scale raid of their own into a gap between several of the forts in an effort to capture the Belgian commander, General Gérard Leman. In the confusion, and no doubt to their surprise, the Germans reached Leman's headquarters but were overpowered by an enraged crowd of Belgian civilians and Civic Guards who rushed to the scene. The latter killed one German officer, two troopers, and took the rest prisoner.
Despite this setback, German cavalry did contribute materially to Liège's fall. Striking out from Visé, hard by the border with the Maastricht Appendix of Holland, they broke into open country west of the Meuse. Leman responded by ordering detachments of his mobile troops—in this case a division and a separate brigade that had come to Liège's defense—to block the German horsemen wheeling to the southwest beyond the river. Unfortunately, Leman's action removed forces intended to hold the gaps between Liège's fortresses and left their garrisons essentially trapped inside. Laboriously, German artillery, including the massive 420-mm Krupp howitzer, came up to batter the works and their unfortunate defenders into submission.
As the German advance gathered force after the fall of Liège, in fact before the city's final submission, the cavalry continued to ride hard ahead. As early as 9 August papers were reporting the presence of mounted reconnaissance patrols, the dreaded and ubiquitous "uhlans" of 1870s vintage, as far south as Dinant. There even appears to have been apprehension that the Germans were poised to launch an all-out cavalry assault on Brussels itself. More concretely, on 12–13 August German cavalry clashed with Belgian troops as the horsemen attempted to seize bridges over the Rivers Demer and Gette at Haelen and the Velpe at Cortenachen. At Haelen, the fighting was particularly intense. There the German cavalry launched mounted attacks against entrenched Belgian infantry supported by machine guns and artillery. The attackers, the 17th (Mecklenburg) Cavalry Brigade, "fell in heaps" with their horses as they stormed the village. Precisely as had happened to the French cavalry in their attack on Morsbronn in Lorraine in 1870, the Mecklenburgers and their mounts were shot down in such numbers that they literally walled up the streets of the village and made further attacks impossible. Around the nearby village of Donck, a similar fate befell the horsemen of the German 3rd Cavalry Brigade's 9th Uhlans and the Cuirassier Regiment _Königin_. In the case of the 9th Uhlans, the attackers charged with couched lances against what the regimental adjutant described as "murderous" small-arms, machine-gun, and artillery fire. Though the lancers rode down several lines of Belgian infantry, continued rifle-fire eventually drove them off. As they retreated via the intervening swale through which they'd attacked, they found it filled with dead comrades and their horses. In the time it took to charge and withdraw, the regiment lost perhaps half its strength: more than 100 men and 250 horses. Among the _Königin_ Cuirassiers only 76 officers and men were lost but fully 270 horses. Meanwhile, to the southwest at Tirlemont (modern Tienen), fully two thousand German horsemen attempted to capture the entire town but were driven off by the defenders. At Èghezèe, on the Belgian right wing near the great fortress-city of Namur on the upper Meuse, the German cavalry fared worse, at least temporarily. Here Belgian horsemen caught German cavalry bivouacked in the town. The latter retreated after a series of sharp, dismounted street fights. In all of these engagements, German cavalry were not always able to accomplish their stated missions. Even where they failed, however, they continued to play the European cavalry's traditional modern role: screening the advance of one's own forces, reconnaissance, and finding and fixing the enemy. The realization slowly dawned that despite great bravery in the attack, such missions as these were all they would very likely be able to do. How far into the future they could carry on doing them remained another question entirely. Nevertheless, in these specific cases, so well did the German cavalry perform in the war's opening days that the Belgian General Staff evidently remained "in total ignorance" of the lumbering approach of the monstrous guns that had demolished the forts of Liège and would soon do the same at Namur.
By 20 August 1914, newspapers reported German cavalry occupying a line running from Diest to Tirlemont. They were said to be pushing out in all directions over the open country of Brabant, terrain that seemed to have been "designed for them." It was at this moment that German horsemen fought an interesting engagement, given what the German army's doctrine would say about cavalry in the interwar period. On 25 August near Ostend, a force of some 150 Belgian gendarmes attacked a superior force of German cavalry approaching down the Bruges road. The German riders were preceded by bicyclists acting as scouts, certainly something of a twist for the normally far-ranging troopers. When the bicycle troops were halted by Belgian rifle-fire, the cavalry dismounted in true dragoon fashion to fight on foot. Then, still more interestingly, they brought up machine guns mounted on automobiles, though reports did not indicate whether these were purpose-built or lash-ups. After about an hour's fighting, a second cavalry detachment managed to arrive and threaten to outflank the Belgians in the old style. Faced with being cut off, the Belgians withdrew into Ostend. While the Germans themselves subsequently failed to break into the town, their employment of what a later generation would call combined arms presaged the post-1918 _Reichswehr's_ doctrine for mounted forces' organization.
As August wore to a close, the French and British armies continued to retreat. Placed in the Allied line between the French Sixth Army to the west and Fifth Army to the east, the British Expeditionary Force (BEF) withdrew in good order after the Battle of Mons and managed to break contact with the pursuing Germans. Attacked but not disabled, the BEF simply "disappeared into the countryside," often leaving the Germans at a loss as to its whereabouts, despite occasional chance collisions such as the one on 22 August when British horsemen of the 2nd Dragoon Guards met a German patrol of the 9th Cavalry Division's 4th Cuirassier Regiment northeast of Mons. Occasionally, the BEF's withdrawal was also helped by the "self-sacrificing" attacks of its own horsemen against the pursuing German First Army. For example, on 24 August at Thulin to the southeast of Condé, the British 2nd Cavalry Brigade under General H. B. de Lisle charged advancing German troops armed with machine guns. Like the _Königin_ Cuirassiers and 9th Uhlans at Haelen earlier that same month, the British horsemen were shot down in large numbers. But in a manner reminiscent of the French cavalry in 1870, they managed to buy a bit of time for the rest of the BEF to continue its withdrawal. Thus, in the pursuit of the BEF, the German cavalry failed in one of its primary missions: maintaining contact with and providing information about the enemy.
The German horsemen made good the failure, however, on 4–5 September. As the Allied armies had retreated and the German armies had advanced, the right flank of General Alexander von Kluck's First Army, itself on the extreme right of the advance, became threatened with being turned by a French counterattack. Here German cavalry served von Kluck and the German cause well. The local German commander, General Hans von Gronau of the IV Reserve Corps, discovered the French threat as a result of his Corps' attached cavalry division. Doing precisely what mounted troops were supposed to do in this sort of situation, namely reconnoitering and screening the advance, Gronau's cavalry alerted him to the impending French attempt to roll up the entire German First Army. Acting decisively on the information, Gronau attacked the French, halted their flanking maneuver, and in the process alerted First Army's commander and the _Oberste Heeresleitung_ (OHL) to the stand being prepared by the Allied armies on the Marne. As Kluck moved forces to counter the French, a gap opened between his First Army and General Karl von Bülow's Second Army immediately to the east-southeast. Into the gap marched the BEF and elements of the adjoining French Fifth Army. The advancing Allied troops collided with two German cavalry corps whose units were screening the German First and Second Armies' fronts. Once again, the German cavalry were performing precisely the sort of job that Moltke the Elder had laid out for Prussian horsemen in the previous century. An admittedly already slow British advance into a vulnerable opening in the German lines was halted by the defensive firepower of the German cavalrymen, firepower made more effective by the German cavalry divisions' own integral _Jäger_ battalions.
Clearly, however, the German horsemen had not had everything their own way. They'd had only two days for rest, farriery, and the refurbishment of weapons before what was soon to be called the Battle of the Marne began. Their horses, worn down by the month-long, fighting advance through Belgium, now also showed the cumulative effects of a lack of ready fodder. Their British and French counterparts, though numerous, did not exploit this situation. The BEF's two cavalry divisions were under orders to maintain contact with the French armies on the British flanks and not go riding off in any putative pursuit of the Germans. For their part, the French cavalry, still commanded by Sordet, had not only used their horses but, in certain respects, had abused them. No watering of mounts, for example, had been permitted in rest halts during the long retreat from Belgium, ostensibly to prevent the columns' being strung out. Furthermore, French troopers evidently saw no need occasionally to dismount and spare their horses, unlike German and British horsemen. As one British second lieutenant caustically noted, and as many a German trooper would doubtless have agreed, "The French cavalrymen was [ _sic_ ] rarely seen off his horse. He had a rooted objection to dismounting." Consequently, 90 percent of French equine losses in 1914 resulted from sickness, and fully 25 percent of all of France's mobilized horses were dead by year's end. The French army's remount system, already skewed by prewar requirements in favor of Anglo-Norman draft horses rather than true cavalry mounts, could not readily make good such losses; and tactical bright spots—such as Sordet's troopers almost capturing von Kluck in his own headquarters on 8 September—could not overcome the French horsemen's deficiencies in the war's opening campaign.
Between the Battle of the Marne and the end of 1914, the Western Front gradually extended itself to the northwest. This was the famous and somewhat erroneously named "Race to the Sea." During these almost three months, both sides attempted to gain room for maneuver by finding and turning the flank of their opponents. Contrary to popular opinion, they did not wish to terminate the lines at the coast of the North Sea, for then maneuver would come to an effective end, as in fact it did by the end of November. "Each [side] became very much aware of the defensive strength the opponent would gain when its northern flank could be anchored against the seacoast, and not [be] subject to the fluctuations of cavalry maneuvers." During this period, cavalry actions occurred as both sides strove to turn the other's flank. On 19 September, for example, German cavalrymen of the 4th Cavalry Division acting as the corps cavalry for the IX Corps maneuvered over what two years later would be the battlefield of the Somme. There they fended off, but evidently did not heavily engage, British cavalry (including attached machine-gunners and bicyclists) advancing from the vicinity of Péronne against the Germans' lines of communication stretching from St. Quentin to Le Cateau. Each of these actions, followed up or accompanied by advancing infantry, extended the lines farther to the northwest. But as the two sides approached the sea, conditions on the ground became ever more congested by the huge masses of troops operating in the semiurban industrial landscape. In early October near Arras elements of at least four German cavalry divisions jostled for passage on the narrow roads of French Flanders. Due north of Arras, around the industrial and coal-mining town of Lens, the cavalry were ordered to fight dismounted. In "bitter street fighting" against Moroccan troops of the French Tenth Army and even against armed civilian miners, cavalrymen of the 4th, 7th, and 9th Cavalry Divisions held those districts until German infantry came up. Of this fighting in Flanders, the later Inspector of Cavalry of the German army wrote:
The opinion of the General Staff...that the army cavalry might be employed in a swift crushing of all resistance to the north of Lens with its full force, for the purpose of taking the enemy wing in rear, was proven impracticable. In the first place, the enemy had extended his northern wing still farther in the direction of La Bassée; also, a speedy passage was impossible over this country, with its railroad embankments, field tramways [and] mine pits, coal shafts, [slag] heaps and rows of workmen's houses, so characteristic of an industrial center, all of which had been merely approximately indicated on the maps. Mounted warfare, such as the army and higher command especially required this day, was simply out of the question. Step by step, the jaegers [ _sic_ ] and cavalry troopers fought their way with carbines for possession of the locality, against an enemy defending from houses and pits; lacking bayonets, no headway could be won against massive factory walls without employment of heavier ammunition.
As a consequence of the front's resulting stabilization in Flanders, the German cavalrymen who'd ridden the entirety of the campaign were gradually converted into infantryman.
Throughout the second half of October and the whole of November 1914, they effectively surrendered their former status and slowly became indistinguishable from the other trench-dwellers populating the lines from Ostend to the borders of Switzerland. The addition of these eight divisions' worth of troops to the trenches was, of course, advantageous for the Germans, even though abandoning their horses often meant that the troopers went into the line without greatcoats, entrenching tools, and other mundane but essential equipment. Many of them, however, evidently insisted on keeping their spurs. As for the matter of what to do with eight cavalry divisions' worth of horses, that remained "a problem well-nigh beyond conception."
Whether they fought mounted or not, German cavalry made a material contribution on the Western Front in 1914, a fact recognized by friend and foe alike. The commander of the German Sixth Army, Bavarian Crown Prince Rupprecht, paid tribute to the horsemen's valor in the trench fighting around Lille in an order of the day late in October. The cavalry, he wrote, "has proven that it could successfully use the carbine against fortified trenches without being driven away and, from the very nature of its organization has been able to cover a wide area of conflict in a series of victories. It has thereby performed a highly meritorious service on a portion of the battlefield. I wish to express warmest thanks and deepest acknowledgment to the troops for their wonderful behavior and exceptional endurance." Of course, one can argue that Rupprecht merely acknowledged the truism that necessity is the mother of invention. Nevertheless, his order took account of the cavalry's real remaining strength: its potential for mobility. On a fluid front, the horsemen could still move in a way that the infantry simply could not.
Nevertheless, from the end of 1914 those German cavalrymen on the Western Front remained essentially what they'd by then become: infantry. That didn't mean, however, that the cavalry disappeared. Instead, from here on out, it was the Allied cavalry that featured prominently, if only by virtue of its relative inactivity. Despite the front's immobility in 1915 and 1916, as well as the tactical dominance of artillery and machine guns, the overall British commander, Sir Douglas Haig, still thought combat horses necessary. A former cavalryman and polo player, his attachment might have appeared merely sentimental. A more practical consideration nevertheless influenced him.
An Allied victory ultimately required driving German forces out of France and Belgium. By contrast, in midsummer 1916 the Germans had less imperative reasons to advance, especially in northwestern France. Given the fighting then still raging at Verdun, as well as Austro-German defensive efforts against the Russians' "Brusilov Offensive" on the Eastern Front, German troops on the Rivers Ancre and Somme could remain precisely where they were. Unlike 1914, the Germans no longer had a real need for mounted troops. If the Allies wanted to come on, let them come. In Haig's calculation, such an offensive stood to accomplish a number of goals. The "Big Push" on the Somme, as it came to be known colloquially, would help relieve pressure on the French at Verdun; it would complicate the Germans' efforts to contain the Russians; and—Haig confidently believed—it would provide the long-awaited opportunity to break the deadlock in the west by shattering the German lines. When that breakthrough occurred, the British cavalry would be waiting to pour through the breach and raise havoc in the German rear all the way to the Belgian border and beyond.
Consequently, as British preparations entered their final stages in the late spring 1916, fully 100,000 horses crowded behind the British front lines. All of them had to be provided with shelter, feed, and fodder, an enormous undertaking; and this number did not include the additional requirements of some 400,000 men. Equally critically, water had to be plentiful as well. To that end, pumping sets were brought in from England, water pipelines were laid, more than three hundred tanker-trucks found work hauling water to the billets, and a number of two thousand-gallon canvas water tanks were set up along the front. In addition to artillery and draft horses, vast numbers were contained in the three British and two Indian cavalry divisions that Haig envisioned sweeping northward through the enemy's rear areas once the German lines had been broken by the British assault. Then, as one noted historian of the battle put it, "the hunt would be on." Haig's critics in the British government would be silenced. The German army would be run to ground and defeated. Northern France and the whole of Belgium would be liberated. The Allies would be crowned with victory.
As is only too well known, the battles on the Somme did not turn out as expected. Losses were unprecedented, little was gained, and the British cavalry never made their sweeping ride. That is not to say, however, that the mounted forces escaped unscathed. They didn't. But their collective fate exemplified what many European detractors of the cavalry—anyone's cavalry—had maintained since 1870. On the modern European battlefield there was no place for a horse, or at least for a horse carrying a combat soldier. No less a critic than then-prime minister Herbert Asquith bluntly told Haig in May 1916 that the British cavalry horses in France were "of no use" and that the British government maintained huge stocks of war horses in France for no purpose. Haig, for his part, dismissed such criticism as the mere carping of ill-informed civilians.
Consequently, when the British offensive opened on 1 July 1916, the British cavalry stood by expectantly. They went on waiting for the next two weeks. In the afternoon of 14 July, however, Haig launched the 7th Dragoon Guards in what he thought might be the beginning of the long anticipated mounted dash through the breach in the German lines. Carrying lances, the cavalrymen charged across open ground near the village of Bazentin-le-Petit on the southeastern end of the British line not far from the boundary with the adjoining French sector of the front. They killed sixteen Germans and captured more than thirty others, only to be withdrawn to the British lines again before dawn the next day. No mounted breakthrough followed because no real breach in the German lines had been opened.
But, like their fathers in 1870, the cavalrymen and artillerymen on the Somme—on both sides of the lines—saw what modern warfare would do to horseflesh; and though the mounted forces didn't suffer the way the French horsemen had at Sedan forty-six years before, the carnage was every bit as sad. A new horror, beyond even poison gas, now confronted the cavalry and other horse-dependent arms: aerial attack. Though still in its infancy, the threat posed by strafing was nonetheless real whether the horses in question were cavalry mounts or ones pulling artillery and wagons. To cite but one example, two British aviators flying a reconnaissance mission over the Somme battlefield attacked German horsemen galloping forward with ammunition caissons in tow. The results were predictable. The lead mount "crumpled up, and the others, with their tremendous momentum, overran him, and whole lot piled up in the ditch, a frenzied tangle of kicking horses, wagons, and men. The second limber, following close behind...could not avoid its leader; its wagons overturned, wheels spinning, and split. Shells rolled over the road. We returned elated [to our airfield]." Not far away, on the same day that the 7th Dragoon Guards rode for Bazentin-le-Petit only a bit over a mile and a half to the west, a squadron of British horsemen was caught in the open by German artillery fire on the road leading to a contested village. "Dead and dying horses, split by shellfire with bursting entrails and torn limbs, lay astride the road that led to battle. Their fallen riders stared into the weeping skies. In front, steady bursts of machine-gun fire vibrated on the air. Caught by a barrage, these brave men and fine horses had been literally swept from the Longueval road." Thus the situation remained for the British cavalry on the Somme for the rest of that months-long battle. They would eventually be withdrawn in early November 1916. As Haig told the commander of the Cavalry Corps, Lieutenant General T. C. McM. Kavanagh, on 5 November, the horsemen no longer had any real hope of a breakthrough "owing to the state of the ground," sodden as it was with early winter rains. Consequently, the mounted forces "might now be withdrawn to more comfortable billets."
In that same year, on the French end of the front at Verdun, not only men but horses too succumbed to the war's carnage in unbelievable numbers. It was noted that on one day of the fighting there fully seven thousand horses were killed by long-range artillery fire, ninety-seven alone by the detonation of a single shell fired by a French naval gun. Still others were caught and killed in agony by the "terrible fumes" of a German barrage using phosgene-filled artillery shells on 22 June in the last major German attempt to capture the town. On the Western Front, as to a certain extent on the Eastern Front, the German army is generally recognized to have fought the war more effectively from a technical point of view, only to lose the _Materialschlacht_. In such a circumstance, cavalry forces, however useful they may still have been in theory, and however effectively they were employed in the war's opening campaigns, could not affect the outcome.
Cavalry on the Eastern Front
Throughout the war on the Eastern Front, the German cavalry played a more active and traditional role than in France. With localized exceptions, World War I from the Baltic coast to Rumania remained a war of movement. It could not be otherwise. Between Riga and the mouth of the Danube lay an airline distance of more than eight hundred miles (nearly 1,300 km), but the front could never be measured in airline distances because it included many hundreds of miles more in twists and turns. One theater of operations that was of central importance to Germany, Austria-Hungary, and Russia alike, namely Russian Poland, by itself measured more than 200 by 250 miles (320 by 400 km). Completely entrenching such vast distances was simply impossible. The front would always be "in the air" somewhere. Consequently, "both sides attempt[ed] vast and daring maneuvers against the enemy's flank and rear, just as they would in a later war from 1941–1945." For the success of any such maneuvers, the cavalry's mobility remained critically important.
At the war's beginning, the Russian army mobilized no fewer than thirty-seven cavalry divisions. On the German side, by dramatic contrast, there was only one, at least in East Prussia. This was the venerable 1st Cavalry Division, whose regiments were based at Königsberg, Insterburg, and Deutsch-Eylau. This division, along with eleven neighboring infantry divisions, comprised about one-tenth of Germany's mobilized strength in 1914. Though the numbers of German cavalry would grow enormously during the war on the Eastern Front, the initial disparity was owing not only to Russia's having to fight both Germany and Austria-Hungary and therefore needing more cavalry but also to the German General Staff's assigning East Prussia a secondary status in prewar planning. Primary attention and the accompanying resources went to the massive attack against France and Belgium in the West. This particular German cavalry division, however, not only comprised storied Prussian regiments; it would also subsequently be maintained as part of the _Reichsheer_ during the interwar period and go to war again on horseback in 1939.
One of the very earliest events on the Eastern Front also involved cavalrymen, though in this case they weren't German. On 6 August 1914, several hundred men of a formation known as Pilsudski's Legion—carrying their saddles—marched across the frontier of Russian Poland from Austrian Galicia near Cracow in the hopes of finding mounts. Wisely, they retreated when they were approached by Cossacks and eventually found their way into the Austrian army. The incident is revealing, for the Cossacks' presence on the Eastern Front from the war's outbreak reinforced the conflict's likely intensity over the whole of that almost immeasurably vast area. From its beginning, the fighting in the east, unlike that in the west, carried overtones of "race war," a feature reaching its gruesome extreme in the Nazis' campaigns between 1941 and 1945. The prejudices between supposedly cultured Germans and supposedly barbarous Russians, with the Poles caught in the middle, were manifested from the beginning of the war of 1914. As early as 11 August, no less an authority than the director of the Prussian Royal Library in Berlin, Adolf von Harnack, pronounced that "Mongolian Muscovite civilization" once again loomed over the eastern horizon to threaten German lands just as had happened in the eighteenth and nineteenth centuries.
This conjuring of the ancestral Western European fear of the horsemen of the steppe could not have been clearer. As it turned out, the very next day Cossacks of Russian general Pavel Rennenkampf's First Army crossed the East Prussian border, sacked the village of Markgrabovo, and ignited precisely the sort of panic that Harnack's "Mongolian Muscovite" hordes had created in generations past. Intensifying the German reaction was the quasi-melding of Prussia's identity with that of Germany as a whole, a process that had begun with Germany's unification under the direction of Otto von Bismarck in 1870–1871. Though certainly not universal, this identification of Prussia with Germany made East Prussia's violation by "asiatics" a national concern, not one limited to East Prussia itself. For a traditionalist unit such as the 1st Cavalry Division, Russian troops' presence on German, and especially East Prussian, soil would pose a grave emotional threat. A prominent later commander in the post-1918 Red Army (and eventual Marshal of the Soviet Union) only reinforced the apprehension accompanying such a threat by evoking the memory of Mongols' style of warfare. "The Russian Army," boasted Mikhail Tukhachevsky, "is a horde, and its strength lies in its being a horde." This image of rampaging barbarians who "would sweep into _deutsches Kulturland_ " was hardly one to reassure East Prussians or other Germans either during World War I, the chaotic later days of the 1920s, or even in the 1930s or 1940s. As it was, the commander of the German I Corps in East Prussia in 1914, General Hermann von François, lamented the sorry plight of the "mad rush" of thousands of civilians away from the Russian horsemen and fretted that the refugees would impede his own armies' efforts to contain the invaders. A senior staff officer who witnessed the invasion and who planned the defenders' operations, Colonel (later General) Max Hoffmann, subsequently noted in his diary that never before had war been waged with such "bestial fury." The Russians, he wrote with brutal succinctness, "are burning everything down." Buildings not burned were plundered. One eyewitness, a captain in the Russian 1st Cavalry Division's _Sumsky_ Hussars, noted that in the campaign's opening days around Markgrabovo, "[the] scene on the German side of the border was quite frightening. For miles, farms, haystacks, and barns were burning. Later on, some apologists...tried to explain these fires by attributing them to the Germans, who were supposed to have started them as signals to indicate the advance of our troops. I doubt this, but even if it were so in some cases, I personally know of many others where fires were started by us." Not surprisingly, Russian cavalrymen, including the captain quoted here, helped themselves to the fine horses of East Prussia when in need of a quick replacement for blown, wounded, lame, or dead Russian mounts. Not a few of these horses came from the Prussian State Stud at Trakehnen, which lay almost directly in the path of the invaders. Some Cossacks also took human hostages from the civilian population, many of whom were deported to the east.
In resisting the Russian invasion, the German armies in East Prussia fought a successful series of battles between 17 and 23 August near Stallupönen and Gumbinnen. These towns lay due east of the provincial capital of Königsberg with Stallupönen being almost literally on the Russian border. Later, around Tannenberg and the Masurian Lakes to the south and southwest, another string of even larger defeats would be inflicted on the Russians. In the fighting near Gumbinnen, the 1st Cavalry Division made a measurable contribution. Though they had sometimes failed to provide accurate intelligence of the Russian advance and been dismissed by the infantry as "frog stickers" because of the lances they still carried, the cavalrymen redeemed themselves. Flanking the Russians in good cavalry fashion, the German horsemen broke clear and played havoc with the Russians' logistics and lines of communication. Having already served with the frontier defense ( _Grenzschutz_ ) before its parent Eighth Army was activated, the 1st Cavalry Division had earlier fought at Stallupönen. Now, near Gumbinnen, it was in its element against a large but lumbering opponent advancing into the acute angle formed by the Gumbinnen-Stallupönen railway line and the River Inster. This opponent was the Russian Imperial Guard Cavalry Corps under the command of the Khan of Nakhitchevan. It had the mission of securing the Russian right wing. Fought to a standstill by German infantry and artillery around the village of Kaushen, the Russian cavalry faltered, and a gap opened in their front. Into that gap plunged the 1st Cavalry Division. The German horsemen broke through, and the ride was on—fully 120 miles (190 km) behind the Russian lines in barely three days' time. It was a cavalryman's waking dream for the division's older officers. The division's commander, General Brecht, had entered the Prussian army in 1867, and two of his brigadiers were well into their fifties. Still, the advance occurred in a fashion never duplicated on the Western Front after the first Battle of the Marne. It also created panic in General Rennenkampf's headquarters. Proving themselves generally better horsemen than their Russian counterparts, the division's troopers moved so far so fast into the Russian rear that they lost contact with their own forces. Consequently, the cavalrymen initially failed to get the subsequent orders for the great redeployment southwestward toward Tannenberg. As that redeployment got underway, however, the division was eventually given the cavalry's other great task: to screen and protect the German movement and prevent the Russians' taking advantage. Despite exhausted mounts, insufficient water, and reduced fighting strength, the horsemen had to harass and confound the Russians to keep Rennenkampf's army from coordinating with General Alexander Samsonov's to the southwest while the Germans pounced on the latter. Even though Rennenkampf continued to advance slowly but successfully toward Königsberg, the 1st Cavalry Division nevertheless managed repeatedly to put itself in the Russians' way. Most importantly, this one cavalry division succeeded in frustrating the larger objectives of an entire enemy field army.
By stunning contrast, the Russian cavalry, three divisions strong between Gumbinnen and Tannenberg, not only failed to take any effective part in the former battle but also failed to exploit the real advantage of its own larger numbers in the latter. Nevertheless, and not a little unusually, it was the Russian 1st Cavalry Division that remained in constant reconnaissance-contact with the German 1st Cavalry Division's horsemen and accompanying bicycle-mounted infantry, and that over a frontage of thirty-five miles. Thus the battles in East Prussia in August and September 1914 not only served to maintain the apparent viability of the German cavalry. They also had a much greater resonance, for they helped propel General Paul von Hindenburg and General Erich Ludendorff to the eventual supreme command of the German armed forces. These victories were ones that, according to one subsequent newspaper account, would for years haunt the children and the grandchildren of the Russian soldiers who'd been so thoroughly defeated there.
Somewhat later, in November 1914, several German cavalry divisions also played prominent roles in the German Ninth Army's offensive into Russian Poland along a line stretching roughly northeast from Posen to Thorn. Aimed at the juncture between the Russian First Army and its neighbor to the southwest, the Second Army, the German offensive intended to relieve pressure on Austro-Hungarian forces to the south and simultaneously to forestall an impending Russian campaign aimed at the rich industrial region of German Silesia. While the Germans' III Cavalry Corps stood in reserve and helped screen the southern end of Ninth Army's line, the I Cavalry Corps comprising the 6th and 9th Cavalry Divisions had been assigned a more active role. Along with the accompanying 3rd Guards Infantry Division, the I Cavalry Corps had the mission of supporting Ninth Army's broad southeasterly advance through the central lowlands along the left bank of the Vistula toward the Polish city of Lodz. Between 11 and 16 November, the Ninth Army, supporting the XXV Reserve Infantry Corps on the right wing of the German advance, covered more than fifty miles (80 km). On 17 November cavalry and reserve infantry were ordered to completely envelop Lodz to the south and west with attacks toward Pabianice. In doing so, they threatened the entire Russian Second Army at Lodz with encirclement and destruction. Unfortunately for the Germans, the Russian Fifth Army executed a heroic march northward to Lodz's relief—two of the Russian infantry corps marched more than seventy miles (112 km) in forty-eight hours—and forced the German cavalry and reserve infantry to fight their way out the way they'd come. While the Russians could claim a victory in saving Second Army from destruction, the Germans could equally claim that Silesia had been preserved from invasion. In that strategic victory the horsemen of the I Cavalry Corps had played no mean part.
In 1915 the cavalry again played a significant role in a major German victory, this time in Lithuania. Having driven the Russians out of East Prussia at the beginning of the year in the Winter Battle of the Masurian Lakes, German armies joined with their Austro-Hungarian allies to expel Russian forces from almost the whole of Poland in a gargantuan offensive during the spring and summer. These offensives included the dispatching of a strong cavalry force into Courland (Latvia) toward Riga in April and May as part of Army Group Lauenstein (later redesignated the Niemen Army after the river of the same name). The cavalry moved ahead with orders to destroy the Russian railways wherever the horsemen found them. Near the town of Mitau (Jelgava), the German riders captured a baggage train, ammunition wagons, and machine guns. To the south, they also cut the Russian railway on both sides of the junction at Shavli (Siauliai) before falling back temporarily. This ride was followed up in early September with a drive farther to the southeast toward Kaunas (Kovno) and Vilnius (Vilna). Three German cavalry divisions participated in this attack on Lithuania's two largest cities. In this offensive, begun on 8–9 September, the German horsemen supported the advance on Grodno, cut the Russian railway linking Vilnius and Riga at Sventsiany, and raided into the Russian rear areas as far as Molodechno and Smorgon, though the Russians subsequently managed to drive them and other German forces back and thus avoid encirclement. Indeed, the first German troops to enter Vilnius were the troopers of the _Death's Head_ Hussars who reminded one native of the Teutonic Knights of five hundred years before, but without the cross.
Similarly, in Rumania in 1916 German and German-led cavalry again had a prominent part to play in a significant victory. In the immediate aftermath of Rumania's declaration of war on the Central Powers in August 1916, Rumanian offensives had not only gained the passes of the Transylvanian Alps but also the easternmost portion of the Great Hungarian Plain. Anticipating such a Rumanian invasion, however, the German and Austro-Hungarian governments, supported by a willing Bulgaria, had already planned an invasion of their own. This took the form of a combined counteroffensive starting on 18 September to drive the Rumanians out of eastern Hungary. That successful effort was followed by a push across the Transylvanian Alps into both Moldavia and Wallachia by German and Austro-Hungarian forces, as well as an invasion across the Danube by German and Bulgarian troops into the southern Dobrudja. Pushing the Rumanians back through the Vulcan, Red Tower, and Predeal Passes, the flank of the descending left hook of German general Erich von Falkenhayn's Ninth Army was covered in part by a mounted corps. On 10 November the force began its advance down the Jiu Valley and into the lowlands of Wallachia north of the Danube. This region of Rumania constitutes the southwestern extension of the Black Sea or Pontic Steppe, a vast, undulating grassland interspersed with trees and stretching all the way to the Volga. In many respects, it was ideal horse country for the cavalry, at least as good as the Polish plains around Lodz. By 21 November the advancing German horsemen and infantry had covered the more than sixty-two miles (100 km) to the important rail junction of Craiova, which quickly fell to the Germans. By 26 November, the German horsemen and infantry had advanced another thirty miles (48 km) and captured the one remaining bridge over the Aluta River (at Stoenesti) not destroyed by the retreating Rumanians. They thereby helped open the way for the drive on Bucharest. They also once again demonstrated the cavalry's utility on the Eastern Front in a fashion impossible in France.
In spite of these successes, however, the Rumanian forces in Wallachia southwest of the capital managed to launch a fairly strong counterattack on 1 December against Falkenhayn's forces and those of General (and _Death's Head_ Hussar) August von Mackensen attacking from below the Danube. Here, too, however, the German cavalry made a signal contribution. To help stem this Rumanian counterattack, Falkenhayn dispatched a combined cavalry-infantry force against the right wing of the Rumanians. The horsemen and their accompanying infantry struck the right flank of the Rumanians, broke through, and got into their rear areas. In true cavalry fashion the German horsemen set about sowing confusion and inflicting heavy casualties on the Rumanians. As a consequence, they created a sense of panic that forced a Rumanian withdrawal. Bucharest fell shortly thereafter, and the Rumanians evacuated the whole of the Dobrudja. Those Rumanian forces still holding the lines in the great bend of the Transylvanian Alps were thus threatened with being cut off from the south. As a result, their position became untenable, and they too were forced to retreat into Moldavia. The setting in of heavy winter rains and snow, however, prevented the Germans from pursuing their defeated enemies. The year 1916 ended with the Rumanians holding a rump territory in Moldavia adjoining the Russian frontier along the River Pruth. Nevertheless, the strategic victory to which the cavalry had contributed its fair share was enormous: Rumania was effectively knocked out of the war; German and Austro-Hungarian forces were released for service on other fronts; and, as in another war one-quarter century later, Germany now enjoyed unfettered access to large reserves of foodstuffs, oil, and other war matériel, including much needed horseflesh.
The enormous haul of goods resulting from the eastern victories of the years 1915 to 1917 was only reinforced in early 1918 by the Treaty of Brest-Litovsk, which the Central Powers (read, Germany) imposed on a Russia already undone by revolution. Whatever else it did, the treaty brought to Germany a seemingly immeasurable area of conquest stretching away to the east and southeast. Included was the bulk of the Black Sea Steppe, while from the newly occupied Ukraine alone "Germany...obtained 140,000 horses during the war." Bearing in mind that the Ukraine really only fell under German occupation as of March 1918, and that the armistice in France brought the fighting officially to a halt in November, the Germans' requisition-process was harsh indeed but necessary in any case. General Erich Ludendorff evidently thought so. Remarking on the acquisition of horses in the newly occupied eastern lands and the protection of that resource by German troops, he said pointedly that Germany could not carry on the war on the Western Front without the horses from the Ukraine. Be that as it may, Germany's armies were nonetheless defeated. However unwillingly, Germany was eventually forced to relinquish all of her conquests and a great deal more once the Allies delivered their own punitive settlement, the Treaty of Versailles.
As regards what was still then called the Great War, one may make several observations about the German cavalry between 1914 and 1918. First of all, by 1914 the German army had more or less taken to heart the elder Moltke's admonition about the cavalry's real mission in an age of increasingly deadly infantry and artillery weapons. That is, the German cavalry had finally come to rely primarily on its own firepower, combined with its inherent mobility, rather than hard-charging speed for battlefield success. At the war's beginning, German horsemen still trained with and carried swords and lances, just as did many of their European counterparts. Nevertheless, they also trained with firearms and used them effectively. Consequently, German cavalrymen could realistically be used as infantry, even if they were only lightly armed. This was precisely the role they played on the Western Front. Against British and French troops in Flanders, the horsemen of Marwitz's cavalry corps found themselves in the trenches alongside their infantry brethren for various lengths of time; and, as noted above, it was precisely that service which was publicly recognized by the Bavarian Crown Prince in October 1914. This fact was perhaps indicative of their future role. That is to say, they would not be full-fledged infantry but neither would they be solely blade-wielding riders. In 1915 or 1916, particularly on the Eastern Front, they still looked quite a lot like their forebears of 1870. Nevertheless, they were already anticipating their descendants of the 1920s and 1930s.
This doctrinal and organizational evolution already showed itself in the German cavalry's organization going into the war of 1914. German (and British) cavalry regiments, for example, were routinely attached to infantry divisions and provided close reconnaissance and communications. Independent cavalry divisions could be attached to infantry corps for the same purposes at that level, and entire cavalry corps of two or three divisions each could be assigned to an army's commander in chief for the vital mission of "deep" reconnaissance and operational exploitation of the enemy's flanks and/or lines of communication. One sees the successful execution of the latter types of missions at Lodz in late 1914 or in Rumania in 1916. Of course, the classic screening function of the cavalry did not disappear. Arguably the best example on the German side was the 1st Cavalry Division's efforts in East Prussia between the battles of Gumbinnen and Tannenberg in late summer 1914.
Enhancing these missions' effectiveness was the fact that the German cavalry divisions between 1914 and 1918 were actually combinedarms units of a sort that would be resurrected after 1919. Each division included at least one _Jäger_ battalion, and sometimes as many as three, providing infantry firepower and a crucial maneuver element. A typical cavalry division also included horse-drawn automatic weapons, three horse-artillery batteries of four guns each, pioneers (i.e., combat engineers), a signals detachment, and a truck column. Provided that all elements were up to establishment—always a crucial consideration—the German cavalry of the war of 1914–1918 was already a far cry from that of the Franco-Prussian War. While certainly not as strong in manpower or firepower as the infantry divisions of the era, the German cavalry showed that they could hold their own when pressed in an infantry fight and, when able to operate in the open on the Eastern Front, could still make a considerable contribution to victory and more than justify their existence. What, then, of swords and lances? Many small engagements occurred on both fronts, especially in the early days of the war, when edged weapons and couched lances were employed, and not only by German horsemen. Nevertheless, the storied steel-on-steel, horse-on-horse collision of cavalry formations was almost entirely a thing of the past. But even in this respect there were exceptions, the most notable of which didn't involve German cavalry at all. In August 1914 more than two thousand Austrian and Russian cavalry fought an old-style cavalry battle near Jaroslawice in the borderlands of Austria's province of eastern Galicia. This was a day-long slugging match whose participants witnessed charges and counter-charges at the gallop, saber-swinging melees, and attacks with couched lances not only against other cavalry but also against machine guns and artillery. The fighting at Jaroslawice did not notably affect the course of the war on the Eastern Front and was, by its very nature, something of an anachronism. Indeed, it was not noticeably different in spirit from the cavalry battles between the Germans and the French around Mars-la-Tour and Vionville in 1870. Anachronism or not, however, it long remained in the memories of those who fought it and managed to survive.
The German cavalry's fate, like that of German army as a whole, was sealed as the war became a _Materialschlacht_. Buttressed after April 1917 by the enormous and untapped resources of the United States, Germany's European enemies were assured of material reinforcement that the Central Powers could not hope to match. Not even Russia's coincidental revolutionary collapse that same spring affected this calculation. The only real hope for the Germans was to win the war in the west, the decisive front, before the Americans really began to arrive in overwhelming numbers. The result was Operation Michael, the great German spring offensive of 1918. Unlike the Germans' opening campaign four years before, the so-called _Kaiserschlacht_ was an affair for the infantry and artillery only. The cavalry had no real role at all, even though the cavalry corps' original commander on the Western Front, General Georg von der Marwitz, commanded the Second Army. But even if the German cavalry played no great part, neither did their eventual successor, the armored forces. The Germans had only ten A7V medium tanks assigned to the battle as against the Allies' eight hundred machines. Despite this almost total lack of horse- or track-mounted units, the spring offensive initially made good ground. Between the offensive's start in March and the end of June, German forces at various points from Flanders to the Marne had once again reached many of the battlefields of September 1914. As in that year, Paris was once again taken under fire by modified German naval guns. After four years of supreme effort, however, the German army simply no longer possessed the reserves of manpower, material, and morale to seal the victory. By the middle of August 1918, the tide on the Western Front had turned irreversibly in the Allies' favor. By November the German forces were in full but orderly retreat. Ominously for the army's fate, however, the government in Berlin had fallen, and a revolution of sorts was under way. A new, republican Germany would have to make whatever peace it could and save whatever could be saved.
How those matters would turn out, no one could really tell as the German forces marched back across the Rhine that autumn. Still, there were indications in the armistice that any eventual peace treaty between Germany and the Allies might well be harsh. Regardless of whether and in what strength the cavalry might still be found in the postwar German army, the mounted units returned to their peacetime quarters with their pride largely intact. They had on the whole conducted themselves honorably and well, if not always with distinction. As winter 1918 came on, there seemed no immediate reason for them to assume that the German horse-mounted soldier would disappear, though he might well soon serve in fewer numbers.
The tributes paid to the returning soldiers were many and fulsome. In Germany that was to be expected, despite, or perhaps because of, the quasi-revolutionary turmoil in Berlin and elsewhere. More interesting was an equally laudatory tribute paid to the German cavalry by a man who was not merely one of Germany's enemies but a Marshal of France, Henri-Philippe Pétain. In an order of the day dated 1 January 1919 on the occasion of the disbanding of the Cavalry Corps of the German army, he credited his erstwhile enemy's horsemen with having made a significant contribution to the Germans' striking success on the Western Front in 1914. "Thrown in again and again on our left wing," he noted, "they successfully lengthened [the front] from the Aisne to the dunes [of the North Sea], during which they anticipated the adversary and permitted the timely deployment of their own infantry." Such an endorsement from such a committed enemy of Germany said much about what Pétain thought of the German cavalry's wartime performance. Regardless of the static situation into which the Western Front eventually devolved, the German mounted formations had continued to bolster the German war effort, particularly on the Eastern Front. Cavalry traditionalists might well bemoan the absence of regular opportunities for the massed charge by war's end; and they did. Nevertheless, when undertaking missions within the cavalry's capabilities, German horsemen (and sometimes British and French, frequently Austrian and Russian) demonstrated that they could still be effective. They showed as much before and after Gumbinnen, in the invasion of Courland, in the campaign against Kaunas and Vilnius, and in the conquest of Rumania. In all of these operations they showed much of the same skill and tenacity as their ancestors of 1870–1871. Notwithstanding the technological changes on the battlefields of World War I, the assumption, albeit fading fast, lingered among some officers that the massed charge with cold steel might just still be possible and necessary under certain circumstances, however unlikely those circumstances might be. Such an attitude did not constitute mere intellectual blindness and doctrinal pig-headedness. Survival in combat, any combat, could sometimes be ascribed to an act of faith. For infantrymen and artillerists, or, later, airmen and tankers, believing that they would not only survive but win sometimes produced precisely that result in the face of apparently certain destruction. It was no different for cavalrymen. Particularly when, as in 1918, the technology did not yet exist that could definitively relegate them to the status of museum-pieces, the horsemen could still envision for themselves a place on future battlefields. Ultimately, the key to the German cavalry's—to any cavalry's—survival going into the post-1918 period lay in such an act of faith. It lay, in other words, "in combining the mental attitude which would encourage horsemen to take advantage of fleeting opportunities with the recognition that fire kills."
CHAPTER 4
FALSE DAWN
THE INTERWAR PERIOD, 1918–1933
In the wake of 1918, the horse still loomed large in European society. Despite the increasing mechanization of all European armies during World War I, horses remained important in all military establishments, and of course in civilian society as well. Clear indications of that importance manifested themselves in the treaty of peace presented to Germany at the end of the Paris Peace Conference in June 1919. Annex IV of the Treaty of Versailles dealt, among other things, with specific numbers, and in some cases specific types, of horses and other livestock that Germany would have to surrender and to whom. The French Government, for example, stood to receive 500 stallions of unspecified breed aged three to seven years, along with 30,000 fillies and mares. In contrast to the stallions (though presumably they would be similar), the fillies and mares had to be specifically of Ardennais, Boulonnais, or Belgian stock, all three types being very large draft-horse breeds weighing up to 2,200 pounds (1,000 kg) and standing between sixteen and seventeen hands. Similarly, the Annex called for Germany to surrender to Belgium 200 stallions aged three to seven years and 10,000 fillies and mares aged eighteen months to three years. In the case of Belgium's haul, all of the horses were to be of "large Belgian type." All animals were to be "of average health and condition." In both the French and Belgian cases, these animals were presumably ones that had been confiscated by the German army during the war. However, "to the extent that animals so delivered cannot be identified as animals taken away or seized [during hostilities], the value of such animals [i.e., those delivered by Germany] shall be credited against the reparation obligations of Germany" as detailed elsewhere in the Treaty.
This clear recognition of the horse's continued value to the civilian economy also found a certain parallel in military circles. The assumption that the horse might still play a part in future wars was not uncommon in Europe and elsewhere after World War I. As late as 1926, British general Sir Douglas Haig wrote that he "believ[ed] that the value of the horse and the opportunity for the horse in the future are likely to be as great as ever. Aeroplanes and tanks...are only accessories to the man and the horse, and I feel sure that as time goes on you will find just as much use for the horse—the well-bred horse—as you have ever done in the past." In the post-1918 U.S. Army, too, occasional voices were raised in favor of the horse cavalry. No less keen an observer of the incipient armored idea than George Patton continued his informed speculation after the Armistice that mounted forces still had a role. In the January 1924 issue of the _Cavalry Journal_ he acknowledged the importance of motorization for the future of warfare; but he also continued—as did his German counterparts—to envision a place on the battlefield for mounted forces. Thus he wrote that armored cars should be equipped to operate alongside, but not to replace, horse-mounted troops. "It is the duty of [the horse] cavalry," he said in a speech of the same year to the 11th Cavalry Regiment in Boston, "and should be its pride to be bold and dashing." Not only was such sentiment inherent in Patton's character, but it had also been reinforced by his two earlier stints at the French Cavalry School in 1912 and 1913. Furthermore, like many of his counterparts in Germany and Great Britain, Patton went into the postwar period still advocating the retention of edged weapons for the cavalry, in his case the colloquially named "Patton sword" (U.S. Saber M-1913). Indeed, as late as 1941, Patton responded affirmatively to the Cavalry Board's enquiry as to whether the saber should be retained. "A cold steel weapon," he maintained, "is not only desirable but vitally necessary." Nevertheless, Patton did not advocate retention of the horse to the exclusion of the U.S. Cavalry's mechanization, as did some others. His equine advocacy was driven by practical considerations such as the limits of technology, as well as a by a certain military romanticism.
Beyond such advocacy, and in terms of actual European military operations, cavalry played a prominent role in the brief but large-scale Russo-Polish War of 1919–1920. Hostilities, in fact, began with a Polish cavalry patrol stumbling upon a Russian camp in Byelorussia early on the morning of 14 February 1919. This conflict, which the Germans no doubt watched intently, was a "vast war of movement" with upwards of two million men sweeping back and forth across the plains of east-central Europe in 1920. One Polish cavalry division in the drive on Kiev in April 1920 covered more than 125 miles (200 km) in thirty-six hours. At Koziatyn on the road linking Kiev and Vinnitsa, later the site of one of Hitler's command posts during World War II, Polish cavalrymen captured, among other things, eight thousand prisoners, three thousand railway cars, five hundred horses, twenty-seven artillery pieces, and—oddly—three airplanes. On 5–6 May, other Polish cavalry actually occupied Kiev itself, admittedly in the absence of effective resistance. Despite these serious setbacks at the Poles' hands, the Red Army's cavalry spearheaded a counteroffensive that drove the Poles out of the whole of the Ukraine in May and June, the Bolshevik commander boasting that his horses would be riding through the streets of Paris before summer's end. In this case the cavalry in question constituted the First Cavalry Army ( _Pervaya Konnaya Armia_ or _Konarmia_ I), a massive force of four full cavalry divisions including some 18,000 mounted troopers, 52 field guns, 5 armored trains, 8 armored cars, and even a squadron of 15 aircraft. Possessing a fearsome reputation born of its Cossack regiments and its service against the Whites in the Russian Civil War, _Konarmia I_ operated under the command of General (later Marshal) Semyon Budenney. It considered itself the elite of the Red Army and produced a generation of Red Army officers believing fiercely in the idea of mobile warfare. In the fighting around Kiev in late May and early June 1920, _Konarmia I_ broke through the Polish defenses and raised havoc in raids on Koziatyn, Berdichev, and Zhitomir. Budenney's horsemen wore down the fiercely resisting Polish Cavalry Division reserve in a number of classic cavalry engagements but also fought dismounted in cooperation with artillery and armored cars. Kiev subsequently fell to the Red Army's infantry, and a general Polish withdrawal from the Ukraine followed. By mid-August, Soviet cavalry had reached the bend on the Vistula at Thorn, only some five days' foot-march—and a much shorter ride—from Berlin itself. There followed, however, the storied "Miracle of the Vistula." Beginning on 15–16 August, a Polish counterattack annihilated three entire Russian armies. It also saw what one writer has called the "last great cavalry battle of European history" in the "Zamosc Ring," also known to history as the Battle of Komarow, on 31 August. Some 20,000 horsemen fought here as the retreating _Konarmia I_ attempted to break free of encircling Polish infantry and cavalry in the stretch of forested hill country between the Rivers Wieprz and Bug. Charging and countercharging, the two cavalry forces clashed in Napoleonic style until the Polish uhlans and lancers carried the day and forced the Russians to withdraw. Moving faster than the Poles could pursue, however, large portions of Budenney's army managed to escape destruction and cross the Bug, thus reaching the safety of Soviet-controlled territory.
With such a conflict raging just to the east of post-1918 Germany's frontiers, officers in Berlin could not help but ponder the matter of what military role the horse might yet play. Not surprisingly, in the German army, too, commanders assumed that the horse would still have a place and not merely as a draft animal. The army ( _das Reichsheer;_ after 1935 _das Heer_ ) constituted the largest element of the now famous "100,000-man _Reichswehr_ " mandated by the Treaty of Versailles. Article 160 of the treaty specifically directed that the army possess no more than seven divisions of infantry and three of cavalry. Such a number constituted a tremendous decline in force-structure for an army that had been one of the world's largest between 1914 and 1918. From an already imposing figure of some 840,000 in 1914, the army in 1917–1918 had numbered some six million, a force of 241 divisions deployed across Europe from the Channel Coast to Courland and the Black Sea. However, the treaty not only specified absolute numbers for the post-1918 army. It also implicitly determined what relative place in the army the various arms would have by prescribing that the army "be devoted exclusively to the maintenance of order within the [national] territory and to the control of the frontiers." General Hans von Seeckt, effectively the army's chief of staff in 1919–1920 and its operational commander in chief as head of the General Troop Office ( _Allgemeines Truppenamt_ ) from 1920 to 1926, played a pivotal role in shaping what post-1945 generations of planners would call the army's "vision." In such a capacity, he and his immediate subordinates necessarily confronted the vexing question of whether and to what extent horse-mounted forces could be retained.
Unsurprisingly, the cavalry divisions under Seeckt's overall command did not initially exhibit tremendous capability. As with the army as a whole, the former Allied Powers wouldn't permit that. On the table of organization and equipment (TOE), each cavalry division of 1919 numbered six mounted regiments. Each regiment included four squadrons, a machine gun section, and a signals platoon. In addition to the mounted regiments, the German cavalry's divisional TOE of 1919 also included a pioneer battalion, a signals battalion, and an artillery battalion of three light horse-drawn batteries. Total divisional establishment was 5,300 men. Still, the question remained: what would the cavalry do? In defining the cavalry's mission, Seeckt drew upon not only his considerable intellect but also his own experiences during World War I, and his observations of the immediate post-1918 period. Having served on the immense Eastern Front, largely devoid of the fixed entrenchments so characteristic of the war in France and Belgium, he appreciated firsthand the continuing, crucial importance of an army's mobility. Whereas the fighting in the west had become largely immobile after September 1914 and was destined to remain so until at least the Battle of Cambrai in November and December 1917, campaigns in the east throughout the war had regularly ranged over scores and hundreds of miles. Generals on the Western Front, particularly Allied generals, had simply not been able to make use of the cavalry forces at their disposal. In the east, generals on both sides never suffered such a restriction. Seeckt therefore retained not only a keen interest in the mobile operations arising from such physical circumstances but also how to execute those operations in a German army functioning under the limitations imposed by Versailles. The upshot of his considerations was that, after 1918, he brought to his new responsibilities a continuing interest in maintaining a strong cavalry component in the army.
Seeckt thought that the cavalry's equine horsepower might still have a place in a world of internal-combustion engines. He reasoned, quite naturally, that the massed charge with cold steel had in all likelihood become irretrievably a thing of the past. In this he shared the presumption of certain other Prussian and German military leaders going back at various points to Helmut von Moltke the Elder and his assessment of the Austro-Prussian War. Even before the war with France in 1870, Moltke had concluded that the cavalry's primary (and perhaps only) real functions would be reconnaissance, screening, security, and pursuit, but always ideally as part of a combined-arms force. As we've seen, those were precisely the German cavalry's roles in 1870–1871, notwithstanding the "Death Ride" at Mars-la-Tour. Again, in 1914–1918 the German cavalry had assumed largely the same missions and, when actual combat occurred, often fought dismounted.
Since 1914, however, European armies had experienced dramatic technological and, in some quarters, organizational changes. Motor vehicles were crucial to these changes but so were weapons possessed of ever-longer ranges and ever-greater lethality. Seeckt nevertheless thought that horse-mounted cavalry, as part of a combined-arms light division, were still capable of executing independent operations. They could play a useful part, he thought, at the level of the military art above the tactical engagement but below the strategic outcome in missions as specified by Moltke. Presuming their own incorporation of newer weaponry and vehicles, horse-mounted troops could even still be a force capable of tactical success at the small-unit level. When they fought, they would do so as infantry, but they would require more infantry combat power to be truly effective. In such light divisions, sometimes also referred to as mixed divisions, cavalry would therefore require the added firepower of organic infantry assets, notably absent from the cavalry's TOE of 1919. That infantry, in turn, would require bicycles, motorcycles, and/or motorized transport in order to keep pace with the horsemen. Trucks and bicycles, however, would have only limited cross-country capability. They were bound to the roads, whether paved or not. Further, since the proposed divisions' objectives would be varied and reached only over long distances, the cavalry would also need mobile artillery, whether horse-drawn or motorized; effective wireless communications; wheeled armored cars; and, perhaps, tanks. Foreseeing things to come, Seeckt also considered cooperation with the air arm to be of particular importance to the cavalry's effectiveness. Therefore, he even advocated placing aerial units under a cavalry division's command. The airplane's primary value to the cavalry lay in its function as a reconnaissance platform. It could extend the cavalry's line of sight by flying over the battlefront well beyond the cavalry troopers' reach. Significantly, however, neither Seeckt nor his Inspector of Cavalry, Lieutenant-General Maximilian von Poseck, saw the airplane replacing the horsemen. On the contrary, Seeckt, Poseck, and others envisioned aircraft as supplementing the cavalry by providing a true over-the-horizon reconnaissance capability. Absent effective air-to-ground radios, however, and in light of the airplane's relative frailty and limited endurance even in the early 1920s, not to mention the vagaries of the weather aloft, the cavalry's champions refused to see aerial observation as inevitably relieving the horseman of the important reconnaissance role.
Indeed, as late as 1930, Seeckt would write that neither aircraft nor motorization had made the cavalry irrelevant. "The solution to the problem [of the cavalry's role]," he wrote, "lies...in making full use of the products of technical science to extend and modernize what already exists, but not by substituting something dead for something alive. The living arm, i.e., our cavalry, should be developed to its fullest perfection on modern lines without loss of its characteristics." Of the latter, the essential one was the cavalry's inherent mobility. That mobility, when combined with newly mobile artillery and motorized infantry, he maintained, still permitted the "modern Seydlitz" not only rapidly to outflank the enemy but now materially to contribute to the "annihilating victory which is the aim of all military thinking."
Notwithstanding becoming proficient with new weapons and vehicles, the cavalryman's real training lay in military equitation and daily horse-care. Otherwise, said Seeckt, all one ended up with was a "mounted yeomanry." Poseck emphatically agreed. He wrote that mounted infantry were nothing more than riflemen riding badly. Their inability at horsemanship would only nullify their mounts' most important advantage: mobility in pursuit, retreat, and surprise attacks on the enemy's flanks and rear. On the contrary, the post-1918 cavalry had to ensure that the trooper always remained not only a good marksman but also a wellschooled rider. Interestingly, Poseck drew upon the post-1918 French cavalry regulations when he affirmed, as they did, that the modern cavalryman had to have all of his military ancestors' skills in equitation while remaining the equal of the infantryman in dismounted combat. The ideal result would be to exploit the horse's natural strengths in effectively bringing modern weapons to bear without the trooper's running the creature into the ground. That could only come from combining firearms training as good as the infantry's with proper equitation, stabling, and regular and competent veterinary care. Thus properly trained, equipped, and mounted, the cavalry's roles would include frontier defense; screening the advance; intelligence-gathering; reconnaissance; dislocation of the enemy's lines of communication and supply; and constant probing for a weak flank to turn or attack.
Certain specific elements of the cavalryman's training aimed at enabling him to accomplish these varied missions. Before 1935, for example, horsemen of the German army were expected to receive as many as three thousand hours of equitation over the course of the twelve-year enlistment specified for common soldiers in the Treaty of Versailles. River crossings also played a major role in the cavalry's training both with and without the support of combat engineers. Consequently, all mounted regiments were expected to conduct at least one large-scale river-crossing exercise per year. Despite such rigorous requirements, and somewhat surprisingly, most personnel in the cavalry and artillery evidently did not earn the badge constituting the outward recognition of the horseman's skill. The Horseman's Badge, depicting a horse and rider performing a classical _levade_ facing the viewer's left and wreathed in oak leaves, was presented by the National Association of Breeding and Testing of German Warm Bloods and worn on the left breast pocket of the uniform blouse. Awarded in bronze, silver, and gold, the Horseman's Badge recognized the recipient's relative level of riding skill and theoretical knowledge on the basis of civilian and military tests. Cavalry and artillery officers holding the rank of captain ( _Rittmeister_ in the mounted regiments and horse-artillery; _Hauptmann_ in the horse-drawn field artillery) were expected to pass the required examinations. Many serving officers in these groups, however, tended not to wear the award even if they'd earned it. Apparently, they considered proper equitation and theoretical knowledge of their horses to be a given in their profession of arms and consequently didn't feel it necessary to wear an outward civilian expression of the fact.
Supported thus by Poseck and others, with a suitable training syllabus in hand, and with a clearly self-confident officer corps at his disposal, Seeckt's considerations resulted in a revised TOE for a cavalry division. The revision appeared in the Army Service Regulation ( _Heeresdienstvorschrift_ ) of 1923 and proposed a significantly different organization from that of four years before. As in 1919, the number of mounted regiments remained at six, but they were now to be grouped in three brigades of two regiments each. Internally, each regiment also now possessed a machine gun company instead of the earlier, smaller machine gun section. To increase each regiment's long-range hitting power, a section of two horse-drawn guns was added. It was other additions to the cavalry division, however, that transformed it in keeping with Seeckt's and his supporters' ideas. The division now also included a separate infantry battalion, a bicycle battalion (of three bicycle companies and two motorized anti-tank batteries of three guns each), and a machine gun battalion. This was the equivalent of adding a full regiment of infantry. Poseck called this addition of "chasseur" or infantry battalions to the cavalry "exceptionally useful." This usefulness, he maintained, had been amply demonstrated between 1914 and 1918, particularly when the infantry was truck-mounted so as to be able to intervene early in an encounter battle. He felt the same could be said of the bicycle-mounted infantry, though he admitted that such had not been the case on the Eastern Front with its paucity of paved roads. The divisional artillery was also augmented to the level of a regiment through the addition of a second, fully motorized battalion of twelve guns. To provide protection against aerial attack, a motorized flak battalion of four batteries was also added. Here again, Poseck's views supported Seeckt's. Poseck maintained that the war of 1914–1918 had clearly shown the importance of the "loyal fraternity of arms" existing between cavalry troopers and the horse-artillery, which had fulfilled all the demands made upon it. He wrote that increasing the number and weight of the cavalry's field pieces and adding anti-aircraft guns would only deepen this fraternity. Similarly, he believed that continuing technical improvements to the cavalry's carbines, machine guns, and artillery, when combined with better "fire training," would steadily increase the cavalry's fighting power just as had been the case in other arms. This fighting power, in turn, would "more surely enable the cavalry divisions to overcome enemy resistance and thus take full advantage of their horses' legs." A battalion of twelve armored cars also found its way into the new TOE, and even a squadron of twelve observation aircraft and a motorcycle platoon were attached to the divisional headquarters. Rounding out the division's strength were the less glamorous but nonetheless critical support elements: a pioneer battalion, a signals battalion, a truck-transport battalion, a horse-drawn wagon battalion, a medical battalion, and a veterinary detail. To the extent that trucks and wagons maintained the cavalry divisions' logistical independence, Poseck deemed them "absolutely indispensable" based upon the lessons learned from World War I.
This organization reflected the experience gained from the occasional attachment of infantry and armored cars to cavalry units in the German Army on the Eastern Front during World War I, a practice Seeckt would have seen firsthand. Interestingly, the Rumanian army had effectively employed cavalry and armored cars together against the Germans in 1916. For their part, the Germans in Rumania had also used armored cars in independent units in the same theater. Meanwhile, in the Baltic States in spring 1919, several _Freikorps_ formations using trains, armored cars, and truck-mounted infantry had executed wide-ranging movements to good effect against the Bolsheviks. While the addition of such motorized units would increase the cavalry's mobility, the doubling of divisional artillery and the addition of large combat-support and logistics elements theoretically permitted the cavalry divisions to execute truly independent operations. The doctrinal development of this new organization would be under the aegis of the Cavalry School in the city of Hannover. Eventually, in 1937, the school would be transferred to Krampnitz just outside of Berlin as the army expanded rapidly under the Nazi regime. The relocation would put the Cavalry School not very far from one of the most famous battlefields and one of the most famous homes in Prussian military history. To the northeast of Krampnitz lay the neighboring towns of Fehrbellin and Wustrau. In June 1675 Fehrbellin had been the site of a victorious battle against the Swedes, an event sometimes regarded as the birthplace of the Prussian army. Interestingly enough, that victory under Elector Friedrich Wilhelm was fought and won almost entirely by the cavalry. For its part, Wustrau was the ancestral home of the Ziethens. Field Marshal Hans Joachim von Ziethen, "Papa" to his men and "ancestor of all hussars," had been one of the most famous cavalrymen in German military history and had commanded the Prussian Life Guard Hussars during the Seven Years' War. No German cavalryman worth his salt could overlook these associations. Be that as it may, even while the Cavalry School was still located at Hannover, the academy's location was not accidental. The Hannoverian State Stud at Celle and its branches in the adjoining regions had been producing high-quality horses for the army and for German agriculture for many generations. Indeed, one of the great European breeds, the Hanoverian, takes its name from that very locale. Therefore, in terms of both personnel and horses, the needs of the army's new-style cavalry divisions would be enormous, given that each mounted regiment alone required more than a thousand mounts, not counting the parent division's needs for horse-drawn transport.
The Cavalry's Role before 1933
Prior to 1933, the cavalry's training was predicated on Seeckt's original idea that the army should be a relatively small, professional force armed with the most modern weapons available to Germany under the restrictions imposed by the settlement of Versailles. While Seeckt initially envisioned an army of 200,000 to 300,000 men, the post-1918 terms limited the force to the ten divisions noted above. The manpower total of some 100,000 officers and men was well below the numbers Seeckt and others deemed useful. Consequently, the army's role had to be altered. By about 1921, Seeckt came to view the army as a sort of cadre serving a dual function. Its primary mission would be to defend the Reich and act as a military striking force. Its secondary but not unimportant function would be to serve as a foundational element for a rapid expansion to a force of twenty-one divisions. Seeckt viewed the latter strength as the minimum necessary to protect Germany from her putative continental enemies.
At the time of Seeckt's command of the army and for years thereafter, those assumed enemies did not include the Soviet Union or Great Britain. Instead, they were limited to Poland and France and perhaps Belgium and Czechoslovakia. German animosity toward Poland throughout the 1920s remained visceral. Arising in the aftermath of Germany's defeat in 1918, Poland had been re-created by the victorious Allies at the Paris Peace Conference. Large swathes of Germany's eastern territories had been ceded under pressure to help constitute the new Polish State, with Poland's eastern provinces coming from Russia. The surrendered German territories included West Prussia, Posen, and portions of Upper Silesia. The city and immediate hinterland of Danzig were also lost. Though not given to Poland, they were administered as a "Free City" under the aegis of the League of Nations. Furthermore, as a result of these cessions, East Prussia was cut adrift from the rest of the Reich and was now separated from it by the so-called Polish Corridor. On Germany's western frontiers, the Saarland, like Danzig, had been placed under the supervision of the League, and the French had been given treaty-based rights of extraction of the region's mineral resources. To the northwest of the Saarland, beyond Luxemburg, the districts of Eupen and Malmedy had gone to Belgium. Adding insult to injury, French, British, Belgian, and American troops had occupied the whole of the region west of the Rhine from Karlsruhe to the borders of the Netherlands. In these Rhenish territories, though they remained integral to the Reich, Germany was forbidden to maintain military forces or build fortifications. To cement these various changes and restrain Germany further, the French Government entered into a number of diplomatic and military alliances after 1919. Of greatest concern to someone in Seeckt's position, as well as to those who commanded the army after 1926, were alliances between France and Belgium in 1920 and between France and Poland in 1921. These, then, were the geostrategic facts confronting the army as it rebuilt under Seeckt's leadership. These, too, were the conditions for which the cavalry trained.
That training centered, of course, on the cavalry's ability to move. As has been noted, Seeckt and others recognized that mobility would be the key to the cavalry's future military viability, provided that motorization and mechanization did not make horsemen superfluous in the interim. Seeckt maintained that the future of armies, of warfare itself, lay in the employment of relatively small, highly mobile, high-quality forces capable of simultaneous mobilization of the whole for either offensive or defensive action. In other words, both tactical and operational mobility were necessary for Germany's military, and therefore political, survival. The high degree of mobility he demanded would be achieved through "numerous and highly efficient cavalry, the fullest possible use of motor transport, and the marching capacity of the infantry." Of course, Seeckt also demanded that the most effective arms be provided and that replacements of men and matériel be continuous. Herein, however, resided what would turn out to be an acute problem for Germany's war-making potential, though the issue was moot in the 1920s. Given modern training, equipment, and leadership, Seeckt saw no reason why the cavalry's days in the army should necessarily be numbered. Any war against the Polish enemy would necessarily be a war of movement. Conversely, a simultaneous or near-simultaneous war with France would be more static and defensive in Seeckt's estimation, at least until the outcome in the east became clear. Thus, the mobility provided by horse-soldiers was still an important element in a planning and training formula that by the mid-1920s regularly included bicyclists, motorcyclists, armored cars, trucks, and early types of tracked vehicles.
To a certain extent, this thinking embodied what might be called the lessons learned from the already-mentioned war of 1920 between Poland and Russia. Though brief, that conflict saw not only the widespread employment of massed cavalry formations but also extensive use of armored cars of varying types. They included proper armored cars made by Austin, Ford, and Renault, but also mere lash-ups constructed by putting plate-armor on converted Fiats (on the Polish side) and Putilovs (on the Russian). In fact, Poseck himself noted the effectiveness armored cars had already shown in sometimes hampering the German cavalry's reconnaissance during World War I. "They [armored cars] proved," he wrote, "that their adoption by us would be very much worthwhile." To the extent that the Poles' and Russians' experiences with armored cars in 1920 became widely known, they would naturally have reinforced his assessment. By contrast with armored cars, however, the tanks used occasionally by the Poles in 1920 were mechanically unreliable, prone to breakdown, and on more than one occasion actually had to be rescued by horse-mounted troopers.
It was with these strictures in mind that the army's cavalry trained throughout the 1920s. Firearms had been part of the cavalry trooper's standard equipment since before World War I, and fighting dismounted had been included in the cavalry's training since the same period. Nevertheless, the army still issued sabers to cavalrymen in the 1920s and even lances until 1927. Notwithstanding Seeckt's thoughts about the massed charge's being a thing of the past, the attraction to cold steel remained strong among the cavalry's senior regimental commanders. According to one authority on the interwar army, all of the latter insisted on the lance's retention despite many junior officers' openly expressed doubts about its usefulness. Similarly, equitation continued to be stressed in troopers' training, both in close formation and in open order. Such formal riding instruction constituted the equivalent of the infantryman's regular drill and ceremony and remained natural to the cavalry for routine movement, despite the recognition that massed mounted attack in formation would likely not recur. Indeed, as early as 1914, brigade and regimental columns had been replaced by the column-of-twos as the cavalry's preferred formation on the move. Also known as the double column, it remained the standard formation throughout the period from 1919 to 1939, although regimental columns were still used in fieldparades and reviews. The principal advantage of the double column and its even narrower variant, the platoon column, was that it reduced a cavalry unit's front while on the march. In the case of the platoon column, for example, the column's front was ideally only about fifteen feet (5 m). In turn, the reduced front allowed the marching cavalry column to make better use of topographical features such as defiles, swales, or forested paths for cover from hostile fire and aerial observation. The double column would continue in use during World War II. The relative traditionalism of the cavalry's training in equitation might in part be explained by the fact that the training manual in use before the middle of the decade had been issued in 1912. That manual was not replaced by a new edition until 1926. Formation riding continued in the training syllabus thereafter for the very simple reason that cavalrymen aren't cavalrymen if they can't ride tolerably well for prolonged periods. Even if they were now almost always to fight dismounted, they'd still have to use their mounts as the means to get to the battle. New training manuals would not change that fact. As a concession to modernization's inevitability, however, Seeckt's replacement as the army's commander in 1927, General Wilhelm Heye—an infantryman—ordered the lance's retirement. The saber, however, remained standard issue as late as 1941.
Cavalry units took full part in the army's major exercises of the mid-1920s, as commanders wrestled with how to include horse-mounted troops. In 1926, for example, elements from all three of the army's cavalry divisions—including a full-mounted brigade—participated in the maneuvers in Group 1, which encompassed East Prussia. In the Group 2 exercises in September of the same year in southern and western areas of the Reich, cavalry again took part. In both sets of exercises, "tactical maneuver and movement were stressed [and] combinations of highly mobile units were tried." Mounted troops were assigned mock tanks in addition to the mobile artillery and reinforced machine-gun elements now carried on the cavalry's TOE. They thus field-tested Seeckt's earlier efforts to maintain the mounted force's mobility while increasing its combat-power. Hearkening back to some of the early advances in the wars of 1870 and 1914, however, cavalrymen did not always ride in mass formations. Instead, they rode in small detachments as screening and reconnaissance assets to the infantry. It cannot be overlooked that this was precisely the function of the German uhlans who had gained such a fearsome reputation over the two preceding generations.
In general terms, the army seems to have been satisfied with the cavalry's performance in the various large-scale maneuvers of the mid-1920s. Tactical adjustments nevertheless occurred as a result of those maneuvers' analysis. Following the exercises of 1927, for example, the Third Cavalry Division's report to Army Headquarters stated bluntly that cavalry combat without tanks was "obsolete." Significantly, however, the same report noted that the cross-country capability of the mock tanks available for the exercises was inadequate to the cavalry's mission. As a result, the vehicles tended to be held back. To remedy the deficiency, Third Division urged the acquisition of sturdier, more capable vehicles for use as mock-tank chassis, in this particular case a machine then being produced by the firm Hanomag. At the time, "these were the only vehicles available with the cross-country capability suitable for cavalry maneuvers." The mid-1920s maneuvers also demonstrated the continuing development of the army's more generalized efforts toward greater motorization and mechanization, a development aimed at least at this stage not so much at abolishing the cavalry as adapting it to new technological conditions. Once again, this evolution reflected Seeckt's earlier efforts.
At this juncture the cavalry appeared briefly on the verge of significant expansion. In 1927 the Interallied Military Control Commission was withdrawn from Germany. This body had been established at the end of World War I for the purpose of ensuring that Germany did not violate Versailles' disarmament clauses. Though never entirely successful in that regard, the inspectors' presence did have the effect of making the army's post-1918 efforts to rebuild more onerous. With the Commission's withdrawal, plans immediately arose to expand the _Reichswehr_ beyond the treaty's limits. An "emergency army" (also called the "A-Army") of sixteen divisions was designated as a sort of way station on the path to an eventual strength of twenty-one divisions. For the A-Army, the Weapons Office, which oversaw arms procurement, recommended increasing the number of cavalry divisions from three to five. This was merely a part of the larger effort as of 1927–1928 to expand not only the army's numbers, but to increase national stockpiles of arms and ammunition for both the army and the navy; increase budgetary authority through greater cabinet-level support; and provide planning guidance for industrial mobilization. Coming as it did from a procurement office known for its innovative approach to technology, such a proposed expansion of the mounted arm can hardly have been an expression of the cavalry's putative obsolescence. Evidently, however, the _Truppenamt_ was not persuaded. It rejected the cavalry's expansion.
Instead, the various cavalry regiments from 1928 to 1933 began to be thought of as adjuncts to the infantry, even if they were not yet actually assigned as such. This process would be both formalized and completed during the army's breakneck expansion after 1933. In the role as it was eventually implemented, the cavalry regiments would no longer be brigaded and then combined into divisions. Instead, with one major exception to be discussed later, they would second ("chop" one would say today) squadron-sized units to the army's infantry divisions. There the horsemen would function as part of the infantry divisions' reconnaissance elements and serve alongside the other arms they'd operated with since 1919 and in some cases before: armored cars, bicycle- and motorcycle-mounted infantry, and combat engineers. Until the early 1930s, the army also envisioned attaching tanks to the cavalry when the latter needed extra firepower to hold positions for follow-on infantry forces. Indeed, from about the time of the maneuvers of 1927–1928, some cavalry officers themselves began to become more enthusiastic advocates for a progressive motorization and mechanization of the mounted arm, a position in step with the views of the then-minister of defense, General Wilhelm Groener, whose term of office had begun in January 1928 and ended in early 1932. Nevertheless, significant opinion still saw a useful place for the military horse in an internal-combustion German army. In the same year that Groener's term of office ended, retired lieutenant general Ernst Kabisch, a military correspondent for the _Kölnische Zeitung_ , wrote to Groener about the cavalry's new capabilities. Citing the military exercises of the autumn of that year, Kabisch maintained that the cavalry had undergone great changes and was no longer what it had been in 1919. It had now, he wrote, become popular among cavalry officers to envision their arm as constituting a force of light divisions. They would retain their horses and the Seeckt-era armored cars and other motorized elements, but they would also now add tank formations. Popular though armored forces were becoming on the eve of the Nazi regime's accession, the tank and the armored car had not yet fully replaced the horse in the cavalry's mind.
CHAPTER 5
THE FIELD OF MARS
CAVALRY EQUIPMENT, HORSES, AND DOCTRINE IN THE 1930s
The characteristic weapon of the German cavalry in the interwar period was the same one with which cavalrymen had been armed for centuries: the saber. The blade was thirty inches (76 cm) long and had a chord or width of one-and-a-quarter inches (3 cm). Normally sheathed in a steel scabbard, the M1916 saber's design very closely adhered to that of the truly fearsome light cavalry saber adopted in Prussia as early as 1796. It had an overall length of just under thirty-seven inches (93.9 cm). Issued to both officers and men until 1941, the saber was slung, edged curve to the rear, in a buckled frog attached to the saddle behind the rider's leg on the off side (the right) for enlisted men and the near side (the left) for officers. At the hilt the saber carried a fist strap tied in a colored knot denoting the trooper's regimental squadron. In addition to the saber, cavalrymen also carried another edged weapon in the form of a fifteen-and-one-quarter-inch-long (38.7 cm) bayonet. The same as that issued to infantrymen, the mounted trooper's bayonet differed only in having a small leather strap and buckle attached to the frog. This addition kept the bayonet from moving excessively when the trooper was in the saddle. The saber's and bayonet's object, of course, was what it had always been among mounted troops: to instill the fear of cold steel. For the same purpose, as already noted, the cavalry also still carried lances, complete with pennants, until 1927. The lance was intended to be at least as psychologically intimidating as the saber and consisted of a piece of tubular steel ten-and-one-half-feet (3.2-m) long. When riding in column, a trooper gripped the lance at its midpoint with his right hand, arm bent, and carried the weapon at an upward diagonal extending over the horse's withers to the near side. Despite their retention of edged weapons, and unlike in some of their grandfathers' experiences in 1870, German cavalrymen of the interwar period wore no armor other than a steel helmet, and their uniforms remained essentially the same as the infantry's. Modeled on the M1918 headgear of World War I, the cavalryman's helmet before the 1930s differed only in having shallow ear cutouts along the bottom edge, supposedly so that bugle calls could be better heard. Possessing a singular, wavy appearance when viewed from the side, this colloquially named "cavalry helmet" was eventually replaced by the standard M1935 helmet familiar to post-1945 generations as the _Stahlhelm_ (even though the term applied also to earlier versions). As for the cavalry's service uniforms, the principal differences from the infantry's dress lay in the cavalrymen's wearing breeches with a full-seat leather addition, the leather overlay helping to keep the rider in the saddle while mounted. Proper riding boots were issued to all ranks. These differed from the infantry's in having a taller leg reaching to just below the knee; additionally, they lacked a hob-nailed sole. This feature allowed for easier use of stirrups. Naturally, mounted troopers also wore spurs. For the cavalry's dress uniform, some small traces of the traditional rider's flair were retained. The dress tunic or blouse, introduced in 1936, somewhat curiously had no pockets, but it did have the addition of decorative, turned-back "Brandenburg cuffs," the whole being trimmed in traditional cavalry-gold piping.
As for firearms, the cavalry trooper's principal individual weapon throughout the 1930s and into the first years of World War II was the Mauser 98k (for _kurz_ , i.e., "short") carbine. This was a 7.92-mm, bolt-action weapon. It carried a five-round clip and weighed between eight-and-a-half and just over nine pounds (4 kg), depending upon the wood used for the stock. Based on an older version of the same rifle from World War I, the 98k was just over three feet (1 m) long. Originally designed for horse-mounted and horse-drawn troops, it was eventually adopted by the entire army owing to its ease of operation and "excellent ballistic characteristics." The carbine originally rode, butt-down, in a boot or scabbard on the near side behind the rider's leg and was attached to the back of his belt by a strap. As happened with the German cavalry in World War I, however, the 98k eventually came to be routinely slung diagonally across the trooper's back in what was called the "Russian style." After 1940, cavalry troopers would see ever more types of semiautomatic weapons and machine-pistols entering service to replace the 98k.
Supplementing the cavalrymen's individual weapons, several types of machine guns supplied heavier firepower between the mid-1920s and 1936. The principal heavy machine gun in service before the 1930s was the "heavy machine gun Model 1908" (sMG 08 or MG 08). As the name indicated, the army had adopted the gun in that year. In its earlier variants, this weapon dated all the way back to Hiram Maxim's original of 1885. It had been used extensively, and to murderous effect, by the German army in World War I, particularly in defense of fixed entrenchments and fortifications. However, the weapon's weight of more than forty pounds (its immediate predecessor, the MG 01, had weighed more than fifty; 18–22.6 kg), not including ammunition and the gallon of water necessary for coolant, dictated that infantrymen simply would not be able to maneuver with it. In 1915 a somewhat lighter variant, the MG 08/15, entered service and was, in its turn, beginning to be replaced by the air-cooled and still lighter MG 08/18 when World War I ended. Despite the weight of all three versions, the cavalry's horsepower eliminated the problem of the MG 08's cumbersomeness, at least until the weapon was dismounted. Therefore the cavalry continued to use it after 1918. The gun had proven itself in mobile combat operations with mounted units as recently as 1920 in the Polish-Russian War. There the Red Army's cavalry had widely used a Russian-built version in the form of the _tachanka_ , consisting of a Maxim gun mounted on a light, horse-drawn buggy or carriage. With the gun aimed to the rear, the Red Army cavalrymen would gallop up, turn and fire, and then, if necessary, gallop away again, firing all the while in a sort of modern-day Parthian shot. Though German cavalry did not employ the MG 08 in this fashion, the Russian experience did show the continuing usefulness of an older technology. When on the march in a German cavalry column, the gun, its ammunition, and its crew all rode on a limbered wagon pulled by a team of six horses.
To overcome the earlier weapon's limitations, the army issued a new set of requirements in the 1920s, and the Dreyse model MG 13 was adopted. The MG 13 accorded with Seeckt's doctrinal emphasis on putting as much firepower as possible in forward units such as the mounted arm. Consequently, he followed the MG 13's development with some attention. At about twenty pounds (9 kg), it not only weighed much less than the MG 08 but also had a higher rate of fire. By 1936, both the remaining MG 08s and the MG 13s were beginning to be replaced, along with the automatic weapons in other arms, by the justly famous MG 34 and, after about 1943, by the even more famous MG 42. Considered perhaps the first "universal" or true multipurpose machine gun, the air-cooled MG 34 was first tested in late 1933 by Mauser and constructed on the basis of an earlier, Danish design. Weighing just over twenty-two pounds (9.9 kg) and measuring slightly more than four feet in length (1.2 m), the MG 34 fired 7.92-mm rounds of drum- or belt-fed ammunition to an effective range of approximately 2,200 yards (2,000 m). With a bipod stabilized cyclic rate of 300–400 rounds per minute, the MG 34 delivered very effective fire support for mounted and other troops not least owing to its rapidly interchangeable barrels. Made as it was, however, from finely machined components and requiring producer-specific ammunition for optimum performance, the MG 34 was relatively expensive and slow of manufacture. Its successor, the MG 42, possessed all of the same basic features and ballistic characteristics but with the advantages that it had a higher standard cyclic rate of 800–900 rounds per minute, was made of stamped metal parts, weighed slightly less, and could be produced somewhat more rapidly and cheaply. Given the larger demands of the _Heer_ after 1943, however, the MG 42 was never as widely used among horse-troopers as the MG 34. In either case, the mounted arm now possessed an easily transportable and deadly automatic weapon. The MG 42 also enjoyed a considerable psychological advantage in the "tearing-silk, buzz-saw sound" it made when fired, a sound that made it impossible to separate the sounds of individual rounds being fired. The MG 42 would go on to acquire a fearsome reputation among Allied troops on both the Western and Eastern Fronts, Russian soldiers even referring to it as the "mincemeat machine." When the cavalry was on the march, the weapon was carried barrel-down in a scabbard on the near side behind the gunner's saddle. The rest of the two- or three-horse load included the other members of the gun crew, extra ammunition, cleaning gear, and a tripod. Some indication of these weapons' success is revealed by the fact that a combined total of perhaps 526,000 MG 34s and MG 42s were produced by 1945.
The heavy weapons of the cavalry throughout the 1930s and at the outbreak of war in 1939 consisted of three main types. The most potent was the 75-mm "light infantry gun" ( _leichtes Infanterie-Geschütz_ 18; le.I.G. 18) developed by the firm Rheinmetall in 1927. Actually a howitzer in American military terms, it carried the designation "cavalry gun" in the German mounted regiments. The le.I.G. 18 was specifically designed to be horse-drawn. Its caisson, originally rolling on two spoked, wooden wheels, gave the cavalry gun units the nickname "gypsy artillery." Later models of the cavalry gun replaced the spoked wheels with steel ones mounting rubber tires. This weapon was well suited to use in mounted units. Its high-trajectory fire could take good advantage of concealment, and its plunging shot was effective against reverse-slope targets. Furthermore, the weapon's typical placement simultaneously offered protection from direct counter-battery fire. Specifically for these reasons, however, the le.I.G. 18 required good spotting of the fall of shot in order to ensure accurate fire. Depending upon a given regiment's designation following the mounted arm's eventual reorganization in 1935–1936, either four or six of these highly accurate weapons appeared on the unit's TOE. Furthermore, and again depending upon the regiment's designation and mission, the le.I.G. 18 would be towed either by a six-horse team or by motorized prime mover. In the latter case, the vehicle in question was the much liked "motor vehicle 69" ( _Kraftfahrzeug_ 69; Kfz 69) built by Krupp. This was the so-called _Krupp-Protze_ (Krupp limber), a six-wheeled, 1.5-ton truck with double rear axles. In addition to the 75-mm howitzer, the cavalry regiments also included a battery of three, towed 37-mm anti-tank guns in the headquarters troop and, at least notionally, a crew-served 50-mm mortar in each bicycle platoon. The bicycle platoon was often equipped with the M1939 _Patria WKC_ bicycle, a simple and rugged machine. It was unadorned except for an occasional headlight and the canvas cyclist's cape carried in a roll slung from the crossbar or handlebars. The _Patria_ took its place in the cavalry's columns along with horse-drawn wagons. Of the latter, the most common was the ubiquitous Army Field Wagon 1 ( _Heeresfeldwagen_ 1; Hf1). Pulled by a team of two horses, the Hf1 weighed 1,430 pounds (650 kg) empty and could transport a useful load of 1,650 pounds (750 kg). The Hf1 also served throughout the rest of the army's non-mechanized formations.
The Cavalry's Horses
In the 1930s, "cavalry" still meant horses. The concept of armored cavalry hadn't yet taken hold, though it loomed on the military horizon. Because the German army couldn't have cavalry (or horse-drawn logistics trains and artillery for that matter) without horses, several major breeding sources assumed particular importance. Given the varying requirements for cavalry mounts and draft horses, as well as mounts for officers in other arms, breeds' different characteristics influenced which horses went to which arm-of-service and in what numbers. Ensuring the supply of sufficient numbers also became critical. The losses of horses in World War I had been enormous, and not only in Germany. In the inter-war period, breed-stock and the overall equine population nevertheless recovered substantially, and between 1929 and 1937 the number of horses in Germany fluctuated only slightly. In the former year statistical estimates indicated that 3,617,000 horses of all breeds were to be found in the country; in the latter year 3,434,000. By 1939 the number had risen to approximately 3,800,000, presumably under the pressure of the army's increased requirements as it expanded. Though several different breeds were native to Germany, certain types, and the regions producing them, excelled in their service to the army both during the interwar period and World War II.
Certainly one of the most famous points of supply for both military and civilian horses in all of Germany was the East Prussian Central State Stud at Trakehnen. Located just off the rail line linking Eydtkuhnen on the Polish border with the World War I battlefields of Gumbinnen and Stallupönen, Trakehnen had been established as a stud between 1726 and 1732 by King Frederick William I, the "Soldier King." Originally a large stretch of low-lying, open, and wet moorland ("Trakehnen" means "great moor") adjoining the Rominten Heath, the Stud eventually encompassed some three thousand hectares of pasture and meadow and an additional three thousand hectares of rich farmland. The processes of canalizing and converting Trakehnen's moors to pasture and farmland were perennial interests of the Hohenzollern princes, as were the horses themselves, 30,000 of which were said to have been required to carry the first Hohenzollern king and his retinue from Berlin to Königsberg for his coronation in 1701. Frederick William I's son, the eventual Frederick the Great, would go on to drain more swamps than any other ruler of the age. In these efforts, pasturage for Prussian horse breeding would have remained important among the latter's concerns, as it had for Frederick William I. Land reclaimed from water-logged moors provided sustenance for horses as well as people, and horses were critical not only for farming but also for the Prussian cavalry. After 1918, the army established a remount depot on the northern side of the railway adjoining the Stud. The depot also encompassed the adjacent village of Kattenau and its surrounding moorland. It was in the remount depot that the army collected and temporarily stabled horses purchased from the Central State Stud and the outlying regional State Studs elsewhere in East Prussia before shipping them to their various units. The army prized these horses, and by 1912 East Prussia produced more than seven thousand remounts per year. By the 1930s, remounts were being purchased at three to four years of age, and all underwent a year of general conditioning at remount parks to bring them up to roughly the same standard of fitness. There followed a year's assignment to their units for a period of initial introduction to the saddle or the harness and a second year's more rigorous training for the horses' respective duties. Only at about six or seven would remounts actually begin an active-duty career generally envisioned to last for about ten years barring permanent injury, incapacitating wounds, and/or premature death. Though the army's demands fell immediately after 1918, by the 1920s the Central State Stud and its satellites were once again producing fine horses. These, however, exhibited a somewhat stockier build than those so heavily influenced by crossbreeding with English and Irish Thoroughbreds before World War I. The post-1918 horses consequently had a greater bone mass and a truer warmblood's disposition but still possessed elegant proportions. The Trakehner thus became the standard riding horse of the interwar German army, as it already had been of the Prussian army before 1914 and as it would remain later for the _Wehrmacht_. The official breeding goals as specified by the army for what it called the East Prussian Horse included, _inter alia:_ "flexibility, toughness, a great galloping ability...endurance, a noble head, a strong elastic back with good saddle positioning, a deep and capacious chest cavity, [and] a high degree of impulsion ( _Schwung_ )." By 1939 standard expectations for a good Trakehner stallion or gelding included heights measuring between 15.3 and 16.2 hands. Suitable girths measured between 75 and 79 inches (190–200 cm). The same measurements for Trakehner mares were 15.2 to 16 hands and 70 to 79 inches (177–200 cm) in girth.
A second major source of horseflesh for the army's needs lay in another traditional region of German horse breeding. This was in Hannover, in what is today Lower Saxony. Centered on Celle—the aforementioned location of the army's Cavalry School—this area stretched northward from the River Aller into the Lüneberg Heath. Downriver it ran northwestward beyond Verden to Bremen and encompassed the flat-lands of East Friesia and Emsland. In these fertile, windswept plains and low, rolling hills originated the renowned Hanoverian horse and its close cousin the Oldenburger. Horses bred in the region were well known as early as the seventeenth century. They had served in the armies of the Swedish kings Gustavus Adolphus and Charles XII. In the 1690s, the English had also acquired cavalry remounts from this source. About 1675, the governor of the Spanish Netherlands wrote that he had never seen finer cavalry horses than those later called Hanoverians. It was only in the 1730s, however, that the breed began to assume its modern characteristics. In 1735 King George II of Great Britain, in his capacity as Elector of Hannover, decreed the establishment of a centralized effort there to "promote horse breeding in our German lands, especially in the duchy of Bremen and the county of Hoya [southeast of Bremen on the River Weser]...until such time as it has been seen what good comes of it for the land as a whole."
After 1815, the establishment of the Kingdom of Hannover in the wake of the Congress of Vienna accelerated the development of a distinct horse-breeding economy in the region. In the second half of the nineteenth century, the Hanoverian finally became a valuable, multipurpose military and civilian horse. Indeed, the breed's military priority was stated explicitly in 1900 by the president of the Central Stud at Celle, who also happened to be a general officer in the army of what by then had become part of Prussia: "A horse suitable for use as a troop horse, heavy cavalry horse, artillery horse, or middle-weight carriage horse." As to the breed's size, the Hanoverian's standard measurements before 1945 tended to be close to those of the Trakehner, though somewhat deeper from the point of withers to the chest and of a slightly heftier overall proportion owing to the lingering influence of the latter's eighteenth-century oriental crosses. As with most horses of predominantly German origin, the Hanoverian's overall standard height would increase somewhat after World War II, while the standard bone mass would drop a bit. The leaner, purpose-bred sporting horse of the postwar era was not the norm in the period from 1920 to 1945 for either the Hanoverian or the Trakehner. Of course, the indicated specifications of the German army's cavalry mounts did not necessarily coincide with those needed for the draft-horse labor of pulling artillery caissons or supply wagons, though cavalry horses were later routinely pressed into draft-horse service. Just as 1920s-era engineers of tanks or armored cars confronted difficult tradeoffs among vehicles' weight, weaponry, armor, and range, so too did horse breeders have to balance size, stamina, brute strength, soundness, and the nonquantifiable but critical factor of "heart" in horses intended for military use.
By 1914, 2,500 Hanoverians were being sold to the army annually. While fewer than the numbers of horses coming from East Prussia, the Hanoverians were at least as high in quality. Such sales were critical to the Central Stud's economic viability. As with the Trakehners of East Prussia, however, sales to the army fell dramatically in the years immediately after 1918 as Versailles' restrictions hit home. Nevertheless, and again like the Trakehners, but also as with the Hanoverians' close relative, the Oldenburger (as well as the somewhat smaller Haflinger of southern Germany and Austria), the currency collapse of 1923–1924 ironically helped stabilize breed-stock as persons still possessing cash looked for investments in property that might hold its value over time, despite the impending age of the automobile. This development stood the breed in good stead, as did the army's gradual and initially surreptitious expansion after the mid-1920s. The breeding goals specified by the army for the Hanoverian included a conformation similar to the Trakehner but with a slightly heavier head. Given the breed's intended military mission, its then heavier bone mass won the army's favor, as did its "outstanding jumping talent [and its] much calmer and more agreeable temperament." While the mass of the army's riding horses continued to come from East Prussia, the Hanoverian was only slightly less prized in both riding and draft roles. Somewhat by contrast, other refined breeds such as Oldenburgers, Holsteiners, and East Friesians served primarily as light draft horses for the army's field artillery as well as for the infantry's service support units such as logistics trains. Other German horse-breeding areas provided the bulk of the army's heavy draft horses. Interestingly, an indication of the army's view of the horse's future may be seen in its willingness to pay more for "coarse" artillery horses than for pure cavalry mounts, the former possessing greater bone-mass and pulling strength than the latter. Whatever the breed of horse, the essence of cavalry warfare at the time lay in the transition from steady, sometimes fast, and often far-ranging riding in any weather and over varied terrain to dismounted infantry fighting. The German cavalryman having essentially become what earlier generations had called the dragoon, the horses selected for service in the mounted arm had to be able to endure routine marches of several days' duration over distances of 30 to 60 miles (48–95 km) per day all the while carrying between 200 and 250 pounds (90–113 kg) of rider and equipment. Throughout the 1920s and 1930s, then, the army still had superb sources for the vast numbers of horses it required, even though it already envisioned a future of increasing motorization and mechanization. Whether cavalry or not, all divisions would include ever larger numbers of vehicles of all kinds (even though the vast majority of units outside the panzer arm never possessed adequate numbers of adequate machines). But until the sometimes facile prewar assumptions about vehicles bore themselves out, horses—Trakehners, Hanoverians, Oldenburgers, Haflingers, and other breeds—would still be needed in extraordinarily large numbers.
Veterinary and Remount Services
By 1939, caring for the hundreds of thousands of horses in the army on the eve of war consumed enormous resources of men and matériel, resources that would be expended in ever-greater amounts as the inevitable attrition of the animals occurred once the shooting started. Hence a sketch of the veterinary and remount services becomes important for a better understanding of the task confronting all mounted units as Hitler unleashed his war. The commander of the Army Veterinary Service was the Veterinarian Inspector General ( _Generaloberstabsveterinär_ ), a post held from 1938 to 1945 by General Curt Schulze. Schulze's command, the _Veterinärinspektion_ , was an Inspectorate of the Armed Forces High Command ( _Oberkommando der Wehrmacht;_ OKW). Thus he served as the commanding technical director in all matters of veterinary medicine and farriery for the entire army, not merely for the cavalry. "During the war, he was responsible for [a nominal strength of] 1,250,000 horses, 37,000 blacksmiths and 125,000 soldiers. The veterinary service was divided into 236 veterinary companies, 48 veterinary hospitals, and 68 horse transport [units]." The hospitals of the veterinary service treated approximately 100,000 horses daily for ailments ranging from general lameness and communicable disease to wounds from bullets and shrapnel after 1939. An indication of the service's success can be seen in a return-to-active-duty rate of approximately 75 percent of all horses treated.
Each cavalry division was assigned two veterinary companies. Infantry, alpine, and _Luftwaffe_ field divisions each were assigned one. Each veterinary company's hospital section provided care to wounded, injured, or sick horses brought in from the company's two collection stations. The latter could also provide the animals first aid. The collection stations were usually located five or six miles (8–10 km) behind the divisional front lines. Recuperating horses were shuttled from the hospital section to the veterinary company's supply section. Following recovery, the horses were first moved back to the collection stations and then out to the division. Horses in need of longer-term care were instead passed back to army-level hospitals and, if necessary, to the level of the appropriate army group. If necessary, they could be sent still further back to the "zone of the interior." Conversely, newly assigned or recovered horses coming from veterinary parks and remount depots at the army and army-group echelon flowed down the same chain. Eventually they reached the company-level supply section and moved through the company's collection stations to the division.
Closely supporting and reinforcing the divisional veterinary service, army-level facilities played a major role. Each army was assigned a hospital for wounded and injured horses. A second hospital treated any horses suffering from infectious diseases, especially captured horses. The hospitals had a rated capacity of 550 horses, but subsequent conditions in Russia drove the numbers into the thousands. Supplementing the army-level hospitals were two motorized veterinary clinics, as well as a motorized veterinary test station capable of executing bacteriological, serological, and chemical examinations. The army-level remount depot, like the hospitals, carried a rated capacity of about five hundred horses but often stabled many more. An interesting feature at this level of command was the motorized veterinary park where horses might recuperate. The veterinary park, however, needed ready access to railway transport owing to its very heavy complement of equipment and, of course, for shipping horses to and receiving them from the zone of the interior. Hospitals could also be located at an intervening army-group echelon, but the principal connections remained those directly linking the zone of the interior and the army-level facilities. As of 1935, the army's veterinarians were trained at the Army Veterinary Academy in Hannover in conjunction with the civilian Veterinary University ( _Hochschule_ ) in the same city. There were also three principal military farriery schools in Hannover, Munich, and Berlin. Given the huge numbers of horses in question, the demands on the skills and physical endurance of the army's veterinarians and farriers may be easily imagined.
Keeping the cavalry and the rest of the army supplied with riding and draft horses was an enormous undertaking. Remounts came from two sources: the army's own remount service and requisitions. Working in the Inspectorate of Riding and Driving of the Army High Command, the remount service's purchasing commissions bought horses both from state establishments such as those at Trakehnen and Celle and from private owners. As before World War I, horses in the Reich were registered as to their potential military suitability and availability. Before 1933, registration took place under the auspices of various civilian agricultural organizations supervised by the army's Inspectorate of Conscription and Recruiting. Horses could thereby be declared militarily indispensable just as could civilians in critical occupations. Ideally aged three or four at the time of purchase, horses initially spent a year in one of fourteen regional remount depots before entering formal training at the Riding and Driving Schools located in the various Military Districts ( _Wehrkreise_ ) of the Reich. In the remount depots, horses were brought to a uniform standard of maintenance. After 1939, the one-year conditioning period was often omitted with horses going directly into training. Even before the war's outbreak, the army's demands grew faster than the Reich's equine population. These demands forced the army to purchase horses abroad as early as 1936. Obviously, by 1939 demands for horseflesh had become voracious. In that year the number of required horses rose at a stroke from some 120,000 to almost 600,000. Domestic and foreign purchase, as well as requisition (whether compensated or not), rose even more dramatically once the war began: 148,000 in 1940; 282,000 in 1941; 400,000 in 1942; and 380,000 in 1943. As far as actual purchases were concerned, Hungary constituted a favored prewar source. That country alone supplied 25,000 horses for the _Heer_ in the year 1934–1935, that is before the expansion of the _Wehrmacht_ even really gathered speed. Rumania, Czechoslovakia, and, until 1939, Ireland also supplied stocks of horses. Eventually, between 1939 and the beginning of 1942, the German armed forces would also acquire 435,000 captured horses from the armies of Poland, France, and the Soviet Union.
In the latter case, the horses in question were the soon to be famous _panye_ or _panje_ horses. Known to the Germans by the same name as the small, two-wheeled wagons and wintertime sledges they pulled, these horses were often colloquially called "steppe ponies." In fact, they were seldom ponies at all in the zoological sense. Of indiscriminate breed, they nevertheless frequently reflected the physiological influence of the horses of the Black Sea and Kazakh Steppe, though one should be cautious in ascribing too much influence here to the taller and more elegant oriental breeds such as the Russian Orlov Trotter, the specifically Turkmen Akhal Teke, or, by extension, the Turkic Karaman. Often measuring only fourteen hands, _panje_ horses tended to be smaller than their German or other Western European counterparts and therefore not as able to bear or pull extremely heavy loads. What they lacked in stature and finesse, however, they more than made up in hardiness. Capable of enormous feats of endurance, the _panje_ horse could travel as much as ninety miles (150 km) in a day and sustain itself on nearly any edible vegetable matter: oats, corn (i.e., maize), barley, hay, grass, straw, weeds, and, when necessary, even roofing thatch and tree bark. Somewhat less efficient in its way of going than its larger western cousins, the _panje_ horse could not easily be incorporated in marching cavalry columns without rapidly becoming exhausted owing to its shorter legs and the more rapid average pace of larger breeds. In horse-drawn logistics or artillery trains, the _panje_ horse did not fit well when coupled to standard _Wehrmacht_ vehicles whose traces and higher centers of gravity made it a much less effective draft animal. When pulling native Ukrainian- or Russian-style carts, wagons, or sledges, however, the _panje_ horse performed well and earned a good reputation among German troops, particularly those outside cavalry and artillery units who often didn't possess good horse-handling skills. Given the smaller size of _panje_ wagons and the numbers of extra personnel who would be required to man ever larger _panje_ columns attempting to move the required amounts of supplies and ammunition, the German army never saw the _panje_ horse as anything more than a temporary expedient, albeit a welcome one, for hard-pressed units, particularly in Russia's winters. Nevertheless, as German soldiers were to discover in the first winter in Russia in 1941–1942, _panje_ horses could survive unsheltered in all but the very coldest conditions; and, as noted, they enjoyed the additional advantage of not only needing less, but also much less refined, feed and fodder than the _Wehrmacht's_ horses. In that respect, the German horses' standard daily ration in 1941 consisted of a whopping eleven pounds (5 kg) of oats for cavalry mounts and light draft horses; fourteen pounds (6.5 kg) for heavy draft horses; and nearly eighteen pounds (8 kg) for very heavy draft horses such as Belgians and Percherons. Of course, it should be borne in mind that uncooked oats in and of themselves do not have the same nutritive value for a horse as they do when rolled, steamed, or cooked as feed-mash. Rolled oats have their hulls cracked in the process, and steamed or cooked oats will tend to slough off their hulls altogether. In these cases, horses will be better able to digest the nutrients in the grain. By contrast, the hulls of the raw seeds make proper digestion more difficult. Consequently, much of the nutritive value of a given ration of raw oats would go in one end of the horse and out the other. Thus a larger amount of raw oats would be necessary to compensate for the relative loss of nutrition if the oats could not first be rolled, steamed, or fully cooked. Obviously, German cavalry, artillery, or draft horses would not always be able to be fed prepared oats in forward areas and/or locations at the far end of, much less beyond the end of, the logistics trains. This very situation would often present itself in Russia, as in the winter of 1941–1942. Greater reliance on local resources—sometimes adequate but all too often not—would necessarily result, with all that that would imply for the condition of the horses. In the mid-1930s, however, such a condition still lay in the realm of the unknown. In the meantime, peacetime rations of grain and sweet feed remained unchanged. It should be added that these rations of oats did not include an additional daily ration of hay varying from eight to eleven pounds (4–5 kg).
Of course, military horses work very hard and need generous rations of feed and fodder regardless of their arm of service. The rations-allowance nonetheless constituted a huge commitment of resources. Given the eventual exigencies of the war, such rations were not always available even when the horses themselves were. Consequently, _panje_ horses not only came to provide a large proportion of remounts and draft horses after 1942, but they also drew less heavily on the army's supply chain. The latter factor assumed ever-greater significance with the increasing intensity of the Allies' strategic bombing campaign of Germany's infrastructure as of mid-1943. Coming from whatever sources they might, however, the army's cavalry remounts and draft horses still had to be found, and found they were. In the end, the total number of horses purchased or requisitioned from all private sources through 1944 reached the amazing figure of 1,645,000. Not counting further requisitions in the chaotic final year of the war, the grand total of horses (and mules) employed by the _Wehrmacht_ during World War II approaches 2,500,000, and one source cites an even higher total of 2,750,000.
On the Brink of War: 1935–1939
As with so many aspects of Germany's national life, the _Reichswehr_ saw itself transformed by the accession of the Nazis to power on 30 January 1933. Throughout his public career before being named Reich Chancellor by President Paul von Hindenburg, Adolf Hitler had made it abundantly clear that history's verdict as handed down in 1918 possessed no validity for him or his party. That verdict would be overturned, and his would be the court to do it. As early as his first speech to assembled military officers on 3 February 1933, Hitler maintained that the prerequisite to regaining the Reich's strength, in fact the essential definition of that strength, was the expansion of the armed forces beyond the limits set by Versailles. For him, the armed forces were the most important institution in the State. No one and nothing could or would be allowed to interfere with their expansion: not the Western Powers, not any bilateral or multilateral international agreements, not general domestic constraints, not his closest associates in the party. On this score he never wavered, though initially he could not say so publicly. Until he could in fact speak publicly about the subject—that is, until rearmament had actually proceeded so far as to make international opposition less likely—his mission would be to mask the expansion process politically.
An important early step in this process was Hitler's decision, supported by various elements in the army and the Foreign Ministry, to withdraw Germany from membership in the League of Nations as well as the League's Disarmament Conference. Taken in the autumn of 1934, this decision scored a huge domestic propaganda success for the Nazi regime. Furthermore, Hitler's determination helped solidify his early support within the armed forces and among the public at large. Not coincidentally, it also occurred at same time as his decision, apparently made in May 1934 and announced privately on 1 October, to begin the secret expansion of the armed forces to three times the size allowed under the Treaty of Versailles. With breathtaking rapidity, this secret decision was followed by Hitler's dramatic public announcement on 16 March 1935 that Germany had once more assumed her rightful sovereignty in military matters. He simultaneously announced the reintroduction of conscription with a one-year period of service effective 1 October 1935 and yet another expansion—a doubling in this case—of the army to a base figure of 600,000 men in 36 divisions. The unveiling of a not-so-secret _Luftwaffe_ followed shortly thereafter. Incidentally, in that same spring and summer of 1935, changes occurred in the nomenclature of the armed forces. The old, Weimar-era designations began officially to fall away. The term _Reichswehr_ came to be replaced officially by _Wehrmacht_. Before 1935, the latter term had referred more or less generically to the military power of the Reich or any other State. Similarly, the post-1918 term _Reichsheer_ , specifically referring to the army, also went by the boards. It was superseded by the simple _Heer_. Seeckt's old billet, the _Truppenamt_ , also officially disappeared. In its place there reappeared the older "General Staff," a term and an office that had been specifically banned by the Treaty of Versailles as reflecting a putatively inherent Prussian militarism.
As much for political reasons as military ones, Hitler demanded that Germany's rearmament be driven forward at breakneck pace. In this demand he overrode or simply ignored occasional opposition from the army's leadership. He also permitted ferocious and ultimately unresolved competition for resources and manpower among the three branches of service— _Heer, Luftwaffe_ , and _Kriegsmarine_. Requirements for the same resources by many grandiose and propagandistically important public works projects only compounded administrative confusion as the Nazi satraps battled each other for prominence and perquisites. This internecine political rivalry further eroded whatever efficiency still existed in the army's expansion. This situation directly affected the cavalry in that the army's programs for motorization and mechanization, already viewed by some in the late 1920s as certain to replace the horsemen, did not proceed effectively. The _Wehrmacht's_ growth "remained a fundamentally unco-ordinated [ _sic_ ] expansion of its individual services. An overall rearmament programme for the Wehrmacht did not exist." Of course, much is often made of the fact that the first three armored divisions were established as early as the autumn of 1935. Furthermore, almost exactly one year later the army had already reached, indeed exceeded, its own and Hitler's goal of a thirty-six-division force, a total that it was originally not supposed to attain until autumn 1939. Nevertheless, throughout the six years of peace remaining between the Nazis' accession to power and the invasion of Poland, the army continued to operate under the guidance of a doctrinal manual still giving horse-mounted troops militarily important, if not critical, tactical and operational roles.
This manual, _Truppenführung_ (Unit Command), appeared in two parts published in 1933 and 1934 as _Heeresdienstvorschrift 300_ (Army Service Regulation 300). In its turn, _Truppenführung_ had succeeded the important manual which had provided the basis of Germany's military doctrine throughout Seeckt's tenure as head of the _Truppenamt_ and afterward. That earlier manual was entitled _Führung und Gefecht der Verbundenen Waffen (Command and Combat of the Combined Arms_ —widely known simply as _das_ FuG). Important for the army as a whole and for the cavalry in particular was the latter's doctrinal restoration of the importance of mobile battle leading to the enemy's annihilation. The _FuG_ 's emphasis on mobile battle resulting in the battle of annihilation ( _Vernichtungsschlacht_ ) attempted to overcome the legacy of the positional warfare ( _Stellungskrieg_ ) earlier waged on the Western Front between 1914 and 1918, despite that legacy's inapplicability to the campaigns on the Eastern Front. Importantly, the _FuG's_ restoration of the centrality of mobile-combat doctrine also returned to an older Prussian-German tradition of emphasizing as much mobility as possible, according to whatever the then-current technology allowed. Such a principle favored military commanders possessed of great initiative and fighting campaigns as briefly and decisively as possible. If anything could be said to constitute a true German way of war, this was it, not the great battles of attrition which had so dominated the image and conduct of the Western Front before 1918.
In _Truppenführung_ , the German army distilled its experiences since 1870, particularly in light of the initial victories and the eventual defeat in the war of 1914–1918. Throughout the manual, the army's doctrinal thinkers continued to envision a prominent role for the cavalry and for horses generally, notwithstanding the advent of the panzer arm. "Combat is the cavalry's principal mission." A clearer statement of doctrinal intent and the cavalry's putative contribution to the army could hardly be imagined. It would be in reconnaissance and screening that most of the cavalry's combat would occur. From these two principal tasks others would derive: movements against the enemy's lines of communication; flank security; pursuit; and delaying actions (706–710). Eventually, after the invasion of the Soviet Union, anti-partisan duties would be added to the list. In clear recognition of the events of 1870 and the Western Front between 1914 and 1918, _Truppenführung_ urged avoidance of mounted attacks against prepared positions. If, however, such efforts were necessary to fix the enemy in place (another classic task of the cavalry, as at Mars-la-Tour in 1870), then the horsemen were to undertake even these (707). In any case, they would normally fight dismounted, dragoon-style, and were to be supported as closely as possible by their own horse-artillery and automatic weapons (716). Mounted combat would be a rarity. If it did actually occur, _Truppenführung's_ authors wrote that such combat would usually result only from a chance engagement of two small units or from the surprising of an enemy formation. Nevertheless, the manual also stressed in very traditional fashion that a mounted attack against a demoralized enemy could have great psychological effect (718). When engaged, the cavalry divisions' own motorized infantry and artillery, as well as their bicycle- and motorcycle-mounted troops, would augment the horsemen's firepower. _Truppenführung_ acknowledged, however, that these units would largely be limited to the roads and be dependent on "other arms" (i.e., horsemen, armored cars, and aircraft) for their reconnaissance and march-security (720–722). In spite of any such limitation, the cavalry divisions were deemed fully capable, along with their infantry counterparts, of independent operations and self-support precisely because of their organic composition (22). Their reconnaissance and screening value was highly regarded because of their all-weather, cross-country capability, a capability not yet matched by motorized vehicles (133). Conversely, horses' marching speeds of perhaps five miles per hour (7 km/h) at the walk and seven (10 km/h) at the trot compared very unfavorably with the eighteen to twenty-five miles per hour (30–40 km/h) of motorized vehicles (292 and footnote there), not to mention the critically important equine requirements of regular rest and watering halts (272–273, 303).
For all that the _Truppenführung_ retained a major role for the cavalry, the writing appeared to be on the wall, or at least it would have been had peace lasted long enough for the army to acquire sufficient numbers of motorized and mechanized vehicles. A second part of the manual, published in 1934, covered the matter of armored combat vehicles ( _gepanzerte Kampffahrzeuge_ ). At the time, this designation included both wheeled armored cars and the tracked progenitors of proper tanks (725–758). While the first panzer divisions would only be established in 1935, the manual clearly foresaw the horse cavalry's replacement. Furthermore, armored cars were explicitly envisioned as assuming the cavalry's reconnaissance functions despite armored cars' then-still-poor, off-road capability (727). For its part, the tank possessed much better cross-country capability but was still hampered by short range and relatively low road speed compared to its wheeled counterpart. Consequently, the gaps in the various vehicles' characteristics still left a place for horse-mounted troops to occupy in the years between 1935 and 1939, even if the evident tendency was toward the horse's retirement from the combat role.
Even if the _Truppenführung_ seemed to herald the end for the German cavalry, nothing was yet a done deal, even in 1935. Similar arguments over the mounted arm's viability occurred at roughly the same time in other European armies as well as in the U.S. Army. To maintain that in 1920, 1930, or even 1935 all officers in every army stood convinced of mechanization's inevitability badly misrepresents the then-prevailing situation. Four of the five cavalry divisions in the French army, for example, were modified into mixed divisions between 1932 and 1939, and three still were still in service when war came. Each of these divisions required more than five thousand horses for its two horse-mounted brigades. Similarly, in the British army, only two of twenty cavalry regiments had been converted from horses to armored cars, much less tanks, by 1933, and major armored experimental exercises had largely ended in 1931. Of course, it remains true that the British army is the only major European army to go to war in 1939 without significant numbers of horses in the active forces. Nevertheless, what was written of the British army applied at least as well to the _Herr_ after 1934. There was no
heroic but vain struggle [in the German army] of a handful of brilliant iconoclasts, who were later proved right, against a compact majority of antediluvian cavalry-loving diehards. Closer inspection...shows that the reality was more complex. The progressives or radicals did not agree with each other on all points and in some respects their predictions proved mistaken or inadequate. Moreover, although diehards or reactionaries certainly existed, the majority of officers...could be described as cautious or moderate progressives; that is, they recognized that machines such as tanks would play an increasingly important part in future war, but they tended to stress the numerous problems and uncertainties. How, for example, would armored forces be supplied and repaired when far from base? Would they not soon be countered by antitank guns? And above all, what part would armored units play in military organization as a whole, given the shortage of funds and equipment, and traditional interservice rivalries?
All of these considerations applied, mutatis mutandis, to the armies that Adolf Hitler was even then preparing to unleash against Europe.
CHAPTER 6
BUCKING THE TREND
THE CAVALRY RIDES TO WAR, 1939–1940
"No one wants to see horses go to war, but they always go just the same to do the work no machine can do, wallow around in the mud, scout through forested and hilly country and bring up supplies where nothing else can get through." Though written in the United States, these words still applied to warfare in Europe in the late 1930s. Despite the experience of World War I and the interwar years, the internal-combustion engine had not yet been married with wheels or tracks sufficiently well or in sufficient numbers to supplant the horse in every circumstance. In Germany and elsewhere, horses still played an important military role, particularly as draft animals. The German army's doctrine throughout the period from 1918 to 1939 certainly envisioned their use to pull artillery, supply wagons, field kitchens, and other vehicles; and even though the _Truppenführung_ still countenanced a horse-mounted cavalry, the question remained whether the latter had any real role to play on the modern battlefield.
In 1939 the head of the Transport Department of the General Staff, Colonel Rudolf Gercke, noted that "as regards transport, Germany is at the moment not ready for war." Coming as it did from the General Staff's expert in the matter, such an assessment could only reinforce the assumption that literal horsepower was going to be crucial for the army's war effort. Of course, Gercke referred only to transport and logistics. Nevertheless, his statement also implied that if motor vehicles couldn't be had in sufficient numbers for the army's logistics trains, then they might also be lacking in the combat arms. If and when a war between Germany and her neighbors became a war of attrition, as indeed it did as of 1942, the matter would only become more acute.
Gercke's concerns touched upon the crucial matter of military means and political ends. Balancing the relationship between military strategy and foreign policy in Hitler's Germany constituted a crucial gamble for the dictator and the Reich. As in any State, Nazi Germany needed sufficient armed strength to make its policy's realization by force credible, if force were required. The _Wehrmacht_ , in turn, required an organizational structure suitable to accomplish whatever missions were determined by the regime's foreign policy. As events developed, serious imbalances between the two elements of this calculation caused equally serious problems for the realization of Hitler's ambitions. The longer Hitler remained firmly in power, the greater his ambitions grew. Similarly, the more he consolidated his authority, the greater became his apparent certainty that he and only he could achieve the goals he set for Germany. By 1939, he would repeatedly tell his military commanders, Foreign Ministry diplomats, and Nazi Party bosses that he was literally irreplaceable: no other German, whether military commander or politician, had ever possessed or would ever again possess his competence and daring. The subsequent early victories between 1939 and 1941 only dramatically increased his hubris. Ultimately, he came firmly to believe in his own infallibility. Surrounded by an inner circle of military officers and party insiders who either could not or would not stand up to him—admittedly no easy thing, wielding as he did all the corrupting and murderous power of a popular, one-party dictatorship—he ultimately plunged headlong into the abyss. With him he took Germany, the rest of Europe, and untold millions of innocents.
Notwithstanding the regime's incessant propaganda, the armed forces generally and the _Heer_ specifically were not the completely modern, battle-ready forces they appeared to be. Their readiness, or the lack thereof, largely depended upon the economic policies pursued after the Nazis' accession to power. To be sure, Hitler inherited an army that was professionally competent and highly motivated. The views of individual officers notwithstanding, the _Heer_ as an institution did not suffer from monarchist longings. At the same time, however, its interwar officer corps had not generally arisen in the same milieu as the civilian leadership of the defunct Weimar Republic. Instead, as a consequence of its now famous status as a "State within the State," the interwar army had focused its professional attention on its institutional competence, especially as regarded armaments and training. In this respect, the _Heer_ that Hitler's government relied upon can perhaps be regarded as the first truly professional army of the modern era. How this professional competence and its attendant readiness would be affected by the new government's drive to war remained to be seen.
In the six years before the invasion of Poland, the Nazi government had achieved apparently spectacular economic successes. Nevertheless, harnessing the national economy efficiently to the production of war matériel never succeeded as thoroughly as the regime asserted. At the time of the Nazis' accession to power, the party had no coherent economic program. Of course, Hitler had campaigned for the chancellorship partly on the basis of his intention to reduce Germany's severe, Depression-era unemployment. He also promised to secure the country's agrarian resources and to restore Germany's military strength. These generalities he made very clear. Attaining these goals was another matter, and he had no fixed plan. To the extent that he did state his government's economic objectives, several seemed inherently contradictory. Since the early 1920s, for example, the Nazi party had stressed a sort of lower-middle-class anti-capitalism and a desire to achieve agricultural autarky. Having now come to power, however, Hitler and his regime clearly required not only the economic but also the political support of Germany's traditionally export-oriented heavy industry. Absent such support, there could be no re-armament. Without re-armament, there could be no reassertion of Germany's role in Europe. Certainly there could be no avenging the loss of World War I, no reacquisition of German lands surrendered in the hated Treaty of Versailles, and no conquest of anything more. Germany would not, so Hitler believed, be able to feed herself, and unemployment (affecting some 5.6 million people by the end of 1932) would not be tamed. Therefore on land, on the sea, and in the air, the aim of re-armament under the Nazis was the alteration to Germany's advantage of the European balance of power, and that as rapidly as possible.
As early as 8 February 1933, Hitler insisted in Cabinet that "every publicly sponsored measure to create employment had to be considered from the point of view of whether it was necessary...to rendering the German people again capable of bearing arms for military service. This had to be the dominant thought, always and everywhere." "The main principal," he went on to emphasize, "was everything for the armed forces. Germany's position in the world depended decisively upon the position of the German armed forces. The position of the German economy in the world was also dependent on that." Hitler flogged rearmament along at a furious pace and not without the support of the generals and admirals. As seen in the preceding chapter, the total strength of the armed forces was something over 100,000 men in January 1933, as mandated by Versailles. By spring 1935, conscription, though banned by the treaty, had been re-introduced, and the ceiling of the army alone had risen to thirty-six divisions. The well-organized police regiments of the Rhineland were incorporated into the army when the Rhineland was remilitarized one year later. Austrian units were added with that country's annexation in early 1938. By the fall of that year, fully 52 percent of the German government's expenditure and a whopping seventeen percent of the Reich's gross national product flowed into armaments. These percentages constituted sums greater than those for Great Britain, France, and the United States combined. In that same year, the German army's nominal strength had risen still further to some forty-two active and twenty-nine reserve and _Landwehr_ (third-line) divisions. Yet, there was still more to come. At the time of the invasion of Poland, the army's order of battle for field divisions was an almost unbelievable 103 divisions of all types. Of course, these figures did not include the extraordinary growth of the _Luftwaffe_ since 1933 or the admittedly slower but nonetheless remarkable increase in the _Kriegsmarine_.
Given these staggering numbers, the retention of a single horse-mounted cavalry brigade (subsequently expanded to a division) on the eve of war would seem superficially unimportant. One must, however, also consider that the _Heer's_ infantry divisions in 1939 still retained horse-mounted cavalry reconnaissance formations, one cavalry troop (essentially equivalent to a company) being assigned to each infantry regiment. Thus each division had three troops of horse-mounted cavalry, as well as a mounted squadron in the divisional reconnaissance element. Furthermore, the infantry divisions also relied on horses to pull artillery pieces, heavy machine guns, ambulances, field kitchens, and supply wagons. Consequently, each infantry division of 17,200 men would include 5,375 horses in its TOE at the time of the invasion of Poland in September 1939.
Of course, as early as 1934 certain officers had noticed the inefficiencies inherent in a too-rapid expansion of the army. Colonel Georg Thomas, then-head of the Defense Economy and Weapons ( _Wehrwirtschaft und Waffenwesen_ ) Bureau, complained directly to Hitler. In a strongly worded memorandum dated 20 June of that year, Thomas complained of the economic friction and industrial wastage caused by the often vicious rivalries among Nazi Party bosses, competing industrial firms, and "misguided interventions and opinions of individuals." The result, as he bluntly put it to the Chancellor, was that "no decisions are made." Consequently, he wrote, the economy, and implicitly Germany herself, would not survive the "coming struggle" if the economic chaos continued. Hitler eventually responded to these and similar concerns not through administrative efficiency but by adding another layer of bureaucracy. Nevertheless, he certainly did not slacken the pace.
If re-armament were to continue and to be more effective, so Hitler reasoned, then greater economic centralization would be necessary, though he envisioned no State ownership of property per se, as in the Soviet Union. Similarly, he came to demand the greatest possible autarky in foodstuffs and strategic raw materials. His demands, however, were driven at least as much by strategic considerations as by mere economic concerns: the Allied blockade during 1914–1918 had demonstrated how vulnerable Germany could be if foreign sources of food and vital materials were interrupted. In August 1936 Hitler ordered the establishment of a new national authority to coordinate the economics of rearmament. This coordination came to be embodied in a program known as the Four-Year Plan. To direct the effort and thus head the office of the Reich Plenipotentiary for the Four-Year Plan, Hitler named one of his oldest henchmen, the commander of the _Luftwaffe_ , Hermann Göring.
With Hitler's constant urging and under Göring's direction, maximum speed and breadth, but not depth, of re-armament remained the watchwords. In theory, nothing would be allowed to hinder rearmament's pace. The plan placed particular emphasis on the greatest possible domestic production of several key industrial raw materials, including synthetic oil, gasoline, and diesel fuel; synthetic rubber; non-ferrous light metals; and iron and steel. Whether the processes were cost-effective or not was irrelevant, at least to Hitler and Göring. Nor was it particularly important to them whether great stocks of surplus matériel were collected before the outbreak of war. What mattered to them was having the weapons and matériel to hand when they decided to go to war. And as Hitler said in concluding the memorandum establishing the Four-Year Plan, the German economy and the German armed forces were to be ready for war by 1940. Besides, as Hitler sometimes pointedly reminded his officers, Germany would go to war more or less when he decided, where he decided, and against whom he decided. As he quite baldly put it to his commanders in chief, everything depended upon him, upon his personality, upon his authority, upon his very existence.
In May 1939 Thomas, by then promoted to general, presented his assessment of the economy and the state of re-armament in a presentation to personnel of the Foreign Ministry. In general terms, he praised the regime's accomplishments regarding re-armament, though he also pointed out that the depth of the regime's re-armament program in spare parts and material reserves did not match the program's breadth. Referring to the army, he noted with evident pride the successful creation of the new armored formations. Interestingly, however, he also specifically noted the establishment of what he called "the modern battle cavalry." Like the armored formations, he said, these cavalry units were "completely new" and had been developed only in the preceding five years. His statement, of course, contradicted the record of steady efforts since Seeckt's day to adapt the cavalry to Germany's post-World War I military doctrine. Nevertheless, Thomas' reference to the cavalry in such a setting constituted at least an indirect indication of horse-mounted forces' continuing operational relevance. Any Foreign Ministry staffers interested in military affairs and worth their pay would have been fully aware that the Polish army at the time fielded some eleven cavalry brigades of its own, while the French army included three full cavalry divisions. Furthermore, such diplomats, like their military counterparts, would have at least assumed that Poland would be Germany's likeliest enemy in the event of war. France would probably follow. Great Britain's inclusion in such calculations remained at that point a matter of some uncertainty. Subsequently, and only fourteen days before the invasion of Poland, General Thomas briefed, among others, the chief of the OKW General Wilhelm Keitel, on the state of Germany's war economy. Thomas stated, evidently rather bluntly, that the Reich's economic preparations for war did not suffice. Germany "could not last through a war on the grounds of its war economy." Insofar as the cavalry was specifically concerned, if Germany went to war with horses still on the army's TOE, and of course they were, then it would in all likelihood still have horses soldiering on when the end came (however that might be) for the good and simple reason that there was no way to replace them with machines.
Thomas' laudatory comments and simultaneous reservations notwithstanding, and despite the Four-Year Plan's relative successes, the Reich's economic mobilization was certainly not complete when Hitler launched the invasion of Poland:
The war that broke out in 1939 cut across all [of Germany's economic] preparations. Neither foundation nor superstructure was complete; despite the growth of state planning and control, the transformation of the German economy into the instrument of super-power status was slower than expected. If war had not started until the mid-1940s Germany might well have proved unstoppable. In 1939 the whole military-industrial complex was still in the throes of expensive and lengthy construction.
This condition had immediate implications for the cavalry's continued existence not least because economic inefficiency affected the army's motorization.
Incomplete motorization on the eve of war was, in turn, a consequence of the _Wehrmacht's_ inefficient growth after 1933. This growth was too rapid to be effectively managed, a situation exacerbated by the inevitable competition for resources among the three services. Hermann Göring and the navy's commander, Admiral Erich Raeder, both enjoyed direct access to Hitler. Ludwig Beck, the chief of staff of the army, did not. He had to go through not only the office of the army's then-commander in chief, General Walther von Brauchitsch, but also, after early 1938, through Keitel and the OKW. By attempting to create the largest and strongest possible _Wehrmacht_ in the shortest possible time, Hitler in effect forced the army to rely upon much greater numbers of horses than might otherwise have been the case had re-armament been pursued with a steadier pace and clearer intent. As it was, shortages of materials and an increasingly fierce struggle among the services for what resources were available prevented a complete motorization and mechanization of the army. When combined with a certain traditionalism on the part of the cavalry itself, these factors resulted in the army's retention of large mounted forces not only for logistics and combat-support but also for the combat arm.
The Army High Command also played a role in the horse's retention. In the period between Hitler's accession and the invasion of Poland, it did not foresee a coming war's being won on the roads of Europe. The war, when it came, "was [envisioned] to be a railway war, just as 1870 and 1914 had been railway wars." Not unreasonably, the army saw railways as the most efficient means of moving the most men and matériel in the least amount of time. This view persisted despite the fact that the railways themselves constituted fixed, high-value targets for long-range interdiction either by fast-moving, horse-mounted or mechanized ground forces or, increasingly, aerial bombardment. Before 1939, when the army's leadership fretted about the transport sector, it was the railways that those leaders had in mind, not canals, air transport, or motorized road-haulage, the latter of which represented an equivalent of about only 0.5 percent of potential rail-borne capacity. Not even Hitler's vaunted project for the _Volkswagen_ , the "People's Car," met the army's motorization requirements. A military prototype entered testing only in 1940, and the army disliked it, even though it eventually entered service known as the _Kübelwagen_ owing to its tub-shaped body. The vehicle lacked sufficient all-terrain capability and was underpowered with its air-cooled engine. The armed forces were also reluctant to commit to a single machine. Instead, they continued to allow automotive manufacturers to produce too many designs of cars and trucks. By 1942 the forces employed at least twenty-nine different types of automobiles and twenty-three types of trucks. This lack of standardization had unfortunate effects in the supply of spare parts and vehicular mass-production, effects that were not even partially remedied before 1944. Whether in laying out the _Autobahnen_ , producing the _Volkswagen_ , or constructing and running the factories to build them, the automotive industry and the army did not coordinate their efforts in any way that corresponded to the Nazi regime's propaganda. On the contrary, "for most of the period [from 1933 to 1942] motorization and rearmament give the impression of being in competition not co-operation." At the same time, there were never sufficient numbers of drivers and mechanics. The regime's two principal means of such training by 1939, the motorized units of the Hitler Youth and the National Socialist Drivers Corps ( _Motorisierte_ HJ and NSKK), never produced enough personnel. Those who did eventually complete the sequential training of the HJ and NSKK were for the most part siphoned off by the army's armored divisions. That demand was compounded after 1940 by the _Waffen_ -SS, whose well-equipped mechanized divisions sucked up even more of the organizations' graduates. Simply put, there were not enough drivers and mechanics left over to meet all of the army's requirements, not to mention those of the navy and the air force.
Thus the army that went to war in 1939 suffered from a great disparity between what it wanted to do and what it could do in terms of motorization and mechanization. This discrepancy only grew between 1939 and 1944, as attested by the chief of the Organization Department of the Army General Staff, Colonel Walter Buhle. In general terms even Germany's victory over Poland, one of the most lopsided campaigns of the war, saw vehicular losses of 50 percent of machines deployed. Though the breathing space preceding the invasion in the west in 1940 allowed some expansion of reserves, especially of armored vehicles, the demands on motorized vehicles of all sorts would always outrun the Reich's ability to replace them.
The army's expansion after 1933, as well as the rapid growth of the air force and the navy after 1936–1937, never constituted the perfectly organized, centrally directed, bureaucratically efficient process trumpeted so loudly by the Nazi regime. Propaganda notwithstanding, the desire to expand the army as rapidly as possible both numerically and qualitatively always remained hamstrung by Nazism's chaotic organization and by Germany's relatively limited resources. These were problems the regime never resolved, particularly when one considers the fact that the expansion of the other armed services and the _Waffen_ -SS, when combined with that of the _Heer_ , more than doubled the raw-materials demands of the _Wehrmacht_ as a whole.
The cavalry's retention, therefore, might reasonably be regarded as an anachronism, though a necessary one. The necessity becomes evident when one considers that the numbers of cavalry, whether in the independent cavalry formations or in the horse-mounted squadrons organic to the infantry divisions' reconnaissance battalions, went up throughout the war. They did not go down. "The speed and almost unrestricted and unco-ordinated [ _sic_ ] armament of the armed services along with political factors led to the complete disregard of the lessons learned from the First World War which had earlier [i.e., in the 1920s] found total acceptance." The lessons referred to here applied to the management of Germany's national economy for war. Nevertheless, the mismanagement accompanying re-armament after January 1933 necessarily affected how the army procured vehicles, what types it procured, and how they might be employed. Thus, in addition to the army's peacetime complement of horses, 14,870 more had to be conscripted for the annexation of Austria in early 1938 and a further 4,539 for the occupation of the Sudetenland later that same year. Not only did the cavalry and horse-drawn transport and logistics still exist at the beginning of 1939, the army could not execute even a major _non_ -combat operation without them. Under the conditions prevailing in that year, horses simply could not be eliminated, and under the eventual demands of the coming war, the need for them would only grow.
Even the later panzer commander Heinz Guderian's detractors were proven wrong by these events. Early in his career when he'd been assigned to the army's Motorized Transport Department, he'd expressed the hope that one day truck-borne troops would become part of the combat arm. His immediate superior, a Colonel von Natzmer, brought Guderian up short. "To hell with combat!" snapped Natzmer. "They're supposed to carry flour!" Not only was Guderian's hope largely thwarted by events, so was Natzmer's. Instead of trucks, it would be the army's horses that would continue to haul flour and a great many other things. Very often those other things included considerable numbers of combat soldiers.
Consequently, the German army would undergo a "de-modernization" during World War II, particularly on the Eastern Front. While the causes were many, including prewar economic mismanagement, combat losses, breakdowns, and eventual disruption of replacements owing to the Allies' strategic bombing offensive, the results for the troops on the ground were profound in any case. "Armoured divisions," writes one noted author, "began the war with 328 tanks apiece; by the summer of 1943 they averaged 73; by the end of the war the figure was 54. The German army fell back on the use of horses. During 1942 German industry turned out only 59,000 trucks for an army of 8 million men, but the same year 400,000 horses were sent to the Eastern Front. The German forces concentrated their air and tank power on a few elite divisions; the rest of the army moved like those of the Great War, by rail, horse, or foot."
Organizing and re-arming the _Heer_ to include a relatively small number of mechanized and motorized formations has been described by another author using what he calls the "Lance Comparison" ( _Lanzenvergleich_ ). That is, the army by May 1940 would possess a "steel point on a wooden shaft." Ten armored divisions would constitute the sharp end. These would be backed up by six mechanized infantry divisions whose personnel would eventually be known as _Panzergrenadiere_. Following on the organizational table and in the field were sixty-one ground-pounding infantry divisions. These, like the armored and mechanized infantry formations, were deemed fully operations-capable ( _voll einsatzfähig_ ) both offensively and defensively. A further twenty-nine infantry divisions were designated as only conditionally capable ( _bedingt einsatzfähig_ ) of offensive and defensive operations, and another twenty-eight divisions were designated for defensive combat only. However, one must always bear in mind that, in addition to the dedicated cavalry brigade still in service in 1939, each infantry division had horses in its organic reconnaissance element, usually referred to as a "detachment" ( _Abteilung_ ). The standard TOE of such a detachment included 623 men, at least 260 horses (most, about 213, for the officers and men of the mounted squadron), 5 horse-drawn vehicles, and 130 motorized vehicles. Of the latter, only 2 or 3 were armored cars. Counting only the fully operations-capable infantry divisions—but not including any of the horses in those divisions' vast logistics trains—the total number of horses reached a nominal sum of 15,860. If one includes the army's 29 conditionally operational infantry divisions' reconnaissance units (again leaving out all draft horses and all of the 1st Cavalry Brigade/Division's horses), one adds a further 7,540 horses for a grand total of at least 23,400 mounts for combat personnel.
Of course, once the war began, and especially in the victorious years between September 1939 and December 1942, Germany and the rest of the world seldom saw the foot-slogging, horse-supported infantry divisions. Instead, the regime's mass-circulation propaganda organs such as the very successful "German Weekly Newsreel" ( _Deutsche Wochenschau_ ) and the lavishly illustrated magazine _Signal_ , typically and effectively depicted just the opposite. One saw seemingly endless columns of "mobile troops" ( _schnelle Truppen_ ) of which the cavalry were a part: trotting or galloping horses and, much more frequently, tracked and wheeled vehicles. Then, too, there were beguiling images of soaring fleets of bombers and fighters, majestic warships, lurking U-boats, audacious infantrymen, and daredevil paratroopers. Whenever the equine reality predominated in still photography or newsreels, it tended toward the sentimental or the militarily romantic: depictions of mounted trumpeters, officers taking victory salutes, or intrepid cavalrymen by twos and threes in the depths of Russia. Notwithstanding such images, the assignment of horse-mounted reconnaissance units to each nonmechanized/motorized infantry division; the continued existence of the 1st Cavalry Brigade/Division; and the hundreds of thousands of horses assigned the mundane but absolutely crucial jobs of pulling everything from ammunition wagons to ambulances to field kitchens, meant that the army simply could not have gone to war without the horse. And in light of the economic problems already discussed, not to mention the wartime dislocation eventually caused by the Allies' strategic bombing campaign, the horse's importance to the army's war effort would only ever increase.
Poland—1939
In September 1939 the army's 1st Cavalry Brigade had a mobilized strength of some 6,700 officers and men and nearly 5,000 horses. Its principal units (as also later in the SS Cavalry Brigade) were two mounted regiments, the 1st and 2nd _Reiter_ Regiments. Stationed respectively at Insterburg and Angerburg in East Prussia, they maintained the unit traditions of such storied Prussian regiments as the 1st Dragoons, 4th Uhlans, and 5th Cuirassiers. As will be seen, these two _Reiter_ regiments had been allowed to remain largely unaffected by the interwar decision to parcel the cavalry regiments out among the army's infantry divisions. Instead, the 1st and 2nd _Reiter_ were tasked with the mission of determining whether and to what extent horse-cavalry could still actually operate in combat as a large maneuver element. Consequently, they were brigaded and had assigned to them a bicycle-mounted infantry battalion, the latter including a motorized signals platoon. The brigade's heaviest weapons lay in a horse-artillery battalion. It comprised three horse-drawn batteries, two of which were activated for the Polish campaign. Each of the horse-artillery batteries included four 75-mm guns. Other heavy-caliber weapons could be found in the brigade's motorized antitank company. It consisted of twelve 37-mm guns and a motorized antiaircraft company of twelve 20-mm guns. Neither of these companies was activated for the invasion. Supply and administration, also not fully activated for Poland, included two light motorized and two horse-drawn supply columns; one light, motorized refueling column; a motorized medical company; ambulance, workshop, and supply platoons; and the critically important veterinary company. In its inclusion of motorized anti-tank, anti-aircraft, and signals elements, cyclist infantry, and so forth, the brigade somewhat resembled certain of the army's light divisions. On the other hand, in its horse-artillery it recalled the army's infantry divisions. Of course, the great difference between the brigade and any other unit in the army was its inclusion of two full, mounted regiments.
For the opening campaign of World War II in Europe, the brigade, commanded by Colonel Kurt Feldt, was assigned to the 12th Infantry Division. The horsemen had the mission of guarding the extreme left (eastern) flank of the division, and thereby the whole of General Georg von Küchler's Third Army's advance southward from East Prussia toward the Polish capital of Warsaw. Prussia—and East Prussia in particular—represented the heartland of the German cavalry tradition. This was the land of the Trakehner horse, and it was here that the German cavalry had executed the famous delaying action in 1914 by screening German forces as they retreated and regrouped in the face of the Russian advance prior to the Battle of Tannenberg. Now, in 1939, the 1st Cavalry Brigade performed the German horsemen's classic missions of screening the advance and serving as flank-guard as set down in the 1860s by the elder Moltke, missions that began on the morning of the invasion's first day, 1 September.
After a preliminary artillery bombardment, 1st Brigade advanced on and seized the small Polish town of Myseinice. At that time, Myseinice lay about twenty miles (32 km) southeast of the Prussian railway junction of Ortelsburg and about five miles inside the Polish frontier. Two Polish cavalry units, the _Novogrodska_ Brigade and, more immediately, the _Mazowiecka_ Brigade, were operating in the German horsemen's vicinity. It was evidently the latter's uhlans whom the 1st Brigade's troopers encountered and drove off in an actual cavalry battle on 3 September near the hamlet of Frankowo. In the same three-day period, the brigade executed another classic mission of the cavalry. Determined Polish forces had unexpectedly held up the advance of Third Army's right wing at Mlawa, a town about fifteen miles (24 km) down the railway running southeastward into Poland from the former Prussian Soldau (Dzialdowo). This resistance, in turn, blocked any further advance by Third Army toward the larger and more important objective of Modlin and thence to Warsaw. Küchler therefore ordered Third Army's left wing to swing southwestward through Przasnysz, take the Polish positions in flank on their (the Poles') right, and then push on to Ciechanów. The country through which the cavalry and other German forces would have to pass lay along both banks of the Omulew and Orzyc Rivers, two southward-flowing tributaries of the Narew. Characterized by scattered, large forests and marshy terrain, the land proved tough going and not merely for the cavalrymen. Nevertheless, the encirclement succeeded. It effectively pried the Polish defenders out of their positions at Mlawa and permitted Third Army's advance on Modlin to continue. Screening and guarding this entire operation, at the furthest and most exposed end of the line, was the 1st Cavalry Brigade. The brigade's campaign, along with that of much of the German effort, effectively ended by mid-September with the horsemen continuing to guard the eastern flank of the corps to which the 12th Infantry Division was attached in the advance up the valley of the Vistula. During the subsequent siege of Warsaw, the cavalrymen effectively used their horses' mobility to patrol the river's eastern bank and intercept scattered Polish units attempting to break through to the southeast.
Map 2. The Invasion of Poland, September 1939
In its initial action, and more particularly in the turning movement near Mlawa, the brigade itself was not the only unit executing a classic cavalry mission. The entire corps (Corps _Wodrig_ , after its commander) was acting, as it were, in the manner of the cavalry: executing wide, rapid, sweeping movements; outflanking the enemy's fortified positions or strongly held fronts; operating against his lines of communication; and either forcing him to withdraw or fixing him in position for the battle of annihilation. The German cavalry had done precisely this in 1870 as the French retreated from Metz. The result was Mars-la-Tour and the "Death Ride" of Bredow's horsemen. Subsequently, in 1914, the German cavalry had tried the same thing on the then-still-fluid Western Front, only to fail in the misnamed "Race to the Sea." Somewhat later, in 1916, German and Austrian mounted and motorized forces, accompanied by hard-marching infantry, had forced the passes of the Carpathians and carried out yet another such movement in the Rumanian lowlands, resulting in the conquest of an entire country in slightly more than a month. Now, in 1939, the army's post-1918 experimentation with what were then new technologies bore fruit in a sort of "cavalryzation" of warfare in Poland. The great and murderous difficulty of World War I, the recurrence of a war of fixed positions ( _Stellungskrieg_ ), seemed finally overcome. War of movement ( _Bewegungskrieg_ ) had apparently returned. The tradition stretching backward from Seeckt, via Alfred von Schlieffen of 1914 fame to von Moltke the Elder was restored. The soon-to-be panzer armies would expand this practice to a vast scale in France in 1940 and, even more dramatically, in Russia in 1941. For the moment, the cavalry brigade seemed to be only a small part of that tradition and a superficially anachronistic one at that. Nevertheless, the cavalrymen would ride on.
Of course, both the German and the Polish armies employed horse-cavalry in 1939. One German panzer commander, F. W von Mellenthin, recounted in a memoir of his wartime experiences that the best units in the Polish army in 1939 were "undoubtedly their cavalry brigades." These, he wrote long after, "fought with magnificent gallantry," and he went on to add, as have many others, that "on one occasion they charged our panzers with drawn sabers," a report also later noted by the commander of the 7th Armored Reconnaissance Regiment, Hans von Luck. Though this oft-told story still makes the rounds in the history of the Polish campaign, it does not appear to accord with the facts on the ground in 1939. Certainly it resonates with the cavalryman's traditional sentiment. Even so, historian M. K. Dziewanowski states flatly that no such attack ever occurred, though something like it may have been the result of a meeting engagement, in other words a chance encounter between Polish cavalrymen and German armored units. Instead, Dziewanowski maintains that the story, however tragically endearing it may be for the tradition of the Polish cavalry, was the product of the Nazi propaganda ministry's efforts to convince neutral European States, as well as Great Britain and France, that further resistance to Germany's military might was futile. This version is supported by a recent history of the war that pays special attention to the Eastern Front. There the tale is of a Polish cavalry regiment hiding in a forest and subsequently attempting to escape but being encircled by German mechanized forces. Unable to break the ring, the Polish riders are decimated by German tank guns, and the account is then spun into the propaganda myth of Polish cavalry stupidly attacking German tanks. This individual incident aside, Dziewanowski nevertheless concedes the obvious: namely that the Polish army of 1939, though unfailingly brave, sorely lacked sufficient motorized transport and possessed "only two recently organized armored brigades." Even so, the Polish cavalry was never so benighted as some have thought. They had, for example, experimented with the mixed divisional organization just as the Germans and French had done between the wars, and in 1939 each Polish cavalry brigade had an armored troop on its TOE.
The tragic consequence of the technological disparity between the Polish army and the _Heer_ , so far as cavalry was concerned, was the destruction of the _Pomorske_ Cavalry Brigade, the very incident referred to by Dziewanowski. Within about three days of the invasion's beginning, the _Pomorske_ Brigade and a number of other units were cut off in the corridor that had separated Germany proper from East Prussia since 1918. Whether the horsemen of this famous unit actually charged German tanks in massed assault or were ground up in a series of meeting engagements is of much less consequence than the final result, for in the general and desperate attempt to break out southward, the brigade was destroyed. Much the same end awaited other Polish cavalry formations, the "pride of the army" as the _New York Times_ described them, as they retreated along with the remaining Polish forces toward the line of the Rivers Narew and Vistula. One of these cavalry formations was the 18th Lancer Regiment commanded by Colonel Kazimierz Mastelarz. In an action that may well have been the genesis of the entire story of horsemen versus tanks, the 18th Lancers attacked and overran a weak German infantry position during the fighting in the Corridor. The lancers rode on, only to encounter several German armored cars whose gunners made short work of the cavalrymen. Notwithstanding such losses, the Polish horsemen during the campaign always hoped to have a better chance at least against German motorized columns and infantry, if not against mechanized formations, in the low-lying, often sodden, and frequently forested watersheds of the Vistula and Narew, precisely the sort of ground that the 1st Cavalry Brigade had successfully negotiated in its advance toward Mlawa. Despite their many reverses, Polish horsemen did occasionally score successes against the invaders. On the night of 1–2 September, for example, Polish cavalry had panicked the staff of the German 2nd Motorized Division into hastily ordering a retreat from its assault into the Tuchel Heath between the Prussian towns of Firchau and Grunau. Only a direct intervention by the corps commander, General Heinz Guderian of XIX Panzer Corps, rectified the situation when he berated the divisional commander and tartly asked whether the latter had ever heard of Pomeranian Grenadiers being broken by enemy cavalry.
In the final analysis, overall German superiority against Poland in 1939 meant that localized Polish successes and occasionally heroic Polish resistance could not stave off German victory, especially given the Soviet invasion of eastern Poland on 17 September under the terms of the Nazi-Soviet Non-Aggression Pact of the preceding August. Thus what the Germans unofficially called the "campaign of eighteen days" didn't really seem to give the German cavalrymen a chance to prove whether they could still fight effectively on the modern battlefield. The _Pomorske_ Brigade's fate seemed to represent the fate of all mounted forces in the dawning age of truly mechanized and motorized warfare. Nevertheless, if that fate were already sealed, then it is worth noting the irony that the invasion "left half the German tanks and motorized vehicles out of action" against a foe not possessing a truly effective anti-armor capability. Under the circumstances, therefore, OKW still did not yet know how the Germans' own cavalry would ultimately fare. Perhaps an expansion of the cavalry was necessary in order to provide the potential for the horsemen's true operational independence. To try to determine the answer to this question, the 1st Cavalry Brigade was reorganized after the end of the fighting in Poland. It was augmented with troopers from other mounted regiments whose personnel had been assigned to various infantry divisions' reconnaissance battalions in keeping with the army's doctrine from the late interwar years. Thus were added two new mounted regiments, eventually designated the 21st and 22nd _Reiter_ Regiments. Additional elements, such as a second horse-artillery battalion, increased the brigade's strength to that of a full division.
The West—1940
Whatever disparities had existed in the Polish campaign between the capabilities of the cavalry and the mechanized and motorized units of the _Heer_ , the mounted brigade's combat effectiveness seemed to warrant expansion to divisional status in the last quarter of 1939. As earlier with the Cavalry Brigade, so now in the 1st Cavalry Division, the principal maneuver elements remained the horse- and bicycle-mounted units, with the horse-artillery still delivering the heaviest offensive punch. The recognition of the cavalry's contribution through its expansion to divisional status was no doubt gratifying to its personnel. It should also be noted, however, that others also watched keenly. The cavalry's relative success to this point in the war was not lost on the Nazi Party's SS formations. Their leadership, too, had already begun deploying horse-mounted units in occupied Poland for rear-area security duties. These units' subsequent organization and mission would in certain respects ape the Cavalry Division's own, even if SS men's ethos didn't.
Be that as it may, by the spring of 1940 and with the impending campaigns in the west, and following the unprovoked invasions of both Denmark and Norway in April, the newly redesignated 1st Cavalry Division was assigned to Küchler's new command, Eighteenth Army of German Army Group B. As had been the case in Poland, the horsemen found themselves on the far end of the line, this time the extreme right (northern) flank of the army. It remains an open question whether the division was assigned there merely to keep it out of the way or so that it could once again perform the traditional mounted troops' mission of providing security and screening the flanks of an advance. Presumably its assignment there was intended further to test the operational viability of a now-division-sized, mounted maneuver-force. However that may be, the cavalrymen's assigned frontage began at the Dollard, a bay of the River Ems lying directly to the south of the German port of Emden. From the Dollard the division's area of operations stretched some sixty-two miles (100 km) to a point just southwest of the German city of Lingen. The horsemen's mission was to break through Dutch frontier obstacles—a zone judged to be about twelve miles (20 km) deep—and occupy the northern provinces of Groningen, Drenthe, and Friesland. The cavalrymen would then face south, be shipped across the Ijsselmeer, and break into "Fortress Holland," the quasi-official Dutch name for the core defensive area of the Netherlands.
Map 3. The First Cavalry Division in the Netherlands, May 1940
The division's command assessed its assigned mission area as "the most unsuitable terrain imaginable" ( _das denkbar Ungünstigste_ ) for mounted operations: numerous rivers, countless bridges, and virtually bottomless footing ( _ungangbare tiefe Gelände_ ) made movement "almost impossible." The fact that the Dutch forces facing them were motorized made the cavalrymen's task of maintaining contact with the enemy even more difficult. Nevertheless, the division reported that on the campaign's first day, 10 May 1940, things had gone well. In several areas the cavalrymen had outflanked Dutch pillboxes at the gallop and had penetrated enemy territory to a depth of several miles within the first twenty minutes of fighting, and this over terrain "impassable for motorized vehicles." By day's end, the initial objective, a line running southwest from the city of Groningen to Meppel and roughly bisecting the whole of northern Holland, had been reached. By the end of 11 May the horses of the division's most advanced elements were being watered on the coast at Harlingen and along the northeastern shores of the Ijsselmeer, follow-on units reaching these areas over the next five days. Remarkably, the division's _Reiter_ regiments had covered some 111 miles (180 km) in two days, a feat indicating less than fanatical Dutch resistance. On the coast, however, the division's troopers ran headlong into strong fortifications along the edge of the Ijsselmeer. At the northern end of the dike separating that body of water from the sea and linking the province of Friesland with North Holland farther to the south stood a number of modern fortifications at Kornwerderzand. Ordered to storm the casemates and gain control of the dike, the division's (cyclist) infantry, dismounted horsemen, and horse-artillery nevertheless failed to overcome the Dutch defenses despite repeated assaults on 12–13 May. Even supporting attacks by _Luftwaffe_ dive-bombers and shelling by a battery of 88-mm guns that appeared on the scene could not blast the defenders from their positions. The point was moot in any case. Too high a sea-state and a galling fire from three Dutch naval vessels would have prevented the division being ferried across the Ijsselmeer to fight in "Fortress Holland" even if Kornwerderzand had fallen to the invaders. Ultimately, however, such reverses for the cavalrymen made no difference. The invasion's outcome was decided much farther to the south where the principal fighting had occurred. On 14 May, Dutch forces capitulated.
After a campaign of only one week, the division received orders on 17 May to return to its original main assembly area around Lingen pending further assignment. By the time the _Reiter_ regiments completed their reassembly three days later, they had ridden almost 435 miles (700 km) in ten days. With the exception of the fighting at Kornwerderzand, all assigned objectives had been achieved, and all of the division's vehicles had been successfully brought along. After-action analysis maintained that the division's advance would not have been possible had it been forced to rely solely on vehicles. Though difficult for horses, the sodden terrain would have been impassable for vehicles, though the same report pointedly failed to explain how the Dutch repeatedly managed to use their own vehicles to escape. On the contrary, the division's horses had made possible the initial skirting and envelopment of Dutch frontier defenses by means of tracks ( _Wege_ ) that only horses could use. Even so, divisional staff deemed it particularly important that the (unspecified) casualties among men and horses be replaced and that all of the latter be reshod before any possible action in France. Where the horsemen had ridden hard-graveled roads ( _Klinkersteinstrassen_ ), they found that their horses' shoes had worn badly. No one wanted worn horseshoes to interfere with the entire division's sole desire: to take part in an invasion of France. To ensure that the point was not lost on higher echelons, and perhaps to stave off army- or army group-level criticism of the division's performance, the report struck a defensive note and argued that the fighting in the Netherlands simply could not be used to determine whether a horse-mounted cavalry division was still effective. Between 24 and 28 May, the 1st Cavalry Division moved by rail to the area around Aachen where it encamped and awaited its next assignment. Twice in this period, Army Group B issued orders that the division would handle the transport of POWs to Germany. The division's records dryly noted that these orders badly lowered morale ( _die Stimmung in der Division_ [ _sank_ ] _erheblich_ ). A reprieve, however, arrived on 28 May. Orders came that day for the division to move as rapidly as possible to Amiens for frontline service with Fourth Army, a movement that required a march of 310 miles (500 km) and an early-morning crossing of the Somme on 7 June.
In France the division was reinforced with an additional bicycle-mounted infantry battalion, motorized howitzer and heavy artillery detachments, and an anti-aircraft machine gun company. It now constituted a sort of reinforced division. The artillery was deemed particularly important in making the division fully operationally independent. But as the army group giveth, so the army group taketh away. Thus, at the same time that the additional units came in, important brigade-level staff elements were ordered seconded to other units, and the division's subsequent operations suffered accordingly.
As in Poland and the Netherlands, the cavalrymen once again received a classic cavalry mission: guarding the flank of an advancing unit, this time the eastern or left flank of XXXVIII Corps. On 7 June divisional elements crossed the Somme just northwest of Amiens. Encountering no significant French forces, the horsemen drove on in a southwesterly direction. In the so-called Poix District, a task force formed around the division's own 22nd _Reiter_ Regiment and the temporarily assigned 21st _Kavallerie_ Regiment rode hard against increasingly heavy French resistance. In this instance the horsemen actually did what virtually everyone assumed would never again be done: they charged the enemy en masse at the gallop. The cavalrymen attacked through artillery-fire, successfully gained defilade in a streambed, and then stormed heavily defended high ground on the other side to capture six hundred French troops. By the end of the next day, divisional elements had reached the area around St. Omer. The following day's operation on 9 June witnessed a bitterly contested, daylong advance of about thirty-seven miles (60 km) resulting in the cavalry's destroying twenty-eight of about thirty French tanks deployed to block their march. Sporadic fighting continued over the next two days, and by the end of 11 June the division had bypassed Paris and had reached a line running roughly southwest from Pontoise to Mantes. No French forces remained north of the Seine in the division's area of operations. At this point the 1st Cavalry Division was transferred to VIII Corps of Eighteenth Army, one of several reassignments during this hectic advance. Simultaneously, it had to give up the motorized artillery and howitzer detachments earlier attached to it. Its flank-guard mission, however, remained unaffected by the removal.
Meanwhile, on 1 June the division's 1st _Reiter_ Regiment had attempted a forced reconnaissance across the Seine by rubber boats, all bridges having been destroyed by retreating French forces. Surprised by concealed enemy artillery on the south bank, the boats were shot to pieces. A second attempt the next morning to send scouting parties across the river succeeded. The French had withdrawn during the night. Nevertheless, the division reported to Eighteenth Army that it would take seventy-two hours for all personnel and horses to cross. The latter—some 12,000 of them—had to swim the river aided by floats ( _die Seine an Floßsäcken durchschwimmen_ ) while the motorized units crossed via a bridge built far to the northwest by German combat engineers at Vernon. Eventually, all divisional elements found themselves on the south bank of the Seine by sunset on 15 June.
Map 4. The First Cavalry Division in France, June 1940
In the week leading to the Franco-German armistice, the Cavalry Division continued its advance to the south while bypassing Chartres on the city's western side as part of the larger German offensive. On 16 June alone, units of the division advanced nearly sixty miles (95 km), the bicycle-battalions in particular seeing occasionally hard fighting. When the next day passed uneventfully, commanders reckoned on the division's being withdrawn from the front in the face of the evident weakening of French resistance. Instead, orders arrived directing the division to advance still farther to the Loire. There the cavalrymen were to throw bridgeheads across the river in the vicinity of Saumur. The historic irony, surely lost on no one, was that Saumur housed the French army's Cavalry School.
At this stage, ad hoc divisional advanced parties of bicyclists and vehicles were followed by the _Reiter_ regiments. French resistance was weak, and large numbers of French prisoners streamed back along the division's line of advance. After a march of nearly 125 miles (200 km) from Chartres, the cavalrymen reached the north bank of the Loire on the afternoon of 19 June. Not surprisingly, most of the bridges around Saumur no longer stood, and when one of the division's bicycle-mounted units tried to a take a remaining span by quick assault, the French blew it up as the first soldier crossed it. Nevertheless, troopers of the division managed to cross the Loire to the east of Saumur, and horsemen of the 22nd _Reiter_ Regiment even seized the bridge at Le Port Boulet when French efforts to demolish it failed. Having thus crossed the great water barrier of the Loire on 20 June, the same regiment pushed on and succeeded in crossing the Vienne, one of the Loire's eastern tributaries, by seizing an intact bridge at Chinon without loss. There they also managed to completely surprise and capture a French cavalry detachment from Saumur consisting of forty officers and two hundred men. In the next two days, the division broke out of the bridgeheads south of the Vienne. Advancing in the direction of La Rochelle, the horsemen repelled several occasionally fierce French counterattacks, including ones employing armored cars, and reached the coast on 23 June. On that day at La Rochelle, news of the armistice reached divisional headquarters. The 1st Cavalry Division's war in the west was over.
In assessing the campaign, divisional commander Kurt Feldt, by this time a major general, maintained that his troopers' greatest successes had come in forcing the heights in the Poix District and seizing the bridges at Le Port Boulet on the Loire and those on the Vienne. By far the most notable combat victory, however, was the destruction of "an entire tank battalion" on 9 June. By temporarily stripping the division of its heavy artillery and repeatedly changing its corps assignments, however, higher commanders had constantly created disadvantages. Nonetheless, the division had "completely accomplished" ( _restlos erfüllt_ ) every mission assigned to it. As for the division's future, Feldt wrote that the cavalry should not be regarded as a "fast force" ( _schnelle Truppe_ ) in an age of motorization, even though the cavalry had been officially regarded as such in 1939. It simply could not maintain contact in places where good roads made an enemy's motorized retreat possible. On the contrary, the cavalry's principal advantage lay in its tactical mobility on the battlefield: it was independent of roads and the limiting factor of weather, bicycles notwithstanding. "As long as technology makes the leadership dependent on weather and the season of the year [Feldt might well have added "the terrain"], the horse has not lost his role." But he also believed that the cavalry's future depended on whom Germany's next continental enemy might be. That enemy, he wrote, would always be looked for "in the East." Given Poland's already occupied status, Feldt could only mean the Soviet Union. There weather conditions and the vast, largely open, and often roadless spaces were made for horsemen. There "the motor is not yet the sole ruler." Even there, however, Feldt maintained that the cavalry made sense only if it were expanded to a corps of two or three divisions. That force-structure and nothing less, Feldt urged, would be capable of the true operational freedom and decisive combat effectiveness that would justify the horsemen's continued retention.
At almost the same time that Feldt wrote these words, his French cavalry counterparts were being destroyed. In May 1940 French general Charles Huntzinger, commander of the French Second Army, had employed several large, forward-deployed cavalry units in the forested heights of the Ardennes. Though the Ardennes' highest elevation reached not much more than 2,500 feet above sea level, these often heavily wooded highlands presented to the armies of any eastern invader a rugged terrain of "deep and meandering defiles cut by myriad creeks and small rivers" that would "sharply restrict and canalize movement either along or across the grain of the land." This was the region, claimed by some German nationalists as an integral part of Germany as early as 1813, where German and French cavalry patrols had hunted each other at the outbreak of World War I. In 1914, French general Charles Louis Marie Lanrezac was reputed to have described the Ardennes as a death trap from which, once entered, no one could return. Whether his observation, if made, had actually typified French attitudes toward the region at that time, the Germans in 1940 had certainly intended to test the theory in a way that they hadn't done in 1914. Rather than marching the bulk of the field armies around the Ardennes as in 1914, OKW sent a massive armored force crashing straight through the hill country in the hope of cutting off and eventually destroying French and British forces marching northeastward to meet the German invaders in the Low Countries.
To some German commanders, the prospect of success, much less a rapid one, in such a daring undertaking had seemed dim. Representing that view, General (later Field Marshal) Wilhelm Ritter von Leeb, commander of German forces facing Alsace-Lorraine, noted in his diary that the French were expecting a drive through Belgium. Surprise was impossible. The invasion, he wrote, could not be waged as in Poland. The army would have to fight a drawn-out ( _langwierig_ ) campaign against much more capable opponents and would suffer extremely heavy losses in the process. Even then, he feared, "the Frenchman would still not be able to be subdued." The campaign came nonetheless.
Unlike 1914, however, French and German cavalry did not hunt each other in the tangles of the Ardennes. Instead, only the French cavalry were preyed upon, and they didn't face their German 1st Cavalry Division counterparts who were in the Netherlands at the time. Instead, they faced German tanks and mechanized infantry. Still, like their German horse-mounted cousins, their mission remained a traditional one for the cavalry arm: screening Huntzinger's main line of resistance running south of the Meuse from the town of Mézièrs, past the battlefields of 1870 at Sedan, and thence to Montmedy where his troops approached the westernmost extremity of the Maginot Line. Units under Huntzinger's overall command included the 2nd and 5th Light Cavalry Divisions (LCD) and the 1st Cavalry Brigade. The French 1st and 4th Light Cavalry Divisions and the 3rd Spahi Brigade were assigned to the adjoining Ninth Army under the command of General Andre Georges Corap. The 2nd and 5th LCDs were mixed divisions, the type with which the German army had experimented in the 1920s. The 5th LCD, for example, included an artillery regiment, a mechanized infantry brigade of two regiments, and a mounted brigade, also of two regiments: the 11th Cuirassier Regiment and the 12th Cavalry Regiment. In the Germans' passage through the Ardennes and their drive to the Meuse crossings at Sedan, these mounted troops were unceremoniously brushed aside by the advancing 1st Panzer Division of XIX Panzer Corps. In all, the French cavalry screen delayed the German advance through the Ardennes by just over two days instead of the five days envisioned by French prewar planning. As General Erwin Rommel wrote, victory in the encounter battles of the Ardennes between the advancing Germans and the Franco-Belgian defenders usually went "to the side that is the first to plaster its opponent with fire. The man who lies low and awaits developments usually comes off second best." Consequently, the results were about the same on both Huntziger's front and on Corap's. Rommel's 7th Panzer Division's tanks and mechanized infantry, for example, often advanced while conducting what U.S. soldiers thirty years later in Viet Nam would call reconnaissance by fire: maintaining as heavy and continuous a fire as possible into the woods on both flanks and to the front of every advancing column. "Into these woods, the French cavalry, horses and tanks mixed up with one another, scattered in disorder." The typical result was poignantly captured on 12 May by Lieutenant Georges Kosak of the 4th LCD of Corap's Ninth Army as French forces retreated to Dinant on the Meuse:
Towards midday, groups of unsaddled horses returned, followed on foot by several wounded cavalrymen who had been bandaged as well as possible; others held themselves in the saddle by a miracle for the honour of being cavalrymen. The saddles and the harnesses were all covered with blood. Most of the animals limped; others, badly wounded, just got as far as us in order to die, at the end of their strength; others had to be shot to bring an end to their sufferings.
Ultimately, a lack of proper coordination between French cavalry and the Belgian light infantry in the Ardennes prevented their using that rugged terrain to best advantage in delaying the German invaders. Conversely, when proper coordination occurred, defensive planning proved too methodical or too weak for the Germans' war of movement. The French cavalrymen, whether horse-mounted, on board vehicles, or dug in, "refrained from holding key positions as long as possible and, therefore, were never in a position to stop the German attackers for any length of time." They "got no rest, neither during the day nor at night [and] their delaying actions frequently degenerated into wild flight." One could scarcely imagine a greater contrast to General Feldt's triumphant summation of the successful campaign of the German 1st Cavalry Division.
Mounted band on parade at an equestrian tournament. Note the ornamental drum-skirts and the kettle drummers' stirrup-reins. The latter permit the rider to guide the drum horse while leaving the hands free. Close examination shows that the kettle drummers and at least one of the buglers wear the pre-1939 "cavalry helmet" with ear cut outs. ( _From the private collection of the author._ )
Hitler presides at an international equestrian tournament in Berlin. Propaganda Minister Joseph Goebbels is at right. ( _From the private collection of the author_.)
German cavalry on parade. The original caption of 1940 referred to the cavalry of the "Army of the Greater German Reich," thus implying a date subsequent to the annexation of the Sudetenland. The venue appears to be the annual Nazi party rally at Nuremberg. However, the Parteitag of 1939 was cancelled owing to the war's outbreak. Therefore, this picture may well date from 1938 or earlier. ( _From the private collection of the author._ )
Undated photo shows the victorious equestrian team of the German army's Cavalry School. The apparently altered background hints strongly at a photomontage. Note the convex or "Roman" nose, accentuated by the white blaze, on the horse at right. ( _From the private collection of the author._ )
A striking, late-1930s example of a Trakehner (East Prussian) mare. The Trakehner was prized by the officers and men of the Prussian cavalry from the days of King Frederick William I to the end of the World War II. Though the breed barely survived Germany's defeat in 1945, Trakehners thrive today as sport horses known for their athleticism. ( _From the private collection of the author._ )
Horse-artillery on the march. Helmet bands, as seen here, were frequently worn on peacetime maneuvers. Note the saddle scabbards for the outriders' Mauser 98k carbines. Spoked, wooden wheels on gun carriages and caissons gave these units their nickname, "gypsy artillery." ( _From the private collection of the author._ )
An SS rider takes a jump. The SS aped the equestrian manners of the army's cavalry units. ( _From the private collection of the author_.)
An evocative pre-1945 depiction of horse-breeding country in the valley of the Weser River. This area, as well as the adjoining region along the Aller River, comprised the ancestral home of Hanoverian horses. The Hanoverian became justly famous as the German army's second great type of cavalry mount. ( _From the private collection of the author._ )
German horsemen laying telephone communications wire. Note the M1925 cavalry saddle and standard-issue, folded, woolen saddle blanket on the horse at left. The "cavalry helmet" with ear cut outs has now been replaced by the standard infantryman's _Stahlhelm. (From the private collection of the author._ )
German cavalrymen, not to mention army officers more generally, were expected to ride and, if possible, to compete. Here horse and rider take a combined fence-and-water obstacle in dramatic fashion. ( _From the private collection of the author._ )
A postage stamp commemorates the "Brown Ribbon of Germany," a major horse race run annually from 1934 to 1944 at Munich-Riem. Riem was the location of the SS Main Riding School. ( _From the private collection of the author._ )
_Reichsführer_ -SS Heinrich Himmler presents awards to a victorious SS equestrian team in this undated photo. Himmler put great emphasis on sporting accomplishments and physical fitness for SS men, though he himself was no athlete, possessed only a weak constitution, and suffered from poor eyesight. ( _From the private collection of the author._ )
First adopted in Prussia in 1796 and modified only slightly in 1916, the Light Cavalry Saber remained standard issue for German cavalrymen until 1941. Troopers of the 1st Cavalry Brigade/Division carried this weapon in Poland in 1939 and the Netherlands and France in 1940. ( _Saber courtesy of Cold Steel, Inc. Photo by the Office of Public Information, Western Carolina University._ )
CHAPTER 7
BARBAROSSA
THE 1ST CAVALRY DIVISION IN RUSSIA, 1941–1942
General Kurt Feldt's assessment from summer 1940 concerning Germany's next enemy was not far from the mark. Though intervening campaigns in spring 1941 led to the conquest of the Balkans and Crete, and though German forces found themselves dispatched to North Africa as well, Hitler's real enemy always lay in the east. The non-aggression pact with Stalin would last not one minute longer than necessary. Along with several million other soldiers, the German cavalrymen would play their part in what would become by far the largest land war in history.
With the invasion of the Soviet Union in June 1941, the 1st Cavalry Division initially found itself operating in the area of the Pripet Marshes. Straddling the borders of prewar Poland and Russia, the marshes stretched away eastward for more than two hundred miles (321 km) from a point near the River Bug and the city of Brest-Litovsk on the border between German- and Soviet-occupied Poland to the Berezina River and the cities of Bobruisk and Gomel. Covering, according to one estimate, a vast area of nearly 100,000 square miles (258,998 sq/km), the region called _Polessia_ ("woodlands") by Poles formed the largest contiguous wetland in Europe. Somewhat reminiscent of the lakes and moor land around the East Prussian State Stud at Trakehnen, though never effectively canalized, the marshes' relatively dry areas were interspersed with sand dunes, innumerable meandering streams, and, occasionally, larger rivers such as the Pripet and Horyn. Indeed, the Pripet River was the only significant stream flowing more or less west to east (i.e., roughly parallel to the path of the German invasion) in the whole of the western Soviet Union. Its and the Horyn's tributaries, and most other rivers in the western Soviet Union, flowed perpendicular to the Germans' axis of advance, thereby creating innumerable large and small riparian barriers to the invaders and requiring "an infinite number of bridges." All of the Pripet Marshes' watercourses drained a shallow basin that at its widest extent stretched some one hundred miles (160 km) north to south. Extensive forested belts included thick stands of willow, birch, and alder. In 1941 the Pripet Marshes sheltered a sparse human population living close to the land on both sides of the border in villages and small towns. Pinsk remained one of the largest of the latter and also served as an administrative center. In addition, the marshes also provided extensive habitat for wolves, wild boar, waterfowl, millions of mosquitoes, and eventually Soviet partisans.
Twice a year the marshes expanded. In the spring, runoff from snow-melt raised water levels and caused rivers and streams to flood. In the fall, for a period of about four weeks, autumnal rains repeated the process until the first hard frost. Cross-country movement for foot soldiers could be agonizingly slow in the marshes. For vehicles it was often literally impossible. The roads that did exist were usually no more than unimproved lanes and usually so narrow that military vehicles could neither detour nor turn around. German combat engineers could, and did, build bridges and corduroy roads in the marshes from readily available logs, but a motorized convoy's speed over such surfaces was limited to about five, kidney-pounding miles per hour (8 km/h). By contrast, horse-mounted columns possessed real advantages. They could move at least as fast as vehicles in the marshes' terrain. They didn't need corduroy roads and could usually ride easily on dirt tracks, wet or dry. They didn't need much, if any, gasoline. If necessary, cavalrymen's horses could forage for their fuel, and they could usually outdistance marching infantry with ease, even at the walk. Mounted or not, however, German soldiers always had to remain watchful, for their maps did not always reveal the true nature of the ground, whether in the Pripet Marshes or in the hundreds of other large, forested wetlands they encountered. What often appeared to be, and what maps often depicted as, meadowlike flats were frequently covered with a sort of dark turf effectively floating on a glutinous, apparently bottomless substratum. The "slightest pressure" would break the turf, and the result would be a "motor vehicle swallow[ed] to its very top" or a horse to its haunches. Such deficiencies in intelligence, whether on maps or in other respects, were later admitted by German veterans to have been a significant contributory factor in their underestimation of the difficulties inherent in conquering a country as immense as Soviet Russia.
As in much of the Soviet Union, the expanses of the Pripet Marshes remained largely bereft of modern transportation arteries. "The entire Soviet Union had only 51,000 miles of railroad, all of a different gauge than those in Germany and eastern Europe. Of 850,000 miles of road, 700,000 were hardly more than cart tracks; 150,000 miles were allegedly all-weather roads, but only 40,000 miles of those were hard surfaced." On the edges of the marshes themselves, roads and railways ran roughly northeast from Brest-Litovsk to Minsk and others almost due east from Brest-Litovsk to Gomel. Connecting the latter two cities was yet another stretch running generally southeast from Minsk via Bobruisk to Gomel. The Pripet Marshes thus found themselves enclosed in a sort of triangle with Minsk at the apex. Within that triangle, however, almost all movement frequently reverted to walking speed at best, particularly in the wet seasons. In the event, that very factor, combined with the marshes' unique topographical features, made the basin of the Pripet River a haven for Red Army soldiers scattered from their units in Operation Barbarossa's initial stages. Eventually added to those stragglers came organized bands of Soviet partisans and civilians fleeing the German invaders. All such persons thus became a major concern for the German 1st Cavalry Division in 1941 (and later for the principal cavalry unit of the _Waffen_ -SS), for the marshes could not be safely bypassed. If they were, then partisans could pounce at will on the logistical columns following in the wake of the then-steadily advancing German frontline formations.
Aggravating this situation was an even greater practical problem. The Pripet Marshes lay astride the operational boundary line dividing the Germans' Army Group Center and Army Group South. Consequently, not only the southern sectors of Army Group Center's axis of advance but also the northern sectors of Army Group South's could be attacked by Soviet partisans holed up in the marshes. Cavalry units might therefore not only be effective in combing the swamps of enemy fighters, as the Germans' situation reports often said. They might also help play the crucial role of operational liaison through the marshes between the two army groups until the German advance reached the Berezina.
This set of problems regarding German motorized and mechanized formations' lack of mobility in the Pripet Marshes casts an interesting light on an issue that had confronted German cavalry in 1914. In that earlier conflict, the great masses of horsemen advancing through Belgium were significantly hampered by the heavily built-up nature of the landscape in Flanders. Their mobility was impeded by numerous villages, walls, fences, hedges, canals, industrial plants, slag heaps, railways, and rivers. In 1941 it was German motorized and mechanized formations that found themselves in an analogous situation regarding the vast extent of the Pripet Marshes and other forested wetlands. The shoe was now on the other hoof: it was the mechanized and motorized forces that either could not move at all or, when they could, then only with difficulty. By marked contrast, the horsemen of the 1st Cavalry Division (and later, once again, the cavalry of the _Waffen_ -SS) could operate more easily in this difficult country and thereby still render valuable service to the German army.
Interestingly enough, that army wasn't alone in having leaders who thought that mounted forces might still play a useful role in such country. The Red Army, too, was aware of the value of horsemen in the region of the Pripet Marshes and elsewhere. Along with partisans, the Red Army sometimes deployed its own cavalry units in what eventually became a sort of front-behind-the-front. To note merely one example, in the fall of 1941 a motorized German supply column of thirty trucks with trailers heading to Bobruisk on the eastern leg of the road-triangle enclosing the marshes ran headlong into elements of a Russian cavalry force of some 2,500 men moving into position in a forested area south of Minsk. Heavy fighting ensued with the Russian horsemen bringing up anti-tank guns. They destroyed several vehicles and the German column managed to extricate itself only with the timely arrival of infantry reinforcements, including units of a machine gun battalion. In this particular battle the Russian cavalry in question may have been a unit isolated in the initial German invasion. Further, there would not be pitched battles in the Pripet Marshes between German and Russian horsemen as such. Nevertheless, whether intentional or not, the very fact of Russian cavalrymen's presence in such numbers both then and later indicated that the Red Army thought mounted combat forces still had a role to play. Indeed, in the summer of 1941, the Red Army undertook a massive expansion of its cavalry arm. In July, precisely when German cavalry were beginning operations in the Pripet River basin, orders went out from the Soviet high command ( _stavka_ ) establishing thirty new light cavalry divisions of 3,447 horsemen each. Later in the same year the number of such divisions rose to eighty-two. Though the expansion may have been driven by alarm as the Soviets attempted to fend off a seemingly unstoppable onslaught, the fact remains that the Red Army high command felt that mounted forces were still useful. As events would show, these same Russian horsemen would prove effective in the long-range, guerrilla warfare that accompanied the immobilization of mechanized forces during the winter of 1941–1942. Furthermore, it is worth noting that in the specific instance cited here, the Russian horsemen executed a traditional mission of the cavalry: interdicting the enemy's supply columns and creating havoc in his rear. The Red Army's cavalry were also deemed to have an advantageous psychological effect upon German infantry. Soviet marshal S. K. Timoshenko, commander of the Red Army's Western Front (i.e., army group) in 1941, noted that German infantry tended to flee before onrushing horsemen and simply go to ground.
Whether in the Pripet Marshes or elsewhere, the war facing the German 1st Cavalry Division and all other German forces in 1941 and afterward in Russia was unlike any in modern European history. For sheer savagery, nothing surpassed it. From the beginning to the end, the war on what the Germans called the Eastern Front was a war of annihilation sparing no one. As is now universally recognized, Hitler's intentions were clear. The war in "the East," as the Soviet Union was so often called in Nazi, and even non-Nazi, parlance, had as its objective not only the conquest and colonization of that country's land but also the suppression and, if necessary, the extermination of her people. Not all German soldiers would have seen the war in that fashion, but all too many did; and while the murderous treatment of the peoples of the Soviet Union was the particular hallmark of the formations of the SS, the German army did not escape complicity.
Some senior army commanders refused to publish orders issued by Hitler via the OKW calling for the summary execution of communist party officials (the "Commissar Order") or effectively exonerating Germans in advance for whatever atrocities they might commit. Still, those same senior commanders were sometimes out of touch with junior officers and enlisted troops who had come of age politically during the Nazi years and who tended to be more ideologically committed to support the regime's heinous policies. In addition, years of nationalist prejudice against Slavs, as already witnessed during the war of 1914–1918, affected how the army behaved:
A large portion of the Wehrmacht regarded the Soviet people as bumbling and potentially treacherous subhumans. In itself this is by no means a unique psychological failing. Soldiers feel the need to dehumanize or demonize their opponents in order to overcome the natural reluctance to kill, and atrocities have all too frequently ensued. In dealing with Soviet prisoners and civilians, however, this unofficial German attitude produced widespread instances of brutality and murder. Quite apart from the moral implications of such conduct, the German behavior served to alienate potential allies and to spark widespread resistance.
The same widespread resistance would greatly augment that of the Red Army and create the very front-behind-the-front whose dangers the men of the German 1st Cavalry Division would so often have to face.
They began this task when the division crossed its start line, the River Bug, on the invasion's first day, 22 June 1941. Along with the 3rd and 4th Panzer Divisions and the 255th Infantry Division, it constituted an element of XXIV Panzer Corps. Not surprisingly accompanied by some confusion, the crossing of the Bug was nevertheless successful everywhere in the division's area of operations around Slawatycze, just south of Brest-Litovsk. Of course, there arose the inevitable unanticipated difficulties. Elements of the division's 1st _Reiter_ Regiment, for example, encountered unexpected resistance from a line of bunkers on the river's eastern bank. In addition, unusable fords and unfavorable riparian shorelines prevented the division's horses being swum across. Consequently, captured bridges had to be used to make the crossing, this being made easier by the fact that "every bridge on all the border rivers from the Baltic to the eastern tip of the Carpathians" was seized within several hours of sunrise.
Initially the division had the mission of guarding the southern flank of XXIV Corps' 4th Panzer Division advancing toward Pinsk, the local administrative center of the western reaches of the Pripet Marshes. The route meant that the cavalrymen would be skirting through the marshes' northern region. Evidently, the mounted elements of the division adequately managed the terrain. The division's motorized vehicles, however, found the sandy soil rough going even though it was still relatively dry, the spring rains and snowmelt having gone, the autumnal rains having not yet begun. This same deep, powdery sand completely exhausted marching infantry, wherever they were found. To the clear frustration of the cavalrymen's commanders, the main paved road running eastward, the so-called _Panzerstraße_ or _Panzer-Rollbahn_ , was reserved for armored and motorized-infantry units. Consequently, the Cavalry Division's vehicles made slow progress and led General Feldt to report that his entire unit would not be "combat capable" ( _Kampffähig_ ) until his vehicles could get back on the road. Nevertheless, throughout the period to 30 June, the division pushed hard to the east, employing combined-arms attacks by its _Reiter_ regiments, the Bicycle Battalion, and motorized reconnaissance elements to fend off sporadic Red Army counterattacks against the armored forces' flank. In this advance, the horsemen kept pace with, and sometimes got ahead of, neighboring armored units, though not without cost. On 24 June, for example, Russian air raids caused "heavy losses" among the division's horses. Nonetheless, three days later the mounted units cut loose from their motorized elements and pushed on without them. The _Reiter_ regiments reached the divisional objective of Siniawka early on 29–30 June, with the motorized elements arriving in the afternoon of the latter day.
As had occurred after the division's campaign in the Low Countries and France in 1940, so now in early July 1941 divisional staff prepared a lessons-learned report on the eastern campaign to that point. Not surprisingly, the report made the case for the unit's continued existence. The Cavalry Division was "particularly suited for flank protection of armored units in roadless areas ( _abseits der Strassen)_ " and for covering gaps in the line. Furthermore, until the armored pursuit of broken enemy forces actually began, the cavalrymen had demonstrated at least as good a marching ability as the mechanized units, even over the most difficult terrain and even on the part of the Bicycle Battalion. Therefore, command concluded that the Cavalry Division should still be included in a panzer corps. Nevertheless, the same report didn't shy away from pointing out problems, and in this respect Feldt seems to have been notably forthright. He observed that the reconnaissance element possessed insufficient personnel, cross-country capability, speed, and uniformity of equipment. Comprised of a collection of horse- and bicycle-mounted troopers, motorized vehicles, and even horse-drawn vehicles, it showed itself insufficiently nimble ( _wendig_ ) and hard to manage. Similarly, divisional artillery and logistics needed greater off-road capability and lighter overall weight. Finally, total losses to 30 June were reported as 77 officers and men killed in action, 289 wounded, and 9 missing.
By 7 July the division had advanced a straight-line distance some 250 miles (402 km) and had reached Bobruisk on the Berezina River where they paused before the next stage in the armored corps' drive on Smolensk. During the rapid advance to the Berezina, threats to the Cavalry Division's horses manifested themselves beyond the combat injuries sustained in the Pripet Marshes. Directives warned, for example, against rabies and glanders, a potentially fatal bacterial disease afflicting a horse's mucous membranes. The directives stipulated that Russian military and civilian horses were not even to be touched if they showed symptoms of nasal drainage or lesions. Russian stalls were not to be used, nor were Russian saddles or tack. Healthy-appearing Russian horses, if requisitioned, were to be screened by the division's veterinary officers before use. To prevent the spread of rabies, no unauthorized dogs were allowed in the division's bivouacs. All strays were to be shot. Despite such warnings about the hazards of employing _panje_ or other Russian horses, particularly for logistical purposes, the necessity for it inevitably arose. As early as 5 July, General Feldt indicated that the troopers would have to "live off the land" owing to difficulties in bringing up re-supply columns. Living off the land obviously implied using not only Russian supplies but also Russian horses. This necessity also reflected the lingering concerns about Barbarossa's logistics, concerns that predated the actual start of the campaign and, because they were never resolved, contributed materially to the Germans' eventual defeat.
Notwithstanding these initial difficulties, normal operations continued. The first week of July saw the cavalrymen providing security in various localities around Bobruisk including, at one point, forward-airfield security for the _Luftwaffe's_ famous 51st Fighter Wing _Mölders_. Reports in this period tartly observed that the division was spread out over an area of fully 463 square miles (1,200 sq/km) of dense forests and was constantly engaged in clearing actions ( _Säuberungsaktionen_ ) against enemy-occupied villages, isolated armored columns, and Soviet paratroopers. Even so, the horsemen were also tasked with helping secure the southeast flank of XXIV Panzer Corps as the latter moved east and slightly north to prepare a forced crossing of the next significant riverine barrier, the Dnepr. The corps' tanks began crossing on 11 July near the town of Novo Bykhov just downstream from the city of Mogilev, and the horsemen's fight was fierce for the next ten days on both sides of the river. While the panzers rolled away eastward from the Dnepr, the Cavalry Division fought off repeated Russian counterattacks against the crossing point from both north and south. Even bypassed Russian troops still on the river's western bank attempted to cut off the eastward-driving panzers by seizing the approaches to the crossing point from the western side of the stream. Thus the cavalrymen, out of contact with their own armor and being only slowly relieved by follow-on German infantry approaching from the west, played a major role in holding open a vital corridor—and at that moment the only bridge across the Dnepr—for General Heinz Guderian's entire 2nd Panzer Group. Gradually, the German 17th and 112th IDs came up in support of the cavalrymen, and German lines on the eastern shore of the Dnepr were gradually pushed out in heavy fighting to the south and southeast.
In reviewing the month's action, divisional command reported with satisfaction that the Cavalry Division's troopers had once again proven themselves. After-action reports struck a slightly newer tone, however. Whereas in Barbarossa's opening weeks the division's principal accomplishments had been rather traditional—hard-riding, rapid advances over long distances in the partisan-infested country of the Pripet Marshes—the battles along the Dnepr seemed to show that the cavalrymen could be equally effective in defensive fighting as dismounted infantry, in a sense a throwback to the concept of the nineteenth-century dragoon. The great superiority of Russian artillery was duly noted, but this was a fact of life for a division whose "gypsy artillery" was never organized or equipped to engage in sustained gun-on-gun combat. This unavoidable disparity, however, was made worse by difficulties in maintaining supplies of ammunition for the horse-artillery batteries. In mid-July alone, for example, the division's guns had expended a number of rounds equal to fifty percent of those used in the entire campaign of 1940. That expenditure, however, was merely one of ammunition. As always, the clearest indicator of the severity of the fighting was the number of losses incurred among divisional personnel. From 22 June to 29 July, the division lost 366 officers and men killed, 1,593 wounded, and 43 missing. Individual mounted squadrons' combat strength had been reduced by as much as 85 percent. After-action summaries made clear that for the division to be once more fully combat-effective, 1,280 noncoms and men and 1,435 remounts would have to be supplied, primarily for the _Reiter_ regiments. These remained the division's cutting edge, and until replacements came up, the _Reiter_ regiments would have to comb divisional and brigade staffs and other sources for spare personnel and horses. In addition to men and horses, however, more prosaic equipment was also badly needed. Among other things, such equipment included 350 bicycles for the Bicycle Battalion and totally unglamorous but absolutely essential items such as bicycle tires and spare parts for automatic weapons.
As for the horses themselves, the division reported a strength of 12,153 on 10 August, of which 1,685 had earlier been captured or requisitioned ( _Beute-und beigetriebene Pferde_ ). When ridden, the requisitioned horses had shown themselves incapable of keeping up with the average German-bred mount and therefore had to be constantly replaced with other captured or requisitioned horses. By contrast, "nine-tenths" of the _panje_ horses taken into the division's service solely as draft animals had held up without apparent difficulty. To date, 3,365 of the Cavalry Division's horses had "fallen out." Of these, 2,551 had been sent back through the veterinary services' network: one-third as a result of wounds suffered; one-quarter owing to general lameness; the rest from various other causes such as exhaustion, colic, nervous stress, or problems with farriery. More than 550 had been killed or had died of natural causes, and 258 had bolted during artillery or aerial attacks and had not been recovered. In order to rebuild the stock of horseflesh, as well as to rest the men and re-equip, the division received orders for another much needed operational pause effective 1 to 11 August.
During the same pause, the hard-hit Bicycle Battalion was reorganized from three squadrons to two owing to its reduced manpower. At this time the division also found itself transferred to Second Army's XIII Corps for a projected drive to the southeast on the city of Gomel. The division's mission would be to block a possible Soviet retreat via that city. In this capacity, the horsemen prepared to participate in the great drive to the south ordered by Hitler. This meant, of course, that the movement toward the Soviet capital would have to wait. The strategic goal in the diversion was to seize the economic resources of the Ukraine before the resumption of the assault on Moscow. The operational objective was to smash the remaining Soviet armies in the Ukraine so that they might not threaten the eventual movement of Army Group Center toward Stalin's center of power.
As the drive on Gomel commenced, the Cavalry Division was ordered to hold up behind the 17th ID that had by now been brought forward. On the contrary, the cavalrymen should have been deployed ahead of the infantry. Coincidentally, this very same foul-up had occurred in the initial Prussian deployment against Austria in 1866. As one would expect, this arrangement generated evident annoyance in the division's command and necessitated a forced march by the horsemen of more than sixty miles (100 km) between midday on 12 August and dusk the following day in order to get out in front; that is before they even came into contact with Soviet forces. Consequently, both the cavalrymen and their horses went into action insufficiently rested. Nevertheless, by 15 August they had cleared the important road junction of Chechersk, north of Gomel, of enemy troops. They had also fended off repeated Russian counterattacks, including ones by armored vehicles, employing the division's own armored reconnaissance and reinforced bicycle detachments. They had further bottled up Soviet troops attempting to avoid encirclement near the Dnepr River town of Rogachev in the face of a riverine assault executed there by XLIII and LIII Infantry Corps. Acidly, divisional after-action reports noted that the German infantry whom the horsemen preceded were able to "march" without fighting all the way to the last prepared Russian positions outside of Gomel. On 16 and 17 August, the Cavalry Division pushed on to Vetka, immediately north of Gomel on the River Sosh. An attack by one of the division's mounted brigades succeeded in capturing the place, and the cavalrymen immediately turned bridge builders. They threw two spans across the river and built up a considerable bridgehead on the eastern bank. Along the way, they just missed capturing the entire staff of the Soviet Twenty-First Army. The horsemen subsequently noted that their action made it possible for the German infantry to avoid assaulting Gomel directly from the west and northwest. Instead, the latter could now attack the city from the northeast by way of Vetka if necessary. There the Russians' defenses were weaker, and on 19 August the German XIII Corps occupied this regional capital.
This remarkable feat was followed by an equally noteworthy one. In a move reminiscent of the uhlans of 1870, the division's 21st _Reiter_ Regiment lunged ( _ausholen_ ) eastward almost sixteen miles (25 km) on the same day Gomel was occupied to reach the bridge at Dobrush on the Iputs River. Though it had to shift slightly to the north to avoid being cut off from the rest of the division, the regiment had single-handedly and considerably extended the Germans' advance in a day. The extension nevertheless forced the 1st _Reiter_ Regiment and the divisional armored reconnaissance detachment to cover the 21st Regiment's northern flank and rear against Soviet counterattacks. This they did successfully. As the cavalrymen had done on the Dnepr, so now they were once more forced to fight on multiple fronts simultaneously. Still, with support from a following infantry regiment, the cavalrymen cleared Dobrush and yet again forced a river-crossing by 22 August in the face of "well positioned and stubbornly resisting enemy forces" ( _geschickt eingebauten und zäh kämpfenden Feind_ ). In the face of two full Soviet infantry divisions dug in on the Iputs' vast and swampy eastern bank, the cavalrymen could not seize their ultimate objective, an area of high ground beyond known as the Korma Heights. In this frustrated operation the troopers replayed, after a fashion but on a larger scale, their failure to seize the fortified Dutch positions at Kornwerderzand in 1940. Nevertheless, they had accomplished much. In four days, they'd materially hastened the capture of a major regional capital and numerous enemy troops between Gomel and the Dnepr. By so doing, they had forced Soviet commanders to withdraw still other formations to avoid the latter's being cut off. In the process, the horsemen had repulsed any number of Soviet counterattacks. Certainly not least, they'd significantly extended the Germans' furthest advance in an important sector of the front. Indeed, said after-action reports, had the 1st Cavalry Division been a cavalry corps instead, it would even have enjoyed the success of seizing the Korma Heights. This was because, had it been a corps, it would have possessed not only a much larger complement of artillery but also guns of heavier caliber. Further, there would have been more numerous armored reconnaissance elements, as well as combat engineers possessing heavier equipment. In the event, the division's 1st and 2nd _Reiter_ Brigades, their four regiments once again united, only seized the Korma Heights on 24 August and then only in the wake of the more slowly advancing infantry of XIII Corps.
Without any real rest, the division subsequently continued its advance as part of XIII Corps' drive to the south toward the city of Chernigov. On 27 August the troopers were transferred once again, this time to the command of XXXXIII Corps. On the same day the cavalrymen crossed the Ukrainian frontier. Throughout the period, division's troops fought a number of limited, though sometimes intense, actions in the vicinity of the small towns of Snovsk and Novi Barovichi, southeast of Gomel. As they had done earlier along the Dnepr, the _Reiter_ regiments proved themselves capable of repeatedly alternating between mounted movement and dismounted combat; and, once again, the armored reconnaissance detachment and the Bicycle Battalion engaged often bitter Soviet resistance. The confusion of combat was exacerbated, however, by yet another transfer, this time to the command of the 2nd Panzer Group. In the view of the division's commanders, the resultant stream of conflicting orders materially complicated operations. Nevertheless, by 2 September this phase of operations was successfully completed, and the Cavalry Division could once again lay claim not only to having hindered a Soviet retreat via Snovsk but also to having blocked reinforcement of those same forces.
Looking back on the campaign since 22 June, General Feldt assessed the division's progress with an unsentimental eye. By some measures, the cavalrymen had performed extremely well. They had, for example, captured 13,872 Soviet soldiers since the start of Barbarossa. They'd also hauled in large quantities of Soviet equipment, including 48 tanks, 73 heavy guns, 230 machine guns and, somewhat improbably, 11 aircraft. In the most recent fighting since 1 August, the division lost 185 officers and men killed, 816 wounded, and 11 missing. Not surprisingly, a significant majority of those losses (79 percent) had been suffered by the four _Reiter_ regiments and the division's horse-artillery regiment. By contrast only 859 personnel had come up from the divisional replacement depots in East Prussia and elsewhere. As a panzer division commander put it in the face of similar losses to his own unit, the German armies in Russia were in danger of winning themselves to death.
For the division's horses Feldt had only unstinting praise and deep sympathy. They had made an immeasurable contribution ( _Unendliches geleistet_ ) to the division's success in spite of what they'd endured. The campaign's pace and bitterness had prevented effective farriery, general care, and the regular delivery of oats. Further, the horses had all too often simply been unable to rest owing to the fighting going on around them, a lament doubtless echoing the concern of cavalry commanders since the dawn of firearms. The division's East Prussian-bred horses—the majority would almost certainly have been Trakehners and crosses bred from them—came in for specific praise. Feldt lauded what he called their "iron" constitution and their ability to keep themselves relatively well nourished even under difficult circumstances ( _sich einigermaßen im Futter hält_ ). Nevertheless, he openly recognized the fact staring him in the face in early September: the condition of the division's horses could "hardly be described as satisfactory" ( _kaum noch als genügend zu bezeichnen_ ).
Just as he did in praising the division's horses, so the commanding general also recognized his men. Here he singled out the division's horse-mounted reconnaissance riders ( _Spähtruppreiter_ ). Whether he intended to or not, Feldt invoked the image of the uhlans of 1870 and 1914 in describing their accomplishments. As he explained, these troopers almost always had to rely solely on themselves while on their missions. They had only poor maps, if they had any at all. Yet they still regularly managed to secure valuable information for divisional commanders. In short, they deserved more recognition. Just as infantrymen, artillerists, and panzer troops had their own specific assault badges, he said, so should the reconnaissance riders. Notwithstanding their achievements and those of the rest of the division's personnel, however, Feldt also recognized that, for the moment at least, the division was played out.
He also confronted other unsavory realities. Even while maintaining that the division had shown itself entirely capable of rendering valuable service in closing the gaps between corps and other units and staying out in front, Feldt asked rhetorically whether a panzer division of some sort might not be more effective in fulfilling the cavalry's mission. The cavalrymen had often been hampered by being assigned to follow infantry divisions rather than leading them, a condition made worse by the division's having been shunted around much too frequently among higherechelon commands. This fact had necessitated not only a great deal of ultimately useless marching and counter-marching. It had also generated mountains of paperwork, the torment of all commanders. A much more pressing concern, however, remained the division's lack of artillery firepower, a concern dating back at least to the division's operations in the Netherlands the year before and one which had arisen again in the fighting on the Iputs River around Dobrush. Trying to get heavy guns from neighboring infantry units caused friction, Feldt opined, but in those instances where assault-gun artillery had been available to the horsemen, cooperation had been good and successes enhanced. Consequently, he urged that heavy motorized (i.e., towed) artillery and self-propelled assault guns be added to the division's establishment.
Similarly, Feldt argued that the motorized vehicles the division did possess had not been sufficiently supported by rear-area commands. There was never enough fuel or oil; there were never enough spare parts; there was never enough time for maintenance. The results were predictable: nearly 40 percent of the division's armored reconnaissance vehicles remained inoperable in early September. More galling still, even the division's bicycles had never been brought back up to strength owing to a pressing lack of spare parts. Clearly exasperated and evidently reflecting the logistical reality in Russia for most German commanders, Feldt wrote that those spare parts, as well as bicycle tires, simply could not be found in Russia.
In this brutal contest in Russia, a contest not yet won in September 1941, the Cavalry Division's mobility, if not its speed, would be crucial in justifying its continued existence. Feldt understood this. Wherever motorized or mechanized columns could operate freely on dry roads, the Cavalry Division seemed to be at a relative disadvantage, and, as already noted, road congestion had plagued the Germans' advance since the start of the invasion. Its troopers—indeed all units in Fourth Army—were therefore ordered to maintain strict march discipline. And however much such an order might tell on any formation having horses, which is to say almost the entirety of Fourth Army outside the mechanized forces, it would disproportionately affect a predominantly horse-mounted unit such as the 1st Cavalry Division. The division's columns were to stick solely to the right shoulder with a ten (horse)-length interval between mounted formations. While single vehicles of other units were always to be given right of way, wheeled or tracked columns from other formations would have to wait for the ten-length gap before intersecting the horsemen's line of march. At that point, ten vehicles could pass. The following mounted unit could then carry on, then ten more vehicles, and so forth. This procedure would spare the division's horses and keep mounted columns intact while simultaneously allowing the vehicular columns to exploit their greater speed. Coincidentally, these orders went out at precisely the time when the annual autumnal rains had begun to fall in Russia, rains that would soon frequently bring much of the army's vehicular traffic to a halt. As would so many German records, an order of the day dated 13 September 1941 commented on the "continuing rain and nearly bottomless tracks (Wege)," conditions demanding the equally continuous employment of the Cavalry Division's combat engineers to help ensure the hoped-for regular delivery of supplies.
Despite such conditions and the increasing psychological and operational intensity of the anti-partisan war being waged behind the front, the Cavalry Division's accomplishments were recognized in at least some higher-echelon commands, no doubt to Feldt's satisfaction. The XIII Corps' commander, General Hans-Gustav Felber, credited the division with making possible XIII Corps' earlier taking of Gomel. The mounted forces' "rapid exploitation of favorable opportunities" ( _schnelles Erfassen günstiger Gelegenheiten_ ) were critical, he wrote, in the successful German advance on that city. He also commended the cavalrymen with containing ( _binden_ ) "strong enemy forces" on the corps' flank in the subsequent advance to the southeast. Securing the flanks of advancing infantry was as traditional a mission for the cavalry as ever there was, even though the troopers also received a rather firm warning about the dangers inherent in allowing their minds to fall into a "partisan psychosis." Such a mental state among the division's troopers might "undermine the pacification of the [occupied] territory" through overly harsh treatment of civilians. Unfortunately, other German units, such as the SS cavalry in Russia, had no such scruples.
Such warnings appeared justified given the horsemen's next mission. In the second half of September, the Cavalry Division was transferred once more, this time from XIII Corps to XXXXVII Panzer Corps, and assigned to relieve 18th Panzer Division. Following a brief stint as army group reserve, the cavalrymen's mission throughout late September and into October was evidently confined to a sort of security role on the front lines and immediately to the rear while the LIII Infantry Corps prepared to assault the major city of Bryansk east of the River Desna. Its area of operations in this period lay east of Gomel around the town of Novgorod Severski and the bridgehead there on the Desna's eastern bank. The cavalrymen also operated southward along the Desna to Trubchevsk, a distance of about fifty miles (80 km). Though Russian cavalry—the 4th and 52nd Cavalry Divisions and the 118th Cavalry Regiment—were reported in the vicinity, there does not appear to have been mounted contact between them and the German horsemen. Just the same, the Cavalry Division's constant operations required continuous replacements of both men and horses. On 28 September divisional staff officers were requisitioning two hundred saddles from rear-area supply depots for an expected shipment of remounts, though tack and saddle blankets were evidently harder to come by. Furthermore, _panje_ horses had to be rounded up for distribution to troopers from the Bicycle Battalion. Because of the large number of mechanical breakdowns already noted and the ever-poorer roads as the division pushed eastward, these troopers had left their two-wheeled mounts behind at various stages in the division's march. As occurred so frequently on the Eastern Front, such troops were effectively immobile until the ubiquitous Russian horses could be collected. Divisional headquarters ordered that this particular situation be rectified by 30 September.
Map 5. Eastern Front, General Situation, Late September 1941
Despite constant references in this period to the threat posed by partisans and the emphatic order that the men should always carry loaded weapons except when in their quarters, divisional command also kept insisting that soldierly discipline be maintained. Unauthorized shooting, apparently widespread, was expressly forbidden because it wasted ammunition and revealed defensive positions. Furthermore, graffiti and stuffed animals were not to be used to decorate the division's vehicles, the stuffed animals evidently being attached to radiator-caps as hood ornaments. Furthermore, men were strictly ordered not to appear out of uniform. Decorations were to be worn and military courtesies always observed. Such observances remained the "calling card of any unit." In sum, as General Feldt wrote, the Cavalry Division was to be known as a unit in which "exemplary discipline reigned." Perhaps he felt that the strain of the campaign could only be rectified by discipline and good order. Perhaps, too, he felt that he had other good grounds for urging such exemplary discipline. As it turns out, in September 1941 the division was in the process of beginning a reorganization, with all of the administrative and personal tumult that a reorganization in wartime brings with it. Fate and the OKW/OKH had decreed that the 1st Cavalry Division was to give up its horses and replace them with tanks. It was to become a panzer division.
In 1939 the ultimate fate of the then-1st Cavalry Brigade had still been uncertain. Even earlier, the German cavalry's corps and divisional structures had been ordered abolished, effective 1936; and, as has been seen, the army's cavalry regiments had been tasked at that time to provide squadrons for the infantry divisions' organic reconnaissance battalions, as well as cavalry troops (i.e., companies) for individual infantry regiments. Only the 1st Cavalry Brigade had remained as an autonomous, dedicated, horse-mounted unit. Following its expansion in the winter of 1940–1941, the Cavalry Division had been ordered, in addition to its actual operational objectives, to determine whether horse-cavalry in the eastern campaign could still execute useful missions or whether its role could be better served by armored formations. Now, in the autumn of 1941, General Feldt submitted his answer to that question.
He began by noting that throughout the eastern campaign, the division had been assigned the following missions: screening the flanks of German armies and panzer groups; closing the gaps between friendly units; pursuing Soviet forces along the flanks and in front of German infantry corps; rapidly seizing and holding sections of the front until follow-on forces came up; attacking the flanks and rear of enemy formations; and executing wholly dismounted defensive infantry operations. With the exception of the last-named task, all of these were traditional functions of the cavalry. In Feldt's view, all of these missions had been completely accomplished. By itself, as he also noted, the Cavalry Division could not contribute decisive combat power, and other German formations had done quite as well with no attached cavalry at all. Nonetheless, he believed that a single cavalry division could do good work even if it was not absolutely necessary to the army's success as a whole.
By contrast, and with what seems to have been more than a touch of bitterness, General Feldt maintained that had his division been a corps instead, then the horsemen could indeed have made decisive contributions to the campaign in Russia, whether in the Pripet Marshes, around Gomel, or even in the massive operational encirclements that were at that moment occurring farther southeast in the Ukraine. Even so, Feldt still recognized that the establishment of a cavalry corps didn't seem likely, since effective cavalry forces couldn't simply be improvised on the spur of the moment. Therefore, if a cavalry corps' establishment were not in the offing, then he felt it would be better simply to disband the division despite its three successful campaigns in Poland, the west, and Russia, and despite the constant demands for its services by other formations, not to mention the troopers' genuine love for their horses.
No sentimentalist, Feldt listed the reasons for his conclusion. The ever-increasing number of motorized and mechanized divisions meant that the Cavalry Division would no longer have a useful place, even if the vast majority of the army's divisions always remained ground-pounding infantry. As he'd already noted, a single cavalry division couldn't deliver decisive, battle-winning combat power at an operational level; and a panzer division could accomplish the same results as the cavalrymen had done and do so more quickly. Evidently believing, as so many German officers still did, that the war in the Soviet Union could have only one outcome—German victory—Feldt maintained that the division would have no reasonable geographic area of operations for its employment beyond the spring of 1942 in any case. More parochially, he pointed out that German youth had increasingly little interest in things equine, the automobile having become their passion. Consequently, younger officers interested in a cavalry career were harder to come by, even as their older cavalry comrades were already finding it difficult to rise to higher command.
Feldt judged that not even the cavalry's vaunted cross-country capability was sufficient to save the horsemen for the simple reason that a purely horse-mounted cavalry was no longer possible. Even the Cavalry Division had ultimately required motorized vehicles for its combat engineers; some of its anti-tank and reconnaissance detachments; and, increasingly, for the all-important logistics elements. When such units had been entirely horse-drawn, they'd simply not been able to keep pace with the hard-riding _Reiter_ regiments. However, when those same supporting elements converted to vehicles in the interest of speed, they lost whatever cross-country capability they'd earlier enjoyed while horse-drawn. Only appropriately outfitted, fast-moving, tracked vehicles, said Feldt, would possess the speed to keep up with the _Reiter_ regiments and the off-road capability of the division's horses. Furthermore, if vehicles ran out of fuel and were idled for a prolonged period, barring mechanical failure or battle-damage, they could nevertheless resume full-speed operations once they were refueled. By contrast, when the division's horses didn't get proper feed and fodder and/or were immobile for an extended period of time, they could well require periods of muscular and cardiovascular reconditioning before they could resume their missions. Facing his own apparently unavoidable conclusions, Feldt nonetheless pleaded that there had to be found a way to keep the art of riding available and accessible to all officers. Precisely because of the army's degree of motorization, one that Feldt evidently presumed to be a matter of endlessly increasing percentages, only _Reitsport_ (though he did not use this word) could provide German officers, especially panzer commanders, with the combination of self-control, daredevil risk-taking, and lightning-fast decision making so essential for modern officers' success. To that end, he urged, every officer in every armored division should have his own mount, at least once the war ended.
All things considered, Feldt clearly believed that he could not avoid recommending that the Cavalry Division be converted to an armored formation. He further recommended that the transformation be quick and en masse so as to avoid disrupting the division's battle-born cohesion. Speeding the conversion would avoid missing the perceived opportunity to acquire up-to-date motorized vehicles. He also urged, however, that rapid conversion should not result in abandoning the cavalry's heritage and traditions such as the cavalry's yellow-gold piping on troopers' uniforms or the "Leaping Horseman" divisional insignia. These, said Feldt, should be transferred to whatever new panzer formation assumed the division's role. Lest they be forgotten, Feldt also addressed the matter of the mounted elements of the infantry divisions' reconnaissance battalions, at least by way of saying that the Cavalry Division did not feel competent to make recommendations for their future status. They, too, he nevertheless pointedly noted, were heirs in fact as well as in name to the German cavalry's traditions and should by no means be ignored. In completing his recommendations, General Feldt fully acknowledged the wide-ranging ramifications of his decision. However much the Cavalry Division had accomplished to that point in the war, and however much the cavalrymen genuinely loved their mounts, the practical experiences to date forced him to make the recommendations he'd made. "The cavalry," read his report's last sentence, "feels compelled to go with the times since no combat arm ( _Waffe_ ) may be allowed to stand still."
Even as Feldt's recommendations went up the line, however, the division's operations continued unabated, especially in the period from 8 to 20 October. Having received one thousand men and four hundred horses as replacements late in the previous month, the cavalrymen along the western bank of the Desna now found themselves fighting not only battles with Russian troops trying to retreat to the east but also large numbers of partisans, both soldiers and civilians, attempting to help their comrades flee. In a phrase sadly reminiscent of orders repeatedly heard elsewhere on the Eastern Front, divisional reports indicated that "this partisan monster ( _Partisanenunwesen_ ) will be fought with the ruthless arrest of all civilian persons suspected of being fit for military service; with [the taking of] hostages; and with the shooting of the guilty." Operating alongside various infantry and panzer divisions, the horse-men also found themselves fighting the weather. On 9 October the autumnal clouds broke once again, and snow now began to mix with the frequently heavy rain that had been regularly noted in the division's war diary since early September. By the next day every dirt track had been turned to bottomless muck, every stream had become a torrent, and even the smallest blown bridge became a matter of long delays in the pursuit of retreating Soviet troops. Even so, the division's bicyclists managed to form a bridgehead across the 300-foot-(100-m)-wide Desna on 11 October and link up with the Germans' 29th ID near Sytenki as the latter attempted to encircle retreating Russians from the eastern bank. This feat the cavalrymen followed up with the construction of a five-ton capacity bridge that allowed the 1st _Reiter_ Brigade to continue the advance to the southeast. As had occurred before, however, the region's horrible roads, particularly at that season, as well as persistent fuel shortages, prevented many of the division's vehicles from following the riders. For the next nine days, the various _Reiter_ regiments and the division's Bicycle Battalion saw almost constant, and sometimes fairly large-scale, action against retreating Russian forces. With the exception of a major battle on 17 October, however, most heavy fighting ended in the immediate aftermath of the crossing of the Desna. At a cost of 122 officers and men killed, 410 wounded, and 8 missing, the division had captured 8,132 Soviet troops by 20 October, the number killed not being indicated. The cavalrymen had also taken 3 tanks, as well as 39 guns, 91 motor vehicles of various types, and some 250 horse-drawn wagons. They'd even managed to capture an entire ammunition train and a Soviet field hospital complete with "six omnibuses." After-action reports proudly noted that the cavalry's mobility on the march and in combat had once again proven itself. "A panzer division," the reports concluded smartly, "could hardly have done better."
Nevertheless, the writing appeared to be on the wall regarding the 1st Cavalry Division's future. Orders arrived at divisional headquarters at the end of the first week of November indicating that the division would be reorganized as the 24th Panzer Division. The cavalrymen were ordered to move by train and motorized columns back to their old home in East Prussia prior to their ultimate relocation for armored-forces training at Ohrdruf in Thuringia. While in East Prussia, the division's units were dispersed for preliminary billeting. Divisional elements such as the combat engineers and logistics units were assigned to Insterburg and Angerapp. The _Reiter_ regiments took over camps in the large (more than 24,000 acres) prewar military training area around the small town of Stablack that lay about twenty miles south of the East Prussian capital of Königsberg. The principal changes in the division's organization saw the 2nd and 21st _Reiter_ Regiments becoming panzer regiments. The 1st and 22nd _Reiter_ Regiments were designated rifle regiments ( _Schützenregimenter_ ). Analogous changes were made for the Bicycle Battalion and other subordinate commands.
Throughout the process of reorganization, however, the division's horses still required their regular attention. Veterinary care and daily maintenance thus continued throughout November and December and was duly noted in activity reports, often with the laconic marginal notation "nothing out of the ordinary" being inserted in the appropriate space on the form. Nonetheless the reduction in the numbers of horses proceeded fairly rapidly. On 1 November, for example, the division's combat-strength in horses was still 8,969. The ration-strength was 10,681. By 21 November, a week before the official date of the division's redesignation, those numbers were 2,498 and 3,899, respectively. Further reductions followed to the end of the year. As deemed necessary by their condition, sick or injured horses were sent back to veterinary hospitals at Potsdam and Neustettin and even as far away as Frankfurt am Main. Others were transferred to infantry and artillery units and various riding and driving schools, such as the one at Soltau in Hannover. In the three Studs under the division's command, at least 428 mares were designated as fit for breeding. Of these, it appears that 153 were sold to the Stud Book Society, the organization then maintaining official breeding standards in Germany. Given the traditional emphasis on quality in modern German horse breeding, this disposition of mares would certainly seem to demonstrate that high-quality breed stock was still to be found in mounted units. At this still successful stage of the war, such a disposition would not seem indicative of a desperate search for brood mares. In addition to those sold to the Society, an unspecified number of other mares were sold directly to the division's personnel. Interestingly, at least one presumably valuable horse from 22nd _Reiter_ Regiment, Sonnengott (Sun God), came in for particular attention. He was sent directly to an unnamed recipient serving on the General Staff of the army and stationed at Angerburg and was the only horse singled out by name in the division's daily activity reports. Finally, most of the division's staff-level veterinary officers were transferred. As a tribute to the new panzer division's heritage, and in keeping with General Feldt's wish, its insignia remained the Cavalry Division's "Leaping Horseman," depicting a rider in profile jumping to the viewer's left over a fence and surrounded by a reversed "C." Whether for reasons of tactical simplicity or mere speed of application, the horse and rider were later sometimes represented by a mere diagonal bar. In whatever form, the "Leaping Horseman" would now find his way onto the hulls of tanks. Though the newly minted panzer troops could not know it at the time, those same tanks and the division's troops would in turn meet their end at Stalingrad one year later.
CHAPTER 8
HELL'S OUTRIDERS
CAVALRY OF THE _WAFFEN_ -SS
In the ideology of the Nazi regime, animals often featured prominently alongside blond-haired, blue-eyed exemplars of the Nordic man. Certain of these animals were considered to embody particularly "Nordic" characteristics. The horse was one such creature, as were hunting dogs, falcons, and eagles. Horses—or at least well-bred and well-groomed ones—possessed nobility and grace, martial strength and courage. In and of itself, such a view is nearly as old as humankind's connection with genus _equus_. Yet, under the Nazis, the aesthetic and practical use of the horse acquired a much more sinister purpose. Along with many other wholly innocuous symbols and objects, horses in the Nazi era came to project an increasingly ominous image, not least because they so frequently carried the paladins of the new regime in the endless parades so dear to Hitler and other Nazi functionaries. To note but one example, the parade celebrating "2000 Years of German Culture" in Munich on 18 July 1937 featured thousands of marchers and legions of horses, many of whose caparisons and armor-plated riders were suitably emblazoned with Nazi insignia. As with all other cultural expressions of the regime, the artistic mission celebrated in this particular instance demanded obeisance not only to lofty sentiment but also to fanatical devotion. In more literally concrete and metallic terms, horses featured prominently in the regime's architecture as well. For example, two monumental bronze horses by Josef Thorak, called "our greatest sculptural talent" by the Nazi government's propaganda minister Joseph Goebbels, adorned the garden entrance of Hitler's new Reich Chancellery that was completed just before war's outbreak in 1939.
Such symbolism was perfectly in keeping with propagandistic agitation and the consequent radicalization of the German people by the Nazi government. To a very great extent, the regime relied upon slogans, monumental architecture, and evocative symbols to express itself and its aims. Of course, the most striking symbol remained the swastika. Though in the early days of the 1920s it served as a sort of "personal totem" for Hitler, the _Hakenkreuz_ (hooked cross) soon became not only the party's emblem but also Germany's national insignia. Besides the swastika, however, there were other equally evocative, and often dreaded, symbols culled from both Germany's actual history and the fevered imaginations of Nazi mythmakers. To such long-established and honored symbols as the Iron Cross, later adulterated through the addition of swastikas, came many others. Neo-peasant clothing styles and domestic buildings were very often, though no less spuriously, cited by the regime as recalling the ideals of a Germanic past. There were oak leaves and eagles in profusion, all intended to evoke past imperial glories. But there was also terribly more sinister Nazi regalia, specifically intended to evoke "an older, darker Germany of forests and hunters." These included "the broad-bladed daggers worn by [the Nazi Party's] men" and commonly seen on the uniforms of so many of its formations. However, the symbol that arguably came to be the most potent of all remained the double, silver-on-black lightning-flash of the SS runes. Adorning SS men's uniforms, their banners, and their units' heraldry, the runes called to mind "that twilight world of ferocious gods and desperate heroes [in] the German romantic imagination." The SS runes also became perhaps the most feared symbol of Nazi Germany's minions, and by 1939 the men who wore them had earned a reputation of absolute obedience to Hitler. Theirs was obedience unto death as they proclaimed in their oath upon being sworn in as members of what their leader, _Reichsführer_ -SS Heinrich Himmler, envisioned as a new order of chivalry. Sometimes described as the praetorians of the Nazi regime, the SS were also its latter-day berserkers, men who took fierce pride in their self-perception of peacetime discipline and honor but who would also remorselessly and without compunction kill enemy soldiers, civilians, prisoners of war, other Germans, and even their own SS comrades. All the while they would glory in the deed and, as they saw things, draw strength from it.
The horse-mounted unit that was eventually to become the 8th _Waffen_ -SS Cavalry Division _Florian Geyer_ fully shared all of these traits. In that respect, it was hardly unique. It was unique, however, in being primarily a horse-mounted formation, even though it was not the only armed formation of the SS to employ horses. Notwithstanding the armed or _Waffen_ -SS' reputation for typically receiving its full complement of tracked and wheeled vehicles, other SS units besides the SS cavalry also used horses. The SS Death's Head ( _Totenkopf_ ) Division, for example, a unit eventually elevated to the status of an armored formation, had regular difficulties before 1939, as did other early _Waffen_ -SS formations, obtaining equipment owing to the army's resistance. In the case of the _Totenkopf_ Division, these difficulties resulted in, among other things, its having to use horse-drawn, World War I-era field bakeries. Given the then-paucity of horses and the fact that the bakeries weighed nearly two-and-a-half tons apiece (they had to be taken off their rail cars by cranes), the latter were practically useless. Only on the eve of the invasion of France and the Low Countries in May 1940 did these difficulties begin to abate for this particular unit. By that time, the German army's own requirements were filled out sufficiently to allow the _Waffen_ -SS divisions to receive surplus gear. Nevertheless, another early _Waffen_ -SS division, this one recruited from police personnel under Himmler's direct law-enforcement control as Chief of the German Police and therefore not subject to the army's recruiting demands, perennially found itself short of motorized and mechanized vehicles, at least until achieving its eventual rank as a _Panzergrenadier_ division. As late as the invasion of France in 1940, this division, named _Polizei_ because of its origins, "remained a marching unit, whose transport and artillery, like that of the bulk of the Wehrmacht, was horsedrawn."
Horses' employment in the SS beyond the draft-animal role, however, had a particular place in the plans of the SS' leadership. Himmler saw a mounted combat unit of the SS providing a useful armed addition to the _Waffen_ -SS while embodying a certain social cachet. The latter consideration had significantly influenced Himmler's thinking since well before the war began. Such a cachet in wartime would, in turn, bolster Himmler's long-standing efforts to attract a more elevated social type into the SS, efforts he'd undertaken even before the Nazis' accession to power in 1933. As early as 1931, for example, he'd raised a mounted SS detachment in Munich, the so-called "Capital of the Movement" (i.e., birthplace of the Nazi Party). In the course of 1932, that detachment was expanded and redesignated as an SS Mounted Company (SS- _Reitersturm_ ). From this early period, and with this nucleus of a mounted arm in hand, Himmler assiduously cultivated the task of recruiting more refined members into the SS, not least to raise his organization's profile ever further away from the plebeian identity of the brown-uniformed, street-brawling elements among the Storm Troopers ( _Sturmabteilungen;_ SA). Himmler found the prestige, leadership qualifications, and, not least, money that he needed to construct his Black Order in the worlds of commerce, finance, and the German nobility. Indeed, since 1933, the roster of new SS men's names sometimes read like a German version of _Debrett's Peerage:_ the Prinz von Hohenzollern-Emden of the House of Sigmaringen; the Grand Duke of Mecklenburg; the hereditary Prince zu Waldeck und Pyrmont; the Princes Christof and Wilhelm of Hesse; Count von der Schulenburg; Count von Rödern; Count Strachwitz; and so on. Just before the war's beginning in 1939, some 20 percent of the SS' nominal senior leadership and about 10 percent of its lower ranks would consist of titled nobility. Of particular relevance to this recruitment but also to the prewar mounted SS and, later, the SS Cavalry was Himmler's success in persuading "all the most important German horse-riding associations, preserves of upper-class sportsmanship and snobbish socializing, to enrol [ _sic_ ] in the SS, irrespective of their political views...so that SS riders regularly won the German equestrian championships." The equestrian associations of Germany's most important horse-breeding regions—East Prussia, Oldenburg, Holstein, Westphalia, Hannover—"put on SS uniform."
While such enforced enrollment was first and foremost a public-relations exercise, it nevertheless helped create an impetus within the prewar SS toward a sort of cementing of the equestrian image within the organization. That image, in turn, would have further impelled Himmler toward the creation of a mounted SS combat unit as war approached, a natural extension, as it were, of the SS' prewar horse-mounted formations. A mounted SS combat unit would also constitute, at least in Himmler's eyes, a clear parallel to what was at that date still the army's 1st Cavalry Brigade. By 1934 the first SS Mounted Regiments ( _Reiterstandarten_ or _Reiter_ -SS) had been established as successors to the original _Reitersturm_. By 1939 they numbered twenty-four and were stationed across the length and breadth of the Reich. Furthermore, in an effort to increase the equestrian proficiency of the SS' mounted personnel, an SS Main Riding School ( _Hauptreitschule_ ) was established in Munich. In 1936 then-SS major ( _Sturmbannführer_ ) Hermann Fegelein was assigned to command the school that happened to be located on his parents' property near the suburb of Riem, eventually the site of Munich's first major post-1945 airport. Not of noble birth himself, Fegelein rose rapidly through the ranks and during the same period became a successful equestrian competitor. His own career and that of the mounted SS combat troops would soon converge.
These combat units arose from early decisions by the Nazi government to elevate selected units of the SS from their original, ostensibly unarmed status to that of full-time, armed formations. As early as 1933, the Nazi regime had begun to establish such armed SS units, and they multiplied with confusing rapidity. First called Special Detachments ( _Sonderkommandos_ ), these units guarded local Nazi bosses and their headquarters. If one of these headquarters guard-detachments grew to more than one hundred men, it could then be redesignated as a Political Readiness Squad ( _Politische Bereitschaft_ ) possessing a military TOE. By 1934 the nationwide network of Political Readiness Squads had effectively been recognized by the Reich Ministry of Defense as a full-time, armed State Police force answering solely to Hitler or, at his discretion, to Himmler. Emerging, in turn, from this force came the SS Special Duty Troops (SS- _Verfügungstruppen;_ SS-VT). Essentially the Nazi Party's standing army, the SS-VT took orders directly from Hitler or his designated lieutenant and remained subject to his exclusive disposition, hence the name. Not regarded by Hitler as belonging either to the army or to the police, they became the immediate organizational ancestors of the _Waffen_ -SS. Arising roughly in parallel to the SS-VT—but initially separate from them organizationally—were the SS Death's Head Units ( _SS-Totenkopfverbände;_ SS-TV). The SS-TV drew personnel from among veterans of the army, active-duty or retired policemen, guard-detachments of the regime's concentration camps, and the General SS ( _Allgemeine_ SS). According to Hitler himself, the Death's Head Units comprised a full-time armed force of the SS for the resolution of "special internal political tasks" of a policing or security nature. As usual, he reserved this force to himself alone or to his appointed subordinate. Regardless of their original tasking, for all practical purposes the SS-TV would be informally incorporated into the _Waffen_ -SS after 1939. Formal amalgamation followed in 1941–1942. It was through these Death's Head Units that many pre-1942 replacements for the SS cavalry would find their way into the larger _Waffen_ -SS structure. Before wartime losses really began to manifest themselves, however, SS cavalrymen came from the pre-existing SS- _Reiterstandarten_. In the invasion of Poland in September 1939, SS horsemen trained and led by Fegelein and drawn from _Reiterstandarten_ 15 (Munich) and 17 (Regensburg) would serve as security forces behind the front lines in the form of the SS Death's Head Horse Regiment (SS _Totenkopf-Reiter-Regiment_ or _Reiterstandarte_ ). They would also participate in "actions" against Jewish and non-Jewish civilians (i.e., summary killings). When this regiment was reorganized as the 1st SS Cavalry Regiment in November and December 1939, Fegelein would be named its commanding officer. When the 1st SS Cavalry Regiment was then assigned to serve as cadre (along with the 2nd Regiment that had been raised in the meantime) for a newly designated SS Cavalry Brigade in occupied Poland, Fegelein was once again assigned to command.
As can be seen in Fegelein's case, as well as the cases of countless other SS men, non-noble birth meant little in terms of potential advancement. What really counted were fanatical loyalty, ruthlessness, and being a Nazi true believer. Still, noble status continued to carry weight in Himmler's and others' eyes. Of course, elevated social standing never really protected SS men, no matter their rank and background, from the murderous intrigue of the Nazi regime generally and within the SS as an organization. An early, aristocratic leader of the SS' mounted arm, Anton _Freiherr_ (Baron) von Hohberg und Buchwald, presented a textbook example of extreme vulnerability despite his being of genuine noble birth. The leading SS horseman of East Prussia, von Hohberg was nevertheless cold-bloodedly gunned down in his own home by a rival in the purge known as the "Night of the Long Knives" in June 1934, ostensibly for having leaked SS secrets to the army. Though he could not know it at the time, Fegelein himself would ultimately share Hohberg's fate in being summarily shot on Hitler's personal orders in the regime's final days in April 1945. Not even Fegelein's marriage to the sister of Hitler's mistress could prevent it.
Cold-blooded murder notwithstanding, Himmler viewed the men admitted to the SS as biologically and, therefore in his view, morally superior. As the best of the "Nordic race," they would be the supreme exemplars of that population's virtues. Like knights errant of old, the mounted SS units—indeed all SS units—were expected to uphold putative chivalric virtues. For his part, Himmler always remained convinced that they could and would do so, or at least could be made to. "Rectitude and chivalry," he said in a wartime speech to new officers of the Replacement Army's grenadier divisions, "were always best bound together in the 'germanic' German [ _sic_ ] man." While he was not here specifically addressing SS personnel, the sentiment most certainly applied to them. "Our _Volk_ ," he added emphatically, "has existed eternally, and the Aryan will go on into eternity.... As long as the Aryan lives, as long as our blood—Nordic German blood—lives, so long will there be order on the Almighty's globe." When specifically addressing men of the _Waffen_ -SS, he was even more emphatic; and Operation Barbarossa, the invasion of Soviet Russia in June 1941, would most effectively allow his SS men to exhibit their chivalric errantry.
In the first month of the invasion, Himmler spoke to reinforcements destined for the SS _Kampfgruppe Nord_ fighting alongside the Finns on the northern extremity of the Russian Front. Though the men of _Kampfgruppe Nord_ were not SS cavalrymen, their mission remained the same in Himmler's eyes. They were going, he said, to defend a Reich that was on the whole "a happy, beautiful world full of culture" whose essence was defined by a National Socialist ideology "based on the value of our Germanic, Nordic blood." On the other side, he went on, "stands a population of 180 million, a mixture of races, whose very names are unpronounceable, and whose physique is such that one can shoot them down without pity or compassion. These animals...you will see for yourself. These people have been welded by the Jews into one religion, one ideology, that is called Bolshevism, with the task: now we have Russia, half of Asia, a part of Europe. Now we will overwhelm Germany and the whole world." And, in a throwback (unwitting perhaps?) to the exhortations made to the German cavalry and other soldiers who'd fought the Russians in 1914–1918, Himmler added for good measure, "When you, my men, fight over there in the East, you are carrying on the same struggle, against the same subhumans, the same inferior races, [as] at one time—1,000 years ago at the time of King Henry and Otto I—under the name of Magyars, another time under the name of Tartars [ _sic_ ], and still another time under the name of Genghis Khan and the Mongols. Today they appear as Russians under the political banners of Bolshevism."
Facing this threat stood the SS man. In Himmler's fantasy world, the SS man, and the SS cavalryman most of all, was lofty of sentiment, knightly in carriage, and possessed of the grim, clear-eyed nature of the predator. As the Italian war correspondent Curzio Malaparte observed the preparations in Rumania for the invasion of the Soviet Union, he wrote that the German fighting man there "had the same clear, lustrous eyes. And there was in them a mysterious, timeless look—a look pregnant with a timeless, mysterious sense of the inexorable." Though not describing the SS, Malaparte captured the precise ideal to which Himmler wanted his SS cavalrymen, indeed all SS men, to aspire. And in the east, the SS cavalrymen would find that domain whose rulers they of the Black Order would be: the same endless reaches from whose depths the feared invaders of World War I had come, the same eastern marches whose defense and expansion the German army had contested in 1914–1918. Those vast spaces, what one recent work has so tellingly called the "bloodlands," would be at the SS' mercy.
In this respect, Himmler was evoking a tradition antedating by decades the beginnings of Nazism. Not merely during World War I but as early as 1879, the renowned journal _Preußische Jahrbücher_ had declared that the east had been Germany's promised land ( _Land der Verheißung_ ) for as long as there had been a German history. The still-largely peaceful nineteenth-century struggle between Germanizing and Slavicizing tendencies had subsequently become open warfare in 1914. That war, in turn, had led inexorably to demands by prominent interest groups and individuals that the "face of Russia be turned back by force [away from Germany] to the East." Those same voices had demanded that Russia's Baltic provinces, along with Russian Poland and White Russia (Belarus), be partially or entirely absorbed into the Reich and Germanized, a process to be executed through the physical expulsion of the inhabitants if necessary. Now, in 1941, the conquest would be driven forward with a ruthlessness and terror perhaps never yet witnessed in history, a ruthlessness necessary, so the Nazis proclaimed, to defend the west against not merely Slavs but "Tartars [ _sic_ ] from the Crimea, the remnants of the Golden Horde; Kurds from Turkestan; and Mongols from the banks of the Don and Volga, from the shores of the Caspian, from the Kirgizian steppes, from the plains of Tashkent and Samarkand."
In the Nazi phantasm, at once self-consciously exotic and malign, these creatures would be driven from all those lands destined to be ruled by the new aristocracy ( _Hochadel_ ) of the SS, the fearless knights ( _wehrhafte Ritter_ ) of a new order of men literally bred to racial purity—just like Hanoverian horses, mused Reich Peasant Leader ( _Reichsbauernführer_ ) Walther Darré several years before. These men would weld together the Germanic peoples, indeed the whole of Europe. The bringing of German _Kultur_ that had recommenced with the war of 1914–1918 after a centuries-long hiatus, but that had been so cruelly ended, as the Nazis saw things, by the reviled Treaty of Versailles, had already been reinitiated in the conquest of Poland in 1939, a country whose very existence had been enshrined in that hated document. Some 900,000 Poles would eventually be driven from their homes in the country's western regions. These areas had been annexed directly into the "Greater German Reich," while Stalin's Russia had gobbled up the eastern portion of the country at the same time under the terms of the Nazi-Soviet Non-Aggression Pact of August 1939. With some of these lands having been part of Germany before World War I, the Nazi government and many ordinary Germans saw their incorporation as a matter of course. The unwanted Poles, and most of all Polish Jews, had subsequently been dumped into a reservation including both Warsaw and the ancient coronation-city of Krakow called the General Government. Now, with the invasion of Russia, the bringing of _Kultur_ and Hitler's drive for living space ( _Lebensraum_ ) in the east would be completed. Himmler's new chivalry would play its assigned part in this gigantic undertaking. "An idealized German _Drang nach Osten_ was mentally spliced with an equally idealized [influence of the frontier]" in the view of many German nationalist writers and thinkers even before the racial ravings of Himmler and the Nazis took firm hold. The result was "an emotion-laden narrative" that had already overturned the verdict of World War I in the Treaty of Versailles, taken lands from Poland, sent the German army and the SS marching eastward, and set the stage for the extermination of whole peoples by the time Himmler unleashed his SS horsemen in Russia.
The SS Cavalry Brigade in Russia, 1941
As early as January 1941 the SS _Reiterstandarten_ were in the process of yet another, and in this instance more overtly military-sounding, reorganization. In that month, orders came from Himmler's command that the old terminology of _Totenkopf-Reiterstandarten_ would be dropped. Henceforth the SS' mounted regiments were to be officially designated as such, namely the 1st and, in the spring of that year, 2nd SS Cavalry Regiments. The change in designation indicated that the older, prewar SS nomenclature no longer quite fit Himmler's or the SS cavalry's own views of the mounted units. The two regiments with all of their support elements continued to train throughout the period from January to the start of Operation Barbarossa in June. Unofficially they were apparently already being referred to collectively as the SS Cavalry Brigade ( _SS-Kavallerie-Brigade_ ) in spring 1941. Orders from Himmler formally establishing the brigade, however, did not reach the regiments until August, by which time the SS cavalrymen had already been in Russia for some time. Depending on losses and other circumstances the brigade's combat-strength in 1941 averaged 3,300–3,500 men; 2,900 horses; and 375 vehicles of all types. The principal maneuver elements of the brigade were the _Reiter_ regiments and a Bicycle Reconnaissance Detachment ( _Radfahrer Auf klärungsabteilung;_ RAA). Essentially a battalion, the RAA consisted of three squadrons of bicycle-mounted infantry supported by several motorized reconnaissance vehicles. Other units organic to the brigade included a veterinary detachment, headquarters and headquarters platoon, logistics elements, and so on. Specifically, as of 3 August 1941, the brigade's combat-strength was 3,364. Its ration-strength was 4,114. Despite the occasionally fierce political differences between the army and the _Waffen_ -SS, the brigade's organization mirrored almost exactly that of the army's earlier 1st Cavalry Brigade in 1939–1940.
In Himmler's initial thinking, the SS Cavalry Brigade was not intended to be a frontline fighting formation, a fact that was crucial to its operational history both initially in Poland and subsequently in Russia and elsewhere. Instead, along with certain other units such as the 1st and 2nd SS Motorized Brigades, he envisioned the Cavalry Brigade as a fully militarized reserve formation to be employed in rear areas for pacification missions. "He had in mind such activities as capturing disorganized Red Army men behind the German lines, combating partisans, and, especially, the shooting of Jews—all in the name of German security." These missions derived from the authority expressly granted to Himmler by Hitler in mid-July 1941 to conduct the "security policing of the newly occupied Eastern territories" ( _polizeiliche Sicherung der neubesetzten Ostgebiete_ ). For this mission, the SS cavalry were placed under the operational command of the Senior SS and Police Commander Center ( _Höhere SS und Polizeiführer_ ; HSSPF) in the area behind Army Group Center or _Russland-Mitte_. At that time, the HSSPF Center was Erich von dem Bach-Zelewski.
The assignment of the SS cavalry to the HSSPF indicated what sort of actual duties the cavalrymen would have. The office of HSSPF had originated within the SS in 1937 with Himmler's intention to incorporate and further centralize the police and security functions of the SS into the overall defense of the Reich. By war's outbreak, the network of HSSPFs had spread across Germany. The number of HSSPFs had expanded with the invasion of Poland, and by the end of 1939 five additional HSSPFs had been assigned to the areas of Poland either annexed into Germany or otherwise occupied. Subsequently, three more HSSPFs, among them Bach, were appointed by Himmler to the occupied Russian territories as the invasion of the Soviet Union rolled forward in 1941. While under Bach's command that summer, the SS cavalrymen would be operating in the area of the Pripet Marshes at roughly the same time as the army's 1st Cavalry Division. Some indication of the ruthlessness with which the SS cavalrymen would carry out their mission may be judged by their previous behavior. In occupied Poland on 6 October 1939, for example, men of the 4th Squadron of the _SS-Totenkopf-Reiterstandarte_ had been ordered to a "special assignment" around Kutno. The phrase already meant the summary killing of Jews and/or suspected partisans or "looters." In November troopers of the 5th Squadron shot 440 of 1,000 prisoners, all of them "while trying to escape." On 14 January men of the 3rd Squadron captured a band of four "well-known thieves" in the town of Stoczek. Three of them were summarily shot. On 15 and 18 January men of the same squadron shot three Poles "while trying to escape" during searches for arms in the villages of Rossa and Russastara. There were other such instances.
Specific orders concerning the SS Cavalry Brigade's mission in the context of Barbarossa arrived at the end of July 1941. On 28 July, over Himmler's signature, a Special Order spelled out which units would participate in the "combing through" of the marshes, those units' combat-performance expectations, and what specific actions those units were to take. All arms would participate: mounted units, motorized detachments, and infantry. The brigade's mounted units were expected to cover twenty-five to thirty-seven miles per day (40–60 km) unless combat and/or local searches were being undertaken. The order acknowledged that the infantry would have a hard time keeping up with the horsemen. By contrast, motorized units might well range farther ahead of them. All villages were to be considered strongpoints, either for the enemy or for the Germans. As a consequence,
if the population serves as the enemy of Germany, is racially or humanly [ _sic_ ] inferior, or indeed, as it often is in the marsh areas, made up of fleeing criminals, then all people that are suspected of helping the partisans are to be shot, while females and children are to be evacuated and cattle and food are to be apprehended and secured. These villages are then to be burned to the ground. Either the villages and settlements are a network of [friendly] strongpoints, whose residents kill partisans and pillagers and inform us of them, or they cease to exist. No enemy will be allowed to find support, food or shelter in these areas.
To reinforce the intent, Himmler evidently met with HSSPF Bach personally on 31 July at the latter's headquarters at Baranovichi lying midway on the route between Brest-Litovsk and Minsk along the marshes' northern fringe, presumably to reiterate at least the order's sense if not also its words. The next day, 1 August, further orders went out via radio that "all Jews must be shot. Drive Jewish women into the marshes." The SS Cavalry Brigade's commander, Hermann Fegelein, later reported that his men had killed 15,000 people between the towns of Pinsk and Baranovichi in the western reaches of the Pripet Marshes but that the water proved too shallow to drown the women. At almost the same date that these orders were going out regarding operations in the marshes, namely on 2 August 1941, the notification came to Bach's headquarters at Baranovichi officially establishing the SS Cavalry Brigade.
Between 29 July and 13 August 1941, the brigade was involved in its first clearing operation of the marshes in an area encompassing approximately the upper third of the triangle formed by the railways linking Brest-Litovsk, Minsk, and Gomel. Periodically, one element of the brigade, namely the Advance Detachment ( _Vorausabteilung_ ), a sort of rapid-response unit, found itself attached to the army's accompanying 162nd ID. In these actions, the SS cavalrymen fought not only partisans but regular Red Army forces, both horse-mounted and infantry. In one instance, for example—and in good cavalry fashion—an SS mounted patrol carried out a reconnaissance of more than 140 miles (230 km) in the four-day period of 28–31 July at the very beginning of the operation. On the basis of reports from the patrol's commander, a combat-engineer platoon leader ( _Obersturmführer_ ) named Karl Fritsche of the 2nd _Reiter_ Regiment, an encirclement of a large number of Red Army forces was accomplished by supporting German units. These included infantry from the 162nd ID, the SS Cavalry Brigade's Advance Detachment, and a German police unit. These troops successfully engaged the Russians, including Russian cavalry and infantry, and repeatedly prevented their breaking out of the pocket thus formed. By the time the clearing operation ended on 13 August, the SS Cavalry Brigade's leaders were reporting great success. They indicated that their enemies had included two Red Army cavalry divisions (36th and 37th) and the 121st Rifle Division.
Further, they reported fully 15,878 enemy personnel killed in action and 830 taken captive at a cost of only 17 SS cavalrymen dead and 36 wounded. The brigade also listed 200 of its horses as having been killed. These, however, the brigade replaced from the haul of more than 800 horses captured. The brigade's summative reports about a month later also recognized the fighting qualities of its opponents. Particular note was made of the Soviet soldiers' and partisans' effective defensive techniques and the concomitant difficulty the SS cavalrymen had in prying them out of their field fortifications even in the face of the Russians' certain death. The relative lightness of the cavalry brigade's indirect-fire weapons might help explain these difficulties, a problem also faced by the army's 1st Cavalry Brigade/Division. The Soviet defenders' desperation might also be explained, however, by the SS men's already established reputation regarding the treatment of prisoners. Furthermore, the Red Army's own very harsh discipline almost certainly played a role in its soldiers' tenacity. It was at just this time, for example, that Soviet dictator Joseph Stalin issued his famous order for the immediate execution of any soldiers of the Red Army who attempted to remove their insignia and surrender. The soldiers' families would be arrested for good measure, with all that such arrest implied for the latter's fate. In addition, any Soviet troops finding themselves encircled and preferring to surrender rather than break out "are to be destroyed by any available means, while their families are to be deprived of all state allowances and assistance." To the extent that Soviet soldiers hiding in the Pripet Marshes had radio contact with command elements farther east, they would presumably have learned of Stalin's directive. If fear of the SS cavalrymen didn't motivate them to fight to the end, fear of Stalin might.
Whatever the principal reason for continued resistance by Red Army soldiers and partisans, the two SS mounted regiments along with other units of the Cavalry Brigade were once again ordered out on a clearing operation after only several days rest and refitting. Between 17 August and 5 September the SS cavalrymen worked their way generally south-southeast through the Pripet Marshes via the town of Starobin, almost dead center in the triangle of roads surrounding the marshes (Starobin was secured by the 1st _Reiter_ Regiment on 22 August), toward the southeastern corner of the triangle. Fierce fighting occurred at Turov on the banks of the Pripet River when other troopers of both _Reiter_ regiments stormed the place on 21 August supported by anti-tank guns, light field artillery, and automatic weapons. Partisans and regular Red Army soldiers fighting from prepared defensive positions responded with heavy small-arms fire, anti-tank guns, and machine guns. The towns of Petrikov and Mozyr, located farther east along the Pripet River, were occupied in the days leading up to 4 September.
In his comprehensive report written early in September, then-Colonel ( _Standartenführer_ ) Fegelein noted that enemy forces were consistently "annihilated" when they comprised regular Red Army soldiers. Partisans posed a greater problem. Indeed, he wrote, they posed the greatest problem for German forces behind the front. They were "coldblooded, brave until annihilated and asiatically [ _sic_ ] cruel." The SS men had to remain constantly on the alert because the partisans could appear anywhere, aided as they were by an apparently excellent communications network and their knowledge of the terrain. They exhibited calm in the difficult fighting in the marshes and continuously obstructed all roads and forest paths by laying mines, destroying bridges, and placing machine gun nests and bunkers at tactically important junctions. Notwithstanding the irony of Fegelein's reference to Russian cruelty, it seems clear that the SS Cavalry Brigade and other German forces in the watershed of the Pripet knew clearly that their enemy was likely to stay there in spite of periodic "pacification" operations. They would have to fight that enemy again and again.
Between 7 September and 1 October, units of the brigade found themselves continuing to do just that. In this period operations were conducted in the marshes south of the road linking Brest-Litovsk and Gomel. The area of operations' western boundary was the Pripet River where that watercourse turned southeast at Mozyr. The River Dnepr formed the sector's eastern boundary. Throughout the period, the partisans' methods remained what they had been before: small-unit attacks on the SS horsemen; the mining and demolition of roads and bridges; and the disruption of German supply lines. On 10 September, 38 Red Army soldiers were captured in fighting around the village of Krassnyi Ostrov. In that same engagement, however, 384 partisans were confirmed killed. Total losses to the SS Cavalry Brigade were 2 horses killed and 19 horses wounded. Given the record of the brigade's personnel to that date, this very disparity (not to mention a complete absence of reported Red Army fatalities) smacks not so much of numerical inflation of enemy casualties but rather an effort to hide the genuine likelihood that most of the partisans in question were innocent civilians. Certainly, losses of some sort were being suffered by the cavalrymen, as 96 replacements arrived on 19 September. Nevertheless, the striking numerical disparities continued in most of the brigade's fighting during the period. On 25–26 September, for example, 280 more partisans and 87 "criminals" were reported shot by the brigade's troopers south and west of Gomel and several ammunition dumps were destroyed.
As the clearing operation around Gomel came to end at the end of September 1941, the SS Cavalry Brigade found itself being eyed for redeployment to the north and east for a rear-area security mission in the forthcoming Operation Typhoon. This operation, it was hoped both in Berlin and German Army Group Center's headquarters, would be the final, successful drive to Moscow. The assault had earlier been postponed in the wake of the destruction of encircled Soviet forces in and around Smolensk in late July and early August. Following that victory, Hitler had ordered Army Group Center's drive on the Soviet capital suspended. To the frustration of many of his commanders, Hitler had the OKW divert significant German armored forces to the south to assist Army Group South in the conquest of Kiev and the western Ukraine. It had been as a small part of that much larger, and spectacularly successful, operational redeployment that the SS Cavalry Brigade itself had moved into the lower southeastern reaches of the Pripet Marshes and crossed the Dnepr in late September. Now, as the Russian autumn came on in earnest, the forces of Army Group Center began preparations for a renewal of the great strategic advance on Moscow. As it had since June, the advance would continue roughly along the axis of the highway running eastward from Brest-Litovsk via Minsk to Moscow. Whether German armies could breach Soviet defenses and reach Moscow before the onset of winter may have troubled the minds of senior commanders. For the SS cavalrymen, as for hundreds of thousands of other German rankers, such larger questions made no difference. They may well have fretted about becoming cold—they and their horses would certainly be cold, bitterly cold, in less than two months' time—but they and their army counterparts would go where they were told for as long as they were told.
On 29 September, the brigade received orders to move to the town of Toropets which lay 124 miles (200 km) north of Smolensk and just above the road running eastward from Velikiye Luki to Rzhev. The redeployment entailed transporting men, horses, and equipment by rail from Gomel to Vitebsk, marching the brigade a further 111 miles (180 km) northward to Nevel, and then entraining again eastward to Toropets. In all, the brigade covered about 310 miles (500 km) and was essentially in place by 15 October. It was subordinated to the 403rd Security Division in the operational area of the German Ninth Army. As the brigade moved closer to the front, operational command-authority for it shifted from the HSSPF to the army, even though the cavalrymen retained their own separate supply chains and replacement pools. By the time the brigade was ready to assume full operations, Typhoon had been under way for about a fortnight. The brigade's headquarters at Toropets and its area of responsibility lay not quite 100 miles (160 km) behind the front lines.
As they had done in the Pripet Marshes, the brigade's troopers spread out in the region between Toropets and Rzhev and carried out security and anti-partisan operations. Rzhev had fallen to German troops on 14 October and was viewed as a likely staging point for a further German advance swinging to the north around Moscow as part of Army Group Center's massive projected encirclement of the Soviet capital. From time to time, the SS men were specifically assigned to guard the supply lines of particular units. In the second half of October, for example, the brigade's 1st _Reiter_ Regiment received orders to guard the supply lines of the Army's 253rd ID while simultaneously executing anti-partisan missions. In the same period, the brigade's Bicycle Reconnaissance Detachment carried out numerous patrols along the road linking Toropets and Yetkino and the railway connecting Velikiye Luki and Rzhev, this corridor being roughly the boundary between Army Group Center and Army Group North. "During these patrols, a combined total of 2,120 partisans and suspicious people were taken prisoner. Fegelein [still the brigade's commander] reported that these prisoners were 'handled in the general way,' although that was not elaborated upon. It is generally assumed that they were executed." Also at this time the autumnal rains fell heavily. Consequently, the brigade's mobility was significantly hampered both for the vehicular units and for the mounted formations. The 1st _Reiter_ Regiment soon reported that its horses had become exhausted by their exertions, particularly in light of the brigade's difficulties with supply lines stretching back, as they did, all the way to Warsaw. Despite such difficulties, however, the brigade's mounts held up fairly well under the prevailing conditions. Brigade-level reports listed a total strength of 3,183 horses in October. Of those, only 4 died of exhaustion between 25 and 31 October. Fifty-one were reported sick and unfit for service. Unfortunately, the more general situation for the army's horses in Russia at this time went from bad to much worse. The standard of equine care and horses' rates of survival in the SS Cavalry Brigade (and the army's 1st Cavalry Division) would presumably have been higher than in the German armies collectively. For the latter, the overwhelming majority of horses were draft animals, and their handlers were not always trained well enough or sufficiently motivated to provide good care. Those animals began to die from exhaustion, poor feed, disease, and wounds at truly alarming rates—as many as one thousand per day—as cold autumnal rains quickly turned Russian roads to bottomless, glutinous sloughs of mud. This was the famed _rasputiza_ , literally the "time without roads." In a situation common to so many German units elsewhere in Russia in 1941, from 21 October all forward progress of the SS Cavalry Brigade as a whole completely stopped, albeit temporarily, owing to the mud, even though the first snows had already fallen before the brigade was actually in place in Ninth Army's area. Even if they weren't actually moving forward, however, the brigade's troopers continued unabated their wideranging security and anti-partisan patrols, sometimes reaching as far northwest as the city of Cholm.
Nevertheless, as winter approached, the nature of the combat facing the brigade's troopers began to change. In early November certain units of the brigade began to find themselves in increasingly heavy fighting with regular Red Army forces. This situation was coincidental with three factors affecting the German offensive generally. First and foremost, the weather was rapidly growing colder. Secondly even as the temperatures dropped, German armies were reaching, or had already reached, the limits of their supply lines. Finally, Russian resistance was stiffening as the invaders approached the region immediately around Moscow, Rzhev being only about 130 miles (209 km) to the northwest. Famed armored commander General Heinz Guderian noted that "the bitterness of the fighting was gradually telling on both our officers and men.... It was indeed startling to see how deeply our best officers had been affected by the latest battles." The cavalrymen of the SS constituted no exception to Guderian's general assessment. They, too, were encountering ever stronger and more formal resistance, even in the rear areas. There also appeared a growing likelihood that they might be called upon by the army for frontline duty. Himmler, however, did not necessarily endorse such a development. As already indicated, the _Reichsführer_ -SS had been entrusted by Hitler with ensuring the proper settling of the "special tasks" attending the "political administration" of the occupied territories of the Soviet Union, tasks recognized officially by the OKW as being entirely Himmler's own independent responsibility. In keeping with the views of the _Reichsführer_ -SS noted earlier in his various addresses to SS men heading for Russia, these "special tasks" would "derive from the decisive struggle that will have to be carried out between the two opposing political systems [of Nazism and communism]." Himmler believed that the war to be waged against partisans, Jews, and other undesirables—whether in the Pripet Marshes or, now, in what OKW touted as the final drive on Moscow—was just as important as the one being fought on the front lines. If his cavalrymen, as well as other SS units, could be kept to their "special tasks," that would be his and their signal contribution to Germany's victory and Europe's future. As early as 15 September, for instance, he had evidently rebuffed suggestions that the SS Cavalry Brigade be assigned to frontline combat. The war's changing nature in late October and early November 1941, however, forced the SS Cavalry Brigade into ever-heavier fighting whether Himmler was comfortable with it or not. In the first week of November elements of the 1st _Reiter_ Regiment fought in a major battle alongside the army's 102nd and 253rd IDs in stopping a Russian assault near the town of Yeltsy. The commander of the 253rd ID subsequently sent formal thanks to the brigade for the _Reiter_ regiment's support. Evidence also continued to indicate that larger and more competent Soviet forces were successfully infiltrating the brigade's area of responsibility. In mid-November, shortly after the 1st _Reiter_ Regiment's battle, troopers of the RAA discovered a munitions dump at the rail junction of Olenino west of Rzhev. They secured almost 500 pounds of explosives, cases of detonators, more than 900,000 rounds of rifle ammunition, and 1,200 rifle grenades. Despite such localized successes, however, the SS cavalrymen might nevertheless have concurred with the views of Army Group Center's commander, Field Marshal Fedor von Bock. In an entry dated 1 December he wrote in his diary: "The fighting of the past 14 days has shown that the notion that the enemy in front of the army group has 'collapsed' was a fantasy.... Halting at the gates of Moscow, where the road and rail net of almost all Eastern Russia converge, is tantamount to heavy defensive fighting against a numerically far superior foe. The forces of the army group are not equal to this, even for a limited time." Somewhat farther south than Olenino, on 4 December, General Guderian recorded that the temperature was minus 32 degrees Fahrenheit. He wrote to his wife: "The enemy, the size of the country and the foulness of the weather were all underestimated, and we are suffering for that now." The SS cavalrymen would suffer right along with the rest.
The Soviet Counteroffensive—Winter 1941–1942
In the first week of December 1941, Soviet armies launched a massive counterattack against Army Group Center and the southernmost elements of Army Group North. Advancing pincer-like, in a negative mirror-image of the "C" formed by the arms of the Germans' farthest advance around Moscow, the offensive's objective was to blunt and, if possible, drive the invaders back from the metropolis. Red Army commanders envisioned a convergence of the arms of their reverse "C" on the highway running westward from the Soviet capital. North and west of Moscow, the Red Army's Kalinin Front (i.e., army group) smashed headlong into the German Ninth Army along a line roughly parallel to, but north of, the area of the SS cavalrymen's earlier security operations between Velikiye Luki and Rzhev. The principal objective of the Fourth Shock Army, Kalinin Front's main force, was to break Ninth Army's lines of supply, the very lines that the SS Cavalry Brigade had spent October and November helping to protect. This was the offensive about which the brigade's men had been getting apparent clues throughout November. An indication of the offensive's seriousness showed clearly in the directives that went out in mid-December to the brigade's rearechelon elements. The brigade's combat engineer company was ordered forward from Warsaw, though why it was still there remains unclear. Even the brigade's rear-area veterinarians were ordered to the front. As with the army at that desperate moment, the cavalry brigade was scouring its rear-echelon formations for all possible replacement personnel. Furthermore, the entire brigade was now placed under the direct operational control of Ninth Army. All security missions along Ninth Army's resupply lines were terminated, though some subsequently had to be resumed, and all available elements of the brigade found themselves in the front lines.
In the first weeks of December the Soviet offensive made good ground. In temperatures of minus 15 degrees Fahrenheit (–26° C) and with snow up to 3 feet (1 m) deep, the entire frontage of Ninth Army along a line east, north, and west of Rzhev gave way. As early as 7 December, Soviet forces overran the headquarters of the LVI Panzer Corps outside Klin on the railway running northwest from Moscow to Kalinin. For a time it seemed that the entire northern wing of Army Group Center might collapse. When Bock soon thereafter recommended withdrawal, Hitler relieved him (Bock was seriously ill with stomach ulcers in any case) and replaced him with General Günther von Kluge. Rather than countenance Bock's suggested retreat, Hitler ordered "fanatical resistance." Gradually, in bitter fighting, the Germans succeeded in hanging on, but the Soviet offensive slowed only slightly. At the end of December elements of the SS Cavalry Brigade's Bicycle Reconnaissance Detachment, temporarily attached to the army's 253rd ID, were conducting patrols on the shores of Lake Volga to the northwest of Rzhev. Attacked on 9 January 1942 by heavy Russian forces supported by tanks, the RAA fought its way out of the lakeside village of Peno where it had been assigned. For the next fortnight, the RAA fought and fell back to the southwest as part of a terrible defensive struggle by elements of Ninth Army trying to hold back the Red Army's attacks. On 17 January the detachment's survivors stumbled into the cavalry brigade's former headquarters town of Toropets. The RAA had suffered 75 percent casualties from its reported December combat-strength of more than six hundred officers and men. For all practical purposes, the Bicycle Reconnaissance Detachment had been destroyed.
The RAA's destruction left the brigade with only two substantial, combat-effective elements, namely the two _Reiter_ regiments. As was the case with the RAA, between 31 December 1941 and 20 January 1942, these regiments found themselves in almost constant combat with strong Russian forces advancing in the Red Army's larger attempt to encircle Rzhev from the northeast. Being forced to leave their horses behind in more sheltered rear areas, the men of the _Reiter_ regiments fought largely as regular infantry during this period, a situation that would repeat itself in the winter of 1942–1943. When they did employ their horses, losses could rise rapidly owing not only to the weather but also to the ferocity of the fighting. By the beginning of February, after nearly four weeks of continuous combat, the two mounted regiments had suffered almost 50 percent casualties in both men and horses. Even as losses mounted alarmingly, the remnants of the brigade found themselves transferred to the XXIII Corps' 1st Panzer Division. They remained attached to that armored unit for most of March. During this period reinforcements of ethnic Germans ( _Volksdeutsche_ ) arrived and brought the brigade's strength temporarily back up to 1,500 men. Nevertheless, by early April, the cavalry brigade's strength had been so reduced that the staff recommended to the _Reichsführer_ -SS that its designation be changed from a brigade to a "weak _Reiter_ regiment" in order to reflect its actual condition.
The designation soon became semiofficial. By April the brigade was being referred to as the SS _Reiter_ Regiment. At about the same time, Ninth Army's new commander, General Walther Model, issued plans that the unit be refurbished (though not necessarily once again expanded to brigade-strength) with better horses to be brought forward from the remount depot at Warsaw. Himmler received notification of the proposal, but it could not be executed in part owing to the Russians' seizure in January of the brigade's principal forward supply base at Toropets. With the town's capture, the brigade lost its supplies of saddles and other equipment. A lack of adequately trained riders also put paid to the effort to re-establish the mounted units at that juncture. By month's end, the SS Cavalry Regiment had been effectively reduced to battalion-strength, approximately seven hundred men. As Fegelein had already returned to Warsaw in anticipation of the unit's probable reorganization, the survivors' then-commander, Major ( _Sturmbannführer_ ) Gustav Lombard, requested of General Model that the remaining elements also be entirely withdrawn from the front and returned to Warsaw. This process gradually occurred between the end of April and the end of May. Only a small, ad hoc battle group remained behind.
As the fitful and rainy Russian spring began to take hold in April and May 1942, the worst of the winter fighting subsided. The SS Cavalry Brigade's former area of operations between Toropets and Rzhev had largely been lost to the Red Army. German forces, however, still held a deep bulge in the Russian lines. Shaped like a right-handed boxing glove, Rzhev lay at the knuckle and Olenino at the fingertips. The thumb was formed by a much narrower salient that German counterattacks had driven northeast from the town of Dukhovshchina to Belyy, which lay to the southwest of Olenino. Very large pockets of Soviet partisans and Red Army regulars, including paratroopers dropped earlier in the winter battles, lay cut off and surrounded east and southeast of Smolensk and stretching along the railway toward Bryansk.
The winter battles had been ferocious, and the brigade's condition reflected in microcosm the exhaustion and losses suffered by the Ninth Army and other German forces. Reporting on the carnage along the Leningrad highway northeast of Moscow, CBS Radio's Larry Lesueur was one of a small group of American correspondents covering the war from the Soviet Union at that time. In his reportage, Lesueur depicted the scene on all the fields of what the Germans were calling the Battle of Rzhev, including those fought over by the SS cavalry: burnt-out villages whose buildings were "now only charred, smoking embers"; battered and blackened German and Soviet vehicles; forests literally stripped bare and blown down as though swept by vast hurricanes; and everywhere the dead—not only dead soldiers but their horses—occasionally covered with what Lesueur called the "merciful cleanliness" of newfallen snow. "The war," he wrote, "was [particularly] hard on horses.... All along the roadside their frozen bodies lay in snow-covered blasted chunks." In the wake of such destruction of man and beast, it remained to be seen how and to what extent the SS Cavalry Brigade would recoup its losses.
CHAPTER 9
PALE HORSEMEN
THE 8TH _WAFFEN_ -SS CAVALRY DIVISION _FLORIAN GEYER_ , 1942–1943
As early as March 1942, even before the final cessation of the bitter winter fighting around Rzhev, the SS Operations Main Office (SS- _Führungshauptamt_ ) was considering the re-establishment of the now badly depleted SS Cavalry Brigade at its bases in occupied Poland. That idea, however, was stillborn. Instead, and in keeping with the expansion of the _Waffen_ -SS as a whole throughout the war, the Main Office decided to enlarge the brigade while rebuilding it. The result was to be a full-fledged SS Cavalry Division. Cadre would be provided by the remnants of the SS Cavalry Brigade as they transferred off the line following the Battle of Rzhev. As so often happened in the German armed forces during World War II, the brigade commander's name had by now become attached to the unit. Thus, cadre was sometimes referred to as coming from the SS Cavalry Brigade _Fegelein_. Beginning with the shattered remnants of the RAA, the brigade's survivors began returning from Russia as early as January. In the spring months, what remained of the _Reiter_ regiments also found their way back. As the units arrived, they were ordered to the SS training area at Debica, about seventy miles (112 km) due east of Krakow. There they were augmented by substantial numbers of replacements. In fairly short order but without much actual training time, the cavalry brigade's remnants incorporated the new arrivals. The new organization would be officially designated the SS Cavalry Division ( _SS-Kavallerie-Division_ ) effective 21 June 1942.
As the reorganization took place, the combat power of the division grew apace. In place of the earlier brigade's two _Reiter_ regiments the division would now have three, though standing up the third evidently took considerable time and effort. Each of the three regiments received an extra staff platoon in addition to four mounted squadrons, a combined machine-gun and mortar squadron, and a motorized heavy squadron. A full artillery regiment was also added, including both horse-drawn and motorized batteries. Further punch was added through the inclusion of a mechanized assault-gun battery and an anti-tank "detachment" (i.e., battalion). The assault-gun battery was later expanded to battalion-strength but was eventually disbanded, its guns being incorporated in the anti-tank battalion. A fully motorized divisional logistics train was also established, along with smaller elements such as bicycle companies, motorized signals platoons, and the all-important veterinary company. As it turned out, these paper arrangements were subject to extraordinarily frequent alteration during the division's lifetime. Just as in the case of the earlier SS Cavalry Brigade, however, subsequent experience saw various elements of the division repeatedly seconded for temporary duty with other formations.
Unfortunately for the new division, most of the incoming replacements were _Volksdeutsche_ from Hungary. Deemed "racially acceptable" from the SS' point of view, they nevertheless brought with them certain problems. Divisional staff found that the new recruits spoke German only haltingly or not at all ( _mangelhaft oder überhaupt nicht_ ). They had had little or no paramilitary training with Nazi formations such as the Hitler Youth, SA, or SS, and the bulk of them had not served with any foreign armed forces. Therefore, even if they could understand their orders, they might not be able to execute them. Consequently, both enlisted men and their NCOs had to be trained simultaneously not only in the German language but also in their military duties. This dual training continued in the field right through the end of 1942 and included, for veterinary personnel, everything from equine first aid, diagnosis, and medicinal treatment to farriery, tacking up, and saddling.
Regardless of the training they conducted, however, the division's officers and NCOs were strongly cautioned not to treat the _Volksdeutsche_ as inferior to Germans from Germany proper ( _Altreich_ ). This was an enduring problem, and the division's commander in April 1943, Colonel ( _Standartenführer_ ) Fritz Freitag, would still be alluding to it more than a year after the division had been formed. Since _Reichsführer_ -SS Heinrich Himmler continued to put great emphasis on ideological indoctrination along with SS cavalry troopers' regular training, the replacements' treatment assumed greater significance than might otherwise have been the case. Thus, for example, the division's leaders were ordered to avoid insulting the Hungarian _Volksdeutsche_ by calling them "gypsies." That would offend their honor, affect their morale, and lessen the ideological indoctrination's impact because gypsies were considered grossly inferior in the Nazis' racial hierarchy. Officers and NCOs using the term would be punished. On the other hand, if it were necessary for disciplinary purposes, a _Volksdeutsch_ recruit could be criticized expressly as a "dirty pig" ( _Dreckschwein_ ) or even a "sad sack of shit" or "limp dick" ( _Schlappschwanz_ ), just not as a gypsy. All personnel from the _Altreich_ were to be instructed in this regard so as to ensure that no mistakes were made.
Seriously complicating the difficulties of incorporating ethnic German replacements was the matter of the new division's horses. Cadre at Debica discovered that the horses shipped to the division had not been sufficiently schooled in the remount system, a fact that could have direct, adverse operational consequences. To expedite the replacements' equestrian training, as well as to further the conditioning of remounts, an SS Cavalry School was established for the division at Zamosc, about fifty miles (80 km) southeast of Lublin. Nevertheless, divisional staff reported that the horses themselves remained "raw," a fact evidently made worse by difficulties in getting enough feed. On 19 September the division's operations section reported to LIX Corps' chief of staff that there were insufficient oats for the horses and no independent means of acquiring any in what was by then the autumnal rainy season. The operations section further reported that the division was short three hundred mounts in any case and warned about the impending likelihood of heavy losses among those horses the division did have. Training difficulties thus multiplied considerably even as the division's units began their deployments. Troopers not yet trained to ride ( _reiterlich nicht vorgebildete Männer_ )—and who might not understand their commanders anyway—were mounted on horses that were not yet trained themselves and that might well have been improperly nourished. Given the urgency of the division's anticipated anti-partisan mission and the short time of only some two-and-half months allotted for the entire training process, this situation remained a recipe for incomplete operational results, not to mention occasional cracked skulls and broken arms and legs.
As it turned out, when the 1st and 2nd _Reiter_ Regiments and the Bicycle Reconnaissance Detachment began their movement earlier than planned, on 27 August 1942, they shipped out not only to fight partisans as they'd been told they would but also to help plug a gap in the German front lines near Vitebsk. They were placed under the command of the 330th ID of LIX Corps. As the last maneuver element to be raised, the 3rd _Reiter_ Regiment, along with the anti-aircraft artillery detachment and the division's combat engineers, remained behind to complete its training. For the 3rd Regiment, this training included squadron versus squadron combat exercises. In the meantime, between 7 and 19 September, the 1st and 2nd _Reiter_ Regiments engaged in regular anti-partisan missions in the 330th ID's area of operations. Their war diaries recorded numerous caches of munitions and other supplies seized, but despite what was occasionally reported as heavy fighting earlier in the month, they recorded only low numbers of casualties inflicted and sustained before the last two weeks of September. The numbers, however, rose fairly significantly in that last fortnight.
Losses of horses were also rather light during the period to 30 September though they, too, rose as the month ended. By that date, divisional veterinary staff recorded 36 horses killed in action and 6 dead from exhaustion or sickness. A further 82 had to be sent to rear-area veterinary hospitals, while another 107 were transferred from the 1st _Reiter_ Regiment to the veterinary company because the horses had become superfluous ( _überzählig_ ) as a result of unspecified reductions, presumably combat casualties, in regimental personnel ( _Mannschaftsausfall_ ).
These horses would be followed a month later by another 203 when, this time specifically on the basis of casualties sustained in October, the 1st _Reiter_ Regiment's 1st Squadron was actually disbanded. While it remains unclear how many of these transferred horses were subsequently sent back to combat units, the veterinary detachment reported to LIX Corps headquarters on 4 October that a total of 118 horses had been treated between the start of the division's operations and 30 September, that is, over a period of about four weeks. Such rates of activity remained constant throughout the period to the end of 1942, with horses being shunted continuously back and forth through the division's veterinary system in numbers ranging from twos and threes to scores and hundreds. The pace of these veterinary transfers persisted well into 1943 and would prove a significant hindrance to the division's operational capability.
Nevertheless, whatever difficulties the troopers were encountering with their supplies of horseflesh, the 330th ID's staff reported that the SS cavalrymen executed their tasks with what it called "noteworthy passion" ( _mit bewundernswerter Passion_ ). Whether that passion was in any way inspired by the fact that the division was now an officially enumerated unit, formally designated as the 8th _Waffen_ -SS Cavalry Division, remains unrecorded. It also remains needless to speculate, though the speculation is unavoidable, about what sort of malevolent excesses might be subsumed under that phrase. Equally chilling was the reported "cleansing" ( _Säuberung_ ) of lines of advance and rear areas at about the same time in the division's area of operations east of Vitebsk, missions undertaken again later, in November 1942, in the region south and west of Smolensk. By that date, the division had been transferred to VI Corps, and that corps' headquarters ordered the division's troopers to clear out ( _räumen_ ) all civilians from an area approximately two miles deep behind the front lines. If necessitated by numbers, the division was to evacuate them to rear areas. Accompanying the directive was the admonitory instruction from VI Corps' staff that proper accommodation would have to be planned in advance of any such evacuation. Otherwise, the "further growth of guerrilla bands ( _Banden_ ) would be unavoidable." The SS men's response to the admonition was not recorded. On the basis of their earlier behavior in Poland and Russia, however, their response may be imagined.
One of the problems encountered by the division in that early autumn of 1942 was the notorious condition of the Russian roads. As in 1941, the division's troopers struggled in the biannual rainy season. Repeatedly, divisional war diaries bemoaned the fact that units, and especially the motorized detachments, simply could not move in the sodden, pudding-like tracks that passed for Russian roads. On 22 September 1942, for example, the divisional war diary recorded that the movements of combat elements were being badly hindered by continuous rain and bottomless mud. Minefields and blown bridges only made matters worse. For the division's supply columns, too, conditions often made motorized movement literally impossible. On 25–26 September, the division's quartermaster was reporting that one brigade had to bring gasoline forward in confiscated Russian _panje_ wagons, presumably drawn by confiscated _panje_ horses, because the supply column's trucks couldn't move in the mud. At one point during the period to 30 September this particular _panje_ -wagon column grew to as many as seventy-six vehicles. Shortly thereafter the tally fell to only thirty-five wagons, but only after the attached veterinary company determined that no more than 50 percent of the draft horses pulling them were deemed fit for service. Initial war-diary entries for the period did not specify whether the division's horse-mounted units made better headway than motorized and mechanized elements. Nevertheless, part of the rationale for retaining horses at all in the Russian campaign was precisely the adverse climatic conditions in which these motorized and mechanized SS detachments were finding themselves forced to operate in 1941 and now again in 1942.
Of course, the difficulties in movement were only beginning; and insofar as the horse-mounted elements of the division were concerned, the coming of the Russian winter would once again make matters much more serious even though lower temperatures made for hardened footing. As early as 7 October, well before the real winter weather had even set in and before the cavalrymen's transfer to VI Corps, the division's veterinary unit had already requested that 80,000 horse-shoe cleats from LIX Corps' logistics be made ready ( _bereitgestellt_ ) for the division's horses. The threaded cleats were screwed into the horse' shoes for added traction in ice and snow. In addition, _panje_ horses continued to be viewed as an increasingly important source of replacements for the approaching winter. A potential requisition of some 200 of them, for example, was discussed with VI Corps veterinary officers in mid-November. Of these, 182 were eventually requested for the formation of an ad hoc battalion of ski troops, most of whom were transferred from the Cavalry Division's artillery but who were further augmented with men from the _Reiter_ regiments. Furthermore, in light of the increasingly severe winter weather, the division's troopers were being warned to ensure that exhausted horses of whatever sort, and specifically those affected by apparently widespread bacterial ailments, be transferred to the veterinary company early enough so that they didn't actually die in transport or immediately upon arrival at the hospital. Though the divisional veterinary staff couldn't foresee it, these steps for the SS men's mounts were being taken mere days before the launching of the massive Russian counterattack far to the southeast at Stalingrad where so many former troopers of the army's 1st Cavalry Division, now 24th Panzer, would find their graves.
For the SS Cavalry Division's men, as well as for their horses, the winter of 1942–1943 would once more pose the fierce challenge that had confronted German troops on the Eastern Front in 1941–1942. Nevertheless, with that first season's experience already under their belts, VI Corps' headquarters confidently pointed out that further weather-related difficulties could be avoided. Experience, winter uniforms, training, and constant supervision by all commanders and among all troops themselves could ward off winter's effects. By contrast, if numerous cases of frostbite did occur, then corps-level commanders would have to draw "certain conclusions" about the effectiveness of the division's training and the level of its morale. Despite the implicit threat of such questions being raised, the division continued to suffer losses to frostbite, combat, and illness. Things could hardly have been otherwise, and by mid-November 1942, the division had been designated as Army Group Reserve with the mission of conducting anti-partisan sweeps and countering any airborne landings. In these missions it operated in support of the 197th ID and the 2nd _Luftwaffe_ Field Division. At that time, the divisional adjutant reported that while the Cavalry Division's ration-strength stood at 10,204 officers and men, its combat-strength was only 5,214. The adjutant's report made no mention of the status of the division's horses.
In mid-December, the division was assigned to yet more antipartisan operations, this time in the region around the town of Baturino about sixty-two miles (100 km) north-northwest of Smolensk. Given the winter conditions and the resulting difficulties of supply, the division's operations officer (the billet's German designation was _Ia_ ) reported on both 19 and 26 December that most of the division's horses had been placed in unspecified winter quarters. Consequently, and at least temporarily, the SS Cavalry Division was now only as mobile as a regular army infantry division. It would move for the time being on foot. The _Ia_ also reported that while the division was only "conditionally capable" ( _bedingt geeignet_ ) of offensive operations, it remained "fully capable" ( _voll geeignet_ ) on the defensive. As it turned out, at the very moment when the division was essentially without its horses, its units also received orders to begin painting the division's newly assigned insignia on divisional vehicles: a horse's head in profile facing the viewer's left with a drawn sword crossing the neck from lower left to upper right, point uppermost.
For the rest of the month of December, in temperatures falling from freezing to about 15 degrees Fahrenheit (–9° C), the troopers of the SS Cavalry Division engaged in sporadic combat with partisans employing not merely small arms but artillery. Divisional reconnaissance elements estimated the combined strength of the partisan _Banden_ to be nearly three thousand. Normally, however, the latter appear to have operated in much smaller groups against both the SS cavalrymen and other units in the area, the latter now including the army's 52nd ID and, once again, the 2nd _Luftwaffe_ Field Division. Divisional daily reports also indicated that these partisans, who had come into the division's area of operations from the north, not only communicated with other Red Army forces via two clandestine radio stations but also had command of an airstrip through which supplies flowed and from which wounded were evacuated. Though the daily reports recorded no specific indication of the partisans' success with the evacuation of their own sick and wounded, the same documents did note at some length an "extraordinary number" ( _ausserordentliche Fülle_ ) of cold-induced illnesses in the division's own three _Reiter_ regiments. As had already occurred so frequently in Russia, these units had occasionally been seconded to other echelons during this period for anti-partisan operations, as had the division's artillery regiment. Ill or not, however, the cavalrymen could not be taken out of the line. Instead, they were treated in place owing to the division's reduced strength.
Notwithstanding its depleted state, the division continued with its grim work of hunting down, capturing, and killing anyone considered a partisan. Some measure of the viciousness of the business is recorded in a "Special Directive to the Forces" ( _Besondere Weisungen an die Truppe_ ) dated 24 December 1942. No civilians were to be "evacuated" from their homes and/or expelled from the division's area of operations without the responsible officers first notifying the divisional Enemy Intelligence Detachment (Ic), implicitly because such unauthorized action would simply create more partisans. More directly sinister ran the following warning: "Any arbitrary killing of partisan prisoners does not serve the interests of the division." Clearly they were being killed arbitrarily, and not only by _Waffen_ -SS units. If for no other reason than to ensure a continued flow of useful tactical and operational intelligence, the Special Directive expressly ordered that prisoners be sent to the divisional _Ic_ for interrogation. What happened to them afterward was another matter.
Orders or no orders, however, the killings went on just as they had since the invasion of Poland in 1939. Indeed, Hermann Fegelein's report from 1941 regarding Himmler's order to shoot or drive into the marshes all the Jews the earlier SS Cavalry Brigade came across clearly bears out the fact that the SS cavalrymen had long been killing persons out of hand, in vast numbers, and on a regular basis. The division's daily activity reports, and occasionally the division's war diary, for the period throughout the first quarter of 1943 are replete with references to prisoners and civilians being shot while trying to escape, wounded prisoners dying after interrogation, and suspected partisans and civilians being shot when found carrying arms. Representative of all of these individual reports was the divisional summary that was eventually compiled in early February 1943 for these continuing anti-partisan operations (code-named Operation Sternlauf). Concerning prisoners taken, the report noted 34 "bandits"; 153 presumably male "civilians"; and 547 women and children, a total of 734. Of persons killed, the report listed 580 "bandits"; 119 "civilians"; and 32 women, a total of 731. The report recorded no figures indicating children who might have been killed during the period. As with all soldiers on the Eastern Front at this time, the SS cavalrymen were being encouraged from the highest levels to be cruel. Hitler himself made this perfectly clear in an Order of the Day of 1 January 1943. It was evident, he said, that the consequence of a victory by Germany's enemies in the east would be Germany's destruction. German soldiers and men of the _Waffen_ -SS already knew that. What German armies were prepared to do to prevent that outcome, he countered, those enemies would soon learn to their sorrow. In Russia, Germany's enemies were already learning it. This kind of encouragement, coming from the führer himself, would obviously override any orders such as the one mentioned above from a mere divisional staff officer urging restraint in the matter.
The bitterness of the resulting anti-partisan warfare was matched by the bitterness of the cold. By January 1943, true winter weather had settled in on the SS cavalry. Temperatures recorded that month in the division's war diary routinely hit as low as minus 15 degrees Fahrenheit (–25° C). Yet the vicious, usually small-scale combats continued. The SS cavalrymen grimly acknowledged the Russian partisans' skill not only in effective daylight cover and concealment but also night fighting. In the vicinity of Simonowo near Baturino, for example, a small scouting party of one officer and twenty-four men from the 1st _Reiter_ Regiment was ambushed and surrounded in the failing afternoon light of 17 January by as many as 180 partisans. In badly cut-up terrain covered in dense brush, the scouting party was smothered with heavy automatic weapons fire and overrun. Only six men, three of them wounded, escaped the hand-to-hand fighting that followed. Upon reaching their own lines, the survivors reported that many of the badly wounded SS men who'd been left behind shot themselves so as to avoid falling into Russian hands. This result, said the division's war diary, was further proof that the 1st Regiment simply could not maintain security across its assigned front of nearly eleven miles (17 km), while at the same time confronting strong partisan bands who were not only fighting to cut the Germans' supply lines but who were also widely supported by the civilian population. Therefore it was imperative that all logistics elements of the division be better trained to resist the partisans' rear-area attacks on the division's supply columns.
Those supply columns, in turn, were at the same time being ordered to make greater use of horse-drawn vehicles because an expected delivery of fuel for the division's vehicles had not arrived. Orders to that effect went out to all units on 23 January 1943. Indeed throughout the entire area of the Army's XXX Corps, to which the division was at that time assigned, horse-drawn vehicles were only to be driven at the walk unless transporting the wounded. This would have to be done in order to spare the horses. Empty horse-drawn vehicles were allowed only one or two passengers besides the driver. Loaded vehicles were not allowed to carry any personnel at all nor were any horses pulling such vehicles to be ridden.
These prevailing winter conditions and the almost constant contact with the enemy resulted in a steady stream of casualties suffered by the division in the first quarter of 1943, a situation reminiscent in its own way of the German cavalry's campaign against _francs-tireurs_ in the bitter French winter 1870–1871. Now in 1942–1943, however, the figures for the numbers of casualties the division's troops inflicted on the enemy—whether partisans, "bandits," or mere civilians—were much higher for the simple reason that the SS cavalrymen had been trained to a level of fanaticism unknown in the German cavalry not only in 1870–1871 but also in 1914–1918. Comparing the numbers cited above, as well as the nature, of Russian prisoners of war and those killed, and even including the last ten days of December 1942, the division's own losses in the period to 4 April 1943, though steady, were relatively minor. They included 154 killed, 498 wounded (74 of whom were not evacuated), and 54 missing.
Of course, whatever losses the division did suffer had to be replaced. The _Reiter_ regiments were given first priority for replacements after the more or less continuous contact with the enemy since the preceding fall and winter. However, insufficient numbers of replacements were available from the division's supply units to bring the _Reiter_ regiments up to their 100 percent wartime establishment ( _Kriegssollstärke_ ). Consequently, only 70 percent of the wartime establishment could be maintained. This stark fact dictated a reduction in the number of mounted squadrons in each _Reiter_ regiment from four to three. Similar conditions also prevailed in several other divisional units. As they had in the winter fighting of 1941–1942, commanders again reached all the way back to the divisional supply detachment based in Warsaw to find the required personnel. In addition, command staff scoured the division's NCO training program for possible replacements. All possible sources found themselves culled of men for the _Reiter_ regiments, the division's anti-tank company, and the pioneer battalion. As a result, the 1st and 3rd _Reiter_ Regiments received some 788 replacements. At 70 percent strength, this meant, for example, that the total number of officers and men in the 1st Regiment fell from 1,416 to 991. By 1 April 1943, the entire division's combat-strength was listed as only 4,585 officers and men, its ration-strength 6,809. Curiously, however, the relevant documents list no horses at all, though the instructions for filling the forms out specifically stated that a unit's horses were always to be included in the reported ration-strength.
Divisional Anti-Partisan Operations—April to August 1943
From early April until the first week of August 1943, the now officially designated 8th _Waffen_ -SS Cavalry Division _Florian Geyer_ was once again assigned to anti-partisan operations under the overall supervision of the same man who had supervised the then-SS Cavalry Brigade's operations in the summer of 1941, HSSPF Erich von dem Bach-Zelewski. Under the circumstances, the moniker _Florian Geyer_ seemed fitting. The Franconian knight who was the division's namesake had been a leader in the Peasants' War (1522–1526). That association fit neatly the crude "blood and soil" ideology of the SS. So too did the fact that Geyer's sixteenth-century followers had traditionally been referred to as the "Black Bands" ( _schwarze Haufen_ ). Black featured very prominently in the SS' uniforms and regalia. As it turns out, when possible divisional names were being solicited, at least one of the division's units, the 3rd _Reiter_ Regiment, suggested that the name of the eighteenth-century Prussian cavalry commander Hans Joachim "Papa" von Ziethen be used. Ziethen's victorious exploits were very widely known, and he'd introduced a death's-head cap-badge similar to, but much larger than, the one worn by the SS.
The cruel, _condottiere_ -like nature of the original Florian Geyer, however, seemed more appropriate than that of a Prussian aristocrat in the conditions prevailing in that perilous spring of 1943 on the Eastern Front. Stalingrad had been lost along with the German Sixth Army in one of the war's greatest battles. Hundreds of thousands of Hungarian, Italian, and Rumanian troops who'd fought the Red Army along the Volga had also been killed or captured. In the retreat that followed, the 2nd SS Panzer Corps had rallied brilliantly and had retaken Kharkov in bitter, close-quarters combat after having been earlier driven out of the city. Further withdrawal would occur only in September in the wake of the Germans' defeat at Kursk in July, Hitler having forbidden such a withdrawal before then. The eventual projected stop line would be the Dnepr. This major north-south waterway was expected to provide a defensible barrier for German forces. The river also happened to be part of the large drainage basin through which flowed the rivers Pripet and Berezina, along whose banks the SS Cavalry Brigade had hunted partisans in 1941. As in that year, so now in 1943 the SS cavalrymen would be under Bach's control. Serving until the end of 1942 as HSSPF- _Russland-Mitte_ , Bach had in the meantime been appointed Chief of Anti-Partisan Combat Units. As such, he now had responsibility for all such tasks on the entire Eastern Front outside of the army's immediate operational areas. Bach had always willingly served his _Reichsführer_ -SS. Now on an even larger scale than in 1941, his primary concern was fulfilling Himmler's orders to ensure that no partisans operating behind German lines hindered the successful retention of the occupied territories or assisted the Red Army.
Unlike 1941, however, Soviet partisans were now better organized, outfitted, and logistically supported by rear-echelon Red Army forces. At this point in the war, Soviet partisans numbered about 250,000 at any given moment. They were potentially capable of severely disrupting German lines of communication and supply in a much more serious way than during the invasion's first year. In addition to appearing for the first time after Stalingrad in the northern Ukraine, large numbers of partisans infiltrated the vast forested areas around Bryansk. They also once again took up station in the middle reaches of the Dnepr watershed and reassembled in the eastern Pripet Marshes. Most importantly, by 1943 partisans had much wider and committed popular support. This support derived not only from the partisans' own brutally successful coercion of Soviet civilians but also because of the invaders' own frightful record of oppression. Consequently, the SS Cavalry Division's efforts entailed an even more intensive effort than in 1941 at rooting out any real or suspected partisans in a given area. In early 1943 that area once more included the region of the eastern Pripet Marshes, but the division also operated far to the east of the River Dnepr in the forests approaching Bryansk as well as southward in the direction of Kursk and Kharkov. As the division's troopers had already experienced (and as had the army's 1st Cavalry Division), the Soviet Union's vast swamps and forests continued to prove, if not an unassailable redoubt, then at least a recurrent refuge for Soviet partisans. Despite recurrent security sweeps, the partisans destroyed rail lines with "clockwork regularity," and, as the German withdrawal began in late summer, in many places they successfully "anticipated the routes...and systematically destroyed every bridge" in the rear areas.
These conditions made the division's anti-partisan operations even grimmer than before, if that were possible. In the view of the division's then-commander, Fritz Freitag, they also required a mental adjustment from the earlier frontline combat around Rzhev. Freitag issued his orders accordingly, even before the Battle of Kursk was fought (and lost) and the German tide in Russia began to run out. All personnel had once again to be aware of the special malice of the partisans so as to avoid unnecessary losses. This was especially the case among troops seizing loot that could be booby-trapped or otherwise wired with explosives. Freitag therefore ordered trophy-hunting strictly prohibited. At the same time, a fundamental suspicion of all persons and places was required. Every civilian could be a "bandit." He cautioned his troopers that once again there would be no recognizable front lines. Partisans could appear anywhere, even in pacified areas. Consequently, precaution and attention to detail would be critical in all things at all times. The combing through of forests and villages would have to be undertaken with the greatest possible care, and even apparently impassable terrain would have to be controlled. The cavalrymen would have to reckon with ambushes everywhere, all the time. The division's personnel would therefore have to respond in kind. They did, and they would continue to do so for the rest of the division's time in Russia and, later still, in the Balkans.
The division's command center for the operations of spring 1943 lay initially at Staryye Dorogi in what is now Belarus about thirty-seven miles (59 km) west of the city of Bobruisk. The latter city was located on the railway linking Minsk and Gomel. The division's three _Reiter_ regiments, as always the principal partisan-hunting units, were based at the village of Lapichi, slightly north and east of the same railway about midway between Minsk and Gomel. Divisional staff reported that the entire area, implicitly all the way from Minsk to Gomel, was infested with "bandits" and that all roads could be traveled only under armed escort. The road linking the divisional headquarters at Staryye Dorogi and the regional SS supply command at Bobruisk, for example, was reported to be so badly threatened that staff orders went out mandating a standing escort of at least twelve riflemen ( _Karabiner_ ) and three vehicles.
Besides the immediate threat posed to the division's mobility by partisan activity, recurrent problems with the division's horses further complicated matters. Always critical to effective anti-partisan warfare in Russia, sound horses were lacking in the division in the spring of 1943. Quite apart from operational implications, for a unit such as the SS Cavalry Division whose honorific title _Florian Geyer_ was now gaining wider circulation and whose commanders really did think of themselves as cavalrymen, lack of sound horses was a very serious problem. It was discovered, for example, that those horses the division did possess had to be treated regularly for scabies-induced mange. Evidently common in Russia at that time, this debilitating ailment had first appeared among the division's mounts in October 1942. At that time in the division's operational area north of Smolensk, mange had been reported as being widespread among civilian livestock. Stabling had been inadequate, and the division's troopers hadn't been able to build proper accommodations owing to constant relocations. Because the mass of the division's personnel had been ordered deployed as infantry that winter, too few qualified personnel had remained to look after too many horses. The situation worsened with the occasional impressment of divisional mounts into regular army units, many of whose men didn't know how to care for them. Horses not properly and regularly groomed were particularly susceptible. Dirty blankets or tack, matted coats left too long uncurried, close exposure to infected animals: all of these could contribute to a major outbreak among the division's horses and evidently did. In early spring 1943, as the division's parent unit, Ninth Army, moved to the rear, provisionally treated horses became reinfected so that by April the majority ( _die Masse_ ) of the division's horses were unfit for service. In the 1st _Reiter_ Regiment fully half of the assigned horses were deemed incapable of operational employment. In the 2nd Regiment every single one fell into that category. For each regiment's horses, the treatment and recovery time was estimated to be at least six weeks. To make matters worse, all three of the division's _Reiter_ regiments were already below strength in horseflesh. Each one carried 1,453 horses on the division's TOE. In reality, the numbers were 1,201 and 933 for the 1st and 2nd Regiments, respectively. Consequently, commanders determined that a fully "cavalry capable" operation at that moment was impossible. Troopers would instead have to be truck-mounted and therefore more or less confined to the region's poor roads. Crew-served weapons, ammunition, and other equipment would be brought along in pack-columns of requisitioned _panje_ horses. Only one full squadron in the 1st _Reiter_ Regiment could remain horse-mounted for the time being. None could be in the 2nd _Reiter_ Regiment. Horses left behind on these operations would be cared for by local volunteers ( _Hilfswillige_ ). The veterinary company would remain at Lapichi to look after the horses but would designate a veterinary squad ( _Staffel_ ) to accompany the troopers forward. Divisional commander Freitag understood the importance of horses in the anti-partisan mission: "It must be guaranteed," he ordered, "that by incorporating every means and making every effort the horses be made operationally effective as soon as possible." Capturing Russian horses was also given a high priority, despite the earlier outbreaks of mange and the fact that _panje_ horses were likely to be infected. Freitag's successor, Hermann Fegelein, who had once again assumed command after a stint in the rear, ordered that the division at least double the number of horses on its TOE through the simple expedient of having "every cavalryman take every horse he sees." This, he said, was a given.
In this reduced condition, the SS Cavalry Division undertook antipartisan operations between 9 and 17 May and between 13 and 16 June 1943 in Operations Weichsel I and II in the roughly triangular area north and south of the town of Rechitsa on the Berezina River extending to the confluence of the Pripet and Dnepr Rivers. For the purpose, it was incorporated in a battle group ( _Kampfgruppe_ ) with the 10th and 11th SS Police Regiments and Assault Battalion "South." Orders indicated that the encirclement and total destruction of partisan bands constituted the standing operational objective. Simply driving them away would accomplish nothing because they would just reappear elsewhere. Partisans' escape through SS lines, especially to the north, was to be avoided under all circumstances.
Until the beginning of June the division's elements operated both east and west of the Pripet River, fighting mostly small engagements and capturing and/or killing many suspected partisans. However, the division's reinforced reconnaissance detachment also fought a sustained, three-hour battle on 17 May near the village of Novoselki. At a cost of 3 dead and 18 wounded, the SS cavalrymen killed more than 150 members of a partisan band armed not only with small arms but also with anti-tank guns and mortars. Despite the heavy rains that followed in the second half of May—rains that, as usual, made the marshes impassable for vehicles—the operations continued until the first week of June. Once again, most engagements were small and losses to the division few. Recorded enemy dead were also relatively few, though numbers of those who were killed were often reported, yet again, as "shot while trying to escape," or, in the case of at least one woman, shot for refusing to answer interrogators' questions. In order to prevent the partisans from re-establishing themselves in the region, the division and its supporting elements "evacuated" all civilians and sent them to unspecified "labor duties." All livestock that could be rounded up was also confiscated. Finally, every village and all individual habitations were burned to the ground ( _restlos niedergebrannt_ ). Between 8 May and 4 June 1943, the division suffered thirty-five casualties, of whom only eleven were killed in action. From the last week of June to the last week of July, similar operations occurred in the old area of activity from 1941 around the city of Mozyr. Interestingly, this latter operation, code-named Seydlitz and occurring between Weichsel I and II, took its name from one of the most famous Prussian cavalrymen of the eighteenth century, Friedrich Wilhelm von Seydlitz. As usual, the division's _Reiter_ regiments and the reconnaissance detachment were the principal partisan hunters. Though riding and marching as much as thirty-seven miles (60 km) per day, the division's units fought only one major battle. Just as it had always done, however, the terrain proved a great difficulty, especially since the partisans had destroyed all the bridges and mined most of the tracks. Consequently, at one point during the operation the 2nd _Reiter_ Regiment had to be resupplied entirely by air—using an airstrip captured from the partisans no less—because motorized logistics vehicles couldn't reach it in the "immense marshes and forests." Combat engineers and even air strikes had to be called in to destroy heavily fortified bunker complexes. All civilian habitations were once again burned down. In keeping with established practice, those civilians who could be rounded up were again dispatched to "labor duties," and all of their livestock was taken. In the period between 25 June and 17 August, the division suffered 26 killed and 56 wounded. The division's strength, not including the 3rd _Reiter_ Regiment or the combat-engineer battalion, stood at 8,890 officers and men on 20 August. The division's horses were not included in the count, and records for the period do not indicate what the number of mounts was.
The End in Russia—September to December 1943
It was at this stage, in the late summer of 1943, that the _Florian Geyer_ Division's time in Russia began to approach its end. In the wake of the fierce fighting at Kursk in early July, German forces began a slow withdrawal to the Dnepr, the same river along whose upper reaches the division had periodically operated for some two years. In the second half of 1943, the division found itself transferred to the Ukraine for anti-partisan and defensive operations in the area south of the city of Kharkov and subsequently around Kremenchug on the lower Dnepr some 140 miles (225 km) to the southwest.
Even as these operations continued, another reorganization of the SS cavalry was impending, one that would eventually and significantly increase at least the nominal strength of the division to some 13,000 men. The _Reiter_ regiments were renumbered and a fourth one was added, so that their designations simultaneously changed from 1st through 4th to 15th through 18th. Furthermore, the assault-gun detachment was expanded to give the division greater punch. This reorganization and expansion featured as part of the much larger expansion of the _Waffen_ -SS as a whole in the war's last two years. Of the thirty-eight _Waffen_ -SS divisions officially established during the war (thirty-nine if one counts an SS mountain division that was subsequently disbanded and its number reassigned), fully thirty of them, starting with the SS Cavalry Division's numerical follow-on, the 9th SS Panzer Division _Hohenstaufen_ , were raised after December 1942.
This vast accretion of strength reflected several important factors. Firstly, Himmler's personal importance to Hitler had grown enormously. Because the SS remained the most ideologically committed arm of a Nazi regime now fighting a purely defensive and increasingly bitter war on all fronts against rising Allied power after July 1943, Himmler could successfully make ever-greater demands for men and equipment. Hitler didn't call Himmler his "Loyal Heinrich" for nothing. Nowhere was this more the case than on the Eastern Front. Secondly, the _Waffen_ -SS constituted some of the most effective combat forces still at Germany's disposal, and it seems clear that the _Florian Geyer_ Division saw itself in that light. Expanding the division would also fit the nature of the war in the region to which it would be transferred in 1944, namely the Balkans. There the war had been nothing but an anti-partisan campaign since the initial German and Italian victories in the spring of 1941. As an experienced and utterly ruthless anti-partisan force, the SS Cavalry Division would be a natural choice for assignment there; and until the division's eventual destruction in Budapest between December 1944 and February 1945, it would see no genuine large-scale combat against regular forces as it certainly had in Russia. Then, too, the extermination of Jews in the Balkans really only occurs during this same period, particularly in Hungary. The division could demonstrate its evil expertise in that enterprise as well. Finally, as a third factor explaining the _Waffen_ -SS' large-scale expansion after December 1942, it should also be noted that the nature of the organization as a whole was fundamentally transformed in the course of 1943–1944 quite aside from numbers. That transformation encompassed the recruitment of very large numbers of personnel who earlier were regarded as strictly off-limits for ideological reasons. Danes, Norwegians, Dutchmen, Flemings—these had always been recruited as racially acceptable SS men. Now, however, in 1943 and 1944, the manpower pool was expanded to include not only "borderline Aryans" such as Latvians, Estonians, and Lithuanians but even Ukrainians and Bosnian Muslims. Hang-dog units these new SS formations may well have been. Evidently, most were. Nevertheless, they represented a radical departure from the perceived racial superiority that certainly still existed in the _Waffen_ -SS when the _Florian Geyer_ Division was established in early 1942, even if many of the division's veterans may have thought less than good things about _Volksdeutsche_ as replacements. This larger environment thus further serves to place the SS Cavalry Division's service in 1944 in a light different from its earlier campaigns between 1939 and the end of 1943.
Ultimately, the _Florian Geyer_ Division did not represent a continuation of any German cavalry tradition between 1939 and the end of 1943 insofar as its racially charged mission is concerned, even though its equipment and organization did actually place it well within the framework of what a German cavalry unit of the time was supposed to look like. Throughout its deployments in Poland in 1939–1940 and in Russia between 1941 and 1943, its primary task always remained rear-area security and anti-partisan warfare. This mission came to it from the very highest levels of the regime. Occasionally, of course, it fought pitched battles with those same partisans. Then, too, it participated in the fierce defensive battles of winter 1941–1942 against the Red Army near Rzhev and 1942–1943 near Smolensk. Nevertheless, the division's _Reiter_ regiments and reconnaissance battalions hunted Jews and other civilians just as often, if not actually more often, than it fought organized enemy formations, whether partisans or otherwise.
This orientation arose directly from the _Waffen_ -SS' origins in the internal security apparatus of the Nazi Party and reflected one of Himmler's most important concerns. Unlike certain other _Waffen_ -SS divisions, that, for all of their murderous excesses, became primarily frontline combat units (e.g., the 1st and 2nd SS Panzer Divisions _Leibstandarte-SS Adolf Hitler_ and _Das Reich_ ), the 8th _Waffen_ -SS Cavalry Division never shed the original security function of its organizational antecedents. Those antecedents had been, and the division always remained, the mounted arm of a Nazi Party formation the enemies of which, real or imagined, were to be ferreted out and exterminated without mercy. Fegelein, Freitag, and others may well have seen themselves as chivalric mounted warriors, but that fact is immaterial. The SS cavalrymen executed few of the modern horse-cavalry's missions that were still being actively trained for in the _Heer_ in the late 1930s and some of which the army's 1st Cavalry Division itself undertook: long-range reconnaissance; turning the enemy's flank; the interdiction of the enemy's logistics and lines of communication; and force protection through screening. On the contrary, the SS cavalrymen, like all SS men, were very much "Hitler's army" to a degree that even the markedly politicized army was not. As in Poland in 1939 and 1940, so too from first to last in Russia, the _Florian Geyer_ Division participated in a war of extermination against largely civilian populations. In stand-up battles against regular troops of the Red Army or well-organized formations of partisans, its men appeared to have fought as tenaciously as any. That fact was in keeping with the cavalrymen's self-perceived warrior's ethos. It was also not uncommon in most units of the _Waffen_ -SS raised before the end of 1942. Nevertheless, most of the division's operations, as well as those of its predecessor, the SS Cavalry Brigade, were conducted against persons who could offer little or no resistance. None of the mounted arm's tradition of true chivalry remained, notwithstanding Fegelein's and the division's aping of those same traditions. To that extent, _Florian Geyer's_ troopers manifested the distinct reality of the _Waffen_ -SS as political soldiers, ever willing to serve the murderous ideological imperatives of the Nazi dictatorship.
CHAPTER 10
LAST RECALL
THE 1ST CAVALRY CORPS, 1943–1945
The dissolution of the German army's 1st Cavalry Division in the autumn of 1941 seemed to mark the end of that army's formal cavalry tradition. True, the mounted squadrons of divisional reconnaissance battalions remained, but the presence of horse-mounted maneuver-units capable of independent action under the control of higher-echelon command appeared over for good. As things turned out, "for good" lasted for about twelve months. In fact, throughout the period from November 1941 to December 1942, various mounted units continued in existence in addition to, and sometimes amalgamated from, divisional reconnaissance squadrons. In July 1942, for example, an improvised cavalry brigade was authorized by the then-commander of Ninth Army, General Walther Model. It had the mission of helping eliminate Red Army forces, some 60,000 strong, still occupying a salient in the dense, swampy forests behind Ninth Army's lines, a result of the latter's near encirclement in the earlier winter fighting in the Battle of Rzhev. Comprised of elements of the reconnaissance battalions of the eight divisions under Model's command, the brigade included three cavalry regiments of one or two horse-mounted troops (companies) and three to four bicycle-mounted troops each. There was also a combat-engineer company, a medical company, and one motorized and one horse-drawn logistics column. Each cyclist troop's immediate supplies were carried in a two-wagon detachment hauled, as usual in Russia, by the ubiquitous _panje_ horses. For their part, the troopers in the mounted elements rode regular military horses. The brigade remained a light formation, however, in that tanks and anti-tank units were seconded to it as required. Its organic artillery consisted of six (per regiment) of the same 75-mm guns typical of the 1st Cavalry Division before 1941. In its one major combat operation, from 2–13 July, the "Cavalry Brigade Model" as it was unofficially known successfully maneuvered and fought its way through more than ten miles of seemingly impenetrable terrain while the tanks of the panzer division to which it was attached sometimes found themselves literally stuck in their tracks. And while postwar German analysis admitted that the operation would likely have been successful even without the brigade's presence, that same analysis concluded that the cost to Ninth Army in men, matériel, and time would almost certainly have been greater owing to the inability of either purely infantry or armored formations to move as effectively as the brigade had done.
The Cavalry Brigade Model's example may also have served as inspiration for another effort to resurrect the mounted arm when, later in 1942, then _Rittmeister_ Georg _Freiherr_ (Baron) von Böselager managed to convince the commander of Army Group Center, Field Marshal Günther von Kluge, that horse-mounted cavalry might still be useful on a fulltime basis, despite the 1st Cavalry Division's having been disbanded the year before. Given the terrible winter of 1941–1942 and the biannual rains, with all of the difficulties in maneuver that such weather always brought the German army on the Eastern Front, Kluge eventually agreed. Possessing a sort of provisional character, this "Cavalry Unit Böselager" was subsequently created by an army-group order in January 1943.
Born in Hesse in 1915, Böselager came from a military family and in his youth became a successful competitive rider. In due course, he found his way into the army, and he became an enthusiastic cavalry officer even as he continued to compete in both show jumping and flat racing. His original regiment, the 15th _Kavallerie-Regiment_ , was one of those that became part of the infantry's reconnaissance forces. In this case, his regiment became the eyes of the 6th ID. It was in his capacity as a squadron commander of the resulting 6th Reconnaissance Battalion that he made his case to Kluge.
By the end of March 1943, this unit was expanded to the size of a regiment and designated, because of its army-group assignment, Cavalry Regiment Center. Army Groups North and South followed suit shortly thereafter. Thus, even though the 1st Cavalry Division had earlier disappeared, the cavalry tradition lived on in at least an ad hoc fashion. That fashion, however, was soon formalized. The then-chief of staff of the army, Colonel-General Kurt Zeitzler, authorized a full-fledged cavalry corps under the initial command of Major General Oswin Grolig and, shortly thereafter, Lieutenant General Gustav Harte-neck, former commander of the 9th Cavalry Regiment in 1939–1940 and chief of staff of Second Army in 1943–1944. Orders establishing the I Cavalry Corps were issued on 25 May 1944, and the corps was supposed to be ready for operations ( _verwendungsbereit_ ) by the beginning of August. Curiously enough, the Cavalry Corps first saw the organizational light of day in the very same region where both the 1st Cavalry Division and the 8th _Waffen_ -SS Cavalry Division _Florian Geyer_ had already served, namely in the neighborhood of Pinsk in the western reaches of the Pripet Marshes. It was here that the Corps received assignment of its principal maneuver elements from the German army: 3rd and 4th Cavalry Brigades (outgrowths of the earlier Cavalry Regiments North, Center, and South). Also assigned was the 1st Royal Hungarian Cavalry Division. That division still carried the designation "royal" in light of the fact that Hungary remained a nominal monarchy, though it had been ruled since the 1920s by a regent, Admiral Miklós Horthy de Nagybánya. It was at Pinsk, too, that General Harteneck assumed command of the Cavalry Corps on 22 June, the third anniversary of the beginning of Operation Barbarossa.
More striking even than the bare fact of the Corps' establishment was the inclusion of an entire Hungarian division. The Hungarian division's heritage derived from the ancient Magyar tradition of the hussars. For centuries, Magyar light horsemen had enjoyed a reputation for dashing intrepidity and equestrian skill. They had even bequeathed the nineteenth-century accoutrement of the fur-trimmed, braid-covered pelisse worn rakishly over one shoulder of the fanciful hussar's dress. By the time of World War II, of course, much had changed. The Hungarian army, though officially fighting alongside Germany's since 1941, had frequently been regarded in Germany largely as an inferior force and, in truth, suffered from inferior equipment and logistics. Furthermore, relations in general between Germans and Hungarians had never been consistently cordial. Nevertheless, the Cavalry Corps' open inclusion of Hungarian horsemen could not be more different from the treatment so frequently meted out not only to Hungarians themselves but also to ethnic _Germans_ from Hungary serving with _Waffen_ -SS' 8th Cavalry Division.
Whether enjoying brotherly relations or not, several of the Corps' units saw action even before its official date for initial operational capability. While the 4th Brigade and the Hungarians still found themselves in the process of establishment and training (with troopers of the 4th Brigade also being assigned the German cavalry's by-now-standard Russian-theater anti-partisan mission), elements of the 3rd Brigade were deemed sufficiently ready to be sent into combat against regular forces of the Red Army. Their presence was sorely needed, as was that of every single German soldier available: June 1944 was the moment of the resumption of the great Soviet drive to the west through Belorussia. In fact the Red Army had opened its offensive the day after Harteneck assumed command of the Cavalry Corps.
At the end of June, troopers from the 3rd Brigade were dispatched to Slutsk, not quite 150 miles (241 km) northeast of Pinsk to help fend off advancing Soviet forces. Tellingly, they were sent without their horses, indeed without most of their supplies. In Slutsk, designated to be held as a sort of hedgehog position ( _fester Platz_ ), the cavalrymen served as part of a blocking force along with an attached assault-gun battery and a company of pioneers. Even though they were eventually reinforced by the bulk of the 4th Brigade and other noncavalry units (ultimately including 4th Panzer Division), and even though they inflicted occasionally heavy losses on advancing Soviet troops, the Germans could not hold out in Slutsk, not least owing to the insufficient defensive works constructed before the cavalrymen arrived. By 30 June the Corps' troopers had been driven out of the town by Soviet mechanized units. As July began, they were forced to retreat farther to the west so as to avoid encirclement. Interestingly enough, the Soviet troops driving them out included cavalry of their own. Part of Lieutenant General I. A. Pliev's cavalry-mechanized group consisting of the 4th Guards Cavalry Corps (including three cavalry divisions) and the 4th Mechanized Corps, the Soviet horsemen were executing the very sorts of reconnaissance and flanking missions, as well as dismounted combat, that the German horsemen themselves had so often conducted since 1940 but now on a much larger scale. Some indication of the intensity of the fighting in the often swampy and heavily forested region west of Slutsk may be gained from the fact that the Corps' own headquarters were directly attacked on 4 July by Russian infantry covered by supporting fire from heavy mortars. Nine staff personnel were killed and wounded and fully nineteen reported missing, presumably taken prisoner, by the time the attackers were driven off. And the Russians very nearly managed to capture or kill the commander of the German Second Army, General Walter Weiß, who happened to be attempting to land at the Cavalry Corps' headquarters at that very moment.
As the Corps slowly withdrew under intense Soviet pressure toward Baranovichi, located at the midpoint on the road linking Brest-Litovsk and Minsk, friction arose between German and Hungarian units. Various reports surfaced in early July that the Hungarian cavalrymen appeared to be folding under the admittedly heavy weight of the Soviet advance. The Hungarian Division had been brought up to the line on 4 and 5 July to support the 4th Cavalry Brigade that had been sent forward earlier. The Hungarians were immediately and heavily attacked by Soviet armored elements. These attacks shook ( _erschüttert_ ) the Hungarians badly, and, as a result, divisional elements began to withdraw, evidently without orders. As the fighting continued on 6 July, the Hungarian Division's commander reported that his unit was, at least temporarily, combat-ineffective ( _keine Kampfkraft mehr besitzt_ ). He said his troops had been without supplies and ammunition for three days, were physically exhausted, and had been unnerved by the Soviet tanks. By way of reply he was told that assault guns from the already hard-pressed German 4th Cavalry Brigade would be sent to support him. Nevertheless, the 4th Brigade's right flank was uncovered by the Hungarian troopers' movement, and the German cavalrymen were threatened with encirclement. Harteneck therefore intervened personally. He went to the Hungarians' command post and peremptorily ordered the Magyar horsemen to stand fast. His mood can be imagined. He had already earlier intervened in a much more formal fashion, writing a two-page letter to the Hungarian divisional commander on 4 July and addressed to "Your Excellency." In that missive, he had pointed out certain disciplinary deficiencies in the Hungarians' logistics elements (the very ones whom the Hungarian commander had said were missing), though he wished the Hungarian cavalrymen well in their "first major action" ( _ersten...einsatz im Großkampf_ ). Evidently that letter had not had the desired effect. For his part, Hungarian commander General Vattay defended himself against accusations of overly rash withdrawal. In a situation report he indicated that because the Corps' headquarters had been temporarily out of communication owing to the attacks on it of 4 July, and beause he'd received the approval of ( _im Einvernehmen mit_ ) the commander of the 4th Brigade, he'd ordered his men back to a more defensible position. In the same report he also incidentally requested rations for 17,000 men and 11,000 horses, as well as 16,000 gallons (60,000 L) of gasoline for his vehicles.
Though it remains unclear from the Corps' records how the Hungarians managed to get their horses forward when so many of the Germans' mounts had initially been left behind, the Corps' defensive positions around Baranovichi were to be defended by more than just horse-mounted troopers. The Cavalry Corps now in fact represented, at least in its TOE, what the commander of the 1st Cavalry Division, General Kurt Feldt, had so consistently advocated in 1940 and 1941: a combinedarms, corps-strength force. On 4 July 1944, just before the fighting around Baranovichi began in earnest, the Corps included not only the 4th Cavalry Brigade and the 1st Royal Hungarian Cavalry Division (both minus certain elements). It also temporarily included the 4th Panzer Division, the 904th Assault Gun Brigade, the 447th Regimental [Combat] Group, a security regiment, an additional field artillery battalion, an anti-aircraft artillery battalion, a combat engineer battalion, an anti-tank detachment, and a separate assault-gun detachment of ten guns. It even included two companies (five vehicles each) of Tiger tanks, the heaviest armored vehicles then available in the German army. Earlier in the war, at the time of the then-1st Cavalry Brigade's campaign in Poland, Feldt had maintained that at least divisional status for his unit would be necessary to prove whether horse-cavalry could still play an effective operational role. That status had been achieved by the time of the invasion of the Low Countries and France in 1940. Subsequently, at the end of the 1st Cavalry Division's campaign in Russia in 1941, Feldt had argued that even his by-then-successful division still didn't possess sufficient artillery firepower and support elements, particularly combat engineers. Cavalry could still be effective, he'd argued, but only if it operated at the organizational level of a corps. On that he'd insisted in his final report as divisional commander. Instead of being reorganized, however, the 1st Cavalry Division had been disbanded and its horses and men dispersed to other formations. Now, in the form of the 1st Cavalry Corps, Feldt's injunctions finally appeared to have been realized, even though he was no longer present. In the great defensive battles that began in the summer of 1944, however, it remained to be seen whether the Corps could hold its own in the face of a surging Red Army that had achieved all-season operational superiority on the Eastern Front.
This superiority evidently continued to make itself felt, and so did the resulting tension in the Corps' command structure. Both the 4th Cavalry Brigade and the Hungarian Division received orders dated 5 July to hold off advancing Soviet forces so as to ensure that the roads running westward from Slutsk through Baranovichi to Bialystok and southwestward from Baranovichi to Brest-Litovsk wouldn't be cut. These perennially important roads had always been critical to the passage of German forces through the northern expanse of the Pripet Marshes. To help ensure that the roads stayed in German hands, all commanders of the Corps' various elements were explicitly ordered by Ninth Army's commander, General Nikolaus von Vormann, to personally lead operations to maintain the link between the Corps' own 4th and the neighboring 12th Panzer Divisions. General Harteneck, in turn, informed the Hungarian Division's General Vattay that Hitler himself had ordered ( _der Führer...ausdrücklich befohlen hat_ ) that Baranovichi's road junction was so important to the situation on the Eastern Front that it simply had to be prevented from falling into Soviet hands, period. Harteneck curtly said that he expected the Hungarians to hold their defensive positions south of the place regardless of their own situation ( _unbedingt gehalten wird_ ).
General Harteneck reiterated these orders on 7 July, as well as similar ones for the 4th Cavalry Brigade but added for emphasis that 4th Brigade had to ensure that the road running southwest from Baranovichi had to be held open until the 4th Panzer Division could be withdrawn along it. He closed by alluding to Field Marshal Model's stated expectation that all commanders lead from the front. As if he'd not said enough on the subject already, Harteneck then sent yet another two-page letter to Vattay on that same day, as well as several additional notes. Among other things, he observed that officers from the German cavalry and armored units felt that their Hungarian counterparts were not behaving in a manner routinely demanded of, and exhibited by, German officers. When properly led, he wrote, the Hungarian cavalrymen were just as worthy as the Germans. When Hungarian officers failed to act as combat leaders ( _Vorkämpfer_ ), however, their troopers "failed immediately" ( _sofort versagen_ ). He coldly pointed out that he'd found the Hungarians' entire 3rd Hussar Regiment in the town of Tartak without their ever having had contact with the enemy, indeed without their having conducted any reconnaissance. He rounded on Vattay for the lack of march discipline in the Hungarian columns and expressly forbade the withdrawal of any Hungarian unit without his (Harteneck's) prior concurrence. Under the enormous pressure of the continuing Soviet attacks, any sloppy behavior on the roads placed both Hungarian and German troops in danger. There was plenty of that in any case, and Harteneck's orders couldn't deflect it: the Soviets succeeded in driving the Cavalry Corps out of the area surrounding Baranovichi by the next day, 8 July.
Harteneck thereupon reported to Second Army that his Corps was executing a fighting withdrawal under extreme pressure from heavy Soviet forces supported by tanks. He also reported "heavy losses" ( _erhebliche Verluste_ ) but did not further specify. At the same time, General Vattay was reporting to him that his own division was "completely exhausted from eleven days' fighting and marching" ( _durch die elf Tage hindurch andauernden hinhaltenden Kampf und Marsch_ ) and that it was momentarily combat-ineffective. He further requested that the Hungarian cavalrymen be withdrawn from the front. As proof of the stress, he reported that his two principal mounted regiments had been reduced from a combined combat-strength of 2,895 men on 24 June to a mere 404 on 8 July. He did not indicate losses of horses. It remains unclear how Vattay's report was received at Harteneck's headquarters. Nevertheless, Vattay's report may not have been exaggerated. Various communications had earlier indicated that the main weight of the Soviet attacks in the Corps' sector had landed squarely on the Hungarian Cavalry Division, so the losses Vattay reported seem reasonable. Then, too, losses to the Corps' German units appeared to be equally severe. For example, one squadron of the 4th Cavalry Brigade's 41st _Reiter_ Regiment was reduced to an effective strength of only 82 men and no heavy weapons at all. The entirety of the 4th Brigade's other mounted regiment, the 5th _Kavallerie_ Regiment, numbered only 674 men, that is essentially one squadron. Neither unit indicated how many horses, if any, they had at that moment. Further, the 5th Regiment's entire complement of heavy weapons consisted of two 37-mm anti-tank guns, weapons that were virtually useless against Soviet armor of vintage 1944. Absent effective anti-tank guns, only German tanks or assault guns might ward off the Soviets' mechanized forces. Unfortunately, the Cavalry Corps' 4th Panzer Division was equally hard-pressed. Its effective combat-strength on 10 July consisted of only a number of Panzer IVs, four assault-guns, and two Tiger tanks. For his part, General Harteneck nevertheless doggedly continued to order "bitter resistance" by his troopers ( _müssen...verbissenen Widerstand lesiten_ ).
He also attempted to instill an even greater will to resistance in his cavalrymen, infantrymen, and tankers by evoking in them a desire to fight for kith and kin. In a directive dated 11 July and sent to the commanders of the all of the Corps' major combat elements (the 4th Cavalry Brigade, the 4th Panzer Division, and the attached 129th ID that had replaced the by-then-withdrawn 1st Royal Hungarian Cavalry Division) Harteneck emphasized that the Corps had now crossed the borders of the Reich in its westward retreat. This was the border between the former _Reichskommissariat Ostland_ —consisting between 1941 and 1944 of Soviet and Baltic territories overrun during the invasion of the Soviet Union—and occupied Poland. In Berlin's view, and therefore Harte-neck's, the latter constituted territory of the Reich proper. The Cavalry Corps was now fighting on soil that was supposed to be settled by Germans. If officers, noncoms, and enlisted men failed to understand this fact and accordingly do their duty as defenders of Germany, then the full severity of military justice would be felt. "I expect," he went on grimly, "that use will be made of armed force and courts-martial where that is necessary" ( _Ich erwarte, daß von Waffengewalt und Kriegsgericht da Gebrauch gemacht wird, wo es notwendig ist_ ).
Clearly, Harteneck deemed such encouragement necessary. The Soviet offensive that had crashed into Army Group Center and the neighboring Army Group North Ukraine continued to roll forward despite the bitter resistance urged by the Corps' commander. In the fighting withdrawal of the Cavalry Corps and other German units, anti-tank weaponry necessarily assumed ever-greater importance. Therefore, the 4th Panzer Division by its very nature, the anti-tank elements of the Cavalry Brigade, and the 129th ID were continually forced to meet advancing Soviet forces head on throughout the month of July. By contrast, references to the Corps' horsemen as horsemen, not to mention their mounts, are almost entirely absent from daily reports in the Corps' war diary for the period. This makes sense, however, in light of the overall operational situation. The Red Army's summer offensive was grinding its way steadily into occupied Poland with Germany as the ultimate prize. In light of repeated orders to the Corps (and all other German troops) to stand fast and hold positions, its cavalry troopers simply did not have much opportunity to employ their mounted skills or their horses against an enemy that was constantly coming straight at them. Given those facts, and until the hold-at-all-costs orders changed, what counted was straightforward defensive firepower as supplied by the cavalrymen's precious tanks, assault guns, and anti-tank guns. The cavalry troopers themselves fought essentially as dismounted infantry. And ferocious though their resistance evidently often was, the Corps' daily reports speak repeatedly, indeed almost every day, of the "retreat continuing."
In these reports, however, one also hears the old refrain, repeated so many times in so many places since 1941, that the terrain over which the Corps was now moving was unsuitable for either tracked or wheeled vehicles beyond the few paved roads in the Corps' operational area. The 4th Panzer Division's headquarters, for example, reported just that condition on 13 July. It warned further that any vehicles that got stuck would likely fall into the Red Army's hands. Such conditions, of course, were precisely one of the original justifications for retaining the cavalry in the first place. A more bitter pill for the German cavalrymen to swallow was the fact the advancing Soviet forces continued to employ their own cavalry for precisely the sorts of fixing-and-holding attacks that the German horsemen had so often undertaken in their own invasion of the Soviet Union three years before. General Harteneck apparently shared this frustration in that he recommended to his own superiors that a more mobile defense be undertaken. The 129th ID, he reported, was practically exhausted in any case and should be transferred to XXIII Corps, thus freeing the 4th Cavalry Brigade and the 4th Panzer Division to use their horses and vehicles, respectively, for the sort of mobile defense that he advocated. He seems to have felt, and implied, that if vehicles got stuck in the process and had to be abandoned, then so be it. He issued orders for just such a mobile defense to the Cavalry Brigade and 4th Panzer on 14 July, even though the 129th ID still remained part of the Corps. The wisdom of Harteneck's orders was borne out the very next day when the Corps' infantry division, weakened and stationary, was very nearly cut off by advancing Soviet forces. It was extricated only with great difficulty through the efforts of a detached armored reconnaissance battalion from the panzer division.
Mobile or not, the Corps' fighting withdrawal continued, as did that of the entire German army on the Eastern Front. By the end of July, the cavalrymen had reached the line of the River Narew, the very stream that the 1st Cavalry Brigade had triumphantly crossed into Poland in 1939. Crossing points and fords were determined, and on 26 July the Cavalry Corps' headquarters requested permission for a withdrawal across the river. The necessary orders were issued two days later. For their part, the horsemen of the 4th Cavalry Brigade were to prevent at all costs ( _mit allen Mitteln zu verhindern_ ) any Soviet breakthrough to the bridge over the Narew at Lapy, about fifteen miles (24 km) southeast of Bialystok. They also ended up defending another crossing point just upstream (i.e., to the south) of Lapy at Suraż against repeated Soviet attacks supported by tanks; and even though the Soviet troops managed to get across the river, subsequent reports commended the cavalrymen's "smartly executed counterattacks" ( _schneidig geführten Gegenangriffen_ ) and their containment of the enemy's bridgehead on the western bank. Nevertheless, another defensive line was already being evaluated, a so-called blue line running partially along the Nurec River that lay another twenty miles (32 km) farther to the south and west. Positions along this river were incorporated into the Cavalry Corps' defensive line by 31 July.
By that point the Cavalry Corps had been in nearly daily contact with advancing Soviet forces for more than five weeks. The fighting had frequently been severe, and the losses told. On the same day, for example, that the defensive positions along the Nurec were being occupied, General Harteneck reported that the 4th Cavalry Brigade had been "burned out by heavy losses" ( _durch starke Verluste ausgebrannt_ ). These losses hadn't been made good with replacements. Consequently, the brigade now consisted really of only one full-strength regiment and assorted bits and pieces. Only the mounted elements specifically remained combat-effective ( _kampfkräftig_ ), but they were short of men and weapons ( _schwach an Zahl und Waffen_ ). The combat-support elements (pioneers, anti-tank units, etc.) were combat-ineffective at that moment. The 4th Brigade's entire strength that day totaled 1,369 officers and men. In other words, its strength represented only about 20 percent of the 1st Cavalry Brigade's numbers when that unit had ridden into Poland five years earlier. Similar conditions obtained for the 129th ID whose combat-effectiveness Harteneck described simply as "zero" ( _gleich Null_ ).
Unfortunately for the 4th Brigade, the Soviets had no intention of granting the horsemen a breathing space. Renewed armored attacks by an entire corps were launched against the cavalry troopers' positions on 3 August. Emerging from the Soviets' bridgehead at Suraż, the attacks inflicted heavy, though unspecified, losses on the brigade, and it should have been overrun. Nonetheless, the cavalrymen's defenses remained steadfast enough to warrant Harteneck's recommendation to Second Army that they be mentioned in OKW's official dispatches. The cavalrymen killed more than seven hundred Soviet troops and captured nearly eighty. Furthermore, they destroyed (among other things) six Soviet assault guns, seventeen anti-tank guns, thirty-nine machine guns, and eight _panje_ wagons loaded with ammunition. In the process, however, the 4th Brigade had reached a stage at which, in Harteneck's assessment, it was "dissolving itself in the aggressive fulfillment of its mission" and was—he reiterated the phrase—"burning itself out" ( _Die ihre Aufgabe stets angriffsweise lösende 4.Kav.Brig. brennt zunehmend aus_ ). Barely three days later, however, the brigade repeated the accomplishment. This time they fended off an attack by a Soviet infantry division. In this battle, the commander of the 41st _Reiter_ Regiment, Lieutenant Colonel Rojahn, and fifteen men of the regimental staff personally succeeded in bringing the entire Russian attack to a standstill ( _den feind[lichen] Ansturm zum Stehen brachte_ ). Similarly, _Rittmeister_ Count Plettenberg, the commander of the Brigade's Heavy Cavalry Detachment, which was essentially a small assault-gun battalion, distinguished himself. Despite having been wounded earlier, Plettenberg led from the front and helped plug gaps in the German lines, thereby re-establishing contact with the brigade's neighboring units. In this fierce fighting, the 4th Brigade's troopers killed another 750 Soviet soldiers. They also destroyed 9 tanks, 6 artillery pieces, and 36 anti-tank guns. For these actions, Harteneck recommended Rojahn for the Knight's Cross. In a period of four days, the brigade's cavalrymen had inflicted casualties on the enemy totaling more than the entire strength of their own unit.
Of course, the 4th Brigade's own casualties also demanded redress. In light of the larger situation facing German armies on the Eastern Front, however, these replacements were increasingly hard to come by. Two hundred fifty stragglers ( _Versprengte_ ) were ordered to be picked up from the rear-area train depot at Zichenau (Ciechanów), about eighty-five miles (137 km) northeast of Warsaw, and assigned to the 4th Brigade. The 69th Cavalry Replacement Detachment was to provide another fifty NCOs and men. A further forty-nine were assigned to the 3rd Cavalry Brigade that only now resurfaced in the Cavalry Corps' order of battle. Interestingly, the Corps' daily reports for early August 1944 also began to note, at least indirectly and really for the first time since the Corps' establishment, the general situation regarding the Corps' horses. General Harteneck, for example, insisted that at least horse-mounted reconnaissance elements remained essential to the Corps' mobile defensive operations. Consequently, the 69th Cavalry Replacement Detachment received orders on 6 August to continue mounted training. Furthermore, horses were now being shifted around within the Corps as various units were cannibalized for personnel to serve as infantry. The 69th Field Security Battalion, for example, was ordered to turn over one hundred horses and their tack to a sister unit of anti-Soviet Cossacks before moving up to the Corps' forward area for frontline duty. To the extent that the Cavalry Corps' horses may at this point have been restricted primarily to the reconnaissance elements, it risked ending up looking like a mere armor-reinforced infantry corps. Again, however, in the defensive fighting on East Prussia's borders, the Corps' horses were not as essential as they would have been in offensive operations had an offensive been possible.
What Harteneck most emphatically did not want, even as replacements, were more Hungarian troops. On 5 August he wrote to Second Army's commander, General Weiß, that his (Harteneck's) experience with the Hungarian troops was the "worst imaginable." All of the Hungarian hussar regiments were failures, he said. They suffered constant shortages of ammunition because they always threw it away when they fled. And if they were equipped with German weapons and ammunition, they'd simply throw that away, too. While recognizing that the Cavalry Corps had been established precisely to bring together German and Hungarian cavalrymen, Harteneck wrote that he "view[ed] with dismay the day when the Hungarian Cavalry Division would once again be at the side of my brave German horsemen." He maintained that if the Hungarian division were again to fight alongside his men, then the Hungarians would not only have to be directly reinforced with German personnel but also be placed directly under German command. He did not, however, think that the Hungarians would agree to these conditions. "As I see it," he added, "the only correct solution is to send the entire Hungarian Cavalry Division, and any other Hungarian troops, back home."
Even as Harteneck was rejecting the possibility of adding more Hungarian troops, the Cavalry Corps underwent yet another reorganization. As August wore on and the Soviet offensive continued, the 4th Panzer Division was withdrawn and redeployed. In its place came two new infantry divisions. In an attempt to mitigate the loss of the armored division, additional miscellaneous mechanized units were assigned to the Corps. When the next push by the Red Army occurred in the cavalrymen's sector of the front on 22 August 1944, the Corps included the 3rd and 4th Cavalry Brigades; the already sorely tested 129th ID; the newly arrived 14th and 102nd IDs; and assorted assault-gun and panzer units, none at the divisional level, though there was evidently some thought of adding the 6th Panzer Division. Interestingly, at this time the Corps' two mounted brigades were placed on the flanks adjoining the operational areas of the XXIII and LV Army Corps, in other words on the weak seams that had frequently been the targets of earlier Soviet attacks. This seems a clear indication of the cavalrymen's reliability, in this case with their being entrusted the crucial mission of maintaining contact with neighboring formations. Their placement also seems to have been a mark of General Harteneck's appreciation of their fighting qualities.
Such tenacity found itself in heavy demand. Throughout the last ten days of August, the Corps fought a series of defensive battles generally east and south of the Narew River and between that stream and the River Bug against large Soviet formations. The infantrymen and cavalry troopers bore the brunt of the bitter combat in the stretch of river running southwest from Łomža past Ostrolenka (renamed Scharfenwiese by the Germans in 1941), a town lying only about twenty-five miles (40 km) from the prewar border of East Prussia. The hard-pressed troops suffered accordingly, both physically and in their morale ( _seelisch_ ). Their condition deteriorated rapidly under the pounding, particularly from the apparently massive preparatory barrages that the Soviets' artillery could bring to bear without fear of German counter-battery fire. Harte-neck reported to Second Army headquarters that the result from this Soviet preponderance in uncontested artillery fire was "disproportionately large human and material losses" for his units. The additional formations that had just been assigned to the Cavalry Corps were clearly inadequate to the task. Therefore, all of the Corps' staffs, logistics units, horse-handler elements, emergency reserves ( _Alarmeinheiten_ ), and irreplaceable maintenance personnel were combed through for additional manpower. These troops were slotted into the front at every opportunity. Furthermore, all riding horses not absolutely required had been sent to rear-area veterinary companies not only to keep valuable horse-flesh out of harm's way but also so that they wouldn't impede the defense. By 1 September, the fifth anniversary of the start of the war, the Corps' troops had destroyed fully 154 enemy tanks and self-propelled guns and 76 pieces of artillery in these river-line battles, and that day passed relatively quietly. Still, in Harteneck's estimation, the Cavalry Corps would simply not be able to prevent a major Soviet breakthrough much longer.
Despite this continuing threat of a breakthrough, the Cavalry Corps earned highest recognition for its accomplishments between the Rivers Bug and Narew. On 2 September the commander of Second Army, General Weiß, forwarded to Harteneck a statement that he (Weiß) had received from the commander of what remained of Army Group Center, _Generaloberst_ (Colonel-General) Georg-Hans Reinhardt. In the ten days preceding the end of August, Second Army had managed, according to Reinhardt, to hold off at least thirty Soviet infantry divisions; three armored brigades; and numerous independent armored and assault-gun regiments. The steadiness and bravery of the German horsemen and infantry was "above all praise" ( _über jedes Lob erhaben_ ). Combat leadership had been excellent, and these qualities manifested themselves even more strongly in light of the fact that air support from the _Luftwaffe_ had been minimal. The Cavalry Corps was therefore singled out for Reinhardt's "unreserved recognition" ( _uneingeschränkte Anerkennung_ ) in that it had carried the heaviest burden of the defensive battles. Nevertheless, Reinhardt also made clear that further fighting awaited the cavalrymen and their comrades: "Inspired by this spirit," he wrote, "we can await the coming battles with confidence." But such confidence demanded a great deal of faith, for on the same day that Reinhardt issued his statement, the Cavalry Corps reported a total combat-strength of a mere 9,022. Fully one third of these (3,504 altogether) were the officers and men of the Corps' principal maneuver elements, the 3rd and 4th Cavalry Brigades.
Following several days of relative quiet on the Corps' sector of the front, the Soviet advance resumed. In the meantime, the elements of the Corps had been withdrawn to "Defensive Position East Prussia II" hard by the Narew. As the name of the new positions indicated, the cavalrymen were now for all practical purposes defending the prewar German homeland. Furthermore, they were also defending the final redoubt of the Prussian-German cavalry tradition. The emotional significance of both facts was surely lost on no one. Not coincidentally, and in order to stoke the defensive effort, the specter was raised once again of threatening Asiatic hordes, a threat overlaid with the veneer of an all-devouring communist menace. On 12 September General Harteneck issued a directive to all commanders aimed at increasing the Corps' defensive effectiveness. The Cavalry Corps now stood on the Reich's very borders in positions dug by German men and women, German boys and girls. They'd dug with the sure hope that Corps' soldiers would protect them and their homes from the Red Terror. Not one trooper or infantryman would be allowed to withdraw without orders, and only then if live enemy fire forced him from his position. Cowardice would be summarily punished with armed force if necessary. Commanders at every level would remain at their posts to the last possible moment, always leading from the front. Every position to a depth of six miles (10 km) behind the lines was to be dug in ( _einzubunkern_ ) for protection from Soviet artillery fire. No man, no horse, no vehicle was to be without a foxhole or antishrapnel revetments. Furthermore, the Corps' horses, such as those of the 4th Brigade that Harteneck noted specifically, were to be kept hidden in woods and not kept in villages, evidently to protect them from aerial attack and long-range artillery-fire. But even as these preparations continued, so too did combat training for unit leaders and technical specialists such as combat engineers, radiomen, and machine-gunners. Almost incredibly, at the same time Harteneck also ordered that even riding, driving, and horse-feeding training was to continue, with 55 officers and men being ordered to Fordon near Bydgoszcz (Bromberg), 160 miles (257 km) in the rear for that purpose.
No doubt surprisingly, the next several weeks passed with relatively little large-scale fighting, though regular and sometimes intense contact with Soviet troops continued. The Corps' positions along the west bank of the Narew came under regular artillery and light weapons fire from Soviet forces on the other side of the river, the latter's objective being to expand several bridgeheads that they'd managed to achieve on that stream's western bank. On the Germans' side, time was spent improving defensive positions and scraping together replacements from stragglers in the rear. In addition, yet another infantry division, the 292nd ID, was attached to the Corps on 22 September. This gave the Corps a total of three nominal infantry divisions on its TOE and brought its total combat-strength at the end of that month to 14,283 (a figure approximately representing the strength of a normal division of the German army in 1939). Since 29 June, the Corps' units had killed a reported total of 8,942 Soviet troops, almost 40 percent of whom had fallen at the hands of the troopers of the 3rd and 4th Cavalry Brigades.
During the period roughly between 1 October and 31 December, the Red Army entered a time of regrouping and resupply following its crushing summer offensive against German Army Group Center. During the Soviets' operational pause, the Cavalry Corps was now at least able to catch its breath, even if replacements continued to be hard to come by. In that late fall of 1944, the Corps continued to hold its positions along a forty-odd-mile stretch of the Narew between Rozan and Nowogrod. Because of the continuing Soviet pressure and the inevitability of a renewal of the Red Army's drive toward Berlin, it also began another reorganization and redeployment of its various elements. One of the most significant of those reorganizations was the detachment of the 3rd Cavalry Brigade. It was moved to Fourth Army whose center of gravity at that moment lay to the east of the Masurian Lakes, the very lakes where Russian armies had come to grief in 1914. This transfer was noteworthy not only in that it meant that the Red Army's forces had reached, and in many places crossed over, the borders of East Prussia. The transfer also meant that the Cavalry Corps had to temporarily relinquish fully half of that operational component that gave the Corps its very name and character, namely horsemen. With the 3rd Brigade's departure for duty with Fourth Army, the Corps retained the 4th Cavalry Brigade as its only dedicated mounted element. The remainder of the Corps' strength consisted in this period of four infantry divisions. They, of course, still had their horse-drawn logistics and artillery trains, along with various mounted reconnaissance elements. Before year's end, other formations, both infantry and armored, would also be assigned to the Corps for various and usually brief periods of time. These included a scratch force named "Combat Group Hannibal." About 1,400 strong, it comprised personnel from the 4th SS Police Regiment. The entire Corps would also be transferred briefly to the command of Fourth Army, even though its defensive positions remained the same.
Throughout the last half of October and the whole of November, Soviet artillery and aerial attacks continued as did ground combat, the latter sometimes intense and often at battalion strength. Though the widely anticipated, front-wide Soviet offensive did not occur, the Corps' units suffered from this near-constant contact. As they had in the fighting retreat to the Narew, the Cavalry Brigades recorded many of those casualties. Alone in the period from 15 to 27 October, for example, the Corps' two mounted brigades suffered 961 troopers of all ranks killed, wounded, and missing. And these losses occurred in a period when the Corps' morning reports very frequently stated that the preceding day or night had passed quietly. The sacrifices in the mounted elements did earn another formal recognition. Second Army's commander explicitly commended the "outstanding service" ( _hervorragende Bewährung_ ) of the 3rd and 4th Cavalry Brigades and wrote further to General Harteneck that the cavalrymen had delivered the "best proofs of the cavalry spirit" ( _Beweise besten Reitergeistes geliefert haben_ ) in the defensive battles. Presumably the cavalrymen appreciated the sentiment. Such glowing praise, however, wouldn't provide the horsemen with fresh mounts or those mounts with feed (deliveries for the entire Fourth Army area had been stopped), much less gasoline for the Corps' vehicles or ammunition for its artillery. Neither could Hitler's ferocious order of 29 October to the _Ostheer_ , which commanded every German soldier to do one of two things: "stand or die." Perhaps, however, the troopers took greater comfort in the fact that dismal autumnal weather was now frequently grounding Soviet fighter-bombers. They certainly needed the respite. Both of the Corps' mounted brigades had been much reduced. By the end of November, the 3rd Brigade's combat-strength (2,054) was only about 41 percent of its total ration-strength (6,055). The 4th Brigade could show a slightly better percentage, namely about 50 percent (2,181 out of 4,350) even though its overall number of personnel was lower. Similar figures (approximately 50 percent) obtained in the Corps' other units such as the now-attached 558th _Volks-grenadier_ -Division.
Even as the Corps attempted to recoup the losses it had suffered since June, General Harteneck evidently attempted, and in keeping with Army-level directives, to ensure that whatever training could be done was done. In one order, for example, he indicated that certain veterinary personnel and feeding specialists were supposed to attend a three-day school for instruction in winterizing the Corps' horses and care and maintenance schedules. Attendees would subsequently act as training cadre for other soldiers. More significantly, he'd also issued a four-page directive entitled "The Basics of Cavalry Leadership" on 5 November. In it Harteneck re-emphasized most of those doctrinal elements of the cavalry that had last been formalized in 1935 in _Truppenführung_ , the same ones that General Kurt Feldt had stressed in his own repeated statements regarding the earlier 1st Cavalry Brigade/Division. Despite nearly six years of war and all of the vicissitudes that the war had brought to the German army's mounted arm, the cavalry's essential characteristics remained the same in Harteneck's view: mobility, flexibility, audacity, tenacity. True, the last physical vestiges of the cavalry's traditional weapons—sabers—had disappeared in 1940–1941. True, as well, at least since the invasion of the Soviet Union if not earlier, the German cavalryman was now essentially a dragoon. He rode to battle but fought dismounted. And now, in 1944, he just as often fought his battles alongside attached armored and infantry formations. Nevertheless, a stubborn cavalry spirit hung on, and Harteneck hammered it home in his directive, not only as regarded his men but also as regarded the Corps' horses. He also made it clear at the end that he expected his commanders to inculcate a National Socialist bearing in the Cavalry Corps. Whether his statement regarding this matter rested on personal conviction or expediency in the wake of the attempted assassination of Hitler on 20 July 1944 cannot be determined from the directive itself. Furthermore, the degree of subsequent Nazi indoctrination of the Corps' various formations after November 1944 cannot be reliably deduced from the documents at hand. Nevertheless, Harteneck's insistence in this regard sheds a disturbing light on the Cavalry Corps' leadership as it faced the chaotic last six months of the war. And in light of his repeated and fairly ruthless advocacy of summary courts-martial, his urging on of stringent political indoctrination at least seems consistent.
As December arrived, the Cavalry Corps received new orders, ones that took it far away from the Soviet avalanche that was to bury East Prussia beginning in January 1945. Unfortunately, those orders took the Corps to Hungary. There a similar fate awaited the remaining horsemen. Between 18 and 23 December 1944, the Cavalry Corps' command elements and logistics units were loaded onto trains at Lyck in the southeastern corner of East Prussia. From there they traveled south. The 4th Cavalry Brigade went as well. There followed a circuitous route by rail through Posen, Beuthen in Silesia, and western Slovakia. Upon arrival in Hungary—from Lyck an airline distance of some five hundred miles (800 km) and much more by train—the Corps was assigned to the area north of the eastern end of Lake Balaton (Plattensee). Orders arrived on Christmas Day placing the Corps under the command of Sixth Army and, shortly thereafter, Second Panzer Army. Three days later the Corps, in turn, received command of the 1st and 23rd Panzer Divisions. From now on, the I Cavalry Corps was essentially an armored formation containing large horse-mounted and horse-drawn components. Its mission was to help stabilize the then S-shaped front stretching northward from Lake Balaton to the borders of prewar Czechoslovakia, to protect a vital oil refinery at Petfürdö and, possibly, to assist in the relief of Budapest, the Hungarian capital having been encircled by advancing Soviet forces on 24 December. By a curious twist of fate, the 8th _Waffen_ -SS Cavalry Division _Florian Geyer_ found itself trapped in the city with the remaining German garrison. It had ended up there following its anti-partisan campaign in the Balkans in 1944 after being earlier withdrawn from Russia. There was even a second, nominal SS cavalry division, the 22nd Volunteer SS Cavalry Division (sometimes carrying the moniker _Maria Theresa_ ) in the city as well. In this last week of December, the Corps was in constant combat with Soviet forces attempting to solidify their lines while other Red Army units lay siege to Budapest, a siege that would continue until the city's fall in February. Though the combat was sometimes heavy—regimental-strength attacks or better by Soviet forces—the cavalrymen and their sister armored divisions held, even though they had to do so without being able to call on reserves. At that moment, there weren't any. As it turned out, the Cavalry Corps and the other German units in Hungary were aided by the fact that Soviet forces there went over to the operational defensive throughout January, February, and into March 1945.
In the meantime, and at that moment unbeknownst to the Corps, Hitler was planning what turned out to be his last operational offensive on the Eastern Front, code-named Operation Spring Awakening. Its objective would consist of the preservation of the oil fields north and south of Lake Balaton by Sixth SS Panzer Army (transferred from the Western Front) and Second Panzer Army, respectively. More airily, Hitler yet dreamed of the recovery of Budapest and the destruction of Soviet forces in Hungary. While the bulk of the offensive's armored strength was comprised of the Sixth SS Panzer Army, the Cavalry Corps also took part. Officially it remained an element of Second Panzer Army but also seems later to have been temporarily assigned to Sixth SS Panzer Army. At some point between January and March, the 3rd Cavalry Brigade—or elements of it—rejoined the Corps. Furthermore, at least on paper, both it and the 4th Cavalry Brigade were re-formed as cavalry divisions by order of OKH effective 23 February 1945. They did not, however, receive reinforcements to flesh out the redesignation. Consequently, whether there would be enough men, equipment, vehicles, and horses for continued effective operations remained an open question. For instance, losses of horses by early 1945 were high enough to have drawn the attention of Hitler himself. In a conference with the head of Army Administration in the Army High Command, SS _Obergruppenführer_ (General) August Frank, on 29 December 1944, Hitler had been informed that the attrition specifically of horses and vehicles could no longer be sustained. For the cavalrymen on the ground, the matter no doubt seemed clear enough, and though the Soviet forces facing them remained on the operational defensive, the fighting nevertheless continued.
For example, in an earlier effort to relieve the garrison encircled in Budapest, the Cavalry Corps' troops had been heavily engaged. On 7 January, in bitterly cold weather, they'd punched a ten-mile-wide (13 km) hole in the Russian lines southeast of the city and had fought their way forward about the same distance. Nevertheless, and despite several days of intense fighting, neither they nor their counterparts to the northeast of Budapest could force their way into the city itself, though unsubstantiated reports maintained that some of the Cavalry Corps' patrols reached the suburbs. By the end of January, the defenders remaining in the fortress of Buda on the Danube's western bank, like their French counterparts in Paris in 1870, were reduced to eating horsemeat and bread. In the final attempted breakout on 11 February, fewer than 700 members of the garrison reached German lines. Total German losses in the city amounted to some 51,000 killed and 92,000 taken prisoner. The suitability of the cavalrymen for such a relief mission may be questioned, though the heavy armor of several of their formations (even if in depleted strength) now theoretically made their employment for such a task conceivable. Nevertheless, Harteneck and other commanders on the scene weren't the only ones wondering whether mounted formations might still be useful. Once again, Hitler and his most senior commanders in Berlin seriously discussed, on 2 March, the defensive use of cavalry on the eve of the spring offensive that began seven days later. Though the Cavalry Corps had clearly shown in occupied Poland and on the borders of East Prussia what the cavalrymen could do on the defensive, the specific suggestion at Hitler's conference—the employment of pro-German Cossacks—seems to have been more or less dismissed.
Ultimately, Operation Spring Awakening began on 5 March. Initially it made headway but in appalling conditions. Hampered from the outset by cold rain and flurrying snow, mud, a serious lack of fuel, and stiffening Russian resistance, the offensive had stuck fast by the middle of that month. The Soviets then responded by launching their own spring counteroffensive in reply. It would roll forward inexorably, and the Germans would retreat just as inexorably until their final surrender. Indicative of the changing fortunes in Hungary, the OKW on 16 March no longer reported news of German attacks but rather news of a "successful defense" and "counterattacks" along Lake Balaton. In other words, Spring Awakening had been stopped, and the German troops in it had gone over to the defensive. By 19 March OKW was reporting a "bitter defense" by German troops in the region. On 24 March the high command's announcements indicated that "'the Bolshevists' forward attack groups had been brought to a standstill on both sides of Veszprém...after heavy enemy losses." Lying just west of Lake Balaton's northern end and on the edge of the Bakony Forest, the city of Veszprém happened to be defended by none other than the now redesignated 3rd and 4th Cavalry Divisions of the Cavalry Corps. Just as had been the case in East Prussia, the cavalrymen's efforts in defense of Veszprém earned them notice, again at the highest levels. At a conference on 23 March, Hitler was specifically informed that the cavalry divisions, along with the 9th SS Panzer Division _Hohenstaufen_ , had at least temporarily and successfully re-established "security" not only east of Veszprém but also along a nearby railway line. Successful or not, however, the Germans simply could not hold off the weight of the Soviets' forward movement. And as the German armies retreated, they began slowly to disintegrate. On 10 April 1945, the _New York Times'_ war correspondent Hanson W. Baldwin reported that an unspecified number of German troops had effectively been isolated in a pocket around Vienna as the German army "melted away," and by the end of that month OKW would have to admit in a public statement that German troops had been forced to withdraw to the southeastern borders of the Reich. These events were accompanied by reports that the Red Army was "storming the last German defenses in Vienna." The city fell on 13–14 April, and the Soviet High Command reported 200,000 German troops killed or captured. Furthermore, other Soviet units had already begun their march up the valley of the Danube toward Bavaria.
As the Soviets advanced and shifted the brunt of their efforts to the drive on Berlin, the German retreat continued and followed two general routes. The Sixth SS Panzer Army and related forces moved more or less to the northwest. The Second Panzer Army, including the Cavalry Corps, moved generally southwest. This route took the Corps through Lower Austria and into the province of Styria. The cavalrymen, like the rest of the Second Panzer Army, were trapped by Russian armies advancing into the eastern province of Burgenland and the British Eighth Army marching steadily northward toward the Italo-Austrian border. Here the Cavalry Corps found itself when Germany's unconditional surrender was signed on 8 May. On 10 May 1945, the Cavalry Corps also officially surrendered to British forces. Its reported total of 22,000 men and 16,000 horses— _The_ (London) _Times_ spoke of the "immense task" of collecting "vast hordes" of both men and horses from many different units—now went into Allied captivity. The day of the German horseman was done.
EPILOGUE
WHITHER THE HORSES?
The day of the European military horseman appears gone forever. Done is the age when horses played a major, even central, role in Europe's wars and affairs of state. Except for ceremonial units in some European countries, many of which antedate World War II in one form or another—for example, the Queen's Household Cavalry in the United Kingdom, the Irish Defense Forces Equitation School, France's Republican Guard, or the King's Royal Guard in Spain—horses in large numbers have not had a significant military or political role for nearly three-quarters of a century. Never again will a monarch travel in a cavalcade requiring 30,000 horses, as Elector Frederick III of Brandenburg is said to have done in 1701 on his way to being crowned King Frederick I in Prussia. Never again will a ruler have 5,000 horses in one stable, as Louis XIV is reputed to have had at Versailles. And certainly never again will a fortress such as the Alcázar of Toledo be built with subterranean stables to house 2,000 cavalry mounts. One might safely paraphrase Shakespeare and say that the horse has done his duty; the horse may go.
One also has to admit, however, that if the history of military equitation begins at a point five thousand or six thousand years ago some-where on the Eurasian steppe, then ending that history's European progress in 1945 is not a bad run. As has been shown, the military horse was written off many times before he actually left the stage. The idea that the horse no longer had a useful place on the modern battlefield arose many times. In the 1890s, in 1918, and in the 1930s, critics dismissed the horse as a romantic obsolescence at best and, at worst, an actual hindrance to military effectiveness. Time and again horses nonetheless proved themselves useful, sometimes exceedingly so. In the case of the German army they were crucial to Prussia's victory over France in 1870, a victory sealing the establishment of a German Empire. In World War I, horses served in every theater where German and other troops took the field. Not least, between 1939 and 1945 they ended up being literally irreplaceable, and they materially helped secure many of the German army's victories before the end of 1942. This remained so throughout the war in the cavalry arm, the logistics trains, and more often than not for the artillery. Whether in Germany or elsewhere, technological promises regarding motorization and mechanization repeatedly showed themselves to be premature. Until vehicles of whatever sort became sufficiently numerous and sufficiently reliable the horse would stay. It made no real difference whether vehicles' deficiencies lay in faulty design, political leaders' ineptitude, military commanders' operational mistakes, combat losses, or domestic industrial bottlenecks caused by strategic aerial bombardment. Whatever the reason, as long as machines could not do the job, horses would have to. In Germany this meant not only that horses would continue to pull supply wagons, field kitchens, ambulances, and guns after 1939 but also that horse-mounted cavalry would soldier on. In confronting this reality, Germany was by no means alone. Poland, France, Rumania, Hungary, Italy, and, most dramatically, Russia all employed horse-cavalry, though not always with the same results.
The military horse thus survived Europe's rapid nineteenth-century industrialization and went on to serve in huge numbers throughout the period from 1900 to 1945. Though their specific cavalry service largely disappeared on the Western Front after the First Battle of the Marne, horses (and mules) nevertheless served valiantly there and in every theater of World War I, particularly in East Prussia, Poland, the Baltic States, and Rumania. They carried on in the interwar period, and they served faithfully once again when a second war came in 1939. Between that year and 1945, approximately seven million horses saw active duty in the armed forces of European countries directly involved in World War II. In the German armed forces alone, the number reached at least 2,800,000 during the same period. Of course, the vast majority was assigned to the army, but the _Luftwaffe's_ field units and even the _Kriegsmarine_ used minimal numbers of horses as well. But just as horses served in unprecedented numbers, so they died. The General Staff of the army reported that for the period between the beginning of the invasion of the Soviet Union on 22 June 1941 and 31 December 1944, the army's and _Luftwaffe's_ average monthly loss of horses to all causes was approximately 30,000, 90 percent of which were horses from the army's units on the Eastern Front, the _Ostheer_. According to the same report, by the latter date the armed forces' total losses in horseflesh amounted to 1,558,508 animals. As demonstrated throughout this history, and particularly in the two world wars, losses could mount with frightening rapidity and in the most appalling fashion. Like their human counterparts, those horses not dying from exposure, illness, or malnutrition were killed and maimed by the bombs and bullets of machines whose triumph no horse-mounted or horse-drawn army could prevent.
In his _Victory Report_ of 1 September 1945, General George C. Marshall, then chief of staff of the U.S. Army, alluded to this sort of technological supremacy when he wrote that "the greatest advantage in equipment the United States has enjoyed on the ground...has been in our multiple-drive motor equipment, principally the jeep and the 2½-ton truck." In the U.S. Army, as well as in other armies (though not always to the same extraordinary degree), these mundane vehicles had given American forces unprecedented mobility. Especially once ashore in France after D-day, the mobility of these vehicles and the troops riding in them became "strikingly clear." The German army, continued Marshall, was "completely outclassed." He went on to observe that "the Germans discovered too late the error of the doctrine a member of their general staff expressed to General [Albert C.] Wedemeyer, then in Berlin, in the late thirties: "The truck has no place on the battlefield." This was, of course, the very same attitude expressed so sharply to a young Heinz Guderian when he was told emphatically that trucks were supposed to carry flour, not troops. It was, however, these same very ordinary jeeps and trucks, possessed of an extraordinary crosscountry capability, that spelled the real doom of the military horse in Europe.
The American automotive industry not only made the horse redundant for all practical purposes in the U.S. Army, it also helped complete the motorization and mechanization of the British ground forces and did a great deal to put Stalin's legions on wheels and tracks. Fully 76,737 jeeps and 98,207 trucks went to the British army and 28,356 jeeps and 218,888 trucks to the Red Army. Still more remarkable was the fact that, in addition to these deliveries and the outfitting of U.S. forces literally around the globe, "almost all of the equipment used by the revitalized French Army, which had 12 fully equipped divisions in action at the time of Germany's surrender, came from the United States." As if such American production alone were insufficient, Great Britain also produced nearly a million wheeled vehicles during the war, not to mention 109,500 armored vehicles. In comparison to these prodigious Allied figures, Germany produced approximately 800,000 military trucks and automobiles of all types and perhaps 68,000 armored vehicles. Though by no means paltry figures, these numbers represent a German industrial base that simply could not keep pace, particularly when wartime attrition, normal wear-and-tear, resistance in the occupied countries, and the Allies' strategic bombing campaign are taken into account. Consequently, even if sentimental attachment contributed to the German army's retention of the cavalry in 1939—and the evidence cited makes clear that this was not a dominant factor—keeping the horse in service after that date in both the combat and noncombat arms became ever more a matter of sheer necessity.
As a semblance of peace returned to the charnel house that was Europe in 1945, no serious thought appears to have been given to retaining the military horse beyond its ceremonial function, if that. Nevertheless, horses lingered on in an official military capacity after V-E Day and would remain on active duty beyond the parade ground in several European armies even at the beginning of the twenty-first century. Interestingly, the U.S. Army itself initially raised a small mounted force in the American Zone of Occupied Germany in the immediate postwar years. As a part of what was called the U.S. Constabulary, this force consisted of both horse-mounted and vehicle-mounted patrols for internal security missions. Furthermore, and more directly germane to the present subject, almost immediately after the Federal Republic of Germany established the Federal Armed Forces ( _Bundeswehr_ ) in 1955, an equine pack-animal unit was raised for the _Heer's_ alpine troops at their training center at Mittenwald in the Bavarian Alps. Another such unit followed in 1960 at Bad Reichenhall near Berchtesgaden. Since 1981, Bad Reichenhall has been the only location where pack animals are based. In 2008 Mountain Pack Animal Company 230 ( _Gebirgstragtierkompanie 230_ ) constituted the sole remaining unit in the _Heer_ with horses and mules on its regular TOE. Training in the operational use and keeping of the animals, both mules and Haflinger horses, is the task of the Pack Animal Mission and Training Center (PAMTC), co-located at Bad Reichenhall and established in 1993. According to the _Bundeswehr_ , all plans for the elimination of the animals from military service have long since been shelved for the same reasons that cavalry originally survived the advent of the first motorization of German armies at the beginning of the twentieth century: horses and mules can frequently go where vehicles can't roll and helicopters can't land, and they're cheaper than either. By the _Bundeswehr's_ current reckoning, each pack horse or mule can do the work of four soldiers. That fact only adds to equids' advantages as force multipliers.
In 2009 the PAMTC included fifty-four mules and Haflinger horses. In a notable development, mounted training has also recently been reinstated for that unit's personnel. The mounted training's purpose is to provide better coordination with the reconnaissance elements of the Army's 23rd Mountain Rifle Brigade ( _Gebirgsjägerbrigade_ 23). Furthermore, and curiously reminiscent of the practice established during World War II, especially on the Eastern Front, German troops of the PAMTC purchased Bosnian "ponies" during their first foreign deployment involving the actual operational use of the animals. This deployment occurred in Kosovo between 2002 and 2004. Though the German troops did not take their own horses, the stock they purchased evidently rendered outstanding service in Kosovo supplying outposts over terrain and in weather conditions that made use of vehicles impossible. This same expertise, though once again not the PAMTC's own animals themselves, subsequently found its way to northern Afghanistan as part of the _Bundeswehr's_ contribution to NATO's International Security Assistance Force (ISAF) operating in that troubled country.
Other European armed forces also recognized the horse's continuing potential utility in the period after 1945. When the Austrian Federal Army ( _Bundesheer_ ) was established in 1958, for example, horses were called upon to outfit three pack-animal companies whose initial missions were planned to be entirely combat service support. As with their German counterparts, these missions included keeping alpine troops equipped with weapons, ammunition, and other supplies. Even though the three companies were eventually reduced in the 1970s to four platoons based at Hochfilzen, Landeck, Lienz, and Spittal/Drau, horses were retained for these tasks. Furthermore, a remount station was established in 1983 to keep the units supplied with trained stock. In 2005 these numbered 47 remounts and foals in training out of a total of 116 horses in the _Bundesheer_ overall. As in the Federal Republic of Germany, the Austrian horses and their handlers also regularly received mounted training. They put this training to use for the first time between fall 1994 and summer 1999 when they executed mounted patrols along Austria's eastern borders, a mission they replicated in 2004 and 2005. Both the German and Austrian armies' employment of horses in such missions, whether in Kosovo or along Austria's own frontiers, were in keeping with what are called the "Petersberg Missions": tasks undertaken by the member States of the Western European Union, a defense association of a number of European countries, to use their armed forces for crisis management, peacekeeping, and humanitarian assistance in loose conjunction with, or even wholly outside of, NATO.
While it may in retrospect seem absurd for any army to have retained horses in the face of motorization and mechanization in the early twentieth century, technology could not always and everywhere fulfill its promise. Such would be the case even at the beginning of the twenty-first century. Interestingly enough, modern technology's limitations in at least one instance were put precisely in equine terms: the aerial Thermal Imaging Airborne Laser Designator (TIALD) pod on the Royal Air Force's Tornado GR4 attack aircraft used in Iraq in 2006–2007 found itself up against certain interesting difficulties in the counterinsurgency campaign there. The TIALD pod, said one RAF officer, could easily designate big, static targets like bunkers, but it couldn't be used "to try to spot a man on a horse with a gun." The point is merely, but importantly, that high-tech systems cannot always surmount challenges presented by decidedly low-tech alternatives, whether in 2007 or circa 1940. And of course everyone is also now familiar with the image of U.S. Special Forces riding into combat against the Taliban in Afghanistan in 2001 alongside as many as two thousand Northern Alliance horsemen. Perhaps not so familiar is the U.S. Army's related horseback training program for Special Forces troops at Ft. Carson, Colorado.
Beyond such technical considerations, any discussion of the German cavalry between 1870 and 1945 must also bear in mind that attitude called the "cavalry spirit," an attitude stressing a commander's initiative, independence, speed, flexibility, and audacity. One noted authority, Robert M. Citino, convincingly maintains that by 1939 that very spirit had been an integral element of a "German way of war" for very nearly three hundred years. In the latter year, it still survived in the concept of _Auftragstaktik_ , the ability of commanders to employ their forces essentially on their own without interference from higher command as long as they accomplished the mission. While implicit at almost every level of command, _Auftragstaktik_ assumed particular importance at corps-level echelons and particularly in the armored forces.
Just as technology spelled the eventual doom of the horse-cavalry, Hitler's stultifying dictatorship did the same for a concept owing so much to the cavalry's traditions. Shortly after becoming chancellor, Hitler issued a "fundamental order" requiring all higher commands to submit a regular blizzard of reports on troop dispositions, supplies, movements, combat actions, status reports, and so on. These higher commands in turn had to request the same of their immediate subordinates, they of theirs, and so on down the line. In this effort, according to Citino, Hitler was aided and abetted by the army's then-chief of staff, General Franz Halder. Such paper-based micromanagement was poison to _Auftragstaktik_. Furthermore, _Auftragstaktik_ by its nature was incompatible with the totalitarian character of Hitler's very much personalized regime, and his "stand fast" order in the Russian winter of 1941 inflicted a mortal wound to what was left of the tradition. His subsequent direct assumption of operational command was the coup de grace. "The dash, the impetuousness, the ability to roam free, away from higher control—all these belonged to a bygone era." The reference here is to an operational concept, but the statement could just as well serve as the German cavalry's epitaph.
In the main, the German army's cavalry in the modern age, including during World War II, fought not only valiantly but honorably, though the latter simply cannot be said of the cavalry of the _Waffen_ -SS. Between 1939 and 1945, the horsemen also fought about as effectively as one could expect in a conflict that came to be dominated ever-more completely by titanic masses of machines. As with so many other formations of the German army, however, the cavalrymen's legacy was fouled by the evil nature and conduct of the régime they served in those years. The horsemen endured much. They and their mounts suffered greatly. But they also willingly inflicted much suffering in Hitler's name. The Nazis' cause condemned them. What was said of another cause eighty years before and an ocean away thus might also apply to that of the German horsemen of World War II: it was one of the worst for which a people ever fought.
Once, some years ago, the author was asked in an interview whether he could name a horse that he regarded as the most famous horse in history. He replied by asking in turn whether he might choose an anonymous one. If so, then he felt compelled to say that the most famous horse in history would have to be the horse of the mounted warrior, whether Scythian, Sarmatian, Hun, Magyar, Mongol, Turk, European knight, U.S. Dragoon, Native American horseman, samurai, or modern cavalryman. This horse carried the fate of empires on his back. Sometimes he carried the fate of civilization itself. One would like to think that he did so largely without complaint. Without doubt, he did so while all too often improperly fed, watered, or cared for. He risked his own life and paid it in full more frequently than did his rider. This horse should be remembered.
Under the western horizon, far below where radiant Venus
Arises, meander pastures that nurture the stallions of lustrous
Sol, not by letting them graze themselves, feeding on grass throughout the night,
But with Ambrosia, restoring strength to legs, setting them aright...
Ovid, _Metamorphoses_ IV.260–263 ( _translation by Cheri S. Dorondo_ )
APPENDIX A
COPY1
Army High Command 4
(Fourth) Army Headquarters 18.9.1941
Abt. Ia/IVc2
ARMY ORDER
Re: Sparing of Horses ( _Pferdematerial_ ).
During the advance to date, it has been observed that completely worn out ( _heruntergekommen_ ) horses and horses reduced to walking skeletons by weight-loss have been delivered to the Army's field veterinary hospitals. That fact, in addition to the marked difference between the numbers of horses dropping out on the march and the total number of horses available in the various units, indicates that understanding of proper horse-management is largely absent.
In view of the increasing demands placed upon the Army's horses, the following is ordered for the sparing and maintaining of same:
1.) _March and driving discipline_ , especially uniform draft-horse usage, is to be constantly observed by all commanders and subordinates.
2.) _Uniform march-tempo at all gaits_. The length of distance marched does not tire horses as much as irregular and unreasonable tempo. All unnecessary trotting is forbidden. Coldbloods [i.e., heavy draft horses] are to be trotted only in exceptional circumstances.
3.) _Sufficient rest_. Whenever possible, the march should initially be paused after one-quarter hour to check the fit of saddles, tack, and harness. Thereafter, a 5–10-minute rest every hour. After the first half of the march, a rest of 1–2 hours. Remove saddles and tack during these rests whenever possible.
4.) To _lighten draft loads_ , it is ordered: a) frequent dismounting of riders and drivers; b) only drivers to sit in the driver's seat ( _Bock);_ other persons permitted only under special _circumstances_ , for a limited time, and by express order of the unit's commander; c) artillerymen and machine-gunners to dismount on uphill inclines and on difficult surfaces (e.g., sand); d) riders and drivers driving from horseback to dismount at every halt, even when the column is simply delayed ( _Marschstockungen_ ).
5.) On steep inclines, brief and preplanned halts. Chock wheels at every halt. Wheel-chocks to be carried.
6.) Use every rest to check _saddles, tack, and shoes_. Loosen girths.
7.) As the march begins, _feed supplies are to be arranged in advance by designated personnel (Vorkommandos_ ) so that horses can be fed sufficient amounts immediately upon arrival or when at a rest-halt. Extended distances from feed supplies or difficulties in acquiring same will not be allowed to result in horses' receiving reduced rations. If necessary, motor vehicles or local transport will be employed [to transport feed]. _Scythes and sickles_ will be carried in vehicles. For the march, hay or green feed will be carried in feedbags. _Horses are to graze at every opportunity_. Clover and alfalfa can be substituted occasionally for hay and oats; in emergencies they can replace them. _Oats on the stalk_ are not dangerous. At every feeding with green feed, water horses beforehand or at most an hour later. In general, _waterhorses_ as often as time and conditions permit. The warmer the weather, the more frequent the watering. As the march begins, ensure that sufficient numbers of _feedbags, mangers_ , and buckets with feed are carried so that they can be employed immediately [at the halt] and not only after large numbers are collected. Allow sufficient time for horses to feed, especially the heavy breeds ( _Schlage_ ).
8.) _No use of horses to extreme limits (bis zum Äussersten_ ). Horses showing signs of exhaustion are to be removed from service in a timely manner. Longer rest periods and timely replacement with fresh horses ( _Vorspann_ ) are to be provided, especially when only two horses are hitched to heavy vehicles.
9.) _No overloading of vehicles_. [There must be] ruthless reduction of loads.
10.) _Distribution of personnel for assisting with haulage (Zieh- und Schiebekommandos_ ). Drag-ropes are to be carried and personnel employed in a timely manner to assist draft horses with haulage. The number of horses dropping out on the march, the number of horses lost, horses' capacity for work ( _Lesitungszustand_ ), and thereby the mobility of the forces depend upon the personal intervention of line officers, as well as veterinary officers and noncoms.
I will observe the execution of the foregoing order throughout the Army's area of operations.
_Signed_ ,
[Günter] von Kluge
_Field Marshal_
* * *
1. NARA Microfilm Publication T-315, Roll 78, Frames 53–54. Translation by the author. A marginal note by the Cavalry Division's commander ordered the transmission of this directive to all squadrons, though cavalrymen presumably felt no need to be instructed in the matter of horsemanship, much less riding and driving.
2. Operations and Veterinary Staffs.
APPENDIX B
Results of SS Cavalry Division's Operations Weichsel I and II
May-June 19431
[Matériel and persons captured, confiscated, and/or destroyed]
Civilians "evacuated": 10,422.
Villages destroyed: 61.
Bandits and suspected bandits "dealt with" ( _erledigt_ , i.e., killed): 1,114.
Bandits and suspected bandits captured: 16.
Forest camps: 12.
Bunkers: 46.
_Panje_ vehicles taken: 162.
Mines discovered: 21.
Explosives: 50 kg + 1 satchel charge [?] ( _Beutel_ ).
Saddles: 5.
Enemy operation center(s): 1 + various medical facilities.
Axes: 15.
Spades: 20 + various hand tools.
Machine-gun ammunition drums: 3 with rounds + large amounts of other ammunition and rifles.
[Livestock taken]
Cattle: 5, 676.
Calves: 1,073.
Sheep: 3,153.
Horses: 1,223.2
Foals: 40.
Swine and piglets: 1,398.
Chickens: 1,588.
Geese and ducks: 633.
Large quantities of rye and flax also seized.
* * *
1. _Zusammenstellung Über_ [ _sic_ ] _Beuteergebnis im Zuge der Unternehmen "Weichsel" im "Nassen Dreieck."_ NARA Microfilm Publication T-354, Roll 642, Frame 1190. Translation by the author.
2. In the general withdrawal of German forces to the Dnepr following the Battle of Kursk in July 1943, more than 150,000 horses, 200,000 head of cattle, and 270,000 sheep were driven ahead of the retreating armies. See Earl F. Ziemke, _The Soviet Juggernaut_ (New York: Time-Life Books, 1980), 74.
APPENDIX C
Order of the Day to the SS Cavalry Division
Brigadier-General ( _Oberführer_ ) Hermann Fegelein
14 May 19431
Men of the SS Cavalry Division!
Men of the _Waffen_ -SS!
By order of the Führer I assumed command of the SS Cavalry Division on 20 April 1943.
The tradition of this division is grounded in the very beginnings of the Guard Echelons ( _Schutzstaffeln;_ SS) of the N.S.D.A.P. The cornerstone was laid in Munich in 1929 by order of the _Reichsführer_ -SS with the establishment of the 3rd Company, 1st Battalion of the 1st Regiment. It was by way of the _Reiter_ companies that the _Reiter_ regiments and _Reiter_ districts ( _Abschnitte_ ) of the General SS were developed following the assumption of power in 1933. With the beginning of this great conflict in September 1939, the first 400 cavalrymen of the reinforced SS-Death's Head Units advanced into Poland and subsequently formed the basis of the 1st SS _Reiter_ Regiment. Disbursed in thirteen garrisons across the General Government,2 this regiment supplied cadre for the formation of the 2nd SS _Reiter_ Regiment and thereby for the SS Cavalry Brigade.
The brigade's march of more than 6,000 kilometers, its renowned engagements and battles, and the crowning achievement of the closing of the gap near Rzhev in the winter battles of 1941/42 are a unique witness to the heroism and bravery of all of the blood-related Reich- and ethnic-German units of the Greater German Reich.3
On the basis of trials in the bitterest offensive and defensive battles, in winter as well as summer, the Führer approved the establishment of the SS Cavalry Division. Once again, in the second winter battles of 1942/43, near Smolensk as well as Orel, the division added imperishable fame to its flags. All of that was accomplished through the efforts of the troops whose bravery, determination, and surpassing heroism overcame even the direst crises. A skilled leadership and the instinct for success in war allowed for an avoidance of severe losses.
Even though we were only established during the course of the war, we proceed from the principle that we are the equal of the other regiments and divisions of the _Waffen_ -SS. It is good when every unit desires to be the best. It is good when regiments strive with one another in their accomplishments. No one need maintain that we suffer from presumption when we say that we-constituted as we are from all of the pedigrees ( _Stämme_ ) of the Reich—desire to be the best men of the whole of Greater Germany. If only others would say the same! Therefore we hold fast to our company ( _Verband_ ) and because of this pride we lay the greatest worth on a doctrine ( _Erziehung_ ) that appropriately meets our objectives. Therefore we know nothing of conceit and presumption, because presumption is merely a disputing over one's lack of worth. The great man is always a natural man; he doesn't forget whence he comes.
Our love is Germany's treasure, our loyalty its protection and security. Manly virtues, with loyalty first of all, must determine the course of how we lead our lives.
The tasks the division now faces have been ordered by the Führer because they help protect and strengthen the entire front in the East. It is an honor for us to prove through superior leadership and hard-bitten effort in marching and fighting that we can exterminate the dangerous enemy who opposes us. Our will to exterminate the enemy must be inexorable, without mercy or compassion. Every weakness means a prolonging of the war.
With exactly the same drive and determination with which I commanded the first _Reiter_ Company of the Guard Echelon, I now assume command of the only cavalry unit of the Greater German Army.4 To you I want to be a good superior and comrade.
We have the holy obligation to continue a tradition of the cavalry stretching back thousands of years, one expressed in the words of the Führer:
Let us more strongly close the ring of our great community in the trust of our _Volk_ , filled with the belief in our mission, ready for any sacrifice that the Almighty might demand of us. Then will Germany—the National Socialist Third Reich—pass through the time of distress and trouble, armed with that metal that alone can preserve the knight, unsullied and fearless, in the battle with death and the devil.
Just as we have withstood our many battles and just as so many of us have calmly looked death in the eye, so too do we love the struggle and thereby life.
Long live the Führer!5
Fegelein
* * *
1. Translation by the author.
2. The General Government was the rump Polish State administered by, but not annexed to, Germany.
3. _"...aller reichs- und volksdetuschen Stammes-Einheiten des Grossdeutschen Reiches."_
4. The Army's 1st Cavalry Division had been officially disbanded in 1941. Ad hoc Army cavalry units, later somewhat more formalized, were nevertheless starting to reappear by the late spring and early summer of 1943.
5. Fegelein did not use the more typical " _Sieg heil_!"
APPENDIX D
The Basics of Cavalry Leadership1
5 November 1944
1. _Be quick! Speed saves blood!_ Speed frequently substitutes for strength! Make decisions quickly. Hesitation and waiting is always bad. Even the best reconnaissance can't provide a completely clear situation in war. Your own action provides the most rapid clarification of the situation.
_Give orders quickly_. Short, clear, thorough orders to saddle up have always been the tools of a good commander of mounted troops. The quicker you give orders, the quicker they'll be executed, and that execution occurs most rapidly in the cavalry. Whoever wants to give orders quickly, however, has to "speak the same language" as his subordinates and be well acquainted with them. Therefore, practice ( _übt_ ) giving orders.
_Transmit orders quickly_. If an old woman ( _Botefrau_ ) can get to the objective first, I don't need a mounted courier. If the radio message takes too long, then a rider or motorcyclist can transmit the information quicker than over the airwaves. Therefore keep your command and control route in mind. Organize it. Accelerate it. Always choose the fastest route, and keep several options open for simultaneous use in difficult conditions.
_Execute orders quickly_. If a _Volkssturm_ battalion can execute orders just as rapidly, then the cavalry has no reason to exist.2 An average march-tempo of five miles per hour (8 km/h) must be maintained in the mounted regiments. Rest periods and enforced halts due to aerial attack or artillery-fire have to be taken into consideration. Speed in assembly hinders, or at least shortens, enemy countermeasures. From the assembly point the mass of troops—not merely reconnaissance and attack elements—moves simultaneously; otherwise the troops are incorrectly assembled. Speed in the attack exposes troops to enemy fire for a shorter time and helps avoid losses.
Speed of execution is limited by the unavoidable fact that all arms must operate together in order to achieve success. But the cavalry commander must be able to organize this coordination with complete thoroughness. That's why his heavy weapons are mobile and move quickly. However, they also require the first orders since they take the longest to prepare to fire.
2. _Be complete (Seiganz_ ). Only complete men ( _Männer_ ) make it through. Only complete men ( _Kerle_ ) are equal to every situation. Only complete leaders make suitable mounted commanders.
Make complete decisions. Whoever tries to cover everything covers nothing. Whoever tries to do everything accomplishes nothing. Collect the strengths you have. Determine a center of gravity ( _Schwerpunkt_ ). Determine it in space and in time. Determine it by combining the fires of all weapons and collecting your entire offensive capability ( _Stosskraft_ ).3 The former is more important than the latter [in light of the defensive battles]. Don't be clumsy ( _klotzen_ ) in approach marches, assembly, or dealing with personnel. That costs blood. Make the enemy your objective ( _Bilde Deinen Schwerpunkt im Feinde_ ). The less you have, the more important the center of gravity becomes. Don't avoid the risk of making yourself weak where you have to in order to build a necessary center of gravity elsewhere. Accomplish one task after another rather than trying to do many things at once. You can do it because on the battlefield the cavalry is always quick and nimble.
Don't divide your heavy weapons. Instead, collect them in a "block" to maximize their effect ( _klotze damit_ ). Since you control decision making, you can always move your center of gravity to the enemy's weak spot.
_Be completely clear in your orders_. Don't push your own responsibilities onto your subordinates.
[...]
3. _Be flexible_. Your speed and your mobility give you the ability to do this. Use them! A cavalryman who can't should join another branch of the service.
[...]
_Only troops who reconnoiter and report properly make flexible command possible_. In the cavalry "reconnaissance" is written in capital letters.
[...]
4. _Stay mobile_. The strength of the cavalry is in its mobility over difficult terrain and on the battlefield.
_Keep yourself mobile_ and lead from the front, so far forward that you can quickly and securely make use of your knowledge of the enemy and the terrain. Only then can you lead correctly and flexibly. If you don't do this, you'll be overtaken by events.
_Keep your troops mobile_. Rapid alteration between fire and maneuver is the essence of cavalry combat. Therefore, keep the led horses4 only as far away as the enemy's fire dictates. Even when you're displacing by truck-transport, bring all horses—or as many as possible—forward so as to keep the reserves and reconnaissance elements mobile.
_Stay mobile on the defensive, too_. If you have to send your led horses [further] to the rear, at least keep your reserves mobile. The reserves' led horses have to be kept as far forward as possible so that the reserves remain mobile even if the dismounted troops are dug in. If led horses can't be kept standing close by, then arrange truck-transport to take troopers to their horses so that they [horses and riders] can be made mobile as quickly as possible. Led horses should also march at five miles per hour. If they lag, drive vehicles through their columns if necessary ( _Fahre dazwischen, wenn sie bummeln_ ).
5. _Take care_ [of your men]. If you don't think ahead about caring for the men, your troopers will quickly lose their edge.
[...]
Ensure that your men have rest whenever there's an opportunity. In addition to combat and marching, a cavalryman can easily be overloaded by the further, required horse-care. A basic rule is that in combat and on the march, every trooper up to and including the platoon leader sees to his own horse. This applies especially to the led horses in the rear.
[...]
_Take care of your horses_. They must be watered and fed _in a timely manner_. Be rigorous about this. Never let them stand idle with tight girths. That causes compression in the gut. Saddles that are kept on but not tightly cinched don't cause problems, and they lend themselves to rapid use. Keep horses dispersed in assembly areas and provide dirt revetments or stone/block stalls for shrapnel protection. Otherwise you'll suffer avoidable losses. A horse can live without oats but not without fodder. It takes time for him to eat the amount he needs, so give him something at every rest-halt. Give him straw if nothing else is available. When green fodder is growing, scythes belong in every squadron, battery, and column so that fodder can be provided at every rest-halt.
6. _If you're attached to other units, then:_
_Ensure that you carry the day (sich durchsetzen_ ). Not just anyone can lead and employ cavalry. You're responsible for appropriate and effective operations. Let yourself take orders, but how you execute them is your concern.
_Ensure that you don't let your forces be divided absent an emergency_.
[...]
_Ensure that you lead_. You have learned how to command cavalry. When combat groups ( _Kampfgruppen_ )5 are formed, strive to be appointed to overall command or at least make sure that your cavalryman's influence isn't ignored by the commander.
7. _Train_. Cavalry training is the most difficult of all arms. Training saves blood.
[...]
Train for combat. The horse is your means of getting to the fight. Your troopers and horses don't need to know anything more than this fact demands. You fight on foot. The bulk of your training time belongs here.
8. [...]
9. Don't just command and train your unit, educate it. Without a clear and tireless National Socialist leadership, you won't have your unit well in hand. Political instruction by the CO ( _der Chef_ ) comes before every other instruction.
_Signed_ ,
Harteneck
* * *
1. _Grundsätzliches über Kavallerieführung_. NARA Microfilm Publication T-314, Roll 27, Frames 499–503. 5 November 1944. Translation by the author. The German familiar form of "you" was used throughout the original. Ellipses indicate omissions of generic statements not specific to the cavalry's ethos, training, or organization
2. The _Volkssturm_ constituted largely useless, scratch formations of over-age civilians raised by the Nazi authorities in a desperate effort to fend off the Soviet invasion of Germany.
3. Nearly illegible in the original, but _Stosskraft_ perfectly fits the context.
4. _Handpferde_ were horses held slightly to the rear by a cavalryman (traditionally every fourth one) while his dismounted comrades were in action. _Handpferd/e_ can also apply to the leading horse(s) in a team.
5. _Kampfgruppen_ were composite formations of varying strength and usually operational for temporary periods.
NOTES
Chapter 1: The Day of the Horseman
1. Norman Davies, _Europe: A History_ (Oxford: Oxford University Press, 1996), 1038. For a general overview of mounted warriors in antiquity, see John Keegan in the video series "War and Civilization."
2. Benno Hubensteiner, _Bayerische Geschichte_ (Munich: Süddeutscher Verlag, 1985), 46.
3. _The Revelation to John_ , VI: 1–4, _The New American Bible_ (Wichita, KS: Catholic Bible Publishers, 1994–1995), 1234; G. Ronald Murphy, S.J., trans., _The Heliand: The Saxon Gospel_ (Oxford: Oxford University Press, 1992), xiv. For the role of cavalry in the age of classical Greece, see Victor Davis Hanson, _The Wars of the Ancient Greeks and Their Invention of Western Military Culture_ (London: Cassell, 2000). On Roman cavalry see Lawrence Keppie, _The Making of the Roman Army From Republic to Empire_ (Norman: University of Oklahoma Press, 1998).
4. Brian M. Fagan, _People of the Earth: An Introduction to World Prehistory_ , 9th ed. (New York: Longman, 1998), 469.
5. John Keegan, _A History of Warfare_ (New York: Alfred A. Knopf, 1993), 181–182.
6. Andrew Jacobs, "Police Turn to the Stable for Crime-Fighting Clout," _New York Times_ , 18 April 2006, 1. See also Steven D. Price, "On Patrol With the NYPD," _Equus_ , 398 (November 2010): 45–48.
7. Ann Hyland, _The Medieval Warhorse From Byzantium to the Crusades_ (Stroud: Sutton Publishing, 1996), xi-xii.
8. Keegan, _A History of Warfare_ , 213, 387.
9. Hyland, _The Medieval Warhorse_ , 11–12. Not all historians agree that Western European cavalry truly acted as a shock force. See Carroll Gillmor, "Cavalry, Ancient and Modern," in Robert Cowley and Geoffrey Parker, eds., _The Reader's Companion to Military History_ (Boston: Houghton Mifflin, 1996), 74–75. On the horse-people's style of warfare see the eloquent description by Marco Polo in his _The Travels of Marco Polo_ , Bk. 1, Ch. 49.
10. Hyland, _The Medieval Warhorse_ , 57–58.
11. Gillmor, "Cavalry, Ancient and Modern," 74–75.
12. Robert L. O'Connell, _Soul of the Sword: An Illustrated History of Weaponry and Warfare from Prehistory to the Present_ (New York: The Free Press, 2002), 123.
13. Andrea Brady, "Dying With Honor: Literary Propaganda and the Second English Civil War," _The Journal of Military History_ , 70, no. 1 (January 2006): 14–15, 18.
14. Ben Cassidy, "Machiavelli and the Ideology of the Offensive: Gunpowder Weapons in _The Art of War," The Journal of Military History_ , 67, no. 2 (April 2003): 392 and note 50 there.
15. Ibid., 394 and note 55.
16. Ibid., 393 and notes 51–53.
17. Louis A. DiMarco, _War Horse: A History of the Military Horse and Rider_ (Yardley, PA: Westholme, 2008), 205–206.
18. Jeremy Black, _European Warfare, 1494–1660_ (London and New York: Routledge, 2002), 208.
19. Ibid., 207.
20. Michael Roberts, "The Military Revolution 1560–1660," in Clifford J. Rogers, ed., _The Military Revolution Debate: Readings on the Transformation of Early Modern Europe_ (Boulder, CO: Westview Press, 1995), 14.
21. David A. Parrott, "Strategy and Tactics in the Thirty Years' War: The 'Military Revolution,'" in Rogers, _The Military Revolution Debate_ , 238.
22. Ibid., 237–239.
23. Vladimir Littauer, _Horseman's Progress: The Development of Modern Riding_ (N.p.: The Long Riders' Guild Press, n.d. [1962]), 92; Louis A. DiMarco, _War Horse_ , 183–190.
24. Littauer, _Horseman's Progress_ , 93.
25. DiMarco, _War Horse_ , 191.
26. Williamson Murray, "Cavalry, 1500–1945," in Cowley and Parker, _The Reader's Companion to Military History_ , 75–76.
27. Ibid. Murray writes that precisely one prepared infantry square was overrun by cavalry attack in the Napoleonic era, and that only because horse and rider were killed before they could turn away. They consequently crashed headlong into the square and broke open its side. It should be noted, however, that the 1st and 2nd Dragoons of the (British) King's German Legion successfully destroyed two French infantry squares at Garcia Hernandez on the day after the Battle of Salamanca.
28. Jeremy Black, "The Military Revolution II: Eighteenth-Century War," in Charles Townshend, ed., _The Oxford Illustrated History of Modern War_ (Oxford: Oxford University Press, 1997), 41.
29. Alan Forrest, "The Nation in Arms I: The French Wars," in Townshend, _The Oxford Illustrated History of Modern War_ , 59.
30. Cf. Martin van Creveld, "Technology and War I: To 1945," in ibid., 181–182, 188.
31. Ibid., 181.
32. The following is drawn from D. R. Dorondo, "Review of _Noble Brutes: How Eastern Horses Transformed English Culture_ , by Donna Landry," _Itinerario_ , no. 2 (2009).
33. Cf. DiMarco, _War Horse_ , 217.
34. Ibid.
Chapter 2: The Legacy of 1870
1. Cf. Lucian K. Truscott Jr., _The Twilight of the U.S. Cavalry: Life in the Old Army, 1917–1942_ (Lawrence: University Press of Kansas, 1989), ix-xx.
2. Hugh Trevor-Roper, _The Rise of Christian Europe_ (New York: Harcourt, Brace, 1965), 77.
3. See general references in G. Ronald Murphy, S.J., trans., _The Heliand: The Saxon Gospel_ (New York: Oxford University Press, 1992). Here see specifically xiv and 135 with the footnote there.
4. Trevor-Roper, _The Rise of Christian Europe_ , 95–98; John Keegan, _A History of Warfare_ (New York: Alfred A. Knopf, 1994), 189.
5. Wolfgang J. Mommsen, _Imperial Germany 1867–1918: Politics, Culture, and Society in an Authoritarian State_ , trans. Richard Deveson (London: Arnold, 1995), 124. See also Koppel S. Pinson, _Modern Germany: Its History and Civilization_ , 2nd ed. (Prospect Heights, ILL: Waveland Press, 1989), 255–273.
6. Michael Howard, _The Franco-Prussian War: The German Invasion of France 1870–1871_ (New York: Dorset Press, 1961), 7–8.
7. Ibid., 60, 350, and note 3 on the latter page. Most of the more than 700 Bismarck memorials built after 1870 loosely depicted the Reich Chancellor either as a knight or in his cuirassier's uniform. See Heinrich August Winkler, _Der lange Weg nach Westen: Deutsche Geschichte_ (München: C. H. Beck Verlag, 2002), I:278–279.
8. Vezio Melegari, _The World's Great Regiments_ (New York: G. P. Putnam's Sons, 1969), 100.
9. On Langensalza and Königgrätz, see Robert M. Citino, _The German Way of War: From the Thirty Years' War to the Third Reich_ (Lawrence: University Press of Kansas, 2005), 153–170. On Moltke's memorandum, see Michael D. Krause, "Moltke and Origins of the Operational Level of War," in Michael D. Krause and R. Cody Phillips, eds., _Historical Perspectives on the Operational Art_ (Washington, D.C.: Center of Military History, 2005), 124–125.
10. A. L. Wagner, ed., _Cavalry Studies from Two Great Wars. Comprising the French Cavalry in 1870 by Lieutenant-Colonel Bonie, the German Cavalry in the Battle of Vionville-Mars-la-Tour by Major Kaehler and the Operations of the Cavalry in the Gettysburg Campaign by Lieutenant-Colonel George B. Davis_ (Kansas City, MO: Hudson-Kimberly Publishing Company, 1896), 5–6.
11. Ibid., 6.
12. Mary Lee Stubbs and Stanley Russell Connor, _Armor-Cavalry_ , part I: _Regular Army and Army Reserve_ (Washington, D.C.: Office of the Chief of Military History Unites States Army, 1969), 4.
13. Edward J. Katzenbach, "The Horse Cavalry in the Twentieth Century: A Study in Policy Response," _Public Policy: A Yearbook of the Graduate School of Public Administration_ , Harvard University, 1958, 128.
14. Bonie cited in Wagner, _Cavalry Studies from Two Great Wars_ , 9. These regulations were presumably the same ones studied by U.S. Cavalry officers sent to France by Secretary of War Joel Poinsette in 1839–1840.
15. Ibid., 12.
16. Cf. Gerhard von Pelet-Narbonne, _Cavalry on Service: Illustrated by the Advance of the German Cavalry Across the Mosel in 1870_ , trans. D'Arcy Legard (London: Hugh Rees, 1906), viii. Pelet-Narbonne's text is a very detailed account of German small-unit cavalry operations from the battles at Wörth and Spicheren to the eve of the clash at Mars-la-Tour. He illustrates clearly the moral superiority of the German cavalry over its French counterparts, particularly at the junior-officer level. He does, however, provide instances of the French cavalry's effective use of firearms, particularly among the dragoons and freely admits (282; see the note there) that their chassepot carbine was "far superior" to the German troopers' weapon. French cavalrymen, he writes, "were much more accustomed to use it than the German cavalry, both on foot and on horseback."
17. Ibid., 15; see the note there.
18. Bonie cited in Wagner, _Cavalry Studies from Two Great Wars_ , 23–28.
19. Geoffrey Wawro, _The Franco-Prussian War: The German Conquest of France in 1870–1871_ (New York: Cambridge University Press, 2003), 132–133.
20. Ibid., 133; Bonie cited in Wagner, _Cavalry Studies from Two Great Wars_ , 23–28; Howard, _The Franco-Prussian War_ , 112.
21. Bonie cited in Wagner, _Cavalry Studies from Two Great Wars_ , 28.
22. Howard, _The Franco-Prussian War_ , 73, 100–101.
23. Pelet-Narbonne, _Cavalry on Service_ , 265. Order from Prussian Royal Headquarters dated 13 August 1870. The original order had the passage underlined.
24. Wawro, _The Franco-Prussian War_ , 141.
25. Ibid., 143.
26. Ibid., 249.
27. Howard, _The Franco-Prussian War_ , 131; Denison, _A History of Cavalry From the Earliest Times_ , 404–407.
28. Pelet-Narbonne, _Cavalry on Service_ , 287–314.
29. Wawro, _The Franco-Prussian War_ , 156; Howard, _The Franco-Prussian War_ , 157; Kaehler as cited in Wagner, _Cavalry Studies from Two Great Wars_ , 167–189. In addition to preparatory fire before a charge, the other principal tasks of the horse-artillery were to prevent enemy cavalry's rallying; to provide a rallying point for one's own cavalry if necessary; and to help secure specific defensive positions.
30. J. F. C. Fuller, _The Conduct of War, 1789–1961_ (New York: Minerva Press, 1968), 105.
31. Ibid., 119.
32. Gordon A. Craig, _The Battle of Königgrätz: Prussia's Victory Over Austria, 1866_ (Philadelphia and New York: Lippincott, 1964), 21.
33. Geoffrey Wawro, _The Austro-Prussian War: Austria's War with Prussia and Italy in 1866_ (New York: Cambridge University Press, 1996), 16–20.
34. Ibid., 17.
35. Ibid., 19.
36. Ibid., 269, 271.
37. Howard, _The Franco-Prussian War_ , 2–4.
38. Ibid., 3.
39. Ibid., 4.
40. P. H. Sheridan, _Personal Memoirs_ , intro. Jeffrey D. Wert (New York: Da Capo Press, 1992), 534.
41. Jonathan M. House, _Towards Combined Arms Warfare: A Survey of 20th-Century Tactics, Doctrine, and Organization_ (Ft. Leavenworth: U.S. Army Command and General Staff College, 1984), Prologue. No page reference in the online edition.
42. Michael Howard, _War in European History_ (Oxford: Oxford University Press, 1976), 104.
43. David Johnson, _Napoleon's Cavalry and Its Leaders_ (New York: Holmes and Meier, 1978), 13.
44. Z. Grbasic and V. Vuksic, _The History of Cavalry_ (New York: Facts On File, 1989), 243.
45. Andrew Roberts, _Waterloo: June 18, 1815. The Battle for Modern Europe_ (New York: HarperCollins, 2005), 66–67, 80.
46. Craig, _The Battle of Königgrätz_ , 17; Katzenbach, "Twentieth Century Horse Cavalry," 122–123; Pelet-Narbonne, _Cavalry on Service_ , 21. See Pelet-Narbonne, 244–246, for use of ferries near Metz in mid-August 1870. In September 1939, horsemen of the German 1st Cavalry Brigade (redesignated 1st Cavalry Division after October 1939) covered as much as forty-five miles per day, fully equipped, in the invasion of Poland.
47. Major F. Maurice, "The Franco-German War (1870–1)," in A. W. Ward et al., eds., _The Cambridge Modern History_ , vol. XI: _The Growth of Nationalities_ (Cambridge: Cambridge University Press, 1969), 580.
48. Geoffrey Wawro, _The Franco-Prussian War: The German Conquest of France in 1870–1871_ (New York: Cambridge University Press, 2003), 86; Bonie, "The French Cavalry in 1870," in Wagner, _Cavalry Studies from Two Great Wars_ , 35–36.
49. Howard, _The Franco-Prussian War_ , 65.
50. Ibid. By way of comparison, in the last sustained cavalry war before 1870, the American Civil War, the Union Army was requiring approximately five hundred remounts _per week_ by mid-1863, a number that could be met only with a remount service that had been operating for more than two years. See Phil Livingston and Ed Roberts, _War Horse: Mounting the Cavalry With America's Best Horses_ (Albany, TX: Bright Sky Press, 2003), 43–45. In the next major European war featuring large numbers of horses, World War I, the French Government was able, but only with greatest effort given its own supreme needs over four years, to supply the American Expeditionary Forces, with some 63,000 equids during the course of 1918. "Final Report of General John J. Pershing: Part III—Supply, Coordination, Munitions, and Administration—'Remounts,'" in Francis J. Reynolds, _The Story of the Great War: History of the European War From Official Sources_ (New York: P. F. Collier and Son Company, 1920), vol. 8, Appendix, xlviii.
51. Howard, _War in European History_ , 84; Bonie, "The French Cavalry in 1870," in Wagner, _Cavalry Studies from Two Great Wars_ , 40.
52. Waitman Beorn, "'Heads Up, By God!' French Cavalry At Eylau, 1807 and Napoleon's Cavalry Doctrine." www.napoleonseries.org/articles/wars/eylau.cfm. 25 January 2005. Beorn also provides examples from Austerlitz, Borodino, and suggests that even at Waterloo the French cavalry's action was not born merely of desperation.
53. Johnson, _Napoleon's Cavalry and Its Leaders_ , 12. See also Grbasic and Vuksic, _The History of the Cavalry_ , 243.
54. Pelet-Narbonne, _Cavalry on Service_ , 248–249, 251.
55. Jay Luvaas, _The Military Legacy of the Civil War_ (Chicago: University of Chicago Press, 1959), 119, 123.
56. Ibid., 124.
57. Ibid., 136. Luvaas does note the exceptions to this way of thinking in the likes of General Friedrich von Bernhardi and the civilian novelist and military writer Karl Bleibtreu.
58. Ibid., 130ff.
59. Howard, _The Franco-Prussian War_ , 8.
60. Grbasic and Vuksic, _The History of Cavalry_ , 250; George T. Denison, _A History of Cavalry From the Earliest Times: With Lessons for the Future_ (London: Macmillan, 1913), 411.
61. Howard, _The Franco-Prussian War_ , 190–194.
62. Ibid., 195, 197–198. Explicit instructions coming from Paris also kept French Marshal MacMahon from ordering a withdrawal away from the Germans.
63. Denison, _A History of Cavalry From the Earliest Times_ , 410–411.
64. Ibid., 411–412. Howard, _The Franco-Prussian War_ , 215, indicates that even before the reserve cavalry's attacks, two squadrons of French lancers had been destroyed attempting to halt the advancing German infantry at nearly the same spot on the same day.
65. Wawro, _The Franco-Prussian War_ , 221.
66. Ibid., 231.
67. Ibid., 241.
68. Ibid., 248.
69. Ibid., 265.
70. Howard, _The Franco-Prussian War_ , 329.
71. Ibid., 379–381; Wawro, _The Franco-Prussian War_ , 279.
72. Wawro, _The Franco-Prussian War_ , 288.
73. Denison, _A History of Cavalry from the Earliest Times_ , 412–413.
74. Sheridan, _Personal Memoirs_ , 495.
75. Luvaas, _The Military Legacy of the Civil War_ , 122–123.
76. Ibid., 123.
77. Ibid., 125.
78. Ibid., 146–151.
Chapter 3: Not Quite Sunset
1. Hew Strachan, _The First World War_ , vol. 1 (Oxford: Oxford University Press, 2001), 231.
2. See Friedrich von Berhardi, _Germany and the Next War_ , trans. Allen H. Powles (n.p., 1912). Here Chapter X. <http://www.gutenberg.org/files/11352/11352-8.txt>
3. Van Michael Leslie, "French, John Denton Pinkstone, Earl of Ypres," in Spencer C. Tucker, ed., _The European Powers in the First World War: An Encyclopedia_ (New York: Garland, 1996), 271.
4. John Ellis, _Eye-Deep in Hell: Trench Warfare in World War I_ (New York: Pantheon Books, 1976), 84.
5. John Keegan, _A History of Warfare_ , 187–188. Here Keegan cites the Marquess of Anglesey's _A History of the British Cavalry_.
6. Ibid., 224.
7. Strachan, _The First World War_ , vol. 1, 232. On British cavalry doctrine after 1902, see Stephen Badsey, "The Boer War (1899–1902) and British Cavalry Doctrine: A ReEvaluation," _The Journal of Military History_ , 71, no. 1 (January 2007): 75–97.
8. Jean Bou, "Cavalry, Firepower, and Swords: The Australian Light Horse and the Tactical Lessons of Cavalry Operations in Palestine, 1916–1918," _The Journal of Military History_ , 71, no.1 (January 2007): 102, and Gervase Phillips, "Scapegoat Arm: Twentieth-Century Cavalry in Anglophone Historiography," in ibid., 39. According to this British analysis, "mounted rifles" differed from "mounted infantry" in that the latter were not expected to use horses for any real purpose other than moving from one position to another. The articles cited do not specify the mounted infantry's training requirement for equitation. Note, however, that novice infantrymen astride anything other than "dead broke" horses would run a very real risk of multiple, and potentially dangerous, falls.
9. Phillips, "Scapegoat Arm," 43–44.
10. Norman Davies, _Europe: A History_ , 902.
11. Keegan, _A History of Warfare_ , 271; Lawrence Keppie, _The Making of the Roman Army: From Republic to Empire_ , 26–32.
12. Keegan, _A History of Warfare_ , 221–234, notes the importance of tradition in helping shape the organization and function of armed forces.
13. Ellis, _Eye-Deep in Hell_ , 84.
14. Henderson had observed the South African campaign from the headquarters of the British commander, Lord Roberts. He is quoted in Peter Paret, ed., _Makers of Modern Strategy from Machiavelli to the Nuclear Age_ (Princeton: Princeton University Press, 1986), 516.
15. Ibid., 519.
16. Keegan, _A History of Warfare_ , 307–308.
17. Edward Spiers, "The Late Victorian Army 1868–1914," in David Chandler, ed., _The Oxford Illustrated History of the British Army_ (Oxford: Oxford University Press, 1994), 212. The British campaign in Palestine is excluded from this discussion merely because of its extra-European geographic context. The British army's use of cavalry there, as well as the Ottomans' both there and in eastern Anatolia and the Caucasus Mountains, remains nonetheless noteworthy.
18. John Keegan, _The First World War_ (New York: Alfred A. Knopf, 1999), 73. Though the British figure seems remarkably low, it is estimated that the British equine population was at least 3.3 million as early as 1901. Monthly averages of 165,000 horses and mules were imported by Great Britain from the United States over the duration of the war. Figures from R. H. C. Davis, _The Medieval Warhorse: Origin, Development and Redevelopment_ (London: Thames and Hudson, 1989), 46.
19. Davis, _The Medieval Warhorse_ , citing a passage from F. Nagle, _Fritz_ (Huntington, WV: n.p., 1981), 15–19. _Bundesrat_ decrees cited in _The Times_ of 1 August 1914 under " _A_ State of War." These decrees applied also to animal products; oil, coal, and their byproducts; automobiles; spare parts, etc.
20. John Singleton, "Britain's Use of Military Horses 1914–1918," _Past and Present_ , 139 (May 1993): 182.
21. Christian Freiherr von Stenglin, _The Hanoverian_ , trans. Christina Belton (London: J. A. Allen, 1990), 36; Eberhard von Velsen and Erhard Schulte, _The Trakehner_ , trans. Christina Belton (London: J. A. Allen, 1990), 12.
22. "The Battle of Big Nations for Thoroughbred Horses," _New York Times Magazine_ , 3 May 1914, SM9 at <http://query.nytimes.com/gst/abstract.html> (18 March 2009).
23. Ibid.
24. Daniel David, _The 1914 Campaign: August-October, 1914_ (New York: Wieser and Wieser, 1987), 19.
25. Ibid.
26. Hew Strachan, "Military Modernization, 1789–1918," in T. C. W. Blanning, ed., _The Oxford Illustrated History of Modern Europe_ (Oxford: Oxford University Press, 1996), 80.
27. "France Invaded"; "Franco-German Fighting"; "Franco-German Encounters"; _Times_ , 2–6 August 1914.
28. Cf. Dennis E. Showalter, "World War I," in Robert Cowley and Geoffrey Parker, eds., _The Reader's Companion to Military History_ (Boston: Houghton Mifflin, 1996), 523. See John Keegan, _The First World War_ , 33–36 for the implications of such density of personnel on the ground. On the flanking mission of the German cavalry see Paret, _Makers of Modern Strategy_ , 513.
29. Robert A. Doughty, "French Strategy in 1914: Joffre's Own," _The Journal of Military History_ , 67, no. 2 (April 2003): 436.
30. Ibid., 437.
31. David, _The 1914 Campaign_ , 122–123, 142,148; Keegan, _The First World War_ , 20, 77, 129; _Handbook of the German Army (Home and Colonial_ ) (London: Imperial War Museum, 1914; Nashville, TN: The Battery Press, 2002), 51–52, 130, 142–143; for cavalry-horse paces, see 239.
32. Keegan, _The First World War_ , 81. For the entire campaign on the Western Front to the end of 1914 see Maximilian von Poseck, _The German Cavalry: 1914 in Belgium and France_ , trans. Alexander C. Strecker et al., ed. Jerome Howe (Berlin: E. S. Mittler und Sohn, 1923). This translation was commissioned by the U.S. Cavalry Association. Poseck held the rank of lieutenant general and Inspector of Cavalry.
33. Doughty, "French Strategy in 1914," 446, 449, 451; Keegan, _The First World War_ , 84; Francis J. Reynolds, ed., _The Story of the Great War_ (New York: P. F. Collier and Son, 1916), ii, 9.
34. Doughty, "French Strategy in 1914," 452; Keegan, _The First World War_ , 92.
35. Reynolds, _The Story of the Great War_ , ii, 18–19.
36. S. L. A. Marshall, _World War I_ (New York: American Heritage Press, 1971), 61.
37. Reynolds, _The Story of the Great War_ , 25; "British and French Join," _New York Times_ (1857-Current file), 10 August 1914; ProQuest Historical Newspapers, _New York Times_ (1851–2002), 2. On the fear of a German cavalry attack on Brussels see ibid. On Haelen and Donck, see Poseck, _The German Cavalry_ , 22–28.
38. "Belgian Peasants Panic," _New York Times_ (1857-Current file), 21 August 1914; ProQuest Historical Newspapers, _New York Times_ (1851–2002), 1.
39. "Save Ostend From Uhlans," _New York Times_ , 26 August 1914; ProQuest Historical Newspapers, _New York Times_ (1851–2002), 2.
40. John Keegan, _The First World War_ , 107; Richard Holmes, "The Last Hurrah: Cavalry on the Western Front, August-September 1914," in Hugh Cecil and Peter Liddle, eds., _Facing Armageddon: The First World War Experienced_ (London: Leo Cooper, 1996), 286.
41. Poseck, _The German Cavalry_ , 53.
42. Keegan, _The First World War_ , 113; Strachan, _The First World War_ , vol. 1, 254.
43. Strachan, _The First World War_ , vol. 1, 254.
44. Ibid., 232, 254, 258.
45. Ibid., 259–260; Martin Gilbert, _The First World War: A Complete History_ (New York: Henry Holt, 1994), 74. British lieutenant quoted in Holmes, "The Last Hurrah," 290. Holmes, 280, 288, further notes that the French alone retained large numbers of breast-plated cuirassiers in 1914; and, given the fact that breastplates were not bulletproof and could be seen for several miles on sunny days, their utility in 1914 was unclear.
46. Trevor N. Dupuy, _1914: The Battles in the West_ (New York: Franklin Watts, 1967), 71.
47. Poseck, _The German Cavalry_ , 160–161.
48. Ibid., 175–176.
49. Ibid., 177.
50. Ibid., 210, 212–213.
51. Ibid., 213.
52. Martin Gilbert, _The Somme: Heroism and Horror in the First World War_ (New York: Henry Holt, 2006), 21–22.
53. Ibid., 27–28; David T. Zabecki, "Somme, Battle of (1 July-19 November 1916)," in Spencer C. Tucker, ed., _The European Powers in the First World War: An Encyclopedia_ (New York: Garland, 1996), 649.
54. Gilbert, _The Somme_ , 27.
55. Ibid., 112–113.
56. Both quotations from ibid., 92–94.
57. Ibid., 228–229.
58. Gilbert, _The First World War_ , 235, 255.
59. Andrew N. Liaropoulus, "Revolutions in Warfare: Theoretical Paradigms and Historical Evidence—The Napoleonic and First World War Revolutions in Military Affairs," _The Journal of Military History_ , 70, no. 2 (April 2006): 377–384.
60. Robert M. Citino, _The German Way of War_ , 223.
61. Keegan, _The First World War_ , 141–142.
62. Ibid., 140; Strachan, _The First World War_ , vol. 1, 316.
63. Davies, _Europe: A History_ , 902.
64. Keegan, _The First World War_ , 83.
65. Citino, _The German Way of War_ , 225. On Prussian/German identification, see Christopher Clark, _Iron Kingdom: The Rise and Downfall of Prussia, 1600–1947_ (Oxford: Oxford University Press, 2006), 607–608. Tukhachevsky quoted in Adam Zamoyski, _The Battle for the Marchlands_ , East European Monographs, No. LXXXVIII (New York: Columbia University Press, 1981), 12, see also 28; "Deutsches Kulturland," in Norman Stone, _The Eastern Front 1914–1917_ (New York: Charles Scribner's Sons, 1975), 49.
66. Gilbert, _The First World War_ , 48–49; Vejas Gabriel Liulevicius, _War Land on the Eastern Front: Culture, National Identity, and German Occupation in World War I_ (Cambridge: Cambridge University Press, 2000), 15. A distinctly pro-Russian account of the Eastern Front is contained in Alexis Wrangel, _The End of Chivalry: The Last Great Cavalry Battles 1914–1918_ (New York: Hippocrene Books, 1982). Wrangel, 70, clearly states that Russian cavalrymen helped themselves in gentlemanly fashion to the relative "affluence" of East Prussian farms, while follow-on forces came "like a swarm of locusts" burning farms and turning villages to wilderness. Wrangel's ancestor, Baron Peter Wrangel, a captain of the Russian Imperial Horse Guards Regiment, fought in East Prussia. Interestingly, Alexander Solzhenitsyn, _August 1914_ , trans. Michael Glenny (New York: Farrar, Straus and Giroux, 1971), 218, has the German general Hermann von François write in an order of the day to the German I Corps of the "Russian hordes who, in defiance of international law, are burning the towns and villages of our homeland." By contrast, Imanuel Geiss, "The Civilian Dimension of the War," _Facing Armageddon_ , 18, says bluntly that stories of Russian depredations were mere German propaganda. The Russian eyewitness account reproduced above is in Vladimir S. Littauer, _Russian Hussar: A Story of the Imperial Cavalry, 1911–1920_ (London: J. A. Allen & Co., 1965; Long Riders' Guild Press, 2007), 138–139, 143–144.
67. Citino, _The German Way of War_ , 227. "Frog stickers" and references to intelligence failures are in Dennis E. Showalter, _Tannenberg: Clash of Empires_ (Hamden, CT: Archon Books, 1991), 189. Intra-service derogation was not unique to the Imperial German Army. British cavalrymen in 1914–1918 were sometimes referred to, not least by ANZAC troops, as "donkey wallopers."
68. Citino, _The German Way of War_ , 229. For the Russian cavalry's encounter with German infantry near Kaushen, see Wrangel, _The End of Chivalry_ , 23–38.
69. Showalter, _Tannenberg_ , 151.
70. Ibid., 210; Citino, _The German Way of War_ , 229.
71. Strachan, _The First World War_ , vol. 1, 321, 327; Littauer, _Russian Hussar_ , 142.
72. "Hindenburg--Falkenhayn," _Vossische Zeitung_ (Berlin), 30 August 1916. <http://www.zld.de/projekte/millenium/original_html/vossische_1916_3008.GIF.html> (9 March 2007).
73. T. Dodson Stamps and Vincent J. Esposito, eds., _A Short Military History of World War I With Atlas_ (West Point: USMAAG Printing Office, 1950), 123–128.
74. Ibid.
75. Marshall, _World War I_ , 216–218; Stone, _The Eastern Front 1914–1917_ , 171–172, 188–189; Liulevicius, _War Land on the Eastern Front_ , 19. On the German cavalry in Courland, see _The Story of the Great War_ , vol. III (New York: P. F. Collier and Son, 1916), 337–339.
76. For this discussion, see Stamps and Esposito, _A Short Military History of World War I_ , 198–203.
77. David Christian, _A History of Russia, Central Asia, and Mongolia_ , vol. 1: _Inner Eurasia from Prehistory to the Mongol Empire_ (Oxford: Blackwell, 1998), 13–16. The Great Hungarian Plain (the _Alföld_ or _puszta_ ), though separated from the Black Sea Steppe by the Carpathian Mountains, constitutes the westernmost extremity of the Eurasian grasslands. The _Alföld_ 's area is some 30,000 square miles.
78. Singleton, "Britain's Use of Military Horses 1914–1918," 189.
79. Ibid.
80. Stamps and Esposito, _A Short Military History of World War I_ , 5.
81. Holmes, "The Last Hurrah," 286.
82. Stamps and Esposito, _A Short Military History of World War I_ , 4.
83. See, for example, those noted throughout by Poseck, _The German Cavalry_.
84. Wrangel, _The End of Chivalry_ , 105–153.
85. Holger Herwig, "The German Victories, 1917–1918," in Strachan, _The First World War_ , 260–263. Herwig also notes that the Germans possessed only 23,000 trucks as against the Allies' 100,000 vehicles.
86. See the useful examination of the armistice's development and final terms in Bullitt Lowry, _Armistice 1918_ (Kent, OH: The Kent State University Press, 1996), here particularly chapters 7 and 8.
87. Poseck, _The German Cavalry_ , 206.
88. Holmes, "The Last Hurrah," 287.
Chapter 4: False Dawn
1. _Peace Treaty of Versailles. Articles 231–247 and Annexes. Reparations_. <http://www.lib.byu.edu/~rdh/wwi/versa/versa7.html> (5 June 2003). The Fourth Hague Convention on the Laws and Customs of War on Land (1907) had earlier reflected the horse's continuing military importance at the beginning of the twentieth century by specifying in Chapter II, Article 4 that all horses, if taken along with POWs, automatically ceased to be the property of enemy individuals or States.
2. John Ellis, _Eye-Deep in Hell_ , 84. This constituted a reiteration of the British army's 1907 _Cavalry Manual's_ assertion that the rifle "cannot replace the effect produced by the speed of the horse, the magnetism of the charge, and the terror of cold steel." Ibid.
3. Carlo D'Este, _Patton: A Genius for War_ (New York: HarperCollins, 1995), 139,162, 334.
4. Quotation in Alexander M. Bielakowski, "General Hawkins's War: The Future of the Horse in the U.S. Cavalry," _The Journal of Military History_ , 71 (January 2007): 137.
5. For the argument over mechanization within the U.S. Cavalry as a whole, see ibid., 127–138.
6. Zamoyski, _The Battle for the Marchlands_ , 44–46.
7. Ibid., 58–61; David M. Glantz and Jonathan House, _When Titans Clashed: How the Red Army Stopped Hitler_ (Lawrence: University Press of Kansas, 1995), 6.
8. Norman Davies, _Europe: A History_ , 935; Zamoyski, _The Battle for the Marchlands_ , 163–174.
9. "Strength of German Army in World War I" in Charles E. Heller and William A. Stofft, eds., _America's First Battles, 1776–1965_ (Lawrence: University Press of Kansas, 1986), 155.
10. Text of the Treaty of Versailles at http:www.FirstWorldWar.com (19 September 2006). See also Wilhelm Deist, _The Wehrmacht and German Rearmament_ , foreword A. J. Nicholls (Basingstoke: The Macmillan Press, 1981), 4. Overview of command structure of the _Reichsheer_ of the Weimer period and the _Heer_ after 1935 at http:www.bundesarchiv.de/php/bestaende_findemittel/bestaendeuebersicht (16 February 2007).
11. Initially each squadron was approximately the size of a U.S. Army infantry company of that period. Later reorganizations increased the size of squadrons to a level comparable with those of the U.S. Cavalry, i.e., roughly the size of U.S. Army battalions. For the German cavalry division's TOE of 1919, see James S. Corum, _The Roots of Blitzkrieg: Hans von Seeckt and German Army Reform_ (Lawrence: University Press of Kansas, 1992), 207 and 46. See also Robert M. Kennedy, _The German Campaign in Poland (1939_ ), Department of the Army Pamphlet No. 20–255 (Washington, D.C.: Department of the Army, 1956), 11. Unlike their infantry counterparts, the cavalry divisions drew personnel from beyond their immediate regional military area ( _Wehrkreis_ ). The 3rd Cavalry Division, for example, though headquartered in Weimar in Thuringia, included a Bavarian mounted regiment. See Kennedy, _The German Campaign in Poland_ , 11.
12. Corum, _Roots of Blitzkrieg_ , 31–32.
13. Von Seeckt, from his _Thoughts of a Soldier_ (London: Ernest Benn, 1930) as quoted in ibid., 32. See also Army Service Regulation ( _Heeresdienstvorschrift_ ) 487, here Part 1, 47 as quoted in ibid., 42. Poseck, _The German Cavalry_ , 231. In the present text, "motorized" refers throughout to armored and unarmored wheeled vehicles; "mechanized" refers to tracked vehicles. On "operational" versus "tactical" see the very useful work by Bruce W. Menning, "Operational Art's Origins," in Michael D. Krause and R. Cody Phillips, eds., _Historical Perspectives of the Operational Art_ (Washington, D.C.: Center of Military History, 2005), here 8–9; note particularly the influence of Soviet thinkers in the 1920s and 1930s. On von Moltke, see Michael D. Krause, "Moltke and the Origins of the Operational Level of War," in Krause and Phillips, here 124–125.
14. Seeckt, _Thoughts of a Soldier_ , 84–86.
15. Ibid., 84, 91–92.
16. Ibid., 92, 95, 99–100; Poseck, _The German Cavalry_ , 233–234.
17. Seeckt, _Thoughts of a Soldier_ , 101–106.
18. Klaus Christian Richter, _Cavalry of the Wehrmacht 1941–1945_ (Atglen, PA: Schiffer, n.d.), 23. The numbers of hours of equitation training would drop dramatically in the wake of the army's rapid expansion beginning under Hitler in 1934–1935. Under the Nazi régime's conscription regulations, the initial term of service was only one year. That fact, when combined with the stipulation that only 10 percent of cavalrymen could be volunteers, meant that the overwhelming majority of horsemen were shortterm draftees who had to be taught hurriedly to ride well and shoot accurately. See Richter, _Cavalry of the Wehrmacht_ , 23. Limiting volunteers to 10 percent of the whole would seem to indicate that the army did not want to allow for a disproportionate share of highly motivated young men going to the mounted arm. Comparisons of volunteer totals for other _Waffengattungen_ , if available, would be useful in this context. The present work draws no conclusions on this point.
19. Ibid.
20. Ibid., 199. In the _levade_ the horse rears to an angle of about 40–45 degrees with the hind feet planted firmly and square. The Horseman's Badge depicted the horse at a somewhat lower degree of inclination.
21. Poseck, _The German Cavalry_ , 226, 229.
22. Corum, _The Roots of Blitzkrieg_ , 46; cavalry's divisional TOE of 1923 on 209; Poseck, _The German Cavalry_ , 227.
23. Ibid., 47, 79, 123–124; <http://www.neufahrland-online.de/seite6.htm> _Chronik des Dorfes Krampnitz_ (7 February 2007).
24. On Hans Joachim von Ziethen, cf. Theodor Fontane, _Wanderungen durch die mark Brandenburg. Erster Band. Die Grafschaft Ruppin_ , ed. Edgar Gross (München: Nymphenburger Verlagshandlung, 1963), 15–25. While "Hannover" is the spelling of the German city's name, the widely accepted (but not universal) English rendering of "Hanoverian" is used throughout to indicate the breed.
25. Corum, _The Roots of Blitzkrieg_ , 69.
26. Matthew Cooper, _The German Army 1933–1945_ (London: Scarborough House, 1978), 135.
27. Ibid., 136.
28. Corum, _The Roots of Blitzkrieg_ , 66. See 55–66 for the various schools of thought in the army regarding the most appropriate form of warfare for Germany. For offensive and defensive war plans, see 172–174.
29. Zamoyski, _The Battle for the Marchlands_ , 25–26; Poseck, _The German Cavalry_ , 231.
30. Corum, _The Roots of Blitzkrieg_ , 71.
31. Ibid.
32. Ibid.
33. Corum, _The Roots of Blitzkrieg_ , 184.
34. Ibid., 186.
35. Ibid., 187.
36. Ibid., 177.
37. Deist, _The Wehrmacht and German Rearmament_ , 10.
38. Ibid., 12–17; Corum, _The Roots of Blitzkrieg_ , 193–194, 196–197.
39. Corum, _The Roots of Blitzkrieg_ , 197.
Chapter 5: The Field of Mars
1. Paul Louis Johnson, _Horses of the German Army in World War II_ (Atglen, PA: Schiffer, 2006), 125. Johnson reproduces the U.S. Army Military History Institute's postwar publication MS#P-090 on horses in the German army. Various period illustrations indicated that the officer's saber had a less dramatic curvature to the blade than the enlisted man's.
2. See the illustration and caption in Klaus Richter, _Weapons and Equipment of the German Cavalry 1935–1945_ (Atglen, PA: Schiffer, 1995), 3. Bayonet in author's collection and illustrated in ibid., 126.
3. Ibid., 5. See also Ian Hogg, "Small Arms," in I. C. B. Dear and M. R. D. Foot, eds., _The Oxford Companion to World War II_ (Oxford: Oxford University Press, 1995), 1013–1016.
4. Richter, _Weapons and Equipment of the German Cavalry 1935–1945_ , 6; Corum, _The Roots of Blitzkrieg_ , 104; Zamoyski, _The Battle for the Marchlands_ , 27. On the MG 01, see _Handbook of the German Army (Home and Colonial) 1912 (Amended to 1914_ ), 118. The all-up weight of the earlier weapon, with water jacket and mounting sledge, was fully 176 pounds. Thus the MG 08 still weighed in at nearly 150 pounds, not including the limber, ammunition, and other ancillary equipment.
5. Corum, _The Roots of Blitzkrieg_ , 104.
6. On the MG 34 and MG 42, see Hogg, "Small Arms," as in note 3 above. Transport details and supporting illustrations in Richter, _Weapons and Equipment of the German Cavalry 1935–1945_ , 7–8.
7. Charles B. MacDonald, _Company Commander_ , intro. Dennis Showalter (New York: History Book Club, 2006), xiv. The quotation is from Showalter.
8. Http://www.lexikon-der-wehrmacht.de/Waffen/Infanteriegeschütze-R.htm (30 January 2007).
9. Ibid.; Richter, _Weapons and Equipment of the German Cavalry 1935–1945_ , 18; David Stone, _Fighting for the Fatherland: The Story of the German Soldier From 1648 to the Present Day_ , foreword Richard Holmes (Washington, D.C.: Potomac Books, 2006), 321. "Gypsy artillery" in Klaus Christian Richter, _Cavalry of the Wehrmacht 1941–1945_ (Atglen, PA: Schiffer, 1995), 114.
10. Ibid. On the cavalry's reorganization, see below.
11. Richter, _Weapons and Equipment of the German Cavalry 1935–1945_ , 18–19.
12. As demonstrated below, bicycle-mounted formations would also become integral to the later cavalry formations of the _Waffen_ -SS. The SS Cavalry Brigade and the subsequent SS Cavalry Division would include an entire bicycle-mounted battalion officially tasked with reconnaissance duties.
13. Table XIII, "Livestock and Dairy Production," in "Germany," _Encyclopedia Britannica_ 10 (Chicago: University of Chicago Press, 1947), 245; Richter, _Cavalry of the Wehrmacht 1941–1945_ , 202. Richter indicates no source for his cited figure of 3,800,000.
14. Velsen-Zerweck and Schulte, _The Trakehner_ , 5–6.
15. Ibid. See also Hans Graf von Lehndorff, _Meschen, Pferde, weites Land: Kindheitsund Jugenderinnerungen_ (München: Verlag C. H. Beck, 2002), 63–64. Reference to Frederick the Great in David Blackbourn, _The Conquest of Nature: Water, Landscape, and the Making of Modern Germany_ (New York: W. W. Norton, 2006), 5, 30. For the reference to 30,000 horses, see Clark, _Iron Kingdom_ , 67. Another translation for "Trakehnen" is "burnt earth," as in land that has been cleared for farming by being burnt over.
16. Lehndorff, _Menschen, Pferde, weites Land_ , 64.
17. Remount progression and Trakehner breeding objectives at _Bundesarchiv_ (hereafter BA), _Pferde im Einsatz bei Wehrmacht und Waffen-SS. Pferderassen: Ostpreuße. Vorschrift: Das Truppenpferd von 1938_. www.bundesarchiv.de/aktuelles/aus_dem_archiv/galerie (3 October 2008).
18. Von Velsen-Zerweck and Schulte, _The Trakehner_ , 15, 38. These heights are somewhat less than those preferred for both stallions/geldings and mares for the breed in the post-1945 period.
19. Sometimes referred to as the _Niedersachsenroß_ ("Lower Saxon Horse"), the Hanoverian's history is so deeply intertwined with the region that, after 1945, it was made the heraldic animal of Lower Saxony. It appears as a white horse rampant on a red field.
20. Stenglin, _The Hanoverian_ , 13.
21. Ibid., 15.
22. Ibid., 34.
23. Ibid., 35–36. Because of its relatively small size of 13–14 hands, the Haflinger was also well-suited as a packhorse for infantry and mountain troops. It would, for example, find a place (along with mules) in the post-1945 armies of the Federal Republic of Germany and Austria. On the Hanoverian's qualities relative to the Trakehner, see BA as in note 17 above.
24. Richter, _Cavalry of the Wehrmacht_ , 23.
25. Stefanie Albrecht, "Prof. Dr. Hans Jöchle (1892–1968)—Ein Leben für den Hufbeschlag. Quellen und Materialien zur Geschichte der tierärztlichen Fakultät der Universität München" (DMV diss., Tierärztliche Hochschule Hannover, 2006), 88. Hereafter _Ein Leben für den Hufbeschlag;_ Johnson, _Horses of the German Army in World War II_ , 55.
26. Johnson, _Horses of the German Army in World War II_ , 55. In reality the total number of horses and mules actually used by the German armed forces of all arms during the course of World War II approached some three million. See the figure in Richter, _Cavalry of the Wehrmacht_ , 202.
27. Johnson, _Horses of the German Army in World War II_ , 55.
28. Ibid, 53. See the chart there.
29. Ibid.
30. Albrecht, _Ein Leben für den Hufbeschlag_ , 166, 168.
31. Richter, _Cavalry of the Wehrmacht_ , 202; Johnson, _Horses of the German Army in World War II_ , 65.
32. Johnson, _Horses of the German Army in World War II_ , 65, for numbers purchased/requisitioned. See Richter, _Cavalry of the Wehrmacht_ , 202, for the latter figure and Johnson, 40, for horses' rations. For the figure of 25,000 Hungarian horses see R. L. DiNardo, _Mechanized Juggernaut or Military Anachronism? Horses and the German Army of World War II_ , foreword Williamson Murray (New York: Greenwood Press, 1991), 12. On the _panje_ horse, see Johnson, 13–14.
33. Cf. excerpts of speech of 3 February 1933 in Jeremy Noakes and Geoffrey Pridham, eds., _Documents on Nazism: 1919–1945_ (New York: Viking, 1974), 508–509. See also Heinrich August Winkler, _Der lange Weg nach Westen_ (München: Verlag C. H. Beck, 2002), 2:50–53; and Deist, _The Wehrmacht and German Rearmament_ , 23.
34. Ian Kershaw, _Hitler 1889–1936: Hubris_ (New York: W. W. Norton, 1998), 490–495.
35. Bruce Condell and David T. Zabecki, eds. and trans., _On the German Art of War: Truppenführung_ (Boulder, CO: Lynne Rienner Publishers, 2001), 5, hereafter _Truppenführung_. Deist, _The Wehrmacht and German Rearmament_ , 36, indicates that Hitler actually decided on the expansion in May 1934 with an effective date of October.
36. Condell and Zabecki, _Truppenführung_ , 5; Deist, _The Wehrmacht and German Rearmament_ , 37.
37. For this discussion, see Deist, _The Wehrmacht and German Rearmament_ , 36–53, and 86–101.
38. Ibid., 38, quotation on 91.
39. _Truppenführung_ , 1–3.
40. "The German way of war" is taken from Robert Citino, _The German Way of War: From the Thirty Years' War to the Third Reich_ (Lawrence: University Press of Kansas, 2005). Throughout his work, Citino maintains that the enduring thought in Prussian and German military history is that those campaigns succeed best that are "short and lively" ( _kurz und vives_ ) and led by the bold.
41. _Truppenführung_ , 184, 187. Subsequent references to the numbered sections of _Truppenführung_ are found in the main body of this text, indicated in parentheses.
42. Brain Bond and Martin Alexander, "Liddell Hart and De Gaulle: The Doctrines of Limited Liability and Mobile Defense," in Peter Paret, ed., _Makers of Modern Strategy from Machiavelli to the Nuclear Age_ (Princeton: Princeton University Press, 1986), 608.
43. Ibid., 606.
44. Ibid., 600.
Chapter 6: Bucking the Trend
1. Editorial by Paul Albert in _Western Horseman_ , November-December 1939, cited in _Western Horseman_ , 71, no. 1 (January 2006): 32. The November-December 1944 (!) issue of the same magazine featured an advertisement depicting the new Model M-5 Type Gas Mask for cavalry use.
2. Matthew Cooper, _The German Army, 1939–1945_ (London: Scarborough House, 1978), 107–108.
3. Cf. P. M. H. Bell, _The Origins of the Second World War in Europe_ (London: Pearson, 2007), 185.
4. See, for example, his remarks to assembled _Wehrmacht_ officers of 22 August and 23 November 1939 in Ian Kershaw, _Hitler 1936–1945: Nemesis_ (New York: W. W. Norton, 2000), 206–208, 276.
5. Michael Geyer, _Deutsche Rüstungspolitik 1860–1890_ , ed. Hans-Ulrich Wehler (Frankfurt am Main: Suhrkamp Verlag, 1984), 125.
6. Ibid.
7. Jeremy Noakes and Geoffrey Pridham, eds., _Documents on Nazism 1919–1945_ (New York: Viking Press, 1975), 375.
8. Ibid., 375–376.
9. Paul Kennedy, _The Rise and Fall of the Great Powers: Economic Change and Military Conflict From 1500 to 2000_ (New York: Random House, 1987), 305. Figures for unemployment and GNP on 306.
10. Noakes and Pridham, _Documents on Nazism 1919–1945_ , 380.
11. Ibid., 381.
12. Kennedy, _Rise and Fall_ , 304–305.
13. Stone, _Fighting for the Fatherland_ , 320. As it has done to this point, the present work treats the army's (and _Waffen_ -SS') maneuver-unit cavalry formations and their various organic elements (brigades, regiments, and other subordinate formations) exclusively. The numerous mounted reconnaissance units of the army's infantry regiments and divisions, though drawn in 1939 from the prewar cavalry regiments, require and deserve a freestanding historical investigation.
14. Noakes and Pridham, _Documents on Nazism 1919–1945_ , 390.
15. Ibid.
16. Ibid., 399.
17. Ibid., 401–412.
18. Louis L. Snyder, ed., _Hitler's Third Reich: A Documentary History_ (Chicago: Nelson-Hall, 1981), 328.
19. Noakes and Pridham, _Documents on Nazism_ , 413.
20. Gervase Phillips, "Scapegoat Arm: Twentieth-Century Cavalry in Anglophone Historiography," _The Journal of Military History_ , 71, no. 1 (January 2007), 51; Keith Sword, "Poland," in I. C. B. Dear and M. R. D. Foot, eds., _The Oxford Companion to World War II_ (Oxford: Oxford University Press, 1995), 900. The Polish army was authorized fourteen cavalry brigades, eleven of which were active in September 1939. See Kennedy, _The German Campaign in Poland (1939_ ), 51–54.
21. Karl-Heinz Frieser, _The Blitzkrieg Legend: The 1940 German Campaign in the West_ , ed. John T. Greenwood (Annapolis: Naval Institute Press, 2005), 17.
22. Richard Overy, _Why the Allies Won_ (New York: W. W. Norton, 1995), 200.
23. Cf. Wilhelm Deist, _The Wehrmacht and German Rearmament_ , 36–53, 91.
24. R. J. Overy, "Transportation and Rearmament in the Third Reich," _The Historical Journal_ , 16 (June 1973), 391.
25. Cf. Martin van Creveld, _The Changing Face of War: Lessons of Combat From the Marne to Iraq_ (New York: Presidio, 2006), 7–8.
26. Overy, "Transportation and Rearmament in the Third Reich," 392.
27. Ibid., 402, 405 and note 89 there.
28. Ibid., 409.
29. DiNardo, _Mechanized Juggernaut or Military Anachronism?_ , 9–10; Gerhard Rempel, _Hitler's Children: The Hitler Youth and the SS_ (Chapel Hill: The University of North Carolina Press, 1989), 22. Interestingly, the Nazi Party also had a formation called the National Socialist Mounted ( _Reiter_ ) Corps (NSRK). It was intended to "train young men to the standard of cavalry recruits of six months' service before they join the army." See _Handbook of the German Army_ (Nashville, TN: The Battery Press, 1996), 216–217. First published in 1940, this source indicates no degree of effectiveness of the NSRK.
30. Frieser, _The Blitzkrieg Legend_ , 23.
31. Ibid., 22.
32. Deist, _The Wehrmacht and German Rearmament_ , 108–109; Geyer, _Deutsche Rüstungspolitik_ , 154.
33. DiNardo, _Mechanized Juggernaut or Military Anachronism?_ , 16–17.
34. Guderian's reminiscence as quoted in Frieser, _The Blitzkrieg Legend_ , 33–34.
35. Richard Overy, "Total War II: The Second World War," in Charles Townshend, ed., _The Oxford Illustrated History of Modern War_ (Oxford: Oxford University Press, 1997), 121.
36. Frieser, _The Blitzkrieg Legend_ , 31–38.
37. Ibid., 33 and the useful diagram there. On reconnaissance detachments' TOEs see BA _Pferde im Einsatz bei Wehrmacht und Waffen-SS. Reiter Schwadron der Aufklärungs-Abteilung 157: Rast auf dem Vormarsch in Kapucany, östliche Slowakei 1939.www.bundesarchiv.de/aktuelles/aus_dem_archiv/galerie/oo172/index. html?index=0&id=3&nr=1. Handbook of the German Army 1940_, 43, 50. Overall, each infantry division's TOE carried approximately 1,743 saddle horses, 3,632 draft horses, and 1,133 horse-drawn vehicles. For these figures see, Johnson, _Horses of the German Army in World War II_ , 148.
38. On the "German Weekly Newsreel" see Robert Edwin Herzstein, _The War ThatHitler Won: The Most Infamous Propaganda Campaign in History_ (New York: G. P. Putnam's Sons, 1978), 223–258. See also _Signal: Years of Triumph 1940–42. Hitler's Wartime Picture Magazine_ , ed. S. L. Mayer (New York: Prentice-Hall, 1978).
39. Mayer, _Signal: Years of Triumph 1940–42_ , no page-numbers indicated.
40. See Kennedy, _The German Campaign in Poland (1939_ ), 11, 23–25 for divisional comparisons. See the brigade's table of organization for September 1939 in Fowler, _Axis Cavalry in World II_ , 7.
41. Kennedy, _The German Campaign in Poland (1939_ ), 61 and the chart, as well as the brigade's initial mission, at 74–75.
42. Ibid., 79–82 for dispositions of the Polish cavalry on Third Army's front.
43. Citino, _The German Way of War_ , 261; Kennedy, _The German Campaign in Poland (1939_ ), 81.
44. Kennedy, _The German Campaign in Poland_ (1939), 98.
45. On this tradition's continuity see Isabel V. Hull, _Absolute Destruction: Military Culture and the Practices of War in Imperial Germany_ (Ithaca: Cornell University Press, 2005), here 163.
46. F. W. von Mellenthin, _Panzer Battles: A Study of the Employment of Armor in the Second World War_ , trans. H. Betzler (Norman: University of Oklahoma Press, 1956), 3. Mellenthin served with the Seventh Cavalry Regiment from 1924 to 1935 and occupied the Intelligence billet in the German Third Corps during the invasion of Poland. He eventually rose to the position of chief of staff of Fourth Panzer Army and was captured by American troops on the Western Front in 1945. Hans von Luck, _Panzer Commander: The Memoirs of Colonel Hans von Luck_ , intro. Stephen A. Ambrose (New York: Dell, 1989), 29.
47. M. K. Dziewanowski, _War At Any Price: World War II in Europe, 1939–1945_ , 2nd ed. (Englewood Cliffs, NJ: Prentice Hall, 1991), 62, 68 and note 11 on the latter page. For a recent example of the story of the Polish cavalry's putative charge against the panzers, see Frieser, _The Blitzkrieg Legend_ , 19. Frieser cites Heinz Guderian's memoirs. On the encircled Polish cavalry, see Norman Davies, _No Simple Victory: World War II in Europe, 1939–1945_ (New York: Viking, 2006), 77–78.
48. See Citino, _The German Way of War_ , 364, note 57. Interestingly, this Polish arrangement would later be echoed in the SS Cavalry Division's retention of an assaultgun detachment on its own TOE.
49. "Germans Cut Off Polish Corridor," _New York Times_ , 5 September 1939, 12.
50. On the Polish 18th Lancers, see Citino, _The German Way of War_ , 259 and note 57 on 364.
51. Heinz Guderian, _Panzer Leader_ , foreword B. H. Liddell Hart, trans. Constantine Fitzgibbon (Costa Mesa, CA: Noontide Press, 1988), 71. Guderian reports (72) that the _Pomorske_ Brigade had attacked German tanks "with lances and swords" and "in ignorance of [the tanks'] nature." He does not, however, cite any source from any unit for this information in the English edition of the work. In many subsequent retellings of Polish cavalry charges against tanks, Guderian's own remarks are cited as the source of the story.
52. Ian Kershaw, _Fateful Choices: Ten Decisions That Changed the World 1940–1941_ (New York: Penguin, 2008), 64.
53. _Erfahrungsbericht der 1. Kavallerie-Division über den Einsatz in Holland und Frankreich, 13 August 1940_ , in NARA Microfilm Publication T-315, Roll 83, Frame 336. Coincidentally, Friesland was the Dutch portion of the ancestral northwestern European breeding area for the Friesian horse. The Friesian was also bred in the adjacent German region of Ostfriesen. Friesians were not unheard-of as light draft horses in the _Heer_ during the World War II.
54. Ibid.
55. Ibid., Frame 337. The division's records do not indicate whether the dive-bombers at Kornwerderzand were Junkers Ju-87 _Stukas_ or Henschel Hs-123 biplanes. The 88-mm gun detachment was not part of the cavalry division's establishment.
56. Ibid., Frame 338.
57. Ibid. The records consulted included no copies of communications from higher echelons criticizing the division's actions in the Netherlands.
58. Ibid., Frame 339.
59. Ibid.
60. Ibid.
61. Ibid., Frame 340. As the _Erfahrungsbericht_ makes no mention of a mounted charge against the heights, the final assault presumably occurred on foot. No German casualties are recorded. _Reiter_ (i.e., "rider" or "horse") regiments were those originally attached to the 1st Cavalry Brigade prior to the outbreak of war. _Kavallerie_ (i.e., "cavalry") regiments were ones normally assigned to provide the mounted reconnaissance squadrons to the army's infantry divisions as per the army's prewar reorganization. Though both types rode, and though both certainly thought of themselves as true cavalrymen, the term _Reiter_ carried a more traditional cachet and can be traced back at least as far as seventeenth-century Prussia. As noted in the previous chapter, the mounted reconnaissance elements of the army's infantry divisions merit a separate history and are not treated in the present work.
62. Ibid., Frame 341.
63. Ibid., Frames 341–342.
64. Ibid., Frame 342.
65. Ibid., Frames 342–343.
66. Ibid., Frame 343.
67. Ibid., Frame 344.
68. Ibid., Frame 345. The division's reports do not mention any resistance being offered by the French cavalrymen or whether any French horses were taken.
69. Ibid., Frame 346.
70. Ibid. The context of Feldt's emphasis in the original might reflect either defensiveness in the face of criticism or simply a cavalryman's enthusiasm. As above, the records consulted include no communications criticizing the division's performance.
71. Ibid.
72. Charles B. MacDonald, _The Last Offensive, United States Army in World War II-- The European Theater of Operations_ , ed. Maurice Matloff (Washington, D.C.: Center of Military History, 1973–1993), 22.
73. Ibid. For German nationalists on the Ardennes, in this case Ernst Moritz Arndt, see Winkler, _Der lange Weg nach Westen_ , I:64–65.
74. _NS Archiv, Dokumente zum Nationalsozialismus: Tagebuch Generaloberst Ritter von Leeb_ , 3 October 1939. <http://www.ns-archiv.de/krieg/1939/leeb/> (12 May 2008).
75. Robert A. Doughty, "French Operational Art 1888–1940," in Krause and Phillips, _Historical Perspectives on the Operational Art_ , 95, 100–101. See the table of organization of the French 5th Light Cavalry Division in Frieser, _The Blitzkrieg Legend_ , 128.
76. Quoted in Alistair Horne, _To Lose a Battle: France: 1940_ (New York: Penguin, 1969), 270–271.
77. Ibid., 271.
78. Ibid., 294–295.
79. Frieser, _The Blitzkrieg Legend_ , 133–136, here specifically 136.
Chapter 7: Barbarossa
1. A very useful general examination of Operation Barbarossa is to be found in Earl F. Ziemke and Magna E. Bauer, _Moscow to Stalingrad: Decision in the East_ (Washington, D.C.: Center of Military History, 1987).
2. Blackbourn, _The Conquest of Nature: Water, Landscape, and the Making of Modern Germany_ , 251. Renderings of "Pripet" are many and varied, "Pripyat" being also commonly used in English.
3. Ibid. See also _Terrain Factors in the Russian Campaign_ , facsimile ed. (Washington, D.C.: Center of Military History, 1982, 1986), 4–5, 28–43; reference to numbers of bridges at 29.
4. National Defense University, "Military Geography for Professionals and the Public: 6. Regional Peculiarities." http://ndu.edu/.../milgeoch6.html (29 October 2008).
5. Ibid.
6. _Terrain Factors in the Russian Campaign_ , 9, 29.
7. Ziemke and Bauer, _Moscow to Stalingrad_ , 14.
8. _Rear Area Security in Russia: The Soviet Second Front Behind German Lines_ , "Chapter 5: The Front Behind the Front." Department of the Army Pamphlet 20–240 (Washington, D.C., 1951). <http://www.history.army.mil/books/wwii/20240/20-2403.html> (4 November 2008).
9. Glantz and House, _When Titans Clashed_ , 66. On the general state of the Red Army in 1941, see Ziemke and Bauer, _Moscow to Stalingrad_ , 7–13.
10. Ibid. Stalin could and did allow his soldiers' lives to be thrown away even more recklessly than Hitler. In a latter-day Charge of the Light Brigade, the 44th Mongolian Cavalry Division attacked the German 106th Infantry Division across an open, snow-covered field near Klin on 16 November 1941 during the Soviet counteroffensive against German forces threatening Moscow. Two thousand cavalrymen and their horses were killed. The Germans suffered no recorded casualties. See ibid., 83.
11. From German translation, dated 15 August 1941, of Timoshenko's _Order to the Troops of the West Front No. 0109_ in NARA Microfilm Publication T-315, Roll 78, Frame 47.
12. Glantz and House, _When Titans Clashed_ , 56.
13. Ibid.
14. Ziemke and Bauer, _Moscow to Stalingrad_ , 3.
15. _Tätigkeitsbericht_ 22–29 June 1941 in NARA Microfilm Publication T-315, Roll 82, Frames 1397–1399. See also the report dated 27 June 1941 at Frame 36. The SS Division _Das Reich_ and the Army's 10th ID (Motorized) were operating in the same area and further added to the traffic congestion on the _Panzer-Rollbahn_. Reference to marching infantry's difficulties in _Terrain Factors in the Russian Campaign_ , 29.
16. _Kurzer Bericht über Aufträge der XXIV.Pz.Korps an 1.K.D. u. ihre Durchführung in der Zeit vom 22.6.-30.6.1941_. NARA Microfilm Publication T-315, Roll 82, Frames 80–87. Subsequent reports hereafter labeled simply _Kurzer Bericht_ or _Bericht_ , as appropriate, with dates and Frame numbers.
17. Ibid.
18. Ibid. Losses among the division's horses were not indicated.
19. Undated [June 1941?] veterinary circular. NARA Microfilm Publication T-315, Roll 82, Frames 1256–1257.
20. _Erfassungskommandos_. 5 July 1941. NARA Microfilm Publication T-315, Roll 82, Frame 1266.
21. Ziemke and Bauer, _Moscow to Stalingrad_ , 14; _Terrain Factors in the Russian Campaign_ , 6–7.
22. _Kurzer Bericht_ , 1.–31.7.1941. NARA Microfilm Publication T-315, Roll 82, Frames 185–197. Bobruisk lies on the northeastern edge of the Pripet Marshes. The surrounding forests in 1941 were thus frequently boggy and subject to localized flooding.
23. Ibid. The cavalrymen readily acknowledged the close-air support (CAS) provided—without being ordered—by the 51st Fighter Wing whose airfield the horsemen had earlier guarded. The troopers noted that such CAS was very useful in suppressing Soviet artillery. Their crossing of the Dnepr was slightly in advance of the one made by XLIII and LIII Infantry Corps farther downstream near Rogachev. On that crossing, see _Terrain Factors in the Russian Campaign_ , 21–27. On 2nd Panzer Group at the Dnepr, see Glantz and House, _When Titans Clashed_ , 58–59.
24. _Kurzer Bericht_ , 1.–31.7.1941. NARA Microfilm Publication T-315, Roll 82, Frames 196–197.
25. _Bericht_ , 1.8.-2.9.1941. NARA Microfilm Publication T-315, Roll 82, Frames 324–343. All details of divisional horses' status on 10 August 1941 taken from Frame 340. Curiously, the _panje_ horses are referred to as having been "hired" ( _ermieteten_ ) before the campaign started. The record does not make clear whether the _panje_ horses in question were from occupied Poland or, much more likely, ones acquired between the operations in the Pripet Marshes and crossing of the Dnepr.
26. Ibid., Frame 324.
27. Ibid.
28. Ibid., Frame 325.
29. Ibid., Frame 327. See note 23 above. The so-called Gomel Pocket had been formed by the combined pressure of XLIII and LIII Infantry Corps attacking eastward across the Dnepr at Rogachev while XIII Corps, now the 1st Cavalry Division's parent formation, drove south and east from Mogilev. Thus Soviet forces were caught between the river, Rogachev, and Gomel.
30. Ibid., Frame 328.
31. Ibid., Frames 329–331. Just as the German cavalrymen were executing this striking advance, so Soviet cavalry were doing the same at about the same time but in the opposite direction. The very Twenty-First Army whose staff the German troopers almost bagged near Gomel had earlier ordered a deep cavalry raid northwesterly across the Denpr in an attempt to disrupt German forces around Bobruisk. See Glantz and House, _When Titans Clashed_ , 59.
32. Ibid., Frame 331.
33. Ibid., Frame 332.
34. Ibid., Frame 334.
35. Ibid., Frames 334–337. The division's reassignment to 2nd Panzer Group was its seventh in total since 31 July.
36. Ibid., Frame 337.
37. All of the following after-action figures and assessments of the division's personnel, horses, equipment, status, and so on, from ibid., Frames 338–343. The report from which this material comes had what appears to have been an unusually large distribution list including, among others: Army High Command (OKH); the OKH's operations staff; Army Group Center; Fourth Army; 2nd Panzer Group; and the School for Rapid Deployment Troops ( _Schule der schnellen Truppen_ ) at Krampnitz. " _Schnell_ " in this case refers to fast-moving, mobile troops, usually motorized or mechanized. Cavalry had been administratively regarded as _schnelle Truppen_. "Winning themselves to death" in Glantz and House, _When Titans Clashed_ , 60. The panzer division in question here was same 18th Panzer that the Cavalry Division had earlier relieved.
38. Feldt wrote specifically of the "East Prussian horse" ( _Ostpreuße_ ) in the singular, but clearly seems to have been referring to all of the division's horses bred there. The division would naturally have drawn most of its mounts and draft horses from the province and _Wehrkreis_ in which it was based.
39. All march-order details dated 13 September 1941 and taken from _Anordnungen für das Überholen und Kreuzen von mot.—und berittenen—Verbänden!_ NARA Microfilm Publication T-315, Roll 78, Frames 4–5. On the larger situation regarding horses' condition on the Eastern Front and efforts to spare them, see the Fourth Army Order of 18 September 1941 in Appendix A.
40. _Tagesbefehl_. 13 September 1941. NARA Microfilm Publication T-315, Roll 78, Frame 13.
41. Untitled report of the divisional IIa (Adjutant) dated 15 September 1941 citing Felber's commendation of 26 August. NARA Microfilm Publication T-315, Roll 78, Frame 29. See Frame 45 for the following quotations regarding partisans and the report dated 18 September 1941.
42. See NARA Microfilm Publication T-315, Roll 78, various Frames from 76 to 171 unless otherwise specified. For general situation at the Desna and in Bryansk's environs, see _Terrain Factors in the Russian Campaign_ , 18–20.
43. NARA Microfilm Publication T-315, Roll 78, Frame 171.
44. Ibid.
45. Ibid, Frames 61, 147–148.
46. Unless otherwise indicated, all of the following material regarding the Cavalry Division's capabilities and organizational fate are taken from _Umwandlung der 1.K.D. zu einer Panzer-Division_ in NARA Microfilm Publication T-315, Roll 82, Frames 344–350.
47. In Directive 32, "Preparations for the Period After Barbarossa," issued 11 June 1941, Hitler had indicated that some sixty divisions would remain for security duty in what would have been former Soviet territory. That sort of static service would likely not have suited the temperament of the Cavalry Division's soldiers or their ethos. Directive cited in Ziemke and Bauer, _Moscow to Stalingrad_ , 15.
48. One has to be careful to avoid reading a feeling of petulance into Feldt's comments at this juncture in the documents.
49. Feldt's advocacy of equestrian sport as effective military training for officers recalls quite similar sentiments regarding polo as expressed at various times before 1939 in the U.S. Army's _Cavalry Journal_.
50. Unless otherwise noted, references to divisional operations and prevailing conditions for the period are taken from _Kurzer Bericht 31.8.-21.10.1941_ , NARA Microfilm Publication T-315, Roll 82, Frames 596–606.
51. Glantz and House, _When Titans Clashed_ , 80, indicate that the weather not only deprived German forces of their vaunted mobility but also forced mechanized divisions (and, by extension, any other motorized or mechanized elements, as in the 1st Cavalry Division) to use up their fuel at three times the planned rate on average.
52. _Tätigkeitsbericht 8.11.1941–29.4.1942_. NARA Microfilm Publication T-315, Roll 83, Frame 217.
53. Ibid., Frames 218–219.
54. Ibid., Frames 237, 240.
55. Ibid., Frames 239–255. Strikingly, even after the division's reorganization as 24th Panzer Division, daily activity reports still regularly indicated nearly 150 horses on the division's ration-strength. These were presumably draft horses.
Chapter 8: Hell's Outriders
1. Ludwig Hollweck, ed., _Unser München: München im 20. Jahrhundert. Erinnerungen und berichte, Bilder und Dokumente von 1900 bis heute_ (München: Süddeutscher Verlag, 1967), 256. For illustrations of the parade's horses and riders see any of the numerous extant video clips on-line. On the Nazi aesthetic, see Joan L. Clinefelter, _Artists for the Reich: Culture and Race from Weimar to Nazi Germany_ (New York: Berg, 2005).
2. Frederic Spotts, _Hitler and the Power of Aesthetics_ (New York: The Overlook Press, 2003), 173, 184, 364.
3. John Keegan, _Waffen-SS: The Asphalt Soldiers_ (New York: Ballantine, 1971), 8.
4. Ibid.
5. On the bonding power of atrocity when collectively committed, see Dave Grossman, _On Killing: The Psychological Cost of Learning to Kill in War and Society_ (New York and Boston: Back Bay Books, 1996), 195–227.
6. On the evolution of the mounted SS, the _Waffen_ -SS, and the honorific _Florian Geyer_ , see below.
7. Charles W. Sydnor, _Soldiers of Destruction: The SS Death's Head Division, 1933–1945_ (Princeton: Princeton University Press, 1977), 43–45 and note 36, 69.
8. Keegan, _Waffen_ -SS, 62.
9. Heinz Höhne, _The Order of the Death's Head: The Story of Hitler's SS_ (New York: Ballantine, 1971), 152–153.
10. Richard Evans, _The Third Reich in Power, 1933–1939_ (New York: Penguin, 2005), 418–419.
11. Höhne, _The Order of the Death's Head_ , 156.
12. _BA Pferde im Einsatz bei Wehrmacht und Waffen-SS. Beiblatt zur Pferde-Einberufung für Gemeinden 1937_ and _SS-Totenkopf Kavallerieregiment 1: Ergänzungs-Einheiten 8.12.1940_ www.bundesarchiv.de/aktuelles/aus_dem_archiv/galerie/00172/index.html?index=0&id=2&nr=1 (3 and 8 October 2008). On the general history of the _Waffen_ -SS see Höhne, _The Order of the Death's Head_ , 493–545, here 495, 513ff. On Fegelein's service see BA _Pferde im Einsatz bei Wehrmacht und Waffen-SS. Beiblatt zur Pferde-Einberufung für Gemeinden 1937_ (2 and 8 October 2008). Different sources provide different dates of the establishment of the _SS-Totenkopf-Reiterstandarte_. Its activation was in any case in the autumn of 1939. For Hitler's statement of the missions of the SS-VT and SS-TV, see Bernd Wegner, "SS," in Dear and Foot, _The Oxford Companion to World War II_ , 1048.
13. See BA link, note 12 above, for Fegelein's service. The more "proletarian" Brown Shirts also had a mounted arm, the _Reiter-SA_. Before his murder on Hitler's orders in 1934, the SA's commander, Ernst Röhm, was frequently to be seen mounted when reviewing Storm Troopers on parade.
14. Evans, _The Third Reich in Power_ , 3; Höhne, _The Order of the Death's Head_ , 156.
15. _Rede des Reichsführers 26 Juli 1944_ , NARA Microfilm Publication T-354, Roll 116, Frames 3750099ff.
16. Ibid.
17. John Keegan, _Waffen_ -SS, 73.
18. Ibid.
19. Curzio Malaparte, _The Volga Rises in Europe_ , trans. David Moore (Edinburgh: Birlinn Limited, 1951), 27.
20. Timothy Snyder, _Bloodlands: Europe Between Hitler and Stalin_ (New York: Basic Books, 2010).
21. Heinrich August Winkler, _Der lange Weg nach Westen_ , I:246, 253–254, 336–343.
22. Malaparte, _The Volga Rises in Europe_ , 78.
23. John M. Steiner, "Über das Glaubensbekenntnis der SS," in Karl Dietrich Bracher, Manfred Funke, Has-Adolf Jacobsen, eds., _Nationalsozialistische Diktatur 1933–1945_ (Bonn: Bundeszentrale für politische Bildung, 1986), 208, 217. Darré's comment in Hermann Rauschning, _Hitler Speaks_ (London: Thornton Butterworth, 1940), 40. On Hitler's and the Nazis' long-standing ambitions in the East, see Kershaw, _Fateful Choices_ , 55–59. Snyder, _Bloodlands_ , xiv, notes a figure of some 200,000 Poles shot by German and Russian occupiers between 1939 and 1941. The figure of 900,000 cited in the text is taken from Nicholas Stargardt, "Hitler in the Driving Seat," _Times Literary Supplement_ , 8 October 2008 at <http://entertainment.timesonline.co.uk/tol/arts_and_entertainment/the_tls> (13 October 2008).
24. Blackbourn, _The Conquest of Nature_ , 295.
25. Ibid.
26. Mark C. Yerger, _Riding East: The SS Cavalry Brigade in Poland and Russia 1939–1942_ (Atglen, PA: Schiffer, 1996), 73ff. Yerger's book provides an interesting examination, usually in tabular fashion, of the early evolution of the SS' mounted formations. The text itself, however, should be used with a certain caution in that it does not always contain the requisite citations of primary sources. Given that caveat, Yerger's work is helpful for the period prior to the actual formation of the SS Cavalry Division in early 1942.
27. Ibid., 88, 91, 94. Yerger refers to "operational strength." His numbers and their context, however, support the view that he is referring to combat-strength only and not ration-strength.
28. Raul Hilberg, Review of _Wegbereiter der Shoah. Die Waffen-SS, der Kommandostab Reichsführer-SS und die Judenvernichtung_ , by Martin Cüppers. <http://www.uni-stuttgart.de/hing/lb/lbrez_hilberg.pdf> (29 October 2008).
29. Ibid.
30. Heinrich August Winkler, _Der lange Weg nach Westen_ , II:88. Winkler, II:88–89, shows clearly the confused lines of authority among high-ranking Nazi officials in the administration of the occupied territories. He also shows, however, the unanimity of purpose among these same officials regarding the extermination of the Jews.
31. Charles W. Sydnor, "On the Historiography of the SS," Simon Wiesenthal Museum of Tolerance Multimedia Learning Center Online. http://motlc.wisenthal.com/site/pp.asp (29 October 2008).
32. Yerger, _Riding East_ , 44, 47, 53.
33. Ibid., 98–100.
34. Ibid. Hilberg, Review of _Wegbereiter der Shoah_ , note 27 above, points out the German verb _abzutransportieren_ ("transport away") in contrast to Yerger's somewhat more benign "evacuate" in discussing the fate of captured women and children.
35. Blackbourn, _The Conquest of Nature_ , 277. See also Hilberg, note 28 above. Winkler, _Der lange Weg nach Westen_ , II:88, notes one of Himmler's meetings on 14–15 August in Minsk where he also referred explicitly to an order directly from Hitler for the shooting of all Jews ("einen 'Führerbefehl über die Erschießung aller Juden'"). On the evolution of Hitler's decision-making process culminating in the "Final Solution," see the excellent treatment in Kershaw, _Fateful Choices_ , 431–470.
36. Ibid. Hilberg, note 28 above, attributes the observation about the marshes' depth to Major ( _Sturmbannführer_ ) Franz Magill, the commander of the 2nd _Reiter_ Regiment's horse-mounted battalion. It is entirely conceivable that Magill's own comment could have found its way up the line via brigade commander Fegelein's report to the _Kommandostab Reichsführer_ -SS.
37. Yerger, _Riding East_ , 91.
38. Ibid., 100–104.
39. Ibid., 101. The HSSPFs in the occupied territories of Eastern Europe often called up policemen in battalion-strength from both the Order Police ( _Ordnungspolizei;_ ORPO) and the Security Police ( _Sicherheitspolizei;_ SIPO). Himmler's authority over both the SS and all German police gave him unfettered access to huge manpower reserves. His place in the Nazi hierarchy close to Hitler also ensured a steady flow of supplies and equipment to SS units, particularly after his appointment in 1944 as Commander of the Replacement Army in the wake of the attempted assassination of Hitler. Hilberg, note 28 above, indicates that police personnel and sometimes local militia were also involved in mass shootings. In his estimation, therefore, it sometimes becomes difficult to determine precisely which members of which units were carrying out extra-judicial killings.
40. Ibid., 104.
41. Ibid., 105.
42. Andrew Nagorski, _The Greatest Battle: Stalin, Hitler, and the Desperate Struggle for Moscow That Changed the Course of World War II_ (New York: Simon & Schuster, 2007), 71. According to Nagorski, 73, as many as 667,364 soldiers of the Red Army had been rounded up by 10 October 1941 for "escaping from the front," 10,201 were shot, and 25,878 were kept under arrest; 632,486 were formed into new units, many of which were penal battalions used for tasks such as clearing mine fields by being marched through them.
43. Yerger, _Riding East_ , 107, 122.
44. Ibid., 126.
45. Ibid.
46. Ibid., 127.
47. Ibid., 126.
48. Ibid., 129.
49. Nagorski, _The Greatest Battle_ , 107–109; Peter Young, ed., _Atlas of the Second World War_ (New York: Paragon, 1979), 86–87. See Yerger, _Riding East_ , 130–131, for illustrations of the barges and the occasionally ramshackle rafts used by the SS horsemen to ferry themselves and their mounts across the Dnepr. Retreating Russian forces had blown up the bridges spanning the river near Gomel.
50. Yerger, _Riding East_ , 130–131.
51. See the map in Young, _Atlas of the Second World War_ , 91.
52. Yerger, _Riding East_ , 133. On Rzhev and its significance, see Nagorski, _The Greatest Battle_ , 130, 134.
53. Yerger, _Riding East_. 134, and footnote 160 on that page.
54. Ibid., 132–133. Equine strength indicated on 141. See there also for the fact that, in the first week of November, the brigade lost another 3 horses dead while 102 were treated for sickness or wounds. Nagorski, _The Greatest Battle_ , 130, notes that the first snows fell on the night of 6–7 October and, on 228, indicates the estimated mortality for draft horses at 1,000 per day. On _rasputiza_ , see Glantz and House, _When Titans Clashed_ , 80.
55. "Special Duties for the SS in 'Operation Barbarossa,' March 13, 1941." Documents of the Holocaust Part III. <http://www.yadvashem.org/about_holocaust/documents/part3/doc169.html> See Guderian's comments in Nagorski, _The Greatest Battle_ , 130.
56. Yerger, _Riding East_ , 128.
57. Ibid., 139–140.
58. Ibid., 142.
59. Nagorski, _The Greatest Battle_ , 255.
60. Ibid., 254.
61. Yerger, _Riding East_ , 150; Young, _Atlas of the Second World War_ , 92–93.
62. Yerger, _Riding East_ , 150, 157.
63. Glantz and House, _When Titans Clashed_ , 89–90; Yerger, _Riding East_ , 160.
64. Yerger, _Riding East_ , 163–170.
65. Ibid., 171–172.
66. Ibid., 183.
67. Ibid., 193, 197. _Volksdeutsche_ were recruited by the SS in areas of occupied Europe outside of Germany as defined by the country's borders of late 1938. Those borders encompassed both pre-1938 Austria and the Sudetenland. Consequently, _Volksdeutsche_ were recruited in Hungary and elsewhere.
68. Ibid., 199.
69. Ibid., 199–201.
70. Ibid., 201. On battle-group ( _Kampfgruppe_ ) "Zehender," see Yerger, 201–206.
71. Young, _Atlas of the Second World War_ , 92.
72. Nagorski, _The Greatest Battle_ , 287, 290–291. Farther to the northwest, the winter battles had also isolated 95,000 German troops and 20,000 horses in a pocket around the city of Demyansk. These forces included the SS Cavalry Brigade's sister unit, the SS Death's Head Division, made up largely of former concentration-camp guards. See Sydnor, _Soldiers of Destruction_ , 190–191.
Chapter 9: Pale Horsemen
1. _Vorgeschichte der SS-Kavallerie-Division_ , NARA Microfilm Publication T-354, Roll 640, Frame 6. Hereafter _Vorgeschichte_ with Frame number. The 3rd _Reiter_ Regiment appears to have been largely absent from the division's operations throughout 1943. It would be used as cadre for a second, nominal SS cavalry division raised in 1944.
2. Yerger, _Riding East_ , 208, shows the division's Order of Battle for September 1942.
3. _Vorgeschichte_ , Frame 7.
4. _Tätigkeitsbericht zum Kriegstagebuch_ , in NARA Microfilm Publication T-354, Roll 641, Frames 958–959, details instructional courses provided in the field to veterinary personnel.
5. "Special Instructions for the Area of Responsibility ( _Arbeitsgebiet_ ) of Detachment VI," 22 April 1943, NARA Microfilm Publication T-354, Roll 642, Frames 913–920. _Abteilung VI_ was the staff designation for the National Socialist Leadership Officer, essentially an indoctrination officer. This billet was set up only in March 1943.
6. Ibid.
7. _Vorgeschichte_ , Frame 7. On the SS Cavalry School at Zamosc, see Yerger, _Riding East_ , 207.
8. _Vorgeschichte_ , Frame 7. Operations staff reports of 19 September 1942 in NARA Microfilm Publication T-354, Roll 641, Frame 6 under _Besprechungspunkte beim Chef des Stabes LIX Armee-Korps_.
9. NARA Microfilm Publication T-354, Roll 640, Frames 8–10.
10. " _Ausstellung der Pferdeausfälle seit Einsatzbeginn bis 30.9.42_ ," NARA Microfilm Publication T-354, Roll 641, Frame 950. Horse transfers indicated by divisional veterinary staff in _Tätigkeitsbericht zum Kriegstagebuch_ of 14 October and 1 December 1942 in ibid., Frames 952 and 958. Of the latter shipment, 183 horses had to be treated in gas tents ( _Gaszellen_ ) for scabies-induced mange ( _Räude_ ).
11. Ibid., Frame 951.
12. NARA Microfilm Publication T-354, Roll 640, Frame 16.
13. Ibid., Frame 64. VI Corps orders of 16 November 1942 in NARA Microfilm Publication T-354, Roll 641, Frame 466: _Generalkommando VI. A.K. Freimachung des Gefechtsgebietes_. The orders from VI Corps refer to a _K.L. [Kampflinie;_ battle line].
14. NARA Microfilm Publication T-354, Roll 640, Frame 40.
15. _Kriegstagebuch. Die Quartiermeisterabteilung/Ib 11.9.1942 bis 20.12.1942_ , NARA Microfilm Publication T-354, Roll 641, Frames 856–858.
16. Ibid., Frame 860.
17. _Tätigkeitsbericht zum Kriegstagebuch_ , 14 October 1942 in ibid., Frame 951.
18. _Tätigkeitsbericht_ , 15 November 1942 in ibid., Frame 957. See also ibid., Frame 958, _Tätigkeitsbericht_ of 1 December 1942, for number of _panje_ horses requested from VI Corps. Regimental personnel transfers in _Aufstellung von winterbeweglichen Verbänden_ in ibid., Frames 467–469. Interestingly, another major combat arm, the _Gebirgsjäger_ , was also requisitioning large numbers of _panje_ horses at about the same time for operations in the Caucasus Mountains. The Alpine Army Corps ( _Gebirgs-Armeekorps_ ) had requisitioned some five-thousand _panje_ horses in August 1942. These were in addition to the alpine troops' already assigned mules and Haflingers. See _Österreichs Bundesheer. Tragtiere im Einsatz;_ www.bmlv.gv.at/truppendienst/ausgaben/artikel.php?id=375 (17 September 2008).
19. _Vermeidung von Kälteschäden_ , in _Tätgkeitsbericht_ , 15 November 1942, Frame 479.
20. _Gefechts- und Verpflegungsstärke SS-Kav. Division am 21. November 1942_ in ibid., Frame 478.
21. _Zustandsmeldungen_ to the Operations Officer of XXX Army Corps of 19 and 26 December 1942 in NARA Microfilm Publication T-354, Roll 642, Frames 422–423 and 430–431. Illustration of divisional emblem in ibid., Frame 428. A directive of 27 December 1942 noted that the divisional command would be requesting from _Reichsführer_ -SS Himmler the awarding of a cuff band for the men's uniforms similar to those already worn by other _Waffen_ -SS divisions. The cuff-title _Florian Geyer_ would appear in 1943.
22. _Kriegstagebuch Nr. 2 SS-Kavallerie-Division_ (hereafter _KTB_ ) of 22–29 December 1942, NARA Microfilm Publication T-354, Roll 642, Frames 10–20. _Terrain Factors in the Russian Campaign_ , 42–43, notes that literally hundreds of thousands of German troops were tied down in anti-partisan operations, and that the partisans in some cases (as in the vicinity of Cholm) were not only able to conduct formal recruiting drives in their respective safe havens but also even live-fire artillery training exercises.
23. "Mit einem eigenmächtigen Umlegen solcher Gefangener ist der Truppe nicht gedient." _Besondere Weisungen an die Truppe_ , 24 December 1942, NARA Microfilm Publication T-354, Roll 642, Frame 426.
24. _Gefangenen und Beute aus Unternehmen "Sternlauf_ ," in daily activity reports to KTB in NARA Microfilm Publication T-354, Roll 642, Frame 758.
25. _Tagesbefehl an die deutschen Soldaten der Wehrmacht_ , 1 January 1943, NARA Microfilm Publication T-354, Roll 644, Frame 538.
26. KTB, 15–17 January 1943, NARA Microfilm Publication T-354, Roll 642, Frames 54–59.
27. Ibid., Frames 54, 60–61.
28. Ibid., Frames 61–62, 73.
29. _Zur Schonung der Pferde, Kraft- und Pferdefahrzeuge_ , 30 December 1942, NARA Microfilm Publication T-354, Roll 644, Frame 540.
30. _Verlustliste SS-Kav. Division (vom 21.12.1942–20.1.1943_ ). This list actually records figures to 15 April 1943. NARA Microfilm Publication T-354, Roll 642, Frames 814–817.
31. _Divisionsbefehl zum Ordnen der Verbände und Umgliederung_ , 18 March 1943, NARA Microfilm Publication T-354, Roll 642, Frames 639–656.
32. _Gefechts- und Verpflegungsstärke der SS-Kav. Division_ , 1 April 1943, NARA Microfilm Publication T-354, Roll 642, Frames 818–819.
33. Michael R. Marrus, _The Nuremberg War Crimes Trial 1945–56_ (New York: Bedford/St. Martin's, 1997), 165.
34. Heinz-Dietrich Löwe, "Resistance," in Dear and Foot, _Oxford Companion to World War II_ , 1241.
35. _Terrain Factors in the Russian Campaign_ , 42–43.
36. All of the following references come from the _Divisionsbefehl für die Vorbereitung des Einsatzes zur Bandenbekämpfung_ , 18 April 1943, NARA Microfilm Publication T-354, Roll 642, Frame 903.
37. _Tätigkeitsberichte_ and divisional orders of 16–17 April 1943 and 17 April 1943, respectively. NARA Microfilm Publication T-354, Roll 642, Frames 900–903.
38. _Stabsbefehl_ of 17 April 1943, NARA Microfilm Publication T-354, Roll 642, Frame 898.
39. See Fegelein's Order of the Day of 14 May 1943 in Appendix C regarding the division's mounted traditions.
40. _Zustandsbericht_ , 26 April 1943, NARA Microfilm Publication T-354, Roll 642, Frames 942–943.
41. Ibid. The 3rd Regiment was still not yet fully organized. Furthermore, its personnel would also subsequently be drawn upon for cadre for other SS units. On the veterinary threats to the division's horses see letter to the author of 20 November 2008 from Ms. Francine Ann Luker. At the time of writing, Luker served as Supervisor of Equine Management at Balsam Mountain Preserve in Jackson County, North Carolina and has more than forty years' experience in equine care.
42. _2. Divisionsbefehl zur Vorbereitung des Einsatzes zur Bandenbekämpfung_ , 24 April 1943, NARA Microfilm Publication T-354, Roll 642, Frame 921. On treating mangy horses, see veterinary staff directives as indicated in note 10 above.
43. Ibid., Frames 922, 924–925.
44. _Divisionsbefehl_ , 25 April 1943, NARA Microfilm Publication T-354, Roll 642, Frame 1093.
45. _Divisions-Befehl_ [ _sic_ ] _für das Unternehmen "Weichsel I_ ," 11 May 1943, NARA Microfilm Publication T-354, Roll 642, Frames 996–999. Two _Einsatzgruppen_ from the _Sicherheitsdienst_ (SD) were also later attached for these operations. Units of the SD and the Security Police (SIPO) had routinely been assigned to shoot Jews ever since the beginning of Barbarossa. The SS Cavalry Division's records do not always indicate numbers of persons shot by the attached _Einsatzgruppen_ or provide complete details of their operations. On the _Einsatzgruppen_ , see Helmut Krausnick and Hans-Heinrich Wilhelm, _Die Truppe des Weltanschauungskrieges. Die Einsatzgruppen der Sicherheitspolizei und des SD 1938–1942_ (Stuttgart: n.p.,1981).
46. See various after-action reports, divisional war-diary entries, etc., for the period in NARA Microfilm Publication T-354, Roll 642, Frames 1096–1188 and specifically the operational summary dated 10 June 1943 at Frame 1188.
47. _Verlustmeldung_ of 10 June 1943 in NARA Microfilm Publication T-354, Roll 642, Frame 1191.
48. _Gefechtsbericht über den Einsatz der SS-Kav-Division im Unternehmen "Seydlitz" vom 25.6. bis 27.7.1943_ , NARA Microfilm Publication T-354, Roll 643, Frames 27–29. For persons and material taken during Weichsel I and II, see Appendix B. The Operation Seydlitz noted here should not be confused with a similarly named operation undertaken by regular army units southwest of Rzhev in July 1942. See _Military Improvisations During the Russian Campaign_ , facsimile ed. (Washington, D.C.: Center of Military History, 1983/1986), 11–17, 107.
49. _Verlustliste der SS-Kavallerie-Division für die Zeit vom 16.4. bis 18.8.1943_ , NARA Microfilm Publication T-354, Roll 643, Frames 282–284.
50. _Tätigkeitsbericht vom 16.4. bis zum 18.8.1943_ , NARA Microfilm Publication T-354, Roll 643, Frame 281. The document was dated 3 September 1943.
51. The division's records for the period from August to December 1943 are curious. While abundant, they lack the degree of completeness shown for similar records covering earlier periods. This is particularly the case regarding after-action reports.
52. See Wenger, "SS," in Dear and Foot, _The Oxford Companion to World War II_ , 1046–1049, and the chart there of all of the _Waffen_ -SS divisions, including units' national composition. The original 3rd _Reiter_ Regiment of the _Florian Geyer_ Division was actually used as cadre for a new SS cavalry division, the 22nd SS Volunteer Cavalry Division _Maria Theresa_. It was ultimately trapped and annihilated in Budapest along with most of the _Florian Geyer_ Division in early 1945.
53. The phrase is Omer Bartov's. See his _Hitler's Army: Soldiers, Nazis, and War in the Third Reich_ (New York and Oxford: Oxford University Press, 1992). On the chivalric presumptions, overlaid with Nazi fervor, of the SS Cavalry's wartime leaders see Appendix C.
Chapter 10: Last Recall
1. All details of the brigade's composition and operations from _Military Improvisations During the Russian Campaign_ , 11–17.
2. Ibid. The brigade's losses in men and horses are not recorded. However, Russian losses included some 50,000 prisoners (approximately 83% of Soviet troops engaged), 230 tanks, 760 artillery pieces, and thousands of small arms. See ibid., 16.
3. Richter, _Weapons and Equipment of the German Cavalry 1935–1945_ , 40. Other sources date the unit's establishment to 12 February 1943.
4. Böselager was killed in action in 1944. During the Cold War, the German _Bundeswehr_ instituted The Boeselager [ _sic_ ] Cup, an annual, later semiannual, competition among NATO armored reconnaissance units. See Lionel Ortiz and Brian Butcher, "Winners! U.S. Cavalry Squadrons Win and Place in Grueling NATO Reconnaissance Competition," _Armor_ (November-December 1987), 21–24. <http://www.benning.army.mil/armor/armormagazine/content/Issues/1987/ArmorNovemberDecember1987web.pdf>
5. Richter, _Weapons and Equipment of the German Cavalry 1935–1945_ , 40–42.
6. _Kriegstagebuch Nr. 1, Kav. Korps_. NARA Microfilm Publication T-314, Roll 25, Frame 7. Records of German Field Commands: Corps.
7. Ibid., Frame 9. The 3rd Cavalry Brigade would not officially be assigned to the Corps until August 1944.
8. By late 1943, the Operations Staff of OKW had drawn up plans for Operation Margarethe calling for the occupation of Hungary by German forces in the event that the country tried to desert the Axis. See Earl F. Ziemke, _Stalingrad to Berlin: The German Defeat in the East_ (Washington, D.C.: Office of the Chief of Military History, 1968), 208, 287–288.
9. _Kriegstagebuch Nr. 1, Kav. Korps_. NARA Microfilm Publication T-314, Roll 25, Frame 9. Records of German Field Commands: Corps.
10. Ibid., Frames 10–11.
11. Ibid., Frames 11–17. On Pliev's forces, see David M. Glantz, _When Titans Clashed_ , 190–191, 208. Pliev's combined cavalry-mechanized force would once more be in the 1st Cavalry Corps' area of operations during the Germans' failed attempt to relieve Budapest in January 1945.
12. _Kriegstagebuch_ as in note 4 above, Frames 41–42.
13. Ibid., Frame 47.
14. Letter of 4 July 1944 in _Anlagen_ [ _sic_ ] _A zum Kriegstagebuch Nr. 2 Gen.-Kdo. Kav.-Korps_ in NARA Microfilm Publication T-314, Roll 25, Frames 360–361.
15. Ibid., Frame 363.
16. Ibid., Frame 362.
17. Ibid., Frames 364–376, 415.
18. Ibid., Frames 423–424. Reference to 3rd Hussar Regiment at Frame 426.
19. Ibid., Frames 455, 461–462. Vattay's desire for his troops' withdrawal may well have been influenced not only by the more or less effective implementation of Operation Margarethe (see note 8 above) but also by the impending Soviet invasion of his country. On Margarethe's activation, see Ziemke, _Stalingrad to Berlin_ , 287–288.
20. Ibid., Frames 508, 598, 600. The Panzer IV, a remarkably versatile prewar design, remained the workhorse of the panzer arm well into 1944.
21. Ibid., Frame 602.
22. Ibid., Frame 658.
23. Ibid., Frames 660–661, 688, 718. _Beurteilung der Lage am 13.7.44_. Orders for mobile defense at Frame 688.
24. Ibid., Frame 893, 920, 937, 948–949.
25. Ibid., Frame 954. Table of combat-strength for the Cavalry Brigade and the rest of the Corps at Frame 944.
26. _Anlagen zum Kriegstagebuch, Kav Korps, Ia. 1 Aug 1944–30 Sep 1944_. NARA Microfilm Publication T-314, Roll 25, Frames 986–988.
27. Ibid., Frame 1023.
28. Ibid., Frame 992.
29. Ibid. and Frames 996, 1001, 1011, 1020.
30. All of Harteneck's comments from his memo of 5 August 1944 in ibid., Frames 1009–1010.
31. _Korpsbefehl für 26.8.44_. NARA Microfilm Publication T-314, Roll 26, Frames 11–12.
32. Ibid., Frames 62–63.
33. Ibid. and Frame 77.
34. Ibid., Frame 88. _Abschrift Fernschreiben An Kav.Korps. 2.9.44_. Combat-strength for 2 September 1944 at Frame 90.
35. All of the following references from _Hebung der Abwehrbereitschaft_ , 12 September 1944, in ibid., Frames 180–185.
36. Reference to equestrian training in ibid., Frame 186. Reference to 4th Brigade's horses at Frame 271.
37. Combat-strength and casualty figures in ibid., Frames 342 and 356, respectively.
38. _Kriegstagebuch Nr.3 mit Anlagen. Generalkdo.Kav.Korps/Ia mit Abt. IIa_. NARA Microfilm Publication T-314, Roll 27. Here Frames 4, 8.
39. Ibid., Frame 621.
40. Ibid., Frames 602–603. Commendatory order of 26 October 1944. For pointed requests for artillery ammunition (none was available) and for statements that units couldn't move for want of fuel, see Frames 18–20. Hitler's order of 29 October 1944 at Frame 578. Reference to feed cutoff at Frame 343.
41. Ibid., Frame 361. Cavalry and infantry strengths as of 20 November 1944.
42. Ibid., Frame 364. Directive of 20 November 1944.
43. See Appendix D.
44. Ibid., Frames 49–50.
45. Ibid. Frames 50–53, 137; Dear and Foot, _The Oxford Companion to World War II_ , 448. The Cavalry Corps' records in the National Archives end 31 December 1944.
46. See date of OKH's orders in Richter, _Weapons and Equipment of the German Cavalry, 1933–1945_ , 42.
47. Helmut Heiber and David M. Glantz, _Hitler and His Generals: Military Conferences 1942–1945_ (New York: Enigma Books, 2003), 581 and note 1570.
48. Ziemke, _Stalingrad to Berlin_ , 435, 448–449. German casualty figures at Budapest in Dear and Foot, _The Oxford Companion to World War II_ , 169.
49. Heiber and Glantz, _Hitler and His Generals_ , 685.
50. Ibid., 1117, note 1968.
51. Ibid., and page 700; Hanson W. Baldwin, "West Front Vanishes," _New York Times_ , 10 April 1945, 6. ProQuest Historical Newspapers The _New York Times_ (1857–2006). http://0-proquest.umi.com.wncln.wncln.org. See also "Round-Up in the Reich" and "Soviet Armies Link Up for Push on Prague as Vienna Falls," _New York Times_ , 11 and 14 April 1945, 22 and 1, respectively, at the same site (2 December 2009). Ziemke, _Stalingrad to Berlin_ , 452–465.
52. Richter, _Weapons and Equipment of the German Cavalry_ , 42; "Cleaning Up In Austria," _The_ (London) _Times_ , 17 May 1945, 4. http://0-infotrac.galegroup.com.wncln.wncln.org (3 December 2009). The records consulted do not indicate the Cavalry Corps' strength at the time of the surrender and are somewhat fragmentary. A nonscholarly source provides the figures noted in the text. At first glance, the number of horses seems high. Nevertheless, the totals correspond to last-ditch efforts to reinforce the southern armies for Operation Spring Awakening. Civilian reportage is corroborative, and Ziemke, _Stalingrad to Berlin_ , 498, notes that German troops in May 1945 still totaled 430,000.
Epilogue
1. Clark, _Iron Kingdom_ , 67; Jan Morris, _Spain_ (New York: Oxford University Press, 1979), 132.
2. "Tragtiere im Einsatz." _Truppendienst_ , Folge 287, Ausgabe 5/2005 (Wien: Bundesministerium für Landesverteidigung, 2005). www.bmlv.gv/truppendienst/ausgaben/artikel.php?id=375 (17 September 2008). Hereafter "Tragtiere im Einsatz."
3. "Pferde im Einsatz bei Wehrmacht und Waffen-SS." www.bundesarchiv.de/aktuelles/aus_dem_archiv/galerie/00172/index.html (17 September 2008).
4. Ibid.
5. George C. Marshall, _Victory Report On the Winning of the War in Europe and the Pacific_ (Washington, D.C.: War Department, 1945), 98.
6. Ibid.
7. Ibid., 99–100.
8. Overy, "Transportation and Rearmament in the Third Reich,"406.
9. Shelby L. Stanton, _Anatomy of a Division: The 1st Cav in Vietnam_ (New York: Warner Books, 1987), 7.
10. "Die Gebirgstragtiere der Bundeswehr." www.bundeswehr.de/portal/a/bwde/streitkraefte/heer (16 September 2008).
11. Letter to the author of 7 July 2009 from the PAMTC's Public Affairs Officer.
12. Ibid.
13. "Tragtiere im Einsatz."
14. Ibid.
15. Douglas Barrie, "Tornado Watch," _Aviation Week and Space Technology_ , 26 February 2007, 68.
16. See Doug Stanton, _Horse Soldiers: The Extraordinary Story of a Band of U.S. Soldiers Who Rode to Victory in Afghanistan_ (New York: Scribner, 2009), here 55.
17. Lance Benzel, "Ft. Carson Special Forces train on horseback," _Army Times_ , 8 September 2010. www.armytimes.com/news/2010/09/ap-special-forces-train-on-horseback-090710/
18. Citino, _The German Way of War_ , 302–303.
19. Ibid., 303.
20. The reference is, of course, to General Ulysses S. Grant's observation on the Confederacy. See E. B. Long, ed., _Personal Memoirs of U.S. Grant_ (New York: Da Capo Press, 1952/1982), 555–556.
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INDEX
A
A-Army,
Afghanistan operations, ,
aircraft: aerial attack impact on cavalry, ; close-air support provided by, 276n23; value to cavalry,
Algerian guerrilla war,
American Army: horseback training program, ; remount requirements, 258n50; support for horse cavalry in, –; technological superiority of, –
American automotive industry,
American Civil War: cavalry role in, , ; railway interdiction in, ; remount requirements of Union Army, 258n50; strategic "rides" in, ; use of railroad in,
American 1st Cavalry Division,
anti-partisan warfare: in France, ; in Russia, , , –, –, –
Ardennes: Battle of the, –; region of France, –
armored cars, , ; in Russo-Polish War,
armored combat vehicles, . _See also specific types_
armored divisions, –
Army Field Wagon 1,
arquebus, –
artillery: mobility of, ; Prussian doctrine on, ; Russian, . _See also_ horseartillery
Asiaticus (pseudonym),
Asquith, Herbert, –
_Auftragstaktik_ , –
Austrian Federal Army,
automotive industry: American, ; German, –
B
Bach-Zelewski, Erich von dem, , , ,
Balck, Wilhelm, –
Baldwin, Hanson W.,
Balkans campaign,
Barb cavalry mount,
"Basics of Cavalry Leadership, The" (Harteneck),
bayonet, , –, –
Bazaine, Achille,
Bazentin-le-Petit, Battle of,
Beck, Ludwig,
Belgium, German advance through, –
Bernhardi, Friedrich von, ,
bicycle M1939 _Patria WKC_ ,
Bicycle Reconnaissance Detachment, –, , –
bicycle troops, , , , , , , ; mobility of, , ; in SS Cavalry Brigade, –, , –, 269n12; supply requirements of, , ; in WWI, ,
Bismarck, Otto von, , ,
Bismarck memorials, 257n7
Black, Jeremy, –
Black Bands,
Blenheim, Battle of,
Bock, Fedor von, ,
Boer War, –,
Book of Revelation, –
Böselager, Georg von, –, 286n4
Brandy Station, Battle of,
Brauchitsch, Walther von,
breastplate, , 263n45
Bredow, Friedrich Wilhelm von, , ,
British cavalry, –, 262n17
British Expeditionary Force (BEF),
British 2nd Cavalry Brigade,
British 7th Dragoon Guards,
British Tornado GR4 attack aircraft, –
British war production,
Brown Shirts, 279n13
Brusilov Offensive,
Budapest, siege of, , –
Budenney, Semyon,
Buford, John,
Buhle, Walter,
Bülow, Karl von,
C
Cambrai, Battle of,
Cannae, Battle of,
_caracole_ , –
cavalry and cavalrymen: cultural perceptions regarding, –; deemphasis of in German propaganda, ; George Patton on, –; in motorized age, –; nobility in, –; post-WWII, ; target profile of, –
cavalry equipment: bayonet, –, –; breastplate, ; firearms, , –; heavy weapons, –; helmet, ; lances, , ; swords, , , , –
cavalry formations, –
cavalry gun,
_Cavalry Journal_ ,
cavalry leadership basics, –
cavalry modernization, –
cavalry mounts: aggressiveness of, –; the Barb, ; breeding programs, , , , , –; feed and fodder requirements for, –, –; French abuse of, –; recognized gaits of, –; remount services, –; rider's bond with, –; veterinary services, –. _See also horse entries_
cavalry officers, –
Cavalry School, –
cavalry spirit,
cavalry swords, , , , –
cavalry training, –, , 267n18
_Cavalry Training Manual_ (British),
cavalry uniforms, ,
_chassepot_ rifle,
_Chassuers d'Afrique_ ,
chivalric ideal, –
chivalric virtues of the SS, , , , ,
Citino, Robert M.,
Combat Group Hannibal,
combined-arms doctrine, , –; cavalry role in,
Commissar Order,
conscription, army, ,
Corap, Andre Georges, ,
Cossacks,
D
Darré, Walther,
Death Ride at Mars-la-Tour, –, , ,
Dnepr River,
Donck, Battle of, –
double column formation, –
draft horses, ,
dragoons, ,
Dreyse model MG 13, –
Dreyse needle gun,
Dziewanowski, M.K., –
E
East Prussian State Stud at Trakehnen, , , –
Eastern Front in WWI, –; distances as factor in, –; elements of race war in, –; German cavalry effectiveness on, –
Eastern Front in WWII. _See_ Operation Barbarossa; _SS entries_
economic centralization, –
economic mobilization,
Èghezèe, Battle of,
8th _Waffen_ -SS Cavalry Division. _See_ SS Cavalry Division _Florian Geyer_
Emmich, Otto von,
equestrian associations,
equine pack-animal units,
equitation, –,
Eylau, Battle of,
F
Falkenhayn, Erich von,
farriery schools,
Federal Armed Forces (Germany),
Fegelein, Hermann, , , , , , , , ,
Fehrbellin, Battle of, –
Felbar, Hans-Gustav,
Feldt, Kurt, , , , , , , , , , ,
First Battle of the Marne,
flanking role of cavalry, –; in WWI, –
_Florian Geyer. See_ SS Cavalry Division _Florian Geyer_
Foix, Gaston de,
formation riding, –
Fort Embourg (Belgium),
Four-Year Plan,
Franco-German Armistice,
François, Hermann von,
Franco-Prussian War, , ; cavalry lessons learned in, –; Death Ride at Mars-la-Tour, –; French cavalry failures in, , , –; French withdrawal from Metz, –; German advance to the Moselle, ; German infantry in, –; German reconnaissance in, –, –; logistics and supply in, –; partisan warfare in, ; railroad transportation in, ; siege of Paris in, , ; strategic "rides" in, –
_francs-tireurs_ ,
Frank, August,
Frankowo, Battle of,
Frederick III of Brandenburg,
Frederick the Great of Prussia, ,
Frederick William I of Prussia,
Freitag, Fritz, , ,
French, John,
French cavalry, ; abuse of mounts in, –; in the Ardennes, –; use of firearms by, 258n16
French Cavalry School, ,
French Ninth Army,
French remount system, ,
French Second Army,
Friesian, 274n53
Fritsche, Karl,
Froeschwiller, Battle of, ,
_Führung und Gefecht der Verbundenen Waffen_ manual,
G
gas warfare,
General Staff,
George II of Great Britain, –
Gercke, Rudolf, –
German A-Army,
German Alpine Army Corps, 283n18
German armed forces nomenclature,
German Army: cavalry divisions in, ; de-modernization of, in WWII, ; inter-war composition, –; inter-war conscription, , ; inter-war expansion, ; interwar reorganization of cavalry, ; limitations imposed on, ; motorization prior to WWII, –; professionalism of,
German army doctrinal manual, –
German army maneuvers, –
German Army of the Meuse,
German Army Service Regulations of 1923,
German Army Veterinary Academy,
German Army Veterinary Service,
German Cavalry Brigade Model,
German cavalry divisions,
German Cavalry School, –
German Cavalry Unit Böselager,
German Cuirassier Regiment _Königin_ , –
German Defense Economy and Weapons Bureau,
German Federal Armed Forces,
German First Army,
German heraldic shields,
German II Cavalry Corps,
German _Kultur_ , –
German 2nd Motorized Division,
German 2nd Panzer Group, ,
German 2nd _Reiter_ Regiment, 1st Cavalry Brigade,
German 22nd _Reiter_ Regiment, 1st Cavalry Brigade, , , ,
German 2nd SS Cavalry Regiment,
German 2nd SS Panzer Corps,
German Ninth Army, , , ,
German 3rd Cavalry Brigade, 9th Uhlans, –
German 3rd Cavalry Division,
German 23rd Mountain Rifle Brigade,
German rearmament, –; economic inefficiencies of, ; in European balance of power, –; Four-Year Plan for, ; Hitler on primacy of,
German Sixth Army,
German SS. _See SS entries_
German 1st Cavalry Brigade: in invasion of Poland, –; units of,
German 1st Cavalry Corps, ; armored attacks against, –; composition of, –; as defenders against Red Terror, ; in defense of Budapest, ; Hungarian division in, –, , , , ; lack of mobility, ; reestablishment of, ; reorganization of, –, –; replacements for, –; surrender of,
German 1st Cavalry Division, , , , ; conversion to panzer division, –; in France, –; in the Netherlands, –; in Operation Barbarossa, , , , , , , ,
German 51st Fighter Wing _Mölders_ , , 276n23
German 1st Panzer Division,
German 1st _Reiter_ Regiment, 1st Cavalry Brigade, , –, ,
German 21st _Reiter_ Regiment, 1st Cavalry Brigade, , , –,
German 1st SS Cavalry Regiment, ,
German 4th Cavalry Brigade, ,
German 4th Cavalry Division,
German 69th Cavalry Replacement Detachment, –
German 12th Infantry Division,
German 17th Infantry Division, ,
German 112th Infantry Division,
German 4th Panzer Division, ,
German 7th Panzer Division,
German 6th Reconnaissance Battalion,
German Third Army,
German Third Cavalry Division,
German war production, –
German Weekly Newsreel,
German XIII Corps, ,
German XXIII Corps,
German XXIV Panzer Corps, ,
German XXXVIII Corps,
German XXXXIII Corps,
German XXXXVII Panzer Corps,
Geyer, Florian,
glanders,
Goebbels, Joseph,
Gomel, Battle of, –
Gomel Pocket, 277n29
Göring, Hermann, ,
Gravelotte, Battle of,
Groener, Wilhelm,
Grolig, Oswin,
Gronau, Hans von,
Gumbinnen, Battle of, –
Guderian, Heinz, –, , , , ,
gunpowder warfare: cavalry role in, , –; cavalry use of firearms in, –; combined-arms doctrine and, ; effect on cavalry charge against infantry, –
Gustavus Adolphus of Sweden,
gypsies,
H
Haelen, Battle of, ,
Haflinger, 269n23
Haig, Douglas, , , , ,
Halder, Franz,
Hanoverian State Stud at Celle, , , , –
Harnack, Adolf von,
Harteneck, Gustav, , , , , , , , , , , , , , ; on cavalry leadership, –
heavy cavalry, –; role of,
_Heer_ ,
_Heliand, The_ , ,
Henderson, G.F.R.,
Heye, Wilhelm,
Himmler, Heinrich, , , , , , , , , , , , , ,
Hindenburg, Paul von,
Hitler, Adolph, , , , , , ; consolidation of authority by, ; economic centralization and, –; micro management by, –; Nazi symbolism and, ; objectives of invasion of Soviet Union, –; Operation Barbarossa and, , ; on primacy of rearmament, ; reliance of on Himmler, ; "stand or die" order, ; on uses of cruelty,
Hitler Youth,
Hoffmann, Max, –
Hohberg, Anton von,
Hohenzollern, Friedrich Karl von, –,
horse breeding programs, , , , –
horse muster commission,
horse registration,
horse-artillery, –, ; in Barbarossa, ; in Franco-Prussian War, , ; in interwar reorganization of cavalry, –; in invasion of Poland, ; principal tasks of, 258n29; in WWI,
Horseman's Badge,
horsemanship. _See_ equitation
horses: army order concerning the sparing of, –; cultural perception of, ; inter-war importance of, –; in Nazi cultural expression, –; _panje_ , –, , –; standard daily rations, –; utility in WWI, –. _See also_ cavalry mounts
horses' gaits, –
Howard, Michael,
HSSPF. _See_ Senior SS and Police Command Center
"Hundred Guards" cuirassiers,
Hungarian Army,
Hungarian 3rd Hussar Regiment,
Hungarian 1st Royal Cavalry Division, , ,
Huntzinger, Charles, , ,
hussars, –
I
Indian Army,
industrial infrastructure, and cavalry operations,
infantry: in American Civil War, ; German, in Franco-Prussian War, –; role in partisan warfare,
infantry phalanx,
_Infantry Regulations_ (British),
infantry square, , , , 256n27
Interallied Military Control Commission,
Iraq,
J
_Jäger zu Pferde_ , –
Japan, ,
Jaroslawice, Battle of,
jeeps, ,
Joffre, Joseph, –
K
Kabisch, Ernst,
_Kaiserschlacht_ ,
Kalinin Front, –
Komarow, Battle of,
Kavanagh, T.C. McM.,
Keegan, John, –
Keitel, Wilhelm,
Kluck, Alexander von, ,
Kluge, Günther von, , ,
_Konarmia_ I, –
Königgrätz, Battle of, , ,
Korma Heights operation,
Kornwerderzand: Battle of, ; fortifications,
Kosak, Georges,
Kosovo operations,
_Kriegsmarine_ , , ,
Krupp howitzer,
Küchler, Georg von,
Kursk, Battle of, , ,
L
lances, , ,
Langensalza, Battle of,
Lanrezac, Charles Louis Marie,
League of Nations,
Leaping Horseman insignia, ,
Leboeuf, Edmond,
Leeb, Wilhelm Ritter von,
Leman, Gérard, ,
Lesueur, Larry,
Liège, Battle of, –
light cavalry: French, in Algeria, ; heavy cavalry and, –; role of,
light divisions, –
light infantry gun,
Lisle, H.B. de,
Lithuania, German offensive in,
Lodz, Battle of,
logistics and supply: French, in Franco-Prussian War, –; French remount system, , ; in Operation Barbarossa, –, –,
Lombard, Gustav,
Louis XIV of France,
Luck, Hans von,
Ludendorff, Erich, ,
_Luftwaffe_ , ,
M
machine guns, –, 268n4
Mackensen, August von,
Magyar horsemen, –
Malaparte, Curzio,
mange, –
Marne, Battle of the, –
Marshall, George C.,
Marwitz, Georg von der,
Marwitz, Johannes Georg von der,
Mars-la-Tour, Battle of, –, , ,
massed cavalry: versus dismounted cavalry role, –; French, in Franco-Prussian War, ; massed infantry and, ; support of infantry by, ; vulnerabilities of, to firearms, –; in WWI,
Mastelarz, Kazimierz,
Masurian Lakes, Battle of the,
_Materialschlacht_ , –
Mauser 98k carbine, –
Mauser M1898,
Mellenthin, F.W. von, , 273n46
_Metamorphoses_ (Ovid),
Miracle of the Vistula,
mixed divisions: French, –; requirements for effectiveness, –
mobility: army doctrine on, ; as basic cavalry doctrine, ; as factor in inter-war planning, –; Operation Barbarossa and, ; Russian roads as hindrance to, –
Model, Walther, , ,
Moltke, Helmut von, the Elder, , , ; memorandum of 1868, –, , ,
Mons, Battle of,
Morgan, John Hunt,
Morsbronn, Battle of, ,
motor vehicle , –
motorization: American, –; in defining cavalry's role, –, , ; driver and mechanic training, ; George Patton on, ; incomplete, prior to WWII, ; lack of standardization in, ; "Lance Comparison" in, ; in Operation Barbarossa, –
Motorized Transport Department, –
Mountain Pack Animal Company ,
mounted infantry, , 261n8
mounted police,
mules, –
N
Nagybánya, Miklós Horthy de,
Napoleon I of France, , ,
Napoleon III of France,
National Association of Breeding and Testing Of German Warm Bloods,
National Socialist Drivers Corps,
National Socialist Mounted Corps, 272n29
Nazi symbolism, –
Nazi-Soviet Non-Aggression Pact, , –
Netherlands campaign, –
Night of the Long Knives,
North European Plain,
O
Occupied Germany, –
100,000-man _Reichswehr_ ,
Operation Barbarossa: anti-partisan warfare in, –, –, –; cavalry role in, , , , , , ; drive on Gomel, –; German losses in, , , ; horsemounted reconnaissance riders in, ; horses' performance in, –, ; logistics difficulties in, –, , ; objectives of, –; rivers and marshes as factors in, –; Russian road conditions as factor in, ; Russian tenacity in response to, ; Soviet cavalry and, ; Soviet counteroffensive to, –, ; SS cavalry in, , –; SS "special assignments" in, , , , –; troop discipline in,
Operation Margarethe, 287n8
Operation Michael,
Operation Seydlitz, –
Operation Spring Awakening, ,
Operation Typhoon,
Operations Weichsel I and II, , –
Ostend, Battle of, –
Ovid,
P
Pack Animal Mission and Training Center (PAMTC), –
_Pallasch_ sword,
_panje_ horse, –, , –
_Panzergrenadiere_ ,
Paris, siege of, ,
Parrott, David A.,
partisan warfare. _See_ anti-partisan warfare
Patton, George, –
Patton sword,
Peasants' War,
Pétain, Henri-Philippe,
Petersberg Missions,
pikemen,
Pilsudski's Legion,
pistoleers,
platoon column formation,
Plettenberg (German commander),
Pliev, I.A.,
Poland: German animosity toward, ; German invasion of, –, –; war with Russia, –, ,
police call-up, –, 281n39
Polish cavalry, ; in alleged charges against tanks, –, 273n51
Polish Corridor,
Polish _Pomorske_ Cavalry Brigade, , , 273n51
Polish 18th Lancer Regiment, –
population density, and cavalry operations,
Poseck, Maximilian von, , , , ,
Pripet Marshes, –, –
R
rabies,
Race to the Sea, –,
racial composition of SS, –
racial warfare, –; on Eastern Front,
Raeder, Erich,
railroad transportation: in Franco-Prussian War, , ; in German inter-war planning, ; of infantry, ; requirements for German corps, –; in Soviet Union,
railway interdiction, –
rearmament in Germany, –; economic inefficiencies of, ; in European balance of power, –; Four-Year Plan for, ; Hitler on primacy of,
reconnaissance and screening, ; in American Civil War, ; French, in Franco-Prussian War, , –; German, in Franco-Prussian War, –, , ; German doctrine on, ; strategic "rides" and, , –; in WWI, ,
reconnaissance by fire,
reconnaissance detachments,
Reich Chancellery,
Reich Plenipotentiary for the Four-Year Plan,
_Reichswehr_ ,
Reinhardt, George-Hans,
remount depot, ,
remount services: American requirements, 258n50; Austrian Federal Army, ; French, , ; German, –
Rennenkampf, Pavel, ,
Riding and Driving Schools,
Riding School, SS Main,
river crossings: Loire, ; Seine,
river-crossing exercises,
Roberts, Michael,
Röhm, Ernst, 279n13
Rojahn (German lieutenant colonel),
Rommel, Erwin,
Rossbach, Battle of, , ,
Rumania, German offensive in, –
Rumanian Army, armored car use by,
Rupprecht of Bavaria, ,
Russia: as German promised land, –; invasion of East Prussia by, –; railroads in, ; Red Army cavalry expansion in, ; rivers and marshes in, –; road conditions in, . _See also_ Operation Barbarossa
Russian cavalry, –, , , 264n66
Russian Fifth Army,
Russian First Army,
Russian First Cavalry Army, –
Russian Imperial Guard Cavalry Corps, –
Russian Red Terror,
Russian 1st Cavalry Division,
Russian 4th Guards Cavalry Corps,
Russian 44th Mongolian Cavalry Division, 275n10
Russo-Japanese War, ,
Russo-Polish War, –, ,
Rzhev, Battle of, ,
S
saber, –
Samsonov, Alexander,
_Schellenbaum_ standard,
Schlieffen, Alfred von,
Schulze, Curt,
Sedan, Battle of, ,
Seeckt, Hans von, –, , , , , , ,
Senior SS and Police Command Center (HSSPF), –, 281n39
Seydlitz, Friedrich Wilhelm von, ,
Sheridan, Philip,
Sherman, William T.,
_Signal_ magazine,
61st Cavalry (India),
Skinner, Robert P.,
Slavic peoples, German attitude toward,
Somme, Battle of the, –
Sonnengott (cavalry mount), –
Sordet, J.F.A.,
Soviet cavalry, –,
Soviet Union. _See_ Russia
Spicheren, Battle of,
SS Bicycle Reconnaissance Detachment, –, , –
SS Cavalry Brigade, ; assignment to front-line combat, –; assignment to HSSPF, ; clearing operations in Pripet Marshes, –; components of, –; horse care in, ; mission in Operation Barbarossa, –; nomenclature of, ; in partisan warfare, –; purpose of,
SS Cavalry Division _Florian Geyer_ : antipartisan operations by, –, –; "cleansing" actions of, ; composition of, ; formation of, –; Hermann Fegelein's assumption of command of, –; horses of, –, –; naming of, ; reorganization of, –; replacements for, –; _Volksdeutsche_ in, –
SS cavalry formations,
SS Cavalry School,
SS Death's Head Division, ,
SS Death's Head Horse Regiment,
SS Death's Head Units,
SS _Kampfgruppe Nord_ , –
SS Main Riding School,
SS Mounted Company,
SS Mounted Regiments, ,
SS Political Readiness Squad,
SS _Polizei_ Division,
SS runes insignia, –
SS ( _Schutzstaffel_ ): chivalric virtues of, , , , , ; racial composition of members, –; recruitment of nobility by, –
SS Special Duties Troops, –
_Stahlhelm_ ,
Stalin, Joseph, , 275n10
Stalingrad, Battle of, ,
Stallupönen, Battle of,
stand or die order,
State studs, , ,
Steinmetz, Karl Friedrich von,
steppe warfare, –, ,
stirrup, –
Storm Troopers,
Stosch, Albrecht von,
strategic "rides," , –
swastika,
swords, , , , –
T
table of organization and equipment of 1919, ,
tanks: as factor in cavalry role, ; in large-scale maneuvers, –; Polish cavalry alleged engagement of, –; in WWI,
Tannenberg, Battle of, ,
Thermal Airborne Laser Designator (TIALD) pod, –
Thomas, Georg, , –
Thorak, Josef,
Timoshenko, S.K.,
Tirlemont, Battle of,
Treaty of Brest-Litovsk (1918),
Treaty of Frankfurt (1871),
Treaty of Versailles (1919), ; disarmament clause oversight, ; on German army composition, –, ; on horses and livestock, –
trench warfare, –,
Trevor-Roper, Hugh,
_Truppenamt_ ,
_Truppenführung_ doctrinal manual, –
Tukhachevsky, Mikhail,
U
uhlan (term), ,
Ukraine,
U.S. Army. _See_ American Army
U.S. Constabulary,
V
Vattay (Hungarian general), , , , , 287n19
Verdun, Battle of, ,
veterinary services, –
_Victory Report_ (Marshall),
Vietnam War,
_Volksdeutsche_ , –
_Volkswagen_ , –
Vormann, Nikolaus von,
W
war economy, ,
war production: American, ; British, ; German, –
Warsaw, siege of,
Waterloo, Battle of,
Wedemeyer, Albert C.,
_Wehrmacht_ , , ,
Weiß, Walter, ,
Werner, Anton von,
Western Front in WWI: Allied cavalry on, ; cavalry forces on, –; German cavalry contribution on, , –; Race to the Sea action and, –
William of Prussia,
Wissembourg, Battle of,
Wilson, Henry,
Winter Battle of the Masurian Lakes,
World War I: Allied cavalry strength in, , ; cavalry in, ; German cavalry mission in, –; German cavalry strength in, –; horse mobilization in, ; materiál as factor in, –; racial element in, –; urbanization's impact on cavalry operations, . _See also_ Eastern Front in WWI; Western Front in WWI
World War II: cavalry retention in, ; German motorization capability as factor in, –; German rearmament prior to, –; German surrender, ; horses in active service during, . _See also_ Operation Barbarossa
Wörth, Battle of,
Wrangel, Gustav,
Z
Zamosc Ring,
Zeitzler, Kurt,
Ziethen, Hans Joachim von, ,
Ziethen, Johan Joachim von,
Zorndorf, Battle of,
ABOUT THE AUTHOR
D. R. Dorondo holds the degree of D. Phil. from the University of Oxford. He is professor of modern German and European military history at Western Carolina University.
**The Naval Institute Press** is the book-publishing arm of the U.S. Naval Institute, a private, nonprofit, membership society for sea service professionals and others who share an interest in naval and maritime affairs. Established in 1873 at the U.S. Naval Academy in Annapolis, Maryland, where its offices remain today, the Naval Institute has members worldwide.
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Q: Using CI collation as case Sensitive in MySQL I want to have both Persian and English letters in one of my Columns,
So I set the collation to utf8mb4_unicode_ci.
But it is case-insensitive ... How can I use it Case-sensitive?
(For example Wordpress uses utf8mb4_unicode_ci but it is still case-sensitive)
A: Change the collation to utf8mb4_bin.
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"redpajama_set_name": "RedPajamaStackExchange"
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{"url":"https:\/\/mathoverflow.net\/questions\/282254\/pi-and-rapid-series-for-the-inverse-tangent\/299382","text":"# $\\pi$ and rapid series for the inverse tangent\n\nI design an educational program to demonstrate convergence rate in computation of $\\pi$ by using the Machin like formula. According to Weisstein, the Hwang\u2019s equation (see eq. (32) in http:\/\/mathworld.wolfram.com\/Machin-LikeFormulas.html): \\begin{align} \\frac{1}{4}\\pi = & 183{{\\cot }^{-1}}239+32{{\\cot }^{-1}}1023-68{{\\cot }^{-1}}5832 \\\\ & +12{{\\cot }^{-1}}110443-12{{\\cot }^{-1}}4841182-100{{\\cot }^{-1}}6826318 \\\\ \\end{align} is the most efficient Machin like formula to compute pi. I used three series for the inverse tangent ($\\cot^{-1}x = \\tan^{-1}\\frac{1}{x}$): $$\\tag{1}\\tan^{-1}x=\\sum\\limits_{n=1}^{\\infty }{\\frac{{{\\left( -1 \\right)}^{n}}{{x}^{2n+1}}}{2n+1}},$$\n\n$$\\tag{2}\\tan^{-1}x=\\sum\\limits_{n=1}^{\\infty }{\\frac{{{2}^{2n}}{{\\left( n! \\right)}^{2}}}{\\left( 2n+1 \\right)!}\\frac{{{x}^{2n+1}}}{{{\\left( 1+{{x}^{2}} \\right)}^{n+1}}}},$$\n\n$$\\tag{3}\\tan^{-1}x=2\\sum\\limits_{n=1}^{\\infty }{\\frac{1}{2n-1}\\frac{{{a}_{n}}\\left( x \\right)}{a_{n}^{2}\\left( x \\right)+b_{n}^{2}\\left( x \\right)}},$$ where \\begin{align} & {{a}_{1}}\\left( x \\right)=2\/x, \\\\ & {{b}_{1}}\\left( x \\right)=1, \\\\ & {{a}_{n}}\\left( x \\right)={{a}_{n-1}}\\left( x \\right)\\left( 1-4\/{{x}^{2}} \\right)+4{{b}_{n-1}}\\left( x \\right)\/x, \\\\ & {{b}_{n}}\\left( x \\right)={{b}_{n-1}}\\left( x \\right)\\left( 1-4\/{{x}^{2}} \\right)-4{{a}_{n-1}}\\left( x \\right)\/x. \\\\ \\end{align}\n\nEq. (1) is the Maclaurin series, eq. (2) is the Euler\u2019s series, eq. (3) is from the paper Abrarov & Quine arXiv:1706.08835. It appears to be that eqs. (2) and (3) are more rapid than eq. (1). I want to ask you two questions. Is eq. (3) more efficient than eq. (2)? Are there other series for the inverse tangent with rapid convergence?\n\n\u2022 Which of several methods is the most efficient generally depends on how clever the coder is and what software and hardware the coder is using, and what the range of the computation is. It's not just a matter of which formula. \u2013\u00a0Gerry Myerson Sep 29 '17 at 3:12\n\u2022 @ Gerry Myerson. Hardware is the desktop computer. My main soft is Maple (or any other). Our goal is to show the best convergence rate by taking appropriate series for $tan^{-1} x$. If the convergence rate is 5 - 10 digits per term, the range of 30 - 50 decimals is enough. One may expect that the Hwang's equation (32) can produce up to 10 new digits of pi per increment. \u2013\u00a0Philip Thomas Sep 29 '17 at 12:46\n\u2022 The software doesn't let you ping two people in one comment. I think that if you want coudy to see your comment, you have to have it as a comment to coudy's answer, not a comment to your question. Also, I think you have to not have a space between the at-sign and the username. \u2013\u00a0Gerry Myerson May 4 '18 at 1:02\n\u2022 @Gerry Myerson. In this paper ijmcs.future-in-tech.net\/13.2\/R-Abrarov.pdf page 167 shows a Mathematica program that generates 17 digits of pi per term. This proves rapid convergence. \u2013\u00a0Philip Thomas May 13 '18 at 23:43\n\nThe Hwang equation is not the most efficient Machin like formula to compute $\\pi$. The following formula\n\n$${\\pi \\over 4} = 32 \\, \\hbox{arctan}({1\\over 40}) - \\hbox{arctan}\\biggl( {38035138859000075702655846657186322249216830232319 \\over 2634699316100146880926635665506082395762836079845121}\\biggr)$$ has a Lehmer's measure around $1.16751$ thus beating Hwang's formula (with Lehmer's measure $1.51240$).\n\nAbrarov and Quine gave a formula with Lehmer's measure $0.245319$ last summer, together with the relevant algorithms, in their paper An iteration procedure for a two-term Machin-like formula for pi with small Lehmer\u2019s measure. That formula provides 16 new digits of $\\pi$ per term increment thus beating the famous Chudnovsky formula.\n\nThe Abrarov-Quine formula won't fit on that small answer box though since one of the fraction has a numerator with $522\\,185\\,807$ digits and a denominator with $522\\,185\\,816$ digits.\n\n\u2022 @ coudy. A fraction where a numerator with 522185807 digits and a denominator with 522185816 digits requires a specific FFT method dealing with long numbers for more efficient computation. As I know this FFT method were used for the Chudnovsky formula for beating the last record. However, this is beyond of my current scope. Thank you anyway! \u2013\u00a0Philip Thomas Sep 28 '17 at 20:34\n\u2022 The aforementioned paper gives numerous Machin formulas together with efficient series to compute the arctangent. You should find something there to your liking. \u2013\u00a0coudy Sep 28 '17 at 20:39\n\u2022 @ coudy. This method is unconventional although it (perhaps) beats the Chudnovsky formula for $\\pi$ in convergence as claimed. I did not verify this claim. However, I verified series (3) from the same paper and confirm that it converges fast. This probably is what I need right now. \u2013\u00a0Philip Thomas Sep 28 '17 at 20:51\n\nThis is not exactly an answer, but R. P. Brent has an excellent survey of computing everything under the Sun (including $\\arctan.$)\n\n\u2022 @ Igor Rivin. Many thanks I will read this paper by R. P. Brent. \u2013\u00a0Philip Thomas Sep 28 '17 at 21:17\n\nThe efficiency of such series depends not just on the rate of convergence but on the bit size of the terms.\n\nIf $\\sum_{k=0}^{\\infty} T(k)$ converges linearly, you can always rewrite it as $\\sum_{k=0}^{\\infty} U(k)$ where $U(k) = \\sum_{j=0}^{N-1} T(N k + j)$ to get a series that converges $N$ times faster. But nothing has really been gained since the terms now cost $N$ times more to compute.\n\n\u2022 @ Fredrik Johansson. This formula is accurate. However it adds many new terms while k increases. Therefore efficiency of computation may be questionable. I will verify it. Thank you for suggestion! \u2013\u00a0Philip Thomas Sep 28 '17 at 21:11\n\nArctangent formula (3) that can also be written as $$\\tan^{-1}\\left(x\\right)=i\\sum\\limits_{n=1}^{\\infty}{\\frac{1}{2n-1}}\\left(\\frac{1}{\\left(1+2i\/x\\right)^{2n-1}}-\\frac{1}{\\left(1-2i\/x\\right)^{2n-1}}\\right)$$ is more rapid in convergence than the equation (2). Therefore, it is more efficient in computation. Here is a link: How to compare convergence of two equations for the arctangent function?","date":"2019-11-14 01:08:30","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 2, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9846329092979431, \"perplexity\": 920.2285849955681}, \"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-2019-47\/segments\/1573496667767.6\/warc\/CC-MAIN-20191114002636-20191114030636-00255.warc.gz\"}"}
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\section{Introduction}
Decomposing a tensors into its components, and determining the number of those (= the rank) is a multidimensional generalization of the singular value decomposition and the matrix rank, and a reoccurring task in all practical sciences, appearing many times under different names; first discovered by Hitchcock~\cite{Hitchcock1927} and then re-discovered under names such as PARAFAC~\cite{Harshman1970} or CANDECOMP~\cite{CarrollChang1970}, it has been applied in many fields such as chemometrics, psychometrics, and signal processing \cite{Bro1997,Parafac2000,NionSidi2009}. An extensive survey of many applications can be found in \cite{Sidi2004,DeLathauwer:2008}.\\
Recently, motivated by real world applications, orthogonality constraints on the decomposition have been studied in the literature, such as the orthogonal rank decomposition and the combinatorial orthogonal rank decomposition, which can be traced back to~\cite{Deni1989,Kolda01orthogonaltensor}, and the orthogonal decomposition in~\cite{Martin06ajacobi-type} and~\cite{Hsu2012}, the latter of which occurs for example in the identification of latent variable models from empirical moments, and several other statistical estimation tasks, see~\cite{Ana2012} for a survey. The orthogonality constraints imposed in these two branches of literature are not the same, as~\cite{Deni1989,Kolda01orthogonaltensor} imposes summand-wise orthogonality, while in~\cite{Martin06ajacobi-type,Hsu2012,Ana2012}, factor-wise orthogonality can be deduced from the model constraints. In~\cite{Martin06ajacobi-type}, a Jacobi-like and heuristic algorithm was described to obtain a close orthogonal decomposition via Jacobi angle optimization for general tensors; in~\cite{Ana2012}, the authors describe a second order fixed point method for obtaining the decomposition.\\
In~\cite{Ish2013ICML,Son2013ICML}, hierarchical tensor decomposition models are discussed in the context of latent tree graphical models, and algorithms for the identification of this decomposition are described. While this is not explicitly done in the language of orthogonal tensor decompositions, the idea of using flattenings is similar to the one presented, and, in the specific context of tree models, a specific instance orthogonal tensor decomposition, as described in~\cite{Ana2012}.\\
In this paper, we study the orthogonal decomposition model, as it occurs in~\cite{Hsu2012,Ana2012}, namely with factor-wise orthogonality constraints. We show that this kind of decomposition can be directly transformed to a set of singular value decompositions, both theoretically and practically. We give identifiability results for this kind of orthogonal decomposition, showing that it is unique\footnote{up to natural symmetries} in case of existence, and we provide algorithms to obtain the orthogonal decomposition, by reducing it to a sequence of singular value decompositions. We apply these algorithms to a latent variable identification problem which was discussed in~\cite{Hsu2012,Ana2012}, reducing it to a series of eigenvalue problems. In particular, by performing the reduction to singular value decomposition, we show that all existing theory on the singular value decomposition, concerning theoretical issues as well as numerical and algorithmical ones, can be readily applied to the orthogonal decomposition problem.
\section{Theoretical Background}
\subsection{Tensors}
\subsubsection{Definition of a Tensor}
While tensors are common objects, their notation diverges throughout the literature. For ease of reading, we provide the basic definitions.
\begin{Def}
A real tensor of size $(n_1\times n_2\times\dots\times n_d)$ and of degree $d$ is an element of the set
$$\ensuremath{\mathbb{R}}^{n_1\times n_2\times \dots \times n_d}=\left\{(a_{i_1\dots i_d})_{\begin{subarray}{l}
1\le i_1\le n_1\\ \vdots \\ 1\le i_d\le n_d
\end{subarray}}\right\}.$$
If $n_1 = n_2 = \dots = n_d,$ we also write $\ensuremath{\mathbb{R}}^{n^{\times d}}:= \ensuremath{\mathbb{R}}^{n_1\times n_2\times \dots \times n_d}.$
\end{Def}
\subsubsection{Linear Transformation}
Let us introduce a useful shorthand notation for linearly transforming tensors.
\begin{Def}
Let $A\in \ensuremath{\mathbb{R}}^{m\times n}$ be a matrix. For a tensor $T\in \ensuremath{\mathbb{R}}^{n^{(\times d)}}$, we denote by $A\circ T$ the application of $A$ to $T$ along all tensor dimensions, that is, the tensor $A\circ T\in \ensuremath{\mathbb{R}}^{m^{(\times d)}}$ defined as
$$\left(A\circ T\right)_{i_1\dots i_d}=\sum_{j_1=1}^n\dots \sum_{j_d=1}^n A_{i_{1}j_{1}}\cdot\ldots\cdot A_{i_{d}j_{d}}\cdot T_{j_1\dots j_d}.$$
\end{Def}
\begin{Rem}
For $T\in \ensuremath{\mathbb{R}}^{n^{(\times d)}}$ and $A\in \ensuremath{\mathbb{R}}^{m\times n}, A'\in \ensuremath{\mathbb{R}}^{m'\times m}$, note that
$$A'\circ (A\circ T) = (A'\cdot A) \circ T.$$
\end{Rem}
\subsubsection{Flattening}
A flattening of a tensor is the tensor obtained from regarding different indices as one index.
\begin{Def}
Denote by $[k]=\{1,2,\dots, k$\}. A surjective map $\sigma:[d]\rightarrow [\tilde{d}]$ is called $d$-to-$\tilde{d}$ \emph{flattening map}.\\
\end{Def}
\begin{Def}
Let $T\in \ensuremath{\mathbb{R}}^{n_1\times \dots \times n_d}$ be a tensor, and let $\sigma$ be a $d$-to-$\tilde{d}$ flattening map. Then, the $\sigma$-flattening of $T$ is the degree $\tilde{d}$ tensor $\sigma\dashv T\in \ensuremath{\mathbb{R}}^{\tilde{n}_1\times \dots \times \tilde{n}_{\tilde{d}}},$ with $\tilde{n}_k=\prod_{\ell\in\sigma^{-1}(k)}n_\ell,$ defined as
$$(\sigma\dashv T)_{j_1\dots j_{\tilde{d}}} := T_{i_1\dots i_d}\quad,\mbox{where}\; j_k=(i_\ell\;:\;\ell\in \sigma^{-1}(k)).$$
Conversely, if $\tilde{T}=\sigma\dashv T$, then we write $T=\sigma\vdash\tilde{T}$ and call $T$ the \emph{unflattening} of $\tilde{T}$.
\end{Def}
Note that the indices of $\sigma\dashv T$ are, as defined, tuples of indices of $T$; however, this does not contradict the definition of tensor since $[n_1]\times[n_2]\times \dots [n_k]$ can be bijectively mapped onto $\left[\prod_{i=1}^k n_i\right].$ It is convenient to choose the lexicographical ordering for the bijection, but it is mathematically not necessary to fix any such bijection.
For unflattening, if only $\tilde{T}$ and $\sigma$ are given, it is not clear what $\sigma\vdash\tilde{T}$ should be without further specification, since the same unflattening can arise from different tensors even if $\sigma$ is fixed. Therefore, we will use it only in the context where a given flattening is being reversed, or partially reversed, therefore making the unflattening well-defined.
\begin{Ex}
Let $T\in \ensuremath{\mathbb{R}}^{n_1\times n_2\times n_3}$ be a tensor, let $\sigma: 1\mapsto 1, 2\mapsto 2, 3\mapsto 2$. The $\sigma$-flattening of $T$ is a $(n_1\times n_2n_3)$-matrix $\tilde{T}:=\sigma\dashv T$. The columns of $\sigma\dashv \tilde{T}$ are all the $n_2n_3$ sub-$(n_1\times 1\times 1)$-tensors of $T$ where second and third index are fixed. The columns of $\sigma\dashv T$ are indexed by the pairs $(k,\ell)$, or, alternatively, by bijection, by the lexicographical index number $(k-1)\cdot n_2 + \ell$. Taking any $(n'_1\times n_2n_3)$-submatrix of $\tilde{T}$, we can unflatten to obtain a $(n'_1\times n_2\times n_3)$-tensor $\sigma\vdash \tilde{T}$.
\end{Ex}
\subsubsection{Outer Product}
Furthermore, we introduce notation for creating tensors of higher order out of tensors of lower order:
\begin{Def}
Let $v^{(1)}\in \ensuremath{\mathbb{R}}^{n_1},\dots, v^{(d)}\in \ensuremath{\mathbb{R}}^{n_d}$. The \emph{outer product} of the $v^{(k)}$ is the tensor $v^{(1)}\otimes \dots \otimes v^{(d)}\in\ensuremath{\mathbb{R}}^{n_1\times \dots n_d}$ defined by
$$(v^{(1)}\otimes \dots \otimes v^{(d)})_{i_1\dots i_d} := \prod_{k=1}^d v^{(k)}_{i_k}.$$
In case that $v=v^{(1)}= \dots = v^{(d)}$, we also write $v^{\otimes d} := v^{(1)}\otimes \dots \otimes v^{(d)}.$\\
Similarly, if $A\in \ensuremath{\mathbb{R}}^{n_1\times \dots\times n_c}$ and $B\in \ensuremath{\mathbb{R}}^{n_{c+1}\times \dots\times n_d}$ are tensors, the outer product of $A$ and $B$ is the tensor $A\otimes B\in \ensuremath{\mathbb{R}}^{n_1\times \dots n_d}$ defined as
$$(A\otimes B)_{i_1\dots i_d} := \prod_{k=1}^c A^{(k)}_{i_1\dots i_c}\cdot \prod_{k=c+1}^d B^{(k)}_{i_{c+1}\dots i_d}.$$
Outer products of several tensors $A_1\otimes \dots\otimes A_k$
by induction on $k$, namely:
$$A_1\otimes \dots\otimes A_k := (A_1\otimes \dots\otimes A_{k-1})\otimes A_k.$$
\end{Def}
A useful calculation rule for linear transformation is the following:
\begin{Lem}
Let $A\in \ensuremath{\mathbb{R}}^{n^{\times d_1}}$ and $B\in \ensuremath{\mathbb{R}}^{n^{\times d_2}},$ let $A\in\ensuremath{\mathbb{R}}^{m\times n}$. Then,
$$P\circ (A\otimes B) = (P\circ A)\otimes (P\circ B).$$
Similarly, if $v\in \ensuremath{\mathbb{R}}^{n}$, then $P\circ \left(v^{\otimes d}\right) = \left(P\circ v\right)^{\otimes d}.$
\end{Lem}
Outer products are also compatible with flattenings:
\begin{Lem}\label{Lem:prodflat}
Let $A\in\ensuremath{\mathbb{R}}^{n_1\times \dots\times n_c}$ and $B\in\ensuremath{\mathbb{R}}^{n_{c+1}\times \dots\times n_d}.$ Let $\tau$ be a $d$-to-$k$-flattening, let $\sigma_1$ be the restriction of $\tau$ to $[c]$, and let $\sigma_2$ be the $(d-c)$-to-$\tilde{k}$-flattening defined by $\sigma(i):=\tau(c+i)$. Then,
$$\tau\dashv(A\otimes B) = (\sigma_1\dashv A)\otimes (\sigma_2\dashv B).$$
\end{Lem}
\subsection{Orthogonality and Duality}
We briefly review the notions of scalar product and some results, which can also be found in~\cite{Kolda01orthogonaltensor} in slightly different formulation and slightly less generality.
\begin{Def}
A \emph{scalar product} is defined on $\ensuremath{\mathbb{R}}^{n_1\times n_2\times \dots \times n_d}$ by
\begin{align*}
\langle .,.\rangle:& \ensuremath{\mathbb{R}}^{n_1\times n_2\times \dots \times n_d}\times \ensuremath{\mathbb{R}}^{n_1\times n_2\times \dots \times n_d}\longrightarrow\ensuremath{\mathbb{R}}\\
& (A,B)\mapsto \sum_{i_1=1}^{n_1}\dots \sum_{i_d=1}^{n_d} A_{i_1\dots i_d}\cdot B_{i_1\dots i_d}
\end{align*}
As usual, $A,B\in \ensuremath{\mathbb{R}}^{n_1\times \dots \times n_d}$ are called orthogonal to each other if $\langle A,B\rangle =0$, and $A$ is called normal if $\langle A,A\rangle = 1$. A set $A_1,\dots, A_r\in \ensuremath{\mathbb{R}}^{n_1\times \dots \times n_d}$ is called orthonormal if $\langle A_i,A_j\rangle =\delta_{ij}$, where $\delta_{ij}$ is the Kronecker-delta.
\end{Def}
By identification of $\ensuremath{\mathbb{R}}^{n_1\times \dots \times n_d}$ with $\ensuremath{\mathbb{R}}^{N}$, where $N=\prod_{i=1}^d n_i$, the scalar product on tensors inherits all properties of the real scalar product.
\begin{Rem}\label{Rem:sctr}
It is seen by checking definitions that the scalar product on matrices is identical to the trace product, i.e., $\langle A,B\rangle =\operatorname{Tr} (A^\top B)$ for $A,B\in\ensuremath{\mathbb{R}}^{m\times n}$.
\end{Rem}
An important property of the scalar product is compatibility with flattenings:
\begin{Lem}\label{Lem:orthflat}
Let $T_1,T_2\in \ensuremath{\mathbb{R}}^{n_1\times n_2\times \dots \times n_d}$, let $\sigma$ be a $d$-to-$\tilde{d}$ flattening map. Then,
$$\langle T_1,T_2\rangle = \langle \sigma\dashv T_1,\sigma\dashv T_2\rangle.$$
In particular, $T_1$ and $T_2$ are orthogonal to each other if and only if $\sigma\dashv T_1$ and $\sigma\dashv T_2$ are.
\end{Lem}
\begin{proof}
A flattening is a bijection on the set of entries, therefore the result of the entry-wise scalar product is not changed by flattening.
\end{proof}
\begin{Prop}\label{Prop:prodorth}
Let $A^{(j)}_1,A^{(j)}_2\in \ensuremath{\mathbb{R}}^{n^{(j)}_1\times \dots\times n^{(j)}_{c_j}}$, for $j=1,\dots, k$. Then,
$$\left\langle A^{(1)}_1\otimes \dots A^{(k)}_1, A^{(1)}_2\otimes \dots A^{(k)}_2\right\rangle = \prod_{j=1}^k\left\langle A^{(j)}_1,A^{(j)}_2\right\rangle.$$
In particular, if there exists $j$ such that $A^{(j)}_1,A^{(j)}_2$ are orthogonal to each other, then the outer products $A^{(1)}_1\otimes \dots \otimes A^{(k)}_1$ and $A^{(1)}_2\otimes \dots \otimes A^{(k)}_2$ are orthogonal to each other.
\end{Prop}
\begin{proof}
By performing induction on $k$, it suffices to prove the statement for $k=2$: Let $A_1,A_2\in \ensuremath{\mathbb{R}}^{n_1\times \dots\times n_c}$ and $B_1,B_2\in \ensuremath{\mathbb{R}}^{n_{c+1}\times \dots\times n_d}.$ Then,
$$\langle A_1\otimes B_1, A_2\otimes B_2\rangle = \langle A_1,B_1\rangle\cdot \langle A_2,B_2\rangle.$$
We proceed to prove this statement. Let $\sigma_1$ be the $c$-to-$1$-flattening, let $\sigma_2$ be the $(d-c)$-to-$1$-flattening. Let $v_i=\sigma_1\dashv A_i$, and $w_i=\sigma_2\dashv B_i$ for $i=1,2$. By Lemma~\ref{Lem:orthflat}, it holds that
$$\langle A_i,B_i\rangle = \langle v_i,w_i\rangle\quad\mbox{for}\;i=1,2.$$
Let $\tau$ be the $d$-to-$2$-flattening defined by $\tau:\{1,\dots, c\}\mapsto \{1\}, \{c+1,\dots, d\}\mapsto \{2\}$. Let $C_i=\tau\dashv (A_i\otimes B_i)$. By Lemma~\ref{Lem:orthflat}, it holds that
$$\langle A_1\otimes B_1, A_2\otimes B_2\rangle = \langle C_1,C_2\rangle.$$
By Lemma~\ref{Lem:prodflat}, it holds that
$$\langle C_1,C_2\rangle = \langle v_1\otimes w_1, v_2\otimes w_2\rangle.$$
Using that scalar product on tensors is the trace product (see~\ref{Rem:sctr}), we obtain
$$\langle v_1\otimes w_1, v_2\otimes w_2\rangle = \operatorname{Tr}(v_1w_1^\top w_2 v_2^\top).$$
The cyclic property of the trace product for matrices yields
$$\operatorname{Tr}(v_1w_1^\top w_2 v_2^\top) = \operatorname{Tr}(w_1^\top w_2 v_2^\top v_1) = w_1^\top w_2v_2^\top v_1 = \langle v_1,v_2\rangle\cdot \langle w_1,w_2\rangle.$$
All equalities put together yield the claim.
\end{proof}
\begin{Cor}
Let $\mu_1,\mu_2\in \ensuremath{\mathbb{R}}^n$, and $d\in\ensuremath{\mathbb{N}}$, such that $\langle \mu_1,\mu_2\rangle = 0$. Then,
$$\left\langle\mu_1^{\otimes d}, \mu_2^{\otimes d}\right\rangle = 0.$$
\end{Cor}
\begin{Def}\label{Def:odec}
Let $T\in \ensuremath{\mathbb{R}}^{n_1\times n_2\times \dots \times n_d}$, let $[d]=S_1\cup S_2\cup\dots \cup S_k$ be a partition. A decomposition
$$T=\sum_{i=1}^r w_i\cdot A^{(1)}_i\otimes \dots \otimes A^{(k)}_i$$
with $w_i\in\ensuremath{\mathbb{R}}$, and $A^{(j)}_i\in \ensuremath{\mathbb{R}}^{\times_{\ell\in S_j} n_\ell}$, such that the set of $A^{(j)}_i$ with fixed $j$ is orthonormal, is called rank-$r$ \emph{orthogonal atomic decomposition} of $T$, with signature $(S_1,\dots, S_k)$. If $k=d$ and $S_i=\{i\}$, then the decomposition is called \emph{orthogonal CP-decomposition}.
\end{Def}
An orthogonal atomic decomposition does not need to exist necessarily. However, if it does, it is compatible with respect to flattenings, as Proposition~\ref{Prop:decompflat} will show. We introduce notation for a more concise statement of the compatibility first:
\begin{Def}
Let $(S_1,\dots, S_k)$ be a partition of $[d]$. We say a $d$-to-$\tilde{d}$-flattening $\sigma$ is \emph{compatible} with the partition $(S_1,\dots, S_k)$, if it holds that $\{i,j\}\in S_\ell$ for some $\ell$ implies $\sigma(i)=\sigma(j)$. We say that $\sigma$ is \emph{strictly compatible} with the partition $(S_1,\dots, S_k)$, if it holds that $\{i,j\}\in S_\ell$ for some $\ell$ if and only if $\sigma(i)=\sigma(j)$.
\end{Def}
\begin{Prop}\label{Prop:decompflat}
Let $T\in \ensuremath{\mathbb{R}}^{n_1\times n_2\times \dots \times n_d}$. Let
$$T=\sum_{i=1}^r w_i\cdot A^{(1)}_i\otimes \dots \otimes A^{(k)}_i$$
be an orthogonal atomic decomposition with signature $(S_1,\dots, S_k),$ let $\sigma$ be compatible with the signature. Then,
$$T=\sum_{i=1}^r w_i\cdot B^{(1)}_i\otimes \dots \otimes B^{(\tilde{d})}_i,\quad\mbox{where}\quad B^{(1)}_i=\sigma\dashv \left(\bigotimes_{j\in\sigma^{-1}(i)}B^{(1)}_j \right),$$
is an orthogonal atomic decomposition of $(\sigma\dashv T)$. In particular, if $\sigma$ is strictly compatible with the signature, then the decomposition is also an orthogonal CP-decomposition.
\end{Prop}
\begin{proof}
This is a direct consequence of Lemma~\ref{Lem:orthflat}, checking compatibility of scalar product and orthogonality with the flattening at each of the sets of indices $S_i$.
\end{proof}
\subsection{Identifiability of the Orthogonal Atomic Decomposition}
The orthogonal decomposition, as given in Definition~\ref{Def:odec}, does not need to exist for a tensor, nor does it need to be unique. We will show that due to the compatibility with flattenings, if it exists, it is unique, if the rank is chosen minimal.
The main ingredient, besides flattenings, is uniqueness of singular value decomposition~\cite{You36}, a classical result, which we state in a convenient form:
\begin{Thm}
Let $A\in\ensuremath{\mathbb{R}}^{m\times n}$, let $r=\operatorname{rank} A$. Then, there is a singular value decomposition (= orthogonal CP-decomposition)
$$A=\sum_{i=1}^r w_i\cdot u_i\cdot v_i^\top\quad\mbox{with}\;u_i\in\ensuremath{\mathbb{R}}^m, v_i\in \ensuremath{\mathbb{R}}^n, w_i\in \ensuremath{\mathbb{R}}$$
such that the $u_i$ are orthonormal, and the $v_j$ are orthonormal. In particular, there is no singular value decomposition of rank strictly smaller than $r$. Moreover, the singular value decomposition of $A$ is unique, up to:
\begin{description}
\item[(a)] the sequence of summation, i.e., up to arbitrary permutation of the indices $i=1,\dots, n$
\item[(b)] the choice of sign of $w_i,u_i,v_i$, i.e., up to changing the sign in any two of $w_i,u_i,v_i$ for fixed $i$
\item[(c)] unitary transformations of the span of $u_i,u_j$ or $v_i,v_j$ such that $|w_i|=|w_j|$
\end{description}
Condition (c) includes (b) as a special case, and (c) can be removed as a condition if no two distinct $w_i,w_j$ have the same absolute value.
\end{Thm}
\begin{Thm}\label{Thm:uniq}
Let $T\in \ensuremath{\mathbb{R}}^{n_1\times n_2\times \dots \times n_d}$, and assume that $T$ has an orthogonal atomic decomposition
$$T=\sum_{i=1}^r w_i\cdot A^{(1)}_i\otimes \dots \otimes A^{(k)}_i$$
of signature $(S_1,\dots, S_k)$, such that $w_i\neq 0$ for all $i$. Then:
\begin{description}
\item[(i)] Denote $N_j=\prod_{i\in S_j} n_i$ for $j=1,\dots, k$. Then, $r\le N_j$ for all $j$.
\item[(ii)] There is no orthogonal atomic decomposition of $T$ with signature $(S_1,\dots, S_k)$, and of rank strictly smaller than $r$.
\item[(iii)] The orthogonal atomic decomposition of $T$ of rank $r$ is unique, up to:
\begin{description}
\item[(a)] the sequence of summation, i.e., up to arbitrary permutation of the indices $i=1,\dots, n$
\item[(b)] the choice of sign of $w_i,A^{(k)}_i$, i.e., up to changing the sign in any two of $w_i$ and the $A^{(k)}_i$ for fixed $i$ and arbitrary $k$
\item[(c)] transformations of factors $A^{(k)}_i,A^{(k)}_j,$ and their respective tensor products, such that $|w_i|=|w_j|$, which induce unitary transformations in all flattenings compatible with the signature $(S_1,\dots, S_k)$.
\end{description}
Condition (c) includes (b) as a special case, and (c) can be removed as a condition if no two distinct $w_i,w_j$ have the same absolute value.
\end{description}
\end{Thm}
\begin{proof}
Fix some arbitrary $j$. Consider the $d$-to-$2$-flattening $\sigma: S_j\mapsto \{1\}, S_i\mapsto \{2\}$ for $i\neq j$, note that $\sigma$ is compatible with the signature. Let $m=N_j, n=\prod_{i\neq j} N_i$, and $A=\sigma\dashv T$. Note that $A$ is a $(m\times n)$-matrix. Let
$$T=\sum_{i=1}^r w_i \cdot A^{(1)}_i\otimes \dots \otimes A^{(k)}_i$$
be the orthogonal atomic decomposition of $T$, and let $u_i=\sigma\dashv A^{(j)}_i$, and $v_i=\sigma\dashv\bigotimes_{k\neq j} A^{(k)}_i$ for all $i$. Note that $u_i$ is an $m$-vector, and $v_j$ is an $n$-vector. By Proposition~\ref{Prop:decompflat},
$$A=\sum_{i=1}^r w_i\cdot u_i\cdot v_i^\top$$
is a singular value decomposition of $A$.\\
(i) In particular, the $u_i$ are a system of $r$ orthonormal vectors in $\ensuremath{\mathbb{R}}^m$. Therefore, $r\le m=N_j$. Since $j$ was arbitrary, statement (i) follows.\\
(ii) Since the $w_i$ are non-zero, it holds that $\operatorname{rank} A = r$. Would there be an orthogonal atomic decomposition of $T$ with signature $(S_1,\dots, S_k)$ of rank strictly smaller than $r$, there would be a singular value decomposition of $A$ of rank strictly smaller than $r$, contradicting Proposition~\ref{Prop:decompflat}.\\
(iii) Observe that the flattening by $\sigma$ induces a bijection between the orthogonal atomic decompositions of $T$, of rank $r$, and the singular value decompositions of $A$, of rank $r$. The statement in (iii) then follows directly from the uniqueness asserted in Proposition~\ref{Prop:decompflat} for the singular value decomposition of $A$.
\end{proof}
Again, we would like to stress that the present orthogonal decomposition model is different from the one in~\cite{Kolda01orthogonaltensor}; ours being factor-wise orthogonal between different summands, while the orthogonal rank decomposition in~\cite{Kolda01orthogonaltensor} being summand-wise orthogonal, and the combinatorial orthogonal rank decomposition enforcing orthogonality of factors in the same summand. Therefore, Theorem~\ref{Thm:uniq} does not contradict Lemma~3.5 in~\cite{Kolda01orthogonaltensor}.\\
Another result which seems to be folklore, but not available in the literature, is that it is a strong restruction for a tensor to assume that it has an orthogonal decomposition. Since it is almost implied by the identifiability Theorem~\ref{Thm:uniq}, we state a quantitative version of this:
\begin{Prop}\label{Prop:notallorth}
The set of tensors $T\in \ensuremath{\mathbb{R}}^{n_1\times n_2\times \dots \times n_d}$, with $d\ge 3$, and $n_j\ge 2$ for all $j$, for which $T$ has an orthogonal CP-decomposition, is a Lebesgue zero set.
\end{Prop}
\begin{proof}
The CP-decomposition can be viewed as an algebraic map
\begin{align*}
\phi: \ensuremath{\mathbb{R}}\times \ensuremath{\mathbb{R}}^{n_1}\times\dots\times\ensuremath{\mathbb{R}}^{n_d}&\rightarrow \ensuremath{\mathbb{R}}^{n_1\times n_2\times \dots \times n_d}\\
(w_i,v^{(j)}_i) &\mapsto \sum_{i=1}^r w_i\cdot v^{(1)}_i\otimes \dots \otimes v^{(k)}_i.
\end{align*}
Since the left hand side is an irreducible variety, the image of the map $\phi$ also is. The orthogonal CP-decompositions form an algebraic subset of the left hand side. Therefore the state follows from the fact that $\phi$ is not surjective. This follows from a degree of freedom resp.~dimension count. One has
\begin{align*}
D_1&:=\dim\ensuremath{\mathbb{R}}^{n_1\times n_2\times \dots \times n_d} =\prod_{i=1}^d n_d,\quad\mbox{and }\\
D_2&:=\dim (\ensuremath{\mathbb{R}}\times \ensuremath{\mathbb{R}}^{n_1}\times\dots\times\ensuremath{\mathbb{R}}^{n_d})^r = r\cdot (n_1+\dots + n_d+1).
\end{align*}
Theorem~\ref{Thm:uniq}~(i) implies
$$D_2\le n_j\cdot (n_1+\dots+n_k+1).$$
An explicit computation shows that $D_1\gneq D_2$, which proves the statement.\\
The proof above can be rephrased in terms of the CP-rank (see~\cite{Cat2002} for an introduction), can be obtained by observing that the generic CP-rank of the tensors in questions must be strictly larger than $\min (n_1,\dots, n_k)$, then proceeding again by arguing that the algebraic set of tensors with orthogonal CP-decompositions must be a proper subset of all tensors with that format, thus a Lebesgue zero set.
\end{proof}
Proposition~\ref{Prop:notallorth} can be extended to orthogonal atomic decompositions with signature $(S_1,\dots, S_k), k\ge 3$, by considering suitable unflattenings.
\subsection{Tensors and Moments}
We briefly show how tensors relate to moments of multivariate real random variables:
\begin{Def}
Let $X$ be a real $n$-dimensional random variable. Then, define:
\begin{align*}
\mbox{the characteristic function of}\; X\;\mbox{as}\quad &\quad\quad \varphi_X(\tau) := \ensuremath{\mathbb{E}} \left[ \exp \left(i \tau X\right) \right], \\
\mbox{the moment generating function of}\; X\mbox{as}\quad &\quad\quad \chi_X(\tau) := \log\ensuremath{\mathbb{E}} \left[ \exp \left(i \tau X\right) \right],
\end{align*}
where $\tau\in \ensuremath{\mathbb{R}}^{1\times n}$ is a formal vector of variables. The $d$-th \emph{moment} (or moment tensor) $\ensuremath{\mathbf{M}}_d(X)\in \ensuremath{\mathbb{R}}^{n^{(\times d)}}$ of $X$, and the $d$-th \emph{cumulant} (or cumulant tensor) $\ensuremath{\kappa}_d(X)\in \ensuremath{\mathbb{R}}^{n^{(\times d)}}$ of $X$ are defined\footnote{in case of convergence} as the coefficients in the multivariate Taylor expansions
\begin{align*}
\varphi_X(\tau) & = \sum_{d=1}^\infty \left(i \tau\right) \circ \frac{\ensuremath{\mathbf{M}}_d(X)}{d!},\\
\chi_X(\tau) & = \sum_{d=1}^\infty \left(i \tau\right) \circ \frac{\ensuremath{\kappa}_d(X)}{d!}.
\end{align*}
\end{Def}
In the following, we will always assume that the moments and cumulants in question exist.
The moments and cumulants of a linearly transformed random variable are the multilinearly transformed
moments.
\begin{Prop}\label{Prop:lintrans}
Let $X$ be a real $n$-dimensional random variable and let $A \in \ensuremath{\mathbb{R}}^{m \times n}.$ Then,
\begin{align*}
\ensuremath{\mathbf{M}}_d(A\cdot X) &= A \circ \ensuremath{\mathbf{M}}_d(X),\\
\ensuremath{\kappa}_d(A\cdot X) &= A \circ \ensuremath{\kappa}_d(X).
\end{align*}
\end{Prop}
\begin{proof}
We prove the statement for cumulants, the proof for moments is completely analogous.
For the cumulant generating functions $\chi_X$ of $X$ and $\chi_{A\cdot X}$ of $A\cdot X$, it holds that
\begin{align*}
\chi_{A\cdot X}(\tau) & = \ensuremath{\mathbb{E}} \left[ \exp \left(i \tau\cdot A\cdot X\right) \right] \\
& = \ensuremath{\mathbb{E}} \left[ \exp \left(i \left(\tau\cdot A \right) \cdot X\right) \right]\\
& = \sum_{d=1}^\infty \left(i \tau\right)\circ \left(A \circ \frac{\ensuremath{\mathbf{M}}_d(X)}{d!}\right).
\end{align*}
The last equality follows from the definition of $\chi_X(\tau)$. But by definition, it also holds that
\begin{align*}
\chi_{A\cdot X}(\tau) & = \sum_{d=1}^\infty \left(i \tau\right) \circ \frac{\ensuremath{\mathbf{M}}_d(A\cdot X)}{d!},
\end{align*}
therefore the statement follows from comparing coefficient tensors.
\end{proof}
\section{Relation to Mixture Models}
\label{sec:estimation}
\subsection{The Estimation Problem}
Throughout the paper, we will consider the following independent rank $1$ mixture model:\\
{\bf Generative Model:} $X_1,\dots, X_r$ are independent, $\ensuremath{\mathbb{R}}^n$-valued random variables, with $r\le n$, and probability/mass density functions $X_i\sim p_i$. Let $w_1,\dots, w_r\in\ensuremath{\mathbb{R}}$ be arbitrary such that $\sum_{i=1}^r w_i = 1$, and let $Y\sim \sum_{i=1}^r w_r p_i$ be the corresponding mixture of the $X_i$. Assume that there are $\mu_1,\dots, \mu_r\in\ensuremath{\mathbb{R}}^n$ with $\|\mu_i\|_2=1$, and random variables $Z_i\in\ensuremath{\mathbb{R}}$, such that $X_i=\mu_i\cdot Z_i$. Assume that the $\mu_i$ are linearly independent, and $\ensuremath{\mathbf{M}}_d(Z_i)=1$ for $d=2,\dots, m$.\\
{\bf Estimation Task:} Given $\ensuremath{\mathbf{M}}_2 (Y),\ensuremath{\mathbf{M}}_3 (Y),\dots,\ensuremath{\mathbf{M}}_m (Y), m\ge 3$, or estimators thereof, determine/estimate $\mu_i$ and $w_i$ for $i=1,\dots, r$.\\
While the above scenario seems very restrictive, several important problems can be reduced to this setting, see for example~\cite{Hsu2012}, or chapter 3 of~\cite{Ana2012}. We recommend the interested reader to read the exposition there.
\subsection{Algebraic Formulation via Moments}
The estimation problem presented above can be reformulated as a purely algebraic problem, see~\cite{Ana2012}. Namely, the $\ensuremath{\mathbf{M}}_i$ are explicitly calculable in terms of the expectations of the $\mu_i$ and $w_i$. Then, Proposition~\ref{Prop:lintrans} implies that $\ensuremath{\mathbf{M}}_d(X_i)= \mu_i^{\otimes d}$ for all $d$, therefore $\ensuremath{\mathbf{M}}_d(Y)=\sum_{i=1}^r w_i\cdot \mu_i^{\otimes d}$ for all $d$, thus yielding the following algebraic version of the estimation problem.\\
{\bf Algebraic Problem:} Let $r\le n$, let $\mu_1,\dots, \mu_r\in\ensuremath{\mathbb{R}}^n$ be linearly independent and $w_1,\dots, w_r\in\ensuremath{\mathbb{R}}$ arbitrary such that $\sum_{i=1}^r w_i = 1$. Given (exact or noisy estimators for)
$$\ensuremath{\mathbf{M}}_d = \sum_{i=1}^r w_i\cdot \mu_i^{\otimes d}\quad\mbox{for}\;d=2,\dots, m,\;\mbox{with}\;m\ge 3,$$
determine the $\mu_i$ and $w_i$.\\
\section{Algorithms}
\subsection{Orthogonal Decomposition of Tensors}
A special case of orthogonal decomposition is singular value decomposition (SVD). There are a huge amount of well-studied methods for obtaining the singular value decomposition, which we will not discuss. However, we will make extensive use of the SVD algorithm, as described in Algorithm~\ref{Alg:SVD} as a black box.
\begin{algorithm}[ht]
\caption{\label{Alg:SVD} \texttt{SVD}. Singular Value Decomposition of Matrices.\newline
\textit{Input:} A matrix $A\in\ensuremath{\mathbb{R}}^{m\times n}$.
\textit{Output:} The singular value decomposition $A= U\cdot \Sigma\cdot V^\top$, with $U\in\ensuremath{\mathbb{R}}^{m\times r}, V\in\ensuremath{\mathbb{R}}^{n\times r}$ orthogonal, $\Sigma\in\ensuremath{\mathbb{R}}^{r\times r}$ diagonal, and the rank $r=\operatorname{rank} A$}
\end{algorithm}
First, for completeness, we treat the trivial case in Algorithm~\ref{Alg:orthdecomp1}.
\begin{algorithm}[ht]
\caption{\label{Alg:orthdecomp1} \texttt{OTD1}. Orthogonal Tensor Decomposition in one factor.\newline
\textit{Input:} A tensor $T\in\ensuremath{\mathbb{R}}^{n_1\times\dots\times n_d}$, a signature $(S_1)$.
\textit{Output:} The orthogonal atomic decomposition $T=\sum_{i=1}^r w_i\cdot A_i$.}
\begin{algorithmic}[1]
\State Return rank $r=1,$ coefficients $w_1=\|T\|,$ factors $A_1=\|T\|^{-1}\cdot T$.
\end{algorithmic}
\end{algorithm}
Now we explicitly describe how to compute the orthogonal decomposition if each summand has two tensor factors. Algorithm~\ref{Alg:orthdecomp2} computes the decomposition by a proper reformatting of the entries, computing the singular value decomposition, then reformatting again.
\begin{algorithm}[ht]
\caption{\label{Alg:orthdecomp2} \texttt{OTD2}. Orthogonal Tensor Decomposition in two factors.\newline
\textit{Input:} A tensor $T\in\ensuremath{\mathbb{R}}^{n_1\times\dots\times n_d}$, a signature $(S_1, S_2)$.
\textit{Output:} The orthogonal atomic decomposition $T=\sum_{i=1}^r w_i\cdot A_i\otimes B_i$ (assumed to exist), including the rank $r$}
\begin{algorithmic}[1]
\State Define $\sigma: [d]\rightarrow [2], S_i\mapsto \{i\}.$
\State Set $A\leftarrow (\sigma\dashv T)$. Note that $A\in\ensuremath{\mathbb{R}}^{m\times n}$, with $m=\prod_{i\in S_1}n_i, n=\prod_{i\in S_2}n_i.$
\State Compute the \texttt{SVD} of $A=U\cdot \Sigma\cdot V^\top$, see Algorithm~\ref{Alg:SVD}.
\State Return rank $r=\operatorname{rank} A$.
\State Return coefficients $w_i=\Sigma_{ii}$ for $i=1,\dots, r$.
\State For all $i$, let $U_i$ be the $i$-th column of $U$, let $V_i$ be the $i$-th columns of $V$.
\State Return factors $A_i=\sigma\vdash U_i, B_i =\sigma\vdash V_i$ for $i=1,\dots, r$.
\end{algorithmic}
\end{algorithm}
The algorithm for the general case, Algorithm~\ref{Alg:orthdecomp}, consists as well of repeated applications of reindexing and singular value decomposition. Variants of singular value decomposition exist with adjustable noise tolerance or singular value thresholding, and can therefore be employed to obtain thresholding and numerically stable variants of Algorithm~\ref{Alg:orthdecomp}. Furthermore, step~\ref{Alg:orthdecomp-step1} allows for an arbitrary choice of $k$-to-$2$-flattening, in each recursion. Since in the presence of noise, the results might differ when taking a different sequence flattenings, the numerical stability can be improved by clustering the results of all possible choices, then averaging.
\begin{algorithm}[ht]
\caption{\label{Alg:orthdecomp} \texttt{OTD}. Orthogonal Tensor Decomposition.\newline
\textit{Input:} A tensor $T\in\ensuremath{\mathbb{R}}^{n_1\times\dots\times n_d}$, a signature $(S_1, \dots, S_k)$.
\textit{Output:} The orthogonal atomic decomposition $T=\sum_{i=1}^r w_i\cdot A^{(1)}_i\otimes \dots \otimes A^{(k)}_i$ (assumed to exist), including the rank $r$}
\begin{algorithmic}[1]
\State \label{Alg:orthdecomp-step1} Choose any $k$-to-$2$-flattening map $\tau$.
\State Set $\tilde{S}_j\leftarrow \cup_{i\in\tau^{-1}(j)}S_i$ for $j=1,2$.
\State Set $\tilde{T}\leftarrow \tau\dashv T$.
\State Use \texttt{OTD2}, Algorithm~\ref{Alg:orthdecomp2}, to compute the orthogonal atomic decomposition
$\tilde{T}=\sum_{i=1}^r w_i\cdot A_i\otimes B_i$
with signature $(\tilde{S}_1,\tilde{S}_2)$.
\State Return the $w_i$ as coefficients and $r$ as the rank for the decomposition of $T$.
\State \label{Alg:orthdecomp-step6} For $i=1,\dots, r$, use the suitable one of \texttt{OTD1},\texttt{OTD2},\texttt{OTD}, i.e., Algorithm~\ref{Alg:orthdecomp1},\ref{Alg:orthdecomp2}, or~\ref{Alg:orthdecomp}, to compute the orthogonal atomic decomposition $(\tau\vdash A_i)=\sum_{i=1}^1 1\cdot\bigotimes_{\tau(j)\in S_1} A^{(j)}_i$, noting that rank is one, and using the signature $(S_j\;:\;\tau(j)\in \tilde{S}_1).$
\State \label{Alg:orthdecomp-step7} For $i=1,\dots, r$, use the suitable one of \texttt{OTD1},\texttt{OTD2},\texttt{OTD}, i.e., Algorithm~\ref{Alg:orthdecomp1},\ref{Alg:orthdecomp2}, or~\ref{Alg:orthdecomp}, to compute the orthogonal atomic decomposition $(\tau\vdash B_i)=\sum_{i=1}^1 1\cdot\bigotimes_{\tau(j)\in S_2} A^{(j)}_i$, noting that rank is one, and using the signature $(S_j\;:\;\tau(j)\in \tilde{S}_2).$
\State Return the $A^{(j)}_i$ as factors for $T$.
\end{algorithmic}
\end{algorithm}
Termination of Algorithm~\ref{Alg:orthdecomp} is implied by the observation that in each recursion, the partition of $[d]$ is made strictly finer. Since $[d]$ has finite cardinality, there is only a finite number of recursions. The fact that the decompositions in steps~\ref{Alg:orthdecomp-step6} and~\ref{Alg:orthdecomp-step7} have rank one, and coefficients $1$, follows from the uniqueness of the orthogonal decomposition guaranteed in Theorem~\ref{Thm:uniq}. Correctness of Algorithm~\ref{Alg:orthdecomp} follows from repeated application of Proposition~\ref{Prop:decompflat}, and the uniqueness of singular value decomposition.
\subsection{An Estimator for the Mixture Model}
For illustrative purposes, we write out Algorithm~\ref{Alg:orthdecomp} for the problem introduced in~\ref{sec:estimation}, which has also extensively been studied in~\cite{Ana2012}:
{\bf Example:} Let $r\le n$, let $\mu_1,\dots, \mu_r\in\ensuremath{\mathbb{R}}^n$ be linearly independent and $w_1,\dots, w_r\in\ensuremath{\mathbb{R}}$ arbitrary such that $\sum_{i=1}^r w_i = 1$. Given (exact or noisy estimators for)
$$\ensuremath{\mathbf{M}}_d = \sum_{i=1}^r w_i\cdot \mu_i^{\otimes d}\quad\mbox{for}\;d=2,3,$$
determine the $\mu_i$ and $w_i$.\\
Algorithm~\ref{Alg:degree3} solves the problem, by reducing it to
\begin{algorithm}[ht]
\caption{\label{Alg:degree3} Model identification.\newline
\textit{Input:} $\ensuremath{\mathbf{M}}_2,\ensuremath{\mathbf{M}}_3$
\textit{Output:} $w_1,\dots, w_r, \mu_1,\dots,\mu_r$.}
\begin{algorithmic}[1]
\State Set $r\leftarrow \operatorname{rank}(\ensuremath{\mathbf{M}}_2)$.
\State Compute the SVD\footnote{Note: since $\ensuremath{\mathbf{M}}_2$ is symmetric, the SVD also is.} $\ensuremath{\mathbf{M}}_2=U\cdot \Sigma\cdot U^\top$.
\State Set $W\leftarrow U\cdot \Sigma^{-\frac{1}{2}}$.
\State \label{Alg:degree3-step4}Set $T:=W^\top\circ \ensuremath{\mathbf{M}}_3$.
\State Define the flattening map $\sigma: 1\mapsto 1,2\mapsto 2, 3\mapsto 2$.
\State Set $\tilde{T}:=\sigma\dashv T.$
\State Compute the rank $r$ SVD $\tilde{T}=\sum_{i=1}^r \tilde{w}_i\cdot \tilde{\mu}^{(1)}_i\cdot v_i^\top$.
\State \label{Alg:degree3-step8} Return $w_i= \tilde{w}_i^{-2}$ for $i=1,\dots, r$.
\State Set $\tilde{A}_i=(\sigma\vdash v_i)$ for $i=1,\dots, r$.
\State \label{Alg:degree3-step10} Compute the rank $1$ SVD $\tilde{A}_i= \tilde{\mu}^{(2)}_i\cdot \left(\tilde{\mu}^{(3)}_i\right)^\top$.
\State \label{Alg:degree3-step11} Set $\tilde{\mu}_i\leftarrow \frac{1}{3}\left(\tilde{\mu}^{(1)}_i+\tilde{\mu}^{(2)}_i+\tilde{\mu}^{(3)}_i\right),$ for $i=1,\dots, r$.
\State \label{Alg:degree3-step12} Compute the pseudo-inverse $B$ of $W'$. Return $\mu_i = B\cdot \tilde{\mu}_i\cdot \tilde{w}_i$, for $i=1,\dots, r.$
\end{algorithmic}
\end{algorithm}
Theorem 4.3 in~\cite{Ana2012} implies that the tensor $T$ obtained in step~\ref{Alg:degree3-step4} has an orthogonal CP-decomposition, and it implies the correctness of steps~\ref{Alg:degree3-step8}, and~\ref{Alg:degree3-step12}. The fact that $\tilde{A}_i$ in step~\ref{Alg:degree3-step8} has rank one, and the coefficients are $1$, follow from the uniqueness of the decomposition guaranteed in Theorem~\ref{Thm:uniq}.
Note that explicit presentation of the algorithm could be substantially abbreviated by applying $\texttt{ODT}$ directly to $\tilde{T}$ in step~\ref{Alg:degree3-step4}, with signature $(\{1\},\{2,3\})$, and then performing the analogues of steps~\ref{Alg:degree3-step8} and~\ref{Alg:degree3-step12}. Furthermore, the accuracy of the estimator in step~\ref{Alg:degree3-step11} can be improved, by repeating the procedure for the three possible signatures $(\{1\},\{2,3\}), (\{2\},\{1,3\}),$ and $(\{3\},\{1,3\})$, then averaging, or weighted averaging, over the nine estimates for each $\tilde{\mu}_i$, making use of the symmetry of the problem.\\
Also, similar to Algorithm~\ref{Alg:orthdecomp}, the presented Algorithm~\ref{Alg:degree3}, while already numerically stable, can be modified to cope better with noise by, e.g., introducing thresholding to the singular value decomposition and rank computations. The numerical stability with respect to noise is governed by the numerical stability of the SVDs performed, and the pseudo-inversion of $W'$ in step~\ref{Alg:degree3-step12}.\\
Algorithm~\ref{Alg:degree3} is also related to Algorithm~1 proposed in ~\cite{Ana2012NIPS}. Namely, $\mbox{Triples}(\eta)$, as defined in section~4.1, is a degree $2$-projection of the tensor $T$, and therefore can be also understood as a random projection of the flattening $\sigma\dashv T$.\\
Furthermore, an estimator for the hierarchical models described in~\cite{Ish2013ICML,Son2013ICML} can be constructed in a similar way.
\section{Conclusion}
We have demonstrated that computing the orthogonal decomposition of an arbitrary degree tensor, symmetric or not, can be reduced to a series of singular value decompositions, and we have described efficient algorithms to do so. This makes orthogonal tensor decomposition approachable by the wealth of theoretical results and existing methods for eigenvalue problems and singular value decomposition. Moreover, we have exemplified our method in the case of identifying components in a low-rank mixture model.
\section*{Acknowledgments}
I thank Arthur Gretton, Zolt\'an Szab\'o, and Andreas Ziehe for interesting discussions. I thank the Mathematisches Forschungsinstitut Oberwolfach for support.
\bibliographystyle{plainnat}
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"redpajama_set_name": "RedPajamaArXiv"
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Q: Swift 4: Firebase Timestamp How should i compare Firebase Timestamp with local device time.
let serverTIme = FIRServerValue.timestamp()
let localtime = Date()
let dbRef = FirebaseServerReference()
here is full code https://pastebin.com/4drtSZA6
Edit:
When i set timestamp in Firebase RealtimeDatabase using FIRServerValue.timestamp it suppose to save timestamp of Server not the my device time.
In my case, it looks like it is depending on my device time. i want to check that if there is a 'Time Difference' in server time & my device time.
So i think that this question might not possible duplicate of Trying to convert Firebase timestamp to NSDate in Swift.
A: You can convert Firebase timestamp to NSDate and then compare Date() later on. Here is a similar question and a possible solution:
Trying to convert Firebase timestamp to NSDate in Swift
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{
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\section{Introduction}
One-point Hermitian ($1$-H) codes are algebraic geometry codes that can be decoded beyond half the minimum distance.
Most of their decoders are conceptually similar to their Reed--Solomon (RS) code analogs, such as the \emph{Guruswami--Sudan} (GS) algorithm \cite{guruswami1998improved} and \emph{power decoding} (PD) \cite{schmidt2010syndrome,kampf2012decoding,nielsen2015sub}.
For both RS and $1$-H codes, PD is only able to correct as many errors as the Sudan algorithm, which is a special case of the GS algorithm.
Recently \cite{nielsen2016power}, PD for RS codes was improved to correct as many errors as the GS algorithm.
In this paper, we combine the idea of improved power decoding (IPD) for RS codes from \cite{nielsen2016power} with the description of PD for $1$-H codes from \cite{nielsen2015sub} in order to obtain an IPD algorithm for $1$-H codes.
Similar to the RS case, we derive a larger system of non-linear key equations (cf.~Section~\ref{sec:key_equations}) than in classical PD and reduce the decoding problem to a linear Pad\'e approximation problem whose solution is likely to agree with the solution of the system of key equations (cf.~Section~\ref{sec:pade_approximation}).
Using a linear-algebraic argument, we derive an upper bound on the maximum number of errors which can possibly be corrected by the decoder (cf.~Section~\ref{sec:decoding_radius}).
In Section~\ref{sec:complexity}, we show that the algorithm can be implemented with sub-quadratic complexity in the code length $n$.
Finally, we present simulation results for various code and decoder parameters which indicate that the new IPD algorithm has a similar failure probability as the GS algorithm for the same parameters and decoding radius (cf.~Section~\ref{sec:numerical_results}).
Besides the theoretical interest in having different decoding paradigms, we see two advantages of the new decoder:
Firstly, the algorithm does not require a root-finding step, which is often considered to be computationally heavy, especially in practical implementations, see e.g.~\cite{ahmed2011vlsi}.
Secondly, the IPD algorithm for RS codes \cite{nielsen2016power} was recently generalized to interleaved RS codes \cite{puchinger2017decoding}, where it improves upon existing decoding algorithms at all rates, including those methods which are based on the GS decoder. It is reasonable to assume that a similar generalization is also possible for $1$-H codes.
\section{Preliminaries}
Let $q$ be a prime power. We follow the notation of \cite{nielsen2015sub}.
The \emph{Hermitian curve} $\H/\mathbb{F}_{q^2}$ is the smooth projective plane curve defined by the affine equation $Y^q+Y=X^{q+1}$.
The curve $\H(\mathbb{F}_{q^2})$ has genus $g = \tfrac{1}{2}q(q-1)$ and $q^3+1$ many $\mathbb{F}_{q^2}$-rational points $\mathcal{P} = \{P_1,\dots,P_{q^3},{P_\infty}\}$, where ${P_\infty}$ denotes the point at infinity.
We define $\mathcal{R} := \cup_{m\geq0} \L(m{P_\infty}) = \mathbb{F}_{q^2}[X,Y]/(Y^q+Y-X^{q+1})$, which has an $\mathbb{F}_{q^2}$-basis of the form $\{ X^iY^j : 0 \leq i, 0 \leq j < q\}$. The order function $\deg_{\mathcal{H}} : \mathcal{R} \to \mathbb{N}_0 \cup \{-\infty\}, f \mapsto -v_{P_\infty}(f)$ is defined by the valuation $v_{P_\infty}$ at ${P_\infty}$. As a result, we have $\deg_{\mathcal{H}}(X^iY^j) = i q + j(q+1)$.
Let $n=q^3$ and $m \in \mathbb{N}$ with $2(g-1) < m <n$. The \emph{one-point Hermitian code} of length $n$ and parameter $m$ over $\mathbb{F}_{q^2}$ is defined by
\begin{align*}
\mathcal{C}_\mathcal{H} = \left\{ \left( f(P_1) , \dots , f(P_n) \right) : f \in \L(m{P_\infty}) \right\}.
\end{align*}
The dimension of $\mathcal{C}_\mathcal{H}$ is given by $k=m-g+1$ and the minimum distance $d$ is lower-bounded by the \emph{designed minimum distance} $d^\ast := n-m$.
\section{System of Key Equations}
\label{sec:key_equations}
In this section, we derive the system of key equations that we need for decoding, using the same trick as \cite{nielsen2016power} for Reed--Solomon codes.
We use the description of power decoding for one-point Hermitian codes as in \cite{nielsen2015sub}.
Suppose that the received word is $\r = \c + \ve e \in \mathbb{F}_{q^2}^n$, consisting of an error $\ve e = (e_1,\dots,e_n)$ and a codeword $\c \in \mathcal{C}_\mathcal{H}$, which is obtained from the \emph{message polynomial} $f \in \L(m{P_\infty})$. We denote the set of \emph{error positions} by $\mathcal{E} = \{i : e_i \neq 0\}$.
In the following sections we show how to retrieve the message polynomial $f$ from the received word $\r$ if the \emph{number of errors}, the Hamming weight $\mathrm{wt}_\mathrm{H}(\ve e) = |\mathcal{E}|$ of the error, does not exceed a certain decoding radius $\tau$, which depends on the parameters of the decoding algorithm.
A non-zero polynomial $\Lambda \in \L\left(-\sum_{i \in \mathcal{E}} P_i + \infty {P_\infty}\right)$ is called \emph{error locator}. It is well-known that there is an error locator of degree $|\mathcal{E}| \leq \deg_{\mathcal{H}} \Lambda \leq |\mathcal{E}|+g$, cf.~\cite{nielsen2015sub}.
and that any error locator fulfills $\deg_{\mathcal{H}}(\Lambda) \geq |\mathcal{E}|$ (cf.~\cite{nielsen2015sub}).
In this section, let $\Lambda$ be some error locator.
\begin{lemma}[\!\!{\cite[Lemma~6]{nielsen2015sub}}]
\label{lem:IP}
There is a polynomial $R \in \mathcal{R}$ with $\deg_{\mathcal{H}}(R)<n+2g$ that satisfies $R(P_i) = r_i$ for all $P_i \in \mathcal{P}^\ast$.
\end{lemma}
In the following, let $R \in \mathcal{R}$ be as in Lemma~\ref{lem:IP} and $G \in \mathcal{R}$ be defined as
\begin{align*}
G = \textstyle\prod_{\alpha \in \mathbb{F}_{q^2}} (X-\alpha) = X^{q^2}-X.
\end{align*}
By \cite[Theorem~24]{nielsen2015sub}, we know that there is a unique \emph{error evaluator polynomial} $\Omega \in \mathcal{R}$ that fulfills $\Lambda(R-f) = \Omega G$.
The following theorem states the system of key equations that we will use for decoding in the next sections.
Note that the formulation is similar to its Reed--Solomon analog \cite[Theorem~3.1]{nielsen2016power}, with the difference that all involved polynomials are elements of the ring $\mathcal{R}$.
\begin{theorem}[System of Key Equations]\label{thm:key_equations}
Let $f$, $\Lambda$, $G$, $R$, and $\Omega$ be as above, and $\ell,s \in \mathbb{N}$ such that $s \leq \ell$. Then, as a congruence over $\mathcal{R}$,
\begin{align}
\Lambda^s f^t &= \sum\limits_{i=0}^{t} \Lambda^{s-i} \Omega^i\tbinom{t}{i} R^{t-i} G^i &&\forall t=1,\dots,s-1, \label{eq:key_eq_equality}\\
\Lambda^s f^t &\equiv \sum\limits_{i=0}^{s-1} \Lambda^{s-i} \Omega^i\tbinom{t}{i} R^{t-i} G^i \mod G^s &&\forall t=s,\dots,\ell. \label{eq:key_eq_congruence}
\end{align}
\end{theorem}
\begin{proof}
We know that $\Omega G = \Lambda\left(f-R\right)$. Thus, for $s,t \in \mathbb{N}$, we have
\begin{align*}
\Lambda^s f^t = \Lambda^s \left(R + \left(f-R\right)\right)^t
= \sum\limits_{i=0}^{t} \tbinom{t}{i} \Lambda^s \left(f-R\right)^i R^{t-i}.
\end{align*}
In all summands with $i<s$, we can rewrite $\Lambda^s \left(f-R\right)^i = \Lambda^{s-i} (\Lambda\left(f-R\right))^i = \Lambda^{s-i} (\Omega G)^i$. If $i \geq s$, $\Lambda^s \left(f-R\right)^i = (\Lambda(f-R))^s (f-R)^i = (\Omega G)^s (f-R)^{s-i}$, so all those summands are divisible by $G^s$, resulting in
\begin{align*}
\Lambda^s f^t = \sum\limits_{i=0}^{\min\{t,s-1\}} \Lambda^{s-i} \Omega^i\binom{t}{i} R^{t-i} G^i + G^s \left(\sum\limits_{i=\min\{t+1,s\}}^{t} \binom{t}{i} \Omega^s (f-R)^{s-i} R^{t-i} \right).
\end{align*}
For $t < s$, we obtain \eqref{eq:key_eq_equality} since the second part of the sum vanishes and for $t \geq s$, \eqref{eq:key_eq_congruence} holds because the latter sum is divisible by $G^s$. \qed
\end{proof}
\section{Solving the System of Key Equations}
\label{sec:pade_approximation}
The idea of decoding is to find the message polynomial $f$ from the known polynomials $R$ and $G$.
Since the system of key equations is non-linear in the unknown polynomials $\Lambda$, $\Omega$, and $f$, we cannot solve it directly.
Instead, we consider the following linearized problem, a \emph{Pad\'e approximation} problem.
\begin{problem}\label{prob:Pade_R}
Let $G$ and $R$ be as in Section~\ref{sec:key_equations}. Given $\tau \in \mathbb{N}$ and
\begin{align*}
\asummands{t,i} := \binom{t}{i} R^{t-i} G^i, \quad
G_t := \begin{cases}
x^{\lfloor \frac{t(n+2g-1)+\tau}{q}\rfloor+1}, &\forall t=1,\dots,s-1, \\
G^s, &\forall t=s,\dots,\ell,
\end{cases}
\end{align*}
for all $i=0,\dots,s-1$ and $t=1,\dots,\ell$, find a vector
\begin{align*}
(\lambdai{0},\dots,\lambdai{s-1},\psii{1},\dots,\psii{\ell}) \in \mathcal{R}^{s+\ell} \setminus \{ \ve 0 \},
\end{align*}
with minimal $\deg_{\mathcal{H}}(\lambdai{0})$ which satisfies
\begin{align}
\sum\limits_{i=0}^{s-1} \lambdai{i} \asummands{t,i} &\equiv \psii{t} \mod G_t \quad &&\forall t=1,\dots,\ell, \label{eq:Pade_R_congruence}\\
\deg_{\mathcal{H}}(\lambdai{i}) &\leq s\tau + i(2g-1) \quad &&\forall i = 0,\dots,s-1, \label{eq:Pade_R_degree_1} \\
\deg_{\mathcal{H}}(\psii{t}) &\leq s\tau + tm \quad &&\forall t = 1,\dots,\ell, \label{eq:Pade_R_degree_2}
\end{align}
where the congruences are over $\mathcal{R}$.
\end{problem}
The following theorem motivates the statement of Problem~\ref{prob:Pade_R} by showing that the polynomials $\Lambda^{s-i}\Omega^i$ and $\Lambda^sf^t$ that occur in the key equation fulfill the congruences and degree constraints of the problem.
The minimality condition ensures that if the problem solution corresponds to an error locator $\Lambda$ of some error vector $\ve e$ (not necessarily the same $\ve e$ as in Section~\ref{sec:key_equations}), i.e., $\lambdai{0} = \Lambda^s$, then it is the one of smallest degree, and thus hopefully the one corresponding to the $\ve e$ of smallest Hamming weight.
The theorem also implies a strategy to obtain $f$ after having solved Problem~\ref{prob:Pade_R}:
If the solution of Problem~\ref{prob:Pade_R} results in $\lambdai{0} = \Lambda^s$ and $\psii{1} = \Lambda^sf$ for some error locator $\Lambda$, we divide $\psii{1}$ by $\lambdai{0}$. See~\cite{nielsen2015sub} for how this division can be performed.
\begin{theorem}\label{thm:decoding_solution_is_Pade_solution}
Let $f$, $\Lambda$, $G$, $R$, and $\Omega$ be as in Section~\ref{sec:key_equations}, and $\ell,s \in \mathbb{N}$ such that $s \leq \ell$.
For $t=1,\dots,\ell$ and $i=0,\dots,s-1$, we define the polynomials (all in $\mathcal{R}$)
\begin{align*}
\Lambdai{i} := \Lambda^{s-i}\Omega^i, \quad
\Psii{t} := \Lambda^s f^t,
\end{align*}
Then, $(\Lambdai{0},\dots,\Lambdai{s-1},\Psii{1},\dots,\Psii{\ell})$ satisfies Conditions \eqref{eq:Pade_R_congruence} - \eqref{eq:Pade_R_degree_2} of Problem~\ref{prob:Pade_R} for any $\tau \geq \deg_{\mathcal{H}}(\Lambda)$.
\end{theorem}
\begin{proof}
Inequality \eqref{eq:Pade_R_degree_1} is fulfilled since
\begin{align*}
\deg_{\mathcal{H}}(\Omega) = \deg_{\mathcal{H}}(\Omega G) - n = \deg_{\mathcal{H}}(\Lambda(f-R)) -n \leq \tau + 2g-1.
\end{align*}
Also, Inequality~\eqref{eq:Pade_R_degree_2} holds due to $\deg_{\mathcal{H}}(f) \leq m$ and
\begin{align*}
\deg_{\mathcal{H}}(\Psii{t}) = \deg_{\mathcal{H}}(\Lambda^s) + \deg_{\mathcal{H}}(f^t) \leq s\tau+tm.
\end{align*}
Condition \eqref{eq:Pade_R_congruence} is satisfied by Theorem~\ref{thm:key_equations} (note that $\asummands{t,i} = 0$ for $i>t$ and that the congruence modulo $x^{\lfloor \frac{t(n+2g-1)+\tau}{q}\rfloor+1}$ in \eqref{eq:Pade_R_congruence} is the same as equality due to the degree restrictions). \qed
\end{proof}
\section{Decoding Radius and Failure Behavior}
\label{sec:decoding_radius}
As any other power decoder, the new decoding algorithm is a partial decoding algorithm, which means that it might fail for certain error patterns.
This failure behavior has many reasons that we would like to discuss in this section.
We start by deriving a bound on the parameter $\tau$ of Problem~\ref{prob:Pade_R} that ensures the problem to have a solution.
\begin{theorem}\label{thm:decoding_radius}
Problem~\ref{prob:Pade_R} is guaranteed to have a solution if
\begin{align*}
\tau \geq \tau_\mathrm{new} := n \left[1-\tfrac{s+1}{2(\ell+1)}\right] -\tfrac{\ell}{2s} m - \tfrac{\ell-s+1}{s(\ell+1)} + \tfrac{g-1}{\ell+1}.
\end{align*}
\end{theorem}
\begin{proof}
Problem~\ref{prob:Pade_R} is guaranteed to have a solution if there at least one vector
\begin{align*}
(\lambdai{0},\dots,\lambdai{s-1},\psii{1},\dots,\psii{\ell}) \in \mathcal{R}^{s+\ell} \setminus \{ \ve 0 \},
\end{align*}
satisfying Conditions \eqref{eq:Pade_R_congruence}, \eqref{eq:Pade_R_degree_1}, and \eqref{eq:Pade_R_degree_2}.
We can find such a solution by solving the following homogeneous linear system of equations in the coefficients of the $\lambdai{i}$, which we consider these coefficients as indeterminates.
Since $\deg_{\mathcal{H}}( \psii{t}) \leq s\tau+tm$ (cf.~\eqref{eq:Pade_R_degree_2}), the coefficients of $\sum_{i=0}^{s-1} \lambdai{i} \asummands{t,i}$ in \eqref{eq:Pade_R_congruence} of degree greater than $s\tau+tm$ and less than
\begin{align*}
T_t := \begin{cases}
t(n+2g-1) + s\tau + 1, &t=1,\dots,s-1, \\
\deg_{\mathcal{H}}(G^s), &t=s,\dots,\ell,
\end{cases}
\end{align*}
must be zero.
Since we require $\deg_{\mathcal{H}} \lambdai{i} \leq s\tau+i(2g-1)$, see \eqref{eq:Pade_R_degree_1}, there at at least $s\tau+i(2g-1)-g+1$ indeterminates for $\lambdai{i}$.
After obtaining non-zero polynomials $\lambdai{i}$, we can find $\psii{t}$ by computing $\sum_{i=0}^{s-1} \lambdai{i} \asummands{t,i}$ modulo $G_t$.
It suffices to show that the described system has a non-zero solution for $\tau \geq \tau_\mathrm{new}$.
The system has at most
\begin{align*}
E &= \textstyle\sum_{t=1}^{\ell} \left[ T_t - (s\tau+tm) - 1 \right] \\
&\leq n s\left[ \ell+1-\tfrac{s+1}{2} \right] - \tfrac{\ell(\ell+1)}{2} m + \tfrac{s(s-1)}{2} (2g-1) + \tau s(s-1-\ell) - (\ell-s+1)
\end{align*}
equations and at least
\begin{align*}
V &= \textstyle\sum_{i=0}^{s-1} \left[ s\tau+i(2g-1) - g + 1 \right] = s^2\tau + \tfrac{s(s-1)}{2} (2g-1) - sg + s
\end{align*}
indeterminates. Thus, it has a non-zero solution if $V \geq 1+ E$, which can be re-written as $\tau \geq \tau_\mathrm{new}$.
\qed
\end{proof}
Theorem~\ref{thm:decoding_radius} can be interpreted as follows.
For some $\tau \in \mathbb{N}$, we denote by $\mathcal{V}_\tau$ the $\mathbb{F}_{q^2}$-vector space consisting of all vectors
\begin{align*}
(\lambdai{0},\dots,\lambdai{s-1},\psii{1},\dots,\psii{\ell}) \in \mathcal{R}^{s+\ell}
\end{align*}
that satisfy the congruences and degree constraints of Problem~\ref{prob:Pade_R} with parameter $\tau$.
If we choose $\tau \geq \tau_\mathrm{new}$, then $\dim_{\mathbb{F}_{q^2}}(\mathcal{V}_\tau) \geq 1$.
In addition, if $\tau \geq \deg_{\mathcal{H}}(\Lambda)$, then
\begin{align*}
(\Lambda^s, \Lambda^{s-1}\Omega,\dots,\Lambda\Omega^{s-1},\Lambda^s f,\Lambda^s f^2,\dots,\Lambda^s f^\ell) \in \mathcal{V}_\tau.
\end{align*}
Hence, if there is a $\tau$ with $|\mathcal{E}| \leq \deg \Lambda \leq \tau \leq \tau_\mathrm{new}$ and $\dim_{\mathbb{F}_{q^2}} (\mathcal{V}_\tau) = 1$, a non-trivial solution of Problem~\ref{prob:Pade_R} must yield a solution $(\Lambda^s, \Lambda^s f)$ of the decoding problem.
Thus, we could expect that at least in some cases, we can decode up to $|\mathcal{E}| \leq \tau_\mathrm{new}$ errors. However, there are several problems that could prevent us from correcting $\tau_\mathrm{new}$ many errors:
\begin{enumerate}[label=\roman*)]
\item The minimal degree of an error locator is greater than $|\mathcal{E}|$. Recall that it is only guaranteed that there is an error locator of $\deg_{\mathcal{H}} \Lambda \leq |\mathcal{E}|+g$.
\item We get $\dim_{\mathbb{F}_{q^2}} (\mathcal{V}_\tau) > 1$ already for some $\tau<\tau_\mathrm{new}$. This can have two reasons:
\begin{itemize}
\item The number of equations is smaller than $E$ (as in the proof of Theorem~\ref{thm:decoding_radius}), which can be the case if $\deg_{\mathcal{H}} R < n+2g-1$.
\item The equations are linearly dependent.
\end{itemize}
\item There is no $\tau$ with $\dim_{\mathbb{F}_{q^2}} (\mathcal{V}_\tau) = 1$ (e.g., if there is a $\tau$ with $\dim_{\mathbb{F}_{q^2}} (\mathcal{V}_\tau) = 0$ and $\dim_{\mathbb{F}_{q^2}} (\mathcal{V}_{\tau+1}) > 1$) and there is a ``smaller'' solution (corresponding to another codeword or a generic one) in $\mathcal{V}_\tau$ than $(\Lambda^s,\dots,\Lambda\Omega^{s-1},\Lambda^s f,\dots,\Lambda^s f^\ell)$.
\end{enumerate}
We will see in the Section~\ref{sec:numerical_results} that in our experiments, for all tested examples, we were able to correct up to $n [1-\tfrac{s+1}{2(\ell+1)}] -\tfrac{\ell}{2s} m - \tfrac{\ell-s+1}{s(\ell+1)} = \tau_\mathrm{new}- \tfrac{g-1}{\ell+1}$ many errors with high probability. This number of errors coincides with the classical power decoding radius for $s=1$, cf.~\cite{nielsen2015sub}.
\section{Complexity}
\label{sec:complexity}
In this section, we show that Problem~\ref{prob:Pade_R} can be solved in sub-quadratic time in the code length $n$. We use the algorithm in~\cite{rosenkilde2017SimHermPade}, which computes, for given $S_{\i,\j} \in \mathbb{F}_{q^2}$, $G_\j \in \mathbb{F}_{q^2}[X]$, $T_\i \in \mathbb{N}$, and $N_\i \in \mathbb{N}$, where $\i \in I$ and $\j \in J$ (index sets), a basis (the solution space is a vector space) of all solutions $\lambda_\i,\psi_\j \in \mathbb{F}_{q^2}[X]$, that fulfill
\begin{align*}
\sum_{\i \in I} \lambda_\i &\equiv \psi_\j \mod G_\j &&\forall \j \in J, \\
\deg \lambda_\i &\leq N_\i &&\forall \i \in I, \\
\deg \psi_\i &\leq T_\i &&\forall \j \in J,
\end{align*}
in $O^\sim\left(|J|^{\omega-1} \cdot |I| \cdot \max_\j\{\deg G_\j\}\right)$ operations over $\mathbb{F}_{q^2}$, where $\omega$ is the matrix multiplication exponent.
We use the $\Fqtwo[X]$-vector representation of an element of $\mathcal{R}$ (cf.~\cite{nielsen2015sub}) to reformulate Problem~\ref{prob:Pade_R} as a problem of the type above. Recall that for $a \in \mathcal{R}$, we can write $a = \sum_{i=0}^{q-1} a_i Y^i \in \mathcal{R}$ with unique $a_i \in \Fqtwo[X]$. Then, the \emph{vector representation} \cite{nielsen2015sub} of $a$ is defined by ${\ve \nu}(a) = (a_0,\dots,a_{q-1}) \in \Fqtwo[X]^q$.
Note that $q\deg(a_i)+i(q+1)\leq \deg_{\mathcal{H}}(a)$. For $a,b \in \mathcal{R}$ it can be shown that
\begin{align*}
{\ve \nu}(a+b) = {\ve \nu}(a) + {\ve \nu}(b), \qquad
{\ve \nu}(ab) = {\ve \nu}(a) {\ve \mu}(b) {\ve \Xi},
\end{align*}
where ${\ve \mu}(b) \in \Fqtwo[X]^{q \times (2q-1)}$ and ${\ve \Xi} \in \Fqtwo[X]^{(2q-1) \times q}$ are defined by
\begin{align*} \small
{\ve \mu}(b) :=
\begin{bmatrix}
b_0 & b_1 & b_2 & \dots & b_{q-1} & & &\\
& b_0 & b_1 & \dots & b_{q-2} & b_{q-1} & &\\
& & \ddots & \ddots & \dots & \ddots & \ddots & \\
& & & b_0 & b_1 & \dots & b_{q-2} & b_{q-1}
\end{bmatrix},
\quad
{\ve \Xi}
:=
\begin{bmatrix}
1 & & & \\
& 1 & & \\
& & \ddots & \\
& & & 1 \\
X^{q+1} & -1 & & \\
& X^{q+1} & -1 & \\
& \ddots & \ddots & \\
& & X^{q+1} & -1 \\
\end{bmatrix}.
\end{align*}
Using this notation, we define $\AM{t,i} := {\ve \mu}(\asummands{t,i}) {\ve \Xi} \in \Fqtwo[X]^{q \times q}$. We are ready to state the final complexity result.
\begin{theorem}\label{thm:complexity}
Problem~\ref{prob:Pade_R} can be solved using the algorithm in \cite{rosenkilde2017SimHermPade} with
\begin{align*}
&I = \left\{(i,j): i \in \{0,\dots,s-1\}, \, j \in \{0,\dots,q-1\} \right\}, &S_{(i,j),(t,r)}& = \AMe{t,i}_{j,r},\\
&J = \left\{ (t,r) : t \in \{1,\dots,\ell\}, \, r \in \{0,\dots,q-1\} \right\}, &G_{(t,r)}& = G_t, \\
&N_{(i,j)} = \tfrac{s \tau+i(2g-1)-j(q+1)}{q}, \quad T_{(t,r)} = \tfrac{s \tau+tm-r(q+1)}{q}
\end{align*}
in $O^\sim( \ell^{\omega-1} s^2 n^{\frac{\omega+2}{3}})$ operations over $\mathbb{F}_{q}$, where the $O^\sim$ hides $\log(ns\ell)$ factors.
\end{theorem}
\begin{proof}
Similar to \cite{nielsen2015sub}, pre-computing the matrices $\AM{t,i}$ is negligible compared to solving the Pad\'e approximation problem.
By the properties of ${\ve \nu}(\cdot)$, it is clear that $\lambdai{i},\psii{t} \in \mathcal{R}$ solve Problem~\ref{prob:Pade_R} if and only if $(\lambda_{(i,0)},\dots,\lambda_{(i,q-1)}) = {\ve \nu}(\lambdai{i})$ and $(\psi_{(t,0)},\dots,\psi_{(t,q-1)}) = {\ve \nu}(\psii{t})$ correspond to a non-zero element in the output of the algorithm in \cite{rosenkilde2017SimHermPade} of minimal $\max_{j \in \{0,\dots,q-1\}}\left\{q \deg(\lambda_{(0,j)}) + (q+1)j \right\}$. Since $\deg G_t \leq \lfloor \frac{t(n+2g-1)+\tau}{q}\rfloor+1 \in O(sn/q)$, a basis of the solution space is found in
\begin{align*}
O^\sim\left( (\ell q)^{\omega-1} (sq) (sn) \right) = O^\sim\left( \ell^{\omega-1} s^2 q^{\omega-1} n \right) = O^\sim\left( \ell^{\omega-1} s^2 n^{\frac{\omega+2}{3}} \right).
\end{align*}
The algorithm in \cite{rosenkilde2017SimHermPade} outputs a reduced basis, so a minimal element is guaranteed to be one of the basis elements. \qed
\end{proof}
Note that for constant parameters $\ell,s$, the complexity in Theorem~\ref{thm:complexity} is sub-quadratic in the code length $n$. We achieve the same complexity\footnote{The exponent of $\ell$ in the complexity statements in \cite{nielsen2015sub} is $\omega$. If we apply the algorithm from~\cite{rosenkilde2017SimHermPade} to these methods, we will also get $\omega-1$.} as the algorithms in \cite{nielsen2015sub}.
\section{Numerical Results}
\label{sec:numerical_results}
In this section, we present simulation results.
We have conducted Monte-Carlo simulations for estimating the failure probability of the new improved power ($\mathrm{\hat{P}}_\mathrm{fail, IPD}$) and the Guruswami--Sudan ($\mathrm{\hat{P}}_\mathrm{fail, GS}$) decoder in a channel that randomly adds $\tau$ errors, using a sample size $N \in \{10^3,10^4\}$.
The decoder was implemented in SageMath~v7.5 \cite{stein_sagemath_????}, based on the power decoder implementation of \cite{nielsen2015sub}. We used the Guruswami--Sudan decoder implementation from \cite{nielsen2015sub}.
\begin{table}[h!]
\caption{Observed failure rate of the improved power ($\mathrm{\hat{P}}_\mathrm{fail, IPD}$) and Guruswami--Sudan ($\mathrm{\hat{P}}_\mathrm{fail, GS}$) decoder. Code parameters $q,m,n,k,d^\ast$. Decoder parameters $\ell,s$. Number of errors $\tau$ ($^\ast$decoding radius as in \eqref{eq:decoding_radius_probably_real}). Number of experiments $N$.}
\label{tab:failure_rate}
\centering
{
\renewcommand{\arraystretch}{1.2}
\setlength{\tabcolsep}{5pt}
\begin{tabular}{c|c||c|c|c||c|c||c|c|c||c}
$q$ & $m$ & $n$ & $k$ & $d^\ast$ & $\ell$ & $s$ & $\tau$ & $\mathrm{\hat{P}}_\mathrm{fail, IPD}$ & $\mathrm{\hat{P}}_\mathrm{fail, GS}$ & $N$ \\
\hline \hline
$4$ & $15$ & $64$ & $10$ & $49$ & $4$ & $2$ & $28$ & $0$ & $0$ & $10^4$ \\
& & & & & & & $\phantom{{}^\ast}29^\ast$ & $0$ & $3.30 \cdot 10^{-3}$ & $10^4$ \\
& & & & & & & $30$ & $9.93 \cdot 10^{-1}$ & $9.39 \cdot 10^{-1}$ & $10^4$ \\
\hline
$5$ & $55$ & $125$ & $46$ & $70$ & $3$ & $2$ & $34$ & $0$ & $0$ & $10^4$ \\
& & & & & & & $35$ & $0$ & $0$ & $10^4$ \\
& & & & & & & $\phantom{{}^\ast}36^\ast$ & $0$ & $4.00 \cdot 10^{-4}$ & $10^4$ \\
\hline
$5$ & $20$ & $125$ & $11$ & $105$ & $5$ & $2$ & $67$ & $0$ & $0$ & $10^3$ \\
& & & & & & & $\phantom{{}^\ast}68^\ast$ & $0$ & $7.00 \cdot 10^{-3}$ & $10^3$ \\
& & & & & & & $69$ & $9.91 \cdot 10^{-1}$ & $9.60 \cdot 10^{-1}$ & $10^3$ \\
\hline
$7$ & $70$ & $343$ & $50$ & $273$ & $3$ & $2$ & $160$ & $0$ & $0$ & $10^3$ \\
& & & & & & & $\phantom{{}^\ast}161^\ast$ & $0$ & $0$ & $10^3$ \\
& & & & & & & $162$ & $9.78 \cdot 10^{-1}$ & $9.86 \cdot 10^{-1}$ & $10^3$ \\
\hline
$7$ & $70$ & $343$ & $50$ & $273$ & $4$ & $2$ & $168$ & $0$ & $0$ & $10^3$ \\
& & & & & & & $\phantom{{}^\ast}169^\ast$ & $0$ & $0$ & $10^3$ \\
& & & & & & & $170$ & $9.79 \cdot 10^{-1}$ & $2.2 \cdot 10^{-2}$ & $10^3$ \\
& & & & & & & $171$ & $1$ & $1$ & $10^3$ \\
\hline
$7$ & $55$ & $343$ & $35$ & $288$ & $4$ & $2$ & $\phantom{{}^\ast}184^\ast$ & $0$ & $0$ & $10^3$ \\
& & & & & & & $185$ & $9.82 \cdot 10^{-1}$ & $1.9 \cdot 10^{-2}$ & $10^3$ \\
& & & & & & & $186$ & $1$ & $1$ & $10^3$
\end{tabular}
}
\end{table}
Table~\ref{tab:failure_rate} presents the simulation results for various code ($q,m,n,k,d^\ast$), decoder ($\ell,s$), and channel ($\tau$) parameters.
It can be observed that both algorithms can almost always correct
\begin{align}
\tau = n \left[1-\tfrac{s+1}{2(\ell+1)}\right] - \tfrac{\ell}{2s} m - \tfrac{\ell-s+1}{s(\ell+1)} \label{eq:decoding_radius_probably_real}
\end{align}
errors, improving upon classical power decoding. Also, none of the two algorithms is generally superior.
\section{Conclusion}
We have presented a new decoding algorithm for one-point Hermitian codes which is based on the improved power decoder for Reed--Solomon codes from~\cite{nielsen2016power}.
Experimental results indicate that the new algorithm has a similar failure probability as the Guruswami--Sudan algorithm at the same decoding radius.
A generalization of the new algorithm to interleaved one-point Hermitian codes, similar to \cite{puchinger2017decoding}, promises improved decoding radii for interleaving degrees $m>1$ compared to existing decoding algorithms, and is work in progress.
\bibliographystyle{IEEEtran}
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Tempsat 1 – amerykański wojskowy satelita technologiczny. Służył badaniom inżynieryjnym nad kontrolą termiczną statków kosmicznych za pomocą czarnych farb. Satelita stanowił zwykłą sferę o średnicy 35 cm (14 cali). Wystrzelony razem z Dodecapole 2.
Satelita pozostaje na orbicie okołoziemskiej, której trwałość szacowana jest na 1000 lat.
Bibliografia
Amerykańskie sztuczne satelity
Satelity technologiczne
Loty kosmiczne w 1965
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а твой красавец опять на мага выходит- вот тогда и посмеешься..
поясни пожалуйста- ты что имеешь в виду??
неужели мы ребенка дурному учить будем??
GRAEME DOTT has endured a year to forget on the snooker table but he hopes the arrival of baby daughter Lucy will see his fortunes turn around.
Scotland`s former world champion has not won a ranking match for 16 months and, by his own admission, he`s got to start delivering the goods to pay for nappies.
Dott, who defeated Englishman Peter Ebdon 18-14 in the 2006 Crucible Theatre showcase in Sheffield , is looking to get his career back on track with a successful run in the Maplin UK Championship which gets underway in Telford a week on Saturday.
Dott, whose wife Elaine gave birth to Lucy on Thursday ,the couple already have a four-year-old son Lewis , said :" I`ve won the world championship so I know what I can do.
"And what I do know is that I have to start winning, especially when you look at the price of nappies these days!
" But Elaine and I are just delighted by the latest addition to the family. Lucy was 12 days overdue but it was worth the wait because she`s a wee cracker."
Dott, recognised as one of the toughest competitors in the game, has fallen to 48 in the provisional world rankings .His last victory in a ranking competition came when he beat Stephen Lee in the quarter-finals of the Shanghai Masters in August last year.
when he thumped another former world champion Shaun Murphy 6-1 in the Berlin final of the non-ranking FSTC Sports Management World Series of Snooker in July.
In a freak accident , however, Dott broke his left wrist while playing football outside the players` , hotel prior to the Shanghai Masters in October , which ruled him out of the event followed by the Royal London Watches Grand Prix in Glasgow.
Dott also missed the Moscow leg of the World Series a couple of weeks ago to be beside pregnant wife Elaine.
He added : "Now that Lucy has arrived that`s me back at the practice table and I`m looking forward to the UK.
"My dad is also coming along with me and hopefully, that will help me as well.
"I`ve been trying everything to get back playing the way I can and while I have fallen down the rankings, the ability is still there. I`ve just got to bring it out."
А у Винни ужо 4:5! Всё только начинается!!!
аня, ну что ж тебе марик так не приглянулся?
а где я хоть раз сказала, что мне Марик не приглянулся?
мы с ними всегда бились и всегда будем. ты что хочешь от них добиться-то?..
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\section{Introduction}\label{intro}
Although modern equipment is capable of significantly reducing the superposition of unwanted external noise on experimental data, still this impact cannot be avoided. Obviously, in order to optimally capture the observed phenomena one would like to lower as much as possible the external noise and the noise from the measurement system. Under the foregoing conditions the studied events are usually recognized as the parts of recorded signal lying above/below the upper/lower threshold level imposed on the base line level (i.e. an idealized signal level in the absence of response signal and all types of noise). In this way, the events, thresholds, and noise are intertwined in each experimental signal.
Measurements on the systems exhibiting avalanche-like relaxation are not an exception. A vast diversity of phenomena dominantly evolving via avalanche-like events can be found in everyday life, e.g. earthquakes \cite{Earthquakes}, neuronal activities \cite{NeuronalAvalanches2012,BrainSignalsPRL2006}, financial markets \cite{Financial2013}, crystalline \cite{Zaiser,Lase2020,Lase2018,Lase2014,SanjaJStat} and amorphous \citep{Budrikis,Sandfeld} solids undergoing plastic deformation, cracks propagating in disordered solids \citep{Santucci,SanjaPRL}, etc. The mentioned type of relaxation can cause the extreme events such as avalanches that span almost the whole system leading to a phase transition in the thermodynamic limit \cite{BelangerNatterman}. Among these phenomena, magnetization and relaxation processes in ferromagnetic materials play a distinguished role \cite{Lieneweg1972,Stanley1996,Durin2000,Kim2003,Shin2007,Ryu2007,Benassi-Zapperi-2011,Lima2017,Bohn2018}.
In order to model and explain the Barkhausen noise that emerges when the ferromagnetic sample is driven by varying external magnetic field, a number of theoretical models were developed \cite{Cizeau1998,Zapperi1998,Sethna2006,SethnaJMMM2001,Bertotti1990,Franz2011,Vives1994,VivesJMMM2000,ABBM,DW,LassePRB2014}. One of the most prominent appears to be the random field Ising model (RFIM), that has been extensively studied in the past few decades \cite{Schulz1988,TadicPRL1996,BalogPRB2018,SethnaPRL93,Vives2001,FytasPRL2013,JSTAT2021}. Renormalization group approach has brought
certain answers regarding the RFIM critical behavior,
but it turned out to be a rather difficult task. The results obtained via perturbative renormalization group showed the limits of this approach as some incorrect predictions in three dimensions arose \cite{YoungPerturbation,ParisiPerturbation,BricmontPerturbation}. This led to the non-perturbative approach that appeared to give better results \cite{ParisiNonPerturbation,TissierNonPerturbation}. Recent numerical investigation of equilibrium version of RFIM offered important information on behavior of this type of model \textcolor{black}{\cite{FytasPRL2016,FytasPRE2017,FytasPRL2019}}.
The nonequilibrium version of RFIM turned out to be more relevant for the correspondence with experiments due to its locally driven dynamics. The nonequilibrium model provides temporal evolution mimicking the response of real ferromagnetic samples to the varying external magnetic field. This model has been studied numerically in a lot of papers. Its critical behavior and scaling laws in the case of equilateral lattices were investigated in \cite{OlgaPRB1999,DahmenPRB1999,RechePRB2003,RechePRB2004,SpasojevicEPL2006,SpasojevicPRL2011,SpasojevicPRE2011,SpasojevicPRE2014, JSTAT2021}, whilst recently the systems with different geometry were studied in \cite{NavasVives,3D-2D2018,BosaSciRep2019,3D-2DHceff,SvetaPRE2020} together with the impact of lattice topology on its criticality \cite{Shukla2013PRE,Shukla2015PRE,Shukla2016,Triangle,Shukla2019}.
So far, very few studies were done on the joint effect that both threshold and noise have on signals obtained from ferromagnetic materials. Recent experimental \cite{BohnExp} and theoretical \cite{SanjaSciRep} studies delivered some important results caused by the implementation of the finite detection threshold when analyzing the original signal.
\textcolor{black}{Effects of thresholding have been considered also, e.g., in the context of fracture \cite{bares2019seismiclike,SanjaPRL}, and argued to be of importance in seismicity \cite{post2021interevent,radiguet2016triggering}. Moreover, the problem has been studied also, e.g., in the case of birth-death processes \cite{font2015perils}. However, the joint effects due to thresholding a crackling noise signal with superimposed additive external white noise remain largely unexplored.}
In the present paper we investigate the joint impact of the external noise and imposed threshold level on the avalanche statistics extracted from the simulations of the nonequilibrium athermal RFIM on equilateral three-dimensional cubic latices of size $L$ containing $L^3$ spins. In this work we used two types of noise. One is \textcolor{black}{uniform} white noise
taken from the uniform distribution (UWN) of width $w$, i.e. the noise that takes with probability density $p(n)=1/2w$ any value $n$ from the interval $[-w,w]$ and has the standard deviation $\sigma=w/\sqrt{3}$.
The other type is the white noise taken from the zero mean Gaussian distribution with standard deviation $\sigma$ (GWN).
\textcolor{black}{
These two theoretically convenient types of noise with flat power spectral density $S(f)=const$ are (almost)
ubiquitous in experiments (e.g. UWN as the quantization noise and GWN as electronic noise arising in amplifiers and detectors),
and in many instances superposed by some $1/f$ noise, having power spectral density $S(f)\propto 1/f^{\alpha_n}$ and
various origins (see e.g. \cite{Weissman}), whose influence we defer for later studies.
}
The paper is organized as follows. Description of the model, together with the simulation details and algorithm description, are given in Section \ref{model}.
In Section \ref{threshold} is explained what is achieved by thresholding of the signal, while in the Section \ref{impact} is shown how addition of external noise affects the properties of the relevant statistics. Finally, in Section \ref{conclusion} we give a discussion and conclusion of this study.
\section{Model}\label{model}
The RFIM is defined as follows. At each site $i$ of the underlying lattice lies the spin $S_i$ having two possible values $\pm 1$. There are three types of interaction to which the spins are exposed: 1) they interact with the nearest neighbours via exchange interaction, 2) there is the interaction between each spin and the applied external magnetic field $H$, and 3) every spin $S_i$ interacts with a local random field $h_i$ at its site. These random field values are chosen independently and without site-to-site correlations from some zero mean distribution so that the average taken over all possible random field configurations satisfies $\langle h_i h_j \rangle = R^2\delta_{i,j}$, where $\delta_{ij}$ is the Kronecker delta function, and $R$ is disorder, i.e. the standard deviation of the employed random field distribution. One such distribution is the Gaussian distribution $$\rho (h)=\frac{1}{\sqrt{2\pi}R}exp\Big(-\frac{h^2}{2R^2}\Big)$$ used in this paper.
Taking all three interactions into account the Hamiltonian of the system reads
\begin{equation}
\mathcal{H}=-J\sum_{\{i,j\}} S_iS_j-H\sum_i {S_i}-\sum_i {h_iS_i}.
\label{Hamiltonian}
\end{equation}
In the first term $J$ represents the strength (1 in this paper) of ferromagnetic coupling between the nearest neighbors, and the summation is performed over all distinct pairs $\{i,j\}$ of such spins.
The system behavior is governed by the local relaxation rule meaning that the spin $S_i$ is stable while its sign is the same as the sign of the \textit{effective field}$$h_i^\mathrm{eff}=\sum_{\langle j\rangle}S_j+H+h_i$$
where the summation in the first term is performed over all nearest neighbors of the spin $S_i$.
All spins that are unstable at the current moment will flip in the next moment of discrete time affecting neighbouring spins in a way that they can become unstable and flip in the next-next moment. This explains the mechanism for creation of avalanches. During the started avalanche the external field is kept constant and afterwards increased in a single step exactly to the value that will flip the least stable spin. This regime is known as \textit{adiabatic}. Each simulation begins with $H=-\infty$ and all spins being $-1$, and stops when all spins have value $+1$. All simulations are done with periodic boundary conditions along all three directions.
As already mentioned, while an avalanche is active we check all the nearest neighbors of the spins flipped at the moment $t$ and those of them that are unstable we flip in the moment $t+1$. In the simulations this a very fast process whereas the finding of the next spin to be flipped once the avalanche is over is the most time-consuming. The so-called brute force algorithm \cite{Kuntz1999} suggests to check all non flipped spins in the system and find which one is the least stable. In big systems, like are ours in this paper, the time needed for such search is extremely large. In order to decrease the time consumption we use, therefore, the sorted list algorithm \cite{OlgaCondMat96,Kuntz1999}, which we implemented in Fortran.
\textcolor{black}{
Our results are obtained in extensive simulations of system sizes up to $L=1024$ for disorders $R$ surpassing the effective critical disorder $R_c^{\mathrm{eff}}(L)$
which provides only the nonspanning avalanches which are mostly encountered in experiments, see \cite{JSTAT2021}.
}
The results gathered from the simulations were analyzed using the proprietary programs coded in Fortran, Visual Basic and Wolfram Mathematica.
\section{Thresholding of the signal}\label{threshold}
\textcolor{black}{The nonequilibrium RFIM systems belongs to the class of systems whose response signal $V(t)$ when driven by the increasing external magnetic field is equal to the number of spins flipped at the moment $t$. In what follows we will limit our analysis to the systems having the response signal $V(t)>0$ while the system is active and $V(t)=0$ otherwise.}
Unlike the signals generated in simulations, where the overall registered signal is $V(t)$, the overall signals registered in experiments contain external noise $n(t)$. In this case, the registered signal is $V(t)+n(t)$, so it becomes much harder to extract events from the registered signal corresponding to individual avalanches.
One of the extraction methods is to impose some threshold level $V_{\mathrm{th}}$ and observe only the activity above $V_{\mathrm{th}}$, dividing the signal into sub-avalanches that are parts of some underlying avalanche. During an avalanche a sub-avalanche starts at the first moment $t_s$ when the signal surpasses the chosen threshold level, $V(t_s)+n(t_s)>V_{\mathrm{th}}$, and ends at the first moment of time when the signal falls below it, i.e. at the moment $t_e$ when $V(t_e)+n(t_e)\le V_{\mathrm{th}}$. The difference between these two moments is the duration of that sub-avalanche, $T=t_\mathrm{e}-t_\mathrm{s}$, while the size of the sub-avalanche is defined as the area that lies between the signal and the imposed threshold, $S=\int_{t_\mathrm{s}}^{t_\mathrm{e}}dt[V(t)-V_{\mathrm{th}}]$.
The thresholding process defined in this way introduces concept of waiting time as the time between two consecutive sub-avalanches.
\textcolor{black}{
In simulations one can recognize two basic kinds of waiting time illustrated in Fig. \ref{Fig1}:
the internal waiting time $T_{\mathrm{w, int}}(V_{\mathrm{th}})$ between two consecutive
sub-avalanches (yellow-colored and labeled by $i_1$ and $i_2$) that belong to the same ongoing avalanche
(labeled by $i$)
and the external waiting time $T_{\mathrm{w, ext}}(V_{\mathrm{th}})$ elapsed between the
end of the last sub-avalanche $i_2$ from the ongoing avalanche $i$ and the start of the next (green-colored) sub-avalanche $j'$
belonging to the first succeeding avalanche $j$ surpassing the threshold, see \cite{AdiabaticReg}.
}
Since it is known that external noise can have considerable impact on the properties defined by the thresholding process, our goal was to investigate that impact. To this end we thresholded generated simulation signals with superimposed noise.
\begin{figure}[htpb]
\centering
\includegraphics[width=0.5\textwidth]{Fig1.eps}
\caption{\textcolor{black}{An example illustrating the imposing of threshold $V_{\mathrm{th}}$ on the response signal, extraction of
sub-avalanches, and definition of internal and external waiting time, $T_{\mathrm{w, int}}(V_{\mathrm{th}})$ and $T_{\mathrm{w, ext}}(V_{\mathrm{th}})$.
Two avalanches (labeled by $i$ and $j$) surpassing the threshold are separated by one (blue-colored) avalanche lying below the threshold. Two
sub-avalanches $i_1$ and $i_2$ are extracted from the avalanche $i$, and a single sub-avalanche $j'$ from the avalanche $j$. Everything lying below $V_{\mathrm{th}}$ including parts of the avalanches $i$ and $j$
lying below the threshold is greyed.}
}
\label{Fig1}
\end{figure}
\section{The effect of adding the noise}\label{impact}
In this section we present the impact of adding two types of white noise, uniform and Gaussian,
on the distributions of average avalanche size and properties of the internal and external waiting times.
\textcolor{black}{
The added noise is of external origin (e.g. noise that in experiments originates from detectors, amplifiers, AD converters, ambient EM interference, etc.)
and is considered here to be much more pronounced than the system's intrinsic (e.g. thermal) noise. This, in particular, means that
the system intrinsic dynamics is (practically) not disturbed by such noise and that the noise solely affects the registered signal by superposing on
the pristine signal, i.e. the signal that would be registered by an ideal experimental system.
}
\subsection{Average avalanche size}\label{gamma}
In Fig. \ref{Fig2} are present against duration $T$ the average size $\langle S\rangle_T$ of avalanches having duration $T$. This is done for added a) UWN and b) GWN to the original signal with the threshold levels $V_{\mathrm{th}}=150$ and $V_{\mathrm{th}}=50$, respectively. Average avalanche size data follow the power law $\langle S\rangle_T\sim T^{\gamma_{S/T}}$
\textcolor{black}{
specified by the universal RFIM exponent $\gamma_{S/T}$ appearing also as the exponent of the power spectral density of the RFIM signals,
$S(f)\propto 1/f^{\gamma_{S/T}}$ \cite{Kuntz2000}.
}
We see that for both types of noise the slope of the $\langle S\rangle_T$ curves decreases as the noise standard deviation grows. This decrease happens because the noise cuts long avalanches into shorter subavalanches. Since those shorter sub-avalanches originate from a longer one having larger average signal $\langle V(t)\rangle=\langle S \rangle /T$, their sizes are likely to be larger than the sizes of the regular avalanches of the same duration. In Fig. \ref{Fig3} we illustrate an example of how the avalanche of the same duration has greater size when the external noise is present in the signal. As the avalanche duration grows the previously explained effect becomes less expressed. Thus, the slope on the log-log plot of $\langle S\rangle$ versus $T$ curve, and therefore the $\gamma_{S/T}$ values, decreases with the increase of the noise standard deviation.
\begin{figure}[htpb]
\centering
\includegraphics[width=0.5\textwidth]{Fig2.eps}
\caption{Average avalanche size for a given avalanche duration for the system of size $1024\times 1024\times 1024$ and disorder $R=2.25$. Detecting threshold level is $V_{\mathrm{th}}=150$ in UWN case when noise function standard deviations ranges from 0 to $69.3$ (panel a), and $V_{\mathrm{th}}=50$ in GWN case when noise function standard deviation ranges from 0 to $17.3$ (panel b).}
\label{Fig2}
\end{figure}
\begin{figure}[htpb]
\centering
\includegraphics[width=0.5\textwidth]{Fig3.eps}
\caption{Example of the noise effect on the $\gamma_{S/T}$ values decrease. On the left side of the figure is presented the part of the signal with added noise (red line), while on the right side is the signal without noise (blue line). At a given threshold level for the same duration we see that the area (i.e. avalanche size) $S_T^n$, when the noise is applied, is larger than the area $S_T$ without the noise.}
\label{Fig3}
\end{figure}
We present how $\gamma_{S/T}$ behaves for various threshold values and noise standard deviations $\sigma$ for UWN in Fig. \ref{Fig4}a) and GWN in Fig. \ref{Fig4}b). In the main panels of both Fig. \ref{Fig4}a) and \ref{Fig4}b) we see that the values of $\gamma_{S/T}$ drop quickly when we increase the threshold from zero, whilst after some value of $V_{\mathrm{th}}$, there is a plateau, i.e. a wide range of $V_{\mathrm{th}}$ for which the values of exponent $\gamma_{S/T}$ remain constant. The plateau is present because at these $V_{\mathrm{th}}$ values the impact of the originally small avalanches cannot be seen \cite{SanjaSciRep}. However, if $\sigma$ is large enough the difference in $\gamma_{S/T}$ disappears even for the small threshold values. The reason for this lies in the fact that regardless the value of $V_{\mathrm{th}}$, the average size of avalanches of small duration is dominantly governed by the noise. The average size of long avalanches in any case is not much affected by threshold or noise. So we expect that for the large noise standard deviation, $\gamma_{S/T}$ remains the same independently of the threshold. This can be observed in the insets of Fig. \ref{Fig4}, where we show how the values of $\gamma_{S/T}$ change for $0\leq \sigma \leq 60$ and $0\leq V_{\mathrm{th}}\leq 90$. We notice that with increase of $\sigma$ the deviations between the $\gamma_{S/T}(V_{\mathrm{th}})$ curves corresponding to different thresholds vanish.
\begin{figure}[htpb]
\centering
\includegraphics[width=0.5\textwidth]{Fig4.eps}
\caption{a) Main panel: values of $\gamma_{S/T}$ versus detecting threshold level for the standard deviation of UWN from $\sigma=0$ to $\sigma=28.9$. Inset: Values of $\gamma_{S/T}$ for wider range of $\sigma$ and for $V_{\mathrm{th}}$ that ranges from 0 to 90 (i.e. the values of $V_{\mathrm{th}}$ before the plateau). b) Main panel: values of $\gamma_{S/T}$ versus detecting threshold level for the standard deviation of GWN from $\sigma=0$ to $\sigma=14.4$. Inset: The same as in inset of panel a).}
\label{Fig4}
\end{figure}
In Fig. \ref{Fig5} we present the values of $\gamma_{S/T}$ at plateaus, denoted by $\gamma_{S/T}^\mathrm{pl}$, for various noise standard deviations for both UWN and GWN. It seems like that there is a wide range of linear decrease of the $\gamma_{S/T}^\mathrm{pl}$ with the $\sigma$ increase in the UWN case, whereas for the GWN case that range is smaller, after which the values of $\gamma_{S/T}^{\mathrm{pl}}$ saturate. Still, we have no explanation for such behavior.
\begin{figure}[htpb]
\centering
\includegraphics[width=0.5\textwidth]{Fig5.eps}
\caption{Plateau values of $\gamma_{S/T}$ obtained from Fig. 4 for various noise standard deviations. Black squares represent the results for UWN, while the red circles are the results for GWN.}
\label{Fig5}
\end{figure}
\subsection{Waiting times}\label{shift}
We start by observing the number of occurrences \textcolor{black}{$n(T_{\mathrm{w}}; R, V_{\mathrm{th}}, \sigma,L)$} \cite{LasseJStat}
of waiting time $T_{\mathrm{w}}$ in one run at the threshold level $V_{\mathrm{th}}$ imposed on the response signal $V(t)$ with
added noise of standard deviation $\sigma$.
In insets of Fig. \ref{Fig6}, \textcolor{black}{showing a representative example obtained
for $T_{\rm w}=1051$, $L=1024$, and $R=2.40$,} one can see that the graphs of these distributions at fixed values of
$\sigma$ shift to the right as $\sigma$ grows.
This happens because on average the added noise increases the threshold
level (at which a certain waiting time is found in a given section of recorded signal) from the value $V_{\mathrm{th}}$ for
the pure signal to $V_{\mathrm{th}}'$ for the signal with noise, as is illustrated in Fig. \ref{Fig7}. In the case of
\textcolor{black}{$n(T_{\mathrm{w}}; R, V_{\mathrm{th}},\sigma,L)$} distribution, we found that the mentioned increase
can be described by a shift parameter $p(\sigma)$ depending on the standard deviation $\sigma$ of the applied noise.
More specifically, if distribution \textcolor{black}{$n(T_{\mathrm{w}}; R, V_{\mathrm{th}}', \sigma,L)$} is translated
along the threshold axis by the amount $p(\sigma)$, it overlaps with the distribution
\textcolor{black}{$n(T_{\mathrm{w}}; R, V_{\mathrm{th}}, \sigma=0,L)$}:
\textcolor{black}{
\begin{equation}
n(T_{\mathrm{w}}; R, V_{\mathrm{th}}'-p(\sigma), \sigma, L)=n(T_{\mathrm{w}}; R, V_{\mathrm{th}}, \sigma=0, L).
\end{equation}
}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{Fig6.eps}
\caption{Number of occurrence of the external waiting time value $T_{\mathrm{w, ext}}=1051$ versus detection threshold level $V_{\mathrm{th}}$ for various standard deviations varying from $\sigma=0$ to $\sigma=57.7$. As shown in the main parts of both panels, the distributions collapse onto a single curve when presented against the threshold displaced by the shift parameter $p(\sigma)$ depending on the noise standard deviation $\sigma$. The data is obtained for the $1024\times1024\times1024$ system at $R=2.40$.}
\label{Fig6}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{Fig7.eps}
\caption{Schematic illustration of the average effect of added noise on the external waiting time: the same value of the external waiting time $T_{\mathrm{w, ext}}$ is found at a higher threshold level $V'_{\rm th}$ in the response signal with added noise (red line) compared to the threshold level $V_{\mathrm{th}}$ corresponding to the same waiting time for the pure response signal (blue line).}
\label{Fig7}
\end{figure}
For the fixed $R>R_c$, where $R_c=2.16$ is the critical disorder in the three-dimensional nonequilibrium RFIM \cite{OlgaPRB1999}, the shift parameter is independent on the lattice size $L$ and on the type \textcolor{black}{and length of waiting time, given that waiting time is long enough}. However, \textcolor{black}{$n(T_{\mathrm{w}}; R, V_{\mathrm{th}}, \sigma,L)$} depends on $L$ and should scale with $L^3$, because the number of peaks in the signal scales in that way with the system size. In the main parts of Fig. \ref{Fig8} we present the collapse of the raw $n(\textcolor{black}{T_{\mathrm{w}}=490; R= 2.40, V_{\mathrm{th}},\sigma, L=1024)}$ data from the insets. The collapses are obtained dividing the distribution data by $L^3$ and translating the threshold values by the shift parameter $p(\sigma)$ corresponding to the UWN data in panel a) and to the GWN data in panel b), respectively. Consequently
\textcolor{black}{
\begin{equation}
n(T_{\mathrm{w}}; R, V_{\mathrm{th}}, \sigma,L)=L^3\tilde{n}(T_{\mathrm{w}}; R, V_{\mathrm{th}}-p(\sigma)),
\label{n(T,V,s,L)}
\end{equation} }
where $\tilde{n}(T_{\mathrm{w}}; V_{\mathrm{th}})$ is the scaling function. Complete collapse can be achieved only for sufficiently long waiting times compared to the noise standard deviation. When the examined waiting time is short and $\sigma$ wide, it may happen that there is no such waiting time in the system at all, although it was present for lower values of $\sigma$.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{Fig8.eps}
\caption{The curves in insets show the distribution of number of occurrences for $T_{\mathrm{w, ext}}=490$ versus $V_{\mathrm{th}}$ for three different systems of linear sizes $L=512$, $L=724$ and $L=1024$ \textcolor{black}{and disorder $R=2.40$}, while in the main panels are presented the collapses of the curves from the insets when (\ref{n(T,V,s,L)}) is applied for the UWN (panel a) and GWN (panel b). Standard deviation ranges from $\sigma=0$ to $\sigma=52$ for UWN, and from $\sigma=0$ to $\sigma=10$ for GWN.}
\label{Fig8}
\end{figure}
It is expected that the shift parameter increases with $\sigma$, since the larger value of threshold is needed to obtain the same value of \textcolor{black}{$n(T_{\mathrm{w}};R,V_{\mathrm{th}},\sigma,L)$} for larger $\sigma$. On the ground of the $p(\sigma)$ values obtained for three different disorders, see Fig. \ref{Fig9}, we assume that the shift parameter obeys a modified power law behavior
\textcolor{black}{
\begin{equation}
p(\sigma) = a + b \sigma^{c}.
\label{shiftfit}
\end{equation}
}
The fitting curves to this function of several sets of the $p(\sigma)$ data are shown in Fig. \ref{Fig9}. The best fits are obtained for the values of parameters given in Table \ref{Table1} for UWN and in Table \ref{Table2} for GWN. Here one can notice that the parameter values are the same within the error bars for various disorders of UWN, whereas for GWN they significantly depend on disorder.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{Fig9.eps}
\caption{Shift parameter values $p(\sigma)$ obtained from $1024\times1024\times1024$ systems with $R=2.25, 2.40, 2.55$ when UWN and GWN are applied. The data fitted to the function (\ref{shiftfit}) with fitting values presented in tables \ref{Table1} and \ref{Table2}.}
\label{Fig9}
\end{figure}
\begin{table}[!hbt]
\centering
\caption{The best fit values of the parameters appearing in (\ref{shiftfit}) in the case of UWN.}
\begin{tabular}{ c|ccc }
\hline
\hline
$R$ & $a$ & $b$ & $c$ \\
\hline
\hline
$2.25$ & $-3.3\pm 0.3$ & $1.33\pm 0.02$ & $1.041\pm 0.002$ \\
$2.40$ & $-3.6\pm 0.3$ & $1.35\pm 0.02$ & $1.038\pm 0.003$ \\
$2.55$ & $-3.6\pm 0.3$ & $1.39\pm 0.02$ & $1.033\pm 0.002$ \\
\hline
\hline
\end{tabular}
\label{Table1}
\end{table}
\begin{table}[!hbt]
\centering
\caption{The same as in Table \ref{Table1}, but for the GWN.}
\begin{tabular}{ c|ccc }
\hline
\hline
$R$ & $a$ & $b$ & $c$ \\
\hline
\hline
$2.25$ & $-3.9\pm 0.9$ & $3.8\pm 0.3$ & $1.20\pm 0.02$ \\
$2.40$ & $-2.6\pm 0.6$ & $3.4\pm 0.2$ & $1.26\pm 0.02$ \\
$2.55$ & $-4.3\pm 0.8$ & $4.8\pm 0.2$ & $1.16\pm 0.02$ \\
\hline
\hline
\end{tabular}
\label{Table2}
\end{table}
\subsection{Scaling properties}\label{SecScaling}
Both types of waiting time, $T_{\mathrm{w, int}}$ and $T_{\mathrm{w, ext}}$, (jointly denoted by $T_{\mathrm {w}}$) follow the scaling properties of temporal correlations (\ref{scaling}) induced by imposing threshold $V_{\mathrm{th}}$ on the signals obtained from systems with disorders $R$, linear lattice size $L$ and without external noise \cite{SanjaSciRep}:
\textcolor{black}{
\begin{equation}
\begin{split}
&V_{\mathrm{th}}^{\frac{\alpha_{\mathrm{int}}\sigma'\nu z}{1-\sigma'\nu z}}D_{T_{\mathrm {w}}}({T_{\mathrm {w}}}; V_{\mathrm{th}}, r, 1/L) = \\
D_{T_{\mathrm {w}}}\Big({T_{\mathrm {w}}}&/V_{\mathrm{th}}^{\frac{\sigma'\nu z}{1-\sigma'\nu z}}; V_{\mathrm{th}}^{\frac{\sigma'^2\nu z}{\sigma'\nu z-1}}r, V_{\mathrm{th}}^{\frac{\sigma'^2\nu^2 z}{\sigma'\nu z-1}}/L\Big).
\end{split}
\label{scaling}
\end{equation}
Here $\alpha$, $\beta$, $\sigma'$, $\nu$, $\delta$, $z$ and $\alpha_{\mathrm{int}}=\alpha+\sigma'\beta\delta/\sigma'\nu z$ are standard RFIM exponents \cite{Sethna2006,OlgaPRB1999,DahmenPRB1999} (note that the standard notation of the, here denoted, exponent $\sigma'$ is $\sigma$, but we choose to denote it here by $\sigma'$ to avoid possible confusion with the noise standard deviation), while $r=(R-R_c)/R$ represents reduced disorder of the system. The exponent $\alpha_{\mathrm{int}}$ is used because the data was gathered from the finite windows of the external magnetic field \cite{SpasojevicPRE2011,SanjaSciRep}.} This means that the distributions $D_{T_{\mathrm {w}}}({T_{\mathrm {w}}}; V_{\mathrm{th}}, r, 1/L)$ of the waiting time, $T_{\mathrm {w}}$ multiplied by \textcolor{black}{$V_{\mathrm{th}}^{\frac{\alpha_{\mathrm{int}}\sigma'\nu z}{1-\sigma'\nu z}}$} and presented versus \textcolor{black}{${{T_{\mathrm {w}}}/V_{\mathrm{th}}^{\frac{\sigma'\nu z}{1-\sigma'\nu z}}}$}, will collapse onto a single curve, if the conditions
\textcolor{black}{
\begin{equation}
V_{\mathrm{th}}^{\frac{\sigma'^2\nu z}{\sigma'\nu z-1}}r=\mathrm{const},\quad V_{\mathrm{th}}^{\frac{\sigma'^2\nu^2 z}{\sigma'\nu z-1}}/L=\mathrm{const},
\label{conditions}
\end{equation}
}
demanding that systems with the lowest size $L$ have the biggest disorders $R$ and the smallest threshold levels $V_{\mathrm{th}}$, are satisfied.
The addition of external noise alters distributions of waiting time in a way that the increase of the added noise is followed by the rise of short waiting times, while the duration of the longest waiting times decreases, see insets in the Fig. \ref{Fig10}. This means that, upon applying (\ref{scaling}), the waiting time distributions will transform differently so that full collapse will not be achieved.
In order to obtain the full collapse of distribution of waiting time curves when the external noise is applied
we propose the shift functions $f(\sigma)$ and $g(\sigma)$ that modify scaling relation (\ref{scaling}) into
\textcolor{black}{
\begin{equation}
\begin{split}
&(V_{\mathrm{th}}-g(\sigma))^{\frac{\alpha_{\mathrm{int}}\sigma'\nu z}{1-\sigma'\nu z}}D_{T_{\mathrm {w}}}({T_{\mathrm {w}}}; V_{\mathrm{th}}, r, 1/L) = \\
D_{T_{\mathrm {w}}}\Big({T_{\mathrm {w}}}&/(V_{\mathrm{th}}-f(\sigma))^{\frac{\sigma'\nu z}{1-\sigma'\nu z}}; V_{\mathrm{th}}^{\frac{\sigma'^2\nu z}{\sigma'\nu z-1}}r, V_{\mathrm{th}}^{\frac{\sigma'^2\nu^2 z}{\sigma'\nu z-1}}/L\Big).
\end{split}
\label{scaling_shift}
\end{equation}}
In this way it is achieved that the transformed distributions overlap with the original distribution obtained without the added noise. In other words, when adequately shifted, noisy distributions behave like the noiseless distributions, see Fig. \ref{Fig10}. There, as in Fig. \ref{Fig13} too, distributions $D_{T_{\mathrm {w}}}({T_{\mathrm {w}}}; V_{\mathrm{th}}, r, 1/L)$ are shortly denoted by $D_{T_{\mathrm{w}}}$.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{Fig10.eps}
\caption{In the main figure of panel a) is presented shifted collapse of the external waiting time distributions obtained from system with $L=2508, R=2.24, V_{\rm {th}}=126$ with added UWN of standard deviation $\sigma$ shown in inset. In panel b) is shown collapse of internal waiting time distributions obtained from the same system with added GWN.}
\label{Fig10}
\end{figure}
An example of shifting functions is presented in Fig. \ref{Fig11}, where one can see that their behavior is not the same for UWN and GWN. \textcolor{black}{The reason for this lies in the difference in the type of noise. Namely, UWN is bounded from the both sides while GWN theoretically can have any value, meaning that in GWN case we need to shift distributions more than in UWN case for the same $\sigma$.} Both $f(\sigma)$ and $g(\sigma)$, when UWN is applied, have a power law behavior $\phi(\sigma)\sim \sigma^s$ ($f(\sigma)$ is almost linear), while in the case when the observed signal contains added GWN their behavior can be described using the error function, $\phi(\sigma)\sim \mathrm{erf}(\sigma)$.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{Fig11.eps}
\caption{Shift functions $f(\sigma)$ (in main parts) and $g(\sigma)$ (in insets) for the distributions of external waiting time in panel a) and internal waiting time in panel b). The values of shift functions are those used to perform the collapsing shown in main panels of Fig. 10 of the distributions from the insets of \textcolor{black}{that} figure. In the UWN case values are fitted to the power law function $\phi(\sigma) \sim \sigma^s$
while in the GWN case they are fitted to the error function, $\phi(\sigma)\sim\mathrm{erf}(\sigma)$.}
\label{Fig11}
\end{figure}
Here it can be seen that after $\sigma$ reaches the value $\sigma_\mathrm{th}\approx V_{\mathrm{th}}/\sqrt{3}$,
there is almost no difference between $f(\sigma)$ functions in the UWN and GWN case (the same for $g(\sigma)$)
and that both tend to the value of threshold level $V_{\mathrm{th}}$.
\textcolor{black}{
In the UWN case, the shift functions reach the threshold value when the noise width $w$ becomes comparable to the value of $V_{\rm th}$
(equivalently, when $\sigma\approx V_{\rm th}/\sqrt{3}$), while in the GWN case this happens for a smaller value of $\sigma$.
}
In the insets of Fig.\ref{Fig12} can be seen that the shift functions $f(\sigma)$ and $g(\sigma)$,
obtained for 4 different systems follow the same rules: the systems in question satisfy conditions
(\ref{conditions}): $L=1448, R=2.27, V_{\mathrm{th}}=75; L=2046, R=2.25, V_{\mathrm{th}}=102;
L=2508, R=2.24, V_{\mathrm{th}}=126$ and $L=3072, R=2.231, V_{\mathrm{th}}=153$. This indicates that
the functions $f(\sigma)$ and $g(\sigma)$, divided by threshold values $V_{\mathrm{th}}$ and presented as
functions of $\sigma/\sigma_{\mathrm{th}}$, collapse onto a single curve, meaning that the behavior of the
shift functions obtained using the set of parameters that satisfy conditions (\ref{conditions}), can be jointly described by
\begin{equation}
\frac{\phi(\sigma)}{{V_\mathrm{th}}}=p+q\Big(\frac{\sigma}{\sigma_{\mathrm{th}}} \Big)^s
\label{shiftfit1}
\end{equation}
when UWN is applied, and by
\begin{equation}
\frac{\phi(\sigma)}{V_{\mathrm{th}}}=i+j\mathrm{erf}\Big(k\frac{\sigma}{\sigma_{\mathrm{th}}}-l\Big)
\label{shiftfit2}
\end{equation}
in the GWN case. Such collapses are presented in the main panels of Fig. \ref{Fig12} alongside the fitting functions obtained for the values of parameters presented in Tables \ref{Table3} and \ref{Table4}.
\begin{table}[!hbt]
\centering
\caption{Values of the best fit parameters for (\ref{shiftfit1}) of the shift functions calculated for internal and external waiting times when the UWN is applied.}
\begin{tabular}{ c|ccc }
\hline
\hline
$$ & $p$ & $q$ & $s$ \\
\hline
\hline
\multicolumn{4}{c}{Internal}\\
\hline
$f(\sigma)$ & $-0.028\pm 0.009$ & $0.83\pm 0.01$ & $1.16\pm 0.03$ \\
$g(\sigma)$ & $ -0.003\pm 0.008$ & $0.77\pm 0.01$ & $1.45\pm 0.05$ \\
\hline
\multicolumn{4}{c}{External}\\
\hline
$f(\sigma)$ & $-0.042\pm 0.007$ & $0.96\pm 0.01$ & $1.08\pm 0.02$ \\
$g(\sigma)$ & $\text{ }0.002\pm 0.008$ & $0.76\pm 0.01$ & $1.44\pm 0.05$\\
\hline
\hline
\end{tabular}
\label{Table3}
\end{table}
\begin{table}[!hbt]
\centering
\caption{Values of the best fit parameters for (\ref{shiftfit2}) of the shift functions calculated for internal and external waiting times when the GWN is applied.}
\begin{tabular}{ c|cccc }
\hline
\hline
$$ & $i$ & $j$ & $k$ & $l$ \\
\hline
\hline
\multicolumn{5}{c}{Internal}\\
\hline
$f(\sigma)$ & $0.391\pm 0.007$ & $0.51\pm 0.01$ & $2.32\pm 0.06$ & $0.94\pm 0.04$ \\
$g(\sigma)$ & $0.387\pm 0.007$ & $0.46\pm 0.01$ & $2.47\pm 0.08$ & $1.09\pm 0.05$ \\
\hline
\multicolumn{5}{c}{External}\\
\hline
$f(\sigma)$ & $0.462\pm 0.004$ & $0.505\pm 0.005$ & $3.09\pm 0.05$ & $1.29\pm 0.03$ \\
$g(\sigma)$ & $0.439\pm 0.004$ & $0.458\pm 0.005$ & $2.96\pm 0.07$ & $1.44\pm 0.04$ \\
\hline
\hline
\end{tabular}
\label{Table4}
\end{table}
As an illustration, in Fig.\ref{Fig13} are given the collapses obtained for several values of $\rho=\sigma/\sigma_{\mathrm{th}}$, where the shift functions are calculated using the fit parameters from Tables \ref{Table3} and \ref{Table4}.
\begin{figure*}
\centering
\includegraphics[width=0.99\textwidth]{Fig12.eps}
\caption{In the insets of panels a) and b) are presented shift functions $f(\sigma)$ and $g(\sigma)$ for the external waiting time distributions obtained for 4 systems such that their dimensions $L$, disorders $R$ and threshold levels $V_{\mathrm{th}}$ satisfy conditions (\ref{conditions}): $L=1448, R=2.27, V_{\mathrm{th}}=75, L=2046, R=2.25, V_{\mathrm{th}}=102, L=2508, R=2.24, V_{\mathrm{th}}=126$, and $L=3072, R=2.231, V_{\mathrm{th}}=153$. On the main panels are shown the collapses of the shift functions divided by $V_{\mathrm{th}}$ as functions of $\sigma/\sigma_{\mathrm{th}}$, fitted to the proposed forms (\ref{shiftfit1}) and (\ref{shiftfit2}). The best fit parameters are given in Tables \ref{Table3} and \ref{Table4}. Panels c) and d): the same as a) and b), but for the internal waiting time.}
\label{Fig12}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=0.99\textwidth]{Fig13.eps}
\caption{Main figures of panels a) and b) show collapses of the distributions of external (a) and internal (b) waiting times obtained from the same systems as the data shown in Fig. \ref{Fig12} (non-collapsed data is shown in insets) but with such a noise that the ratio between standard deviation of added UWN and imposed threshold $\rho$ ranges from 0 to 0.6. Values of shift functions are calculated using fitting parameters given in Tables \ref{Table3} and \ref{Table4}. The same is presented in panels c) and d), but for GWN and $\rho$ in the range from 0 to 0.3.}
\label{Fig13}
\end{figure*}
The distribution of avalanche duration $T$ also follows the scaling properties (\ref{scaling}), (with a change in notation $T_{\mathrm{w}}\rightarrow T$) which are affected when the external noise is added, see insets in Fig. \ref{Fig14}. One can see that adding of noise doesn't change significantly these distributions as external noise grows like in the case of waiting times. This holds as long as the threshold level was chosen so that the majority of signal stays above the $V_{\mathrm{th}}$ and the standard deviation of the external noise, comparable to $V_{\mathrm{th}}$, is much smaller than the amplitude of the signal, as is usually accomplished in experiments. In this way the presence of external noise can not significantly decrease the number of events of long duration. Still, the symmetry between distributions of duration and internal waiting time is present since both distributions follow the power-laws, $D_T \sim T^{-\tau_T}$ and $D_{T_{\mathrm{w, int}}}\sim {T_{\mathrm{w, int}}}^{-\tau_{T_{\mathrm{w, int}}}}$ with the same value of exponent $\tau=\tau_T=\tau_{T_{\mathrm{w, int}}}\approx 1.62$ \cite{SanjaPRL}, which is unaffected when the external noise is added. The shifting (\ref{scaling_shift}) can also be applied in the case of duration distributions, see main panels in Fig. \ref{Fig14}, but now the shifting functions behave differently - $g(\sigma)$ is zero, while $f(\sigma)$ is nearly linear for both UWN and GWN, as can be seen in Fig. \ref{Fig15}.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{Fig14.eps}
\caption{Main figures of panels a) and b) show shifted collapse of the distributions of duration obtained from the same system as the data shown in Fig. \ref{Fig10} when UWN (a) and GWN (b) is added, while non-collapsed
\textcolor{black}{distributions, much less affected than the waiting time distributions by the presence of noise from the employed range, are} shown in insets.}
\label{Fig14}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{Fig15.eps}
\caption{Shift function $f(\sigma)$ for the distributions of duration. The values of shift functions are those used to perform the collapsing shown in main panels of Fig. \ref{Fig14}. In both UWN and GWN cases the values of shift function are fitted to the power law function $\phi(\sigma) \sim \sigma^s$. The values of parameter $s$ are $1.04\pm0.05$ in UWN and $0.96\pm0.06$ GWN case}
\label{Fig15}
\end{figure}
\section{Discussion and Conclusion}\label{conclusion}
In experimental research the impact of external noise and detection threshold is inevitably present. Still, it is often common to pay a minor attention to those impacts in theoretical and numerical analysis of the experimental results. For example, in paper \cite{SanjaSciRep} was shown that the reason for the difference in the experimentally and numerically obtained values of exponent $\gamma$ lies possibly in the effects of nonzero threshold level. Thus, in the present paper we focused on the particular features that are affected by the introduced threshold level and external noise. We observe that the external noise may cause the exponent $\gamma$ to reach its plateau value for lower threshold levels than in the case when there is no external noise. This means that in experiments it would be very difficult to detect the higher values of $\gamma$ before the plateau because the external noise sets a very low threshold level above which the plateau appears. On the other hand, the different noise level impacts the number of occurrences of the given waiting time in the system, see Fig. \ref{Fig6}. Here we see that the interplay between the threshold level and the noise can explain the potential disagreements in the experimental curves of the same type.
In this paper we didn't present the results that correspond to a very large noise, i.e. the noise having magnitude comparable to the average signals. Still, we can see some of the consequences that arise from the large noise, for example in Fig. \ref{Fig5} in the GWN case, where the $\gamma$ values decrease linearly for smaller $\sigma$, but after some value of noise standard deviation the analytical type of decrease changes. This does not happen in the case of UWN, due to the fact that UWN is bounded from both sides, while in GWN case there are no boundaries for the noise values. Thus, although the standard deviations are the same, there is a larger probability of getting greater values for the noise in the GWN case. We expect that the linear drop in the UWN case, presented in Fig. \ref{Fig5}, also proceeds to some other type of decrease, but with much larger values of $\sigma$. Although the effects created by noise would possibly lead to some substantial differences in the presented quantities, it is still not of interest to investigate such noise, because the main motivation for the present research came from the experimental studies, and experimental study of any phenomenon becomes useless if the external (unwanted) noise is that large.
The collapses presented in Section \ref{SecScaling} also show in what way the applied noise disturb the original system's response. We see that the curves of the distributions of waiting time and duration can also collapse onto a single curve when the external noise is applied, but one has to be careful \textcolor{black}{and to modify equation (\ref{scaling}) by adding the shifting functions like in (\ref{scaling_shift})}. Although different types of noise bring quantitatively different effects, these effects are qualitatively the same. This is expected due to the above explained reasons and the manner how the noise affects the signal.
To conclude, in this paper we examined the joint impact of the external noise and detection threshold level on the response of the externally driven nonequilibrium athermal RFIM. We showed that both noise and threshold level significantly affect the behavior of the signal properties and scaling relations. Thus, the inevitable experimental occasions indeed influence the studied phenomena and should be adequately treated in order to obtain adequate results.
\begin{acknowledgments}
This work was supported by the Serbian Ministry of Education, Science and Technological development.
\end{acknowledgments}
\section*{References}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,247
|
\section{Conclusion and Future Work}
\label{sec:future-work-concl}
This article defined a type-directed dictionary-passing translation from Featherweight Generic Go (FGG)
to an extension of the untyped $\lambda$-calculus. The translation represents a value
at the type of an FGG interface as an existential combining a concrete struct value
with a dictionary for all methods of the interface. Bounded type parameters in FGG become
extra function arguments in the target. These extra arguments are coercions from the
instantiation of a type variable to its upper bound.
The formal translation covers all features of FGG except type assertions,
which we treated only informally.
Every program in the image of the translation has the same dynamic
semantics as its source FGG program. The proof of this result is based
on a syntactic, step-indexed logical relation. The step-index ensures a
well-founded definition of the relation in the presence of recursive
interface types and recursive methods. We also reported on an implementation
of the translation.
In this article, we relied on FGG as defined by Griesemer and
coworkers \citeyearpar{FeatherweightGo}, without reconsidering design
decisions. But our translation raises several questions with respect to the
design of generics in Go. For example, the translation clearly shows that type bounds
in structs and interfaces have no operational meaning. Should we eliminate these type bounds?
Or should we give them a meaning inspired by Haskell's type class mechanism?
Further, a method declaration in full Go must reuse the type bounds of its struct and must
be defined in the same package as the struct. Clearly, this limits
extensibility and flexibility. Can we provide a more flexible design to solve
the expression problem~\citep{ExpressionProblemWadler} in Go, without resorting to
unsafe type assertions?
We would like to use the insights gained through this article to answer these and similar questions in future
work.
A somewhat related point is performance. As explained earlier,
generics in Go are compiled by monomorphization. This gives the best possible performance because
the resulting code is specialized for each type argument. However, not all programs can be monomorphized
and the increase in code size is often considered problematic. This raises another interesting question for
future work. Could selective monomorphization or specialization offer a viable trade-off between
performance, code size, and the ability to compile Go programs which are not monomorphizable?
A statically-typed target language typically offers more room for compiler optimization. Thus, another
interesting direction for future work is a translation to a typed backend, for example
System F$_{\mathrm{C}}$~\citep{conf/tldi/SulzmannCJD07}.
\section{Implementation and Type Assertions}
\label{sec:discussion}
We provide an implementation of the translation\footnote{\ImplementationURL}
written in Haskell~\citeyearpar{Haskell}.
The target language of the translation is Racket~\citeyearpar{Racket}.
The implementation supports the source language from \Cref{sec:featherweight-generic-go},
extended with type assertions, generic functions, and several base types (integers, characters, strings, and booleans).
All extensions except type assertions are straightforward to support.
Thus, the remainder of this section explains how to add type assertions
to the translation.
Type assertions are Go's notion for dynamic type casts. If the Go compiler sees a type assertion,
written $e.(\fgType)$ for some expression $e$ and type $\fgType$, it treats the whole expression
as having type $\fgType$ and inserts a runtime conversion that fails if the type of $e$ is not
a subtype of $\fgType$. Type assertions are quite expressive: the asserted type $\fgType$ might be a
(generic) interface type or even a type variable.
We have extended our translation with support for type assertions, and we also have a rough
proof sketch to verify that the extended translation preserves the dynamic semantics.
However,
the translation rules for type assertions are quite verbose and contain a lot of boilerplate.
Further, their treatment is largely orthogonal to the main aspect of this article, namely
the handling of structural subtyping through a dictionary-passing translation. Thus,
we only provide an informal explanation how to extend the translation with type assertions.
\subsection{Example for Type Assertions}
\label{sec:example-type-assert}
To get an intuition for the treatment of type assertions, we start with a simple example
and ignore generics. The
later sections then fill the pieces missing.
Consider the following FGG code in the context of \Cref{f:fgg-format}
and \Cref{f:fgg-format2} that converts an arbitrary value into a value of interface
type \go{Format}. If the conversion fails, a runtime error occurs.
\begin{lstlisting}[language=golang]
func asFormat(x Any) Format {
return x.(Format)
}
\end{lstlisting}
To translate the Go code, the representation of a struct value carries a tag specific to the struct.
Each interface value carries the tag \hs{Iface} to distinguish interface from struct values.
\[
\begin{array}{r@{\quad}c@{\quad}l}
\texttt{Num\{1\}}\textrm{~at~type~}\texttt{Num}
&
\leadsto
&
\texttt{(K}_{\sub{\texttt{Num}}}\texttt{, 1)}
\\
\texttt{Num\{1\}}\textrm{~at~type~}\texttt{Format}
&
\leadsto
&
\texttt{Iface (K}_{\sub{\texttt{Num}}}\texttt{, 1)}~\texttt{format}_{\sub{\texttt{Num}}}
\end{array}
\]
To construct method dictionaries dynamically, we need an encoding for method signatures. This encoding
consists of a constructor for the method name,
a tuple with the argument types, and the result type. (We ignore generics for the moment, so
we do not need to represent type parameters.) For example, the encoding
of the signature \go{format()string} is
$\texttt{K}_{\sub{\texttt{format}}}\texttt{ () K}_{\sub{\texttt{string}}}$.
The output of the translation then contains a small \enquote{runtime system}.
Function \hs{getDictEntry} matches on the tag of a struct and the encoding of
a method signature to return an appropriate dictionary entry. If no method
declaration is found, an error is raised.
\begin{lstlisting}[language=myhaskell,mathescape=true]
getDictEntry K$_{\sub{Num}}$ (K$_{\sub{format}}$ () K$_{\sub{string}}$) = format$_{\sub{Num}}$
-- other cases for getDictEntry omitted
getDictEntry _ _ = error "type assertion failed"
\end{lstlisting}
Function \hs{dynAssert} takes a struct value and a list of method
signatures and tries to construct an interface value for the method signatures
given. The \hs{map}-function is defined in the usual way.
\begin{lstlisting}[language=myhaskell,mathescape=true]
dynAssert (tag, fields) sigs =
((tag, fields), map (getDictEntry tag) sigs)
\end{lstlisting}
The translation of \go{asFormat} is now straightforward. We use
\hs{[...]} to denote a list of values.
\begin{lstlisting}[language=myhaskell,mathescape=true]
asFormat x =
case x of
Iface y () -> dynAssert y [K$_{\sub{format}}$ () () K$_{\sub{string}}$]
\end{lstlisting}
\subsection{Runtime-representation of Types}
We have already seen to we need to encode FGG's types and method signatures
at the expression-level of the target language.
For a struct or interface types $t[\Multi{\fgType}]$, the encoding
is a unique constructor $K_t$ applied to the encodings
of $\Multi{\fgType}$.
The encodings of a method signature includes the method name, the bounds of its type parameters,
the argument types, and the result type.
Type parameters local to a method are represented as de Bruijn indices,
so the encoding of method signatures is invariable under renaming of type variables.
The encoding of type variables not local to a method signature or
appearing outside of a method signature are
passed as additional parameters. In fact, a type variable instantiated with an interface
type requires two encodings. The first is the encoding of the interface type, and the second
is a list with the encodings of all method signatures of the interface.
Thus, a type variable now gives raise to three additional
parameters: two encodings and the coercion to its upper bound.
\subsection{Enforcing Type Assertions}
The translation of a type assertion $e.(\fgType)$
is a $\lambda$-expression that converts the translation of $e$
to a representation of the asserted type $\fgType$.
Clearly, such a conversion might fail at runtime.
To realize type assertions, we assume that we can distinguish struct and interface values in the target,
and that each struct value carries its type encoding.
In the following, we describe how a type assertion with asserted type $\fgType$
is enforced on some value $V$ of the target language. Three different cases may arise:
\begin{enumerate}
\item Asserted type $\fgType$ is a \emph{struct type}. Such a type assertion only succeeds
if $V$ is a value of the same struct. Thus, we dynamically check if $V$ is a struct value
and if the type encoding carried by $V$ is equal to the encoding of $\fgType$.
If yes, the type assertion yields $V$. Otherwise, a runtime-error arises.
\item Asserted type $\fgType$ is an \emph{interface type}. Thus, we need to
construct a new interface value by dynamically creating
a dictionary for $\fgType$.
To support dynamic dictionary creation, the output of the translation provides
a global lookup table. For each method declaration, the
table maps the encoding of the receiver type and the encoding of the method
signature to a dictionary entry for the method. In the example of \Cref{sec:example-type-assert},
the global lookup table has been realized as the \hs{getDictEntry} function.
For the type assertion to succeed, $V$ can be either a struct value or an interface value
with an embedded struct value. In any case, we have a type encoding of a struct value.
Then, for each method signature $R$ of
$\fgType$, we use the type encoding and the encoding of $R$ to extract a dictionary entry for the method.
If any of these lookup operations fails, the whole type assertions fails. Otherwise,
we are able to construct an interface value for $\fgType$.
\item Asserted type $\fgType$ is a \emph{type variable}. We proceed as in the two preceding
cases, depending on whether the type variable has been instantiated with a struct or interface type.
\end{enumerate}
The lookup in the global table performs an equality
comparison on encodings of method signatures. This simple approach works because a struct
implements a method of some interface only if the method signatures in the method declaration
and in the interface match exactly (apart from names for type and argument variables).
For example, it is not allowed to provide a more general signature in the method declaration.
\section{Overview}
\label{sec:examples}
This section introduces Featherweight Generic Go \cite[FGG,][]{FeatherweightGo} and
our type-directed dictionary-passing translation through a
series of examples.
FGG is a tiny model of Go that includes essential typing features
such as method overloading, structs, interfaces, structural subtyping, and the extension with generics.
An FGG program consists of declarations for structs, interfaces, methods, and a main function. Function and method bodies
only contain a single return statement, all expression are free from side effects.
For the examples in this section, we extend FGG with primitive types for integers and strings, with
an operator \go{+} for string concatenation and a builtin function \go{intToString},
with definitions of local variables, and with function definitions.
We will first consider FGG without generics to highlight the idea behind our type-directed
dictionary-passing translation scheme.
We refer to FGG without generics as Featherweight Go (FG).
Then, we show how the translation scheme for FG can be adapted to deal with the addition of generics.
All examples have been checked against our implementation\footnote{\ImplementationURL}
of the translation.
\subsection{Featherweight Go}
\boxfig{f:fgg-format}{String-formatting and its translation}{
\renewcommand\subColored{yes}
\vspace{-1ex}
\lstinputlisting[name=goexample,numbers=left,escapechar=@,language=Golang,mathescape=true]{example-format.go}
\hrule{}
\lstinputlisting[name=hsexample,firstnumber=15,numbers=left,escapechar=@,language=myhaskell,mathescape=true,lastline=23]{example-format.hs}
\vspace{-1ex}
\renewcommand\subColored{no}
}
The upper part of \Cref{f:fgg-format} shows an FG program for formatting values as strings. The code does not use
generics yet.
Structs in Go are similar to structs in C,
a syntactic difference is the Go convention that field or variable names precede their types.
Here, struct \go{Num} has a single field \go{val} of type \go{int}, so it simply
acts as a wrapper for integers.
Methods and functions are introduced via the keyword \go{func}.
A method can be distinguished from a
function as the receiver
argument always precedes the method name.
Methods can be overloaded on the receiver type.
In lines 5 and 6, we find methods \go{format} and \go{pretty}, respectively, for receiver type
\go{Num}. Lines 8 and 10 defines two functions.
Interfaces in Go declare sets of method signatures sharing the same receiver
where method names must be distinct and the receiver is left implicit.
Interfaces are types and describe all receivers that implement the methods declared by the interface.
In our example, interface \go{Format} declares a method \go{format} for rendering its receiver as a string.
The second interface \go{Pretty} also declares \go{format}, but adds
a second method \go{pretty} with the intention to produce a visually more attractive output.
Interfaces and method definitions imply structural subtype relations.
Interface \go{Format} contains a subset of the methods declared by interface \go{Pretty}.
Hence, \go{Pretty} is a structural subtype of \go{Format}, written (1) $\subtypeOfTT{Pretty}{Format}$.
Line~5 defines a method \go{format}
for receiver type \go{Num}, where the method body assumes a builtin function \go{intToString}
for converting integers to strings.
We say that \go{Num} implements method \go{format}.
Hence, \go{Num} is a structural subtype of \go{Format}, written (2) $\subtypeOfTT{Num}{Format}$.
Receiver \go{Num} also implements the \go{pretty} method, see line~6.
Hence, we also find that (3) $\subtypeOfTT{Num}{Pretty}$.
Structural subtype relations play a crucial role when typechecking programs.
For example, consider the function call \go{formatSome(Num\{1\})} in line~11.
From above we find that (2) $\subtypeOfTT{Num}{Format}$.
That is, \go{Num} implements the \go{Format} interface and therefore the function call typechecks.
Consider the variable declaration and assignment in line~12.
Value \go{Num\{2\}} is assigned to a variable of interface type \go{Pretty}.
Based on the subtype relation (3) $\subtypeOfTT{Num}{Pretty}$ the assignment typechecks.
Consider the function call \go{formatSome(pr)} in line~13.
Based on the subtype relation (1) $\subtypeOfTT{Pretty}{Format}$ the function call typechecks.
In~\cite{FeatherweightGo}, the dynamic behavior of programs is explained via runtime lookup of methods, where
based on the receiver's runtime type the appropriate method definition is selected.
The Go conditions demand that for each method name and receiver type there can be at most one definition.
This guarantees that method calls can be resolved unambiguously.
\subsection{Type-Directed Translation}
We explain the meaning of Go programs by means of translation to
an untyped $\lambda$-calculus with recursive top-level definitions,
let-bindings, pattern matching,
integers, strings,
an operator \texttt{++} for string concatenation, and
a builtin function \texttt{intToString}.
We will use a Haskell-style notation.
Method definitions belonging to an interface are grouped together in a \emph{dictionary} of methods.
Thus, method calls can be turned into primitive function calls by simply looking up
the appropriate method in the dictionary.
Structural subtype relations are turned into coercion functions that
transform, for example, a struct value into an interface value to
make sure that the appropriate dictionaries are available.
Where to insert dictionaries and coercions in the program is guided
by the typechecking rules.
Hence, the translation is type-directed.
Our translation strategy can be summarized as follows:
\begin{description}
\item[Struct.] An FGG value at the type of a struct with $n$ fields is represented by an $n$-tuple
holding the values of the fields. We call such an $n$-tuple a \emph{struct value}.
\item[Interface.] An FGG value at the type of an interface is represented as a pair
$(V, \mathcal{D})$, where $V$ is a struct value
and $\mathcal D$ is a \emph{method dictionary}.
Such a method dictionary is a tuple holding implementations of all interface methods for $V$,
in order of declaration in the interface.
We call the pair $(V, \mathcal{D})$ an \emph{interface value}.
\item[Coercion.] A structural subtype relation $\subtypeOf{\fgType}{\fgTypeAux}$ implies a \emph{coercion function}
to transform the target representation of an FGG value of type $\fgType$
into a representation at type $\fgTypeAux$.
\end{description}
The lower part of \Cref{f:fgg-format} gives the translation of our running example.
In this overview section, we identify a $1$-tuple with the single value it holds.
For each field name, we assume a helper function to access the field component,
see line~16.
Method calls on interface values lookup the respective method definition in the dictionary
and apply it to the struct value embedded inside the interface value.
See lines~19-21.
Method definitions translate to plain functions,
see lines~24-25. Recall that
for each method name and receiver type there can be at most one definition.
Hence, the generated function names are all distinct.
Structural subtype relations translate to coercions,
see lines~28-30.
For example, (2) $\subtypeOfTT{Num}{Format}$
translates to the $\mathtt{toFormat}_{\sub{Num}}$ coercion.
Input parameter \go{x} represents a target representation of a \go{Num} value.
The output $\Pair{\mathtt{x}}{\mathtt{format}_{\sub{Num}}}$ is an interface value
holding the receiver and the corresponding method definition.
Coercion $\mathtt{toPretty}_{\sub{Num}}$ corresponds to (3) $\subtypeOfTT{Num}{Pretty}$
and coercion $\mathtt{toFormat}_{\sub{Pretty}}$
to (1) $\subtypeOfTT{Pretty}{Format}$.
The translation of the main function, starting at line~35,
is guided by the typechecking of the source program.
Each application of a structural subtype relation leads to the insertion
of the corresponding coercion function in the target program.
For example, the function call \go{formatSome(Num\{1\})}
translates to $\texttt{formatSome}~\texttt{(toFormat}_{\sub{Num}}~\texttt{1)}$
because typing of the source requires (2) $\subtypeOfTT{Num}{Format}$.
The other coercions arise for similar reasons.
\subsection{Adding Generics}
We extend our running example by including pairs, see \Cref{f:fgg-format2}.
The struct type \texttt{Pair[T Any, U Any]} is \emph{generic}
in the type of the pair components, \go{T} and \go{U} are \emph{type variables}.
When introducing type variables we must also specify an upper \emph{type bound} to constrain the set of concrete types
that will replace type variables.
The \emph{bounded type parameter} \texttt{T Any} can therefore be interpreted as
$\forall \texttt{T}. \subtypeOf{\texttt{T}}{\texttt{Any}}$.
Upper bounds are always interface types.
The upper bound \texttt{Any} is satisfied by any type because the set of methods that need to be implemented is empty.
\boxfig{f:fgg-format2}{String-formatting with generics (extending code from \Cref{f:fgg-format})}{
\renewcommand\subColored{yes}
\vspace{-1ex}
\lstinputlisting[name=goexample,numbers=left,escapechar=@,language=Golang,mathescape=true]{example-format2.go}
\hrule{}
\lstinputlisting[name=hsexample,firstnumber=13,numbers=left,escapechar=@,language=myhaskell,mathescape=true,lastline=16]{example-format2.hs}
\vspace{-1ex}
\renewcommand\subColored{no}
}
To format pairs, we need to format the left and right component that are of generic type~\go{T} and~\go{U}.
Hence, the method definition in line~4 states the type bound \go{Format} for type variables~\go{T} and~\go{U}.
In general, bounds of type parameters for the receiver struct of a method declaration must be in a covariant subtype relation
relative to the bounds in the struct declaration.
This is guaranteed in our case as we find $\subtypeOfTT{Format}{Any}$.
Importantly, the type bounds in line~4 imply the subtype relations (4) $\subtypeOfTT{T}{Format}$
and (5) $\subtypeOfTT{U}{Format}$.
Thus, we can show that the method body typechecks.
For example, expression \go{this.left} is of type \go{T}.
Based on (4), this expression is also of type \go{Format} and therefore the method call in line~5.
\go{this.left.format()} typechecks.
We consider typechecking the main function.
Instances for generic type variables must always be explicitly supplied.
Hence, when constructing a pair that holds number values, see line~9, we find \texttt{Pair[Num, Num]}.
Consider the method call \go{p.format()} in line~10.
The receiver struct \go{Pair[T Format, U Format]} of the method definition in line~4 matches
\go{p}'s type \go{Pair[Num, Num]}
by replacing \go{T} and \go{U} by \go{Num}.
The type bounds in the receiver type are satisfied as we know from above that (2) $\subtypeOfTT{Num}{Format}$.
Hence, the method call typechecks.
By generalizing the above argument we find that
\begin{center}
(6) \quad $ \{\subtypeOfTT{T}{Format}, \subtypeOfTT{U}{Format} \} \, \vdash \, \subtypeOfTT{Pair[T, U]}{Format}$.
\end{center}
That is, under the assumptions $\subtypeOfTT{T}{Format}$ and $\subtypeOfTT{U}{Format}$
we can derive that $\subtypeOfTT{Pair[T, U]}{Format}$.
In particular, we find that $\subtypeOfTT{Pair[Num, Num]}{Format}$.
Hence, the function call \go{formatSome(p)} in line~11 typechecks.
Extending our type-directed translation scheme to deal with generics turns out to be fairly straightforward.
\begin{description}
\item[Bounded type parameter.] A bounded type parameter \go{T Ifce} where \go{T} is a type variable
and \go{Ifce} is an interface type becomes a coercion parameter \go{toIfce}$_{\sub{T}}$.
At instantiation sites, coercions need to be inserted.
\end{description}
The lower part of \Cref{f:fgg-format2} shows the translated program.
Starting at line~18 we find the translation of the definition of method \go{format} for pairs.
Each bounded type parameter \go{T Format} and \go{U Format} is turned into a coercion parameter
\texttt{toFormat$_{\sub{T}}$} and \texttt{toFormat$_{\sub{U}}$}.
In the target, we use a curried function definition where coercion parameters are collected in a tuple.
A method call of \go{format} needs to supply concrete instances for these coercion parameters.
See line~27 which is the translation of calling \go{format} on receiver type \go{Pair[Num,Num]}.
Hence, we must pass as the first argument
the tuple of coercions \texttt{(toFormat$_{\sub{Num}}$, toFormat$_{\sub{Num}}$)}
to \texttt{format$_{\sub{Pair}}$}.
Subtype relation (6) implies the (parameterized) coercion \texttt{toFormat$_{\sub{Pair}}$} in line~22.
Given coercions \texttt{toFormat$_{\sub{T}}$} and \texttt{toFormat$_{\sub{U}}$}
we can transform a pair \go{p} into an interface value for \go{Format},
where the method dictionary consists of
the partially applied translated method definition
\texttt{format$_{\sub{Pair}}$}.
We make use of \texttt{toFormat$_{\sub{Pair}}$}
in the translation of the function call \go{formatSome(p)}, see line~28.
Based on the specific coercion \texttt{toFormat$_{\sub{Num}}$},
the call \texttt{toFormat$_{\sub{Pair}}$} transforms the pair value \go{p} into
the interface value \texttt{(p, format$_{\sub{Pair}}$ (toFormat$_{\sub{Num}}$,toFormat$_{\sub{Num}}$))}.
Then, we call \go{formatSome} on this interface value.
\subsection{Bounded type parameters of structs and interfaces}
We could strengthen the type bound in the definition for pairs by replacing \go{Any} with \go{Format}.
\begin{lstlisting}[name=goexample,escapechar=@,language=Golang,mathescape=true]
type PPair$\tyarg{[T Format, U Format]}$ struct { left T; right U }
\end{lstlisting}
Such stronger type bounds only serve a purpose to rule out more programs (statically).
For example, the following program will not typecheck because \go{Int} does not implement
the \go{Format} interface.
\begin{lstlisting}[name=goexample,escapechar=@,language=Golang,mathescape=true]
type Int struct { iVal int }
func main2b() {
// Will not typecheck because Int <: Format does not hold
var p PPair$\tyarg{[Int, Int]}$ = PPair$\tyarg{[Int, Int]}${ Int{1}, Int{2} }
}
\end{lstlisting}
In general, type bounds in the definition of structs have no meaning at runtime.
Hence, in our translation scheme such type bounds can effectively be ignored.
The same applies to type bounds that arise in the definition of generic interfaces.
We will see an example of a generic interface in the later \Cref{sec:example-translation}.
\subsection{Bounded type parameters of methods}
\boxfig{f:fgg-format3}{Bounded type parameters of methods (extending code from \Cref{f:fgg-format2})}{
\renewcommand\subColored{yes}
\vspace{-1ex}
\lstinputlisting[name=goexample,numbers=left,escapechar=@,language=Golang,mathescape=true]{example-format3.go}
\hrule{}
\lstinputlisting[name=hsexample,firstnumber=16,numbers=left,escapechar=@,language=myhaskell,mathescape=true,lastline=25]{example-format3.hs}
\vspace{-1ex}
\renewcommand\subColored{no}
}
There may be bounded type parameters local to methods.
Consider Figure~\ref{f:fgg-format3} where we further extend our running example.
Starting at line~1 we find a definition of method \go{formatSep} for pairs.
This method takes an argument \go{s} that acts as a separator when formatting pairs.
Argument \go{s} is of the generic type \go{S} constrained by the type bound \go{S Format}.
Type parameter \go{S} is local to the method and not connected to the receiver struct.
Type arguments for \go{S} must also be explicitly specified in the program text,
see method calls in lines 8 and 13.
In the translation, bounded type parameters of methods simply become additional coercion parameters.
Consider the translation of \go{formatSep} defined on pairs starting at line~21.
The translated method definition first expects the coercion parameters
\texttt{(toFormat$_{\sub{T}}$, toFormat$_{\sub{U}}$)} that result from the bounded type
parameters \texttt{T~Format} and \texttt{U~Format} of the receiver.
Then, we find the receiver argument \go{this} followed
by the coercion parameter \texttt{toFormat$_{\sub{S}}$} resulting from \texttt{S~Format},
and finally the method argument \go{s}.
The translation of the method body follows the scheme we have seen so far, see lines~22-24.
When calling method \go{formatSep} on a pair we need to provide
the appropriate coercions, see line~37.
From the method definition of \go{formatSet} for pairs and from the definition of interface
\go{FormatSep}, we find that the following subtype relation holds:
\begin{center}
(7)\quad $ \{\subtypeOfTT{T}{Format}, \subtypeOfTT{U}{Format} \} \, \vdash \, \subtypeOfTT{Pair[T, U]}{FormatSep}$.
\end{center}
Subtype relation (7) implies the coercion \texttt{toFormatSep$_{\sub{Pair}}$} in line~28.
Thus, the function call of \go{formatSepSome} from line 14
translates to the target code starting in line~38.
The point to note is that a coercion parameter corresponding to a bounded type parameter of a method is
not part of the dictionary; it is only supplied at the call site of the method.
Consider the call \go{x.formatSep[Format](s)} in line~8.
In the translation (line 33), we first partially apply the respective dictionary entry on the receiver.
This is done via the target expression \texttt{(formatSep$_{\sub{FormatSep}}$ x}\texttt{)}.
Type \go{Format} is a valid instantiation for
type parameter \go{S} of \go{formatSep} because
$\subtypeOfTT{Format}{Format}$ in FGG. In the translation, this
corresponds to the (identify) coercion \texttt{toFormat$_{\sub{Format}}$}.
Hence, we supply the remaining arguments \texttt{toFormat$_{\sub{Format}}$} and \go{s}.
\subsection{Outlook}
Next, we formalize FGG following the description by~\cite{FeatherweightGo} (\Cref{sec:featherweight-generic-go}).
Then, we give the details of our type-directed translation scheme (\Cref{sec:type-direct-transl})
and establish that the meaning of FGG programs is preserved (\Cref{sec:formal-properties}).
\section{Featherweight Generic Go}
\label{sec:featherweight-generic-go}
Featherweight Go \cite[FG,][]{FeatherweightGo} is a small subset of the full Go language~\citeyearpar{golang}
supporting only essential features such as structs, interfaces, method
overloading and structural subtyping.
In the same article, the authors add generics to FG with the goal to scale
the design to full Go. The resulting
calculus is called Featherweight Generic Go (FGG). Since version 1.18, full Go includes
generics as well, but with limited expressivity compared to the FGG proposal
(see \Cref{sec:generics-go}).
For the translation presented in this article, we stick to the original FGG language
with minor differences in presentation but excluding dynamic type assertions.
\Cref{sec:discussion} sketches how to extend the translation with dynamic type assertions.
The next two subsections introduce the syntax and the dynamic semantics of FGG. We defer
the definition of its static semantics until \Cref{sec:type-directed}, where we specify
it as part of the type-directed dictionary-passing translation.
\subsection{Syntax}
\label{sec:syntax}
\boxfig{f:fgg}{Syntax of Featherweight Generic Go (FGG) without type assertions}{
\vspace{-1ex}
\[\ba{ll}
\begin{array}{lr@{~}l}
\GoSynCatName{Struct name} & t_S, u_S & \in \structNameSet
\\ \GoSynCatName{Interface name} & t_I, u_I & \in \ifaceNameSet
\\ \GoSynCatName{Type variable name} & \fgTyVar, \fgTyVarAux & \in \tyvarSet
\\[\medskipamount]
\GoSynCatName{Type name} & t, u & ::= t_S \mid t_I
\\ \GoSynCatName{Type} & \fgType, \fgTypeAux & ::= \fgTyVar \mid t[\ForeachN{\fgType}]
\end{array}
&
\begin{array}{lr@{~}l}
\GoSynCatName{Field name} & f & \in \fieldNameSet
\\ \GoSynCatName{Method name} & m & \in \methodNameSet
\\ \GoSynCatName{Variable name} & x,y & \in \expvarSet
\\[\medskipamount]
\GoSynCatName{Struct type} & \fgType_S, \fgTypeAux_S & ::= t_S[\ForeachN{\fgType}]
\\
\GoSynCatName{Interface type} & \fgType_I, \fgTypeAux_I & ::= t_I[\ForeachN{\fgType}]
\end{array}
\ea\]
\[\ba{lr@{~}r@{~}l}
\GoSynCatName{Expression} & e, g & ::= &
x \mid e.m[\Multi{\fgType}](\overline{e})
\mid \fgType_S \{ \Multi{e} \} \mid e.f
\\
\GoSynCatName{Method signature} & R & ::= &
m[\Multi{\fgTyVar\,\fgType_I}](\overline{x \ \fgType}) \ \fgType
\\
\GoSynCatName{Declaration} & D & ::= &
\mbox{\kw{type}}\ t_S[\Multi{\fgTyVar\,\fgType_I}] \ \mbox{\kw{struct}}\ \{ \overline{f \ \fgType} \} ~~\mid~~
\mbox{\kw{type}}\ t_I[\Multi{\fgTyVar\,\fgType_I}] \ \mbox{\kw{interface}}\ \{ \overline{R} \} \\
&& \mid & \mbox{\kw{func}}\ (x \ t_S[\Multi{\fgTyVar\,\fgType_I}])\,R\, \{ \mbox{\kw{return}}\ e \}
\\
\GoSynCatName{Program} & P & ::= & \overline{D} \ \mbox{\kw{func}}\ \mbox{\mathem main} () \{ \_ = e \}
\ea\]
}
\Cref{f:fgg} introduces the syntax of FGG without dynamic type assertions.
We assume several countably infinite, pairwise disjoint sets
for names, ranged over by $\NameSetFGG$ with some subscript (upper part of the figure).
Meta variables $t_S$ and $u_S$ denote struct names,
$t_I$ and $u_I$ interface names,
$\alpha$ and $\beta$
type variables,
$f$ field names, $m$ method names,
and $x, y$ denote names for variables in expressions.
Overbar notation $\sOver[n]{\mathfrak s}$ is a shorthand for the
sequence $\mathfrak s_1 \ldots \mathfrak s_n$ where $\mathfrak s$ is some
syntactic construct. In some places, commas separate the sequence
items.
If irrelevant, we omit the $n$ and simply write
$\sOver{\mathfrak s}$. Using the index variable $i$ under an overbar
marks the parts that vary from sequence item to sequence
item; for example, $\sOver[n]{\mathfrak s'\,\mathfrak s_i}$ abbreviates
$\mathfrak s'\,\mathfrak s_1\ldots\mathfrak s'\,\mathfrak s_n$
and $\sOver[q]{\mathfrak s_j}$ abbreviates
$\mathfrak s_{j1}\,\ldots\,\mathfrak s_{jq}$.
The middle part of \Cref{f:fgg} shows the syntax of types in FGG.
A type name $t,u$ is either a struct or interface name. Types
$\fgType, \fgTypeAux$ include types variables $\fgTyVar$ and instantiated types
$t[\Multi{\fgType}]$.
For non-generic structs or interfaces, we often
write just $t$ instead of $t[]$.
Struct types $\fgType_S$, $\fgTypeAux_S$ and interface types $\fgType_I$, $\fgTypeAux_I$
denote syntactic subsets of the full type syntax.
The lower part of \Cref{f:fgg} defines the syntax of FGG expressions, declarations, and programs.
Expressions, ranged over by $e$ and $g$, include variables $x$, method calls, struct literals,
and field selections.
A method call $e.m[\Multi{\fgType}](\Multi{e})$ invokes method $m$ on receiver $e$ with type
arguments $\Multi{\fgType}$ and arguments $\Multi{e}$. If $m$ does not take type arguments,
we often write just $e.m(\Multi{e})$. A struct literals $\fgType_S\{\Multi[n]{e}\}$
creates an instance of a struct with $n$ fields,
the arguments $\Multi[n]{e}$ become the values of the fields in order of appearance
in the struct definition.
A field selection $e.f$ projects the value of some struct field $f$ from
expression $e$.
A method signature $R ::= m[\Multi{\fgTyVar\,\fgType_I}](\Multi{x\ \fgType})\,\fgType$
consists of a name $m$, bounded type parameters $\fgTyVar_i$ with interface type $\fgType_{Ii}$
as upper bounds, parameters $x_i$ of type $\fgType_i$,
and return type $\fgType$. It binds $\Multi{\fgTyVar}$ and $\Multi{x}$.
The scope of a type variable $\fgTyVar_i$ is $\Multi{\fgType}$, $\fgType$, and all upper bounds
$\Multi{\fgType_I}$, so FGG supports F-bounded quantification \citep{Canning1989}.
For non-generic methods, we often write just $m(\Multi{x_i\ \fgType_i})\,\fgType$.
A declaration $D$ comes in three forms: a struct
$\mbox{\kw{type}}\ t_S[\Multi{\fgTyVar\,\fgType_I}] \ \mbox{\kw{struct}}\ \{ \overline{f \ \fgType} \}$
with fields $f_i$ of type $\fgType_i$; an interface
$\mbox{\kw{type}}\ t_I[\Multi{\fgTyVar\,\fgType_I}] \ \mbox{\kw{interface}}\ \{ \overline{R} \}$
with method signatures $\Multi{R}$; or a method
$\mbox{\kw{func}}\ (x \ t_S[\Multi{\fgTyVar\,\fgType_I}])\,R\, \{ \mbox{\kw{return}}\ e \}$
providing an
implementation of method $R$ for struct $t_S$.
All three forms bind the type variables $\Multi{\fgTyVar}$, a method implementation additionally binds
the receiver parameter $x$.
The scope of a type variable $\fgTyVar_i$ includes all upper bounds
$\Multi{\fgType_I}$, the body of the declaration enclosed in $\{\ldots\}$, and for method declarations also the
signature $R$. We omit the $[\Multi{\fgTyVar\,\fgType_I}]$ part completely
if $\Multi{\fgTyVar\,\fgType_I}$ is empty.
Finally, a program $P$ consists of a sequence of declaration together
with a main function. Method and function bodies only contain a single expression.
We follow the usual convention and identify syntactic constructs up to renaming of bound variables or
type variables.
The syntax of FGG as presented here differs slightly from its original form~\citep{FeatherweightGo}.
The original article encloses
type parameters in parenthesis, an additional $\mbox{\kw{type}}$ keyword
starts a list of type parameters. Here,
we follow the syntax of full Go and use square brackets without any keyword.
Further, the original article prepends \kw{package main} to each program, something
we omit for succinctness. Finally, we reduce the number of syntactic meta-variables to improve readability.
\subsection{Dynamic Semantics}
\label{sec:dynamic-semantics}
\boxfig{f:fgg-dynamic}{Dynamic semantics of FGG}{
\vspace{-1ex}
\[\ba{lr@{~}r@{~}l}
\sSynCatName{Value} &
v, u, w & ::= & \fgType_S \{ \ForeachN{v} \}
\\
\sSynCatName{Evaluation context} &
{\mathcal E} & ::= & \Hole
\mid \fgType_S \{ \ForeachN{v}, {\mathcal E}, \ForeachN{e} \}
\mid {\mathcal E}.f
\mid {\mathcal E}.m[\Multi{\fgType}](\ForeachN{e})
\mid v.m[\Multi{\fgType}](\ForeachN{v}, {\mathcal E}, \ForeachN{e})
\\
\sSynCatName{Value substitution} &
\vbFG & ::= & \Angle{\ForeachN{\subst{x}{v}}}
\\
\sSynCatName{Type substitution} &
\tbFG & ::= & \Angle{\ForeachN{\subst{\fgTyVar}{\fgType}}}
\ea\]
\sTransSection{
\ruleform{\reduce{e}{e}}
}{
Reductions
}
\begin{mathpar}
\inferrule[fg-context]
{\reduce{e}{e'}
}
{\reduce{{\mathcal E} [e]}{{\mathcal E} [e']}
}
\inferrule[fg-field]
{ \mbox{\kw{type}}\ t_S[\Multi{\fgTyVar\,\fgType_I}] \ \mbox{\kw{struct}}\ \{ \Foreach{f \ \fgTypeAux}{n} \} \in \ForeachN{D}\\
}
{ \reduce{t_S[\Multi{\fgType}] \{ \Foreach{v}{n} \}.f_i}{v_i}
}
\inferrule[fg-call]
{ v = t_S[\Multi{\fgType}] \{ \ForeachN{u} \}
\\ \mbox{\kw{func}}\ (x \ t_S[\Multi{\fgTyVar\,\fgType_I}]) \ m[\Multi{\fgTyVar'\,\fgType_I'}]
(\ForeachN{x \ \fgTypeAux}) \ \fgTypeAux \ \{ \mbox{\kw{return}}\ e \} \in \ForeachN{D}
}
{\reduce{v.m[\Multi{\fgType'}](\ForeachN{v})}
{\Angle{\subst{x}{v}, \ForeachN{\subst{x}{v}}}
\Angle{\Multi{\subst{\fgTyVar}{\fgType}}, \Multi{\subst{\fgTyVar'}{\fgType'}}}
e}
}
\end{mathpar}
}
\Cref{f:fgg-dynamic} defines a call-by-value dynamic semantics for FGG using a small-step reduction semantics
with evaluation contexts. The definition is largely taken from \citet{FeatherweightGo}.
We use $v, u, w$ to denote values, where
a value is a struct literal with all fields being values.
A call-by-value evaluation context ${\mathcal E}$ is an expression with a hole $\Hole$ such that
the hole marks the point where the next evaluation step should happen.
We write ${\mathcal E}[e]$ to denote the replacement of the hole in ${\mathcal E}$ with expression $e$.
A value substitution $\vbFG$ is a finite mapping $\Angle{\Multi{\subst{x}{v}}}$ from variables
to values, whereas a type substitution $\tbFG$ is a finite mapping
$\Angle{\Multi{\subst{\fgTyVar}{\fgType}}}$ from type variables to types.
The (type) variables in the domain of a substitution must be distinct.
Substitution application, written in prefix notation as
$\vbFG e$ or $\tbFG e$ or $\tbFG \fgType$, is defined in the usual, capture-avoiding way.
When combining two sequences, we implicitly assume that both sequences have the same length.
For example, combining variables $\Multi{x}$ and values $\Multi{v}$ to a substitution
$\Angle{\Multi{\subst{x}{v}}}$ implicitly assumes that there are as many variables as values.
The reduction relation $\reduce{e}{e'}$ denotes that expression $e$ reduces to expression $e'$.
To avoid clutter, the sequence of declarations $\fgDecls$ of the underlying program is implicitly available
in the rules defining this reduction relation. Rule \Rule{fg-context} applies a reduction step
inside an expression. Rule \Rule{fg-field} reduces a field selection $t_S[\Multi{\fgType}]\{\Multi{v}\}.f_i$
by extracting value $v_i$ corresponding to field $f_i$ from the struct literal.
Rule \Rule{fg-call} reduces a method call $t_S[\Multi{\fgType}]\{\Multi{u}\}.m[\Multi{\fgType'}](\Multi{v})$.
It retrieves a method definition for $m$ and $t_S$ and substitutes
type arguments, receiver, and value arguments in the method body.
Reduction in FGG is deterministic (see \Cref{lem:determ-eval-fgg} in \Cref{sec:determ-eval-fgg}
for a formal proof), assuming the following three
restrictions:
\begin{description}
\item[\Restriction{fgg-unique-structs}] Each struct $t_S$ is defined at most once in the program.
\item[\FGGUniqueFields] For each struct definition
$\mbox{\kw{type}}\ t_S[\Multi{\fgTyVar\,\fgType_I}]\ \mbox{\kw{struct}}\ \{ \overline{f \ \fgType} \}$,
field names $\Multi f$ are distinct.
\item[\FGGUniqueReceiver] Each method definition
$\mbox{\kw{func}}\ (x \ t_S[\Multi{\fgTyVar\,\fgType_I}]) \ m[\Multi{\fgTyVar'\,\fgType_I'}]
(\ForeachN{x \ \fgTypeAux}) \ \fgTypeAux \ \{ \mbox{\kw{return}}\ e \}$
is uniquely identified by struct name $t_S$ and method name $m$.
\end{description}
The first two restrictions ensures that the value for a field in rule \Rule{fg-field} is
unambiguous. The third restriction avoids multiple matching method definitions in
rule \Rule{fg-call}.
\section{Formal Properties}
\label{sec:formal-properties}
In this section, we establish that the type-directed translation from
\Cref{sec:type-directed} preserves the static and dynamic semantics of FGG programs.
Detailed proofs for all lemmas and theorems are given in the appendix.
\subsection{Preservation of Static Semantics}
\label{sec:pres-stat-semant-1}
It is straightforward to verify that the type system originally defined for FGG
is equivalent to the type system induced by the type-directed translation presented
in \Cref{sec:type-directed}.
In the following, we write $\Delta \, \vdash_{\mathsf{G}} \, \subtypeOf{\fgType}{\fgTypeAux}$ for FGG's subtyping relation,
$\Delta; \Gamma \, \vdash_{\mathsf{G}} \, e : \fgType$ for its typing relation on expressions, and
$\, \vdash_{\mathsf{G}} \, \fgOk{P}$ for the FGG typing relation on programs. These three relations were specified
by \citet{FeatherweightGo}. The original article on FGG also includes support for dynamic type assertions,
something we do not consider for our translation. Hence, we assume that FGG expressions do not
contain type assertions.
\begin{lemma}[FGG typing equivalence]
\label{thm:fgg-typing-equiv}
Typing in FGG is equivalent to the type system
induced by the translation.
\begin{EnumAlph}
\item If $\Delta \, \vdash_{\mathsf{G}} \, \subtypeOf{\fgType}{\fgTypeAux}$ then
either $\tdICons{\Delta}{\fgType}{\fgTypeAux}{V}$ for some $V$ or
$\fgTypeAux = \fgType$ and $\fgType$ is not an interface type.
\item If $\tdICons{\Delta}{\fgType}{\fgTypeAux}{V}$ then
$\Delta \, \vdash_{\mathsf{G}} \, \subtypeOf{\fgType}{\fgTypeAux}$.
\item If $\Delta; \Gamma \, \vdash_{\mathsf{G}} \, e : \fgType$ then
$\tdExpTrans{\pair{\Delta}{\Gamma}}{e : \fgType}{E}$ for some $E$.
\item If $\tdExpTrans{\pair{\Delta}{\Gamma}}{e : \fgType}{E}$ then
$\Delta; \Gamma \, \vdash_{\mathsf{G}} \, e : \fgType'$ for some $\fgType'$ and $\Delta \, \vdash_{\mathsf{G}} \, \subtypeOf{\fgType'}{\fgType}$.
\item $\, \vdash_{\mathsf{G}} \, \fgOk{P}$ iff $\tdProgTrans{P}{\mathit{Prog}}$.
\end{EnumAlph}
\end{lemma}
Claims (a) and (b) state that structural subtyping in FGG is equivalent to the relation
from \Cref{f:upcast}, except that the latter is not reflexive for type variables and struct types.
Claims (c) and (d) establish that expression typing in FGG and our expression typing
from \Cref{f:trans-exp1} are equivalent modulo subtyping.
FGG's expression typing rules do not have a general subsumption rule, so the type computed
by the original rules for FGG might be a subtype of the type deduced by our system.
FGG enjoys type soundness (see Theorem~4.3 and~4.4 of \citealt{FeatherweightGo}).
With \Cref{thm:fgg-typing-equiv}, we get the following type soundness result for our type system:
\begin{corollary}
Assume $\tdExpTrans{\pair{\EmptyFgEnv}{\EmptyFgEnv}}{e : \fgType}{E}$ for some $e$, $\fgType$, and $E$.
Then either $e$ reduces to some value of type $\fgType$ or $e$ diverges.
\end{corollary}
\subsection{Preservation of Dynamic Semantics}
\label{sec:pres-dynam-semant-1}
This section proves that evaluating a well-typed FGG program yields the same behavior as evaluating its
translation. Thereby, we consider all possible outcomes of evaluation: reduction to a value or divergence.
The proof of semantic equivalence is enabled by a syntactic, step-indexed logical relation that
relates an FGG expression and a TL expression at some FGG type.
We write $\reducek{k}{e}{e'}$ if $e$ reduces to $e'$ in \emph{exactly} $k \in \Nat$ steps,
where $\Nat$ denotes the natural numbers including zero.
By convention, we write $\reducek{0}{e}{e'}$ to denote $e = e'$.
The notation $\reduceStar{e}{e'}$ states that $\reducek{k}{e}{e'}$ for some
unknown $k \in \Nat$.
We write $\Diverge{e}$ to denote that $e$ does not terminate; that is, for all
$k \in \Nat$ there exists some $e'$ with $\reducek{k}{e}{e'}$.
The same definitions apply analogously to reductions in the target language.
\subsubsection{The Logical Relation}
\boxfig{f:reduce-rel-fg-tl-exp}{Relating FGG to \TL\ expressions}{
\sTransSection{
\ruleform{\LREquiv{e}{E}{\fgType}{k}}
}{
Expressions
}
\begin{mathpar}
\inferrule[equiv-exp]{
{
\begin{array}{c}
(
\forall k' < k,
v ~.~ \reduceFGk{e}{\kA}{v} \implies \exists \uT.
\reduceTLN{\mu}{E}{\uT} \wedge \LREquivVal{v}{\uT}{\fgType}{k - \kA}
)
\\
(
\forall k' < k,
e' ~.~ \reduceFGk{e}{k'}{e'} \wedge \Diverge{e'}
\implies
\Diverge{E}
)
\end{array}
}
}{
\LREquiv{e}{E}{\fgType}{k}
}
\end{mathpar}
\sTransSection{
\ruleform{\LREquivVal{v}{\uT}{\fgType}{k}}
}{Values}
\begin{mathpar}
\inferrule[equiv-struct]{
\mbox{\kw{type}}\ t_S[\Multi{\fgTyVar\,\fgType_I}] \ \mbox{\kw{struct}}\ \{ \Foreach{f \ \fgTypeAux}{n} \} \in \ForeachN{D} \\
\forall i \in [n] . \LREquivVal{v_i}{V_i}{\MultiSubst{\fgTyVar}{\fgType} \fgTypeAux_i}{k}
}{
\LREquivVal{t_S[\Multi{\fgType}] \{ \Foreach{v}{n} \}}{\Tuple{\Foreach{V}{n}}}{t_S[\Multi{\fgType}]}{k}
}
\inferrule[equiv-iface]{
\exists \fgTypeAux_S . \forall k_1 < k . \LREquivVal{v}{U}{\fgTypeAux_S}{k_1} \\
\methodSpecifications{\fgType_I} = \{ \Foreach{R}{n} \} \\
\forall i \in [n], k_2 < k \,.\,
\LREquiv{\methodLookup{\methodName{R_i}}{\fgTypeAux_S}}{\uT_i}{R_i}{k_2}
}
{ \LREquivVal{v}{\Pair{U}{\Tuple{\Foreach{\uT}{n}}}}{\fgType_I}{k}
}
\end{mathpar}
\sTransSection{
\ruleform{\methodLookup{m}{\fgType_S} = \MLookupRes{x}{\fgType_S}{R}{e}}
}{
Method lookup
}
\begin{mathpar}
\inferrule[method-lookup]{
\mbox{\kw{func}}\ (x \ t_S[\Multi{\fgTyVar\,\fgType_I}]) \ R \ \{ \mbox{\kw{return}}\ e \} \in \ForeachN{D}\\
m = \methodName{R}\\
\fgTySubst = \MultiSubst{\fgTyVar}{\fgType}
}{
\methodLookup{m}{t_S[\Multi{\fgType}]} = \MLookupRes{x}{t_S[\Multi{\fgType}]}{\fgTySubst R}{\fgTySubst e}
}
\end{mathpar}
\sTransSection{
\ruleform{
\LREquiv{\MLookupRes{x}{\fgType_S}{R}{e}}{U}{R}{k}
}
}{
Method dictionary entries
}
\begin{mathpar}
\inferrule[equiv-method-dict-entry]{
\forall \kA \leq k, \Multi[p]{\fgType}, W, v, V, \Multi[n]{v}, \Multi[n]{V}.\\\\
(\fgTySubst = \MultiSubst[p]{\fgTyVar}{\fgType} \wedge
\LREquiv{\Multi[p]{\fgType}}{W}{\Multi[p]{\fgTyVar\,\fgType_I}}{\kA} \wedge
\LREquiv{v}{V}{\fgType_S}{\kA}
\wedge (\forall i \in [n]. \LREquiv{v_i}{\uT_i}{\fgTySubst \fgTypeAux_i}{\kA})) \\\\
\implies \LREquiv{
\Angle{\subst{x}{v},\Multi[n]{\subst{x}{v}}} \fgTySubst e
}{
U~\Triple{V}{W}{\Tuple{\Multi[n]{V}}}
}{\fgTySubst \fgTypeAux}{\kA}
}
{
\LREquiv{
\MLookupRes{x}{\fgType_S}{
m[\Multi[p]{\fgTyVar\,\fgType_I}](\Foreach{x \ \fgTypeAux}{n}) \, \fgTypeAux
}{e}
}{U}{
m[\Multi[p]{\fgTyVar\,\fgType_I}](\Foreach{x \ \fgTypeAux}{n}) \, \fgTypeAux
}{k}
}
\end{mathpar}
\sTransSection{
\ruleform{
\LREquiv{\Multi{\fgTypeAux}}{V}{\Multi{\fgTyVar\,\fgType_I}}{k}
}
}{
Bounded type parameters
}
\begin{mathpar}
\inferrule[equiv-bounded-typarams]{
\fgTySubst = \Angle{\Multi[n]{\subst{\fgTyVar}{\fgTypeAux}}}\\
\forall k' \leq k, i \in [n], u_i, U_i \,.\,
\LREquiv{u_i}{U_i}{\fgTypeAux_i}{k'} \implies
\LREquiv{u_i}{V_i\ U_i}{\fgTySubst \fgType_{Ii}}{k'}
}{
\LREquiv{\Multi[n]{\fgTypeAux}}{\Tuple{\Multi[n]{V}}}{\Multi[n]{\fgTyVar\ \fgType_I}}{k}
}
\end{mathpar}
}
The definition of the logical relation spreads over two figures~\ref{f:reduce-rel-fg-tl-exp}
and~\ref{f:reduce-rel-fg-tl-decl}.
In these figures, we assume that the declarations $\fgDecls$ of the FGG program being translated
are implicitly available in all rules. Also, we assume that an arbitrary but fixed method
substitution $\tlMethTable$ is implicitly available to all rules.
We now
explain the logical relation on expressions, see \Cref{f:reduce-rel-fg-tl-exp}.
The relation $\LREquiv{e}{E}{\fgType}{k}$ denotes that FGG expression $e$
and TL expression $E$ are equivalent at type $\fgType$ for at most $k$ reduction
steps. We call $k$ the \emph{step index}.
Rule \Rule{equiv-exp} has two implications as its premises. The first states that if $e$ reduces to a value $v$
in $k' < k$ steps, then $E$ reduces to some value $V$ in an arbitrary number
of steps and $v$ is equivalent to $V$ at type $\fgType$ for the remaining $k - k'$ steps.
The second premise is for diverging expressions: if $e$ reduces in less than $k$ steps to
some expression $e'$ and $e'$ diverges, then $E$ diverges as well.
The relation $\LREquivVal{v}{\uT}{\fgType}{k}$ defines equivalence of FGG value $v$ and
TL value $V$ at type $\fgType$ with step index $k$.
Rule \Rule{equiv-struct} handles the case where $\fgType$ is a struct type. Then
$v$ must be a value of this struct type and $V$ must be a struct value
such that all field values of $v$ and $V$ are equivalent.
Rule \Rule{equiv-iface} deals with the case that $\fgType$ is an interface type.
Hence, $V$ must be an interface value $\Pair{U}{\Tuple{\Multi{V}}}$
with two requirements. First, $v$ and $U$ are equivalent for all step indices $k_1 < k$ at some struct type
$\fgTypeAux_S$. Second, $\Tuple{\Multi{V}}$ must be
an appropriate dictionary for the methods of the interface
with receiver type $\fgTypeAux_S$.
To check this requirement, rule \Rule{method-lookup} defines the auxiliary
$\methodLookup{m_i}{\fgTypeAux_S}$ to
retrieve a quadruple $\MLookupRes{x}{\fgTypeAux_S}{R}{e}$ from the declaration
of $m_i$. This quadruple has to be equivalent to dictionary entry $V_i$ for all step indices $k_2 < k$
at the signature of the method.
Equivalence between such a quadruple and dictionary entry $U$
is written $\LREquiv{\MLookupRes{x}{\fgType_S}{R}{e}}{U}{R}{k}$. Rule \Rule{equiv-method-dict-entry}
defines this equivalence such that method body $e$ and $U$ take related
arguments to related outputs. Thus, the premise of the rule requires for all step indices $k' \leq k$,
all related type parameters $\Multi{\fgType}$ and $W$, all related receiver values $v$ and $V$,
and all related arguments $\Multi{v}$ and $\Multi{V}$ that $e$ and $U$ yield related
results when applied to the respective arguments.
The judgment $\LREquiv{\Multi{\fgTypeAux}}{V}{\Multi{\fgTyVar\,\fgType_I}}{k}$ denotes equivalence between
concrete type arguments
$\Multi{\fgTypeAux}$ and their TL realization $V$ when checking the bounds of type parameters $\Multi{\fgTyVar\,\fgType_I}$.
The definition in rule \Rule{equiv-bounded-typarams} relies on our translation strategy that
bounded type parameters are represented by coercions.
Having explained all judgments from \Cref{f:reduce-rel-fg-tl-exp}, we verify that the recursive definitions
of $\LREquiv{e}{E}{\fgType}{k}$ and $\LREquivVal{v}{\uT}{\fgType}{k}$ are well-founded.
Often, logical relations are defined by induction on the structure of
types. In our case, this approach does not work because interface types in FGG might be recursive,
see our previous work \citep{SulzmannWehr-mpc2022} for an example.
Thus, we use the step index as part of a decreasing measure $\Measure$.
Writing $\Size{V}$ for the size of some target value $V$, we define
$\Measure(\LREquiv{e}{E}{\fgType}{k}) = (k, 1, 0)$ and
$\Measure(\LREquivVal{v}{\uT}{\fgType}{k}) = (k, 0, \Size{V})$.
In \Rule{equiv-exp}, either $k$ decreases or stays constant but the second component of $\Measure$ decreases.
In \Rule{equiv-struct}, $k$ and the second component
stay constant but $\Size{V}$ decreases, and
in \Rule{equiv-iface} together with \Rule{equiv-method-dict-entry} and \Rule{equiv-bounded-typarams} step index $k$
decreases. Note that \Rule{equiv-method-dict-entry} and \Rule{equiv-bounded-typarams} only require $k' \leq k$.
This is ok because we already have $k_2 < k$ in \Rule{equiv-iface}.
\boxfig{f:reduce-rel-fg-tl-decl}{Relating FGG to \TL\ substitutions and declarations}{
\sTransSection{
\ruleform{
\LREquiv{\fgTySubst}{\rho}{\Delta}{k}\qquad
\LREquiv{\vbFG}{\rho}{\Gamma}{k}
}
}{Substitutions}
\begin{mathpar}
\inferrule[equiv-ty-subst]{
\LREquiv{
\Multi{\fgTySubst \fgTyVar_i}
}{
\Tuple{\Multi{\tlSubst X_{\fgTyVar_i}}}
}{\Multi{\fgTyVar\ \fgType}}{k}
}
{
\LREquiv{
\fgTySubst
}{
\tlSubst
}{\{\Multi{\fgTyVar : \fgType}\}}{k}
}
\inferrule[equiv-val-subst]
{ \forall (x : \fgType) \in \Gamma .\
\LREquiv{\vbFG(x)}{\rho(X)}{\fgType}{k}
}
{
\LREquiv{\vbFG}{\rho}{\Gamma}{k}
}
\end{mathpar}
\sTransSection{
\ruleform{
\LREquivNoTy{\mbox{\kw{func}}\ (x \ t_S[\Multi{\fgTyVar\,\fgType_I}]) \ R \ \{ \mbox{\kw{return}}\ e \}}{k}{X}
}
}{
Method declarations
}
\begin{mathpar}
\inferrule[equiv-method-decl]{
\forall k' < k, \Multi[p]{\fgType}, \Multi[q]{\fgType'}, \Multi[p]{W},
\Multi[q]{W'}, v, V, \Multi[n]{v}, \Multi[n]{V} .\\
\fgTySubst = \Angle{\Multi[p]{\subst{\fgTyVar}{\fgType}}, \Multi[q]{\subst{\fgTyVar'}{\fgType'}}} \wedge
\LREquiv{
\Multi[p]{\fgType} \Multi[q]{\fgType'}
}{
\Tuple{\Multi[p]{W}, \Multi[q]{W'}}
}{\Multi[p]{\fgTyVar\,\fgType_I}, \Multi[q]{\fgTyVar'\,\fgType_I'}}{k'}
\wedge\\
\LREquiv{v}{V}{t_S[\fgTySubst \Multi[p]{\fgTyVar}]}{k'} \wedge
(\forall i \in [n] . \LREquiv{v_i}{V_i}{\fgTySubst \fgTypeAux_i}{k'})
\implies\\
\LREquiv{
\Angle{\subst{x}{v}, \Multi[n]{\subst{x}{v}}} \fgTySubst e
}{
X~\Quadr{\Tuple{\Multi[p]{W}}}{V}{\Tuple{\Multi[q]{W'}}}{\Tuple{\Multi[n]{V}}}
}{
\fgTySubst \fgTypeAux
}{k'}
}{
\LREquivNoTy{\mbox{\kw{func}}\ (x \ t_S[\Multi[p]{\fgTyVar\,\fgType_I}]) \
m[\Multi[q]{\fgTyVar'\,\fgType_I'}](\Multi[n]{x\ \fgTypeAux})\, \fgTypeAux \
\{ \mbox{\kw{return}}\ e \}}{k}{X}
}
\end{mathpar}
\sTransSection{
\ruleform{\LREquivNoTy{\Multi{D}}{k}{\tlMethTable}}
}{
Programs
}
\begin{mathpar}
\inferrule[equiv-decls]{
\Multi{D}, \tlMethTable \textrm{~are implicitly available in all subderivations}\\
\forall D_i \in \Multi{D} .
D_i = \mbox{\kw{func}}\ (x \ t_S[\Multi{\fgTyVar\,\fgType}]) \ m M \ \{ \mbox{\kw{return}}\ e \}
\implies \LREquivNoTy{D_i}{k}{\mT{m}{t_S}}
}{
\LREquivNoTy{\Multi{D}}{k}{\tlMethTable}
}
\end{mathpar}
}
\Cref{f:reduce-rel-fg-tl-decl} extends the logical relation to whole programs.
Judgment $\LREquiv{\fgTySubst}{\rho}{\Delta}{k}$ denotes how a FGG type
substitution $\fgTySubst$ intended to substitute the type variables from
$\Delta$ is related to a TL substitution $\rho$.
The definition in rule \Rule{equiv-ty-subst} falls back to equivalence of type parameters.
Judgment $\LREquiv{\vbFG}{\rho}{\Gamma}{k}$ similarly relates a FGG value substitution $\fgSubst$
intended for value environment $\Gamma$ with a TL substitution $\rho$. See rule
\Rule{equiv-val-subst}.
Judgment $\LREquivNoTy{\mbox{\kw{func}}\ (x \ t_S[\Multi{\fgTyVar\,\fgType_I}]) \ R \ \{ \mbox{\kw{return}}\ e \}}{k}{X}$
states equivalence of a function declaration with a TL variable $X$.
Rule \Rule{equiv-method-decl} takes an approach similar as in rule \Rule{equiv-method-dict-entry}:
method body $e$ and variable $X$ must yield related outputs when applied to
related arguments. Thus, for all related type arguments $\Multi{\fgType}$, $\Multi{\fgType'}$ and
$\Pair{\Multi{W}}{\Multi{W'}}$, all related receiver values $v$ and $V$, and
all related arguments $\Multi{v}$ and $\Multi{V}$, the expression $e$ and variable
$X$ must be related when applied to the appropriate arguments.
However, different than in \Rule{equiv-method-dict-entry}, we only requires
this to hold for all $k' < k$.
Judgment $\LREquivNoTy{\Multi{D}}{k}{\tlMethTable}$ defines equivalence between
FGG declarations $\fgDecls$ and TL method substitution $\tlMethTable$. The definition in
rule \Rule{equiv-decls} is straightforward: each method declaration for some
method $m$ and struct $t_S$ must be equivalent to variable $\mT{m}{t_S}$.
\subsubsection{Results}
To establish the desired result of semantic equivalence,
we implicitly make the following assumptions about the globally available declarations
$\fgDecls$ and method substitution $\tlMethTable$.
\begin{assumption}\label{conv:fg-decls}
We assume that the globally available declarations $\fgDecls$ are well-formed; that is,
$\fgDeclOk{D_i}$ for all $D_i \in \Multi{D}$
and $\tdMethTrans{D_i'}{X_i = V_i}$ for some $X_i$ and $V_i$ and all $D_i' = \mbox{\kw{func}}\,\ldots \in \Multi{D}$.
Further, we assume that the globally available method substitution $\mu$
has only variables of the form $\mT{m}{t_S}$ in its domain.
\end{assumption}
Several basic properties hold for our logical relation. For example,
monotonicity gives us that with $\LREquiv{e}{E}{\fgType}{k}$ and $k' \leq k$
we also have $\LREquiv{e}{E}{\fgType}{k'}$.
Another property is how target and source reductions preserve equivalence:
\begin{lemma}[Target reductions preserve equivalence]\label{lem:target-reduce}
If $\LREquiv{e}{E}{\fgType}{k}$ and $\reduceStar{E_2}{E}$ then
$\LREquiv{e}{E_2}{\fgType}{k}$.
\end{lemma}
\begin{lemma}[Source reductions preserve equivalence]\label{lem:source-reduce}
If $\LREquiv{e}{E}{\fgType}{k}$ and $\reduce{e_2}{e}$ then
$\LREquiv{e_2}{E}{\fgType}{k+1}$.
\end{lemma}
The lemmas for monotonicity and several other properties
are stated in \Cref{sec:pres-dynam-semant}, together with all proofs.
We can then establish that an FGG expression $e$ is semantically equivalent
to its translation $E$.
\begin{lemma}[Expression equivalence]\label{lem:exp-equiv}
Assume $\LREquivNoTy{\fgDecls}{k}{\tlMethTable}$ and
$\LREquiv{\fgTySubst}{\tlSubst}{\Delta}{k}$ and
$\LREquiv{\fgSubst}{\tlSubst}{\fgTySubst \Gamma}{k}$.
If $\tdExpTrans{\pair{\Delta}{\Gamma}}{e : \fgType}{E}$
then $\LREquiv{\fgSubst \fgTySubst e}{\tlSubst E}{\fgTySubst \fgType}{k}$.
\end{lemma}
The proof is by induction on the derivation of
$\tdExpTrans{\pair{\Delta}{\Gamma}}{e : \fgType}{E}$, see \Cref{sec:proof-exp-equiv}
for the full proof.
We next establish semantic equivalence for method
declarations.
\begin{lemma}[Method equivalence]\label{lem:method-equiv}
Let $\fgDecls$ and $\mu$ such that for each
$D = \mbox{\kw{func}}\ (x\, t_S[\Multi{\fgTyVar\,\fgType_I}])\,R\,\{\mbox{\kw{return}}\, e\} \in \fgDecls$ with $m = \methodName{R}$ we have
$\tdMethTrans{D}{\mT{m}{t_S} = V}$ and
$\mu(\mT{m}{t_S}) = V$ for some $V$.
Then $\LREquivNoTy{\fgDecls}{k}{\mu}$ for any $k$.
\end{lemma}
The proof of this lemma is by induction on $k$, see \Cref{sec:lem:method-equiv} for the full proof.
Finally, the following theorem states our desired result:
semantic equivalence between an FGG program and its translation.
\begin{theorem}[Program equivalence]\label{thm:prog-equiv}
Let $\tdProgTrans{\ForeachN{D} \ \mbox{\kw{func}}\ \mbox{\mathem main} () \{ \_ = e \}} {\mbox{\kw{let}}\ \Multi{X_i = V_i} \ \mbox{\kw{in}}\ E}$
where we assume that $e$ has type $\fgType$.
Let $\mu = \Angle{\Multi{\subst{X_i}{V_i}}}$.
Then both of the following holds:
\begin{enumerate}
\item If $\reduceStar{e}{v}$ for some value $v$ then there exists
a target language value $V$ such that
$\reduceTLN{\mu}{E}{V}$ and
$\LREquivVal{v}{V}{\fgType}{k}$ for any $k$.
\item If $e$ diverges then so does $E$.
\end{enumerate}
\end{theorem}
Obviously, $\fgDecls$ and $\tlMethTable$
meet the requirements of Assumption~\ref{conv:fg-decls}.
The theorem then follows from \Cref{lem:exp-equiv} and
\Cref{lem:method-equiv}. See \Cref{sec:proof-crefthm:pr-equ}
for the full proof.
\subsection{Getting the step index right}
\label{sec:step-index}
At some places, we require the step index in the premise to be strictly
smaller than in the conclusion ($<$), other places require only less-than-or-equal ($\leq$).
In \Rule{equiv-exp}, we have $<$ to keep the definition of the LR well-founded.
The $<$ in rule \Rule{equiv-method-decl} is required for the inductive argument in the proof of
\Cref{lem:method-equiv}.
Rule \Rule{equiv-iface} also has $<$, but rule
\Rule{equiv-method-dict-entry} only requires $\leq$. For well-foundedness, it is crucial
that one of these rules decreases the step index. However, equally important is that the step index
is not forced to decrease more than once, so we need $<$ in one rule and $\leq$ in the other.
If both rules had $<$, then the proof of \Cref{lem:exp-equiv} would not go through for case \Rule{call-iface}.
\renewcommand\LabelQualifier{lr-design-choices}
Consider the following example in the context of \Cref{f:trans-example-src}:
\[
\begin{array}{r@{\quad}c@{\quad}l}
w_1 = \Num\{1\} \textrm{~at type~} \Eq[\Num] &
\leadsto
&
W_1 = \Pair{\Tuple{1}}{\Tuple{U}}\\
&& \quad \textrm{where~} U = \tllambda{\TripleY} X_{\eq,\Num}\,\QuadrY{\Void}
\\
w_2 = \Num\{2\} \textrm{~at type~} \Num &
\leadsto
&
W_2 = \Tuple{\Tuple{2}}
\\
w_1.\eq(w_2)
&
\leadsto
&E = \mbox{\kw{case}}\,W_1\,\mbox{\kw{of}}\,\Pair{Y}{\Tuple{X_1}} \to X_1\,\Triple{Y}{\Void}{\Tuple{W_2}}
\end{array}
\]
For values $w_1$ and $w_2$, we may assume \TextLabel{eq:equiv-val} $\LREquivVal{w_1}{W_1}{\Eq[\Num]}{k}$
and $\LREquiv{w_2}{W_2}{\Num}{k}$ for some $k$.
To verify that the translation yields related expressions,
we must show
\begin{igather}
\LREquiv{w_1.\eq(w_2)}{E}{\bool}{k} \QLabel{eq:toshow}
\end{igather}
From \QRef{eq:equiv-val}, via inversion of rule \Rule{equiv-iface}, we can derive
\begin{igather}
\LREquiv{\methodLookup{\eq}{\Num}}{U}{\eq(\that\,\Num)\,\bool}{k-1} \QLabel{eq:methodLookup}
\end{igather}
because the premise of the rule requires this to hold for all $k_2 < k$.
Let $e$ be the body of the method declaration of $\eq$ for $\Num$.
Inverting rule \Rule{equiv-method-dict-entry} for \QRef{eq:methodLookup} yields
\begin{igather}
\LREquiv{
\Angle{\subst{\this}{w_1}, \subst{\that}{w_2}} e
}{
U\,\Triple{\One}{\Void}{\Tuple{\Two}}
}{
\bool
}{
k'
}\QLabel{eq:equiv-W}
\end{igather}
for $k' = k - 1$ because rule \Rule{equiv-method-dict-entry} has $\leq$
in its premise. Also, we have
$\reducek{1}{w_1.\eq(w_2)}{\Angle{\subst{\this}{w_1}, \subst{\that}{w_2}}} e$ and
$\reduceStar{E}{U\,\Triple{\One}{\Void}{\Tuple{\Two}}}$.
Thus, with \QRef{eq:equiv-W}, \Cref{lem:target-reduce}, and \Cref{lem:source-reduce} we
get $\LREquiv{w_1.\eq(w_2)}{E}{\bool}{k' + 1}$. For $k' = k - 1$, this is exactly
\QRef{eq:toshow}, as required. But if rule \Rule{equiv-method-dict-entry}
required $<$ in its premise, then \QRef{eq:equiv-W} would only hold
for $k' = k - 2$ and we could not derive \QRef{eq:toshow}.
Whether we have $<$ in \Rule{equiv-iface} and $\leq$ in \Rule{equiv-method-dict-entry} or vice versa
is a matter of taste.
In our previous work at MPC \citep{SulzmannWehr-mpc2022}, we established a
dictionary-passing translation for Featherweight Go without generics. The situation
is slightly different there. With generics, we need two rules with respect to methods:
\Rule{equiv-method-decl} for method declarations and
\Rule{equiv-method-dict-entry} for dictionary entries where the coercions for the bounds of the
receiver's type parameters have already been supplied. Without generics, there are no type parameters,
so a single rule suffices (rule \Rule{red-rel-method} in MPC). So in the article at MPC,
we use $<$ for rule \Rule{red-rel-method} and $\leq$ for rule \Rule{red-rel-iface},
the pendant to rule \Rule{equiv-iface} of the current article.
\section{Introduction}
\label{sec:intro}
Go~\citeyearpar{golang} is a statically typed programming language introduced by
Google in 2009.
It supports method overloading by allowing multiple declarations of the same method signature
for different receivers. Receivers are structs, similar to structs in C.
The language also supports interfaces; as in many object-oriented languages, an interface
consists of a set of method signatures. But unlike in many object-oriented
languages, subtyping in Go is structural not nominal.
Earlier work by \citet{FeatherweightGo}
introduces Featherweight Go (FG), a minimal core calculus that
covers method overloading, structs, interfaces and structural subtyping.
Their work specifies static typing rules and a dynamic semantics for FG based on runtime method lookup.
However, the actual Go implementation appears to employ a different dynamic semantics.
Quoting Griesemer and co-workers:
\enquote{\emph{Go is designed to enable efficient implementation. Structures are laid out in memory as a sequence of fields,
while an interface is a pair of a pointer to an underlying structure and a pointer to a dictionary of methods.}}
In our own prior work~\citep{SulzmannWehr-aplas2021,SulzmannWehr-mpc2022},
we formalize a type-directed dictionary-passing translation for FG
and establish its semantic equivalence with FG's dynamic semantics.
Griesemer and coworkers also introduce Featherweight Generic Go (FGG), an extension of FG with generics.
In this work, we extend our translation approach to FGG.
Our contributions are as follows:
\begin{itemize}
\item We specify the translation of source FGG programs without type assertions to
an untyped $\lambda$-calculus with recursive let-bindings, constructors and pattern matching.
We employ a dictionary-passing translation scheme \`a la type classes~\citep{Hall:1996:TCH:227699.227700}
to statically resolve overloaded method calls. The translation
is guided by the typing of the FGG program.
\item We establish the semantic correctness of the dictionary-passing translation.
The result relies on a syntactic, step-indexed logical relation to ensure well-foundedness of
definitions in the presence of recursive interface types and recursive methods.
\item We explain how to extend the translation to support type assertions.
\item We present an implementation of the translation, including support for type assertions.
\end{itemize}
The upcoming \Cref{sec:examples} gives an overview of our translation by example.
\Cref{sec:featherweight-generic-go} gives a recap of the source language FGG, whereas
\Cref{sec:type-direct-transl} defines the target language and the translation itself.
Next, \Cref{sec:formal-properties} establishes the formal properties of the translation,
rigorous proofs of our results can be found in the Appendix.
\Cref{sec:discussion} presents the implementation and explains the handling of type assertions.
\Cref{sec:related-work} covers related work.
Finally, \Cref{sec:future-work-concl} summarizes this work and points out directions for future work.
\section{Proofs}
\label{sec:more-form-prop}
\subsection{Deterministic Evaluation in FGG and TL}
\label{sec:determ-eval-fgg}
\begin{lemma}[Deterministic evaluation in FGG]\label{lem:determ-eval-fgg}
If $\reduce{e}{e'}$ and $\reduce{e}{e''}$ then $e' = e''$.
If $\reduce{E}{E'}$ and $\reduce{E}{E''}$ then $E' = E''$.
\end{lemma}
\begin{proof}
We first state and prove three sublemmas:
\begin{EnumAlph}
\item If $e = {\mathcal E}_1[{\mathcal E}_2[e']]$ then there exists ${\mathcal E}_3$ with $e = {\mathcal E}_3[e']$.
The proof is by induction on ${\mathcal E}_1$.
\item If $\reduce{e}{e'}$ then there exists a derivation of $\reduce{e}{e'}$ that ends with
at most one consecutive application of rule \Rule{fg-context}. The proof is by induction
on the derivation of $\reduce{e}{e'}$. From the IH, we know that this derivation ends with
at most \emph{two} consecutive applications of rule \Rule{fg-context}. If there are
two such consecutive applications, (a) allow us to merge the two
evaluation contexts involved, so that we need only one consecutive application
of \Rule{fg-context}.
\item We call an FGG expression \emph{directly reducible} if it reduces but not by rule \Rule{fg-context}.
If $e_1$ and $e_2$ are now directly reducible and ${\mathcal E}_1[e_1] = {\mathcal E}_2[e_2]$ then ${\mathcal E}_1 = {\mathcal E}_2$
and $e_1 = e_2$. For the proof, we first note that ${\mathcal E}_1 = \Hole$ iff ${\mathcal E}_2 = \Hole$. This
holds because directly reducible expressions have no inner redexes. The rest of the
proof is then a straightforward induction on ${\mathcal E}_1$.
\end{EnumAlph}
Now assume $\reduce{e}{e'}$ and $\reduce{e}{e''}$. By (b) we may assume that both derivations
ends with at most one consecutive application of rule \Rule{fg-context}. It is easy to see
(as values do not reduce) that both derivations must end with the same rule. If this rule
is \Rule{fg-field}, then $e' = e''$ by restrictions \Restriction{fgg-unique-structs}{} and
\FGGUniqueFields{}. If this rule is \Rule{fg-call}, then $e' = e''$ by
\FGGUniqueReceiver{}.
If the rule is \Rule{fg-context},
we have the following situation with $R_1 \neq \Rule{fg-context}$ and
$R_2 \neq \Rule{fg-context}$:
\begin{mathpar}
\inferrule*[left=fg-context]{
\inferrule*[left=$R_1$]{
}{
\reduce{g_1}{g_1'}
}
}{
\reduce{\BraceBelow{{\mathcal E}_1[g_1]}{= e}}{\BraceBelow{{\mathcal E}_1[g_1']}{= e'}}
}
\inferrule*[right=fg-context]{
\inferrule*[right=$R_2$]{
}{
\reduce{g_2}{g_2'}
}
}{
\reduce{\BraceBelow{{\mathcal E}_2[g_2]}{= e}}{\BraceBelow{{\mathcal E}_2[g_2']}{= e''}}
}
\end{mathpar}
As neither $R_1$ nor $R_2$ are \Rule{fg-context}, we know that $g_1$ and $g_2$ are directly
reducible. Thus, with ${\mathcal E}_1[g_1] = {\mathcal E}_2[g_2]$ and (c) we get ${\mathcal E}_1 = {\mathcal E}_2$ and
$g_1 = g_2$. With $R_1$ and $R_2$ not being \Rule{fg-context}, we have $g_1' = g_2'$, so
$e' = e''$ as required. \MyQED
\end{proof}
\begin{lemma}[Deterministic evaluation in TL]\label{lem:determ-eval-tl}
If $\reduceExpTL{\tlMethTable}{E}{E'}$ and $\reduceExpTL{\tlMethTable}{E}{E''}$ then $E' = E''$.
Further, if $\reduce{E}{E'}$ and $\reduce{E}{E''}$ then $E' = E''$.
\end{lemma}
\begin{proof}
\renewcommand\LabelQualifier{lem:determ-eval-tl}
We first prove the first implication of the lemma
\begin{igather}
\forall E, E', E'', \tlMethTable ~.~
\reduceExpTL{\tlMethTable}{E}{E'} \wedge \reduceExpTL{\tlMethTable}{E}{E''} \implies E' = E''
\QLabel{main}
\end{igather}
There are three sublemmas, analogously to the proof of \Cref{lem:determ-eval-fgg}.
\begin{EnumAlph}
\item If $E = {\mathcal R}_1[{\mathcal R}_2[E']]$ then there exists ${\mathcal R}_3$ with $E = {\mathcal R}_3[E']$.
\item If $\reduceExpTL{\tlMethTable}{E}{E'}$ then there exists a derivation of
$\reduceExpTL{\tlMethTable}{E}{E'}$ that ends with
at most one consecutive application of rule \Rule{tl-context}.
\item We call a target-language expression \emph{directly reducible} if it reduces but not by rule \Rule{tl-context}.
If $E_1$ and $E_2$ are now directly reducible and ${\mathcal R}_1[E_1] = {\mathcal R}_2[E_2]$ then ${\mathcal R}_1 = {\mathcal R}_2$
and $E_1 = E_2$.
\end{EnumAlph}
The proofs of these lemmas are similar to the proofs of the sublemmas in \Cref{lem:determ-eval-fgg}.
Then \QRef{main}
follows with reasoning similar to the proof of \Cref{lem:determ-eval-fgg}.
If the derivations of
$\reduceExpTL{\tlMethTable}{E}{E'}$ and $\reduceExpTL{\tlMethTable}{E}{E''}$ both end
with rule \Rule{tl-case}, then our assumption that the constructors of a case-expression are
distinct ensures determinacy.
The second claim of the lemma
($\reduce{E}{E'}$ and $\reduce{E}{E''}$ imply $E' = E''$) then follows directly from \QRef{main}.
Our assumption that the variables of a top-level let-binding are distinct
ensures that the substitution $\tlMethTable$ built from the
top-level let-bindings is well-defined.
\MyQED
\end{proof}
\subsection{Preservation of Static Semantics}
\label{sec:pres-stat-semant}
\begin{proof}[Proof of \Cref{thm:fgg-typing-equiv}]
We prove (a) and (b) by case distinctions on the last rule of the given derivations;
(c) and (d) follow by induction on the derivations, using (a) and (b). Claim (e) then
follows by examining the typing rules, using (c) and (d). \MyQED
\end{proof}
\subsection{Preservation of Dynamic Semantics}
\label{sec:pres-dynam-semant}
\begin{convention}
We omit $\tlMethTable$ from reductions in the target language, writing
$\reduce{E}{E'}$ instead of $\reduceExpTL{\tlMethTable}{E}{E'}$.
\end{convention}
\newcommand\fgMakeSubst[3]{\subst{#1}{#2} : #3}
\newcommand\fgTyFormals{\Phi}
\newcommand\fgTyFormalsAux{\Psi}
\newcommand\fgTyActuals{\phi}
\newcommand\fgTyActualsAux{\psi}
\makeatletter
\define@key{fgSyn}{recv}{\def\fgSyn@recv{#1}}
\define@key{fgSyn}{spec}{\def\fgSyn@spec{#1}}
\define@key{fgSyn}{body}{\def\fgSyn@body{#1}}
\newcommand{\fgFunc}[1][]{%
\begingroup%
\setkeys{fgSyn}{recv={x\ t_S[\fgTyFormals]}, spec={mM}, body=e}%
\setkeys{fgSyn}{#1}%
\mbox{\kw{func}}\,(\fgSyn@recv)\,\fgSyn@spec\,\{ \mbox{\kw{return}}\,\fgSyn@body \}%
\endgroup%
}
\makeatother
\begin{definition}
We make use of some extra metavariables and notations.
\begin{itemize}[noitemsep,topsep=0pt,parsep=0pt,partopsep=0pt]
\item $\fgTyFormals, \fgTyFormalsAux$ denote formal type parameters $\ForeachN{\fgTyVar\,\fgType_I}$.
\item $\fgGetTyVars{\fgTyFormals}$ denotes the type variables of $\fgTyFormals$;
that is, if $\fgTyFormals = \Multi{\fgTyVar\,\fgType_I}$ then $\fgGetTyVars{\fgTyFormals} = \Multi{\fgTyVar}$.
\item $\fgTyActuals, \fgTyActualsAux$ denote actual type arguments $\ForeachN{\fgType}$.
\item $M ::= [\fgTyFormals](\overline{x \ \fgType}) \ \fgType$ denotes the type-part of a method signature $R$.
\item $L$ denotes a type literal
$\mbox{\kw{struct}}\ \{ \overline{f \ \fgType} \}$ or
$\mbox{\kw{interface}}\ \{ \overline{R} \}$.
\item $\fgMakeSubst{\fgTyFormals}{\fgTyActuals}{\fgTySubst}$ create a type substitution
$\fgTySubst$ form type parameters $\fgTyFormals$ and arguments $\fgTyActuals$. It
is defined like this:
$\fgMakeSubst{\Foreach{\fgTyVar\, \fgType}{n}}{\Foreach{\fgTypeAux}{n}}{\Angle{\Foreach{\subst{\fgTyVar_i}{\fgTypeAux_i}}{n}}}$
\end{itemize}
\end{definition}
\begin{lemma}[Monotonicity for expressions]\label{lem:mono-exp}
Assume $k' \leq k$.
If $\LREquiv{e}{E}{\fgType}{k}$ then $\LREquiv{e}{E}{\fgType}{k'}$.
If $\LREquivVal{v}{V}{\fgType}{k}$ then $\LREquivVal{v}{V}{\fgType}{k'}$.
\end{lemma}
\begin{proof}
\renewcommand\LabelQualifier{lem:mono-exp}
We proceed by induction on $(k, s)$ where $s$ is the combined size of $v, V$.
\begin{CaseDistinction}{on the last rule used in the two derivations}
\Case{rule \Rule{equiv-exp}}
We label the two implications in the premise of the rule as (a) and (b).
\begin{EnumAlph}
\item Assume $k'' < k'$ and $\reducek{k''}{e}{u}$ for some value $u$.
From $\LREquiv{e}{E}{\fgType}{k}$
\begin{igather}
\exists U \DOT \reduceStar{E}{U} \\
\LREquivVal{u}{U}{\fgType}{k - k''} \QLabel{red-u}
\end{igather}
If $k = k'$ then $\LREquivVal{u}{U}{\fgType}{k'-k''}$. Otherwise, $k' - k'' < k - k''$, so the IH
(induction hypothesis)
applied to \QRef{red-u} also yields $\LREquivVal{u}{U}{\fgType}{k'-k''}$.
This proves implication (a).
\item Assume $k'' < k'$ and $\reducek{k''}{e}{e'}$ and $\Diverge{e'}$.
Then we get with $\LREquiv{e}{E}{\fgType}{k}$ and $k' \leq k$ that
$\Diverge{E}$.
\end{EnumAlph}
\Case{rule \Rule{equiv-struct}} Follows from IH.
\Case{rule \Rule{equiv-iface}} Obvious.
\end{CaseDistinction} \MyQED
\end{proof}
\begin{lemma}[Monotonicity for method dictionaries]\label{lem:mono-dict}
If $\LREquiv{\Angle{x, \fgType_S, R, e}}{V}{R'}{k}$ and $k' \leq k$ then
$\LREquiv{\Angle{x, \fgType_S, R, e}}{V}{R'}{k'}$.
\end{lemma}
\begin{proof}
Obvious. \MyQED
\end{proof}
\begin{lemma}[Monotonicity for type parameters]\label{lem:mono-typarams}
If $\LREquiv{\fgTyActuals}{V}{\fgTyFormals}{k}$ and $k' \leq k$ then
$\LREquiv{\fgTyActuals}{V}{\fgTyFormals}{k'}$.
\end{lemma}
\begin{proof}
Obvious. \MyQED
\end{proof}
\begin{lemma}[Monotonicity for method declarations]\label{lem:mono-mdecl}
Assume declaration $D = \fgFunc[spec=R]$ and $k' \leq k$. If $\LREquivNoTy{D}{k}{X}$ then
$\LREquivNoTy{D}{k'}{X}$.
\end{lemma}
\begin{proof}
Obvious. \MyQED
\end{proof}
\begin{lemma}[Monotonicity for programs]\label{lem:mono-prog}
If $\LREquivNoTy{\Multi{D}}{k}{\tlMethTable}$ and $k' \leq k$ then
$\LREquivNoTy{\Multi{D}}{k'}{\tlMethTable}$.
\end{lemma}
\begin{proof}
Follows from \Cref{lem:mono-mdecl}. \MyQED
\end{proof}
\begin{proof}[Proof of \Cref{lem:target-reduce}]
Straightforward.
\end{proof}
\begin{proof}[Proof of \Cref{lem:source-reduce}]
We label the two implications in the premise of rule \Rule{equiv-exp} with (a) and (b).
\begin{EnumAlph}
\item Assume $k' < k+1$ and $\reducek{k'}{e_2}{v}$.
Then by \Cref{lem:determ-eval-fgg} $e_2 \longrightarrow \reducek{k'-1}{e}{v}$.
Noting that $k' -1 < k$ we get with the assumption $\LREquiv{e}{E}{\fgType}{k}$
\begin{igather}
\exists V \DOT \reduceStar{E}{V} \wedge \LREquivVal{v}{V}{\fgType}{k+1-k'}
\end{igather}
But this is exactly what is needed to prove implication (a) for
$\LREquiv{e_2}{E}{\fgType}{k+1}$.
\item Assume $k' < k+1$ and $\reducek{k'}{e_2}{e'}$ and $\Diverge{e'}$.
Then by \Cref{lem:determ-eval-fgg} $e_2 \longrightarrow \reducek{k'-1}{e}{e'}$.
Noting that $k' -1 < k$ we get with the assumption $\LREquiv{e}{E}{\fgType}{k}$
that $\Diverge{E}$. This proves implication (b).
\MyQED
\end{EnumAlph}
\end{proof}
\begin{lemma}[Expression equivalence implies value equivalence]\label{lem:exp-equiv-implies-val-equiv}
If $k \geq 1$ and $\LREquiv{v}{V}{\fgType}{k}$ then
$\LREquivVal{v}{V}{\fgType}{k}$.
\end{lemma}
\begin{proof}
From the first implication of rule \Rule{equiv-exp} we get for $k' = 0 < k$ and with
$\reducek{0}{v}{v}$ that
$\exists V' \DOT \reduceStar{V}{V'} \wedge \LREquivVal{v}{V'}{\fgType}{k}$.
But V is already a value, so $V' = V$.\MyQED
\end{proof}
\begin{lemma}[Value equivalence implies expression equivalence]\label{lem:val-equiv-implies-exp-equiv}
If $\LREquivVal{v}{V}{\fgType}{k}$ then
$\LREquiv{v}{V}{\fgType}{k}$ for any $k$.
\end{lemma}
\begin{proof}
We have $\reducek{0}{v}{v}$, so we get the first implication of rule \Rule{equiv-exp} by
setting $E = V$ and by assumption $\LREquivVal{v}{V}{\fgType}{k}$.
The second implication holds vacuously because values do not diverge. \MyQED
\end{proof}
\begin{lemma}
\label{lem:method-decl-equiv-implies-dict-equiv}
Assume $\LREquivNoTy{\fgFunc}{k}{X}$.
Then the following holds:
\begin{mathparNarrow}
\forall k' < k, \fgTyActuals, W \DOT \LREquiv{\fgTyActuals}{W}{\fgTyFormals}{k'}
\implies
\fgMakeSubst{\fgTyFormals}{\fgTyActuals}{\fgTySubst} \wedge\\
\LREquiv{
\Angle{x, t_S[\fgTyActuals], \fgTySubst mM, \fgTySubst e}
}{
\tllambda{\Triple{Y_1}{Y_2}{Y_3}} X\,\Quadr{W}{Y_1}{Y_2}{Y_3}
}{
\fgTySubst mM
}{k'}
\end{mathparNarrow}
\end{lemma}
\begin{proof}
\renewcommand\LabelQualifier{lem:method-decl-equiv-implies-dict-equiv}
Let $M = [\fgTyFormals'](\MultiN{x_i\ \fgType_i})\,\fgType$ and assume
for any $k', \fgTyActuals, W$
\begin{igather}
k' < k \QLabel{kprime}\\
\LREquiv{\fgTyActuals}{W}{\fgTyFormals}{k'} \QLabel{phi-equiv-W}
\end{igather}
Obviously
\begin{igather}
\fgMakeSubst{\fgTyFormals}{\fgTyActuals}{\fgTySubst} \QLabel{def-eta}
\end{igather}
To show
that
\begin{igather}
\LREquiv{
\Angle{x, t_S[\fgTyActuals], \fgTySubst m[\fgTyFormals'](\MultiN{x_i\ \fgType_i})\,\fgType, \fgTySubst e}
}{
\tllambda{\Triple{Y_1}{Y_2}{Y_3}} X\,\Quadr{W}{Y_1}{Y_2}{Y_3}
}{
\fgTySubst mM
}{k'}
\end{igather}
holds, we assume the left-hand side of the implication in the premise of rule \Rule{equiv-method-dict-entry}
for some $k'', \fgTyActuals', W', u, U, \Multi[n]{v}, \Multi[n]{V}$:
\begin{igather}
k'' \leq k' \QLabel{def-kpprime}\\
\fgMakeSubst{\fgTySubst \fgTyFormals'}{\fgTyActuals'}{\fgTySubst'} \QLabel{def-eta-prime}\\
\LREquiv{\fgTyActuals'}{W'}{\fgTySubst \fgTyFormals'}{k''} \QLabel{phi-equiv-W-prime}\\
\LREquiv{u}{U}{t_S[\fgTyActuals]}{k''} \QLabel{u-equiv-U}\\
(\forall i \in [n]) \DOT \LREquiv{v_i}{V_i}{\fgTySubst' \fgTySubst \fgType_i}{k''} \QLabel{vi-equiv-Vi}
\end{igather}
We then need to prove \QRef{goal} to show the overall goal.
\begin{igather}
\LREquiv{
\fgSubst\fgTySubst'\fgTySubst e
}{
V\,\Triple{U}{W'}{\Tuple{\MultiN{V}}}
}{
\fgTySubst' \fgTySubst \fgType
}{k''} \QLabel{goal}
\\
\fgSubst = \Angle{\subst{x}{u}, \MultiN{\subst{x_i}{v_i}}}
\\
V = \tllambda{\TripleY} X\, \Quadr{W}{Y_1}{Y_2}{Y_3} \QLabel{def-V}
\end{igather}
Let $\fgTyFormals = \Multi[p]{\fgTyVar\,\fgTypeAux}$,
$\fgTyFormals' = \Multi[q]{\fgTyVarAux\,\fgTypeAux'}$,
$\fgTyActuals = \Multi[p]{\fgTypeAux''}$,
$\fgTyActuals' = \Multi[q]{\fgTypeAux'''}$.
Then by \QRef{def-eta} and \QRef{def-eta-prime}
\begin{igather}
\fgTySubst = \Angle{\Multi[p]{\subst{\fgTyVar_i}{\fgTypeAux''_i}}} \QLabel{def2-eta}\\
\fgTySubst' = \Angle{\Multi[q]{\subst{\fgTyVarAux_i}{\fgTypeAux'''_i}}} \QLabel{def2-eta-prime}
\end{igather}
Define
\begin{igather}
\fgTyFormalsAux := \Multi[p]{\fgTyVar_i\,\fgTypeAux_i} \Multi[q]{\fgTyVarAux_i\,\fgTypeAux'_i}\\
\fgTyActuals'' := \Multi[p]{\fgTypeAux''} \Multi[q]{\fgTypeAux'''}\\
\fgTySubst'' := \Angle{\Multi[p]{\subst{\fgTyVar_i}{\fgTypeAux''_i}} \Multi[q]{\subst{\fgTyVarAux_i}{\fgTypeAux'''_i}}}
\QLabel{def2-eta-pprime}
\end{igather}
Then
\begin{igather}
\fgMakeSubst{\fgTyFormalsAux}{\fgTyActuals''}{\fgTySubst''} \QLabel{subst-Psi}
\end{igather}
The $\Multi[q]{\fgTyVarAux}$ are sufficiently fresh, so
$\ftv{\Multi[p]{\fgTypeAux''}} \cap \Multi[q]{\fgTyVarAux} = \emptyset$. Hence
by \QRef{def2-eta}, \QRef{def2-eta-prime}, \QRef{def2-eta-pprime}
\begin{igather}
\fgTySubst'\fgTySubst \Multi[q]{\fgTypeAux'_i} = \fgTySubst'' \Multi[q]{\fgTypeAux'_i} \quad
\fgTySubst'\fgTySubst \Multi[n]{\fgType_i} = \fgTySubst'' \Multi[n]{\fgType_i} \quad
\fgTySubst'\fgTySubst \fgType = \fgTySubst'' \fgType \quad
\fgTySubst'\fgTySubst e = \fgTySubst'' e \QLabel{subst-eqs}
\end{igather}
We have from \QRef{phi-equiv-W} and \QRef{phi-equiv-W-prime}
\begin{igather}
W = \Tuple{\Multi[p]{W}} \QLabel{def-W}\\
W' = \Tuple{\Multi[q]{W'}} \QLabel{def-W-prime}
\end{igather}
We now prove
\begin{igather}
\LREquiv{\fgTyActuals''}{\Pair{\Multi[p]{W}}{\Multi[q]{W'}}}{\fgTyFormalsAux}{k''} \QLabel{phi-equiv-pprime}
\end{igather}
by verifying the implication in the premise of rule \Rule{equiv-bounded-typarams}.
We consider two cases for every $\ell \leq k''$.
\begin{CaseDistinction}{whether $i$ in $[p]$ or in $[q]$}
\Case{$i \in [p]$} We need to prove
$\forall u, U \DOT \LREquiv{u}{U}{\fgTypeAux''_i}{\ell} \implies
\LREquiv{u}{W_i\,U}{\fgTySubst'' \fgTypeAux_i}{\ell}$.
From \QRef{phi-equiv-W} we get with $ \LREquiv{u}{U}{\fgTypeAux''_i}{\ell}$
and $\ell \leq k'' \ReasonAbove{\leq}{\QRef{def-kpprime}} k'$ that
$\LREquiv{u}{W_i\,U}{\fgTySubst \fgTypeAux_i}{\ell}$.
By assumption~\ref{conv:fg-decls} $\ftv{\fgTypeAux_i} \subseteq \Multi[p]{\fgTyVar}$, so
$\fgTySubst'' \fgTypeAux_i = \fgTySubst \fgTypeAux_i$ by \QRef{def2-eta} and \QRef{def2-eta-pprime}.
\Case{$i \in [q]$} We need to prove
$\forall u, U \DOT \LREquiv{u}{U}{\fgTypeAux'''_i}{\ell} \implies
\LREquiv{u}{W'_i\,U}{\fgTySubst'' \fgTypeAux'_i}{\ell}$.
From \QRef{phi-equiv-W-prime} we get with $\LREquiv{u}{U}{\fgTypeAux'''_i}{\ell}$ that
$\LREquiv{u}{W'_i\,U}{\fgTySubst'\fgTySubst \fgTypeAux'_i}{\ell}$.
Also, $\fgTySubst'\fgTySubst \fgTypeAux'_i = \fgTySubst'' \fgTypeAux'_i$
by \QRef{subst-eqs}.
\end{CaseDistinction}
This finishes the proof of \QRef{phi-equiv-pprime}.\\[\smallskipamount]
From \QRef{u-equiv-U} and \QRef{def2-eta-pprime} we have
\begin{igather}
\LREquiv{u}{U}{\fgTySubst'' t_S[\Multi[p]{\fgTyVar}]}{k''} \QLabel{u-equiv-U2}
\end{igather}
From \QRef{vi-equiv-Vi} and \QRef{subst-eqs}
\begin{igather}
\LREquiv{v_i}{V_i}{\fgTySubst'' \fgType_i}{k''} \QLabel{vi-equiv-Vi2}
\end{igather}
From the assumption $\LREquivNoTy{\fgFunc}{k}{X}$, we can invert rule
\Rule{equiv-method-decl}. Noting that $k'' \ReasonAbove{\leq}{\QRef{def-kpprime}} k' \ReasonAbove{<}{\QRef{kprime}} k$
and that \QRef{subst-Psi}, \QRef{phi-equiv-pprime},
\QRef{u-equiv-U2}, \QRef{vi-equiv-Vi2} give us the left-hand side of the implication
in the premise of the rule, we get by the right-hand side of the implication
\begin{igather}
\LREquiv{
\fgSubst \fgTySubst'' e
}{
X\ \Quadr{\Tuple{\Multi[p]{W}}}{U}{\Tuple{\Multi[q]{W'}}}{\Tuple{\Multi[n]{V}}}
}{
\fgTySubst'' \fgType
}{k''}
\end{igather}
With \QRef{subst-eqs}, \QRef{def-W}, \QRef{def-W-prime}
\begin{igather}
\LREquiv{
\fgSubst \fgTySubst'\fgTySubst e
}{
X\ \Quadr{W}{U}{W'}{\Tuple{\Multi[n]{V}}}
}{
\fgTySubst' \fgTySubst \fgType
}{k''}
\QLabel{e-equiv}
\end{igather}
We have by \QRef{def-V}
\begin{igather}
V~\Triple{U}{W'}{\Tuple{\Multi[n]{V}}} =
(\tllambda{\TripleY} X\, \Quadr{W}{Y_1}{Y_2}{Y_3})~\Triple{U}{W'}{\Tuple{\Multi[n]{V}}}
\end{igather}
so
\begin{igather}
V~\Triple{U}{W'}{\Tuple{\Multi[n]{V}}} \reduceSym^*
X\ \Quadr{W}{U}{W'}{\Tuple{\Multi[n]{V}}}
\end{igather}
Thus, with \QRef{e-equiv} and \Cref{lem:target-reduce}
\begin{igather}
\LREquiv{
\fgSubst\fgTySubst'\fgTySubst e
}{
V\,\Triple{U}{W'}{\Tuple{\MultiN{V}}}
}{
\fgTySubst' \fgTySubst \fgType
}{k''}
\end{igather}
as required to prove \QRef{goal}. \MyQED
\end{proof}
\begin{definition}[Domain]
We write $\dom{\cdot}$ for the domain of a substitution $\fgTySubst$, $\fgSubst$,
$\tlSubst$ or $\tlMethTable$, of a type environment $\Delta$,
of a value environment $\Gamma$, or some type parameters $\fgTyFormals$.
\end{definition}
\begin{definition}[Free variables]
We write $\fv{\cdot}$ for the set of free term variables,
and $\ftv{\cdot}$ for the set of free type variables.
\end{definition}
\begin{lemma}[Subtyping preserves equivalence]\label{subtyping-equiv}
Let $\LREquivNoTy{\fgDecls}{k}{\tlMethTable}$. Assume
$\tdICons{\Delta}{\fgType}{\fgTypeAux}{V}$ and $\LREquiv{\fgTySubst}{\tlSubst}{\Delta}{k}$
and $\LREquiv{e}{E}{\fgTySubst\fgType}{k}$.
Then $\LREquiv{e}{(\tlSubst V)\,E}{\fgTySubst \fgTypeAux}{k}$.
\end{lemma}
We prove \Cref{subtyping-equiv} together with the following two lemmas.
\begin{lemma}\label{lem:method-lookup}
Assume $\LREquivNoTy{\fgDecls}{k}{\tlMethTable}$ and
$\LREquiv{\fgTySubst}{\tlSubst}{\Delta}{k}$.
Let $ \tdMethods{R}{V}{\Delta}{t_S[\fgTyActuals]}$ and
define $U = \tllambda{\TripleY}{\mT{m}{t_S}}\,\Quadr{\tlSubst V}{Y_1}{Y_2}{Y_3}$.
Then we have for all $k' < k$ that
$\LREquiv{\methodLookup{m}{\fgTySubst t_S[\fgTyActuals]}}{U}{\fgTySubst R}{k'}$.
\end{lemma}
\begin{lemma}[Substitution preserves equivalence]\label{lem:subst-equiv}
Assume $\LREquivNoTy{\fgDecls}{k}{\tlMethTable}$ and
$\LREquiv{\fgTySubst}{\tlSubst}{\Delta}{k}$.
If $\tdCheckSubst{\Delta}{\fgTyFormals}{\fgTyActuals}{\fgTySubst'}{V}$
then $\LREquiv{\fgTySubst \fgTyActuals}{\tlSubst V}{\fgTySubst \fgTyFormals}{k}$.
\end{lemma}
\begin{proof}[Proof of Lemmas~\ref{subtyping-equiv}, \ref{lem:method-lookup}, and~\ref{lem:subst-equiv}]
\renewcommand\LabelQualifier{subtyping-equiv}
We show the three lemmas by induction on the combined height of the derivations for
$\tdICons{\Delta}{\fgType}{\fgTypeAux}{V}$ and
$\tdMethods{R}{V}{\Delta}{t_S[\fgTyActuals]}$ and
$\tdCheckSubst{\Delta}{\fgTyFormals}{\fgTyActuals}{\fgTySubst'}{V}$.
We start with the proof for \emph{\Cref{subtyping-equiv}}. We have from the assumptions
\begin{igather}
\LREquiv{e}{E}{\fgTySubst\fgType}{k} \QLabel{e-equiv-E}
\end{igather}
Assume $k' < k$ and $\reducek{k'}{e}{e'}$.
The second implication in the premise of rule \Rule{equiv-exp} holds obviously, because
with $\Diverge{e'}$ we get from \QRef{e-equiv-E} $\Diverge{E}$, so also $\Diverge{(\tlSubst V)\ E}$.
Thus, we only need to prove the first implication. Assume that $e' = v$ for some value $v$.
Then via \QRef{e-equiv-E} for some $U$
\begin{igather}
\reduceStar{E}{U} \QLabel{eval-E}\\
\LREquivVal{v}{U}{\fgTySubst \fgType}{k-k'} \QLabel{v-equiv-U}
\end{igather}
We then need to verify that $\reduceStar{(\tlSubst V)\ U}{U'}$ for some $U'$ with
$\LREquivVal{v}{U'}{\fgTySubst \fgTypeAux}{k - k'}$.
In fact, $k' < k$, so with \Cref{lem:exp-equiv-implies-val-equiv}
it suffices to show that
$\LREquiv{v}{U'}{\fgTySubst \fgTypeAux}{k - k'}$.
\begin{CaseDistinction}[repeat]{on the last rule in the derivation of $\tdICons{\Delta}{\fgType}{\fgTypeAux}{V}$}
\Case{\Rule{coerce-tyvar}}
\begin{mathpar}
\inferrule{
X \textrm{~fresh}\\
(a : \fgType_I) \in \Delta\\
\tdICons{\Delta}{\fgType_I}{\fgTypeAux}{W}
}{
\tdICons{\Delta}{\EqualBelow{\fgTyVar}{\fgType}}{\fgTypeAux}{\BraceBelow{\tllambda{X}{W\ (\xTy{\fgTyVar}\ X)}}{= V}}
}
\end{mathpar}
Our goal to show is
\begin{igather}
\LREquiv{e}{(\tlSubst \xTy{\fgTyVar})\ U}{\fgTySubst \fgType_I}{k} \QLabel{goal}
\end{igather}
With \QRef{goal} and the IH for \Cref{subtyping-equiv} we then get
\begin{igather}
\LREquiv{e}{(\tlSubst W)\ ((\tlSubst \xTy{\fgTyVar})\ U)}{\fgTySubst \fgTypeAux}{k}
\end{igather}
Then, with $\reducek{k'}{e}{v}$, we get
$(\tlSubst V)\ U \longrightarrow \reduceStar{(\tlSubst W)\ ((\tlSubst \xTy{\fgTyVar})\ U)}{U'}$ for some $U'$ with
$\LREquivVal{v}{U'}{\fgTySubst \fgTypeAux}{k - k'}$.
\\
We now prove \QRef{goal}. From the assumption $\LREquiv{\fgTySubst}{\tlSubst}{\Delta}{k}$ we have
\begin{igather}
\Delta = \Multi[n]{\fgTyVar_i : \fgType_i}\\
\fgTySubst = \Angle{\Multi[n]{\subst{\fgTyVar_i}{\fgTypeAux_i}}}\\
\tlSubst = \Angle{\Multi[n]{\subst{\xTy{\fgTyVar_i}}{V_i}}}\\
\LREquiv{\Multi[n]{\fgTypeAux}}{\Multi[n]{V}}{\Multi[n]{\fgTyVar_i\ \fgType_i}}{k} \QLabel{eq-typarams}
\end{igather}
such that $\fgTyVar = \fgTyVar_j$ and $\fgType_I = \fgType_j$ for some $j \in [n]$.
Inverting rule \Rule{equiv-bounded-typarams} on \QRef{eq-typarams} yields
\begin{igather}
\forall k'' \leq k, w, W' \DOT
\LREquiv{w}{W'}{\fgTypeAux_j}{k''} \implies
\LREquiv{w}{V_j\ W'}{\fgTySubst \fgType_j}{k''} \QLabel{forall-kpprime}
\end{igather}
From \QRef{v-equiv-U} by $\fgTySubst \fgType = \fgTypeAux_j$ then
$\LREquivVal{v}{U}{\fgTypeAux_j}{k - k'}$. Thus with \QRef{forall-kpprime} and \Cref{lem:val-equiv-implies-exp-equiv}
\begin{igather}
\LREquiv{v}{V_j\ U}{\fgTySubst \fgType_j}{k - k'}
\end{igather}
With \Cref{lem:source-reduce} and $\reducek{k'}{e}{v}$ then
\begin{igather}
\LREquiv{e}{V_j\ U}{\fgTySubst \fgType_j}{k}
\end{igather}
But $\fgType_j = \fgType_I$ and $V_j = \tlSubst \xTy{\fgTyVar}$, so this proves \QRef{goal}.
\Case{\Rule{coerce-struct-iface}}
\begin{mathpar}
\inferrule{
\LabeledRule{
X, Y_1, Y_2, Y_3~\textrm{fresh}\\
\mbox{\kw{type}}\ t_I[\fgTyFormals] \ \mbox{\kw{interface}}\ \{ \Foreach{mM}{n} \} \in \ForeachN{D} & \TextLabel{def-I}\\
\fgMakeSubst{\fgTyFormals}{\fgTyActuals}{\fgTySubst'} & \TextLabel{def-eta-prime}\\
\tdMethods{\fgTySubst'(m_i M_i)}{V_i}{\Delta}{t_S[\fgTyActualsAux]} & \TextLabel{methods}\\
V_i' = \PartialM{\mT{m_i}{t_S}}{V_i}
\quad\noteForall{i \in [n]} & \TextLabel{def-V-prim}
}
}{
\tdUpcast{\Delta}{\subtypeOf{\EqualBelow{t_S[\fgTyActualsAux]}{\fgType}}{\EqualBelow{t_I[\fgTyActuals]}{\fgTypeAux}}}
{\BraceBelow{\tllambda{X} \Tuple{X\Comma \Tuple{\Foreach{V_i'}{n}}}}{= V}}
}
\end{mathpar}
Hence $\reduce{(\tlSubst V)\ U}{\Pair{U}{\tlSubst\Tuple{\MultiN{V'}}}}$ and
$U' := \Pair{U}{\tlSubst\Tuple{\MultiN{V'}}}$ is a value.
We now want to show $\LREquiv{v}{U'}{\fgTySubst \fgTypeAux}{k - k'}$ via
rule \Rule{equiv-iface}. Define the $\fgTypeAux_S$ in the premise of \Rule{equiv-iface}
as $\fgTySubst \fgType = t_S[\fgTySubst \fgTyActualsAux]$.
The first premise of \Rule{equiv-iface}
\begin{igather}
\forall k_1 < k - k' \DOT \LREquivVal{v}{U}{\fgTySubst \fgType}{k_1} \QLabel{premise1}
\end{igather}
follows from \QRef{v-equiv-U} and \Cref{lem:mono-exp}. From \QRef{def-I} and \QRef{def-eta-prime} we get with
$\fgTypeAux = t_I[\fgTyActuals]$
the second premise as
\begin{igather}
\methodSpecifications{\fgTySubst \fgTypeAux} = \fgTySubst \fgTySubst' \MultiN{mM} \QLabel{premise2}
\end{igather}
We next prove the third premise of \Rule{equiv-iface}. Pick some $j \in [n]$ and $k_2 < k -k'$.
With the assumptions $\LREquivNoTy{\fgDecls}{k}{\tlMethTable}$ and $\LREquiv{\fgTySubst}{\tlSubst}{\Delta}{k}$
and with \QRef{methods}, \QRef{def-V-prim}, and the IH for \Cref{lem:method-lookup} we get
\begin{igather}
\LREquiv{
\methodLookup{m_j}{\fgTySubst t_S[\fgTyActualsAux]}
}{
\tlSubst V_j'
}{
\fgTySubst \fgTySubst' m_j M_j
}{k_2} \QLabel{premise3}
\end{igather}
\QRef{premise1}, \QRef{premise2}, \QRef{premise3}, and the definition of $U'$
are the pieces required to derive
$\LREquiv{v}{U'}{\fgTySubst \fgTypeAux}{k - k'}$ via rule \Rule{equiv-iface}.
\Case{\Rule{coerce-iface-iface}}
\begin{mathpar}
\inferrule{
\LabeledRule{
Y,\Foreach{X}{n}~\textrm{fresh} \qquad
\mapPerm : [q] \to [n]~\textrm{total}\\
\mbox{\kw{type}}\ t_I[\fgTyFormals_1] \ \mbox{\kw{interface}}\ \{ \Foreach{R}{n} \} \in \ForeachN{D} & \TextLabel{def-tI}\\
\mbox{\kw{type}}\ u_I[\fgTyFormals_2] \ \mbox{\kw{interface}}\ \{ \Foreach{R'}{q} \} \in \ForeachN{D} & \TextLabel{def-uI}\\
\fgMakeSubst{\fgTyFormals_1}{\fgTyActuals_1}{\fgTySubst_1} \qquad
\fgMakeSubst{\fgTyFormals_2}{\fgTyActuals_2}{\fgTySubst_2}\\
\fgTySubst_2 R'_i = \fgTySubst_1 R_{\mapPerm(i)} \quad\noteForall{i \in [q]} & \TextLabel{Si-eq-Ri}
}
}{
\tdUpcast{\Delta}{\subtypeOf{\EqualBelow{t_I[\fgTyActuals_1]}{\fgType}}{\EqualBelow{u_I[\fgTyActuals_2]}{\fgTypeAux}}}
{ \BraceBelow{
\tllambda{\Tuple{Y\Comma \Tuple{\Foreach{X}{n}}}}
\Tuple{Y\Comma \Tuple{X_{\mapPerm(1)}\Comma \ldots\Comma X_{\mapPerm(q)}}}
}{= V ~~~~\TextLabel{def-V}}
}
}
\end{mathpar}
As $\fgTySubst \fgType = \fgTySubst t_I[\fgTyActuals_1]$ is an interface type, we get from \QRef{v-equiv-U} by inverting
rule \Rule{equiv-iface} for some $W, \fgTypeAux_S, \MultiN{W}$ that
\begin{igather}
\forall k_1 < k - k' \DOT \LREquiv{v}{W}{\fgTypeAux_S}{k_1} \QLabel{impl1}\\
\methodSpecifications{\fgTySubst \fgType} = \fgTySubst \fgTySubst_1 \MultiN{R} = \MultiN{mM} \QLabel{methods-tau}\\
\forall i \in [n], k_2 < k - k' \DOT
\LREquiv{\methodLookup{m_i}{\fgTypeAux_s}}{W_i}{m_i M_i}{k_2} \QLabel{impl2}\\
U = \Pair{W}{\Tuple{\MultiN{W}}} \QLabel{def-U}
\end{igather}
Our goal is to show $\reduceStar{(\tlSubst V)\ U}{U'}$ for some $U'$ with
$\LREquiv{v}{U'}{\fgTySubst \fgTypeAux}{k - k'}$. Via \QRef{def-V} and \QRef{def-U}
\begin{igather}
(\tlSubst V)\ U = V\ U \reduceSym^*
\Pair{W}{\Tuple{W_{\mapPerm(1)}\Comma \ldots \Comma W_{\mapPerm(q)}}} =: U'
\QLabel{def-Uprime}
\end{igather}
From \QRef{def-tI}, \QRef{def-uI}, \QRef{Si-eq-Ri}, and \QRef{methods-tau} we have
\begin{igather}
\begin{array}[t]{@{}r@{~}l@{}}
\Multi[q]{m' M'}
= \fgTySubst \fgTySubst_2 \Multi[q]{R'}
&= \fgTySubst \fgTySubst_1 R_{\mapPerm(1)}, \ldots, \fgTySubst \fgTySubst_1 R_{\mapPerm(q)}\\
&= m_{\mapPerm(1)} M_{\mapPerm(1)}, \ldots, m_{\mapPerm(q)} M_{\mapPerm(q)}
\end{array} \QLabel{eq-M}
\\
\methodSpecifications{\fgTySubst \fgTypeAux} = \{\Multi[q]{m' M'} \} \QLabel{methods-sigma}
\end{igather}
Pick $j \in [q]$. Then via \QRef{eq-M}
\begin{igather}
\methodLookup{m_j'}{\fgTypeAux_S} =
\methodLookup{m_{\mapPerm(j)}}{\fgTypeAux_s}
\end{igather}
Hence with \QRef{impl2} and \QRef{eq-M}
\begin{igather}
\forall j \in [q], k_2 < k - k' \DOT
\LREquiv{
\methodLookup{m_j'}{\fgTypeAux_S}
}{
W_{\mapPerm(j)}
}{m_j' M_j'}{k_2} \QLabel{impl22}
\end{igather}
With \QRef{impl1}, \QRef{impl22}, \QRef{methods-sigma} and the definition of $U'$ in \QRef{def-Uprime} we then
get by applying rule \Rule{equiv-iface}
$\LREquiv{v}{U'}{\fgTySubst \fgTypeAux}{k - k'}$ and with \QRef{def-Uprime} also
$\reduceStar{(\tlSubst V)\ U}{U'}$.
\end{CaseDistinction}
\\
This finishes the proof of \Cref{subtyping-equiv}.
\renewcommand\LabelQualifier{lem:method-lookup}
We next prove \Cref{lem:method-lookup}. By inverting rule \Rule{methods-struct} for the assumption
$\Angle{R,V} \in \methodSpecifications{\Delta, t_S[\fgTyActuals]}$ we get
\begin{igather}
\fgFunc \in \fgDecls \QLabel{func} \\
\tdCheckSubst{\Delta}{\fgTyFormals}{\fgTyActuals}{\fgTySubst'}{V} \QLabel{subst}\\
R = \fgTySubst' m M \QLabel{R}
\end{igather}
Inverting \QRef{subst} yields
\begin{igather}
\fgTyFormals = \MultiN{\fgTyVar\ \fgType}\\
\fgTySubst' = \Angle{\Multi[n]{\subst{\fgTyVar_i}{\fgTypeAux_i}}}\\
\fgTyActuals = \MultiN{\fgTypeAux}\\
\tdICons{\Delta}{\fgTypeAux_i}{\fgTySubst' \fgType_i}{V_i} \quad (\forall i \in [n])\\
V = \Tuple{\MultiN{V}}
\end{igather}
Define $\fgTySubst'' = \Angle{\MultiN{\subst{\fgTyVar_i}{\fgTySubst\fgTypeAux_i}}}$.
Then by rule \Rule{method-lookup} and \QRef{func}
\begin{igather}
\methodLookup{m}{\fgTySubst t_S[\fgTyActuals]} =
\Angle{x, \fgTySubst t_S[\fgTyActuals], \fgTySubst'' mM, \fgTySubst'' e}
\QLabel{method-lookup}
\end{igather}
By assumption~\ref{conv:fg-decls}, the $\MultiN{\fgTyVar}$ can be assumed to be fresh,
$\ftv{\fgTyFormals} \subseteq \MultiN{\fgTyVar}$, and
$\fgTySubst \fgTyFormals = \fgTyFormals$.
Applying the IH for \Cref{lem:subst-equiv} on \QRef{subst} yields
$\LREquiv{\fgTySubst \fgTyActuals}{\tlSubst V}{\fgTySubst \fgTyFormals}{k}$.
\begin{igather}
\LREquiv{\fgTySubst \fgTyActuals}{\tlSubst V}{\fgTyFormals}{k} \QLabel{phi-equiv-V}
\end{igather}
From the assumption $\LREquivNoTy{\fgDecls}{k}{\tlMethTable}$ we get with \QRef{func}
\begin{igather}
\LREquivNoTy{\fgFunc}{k}{\mT{m}{t_S}}
\end{igather}
Then for any $k' < k$ by
\Cref{lem:method-decl-equiv-implies-dict-equiv}, where \QRef{phi-equiv-V} and
\Cref{lem:mono-typarams} give the left-hand side of the implication
\begin{igather}
\fgMakeSubst{\fgTyFormals}{\fgTySubst \fgTyActuals}{\fgTySubst''}\\
\LREquiv{
\Angle{x, \fgTySubst t_S[\fgTyActuals], \fgTySubst'' m M, \fgTySubst'' e}
}{~\\
\BraceBelow{\tllambda{\TripleY}{\mT{m}{t_S}\ \Quadr{\tlSubst V}{Y_1}{Y_2}{Y_3}}}{= U}
}{
\fgTySubst'' m M
}{k'}
\QLabel{equiv}
\end{igather}
We have $\fgTySubst R \ReasonAbove{=}{\QRef{R}} \fgTySubst \fgTySubst' mM = \fgTySubst'' mM$,
where the last equality holds because $\ftv{mM} \subseteq \MultiN{\fgTyVar}$ and $\MultiN{\fgTyVar}$
fresh by assumption~\ref{conv:fg-decls}.
Hence \QRef{method-lookup} and \QRef{equiv} give
the desired claim.
Finally, we prove \Cref{lem:subst-equiv}. By inverting rule \Rule{type-inst-checked} for the
assumption $\tdCheckSubst{\Delta}{\fgTyFormals}{\fgTyActuals}{\fgTySubst'}{V}$ we get
\begin{igather}
\fgTyFormals = \MultiN{\fgTyVar\,\fgType}\\
\fgTyActuals = \MultiN{\fgTypeAux}\\
\fgTySubst' = \Angle{\MultiN{\subst{\fgTyVar_i}{\fgTypeAux_i}}}\\
\tdICons{\Delta}{\fgTypeAux_i}{\fgTySubst' \fgType_i}{V_i} \quad (\forall i \in [n]) \QLabel{icons}\\
V = \Tuple{\MultiN{V}}
\end{igather}
Define $\fgTySubst'' = \Angle{\MultiN{\subst{\fgTyVar_i}{\fgTySubst \fgTypeAux_i}}}$.
To prove $\LREquiv{\fgTySubst \fgTyActuals}{\tlSubst V}{\fgTySubst \fgTyFormals}{k}$
we need to show the implication
$\forall j \in [n], k' \leq k \DOT \LREquiv{u}{U}{\fgTySubst \fgTypeAux_j}{k'} \implies
\LREquiv{u}{(\tlSubst V)\ U}{\fgTySubst''\fgTySubst\fgType_j}{k'}$ from the premise
of rule \Rule{equiv-bounded-typarams}.
Assume $j \in [n], k' \leq k$, and $\LREquiv{u}{U}{\fgTySubst \fgTypeAux_j}{k'}$. Applying
the IH for \Cref{subtyping-equiv} on \QRef{icons} yields together with
\Cref{lem:mono-typarams} and \Cref{lem:mono-prog} that
\begin{igather}
\LREquiv{u}{(\tlSubst V_j)\ U}{\fgTySubst\fgTySubst'\fgType_j}{k'} \QLabel{u-equiv-VU}
\end{igather}
As the $\Multi{\fgTyVar}$ are bound in $\fgTyFormals$, we may assume that
$\dom{\fgTySubst} \cap \Multi{\fgTyVar} = \emptyset = \ftv{\fgTySubst} \cap \Multi{\fgTyVar}$.
We now argue that
\begin{igather}
\fgTySubst\fgTySubst'\fgType_j = \fgTySubst''\fgTySubst\fgType_j \QLabel{type-eq}
\end{igather}
by induction on the structure of $\fgType_j$. The interesting case is were $\fgType_j$ is
a type variable (otherwise the claim follows by the IH). If $\fgType_j \in \Multi{\fgTyVar}$ then
\begin{igather}
\fgTySubst\fgTySubst' \fgType_j \ReasonAbove{=}{\textrm{def. of~}\fgTySubst''}
\fgTySubst'' \fgType_j \ReasonAbove{=}{\dom{\fgTySubst} \cap \Multi{\fgTyVar} = \emptyset}
\fgTySubst'' \fgTySubst \fgType_j
\end{igather}
If $\fgType_j \in \dom{\fgTySubst}$ then
\begin{igather}
\fgTySubst\fgTySubst' \fgType_j \ReasonAbove{=}{\dom{\fgTySubst} \cap \Multi{\fgTyVar} = \emptyset}
\fgTySubst \fgType_j \ReasonAbove{=}{\ftv{\fgTySubst} \cap \Multi{\fgTyVar} = \emptyset}
\fgTySubst'' \fgTySubst \fgType_j
\end{igather}
If $\fgType_j$ is some other type variable, \QRef{type-eq} holds obviously.
With \QRef{u-equiv-VU} and \QRef{type-eq} we get
$\LREquiv{u}{(\tlSubst V_j)\ U}{\fgTySubst''\fgTySubst\fgType_j}{k'}$ as required.
\MyQED
\end{proof}
\begin{lemma}[Free variables of coercion values]\label{lem:free-icons}
If $\tdICons{\Delta}{\fgType}{\fgTypeAux}{V}$
then
$\fv{V} \subseteq \{ \xTy{\fgTyVar} \mid \fgTyVar \in \dom{\Delta} \} \cup \mathcal X$
where $\mathcal X = \{ \mT{m}{t_S} \mid m \textrm{~method name}, t_S \textrm{~struct name}\}$.
\end{lemma}
\begin{proof}
By straightforward induction on the derivation of $\tdICons{\Delta}{\fgType}{\fgTypeAux}{V}$.
\MyQED
\end{proof}
\subsubsection{Proof of \Cref{lem:exp-equiv}}
\label{sec:proof-exp-equiv}
By induction on the derivation of
$\tdExpTrans{\pair{\Delta}{\Gamma}}{e : \fgType}{E}$.
\begin{CaseDistinctionExplicit}{on the last rule in the derivation}{ on the last rule in the derivation of $\tdExpTrans{\pair{\Delta}{\Gamma}}{e : \fgType}{E}$}
\renewcommand\LabelQualifier{exp-equiv-var}
\Case{\Rule{var}}
\begin{mathpar}
\inferrule{
(x : \fgType) \in \Gamma
}{
\tdExpTrans{\pair{\Delta}{\Gamma}}{x : \fgType}{X}
}
\end{mathpar}
with $\fgSubst\fgTySubst e = \fgSubst x$ and
$\tlSubst E = \tlSubst X$. From the assumption
$\LREquiv{\fgSubst}{\tlSubst}{\fgTySubst \Gamma}{k}$ we get
$\LREquiv{\fgSubst x}{\tlSubst X}{\fgTySubst \fgType}{k}$ as required.
\renewcommand\LabelQualifier{exp-equiv-struct}
\Case{\Rule{struct}}
\begin{mathpar}
\inferrule{
\LabeledRule{
\fgTyOk{\Delta}{t_S[\fgTyActuals]}\\
\mbox{\kw{type}}\ t_S[\fgTyFormals] \ \mbox{\kw{struct}}\ \{ \Foreach{f \ \fgType}{n} \} \in \ForeachN{D} & \TextLabel{def-struct}\\
\fgMakeSubst{\fgTyFormals}{\fgTyActuals}{\fgTySubst'} & \TextLabel{def-eta-prime}\\
\tdExpTrans{\pair{\Delta}{\Gamma}}{e_i : \fgTySubst'\fgType_i}{E_i}
\quad\noteForall{i \in [n]} & \TextLabel{trans-ei}
}
}{
\tdExpTrans{\pair{\Delta}{\Gamma}}
{\BraceBelow{
t_S[\fgTyActuals] \{ \Foreach{e}{n} \}
}{= e}
: \BraceBelow{t_S[\fgTyActuals]}{= \fgType}
}{
\BraceBelow{\Tuple{\Foreach{E}{n}}}{= E}
} \qquad \TextLabel{conclusion}
}
\end{mathpar}
Applying the IH to \QRef{trans-ei} yields
\begin{igather}
\LREquiv{
\fgSubst\fgTySubst e_i
}{
\tlSubst E_i
}{
\fgTySubst\fgTySubst' \fgType_i
}{k} \quad(\forall i \in [n]) \QLabel{equiv-ei}
\end{igather}
We now consider the two implications in the premise of rule \Rule{equiv-exp}
\begin{EnumAlph}
\item Assume $k' < k$ and $\reducek{k'}{\fgSubst\fgTySubst e}{v}$ for some value $v$.
The goal is to show that there exists some value $V$ with $\reduceStar{\tlSubst E}{V}$
and $\LREquivVal{v}{V}{\fgTySubst\fgType}{k - k'}$.\\
With $\reducek{k'}{\fgSubst\fgTySubst e}{v}$ there must
exist values $\MultiN{v}$ such that
\begin{igather}
\fgSubst\fgTySubst e_i \longrightarrow^{k_i} v_i \qquad (\forall i \in [n])\\
k_i \leq k' \qquad (\forall i \in [n]) \QLabel{ki-leq-k}\\
v = t_S[\fgTySubst\fgTyActuals]\{\MultiN{v}\} \QLabel{shape-v}
\end{igather}
Via \QRef{equiv-ei} and $k_i \leq k' < k$ then for all $i \in [n]$
\begin{igather}
\tlSubst E_i \reduceSym^* V_i ~\textrm{for some}~V_i \quad(\forall i \in [n]) \QLabel{reduce-Ei}\\
\LREquivVal{v_i}{V_i}{\fgTySubst\fgTySubst'\fgType_i}{k - k_i} \quad(\forall i \in [n]) \QLabel{equiv-vi}
\end{igather}
We have $k - k' \leq k - k_i$ for all $i \in [n]$ by \QRef{ki-leq-k}. Thus with \QRef{equiv-vi}
and \Cref{lem:mono-exp}
\begin{igather}
\LREquivVal{v_i}{V_i}{\fgTySubst\fgTySubst'\fgType_i}{k - k'} \qquad (\forall i \in [n]) \QLabel{equiv-vi2}
\end{igather}
We also have with \QRef{reduce-Ei} and the definition of $E$ in \QRef{conclusion}
\begin{igather}
\tlSubst E \reduceSym^* \Tuple{\MultiN{V}} \QLabel{reduce-E}
\end{igather}
Assume $\fgGetTyVars{\fgTyFormals} = \MultiP{\fgTyVar}$ and
$\fgTyActuals = \MultiP{\fgTypeAux}$. Then by \QRef{def-eta-prime}
$\fgTySubst' = \Angle{\MultiP{\subst{\fgTyVar_i}{\fgTypeAux_i}}}$ and for
$\fgTySubst'' = \Angle{\MultiP{\subst{\fgTyVar_i}{\fgTySubst\fgTypeAux_i}}}$
we have
\begin{igather}
\fgMakeSubst{\fgTyFormals}{\fgTySubst \fgTyActuals}{\fgTySubst''} \QLabel{subst2}
\end{igather}
By assumption~\ref{conv:fg-decls} we have $\ftv{\MultiN{\fgType}} \subseteq \{\Multi{\fgTyVar}\}$, so
\begin{igather}
\fgTySubst\fgTySubst'\fgType_i = \fgTySubst'' \fgType_i \qquad (\forall i \in [n]) \QLabel{tau-eq}
\end{igather}
With \QRef{def-struct}, \QRef{equiv-vi2}, \QRef{shape-v}, \QRef{subst2}, \QRef{tau-eq}, and rule
\Rule{equiv-struct} then
\begin{igather}
\LREquivVal{v}{\Tuple{\MultiN{V}}}{t_S[\fgTySubst\fgTyActuals]}{k - k'}
\end{igather}
Together with \QRef{reduce-E}, this finishes subcase (a) for $V = \Tuple{\MultiN{V}}$.
\item Assume $k' < k$ and $\reducek{k'}{\fgSubst\fgTySubst e}{e'}$ and $\Diverge{e'}$. Then $\Diverge{e_j}$
for some $j \in [n]$, so with \QRef{equiv-ei} and \QRef{ki-leq-k} also $\Diverge{\tlSubst E_j}$. Thus, by definition of $E$ in
\QRef{conclusion}, $\Diverge{\tlSubst E}$ as required.
\end{EnumAlph}
\renewcommand\LabelQualifier{exp-equiv-access}
\Case{\Rule{access}}
\begin{mathpar}
\inferrule{
\LabeledRule{
\tdExpTrans{\pair{\Delta}{\Gamma}}{e' : t_S[\fgTyActuals]}{E'} & \TextLabel{type-eprime}\\
\mbox{\kw{type}}\ t_S[\fgTyFormals] \ \mbox{\kw{struct}}\ \{ \Foreach{f \ \fgType}{n} \} \in \ForeachN{D}\\
\fgMakeSubst{\fgTyFormals}{\fgTyActuals}{\fgTySubst'}
}
}{
\tdExpTrans{\pair{\Delta}{\Gamma}}
{\BraceBelow{e'.f_j}{= e} : \BraceBelow{\fgTySubst' \fgType_j}{= \fgType}}
{ \BraceBelow{\mbox{\kw{case}}\ E'\ \mbox{\kw{of}}\ \Tuple{\Foreach{X}{n}} \rightarrow X_i}{= E}}
\qquad \TextLabel{conclusion}
}
\end{mathpar}
Applying the IH to \QRef{type-eprime} yields
\begin{igather}
\LREquiv{
\fgSubst\fgTySubst e'
}{
\tlSubst E'
}{
t_S[\fgTySubst\fgTyActuals]
}{k} \QLabel{IH}
\end{igather}
We now consider the two implications in the premise of rule \Rule{equiv-exp}
\begin{EnumAlph}
\item Assume $k' < k$ and $\reducek{k'}{\fgSubst\fgTySubst e}{v}$ for some value $v$.
The goal is to show that there exists some value $V$ with $\reduceStar{\tlSubst E}{V}$
and $\LREquivVal{v}{V}{\fgTySubst\fgType}{k - k'}$.\\
With $\reducek{k'}{\fgSubst\fgTySubst e}{v}$ then
$\reducek{k''}{\fgSubst\fgTySubst e'}{v'}$ for some $v'$ and $k'' < k'$. With \QRef{IH} then
for some $V'$
\begin{igather}
\tlSubst E' \reduceSym^* V' \QLabel{reduce-Eprime} \\
\LREquivVal{v'}{V'}{t_S[\fgTySubst \fgTyActuals]}{k - k''} \QLabel{equiv-vprime}
\end{igather}
Inverting rule \Rule{equiv-struct} on \QRef{equiv-vprime} yields
\begin{igather}
v' = t_S[\fgTySubst \fgTyActuals]\{\MultiN{v}\} \QLabel{shape-vprime}\\
V' = \Tuple{\MultiN{V}} \quad \textrm{for some}~\MultiN{V}\\
\LREquivVal{v_i}{V_i}{\fgTySubst''\fgType_i}{k - k''} \quad (\forall i \in [n]) \QLabel{equiv-vi}
\end{igather}
where $\fgTySubst'' = \Angle{\MultiP{\subst{\fgTyVar_i}{\fgTySubst\fgTypeAux_i}}}$, assuming
$\fgGetTyVars{\fgTyFormals} = \MultiP{\fgTyVar}$ and $\fgTyActuals = \MultiP{\fgTypeAux}$.
By assumption~\ref{conv:fg-decls} we have $\ftv{\fgType_j} \subseteq \{\Multi{\fgTyVar}\}$.
Thus, $\fgTySubst''\fgType_j = \fgTySubst\fgTySubst'\fgType_j \ReasonAbove{=}{\QRef{conclusion}} \fgTySubst\fgType$.
Also, $k'' \leq k'$, so $k - k' \leq k - k''$.
Hence with \QRef{equiv-vi} and \Cref{lem:mono-exp}
\begin{igather}
\LREquivVal{v_j}{V_j}{\fgTySubst\fgType}{k - k'} \QLabel{equiv-vj}
\end{igather}
With \QRef{reduce-Eprime} and the definition of $E$ in \QRef{conclusion} we get
\begin{igather}
\tlSubst E \reduceSym^* V_j \QLabel{reduce-E}
\end{igather}
With $\reducek{k'}{\fgSubst\fgTySubst e}{v}$ and
$\reducek{k''}{\fgSubst\fgTySubst e'}{v'}$ and the form of $v'$ in \QRef{shape-vprime}, we get
$\reducek{k'}{\fgSubst\fgTySubst e}{v_j}$ and $v = v_j$ by rule \Rule{fg-field}.
Define $V = V_j$ and we are done with subcase (a) by \QRef{equiv-vj} and \QRef{reduce-E}.
\item Assume $k' < k$ and $\reducek{k'}{\fgSubst\fgTySubst e}{e''}$ and $\Diverge{e''}$.
Then we must have that $\reducek{k''}{\fgSubst\fgTySubst e'}{e'''}$ for some $k'' < k'$
and some $e'''$. Thus, $\Diverge{e'''}$ by the definition
of $e$ in \QRef{conclusion} and the evaluation rules for FGG. With \QRef{IH} then
$\Diverge{\tlSubst E'}$. By definition of $E$ in \QRef{conclusion} then $\Diverge{\tlSubst E}$
as required.
\end{EnumAlph}
\renewcommand\LabelQualifier{exp-equiv-call-struct}
\Case{\Rule{call-struct}}
\begin{mathpar}
\inferrule{
\LabeledRule{
\tdExpTrans{\pair{\Delta}{\Gamma}}{g : t_S[\fgTyActuals]}{G} & \TextLabel{deriv-g} \\
\tdMethods{m[\fgTyFormalsAux](\Foreach{x\,\fgTypeAux}{n})\fgTypeAux}{W}{\Delta}{t_S[\fgTyActuals]} & \TextLabel{methods}\\
\tdCheckSubst{\Delta}{\fgTyFormalsAux}{\fgTyActualsAux}{\fgTySubst_1}{W'} & \TextLabel{subst}\\
\tdExpTrans{\pair{\Delta}{\Gamma}}{e_i: \fgTySubst_1 \fgTypeAux_i}{E_i}
\quad(\forall i \in [n]) & \TextLabel{deriv-ei}
}
}{
\tdExpTrans{\pair{\Delta}{\Gamma}}{
\BraceBelow{g.m[\fgTyActualsAux](\Foreach{e}{n})}{= e} : {\BraceBelow{\fgTySubst_1 \fgTypeAux}{=\fgType}}
}
{\BraceBelow{\mT{m}{t_S} \ \Quadr{W}{G}{W'}{\Tuple{\Foreach{E}{n}}}}{= E}}
\qqquad \TextLabel{conclusion}
}
\end{mathpar}
From the IH applied to \QRef{deriv-g} and \QRef{deriv-ei}
\begin{igather}
\LREquiv{
\fgSubst \fgTySubst g
}{
\tlSubst G
}{\fgTySubst t_S[\fgTyActuals]}{k} \QLabel{g-equiv}\\
\LREquiv{
\fgSubst \fgTySubst e_i
}{
\tlSubst E_i
}{\fgTySubst \fgTySubst_1 \fgTypeAux_i}{k} \quad (\forall i \in [n]) \QLabel{ei-prime-equiv}
\end{igather}
Assume $\reducek{k'}{\fgSubst \fgTySubst e}{e'}$ for some $k' < k$. We first consider the following situation
for some values $u, \MultiN{v}$:
\begin{igather}
\reducek{k''}{\fgSubst \fgTySubst g}{u} \QLabel{reduce-g}\\
\reducek{k_i}{\fgSubst \fgTySubst e_i}{v_i} \QLabel{reduce-ei}\\
\fgSubst \fgTySubst e \longrightarrow^{k''+\smallsum{k_i}} u.m[\fgTySubst \fgTyActualsAux](\MultiN{v})
\longrightarrow^{k' - k'' - \smallsum{k_i}} e' \QLabel{reduce-e}
\end{igather}
with $k'' + \smallsum{k_i} \leq k'$. \QRef{reduce-g}, \QRef{reduce-ei}, and \QRef{reduce-e} are intermediate
assumptions, which become true when we later prove the two implications of rule \Rule{equiv-exp}.\\
We have from \QRef{g-equiv}, \QRef{reduce-g}, \QRef{ei-prime-equiv}, and \QRef{reduce-ei}
\begin{igather}
\reduceStar{\tlSubst G}{U} \textrm{~for some~} U \textrm{~with~}
\LREquivVal{u}{U}{\fgTySubst t_S[\fgTyActuals]}{k - k''} \QLabel{reduce-G}\\
(\forall i \in [n])~\reduceStar{\tlSubst E_i}{V_i} \textrm{~for some~} V_i \textrm{~with~}
\LREquivVal{v_i}{V_i}{\fgTySubst\fgTySubst_1 \fgTypeAux_i}{k - k_i} \QLabel{reduce-Ei}
\end{igather}
From \QRef{methods} we get by inverting rule \Rule{methods-struct}:
\begin{igather}
\fgFunc[spec=m{[\fgTyFormalsAux']}(\MultiN{x\ \fgTypeAux'})\,\fgTypeAux', body=g'] \in \fgDecls \QLabel{func-def}\\
\tdCheckSubst{\Delta}{\fgTyFormals}{\fgTyActuals}{\fgTySubst_2}{W} \QLabel{subst2}\\
m[\fgTyFormalsAux](\MultiN{x\ \fgTypeAux})\,\fgTypeAux =
\fgTySubst_2(m[\fgTyFormalsAux'](\MultiN{x\ \fgTypeAux'})\,\fgTypeAux') \QLabel{eq-m}
\end{igather}
From the assumption $\LREquivNoTy{\fgDecls}{k}{\tlMethTable}$ and \QRef{func-def}
\begin{igather}
\LREquivNoTy{
\fgFunc[spec=m{[\fgTyFormalsAux']}(\MultiN{x\ \fgTypeAux'})\,\fgTypeAux', body=g']
}{k}{
\mT{m}{t_S}
} \QLabel{m-equiv}
\end{igather}
Define
\begin{igather}
k''' := min(k - k'', k - \smallsum{k_i}) - 1 \QLabel{def-k3}
\end{igather}
We have $k' - k'' - \smallsum{k_i} < k''' + 1$ by the following reasoning:
\begin{igather}
\begin{array}{@{}r@{}c@{}l@{}}
k''' + 1 & \ReasonAbove{=}{\QRef{def-k3}} & min(k - k'', k - \smallsum{k_i}) \\
& = & k - max(k'', \smallsum{k_i}) \\
& \geq & k - k'' - \smallsum{k_i} \\
& \ReasonAbove{>}{k' < k} & k' - k'' - \smallsum{k_i}
\end{array} \QLabel{ineq-k3}
\end{igather}
With \QRef{m-equiv} and $k''' < k$, we now want to use
the implication from the premise of rule \Rule{equiv-method-decl}. We instantiate
the universally quantified variables of the implication as follows:
$k' = k''', \fgTyActuals = \fgTySubst(\fgTyActuals, \fgTyActualsAux), \MultiP{W} = \tlSubst W, \MultiQ{W'} = \tlSubst W', v=u,
V=U, \MultiN{v} = \MultiN{v}, \MultiN{V} = \MultiN{V}$.
Next, we prove the
left-hand side of the implication.
But first assume (see \QRef{subst2}, \QRef{subst}, \QRef{eq-m})
\begin{igather}
\fgTyFormals = \MultiP{\fgTyVar\,\fgType} \qquad
\fgTyActuals = \MultiP{\fgType'} \qquad W = \MultiP{W} \QLabel{def-typarams1}\\
\fgTyFormalsAux' = \MultiQ{\fgTyVarAux\,\fgType''} \qquad
\fgTyActualsAux = \MultiQ{\fgType'''} \qquad W' = \MultiQ{W'} \QLabel{def-typarams2}\\
\end{igather}
and define
\begin{igather}
\fgTySubst_3 = \Angle{\MultiP{\subst{\fgTyVar_i}{\fgTySubst \fgType'_i}}, \MultiQ{\subst{\fgTyVarAux_i}{\fgTySubst\fgType'''_i}}} \QLabel{def-eta3}
\end{igather}
\begin{itemize}
\item We start by showing the first two conjuncts of the implication's left-hand side.
\begin{igather}
\fgMakeSubst{\fgTyFormals, \fgTyFormalsAux'}{\fgTySubst(\fgTyActuals, \fgTyActualsAux)}{\fgTySubst_3} \wedge
\LREquiv{
\fgTySubst(\fgTyActuals,\fgTyActualsAux)
}{
\tlSubst \Pair{\Multi{W}}{\Multi{W'}}
}{
\fgTyFormals,\fgTyFormalsAux'
}{
k'''
} \QLabel{lhs1}
\end{igather}
The left part of the conjunction follows from \QRef{def-eta3}.
We then show
$\LREquiv{
\fgTySubst(\fgTyActuals,\fgTyActualsAux)
}{
\tlSubst \Pair{\Multi{W}}{\Multi{W'}}
}{
\fgTyFormals,\fgTyFormalsAux'
}{
k
}$ by proving the two implications required to fulfill the premise of rule
\Rule{equiv-bounded-typarams}. The right part of the conjunction in \QRef{lhs1} then follows via \Cref{lem:mono-typarams}.
\begin{itemize}
\item First implication:
$\LREquiv{u_j}{U_j}{\fgTySubst \fgType_j'}{k} \implies \LREquiv{u_j}{(\tlSubst W_j)\,U_j}{\fgTySubst_3 \fgType_j}{k}$
for all $j \in [p]$ and all $u_j, U_j$.\\
From \QRef{subst2} and \Cref{lem:subst-equiv} we have
$\LREquiv{\fgTySubst\fgTyActuals}{\tlSubst W}{\fgTySubst\fgTyFormals}{k}$
Hence, with $\LREquiv{u_j}{U_j}{\fgTySubst \fgType_j'}{k}$ and the implication in the premise of rule
\Rule{equiv-bounded-typarams}, we have
$\LREquiv{u_j}{(\tlSubst W_j)\,U_j}{\Angle{\Multi{\subst{\fgTyVar_i}{\fgTySubst \fgType'_i}}}\fgTySubst\fgType_j}{k}$.
From assumption~\ref{conv:fg-decls}, \QRef{func-def}, and \QRef{def-typarams1},
we know that $\ftv{\fgType_j} \subseteq \{\Multi{\fgTyVar}\}$ and $\Multi{\fgTyVar}$ fresh, so
$\Angle{\Multi{\subst{\fgTyVar_i}{\fgTySubst\fgType'_i}}}\fgTySubst\fgType_j = \fgTySubst_3 \fgType_j$.
Thus
$\LREquiv{u_j}{(\tlSubst W_j)\,U_j}{\fgTySubst_3 \fgType_j}{k}$ as required.
\item Second implication:
$\LREquiv{u_j}{U_j}{\fgTySubst \fgType_j'''}{k} \implies \LREquiv{u_j}{(\tlSubst W'_j)\,U_j}{\fgTySubst_3 \fgType''_j}{k}$
for all $i \in [q]$ and all $u_j, U_j$.\\
From \QRef{subst} and \Cref{lem:subst-equiv} we have
$\LREquiv{\fgTySubst\fgTyActualsAux}{\tlSubst W'}{\fgTySubst\fgTyFormalsAux}{k}$. Hence, with
$\LREquiv{u_j}{U_j}{\fgTySubst \fgType_j'''}{k}$, the implication
in the premise of rule \Rule{equiv-bounded-typarams}, and \QRef{eq-m} then
$\LREquiv{u_j}{(\tlSubst W_j')\,U_j}{\Angle{\Multi{\subst{\fgTyVarAux_i}{\fgTySubst\fgType_i'''}}}\fgTySubst\fgTySubst_2\fgType_j''}{k}$.
We have with \QRef{subst2} and \QRef{def-typarams1} that $\fgTySubst_2 = \Angle{\Multi{\subst{\fgTyVar_i}{\fgType_i'}}}$.
Because of assumption~\ref{conv:fg-decls}, \QRef{func-def}, and \QRef{def-typarams2}, we know that
$\ftv{\fgType_j''} \subseteq \{\Multi{\fgTyVar}, \Multi{\fgTyVarAux}\}$ and $\Multi{\fgTyVar}, \Multi{\fgTyVarAux}$ fresh.
Hence, $\Angle{\Multi{\subst{\fgTyVarAux_i}{\fgTySubst\fgType_i'''}}}\fgTySubst\fgTySubst_2\fgType_j'' =
\Angle{\Multi{\subst{\fgTyVarAux_i}{\fgTySubst\fgType_i'''}}}\Angle{\Multi{\subst{\fgTyVar_i}{\fgTySubst\fgType_i'}}}\fgType_j''
\ReasonAbove{=}{\QRef{def-eta3}} \fgTySubst_3\fgType_j''$.
Thus
$\LREquiv{u_j}{(\tlSubst W_j')\,U_j}{\fgTySubst_3\fgType_j''}{k}$ as required.
\end{itemize}
This finishes the proof of \QRef{lhs1}.
\item We next show the third conjunct of the implication's left-hand side.
\begin{igather}
\LREquiv{u}{U}{t_S[\fgTySubst_3 \Multi{\fgTyVar}]}{k'''} \QLabel{lhs2}
\end{igather}
We have $t_S[\fgTySubst_3 \Multi{\fgTyVar}] = t_S[\fgTySubst\fgTyActuals]$ by \QRef{def-eta3} and \QRef{def-typarams1}.
Hence, with \QRef{reduce-G}, \Cref{lem:val-equiv-implies-exp-equiv}, and \Cref{lem:mono-exp}, it suffices
to show that $k''' \leq k - k''$. But this follows from construction of $k'''$ in \QRef{def-k3}.
\item Finally, we show the fourth conjunct:
\begin{igather}
\LREquiv{v_i}{V_i}{\fgTySubst_3\fgTypeAux_i'}{k'''} (\forall i \in [n]) \QLabel{lhs3}
\end{igather}
By \QRef{subst}, \QRef{eq-m}, \QRef{def-typarams2} we have $\fgTySubst_1 = \Angle{\MultiQ{\subst{\fgTyVarAux_i}{\fgType'''_i}}}$.
By \QRef{subst2} and \QRef{def-typarams1} we have $\fgTySubst_2 = \Angle{\MultiP{\subst{\fgTyVar_i}{\fgType'_i}}}$.
Thus,
\begin{igather}
\fgTySubst\fgTySubst_1\fgTypeAux_i \ReasonAbove{=}{\QRef{eq-m}}
\fgTySubst\fgTySubst_1\fgTySubst_2\fgTypeAux_i' \ReasonAbove{=}{\QRef{def-eta3}}
\fgTySubst_3\fgTypeAux_i'
\end{igather}
For the last equation, note that $\ftv{\fgTypeAux_i'} \subseteq \{\Multi{\fgTyVar}, \Multi{\fgTyVarAux}\}$
by assumption~\ref{conv:fg-decls} and \QRef{func-def}, \QRef{def-typarams1}, \QRef{def-typarams2}.
Hence, with \QRef{reduce-Ei}, \Cref{lem:val-equiv-implies-exp-equiv}, and \Cref{lem:mono-exp}, it suffices
to show that $k''' \leq k - k_i$. But this follows from construction of $k'''$ in \QRef{def-k3}.
\end{itemize}
Now \QRef{lhs1}, \QRef{lhs2}, and \QRef{lhs3} are the left-hand side of the implication of rule
\Rule{equiv-method-decl}, which we get from \QRef{m-equiv}. The right-hand side of the implication then yields
\begin{igather}
\LREquiv{
\BraceBelow{\Angle{\subst{x}{u}, \MultiN{\subst{x_i}{v_i}}}}{=: \fgSubst'}\fgTySubst_3 g'
}{
\mT{m}{t_S}\,\Quadr{\tlSubst W}{U}{\tlSubst W'}{\Tuple{\Multi{V}}}
}{
\fgTySubst_3\fgTypeAux'
}{
k'''
} \QLabel{equiv-call1}
\end{igather}
From \QRef{reduce-G} we have $u = t_S[\fgTySubst\fgTyActuals]$ by inverting rule \Rule{equiv-struct}. Hence by
\QRef{func-def}, \QRef{def-eta3}, and rule \Rule{fg-call}
\begin{igather}
u.m[\fgTySubst \fgTyActualsAux](\Multi{v}) \longrightarrow \fgSubst'\fgTySubst_3 g' \QLabel{reduce-call}
\end{igather}
Also we have
\begin{igather}
\fgTySubst \fgType \ReasonAbove{=}{\QRef{conclusion}}
\fgTySubst\fgTySubst_1\fgTypeAux \ReasonAbove{=}{\QRef{eq-m}}
\fgTySubst\fgTySubst_1\fgTySubst_2\fgTypeAux'
= \fgTySubst_3\fgTypeAux'
\end{igather}
where the last equation follows from
\QRef{subst}, \QRef{eq-m}, \QRef{def-typarams2}, \QRef{subst2}, \QRef{def-typarams1} and
$\ftv{\fgTypeAux'} \subseteq \{\Multi{\fgTyVar}, \Multi{\fgTyVarAux}\}$ with
assumption~\ref{conv:fg-decls}.
Thus, with \QRef{equiv-call1}, \QRef{reduce-call}, and \Cref{lem:source-reduce}
\begin{igather}
\LREquiv{
u.m[\fgTySubst \fgTyActualsAux](\Multi{v})
}{
\mT{m}{t_S}\,\Quadr{\tlSubst W}{U}{\tlSubst W'}{\Tuple{\Multi{V}}}
}{
\fgTySubst\fgType
}{
k''' + 1
} \QLabel{equiv-call2}
\end{igather}
By definition of $E$ in \QRef{conclusion} and with \QRef{reduce-G} and \QRef{reduce-Ei} we have
\begin{igather}
\tlSubst E \reduceSym^*
\tlMethTable(\mT{m}{t_S})\,\Quadr{\tlSubst W}{U}{\tlSubst W'}{\Tuple{\Multi{V}}} \QLabel{eval-E}
\end{igather}
Also, we have by rules \Rule{tl-context} and \Rule{tl-method}
\begin{igather}
\mT{m}{t_S}\,\Quadr{\tlSubst W}{U}{\tlSubst W'}{\Tuple{\Multi{V}}} \longrightarrow
\tlMethTable(\mT{m}{t_S})\,\Quadr{\tlSubst W}{U}{\tlSubst W'}{\Tuple{\Multi{V}}} \QLabel{eval-Xmts}
\end{igather}
So far, we proved everything under the assumptions
\QRef{reduce-g}, \QRef{reduce-ei}, \QRef{reduce-e}. We next consider
the two implications of rule \Rule{equiv-exp}.
\begin{EnumAlph}
\item Assume $e' = v$ for some value $v$. Our goal is to prove that there exists some value $V$
such that $\reduceStar{\tlSubst E}{V}$ and $\LREquivVal{v}{V}{\fgTySubst\fgType}{k-k'}$.
Noting that \QRef{reduce-g}, \QRef{reduce-ei}, \QRef{reduce-e} hold, we have together with \QRef{reduce-call}
\begin{igather}
\fgSubst\fgTySubst e \longrightarrow^{k'' + \smallsum{k_i}}
u.m[\fgTySubst \fgTyActualsAux](\Multi{v}) \longrightarrow^{k' - k'' - \smallsum{k_i}} v \QLabel{reduce-steps}
\end{igather}
with $k'' + \smallsum{k_i} < k'$. We have $k' - k'' - \smallsum{k_i} < k''' + 1$ by \QRef{ineq-k3}.
Hence with \QRef{equiv-call2} and \QRef{reduce-steps}
we know that there exists some value $V$ with
\begin{igather}
\mT{m}{t_S}\,\Quadr{\tlSubst W}{U}{\tlSubst W'}{\Tuple{\Multi{V}}} \reduceSym^* V \QLabel{eval-Xmts2}\\
\LREquivVal{v}{V}{\fgTySubst\fgType}{k''' + 1 - k' + k'' + \smallsum{k_i}} \QLabel{equiv-v-V}
\end{igather}
We have $k - k' \leq k''' + 1 - k' + k'' + \smallsum{k_i}$ by the following reasoning:
\begin{igather}
\begin{array}{@{}r@{}c@{}l@{}}
k''' + 1 - k' + k'' + \smallsum{k_i}
& \ReasonAbove{=}{\QRef{def-k3}} & min(k - k'', k - \smallsum{k_i}) - k' + k'' + \smallsum{k_i} \\
& = & k - max(k'', \smallsum{k_i}) - k' + k'' + \smallsum{k_i} \\
& \geq & k - k'' - \smallsum{k_i} - k' + k'' + \smallsum{k_i} \\
& = & k - k'
\end{array}
\end{igather}
With \QRef{equiv-v-V} and \Cref{lem:mono-exp} then
$\LREquivVal{v}{V}{\fgTySubst\fgType}{k - k'}$.
And from \QRef{eval-E}, \QRef{eval-Xmts}, \QRef{eval-Xmts2}, and \Cref{lem:determ-eval-tl} we have that
$\reduceStar{\tlSubst E}{V}$.
\item Assume $\Diverge{e'}$. We then have to show $\Diverge{\tlSubst E}$.
\begin{CaseDistinction}{whether receiver, argument or method call diverges}
\Case{receiver diverges} Then
$\reducek{k'}{\fgSubst\fgTySubst g}{g''}$ and $\Diverge{g''}$.
With \QRef{g-equiv} and $k' < k$ then $\Diverge{\tlSubst G}$,
so by the definition of $E$ in \QRef{conclusion} we get
$\Diverge{\tlSubst E}$.
\Case{$j$-th argument diverges} Then
$\reducek{k''}{\fgSubst\fgTySubst g}{u}$ and
$\reducek{k_i}{\fgSubst\fgTySubst e_i}{v_i}$ for all $i < j$ and
$\reducek{k_j}{\fgSubst\fgTySubst e_j}{e''}$ and
$\Diverge{e''}$. With \QRef{ei-prime-equiv} and $k_j \leq k' < k$ we get $\Diverge{\tlSubst E_j}$.
By definition of $E$ in \QRef{conclusion} then $\Diverge{\tlSubst E}$.
\Case{method call diverges} Then we are in the situation that \QRef{reduce-g}, \QRef{reduce-ei}, and
\QRef{reduce-e} hold. We then have
\begin{igather}
u.m[\fgTySubst \fgTyActualsAux](\MultiN{v}) \longrightarrow^{k' - k'' - \smallsum{k_i}} e'
\end{igather}
Hence, with \QRef{ineq-k3}, \QRef{equiv-call2}, and the second implication in the premise of rule
\Rule{equiv-exp}, we have that $\Diverge{\mT{m}{t_S}\,\Quadr{\tlSubst W}{U}{\tlSubst W'}{\Tuple{\Multi{V}}}}$.
With \QRef{eval-E} and \QRef{eval-Xmts} and \Cref{lem:determ-eval-tl} then also $\Diverge{\tlSubst E}$ as required.
\end{CaseDistinction}
\end{EnumAlph}
This finishes the proof for rule \Rule{call-struct}.
\renewcommand\LabelQualifier{exp-equiv-call-iface}
\Case{\Rule{call-iface}}
\begin{mathpar}
\inferrule{
\LabeledRule{
\tdExpTrans{\pair{\Delta}{\Gamma}}{g : \fgType_I}{G} &\TextLabel{deriv-eprime}\\
\methodSpecifications{\fgType_I} = \Foreach{R}{q} &\TextLabel{methods}\\
R_j = m[\fgTyFormalsAux](\Foreach{x\,\fgTypeAux}{n})\fgTypeAux \quad(\textrm{for some}~j \in [q]) &\TextLabel{def-Sj}\\
\tdCheckSubst{\Delta}{\fgTyFormalsAux}{\fgTyActualsAux}{\fgTySubst_1}{V} & \TextLabel{subst}\\
\tdExpTrans{\pair{\Delta}{\Gamma}}{e_i : \fgTySubst_1 \fgTypeAux_i}{E_i}
\quad\noteForall{i \in [n]} & \TextLabel{deriv-ei}\\
Y, \Foreach{X}{q}\textrm{~fresh}
}
}{
\LabeledRule{
\tdExpTrans{\pair{\Delta}{\Gamma}}
{\BraceBelow{g.m[\fgTyActualsAux](\Foreach{e}{n})}{= e} : \BraceBelow{\fgTySubst_1 \fgTypeAux}{= \fgType}}
{E}
&\TextLabel{conclusion}
}
}
\end{mathpar}
with
\begin{igather}
E = \mbox{\kw{case}}\ G \ \mbox{\kw{of}}\ \Tuple{Y\Comma \Tuple{\Foreach{X}{q}}} \to X_j\ \Triple{Y}{V}{\Tuple{\Foreach{E}{n}}} \QLabel{def-E}
\end{igather}
From the IH applied to \QRef{deriv-eprime}, \QRef{deriv-ei}
\begin{igather}
\LREquiv{
\fgSubst \fgTySubst g
}{
\tlSubst G
}{\fgTySubst \fgType_I}{k} \QLabel{g-equiv}\\
\LREquiv{
\fgSubst \fgTySubst e_i
}{
\tlSubst E_i
}{\fgTySubst \fgTySubst_1 \fgTypeAux_i}{k} \quad (\forall i \in [n]) \QLabel{ei-prime-equiv}
\end{igather}
Assume $\reducek{k'}{\fgSubst \fgTySubst e}{e'}$ for some $k' < k$. We first consider the following situation
for some values $u, \MultiN{v}$:
\begin{igather}
\reducek{k''}{\fgSubst \fgTySubst g}{u} \QLabel{reduce-g}\\
\reducek{k_i}{\fgSubst \fgTySubst e_i}{v_i} \QLabel{reduce-ei}\\
\fgSubst \fgTySubst e \longrightarrow^{k''+\smallsum{k_i}} u.m[\fgTySubst \fgTyActualsAux](\MultiN{v})
\longrightarrow^{k' - k'' - \smallsum{k_i}} e' \QLabel{reduce-e}
\end{igather}
with $k'' + \smallsum{k_i} \leq k'$. \QRef{reduce-g}, \QRef{reduce-ei}, and \QRef{reduce-e} are intermediate
assumptions, which become true when we later prove the two implications of rule \Rule{equiv-exp}.\\
We have from \QRef{g-equiv}, \QRef{ei-prime-equiv}, \QRef{reduce-g}, and \QRef{reduce-ei}
\begin{igather}
\reduceStar{\tlSubst G}{U} \textrm{~for some~} U \textrm{~with~}
\LREquivVal{u}{U}{\fgTySubst \fgType_I}{k - k''} \QLabel{reduce-G}\\
(\forall i \in [n])~\reduceStar{\tlSubst E_i}{V_i} \textrm{~for some~} V_i \textrm{~with~}
\LREquivVal{v_i}{V_i}{\fgTySubst\fgTySubst_1 \fgTypeAux_i}{k - k_i} \QLabel{reduce-Ei}
\end{igather}
From \QRef{reduce-G} and \QRef{methods} we get by inverting rule \Rule{equiv-iface}
\begin{igather}
\exists \fgTypeAux_S = t_S[\fgTyActuals] \QLabel{def-tauS}\\
U = \Pair{U'}{\Tuple{\MultiQ{U}}} \QLabel{U-eq}\\
\forall \ell_1 < k - k'' \DOT \LREquivVal{u}{U'}{\fgTypeAux_S}{\ell_1} \QLabel{u-equiv}\\
\forall \ell_2 < k - k'' \DOT \LREquiv{\methodLookup{m_j}{\fgTypeAux_S}}{U_j}{\fgTySubst R_j}{\ell_2} \QLabel{methodLookup-equiv}
\end{igather}
Hence we have by \QRef{def-tauS}, \QRef{methodLookup-equiv}, \QRef{def-Sj}, and rule \Rule{method-lookup}
\begin{igather}
\fgFunc[spec={\BraceBelow{m[\fgTyFormalsAux'](\MultiN{x\ \fgTypeAux'})\,\fgTypeAux'}{=: R'}}, body=e''] \in \fgDecls \QLabel{func}\\
\fgMakeSubst{\fgTyFormals}{\fgTyActuals}{\fgTySubst_2} \QLabel{def-eta2}\\
\methodLookup{m_j}{\fgTypeAux_S} = \Angle{x, t_S[\fgTyActuals], \fgTySubst_2 R', \fgTySubst_2 e''} \QLabel{methodLookup-eq}\\
\fgTySubst_2 R' = \fgTySubst R_j = \fgTySubst (m[\fgTyFormalsAux](\MultiN{x_i\ \fgTypeAux_i})\,\fgTypeAux) \QLabel{Sprime-eq-Sj}
\end{igather}
Then by \QRef{methodLookup-equiv} and \QRef{methodLookup-eq}
\begin{igather}
\LREquiv{\Angle{x, t_S[\fgTyActuals], \fgTySubst_2 R', \fgTySubst_2 e''}}{U_j}{\fgTySubst R_j}{k - k'' - 1}
\QLabel{star}
\end{igather}
Define $k''' := \min(k - k'' - 1, k - \smallsum{k_i} - 1)$. Then
\begin{igather}
k''' \leq k - k'' - 1 \QLabel{k3-ineq0}\\
k''' < k \QLabel{k3-ineq1}\\
k''' < k - k'' \QLabel{k3-ineq2}\\
k''' < k - k_i \quad(\forall i \in [n]) \QLabel{k3-ineq3}\\
k' - k'' - \smallsum{k_i} < k''' + 1 \QLabel{k3-ineq4}
\end{igather}
The first four of these claims are straightforward to verify. The last can be shown with
the following reasoning:
\begin{igather}
\begin{array}{@{}r@{}c@{}l@{}}
k''' + 1
& = & k - \max(k'' + 1, \smallsum{k_i} + 1) + 1 \\
& > & k - (k'' + 1 + \smallsum{k_i} + 1) + 1\\
& = & k - 1 - k'' - \smallsum{k_i} \\
& \ReasonAbove{\geq}{k' < k} & k' - k'' - \smallsum{k_i}
\end{array}
\end{igather}
From \QRef{star} we get the implication in the premise of rule \Rule{equiv-method-dict-entry}.
We now show that the left-hand side of the implication holds.
The universally quantified variables of the rule's premise are instantiated as follows:
$k' = k''', \fgTyActuals = \fgTySubst \fgTyActualsAux, W = \tlSubst V, v = u, V = U', \MultiN{v} = \MultiN{v}, \MultiN{V} = \MultiN{V}$.
The variables in the conclusion are instantiated as follows:
$x = x, \fgType_S = t_S[\fgTyActuals], m[\fgTyFormals](\MultiN{x_i\ \fgType_i})\,\fgType = \fgTySubst_2 R', e = \fgTySubst_2 e''$.
The requirement $k''' \leq k - k'' - 1$ follows from \QRef{k3-ineq0}.\\
We have from \QRef{subst} and \QRef{Sprime-eq-Sj} the first conjunct:
\begin{igather}
\fgMakeSubst{\fgTySubst \fgTyFormalsAux}{\fgTySubst \fgTyActualsAux}{
\BraceBelow{\Angle{\Multi{\subst{\fgTyVar}{\fgTySubst\fgType}}}}{= \fgTySubst_4}
} ~(\textrm{assuming } \fgTySubst_1 = \Angle{\Multi{\subst{\fgTyVar}{\fgType}}},
\fgGetTyVars{\fgTyFormalsAux} = \Multi{\fgTyVar}, \fgTyActualsAux = \Multi{\fgType})\QLabel{eta4}
\end{igather}
From \QRef{subst} we get the second conjunct by \Cref{lem:subst-equiv}, \QRef{k3-ineq1}, by the assumptions
$\LREquivNoTy{\fgDecls}{k}{\tlMethTable}$ and
$\LREquiv{\fgTySubst}{\tlSubst}{\Delta}{k}$, and by \Cref{lem:mono-typarams}:
\begin{igather}
\LREquiv{
\fgTySubst \fgTyActualsAux
}{
\tlSubst V
}{
\fgTySubst \fgTyFormalsAux
}{k'''}
\end{igather}
With \QRef{k3-ineq2}, \QRef{u-equiv}, \QRef{def-tauS},
and \Cref{lem:val-equiv-implies-exp-equiv} we get the third conjunct:
\begin{igather}
\LREquiv{u}{U'}{t_S[\fgTyActuals]}{k'''}
\end{igather}
With \QRef{k3-ineq3}, \QRef{reduce-Ei}, \Cref{lem:val-equiv-implies-exp-equiv}, and \Cref{lem:mono-exp} we have
\begin{igather}
\LREquiv{v_i}{V_i}{\fgTySubst\fgTySubst_1\fgTypeAux_i}{k'''} \quad (\forall i \in [n]) \QLabel{vi-equiv-Vi-1}
\end{igather}
We next prove
\begin{igather}
\fgTySubst \fgTySubst_1 \fgTypeAux_i = \fgTySubst_4 \fgTySubst \fgTypeAux_i \quad (\forall i \in [n]) \QLabel{subst-eq1}\\
\fgTySubst \fgTySubst_1 \fgTypeAux = \fgTySubst_4 \fgTySubst \fgTypeAux \QLabel{subst-eq2}
\end{igather}
by induction on $\fgTypeAux_i$ or $\fgTypeAux$.
The interesting case is where $\fgTypeAux_i$ or $\fgTypeAux$ is a type variable $\fgTyVar \in \dom{\fgTySubst_1} \cup \dom{\fgTySubst}$.
As the $\Multi{\fgTyVar} = \dom{\fgTySubst_1} = \dom{\fgTySubst_4}$ are bound in $\fgTyFormalsAux$
(see \QRef{eta4}), we may assume that
$\Multi{\fgTyVar} \cap \dom{\fgTySubst} = \emptyset = \Multi{\fgTyVar} \cap \ftv{\fgTySubst}$.
If $\fgTyVar \in \dom{\fgTySubst_1}$ then
\begin{igather}
\fgTySubst \fgTySubst_1 \fgTyVar \ReasonAbove{=}{\QRef{eta4}} \fgTySubst_4 \fgTyVar
\ReasonAbove{=}{\dom{\fgTySubst_1} \cap \dom{\fgTySubst} = \emptyset} \fgTySubst_4 \fgTySubst \fgTyVar
\end{igather}
If $\fgTyVar \in \dom{\fgTySubst}$ then
\begin{igather}
\fgTySubst\fgTySubst_1 \fgTyVar \ReasonAbove{=}{\dom{\fgTySubst} \cap \dom{\fgTySubst_1} = \emptyset}
\fgTySubst \fgTyVar \ReasonAbove{=}{\dom{\fgTySubst_1} \cap \ftv{\fgTySubst} = \emptyset}
\fgTySubst_4\fgTySubst\fgTyVar
\end{igather}
We now get with
\QRef{Sprime-eq-Sj} and \QRef{subst-eq1}
that $\fgTySubst\fgTySubst_1 \fgTypeAux_i = \fgTySubst_4\fgTySubst_2\fgTypeAux_i'$. Hence
with \QRef{vi-equiv-Vi-1} the fourth conjunct:
\begin{igather}
(\forall i \in [n])\quad \LREquiv{v_i}{V_i}{\fgTySubst_4\fgTySubst_2\fgTypeAux'_i}{k'''}
\end{igather}
Now the right-hand side of the implication of rule \Rule{equiv-method-dict-entry} yields with \QRef{star}
\begin{igather}
\LREquiv{
\Angle{\subst{x}{u}, \MultiN{\subst{x_i}{v_i}}}\fgTySubst_4\fgTySubst_2 e''
}{
U_j\ \Triple{U'}{\tlSubst V}{\Tuple{\MultiN{V}}}
}{
\fgTySubst_4 \fgTySubst_2 \fgTypeAux'
}{k'''} \QLabel{eta4-eta2-epprime}
\end{igather}
Define $\fgTySubst_3$ such that
\begin{igather}
\fgMakeSubst{\fgTyFormals,\fgTyFormalsAux'}{\fgTyActuals, \fgTySubst\fgTyActualsAux}{\fgTySubst_3} \QLabel{def-eta3}
\end{igather}
Then with \QRef{def-eta2} and \QRef{eta4}
\begin{igather}
\fgTySubst_4\fgTySubst_2 e'' = \fgTySubst_3 e'' \QLabel{eq-epprime}
\end{igather}
by induction on $e''$. The interesting case is the one for a type variable $\fgTyVar$. By assumption~\ref{conv:fg-decls}
and \QRef{func}, we know that $\fgTyVar \in \fgGetTyVars{\fgTyFormals} \cup \fgGetTyVars{\fgTyFormalsAux'}$.
Further we may assume that the type variables $\fgGetTyVars{\fgTyFormalsAux'}$ are fresh, and we have
$\dom{\fgTySubst_2} = \fgGetTyVars{\fgTyFormals}$ by \QRef{def-eta2} and
$\dom{\fgTySubst_4} = \fgGetTyVars{\fgTyFormalsAux'}$ by \QRef{eta4}.
Thus, if $\fgTyVar \in \fgGetTyVars{\fgTyFormals}$ then $\fgTySubst_4\fgTySubst_2\fgTyVar = \fgTySubst_2\fgTyVar$
because $\fgGetTyVars{\fgTyFormalsAux'}$ fresh, and $\fgTySubst_3\fgTyVar = \fgTySubst_2\fgTyVar$ by
\QRef{def-eta2} and \QRef{def-eta3}. If $\fgTyVar \in \fgGetTyVars{\fgTyFormalsAux'}$ then
$\fgTySubst_4\fgTySubst_2\fgTyVar = \fgTySubst_4\fgTyVar$ because $\fgGetTyVars{\fgTyFormalsAux'}$ fresh, and
$\fgTySubst_3\fgTyVar = \fgTySubst_4\fgTyVar$ by \QRef{def-eta3} and \QRef{eta4}.
\\
With \QRef{subst-eq2} and \QRef{Sprime-eq-Sj} $\fgTySubst_4\fgTySubst_2 \fgTypeAux' = \fgTySubst\fgTySubst_1\fgTypeAux$.
Hence we have with \QRef{eq-epprime}, \QRef{eta4-eta2-epprime}
\begin{igather}
\LREquiv{
\Angle{\subst{x}{u}, \MultiN{\subst{x_i}{v_i}}}\fgTySubst_3 e''
}{
U_j\ \Triple{U'}{\tlSubst V}{\Tuple{\MultiN{V}}}
}{
\fgTySubst\fgTySubst_1 \fgTypeAux
}{k'''} \QLabel{equiv-body}
\end{igather}
From \QRef{def-tauS} and \QRef{u-equiv} we get by inverting rule \Rule{equiv-struct} that
$u = t_S[\fgTyActuals]\{\ldots\}$.
Hence by rule \Rule{fg-call} with \QRef{func} and \QRef{def-eta3}
\begin{igather}
u.m[\fgTySubst\fgTyActualsAux](\MultiN{v}) \longrightarrow
\Angle{\subst{x}{u}, \MultiN{\subst{x_i}{v_i}}}\fgTySubst_3 e''
\end{igather}
Then with \QRef{equiv-body} and \Cref{lem:source-reduce}
\begin{igather}
\LREquiv{
u.m[\fgTySubst\fgTyActualsAux](\MultiN{v})
}{
U_j\ \Triple{U'}{\tlSubst V}{\Tuple{\MultiN{V}}}
}{
\fgTySubst\fgTySubst_1 \fgTypeAux
}{k'''+1} \QLabel{equiv-call}
\end{igather}
We also have
\begin{igather}
\tlSubst E
\ReasonAbove{\reduceSym^*}{\QRef{reduce-G}, \QRef{def-E}}
\mbox{\kw{case}}\ U \ \mbox{\kw{of}}\ \Tuple{Y\Comma \Tuple{\Foreach{X}{q}}} \to X_j\ \Triple{Y}{V}{\Tuple{\Foreach{E}{n}}}\\
\ReasonAbove{\longrightarrow}{\QRef{U-eq}}
U_j\ \Triple{U'}{\tlSubst V}{\tlSubst \Tuple{\MultiN{E}}} \\
\ReasonAbove{\reduceSym^*}{\QRef{reduce-Ei}}
U_j\ \Triple{U'}{\tlSubst V}{\Tuple{\MultiN{V}}}
\QLabel{eval-E}
\end{igather}
So far, we proved everything under the assumptions
\QRef{reduce-g}, \QRef{reduce-ei}, \QRef{reduce-e}. We next consider
the two implications of rule \Rule{equiv-exp}.
\\
\begin{EnumAlph}
\item Assume $e' = v$ for some value $v$.
Then \QRef{reduce-g}, \QRef{reduce-ei}, and \QRef{reduce-e} hold.
We now need to show that there exists some $W$ with
$\tlSubst E \reduceSym^* W$ and $\LREquivVal{v}{W}{\fgTySubst \fgType}{k - k'}$.
We have $k' - k'' - \smallsum{k_i} < k''' + 1$ by \QRef{k3-ineq4}
Also, we have with \QRef{reduce-e} that
\begin{igather}
u.m[\fgTySubst \fgTyActualsAux](\MultiN{v}) \longrightarrow^{k' - k'' - \smallsum{k_i}} v
\end{igather}
Hence, \QRef{equiv-call} gives us the existence of some $W$ such that
\begin{igather}
U_j\ \Triple{U'}{\tlSubst V}{\Tuple{\MultiN{V}}} \reduceSym^* W \QLabel{eval-to-W}\\
\LREquiv{v}{W}{\fgTySubst \fgTySubst_1 \fgTypeAux}{k''' + 1 - (k' - k'' - \smallsum{k_i})}
\end{igather}
We get $k - k' \leq k''' + 1 - (k' - k'' - \smallsum{k_i})$ by
\begin{igather}
\begin{array}{@{}r@{~}l@{}}
& k''' + 1 - (k' - k'' - \smallsum k_i) \\
= & k - max(k'' + 1, \smallsum{k_i} + 1) + 1 - k' + k'' + \smallsum{k_i} \\
= & k - max(k'', \smallsum{k_i}) - k' + k'' + \smallsum{k_i} \\
\geq & k - (k'' + \smallsum{k_i}) - k' + (k'' + \smallsum{k_i}) \\
= & k - k'
\end{array}
\end{igather}
Hence by \Cref{lem:mono-exp}
\begin{igather}
\LREquiv{v}{W}{\fgTySubst \fgTySubst_1 \fgTypeAux}{k - k'}
\end{igather}
By \QRef{conclusion} $\fgTySubst_1 \fgTypeAux = \fgType$ so
$\LREquiv{v}{W}{\fgTySubst \fgType}{k - k'}$ and with
\QRef{eval-E} and \QRef{eval-to-W} $\tlSubst E \reduceSym^* W$.
\item Assume $\Diverge{e'}$. We then have to show $\Diverge{\tlSubst E}$.
\begin{CaseDistinction}{whether receiver, argument or method call diverges}
\Case{receiver diverges} Then
$\reducek{k'}{\fgSubst\fgTySubst g}{g'}$ and $\Diverge{g'}$.
With \QRef{g-equiv} and $k' < k$ then $\Diverge{\tlSubst G}$,
so by the definition of $E$ in \QRef{def-E} we get
$\Diverge{\tlSubst E}$.
\Case{$j$-th argument diverges} Then
$\reducek{k''}{\fgSubst\fgTySubst g}{u}$.
By \QRef{g-equiv} and rule \Rule{equiv-iface} we know that
$U = \Pair{U'}{\MultiQ{U}}$ for some $U', \MultiQ{U}$.
Hence
\begin{igather}
\tlSubst E \reduceSym^* U_j\ \Triple{U'}{\tlSubst V}{\tlSubst \Tuple{\MultiN{E}}}
\QLabel{eval-subst-E}
\end{igather}
Because the $j$-th argument diverges, we also have
$\reducek{k_i}{\fgSubst\fgTySubst e_i}{v_i}$ for all $i < j$ and
$\reducek{k_j}{\fgSubst\fgTySubst e_j}{e''}$ and
$\Diverge{e''}$.
With \QRef{ei-prime-equiv} we get $\Diverge{\tlSubst E_j}$, so with \QRef{eval-subst-E} also
$\Diverge{\tlSubst E}$.
\Case{method call diverges} Then we are in the situation that \QRef{reduce-g}, \QRef{reduce-ei}, and
\QRef{reduce-e} hold. Thus, we get with \QRef{reduce-e}, \QRef{equiv-call}, \QRef{k3-ineq4}
\begin{igather}
u.m[\fgTySubst \fgTyActualsAux](\MultiN{v}) \longrightarrow^{k' - k'' - \smallsum{k_i}} e'\\
\LREquiv{
u.m[\fgTySubst\fgTyActualsAux](\MultiN{v})
}{
U_j\ \Triple{U'}{\tlSubst V}{\Tuple{\MultiN{V}}}
}{
\fgTySubst\fgTySubst_1 \fgTypeAux
}{k'''+1}
\\
k' - k'' - \smallsum{k_i} < k''' + 1
\end{igather}
Hence $\Diverge{U_j\ \Triple{U'}{\tlSubst V}{\Tuple{\MultiN{V}}}}$ by the implication in the
premise of rule \Rule{equiv-exp}. So by \QRef{eval-E} also $\Diverge{\tlSubst E}$ as required.
\end{CaseDistinction}
\end{EnumAlph}
\renewcommand\LabelQualifier{exp-equiv-sub}
\Case{\Rule{sub}}
\begin{mathpar}
\inferrule[sub]{
\tdExpTrans{\pair{\Delta}{\Gamma}}{e : \fgTypeAux}{E'}\\
\tdUpcast{\Delta}{\subtypeOf{\fgTypeAux}{\fgType}}{V}
}{
\tdExpTrans{\pair{\Delta}{\Gamma}}{e : \fgType}{\BraceBelow{V \ E'}{= E}}
}
\end{mathpar}
From the IH then $\LREquiv{\fgSubst\fgTySubst e}{\tlSubst E'}{\fgTySubst\fgTypeAux}{k}$.
From \Cref{subtyping-equiv} we get
$\LREquiv{\fgSubst\fgTySubst e}{(\tlSubst V)\,\tlSubst E'}{\fgTySubst\fgType}{k}$
with $\tlSubst E = (\tlSubst V)\,(\tlSubst E')$ as required.
\end{CaseDistinctionExplicit} \RealQED
\subsubsection{Proof of \Cref{lem:method-equiv}}
\label{sec:lem:method-equiv}
\renewcommand\LabelQualifier{lem:method-equiv}
We proceed by induction on $k$. For $k = 0$,
we first note that $\LREquiv{e}{E}{\fgType}{0}$ holds for any $e, E, \fgType$ because
the two implications in the premise of rule \Rule{equiv-exp} hold trivially.
Thus, we get
$\LREquivNoTy{D}{0}{\mT{m}{t_S}}$ for all $D = \fgFunc[spec=mM] \in \fgDecls$
by rule \Rule{equiv-method-decl}.
Hence $\LREquivNoTy{\fgDecls}{0}{\tlMethTable}$ by rule \Rule{equiv-decls}.
Now assume
$\LREquivNoTy{\fgDecls}{k}{\mu}$ (IH)
for some $k$ and prove $\LREquivNoTy{\fgDecls}{k+1}{\mu}$.
By rule \Rule{equiv-decls}, we need to show $\LREquivNoTy{D}{k+1}{\mT{m}{t_S}}$ for all
\begin{igather}
D = \fgFunc[spec=m{[\fgTyFormals']}(\MultiN{x\ \fgType})\,\fgType] \in \fgDecls \QLabel{D}
\end{igather}
Thus, we assume the left-hand side of the implication in the premise of rule \Rule{equiv-method-decl}
and then show the right-hand side of the implication. More specifically, let
\begin{igather}
\fgTyFormals = \MultiP{\fgTyVar_i\,\fgTypeAux_i} \qquad
\fgTyFormals' = \MultiQ{\fgTyVarAux_i\,\fgTypeAux'_i} \qquad
\fgTyFormalsAux = \fgTyFormals, \fgTyFormals' = \MultiP{\fgTyVar_i\,\fgTypeAux_i}\,\MultiQ{\fgTyVarAux_i\,\fgTypeAux'_i}
\end{igather}
and assume for arbitrary
$k' < k + 1, \fgTyActuals = \MultiP{\fgTypeAux''}, \fgTyActuals' = \MultiQ{\fgTypeAux'''}, \MultiP{W}, \MultiQ{W'}, u, U, \MultiN{v}, \MultiN{V}$
the left-hand side of the implication:
\begin{igather}
\fgMakeSubst{\fgTyFormalsAux}{\fgTyActuals, \fgTyActuals'}{\fgTySubst}
\textrm{~with~}
\fgTySubst = \Angle{\MultiP{\subst{\fgTyVar_i}{\fgTypeAux_i''}}\,\MultiQ{\subst{\fgTyVarAux_i}{\fgTypeAux_i'''}}} \QLabel{def-eta}\\
\LREquiv{\fgTyActuals,\fgTyActuals'}{\Pair{\MultiP{W}}{\MultiQ{W'}}}{\fgTyFormalsAux}{k'} \QLabel{typarams-equiv}\\
\LREquiv{u}{U}{t_S[\fgTySubst\MultiP{\fgTyVar}]}{k'} \QLabel{u-equiv}\\
\LREquiv{v_i}{V_i}{\fgTySubst \fgType_i}{k'} \quad (\forall i \in [n]) \QLabel{vi-equiv}
\end{igather}
From this we need to prove the following goal:
\begin{igather}
\LREquiv{
\BraceBelow{\Angle{\subst{x}{u}, \MultiN{\subst{x_i}{v_i}}}}{=: \fgSubst} \fgTySubst e
}{
\mT{m}{t_S}\ \Quadr{\Tuple{\MultiP{W}}}{U}{\Tuple{\MultiQ{W'}}}{\Tuple{\MultiN{V}}}
}{
\fgTySubst\fgType
}{
k'
} \QLabel{goal}
\end{igather}
Define
\begin{igather}
\tlSubst = \Angle{\MultiP{\subst{\xTy{\fgTyVar_i}}{W_i}}, \MultiQ{\subst{\xTy{\fgTyVarAux_i}}{W_i'}},
\subst{X}{U}, \MultiN{\subst{X_i}{V_i}}} \QLabel{def-rho} \\
\Delta = \{ \MultiP{\fgTyVar_i : \fgTypeAux_i},\,\MultiQ{\fgTyVarAux_i : \fgTypeAux_i'} \} \\
\Gamma = \{ x: t_S[\MultiP{\fgTyVar}], \MultiN{x_i : \fgType_i} \}
\end{igather}
Then with \QRef{def-eta}, \QRef{typarams-equiv}, and rule \Rule{equiv-ty-subst}
\begin{igather}
\LREquiv{\fgTySubst}{\tlSubst}{\Delta}{k'} \QLabel{tyenv-equiv}
\end{igather}
And with \QRef{u-equiv}, \QRef{vi-equiv}, the definition of $\fgSubst$ in \QRef{goal}, and rule \Rule{equiv-val-subst}
\begin{igather}
\LREquiv{\fgSubst}{\tlSubst}{\fgTySubst \Gamma}{k'} \QLabel{valenv-equiv}
\end{igather}
From the assumption $\tdMethTrans{D}{\mT{m}{t_S} = V}$ we get by inverting rule \Rule{method}
\begin{igather}
\tdExpTrans{\Angle{\Delta,\Gamma}}{e : \fgType}{E} \QLabel{type-e}\\
V = \tllambda{\Quadr{\Tuple{\MultiP{\xTy{\fgTyVar_i}}}}{X}{\Tuple{\MultiQ{\xTy{\fgTyVarAux_i}}}}{\Tuple{\MultiN{X}}}}{E} \QLabel{def-V}
\end{igather}
With $k' < k+1$ we have $k' \leq k$. With the IH and~\Cref{lem:mono-prog} then
\begin{igather}
\LREquivNoTy{\fgDecls}{k'}{\tlMethTable} \QLabel{prog-equiv}
\end{igather}
\QRef{prog-equiv}, \QRef{tyenv-equiv}, \QRef{valenv-equiv} and \QRef{type-e} are the requirements
of \Cref{lem:exp-equiv}. The lemma then yields
\begin{igather}
\LREquiv{\fgSubst\fgTySubst e}{\tlSubst E}{\fgTySubst\fgType}{k'} \QLabel{equiv-E}
\end{igather}
We also have
\begin{igather}
\mT{m}{t_S}\ \Quadr{\Tuple{\MultiP{W}}}{U}{\Tuple{\MultiQ{W'}}}{\Tuple{\MultiN{V}}} \longrightarrow
V\ \Quadr{\Tuple{\MultiP{W}}}{U}{\Tuple{\MultiQ{W'}}}{\Tuple{\MultiN{V}}} \reduceSym^*
\tlSubst E
\end{igather}
where the first reduction follows from assumption $\tlMethTable(\mT{m}{t_S}) = V$ and rule \Rule{tl-method},
the remaining steps by \QRef{def-V} and \QRef{def-rho}.
With \QRef{equiv-E} and \Cref{lem:target-reduce} we then get \QRef{goal} as required. \RealQED
\subsubsection{Proof of \Cref{thm:prog-equiv}}
\label{sec:proof-crefthm:pr-equ}
We first prove that the assumptions of the theorem imply $\LREquiv{e}{E}{\fgType}{k}$ for any $k$.
$\fgDecls$ and $\tlMethTable$ are the declarations and the substitution whose existence we assumed
globally. Obviously, they meet the requirements of Assumption~\ref{conv:fg-decls}.
Assume $k \in \Nat$.
From \Cref{lem:method-equiv} we get $\LREquivNoTy{\fgDecls}{k}{\mu}$. By the assumption
$\tdProgTrans{\ForeachN{D} \ \mbox{\kw{func}}\ \mbox{\mathem main} () \{ \_ = e \}} {\mbox{\kw{let}}\ \Multi{X_i = V_i} \ \mbox{\kw{in}}\ E}$,
by inverting rule \Rule{prog}, and by the assumption that $e$ has type $\fgType$, we find
$\tdExpTrans{\pair{\EmptyFgEnv}{\EmptyFgEnv}}{e : \fgType}{E}$.
\Cref{lem:exp-equiv} then yields $\LREquiv{e}{E}{\fgType}{k}$ as required.
From $\LREquiv{e}{E}{\fgType}{k}$ for any $k$ and the two implications in the premise
of rule \Rule{equiv-exp}, we then get the two claims needed to show. \MyQED
\section{Related Work}
\label{sec:related-work}
The related work section covers generics in Go, type classes in Haskell, logical relations,
and a summary of our own prior work. At the end, we give an overview of the existing
translations with source language Featherweight Generic Go.
\subsection{Generics in Go}
\label{sec:generics-go}
The results of this work rest on the definition of Featherweight Generic Go (FGG)
provided by Griesemer and colleagues~\citeyearpar{FeatherweightGo}.
FGG is a minimal core calculus modeling the
essential features of the programming language Go \citeyearpar{golang}.
It includes support for overloaded methods, interface types, structural subtyping,
generics, and type assertions.
Our formalization of FGG ignores dynamic type assertions but otherwise sticks to
the original definition of FGG, apart from some minor cosmetic changes in
presentation. We prove that the type system implied by our
translation is equivalent to the original type system
of FGG, and that translated programs behave the same way as under the original
dynamic semantics.
The original dynamic semantics of FGG uses
runtime method lookup, in the same way as we did
in \Cref{sec:featherweight-generic-go}. The authors define an alternative
semantics via monomorphisation; that is, they specialize generic code for
all type arguments appearing in the program.
This alternative semantics is equivalent
to the one based on runtime method lookup, but there exist type-correct FGG programs
that cannot be monomorphized. Further, monomorphization
often leads to a blowup in code size. In contrast, our translation handles all
type-correct FGG programs, and instantiations of generic code with different
type arguments do not increase the code size. However, we expect that monomorphized code
will offer better performance than code generated by our dictionary-passing translation,
because method dictionaries imply several indirections not present in
monomorphized programs.
The current implementation of
generics~\citep{GoGenerics2021} in Go versions 1.18 and 1.19~\citeyearpar{golang}
differs significantly from the formalization in FGG.
For example, full Go requires a method declaration for a generic struct to have
exactly the same type bounds as the struct.
In FGG, bounds of the receiver struct in a method declaration
might be stricter than the bounds in
the corresponding struct declaration. In \Cref{f:trans-example-src}, we used this feature
to implement equality on the fully generic \go{Box} type, provided the
type parameter can be compared for equality. Go cannot express this
scenario without falling back to dynamic type assertions.
\cite{Ellis2022} formalize a dictionary-passing translating from a restricted
subset of FGG to FG. The restriction for FGG is the same as previously explained for
full Go: a method declaration must have the same type bounds as its receiver struct.
The translation utilizes this restriction to translate an FGG struct together with all its methods
into a single FG struct (dictionary). This approach would scale to full Go even with
separate compilation because a struct and all its methods must be
part of the same package. Further, the translation of Ellis and coworkers
replaces all types in method signatures
with the top-type \go{Any}, relying on dynamic type assertions to enable
typechecking of the resulting FG program. The authors provide a working implementation
and a benchmark suite to compare their translation against several other approaches,
including the current implementation of generics in full Go.
Our translation targets an extended $\lambda$-calculus and
does not restrict the type bounds of the receiver struct in a method declaration.
We also provide an implementation but no evaluation of its performance.
Method dictionaries bear some resemblance to virtual method tables
(vtables) used to implement virtual method dispatch in
object-oriented languages~\citep{DBLP:conf/oopsla/DriesenH96}.
The main difference between vtables and
dictionaries is that there is a fixed connection between an object and its
vtable (via the class of the object), whereas the connection between a
value and a dictionary may change at runtime, depending on the type
the value is used at. Dictionaries allow access to a method at a fixed
offset, whereas vtables in the presence of multiple inheritance require
a more sophisticated lookup algorithm~\citep{DBLP:conf/oopsla/AlpernCFGL01}.
A possible optimization to the dictionary-passing
translation is selective code specialization~\citep{conf/pldi/DeanCG95}. With this
approach, the dictionary-passing translation generates code
that runs for all type arguments. In addition, specialized code is generated
for frequently used combinations of type arguments. This approach allows
to trade code size against runtime performance. The GHC compiler for Haskell
supports a \texttt{SPECIALIZE} pragma~\citep[][\S\,6.20.11.]{GHC941}
that allows developers to specialize a polymorphic function to a particular type.
The specialization also supports type-class dictionaries.
\subsection{Type Classes in Haskell}
The dictionary-passing translation is well-studied in the context of
Haskell type classes
\citep{Wadler:1989:MAP:75277.75283,Hall:1996:TCH:227699.227700}.\footnote{%
Several points in the following discussion were already included
in own prior work \citep{SulzmannWehr-aplas2021,SulzmannWehr-mpc2022}.}
A type-class
constraint translates to an extra function parameter, constraint resolution
provides a dictionary with the methods of the type class for this parameter.
In FGG, structural subtyping relations imply coercions and bounded type parameters translate to coercion parameters.
An interface value pairs a struct value with a dictionary for the methods
of the interface.
Thus, interface values can be viewed as
representations of
existential types~\citep{DBLP:journals/toplas/MitchellP88,laeufer_1996,ThiemannWehr2008}.
Another important property in the type class context is coherence.
Bottu and coworkers~\citeyearpar{10.1145/3341695} make use of logical relations
to state equivalence among distinct target terms resulting
from the same source type class program.
Thanks to our main result (\Cref{thm:prog-equiv}), we get coherence for free.
We believe it is worthwhile to establish a property similar to this theorem for type classes.
We could employ a simple denotational semantics
for source type class programs
similar as \cite{DBLP:conf/lfp/Thatte94} or \cite{10.1145/2633357.2633364},
which would then be related to target programs obtained via the dictionary-passing translation.
\subsection{Logical Relations}
Logical relations have a long tradition of proving properties of
typed programming languages.
Such properties include termination~\citep{DBLP:journals/jsyml/Tait67,journals/iandc/Statman85},
type safety~\citep{Skorstengaard}, and program equivalence~\citep[Chapters~6,~7]{ATAPL}.
A logical relation (LR) is often defined inductively, indexed by type.
If its definition is based on an operational semantics, the LR is called
syntactic \citep{conf/icalp/Pitts98,journals/entcs/CraryH07}.
With recursive types, a step-index \citep{journals/toplas/AppelM01,10.1007/11693024_6}
provides a decreasing measure
to keep the definition well-founded.
See \citet[Chapter~8]{books/daglib/0085577} and \citet{Skorstengaard} for introductions to the topic.
LRs are often used to relate two terms of the same language. For our translation, the two terms
are from different languages, related at a type from the source language.
\citet{conf/icfp/BentonH09} prove correctness of compiler transformations.
They use a step-indexed LR
to relate a denotational semantics of the $\lambda$-calculus with recursion to configurations of a SECD-machine.
\citet{conf/popl/HurD11} build on this idea to show equivalence between
an expressive source language (polymorphic $\lambda$-calculus with references, existentials, and recursive types)
and assembly language. Their biorthogonal, step-indexed Kripke LR does not directly relate the two languages
but relies on abstract language specifications.
Our setting is different in that we consider a source language with support for overloading.
Besides structured data and functions, we need to cover recursive interface values.
This leads to some challenges to get the step index right \citep{SulzmannWehr-mpc2022}.
Simulation or bisimulation (see e.g. \citealt{journals/jacm/SumiiP07}) is another common technique for showing
program equivalences. In our setting, using this technique amounts to proving that reduction and translation
commutes:
if source term $e$ reduces to $e'$ and translates to target term $E$, then $e'$ translates
to $E'$ such that $E$ reduces to $E''$ (potentially in several steps) with $E' = E''$.
One challenge is that the two target
terms $E'$ and $E''$ are not necessarily syntactically equal but only semantically.
In our setting, this might be the case if $E'$ and $E''$ contain coercions for structural subtyping.
Even if such coercions behave the same, their syntax might be different.
With LR, we abstract away certain details of single step reductions, as we only compare values, not intermediate
results. A downside of the LR is that getting the step index right is sometimes not trivial.
\citet{journals/pacmpl/Paraskevopoulou21a} combine simulation
and an untyped, step-indexed LR \citep{conf/popl/AcarAB08}
to relate the translation of a reduced expression (the $E'$ from the preceding paragraph)
with the reduction result of the translated expression (the $E''$). They use this technique
to prove correctness of CPS transformations using small-step and big-step operational
semantics. Resource invariants connect the number of steps
a term and its translation might take, allowing them to prove that
divergence and asymptotic runtime is preserved by the transformation. Our LR
does not support resource invariants but includes a case for divergence directly.
\subsection{Prior Work}
Our own work published at APLAS \citep{SulzmannWehr-aplas2021} and MPC
\citep{SulzmannWehr-mpc2022} laid the foundations for the dictionary-passing
translation and its correctness proof of the present article.
For the APLAS paper, we defined a dictionary-passing translation for Featherweight Go
\citep[FG,][]{FeatherweightGo}, the non-generic variant of FGG. That translation
is similar in spirit to the translation presented here, it supports type assertions but not
generics. The APLAS paper includes a proof for
the semantic equivalence between the source FG program and its translation. The result is,
however, somewhat limited as semantic equivalence only holds for terminating programs
whose translation is also known to terminate.
In the MPC paper, we addressed the aforementioned limitation by extending the proof
of semantic equivalence to all possible outcomes of an FG program: termination,
panic (failure of a dynamic type assertion), and divergence. The proof
uses a logical relation similar to the one used here, but without support for generics.
We have already shown more differences in \Cref{sec:step-index}.
\subsection{Summary of Translations}
\begin{figure*}
\Fbox{
\begin{minipage}{\textwidth}
\begin{center}
\begin{tikzpicture}
\matrix [column sep=25mm, row sep=12mm] {
&
\node (pfg1) {$\PFG'$}; &
\node (ptl1) {$\PTL'$}; \\
\node (pfgg) {$\PFGG$}; &
\node (pfg) {$\PFG$}; &
\node (ptl) {$\PTL$}; \\
&
&
\node (ptl2) {$\PTL''$};
\\
};
\draw[->, thick] (pfg1) -- (ptl1) node[above,midway] {\DiagLabel{MPC 2022}};
\draw[->, thick] (pfgg) -- (pfg) node[above,midway] {\DiagLabel{OOPSLA 2020}};
\draw[->, thick] (pfg) -- (ptl) node[above,midway] {\DiagLabel{MPC 2022}};
\draw[->, thick] (pfgg) -- (ptl2) node[midway,sloped,below] {\DiagLabel{this article}};
\draw[->, thick] (pfgg) -- (pfg1) node[midway,sloped,above] {\DiagLabel{OOPSLA 2022}};
\end{tikzpicture}
\end{center}
\end{minipage}
}
\caption{Summary of translations. Arrows represent translations,
$P_{\ell}$ is a program in language $\ell$.
Program $\PFGG$ is subject to certain restrictions, depending on the translation being performed.
}
\label{f:summary-diag}
\end{figure*}
The diagram in \Cref{f:summary-diag} summarizes the existing translations
by \citet[OOPSLA][]{FeatherweightGo},
by \citet[OOPSLA][]{Ellis2022}, from our MPC 2022 paper
\citep{SulzmannWehr-mpc2022}, and from the article at hand.
The three resulting target language programs $\PTL$, $\PTL'$, and $\PTL''$ are
semantically equivalent because all translations preserve the dynamic semantics.\footnote{%
Strictly speaking, the MPC 2022 paper uses a target language slightly
different from the one presented in this article. The differences, however,
are straightforward to level out.}
Each translation with $\PFGG$ as its source has different restrictions.
OOPSLA 2022 requires the receiver struct of some method declaration to have exactly the same
type bounds as the struct declaration itself.
OOPSLA 2020 requires $\PFGG$ to be monomorphizable, checked by a simple syntactic condition.
The formal translation of this article does not support type
assertions, but we informally explained in \Cref{sec:discussion} how to extend
the translation in this direction.
\section{Type-directed translation}
\label{sec:type-direct-transl}
This section defines a type-directed, dictionary-passing translation from FGG to an untyped $\lambda$-calculus
extended with recursive let-bindings, constructors and pattern matching. We first introduce
the target language, then specify the translation itself, and last not least
give some examples. Formal properties of the translation
are deferred until~\Cref{sec:formal-properties}.
\subsection{Target Language}
\label{sec:target-language}
\boxfig{f:target-lang}{Target language (TL)}{
\vspace{-1ex}
\[\ba{ll}
{
\begin{array}[t]{lr@{~}l}
\sSynCatName{Variable} & X,Y & \in \TLVarSet
\\
\sSynCatName{Constructor} & K & \in \TLConsSet
\\
\sSynCatName{Expression} & E, G & ::=
X \mid K \mid E \ E \mid
\lambda X. E
\\ & & \phantom{::=}
\mid \tcaseof{E}{\ForeachN{\mathit{Cls}}}
\end{array}
}
&
{
\begin{array}[t]{lr@{~}l}
\sSynCatName{Pattern clause} &
\mathit{Cls} & ::= \mathit{Pat} \rightarrow E
\\
\sSynCatName{Pattern} &
\mathit{Pat} & ::= \kT \ \ForeachN{X}
\\
\sSynCatName{Program} &
\mathit{Prog} & ::=
\mbox{\kw{let}}\ \ForeachN{X = V} \ \mbox{\kw{in}}\ E
\end{array}
}
\ea\]
\[\ba{l@{\quad}r@{~}r@{~}l}
\sSynCatName{Value} &
\uT, U, W & ::= & \kT\ \ForeachN{\uT} \mid \lambda X . E
\\
\sSynCatName{Evaluation context} &
\REvCtxT & ::= & \Hole
\mid \tcaseof{\REvCtxT}{\Multi{\mathit{Cls}}}
\mid \REvCtxT\ E
\mid \uT\ \REvCtxT
\\
\sSynCatName{Substitution} &
\rho, \mu & ::= & \Angle{\ForeachN{\subst{X}{\uT}}}
\ea\]
\vspace{1ex}
\sTransSection{
\ruleform{
\reduceExpTL{\mu}{E}{E'}
}
}{
TL expression reductions
}
\vspace{-0.5ex}
\begin{mathpar}
\inferrule[tl-context]
{\reduceExpTL{\mu}{E}{E'}
}
{\reduceExpTL{\mu}{\REvCtxT [E]}{\REvCtxT [E']}
}
\inferrule[tl-lambda]
{}
{ \reduceExpTL{\mu}{(\lambda X. E) \ \uT}{\Angle{\subst{X}{\uT}} E}
}
\inferrule[tl-case]
{ \kT \ \Multi[n]{X} \rightarrow E \in \Multi{\mathit{Cls}}
}
{\reduceExpTL{\mu}{\tcaseof{\kT \ \Multi[n]{\uT}}{\Multi{\mathit{Cls}}}}
{\Angle{\Multi[n]{\subst{X}{\uT}}} E}
}
\inferrule[tl-method]{}{
\reduceExpTL{\mu}{X}{\mu(X)}
}
\end{mathpar}
\vspace{1ex}
\sTransSection{
\ruleform{\reduceTL{\mathit{Prog}}{\mathit{Prog}'}}
}{
TL reductions
}
\vspace{-0.5ex}
\begin{mathpar}
\inferrule[tl-prog]{
\mu = \Angle{\ForeachN{\subst{X}{V}}}\\
\reduceExpTL{\mu}{E}{E'}
}{
\reduceTL{\tletrecin{\ForeachN{X = V}}{E}}{\tletrecin{\ForeachN{X = V}}{E'}}
}
\end{mathpar}
} %
\Cref{f:target-lang} defines the syntax and the call-by-value dynamic semantics of the target language (TL).
We use uppercase letters for constructs of the target language.
Variables $X, Y$ and constructors $K$ are drawn from countably infinite, pairwise disjoint sets
$\TLVarSet$ and $\TLConsSet$, respectively.
Expressions, ranged over by $E$ and $G$, include variables $X$, constructors $K$, function applications $E\,E'$,
$\lambda$-abstractions $\lambda X.E$, and pattern matching via case-expressions
$\tcaseof{E}{\ForeachN{\mathit{Pat} \to E}}$. Patterns $\mathit{Pat}$ have the form $K\,\Multi{X}$, they do not nest.
We assume that all constructors
in $\Multi{\mathit{Pat}}$ are distinct. To avoid some parenthesis, we use the conventions that application
binds to the left and that the body of a $\lambda$ extends to the right as far as possible.
A program $\mbox{\kw{let}}\ \ForeachN{X = V} \ \mbox{\kw{in}}\ E$
consists of a sequence of (mutually recursive) definitions and a (main) expression,
where we assume that the variables $\Multi X$ are distinct.
In the translation from FGG, the values $\Multi V$ are always
functions resulting as translations of FGG methods.
We identify expressions, pattern clauses and programs up to renaming of bound variables.
Variables are bound by $\lambda$ expressions, patterns, and let-bindings of programs.
Some syntactic sugar simplifies the construction of patterns, expressions and programs.
(a) We use nested patterns to abbreviate nested case-expressions.
(b) We assume data constructors for tuples up to some fixed but arbitrary size. The syntax
$\Tuple{\Foreach{E}{n}}$ constructs an $n$-tuple when used as an expression, and
$\Tuple{\Multi[n]{\mathit{Pat}}}$ deconstructs it when used in a pattern context.
(c) We use patterns in $\lambda$-expressions; that is, the notation $\lambda \mathit{Pat}. E$ stands for
$\lambda X . \tcaseof{X}{\mathit{Pat} \rightarrow E}$ where $X$ is fresh.
Target values $V,U,W$ are either $\lambda$-expressions or constructors applied to values.
A constructor value $K\,\Foreach{V}{n}$ is short for $(\ldots (K\,V_1) \ldots)\,V_n$.
A call-by-value evaluation context $\REvCtxT$ is an expression with a hole $\Hole$ such that
the hole marks the point where the next evaluation step should happen.
We write $\REvCtxT[E]$ to denote the replacement of the hole in $\REvCtxT$ with expression $E$.
A substitution $\rho, \mu$ is a finite mapping $\Angle{\Multi{\subst{X}{V}}}$ from variables
to values. The variables $\Multi{X}$ in the domain must be distinct.
Substitution application, written in prefix notation
$\rho E$, is defined in the usual, capture-avoiding way.
We use two different meta variables $\mu$ and $\rho$ for substitutions in the target language
with the convention that
the domain of $\mu$ contains only top-level variables bound by $\mbox{\kw{let}}$.
As top-level variables result from translating FGG methods,
we sometimes call $\mu$ a \emph{method substitution}.
The reduction semantics for the target language is defined by two relations:
$\reduceExpTL{\mu}{E}{E'}$ reduces expression $E$ to $E'$ under method substitution $\mu$,
and $\reduceTL{\mathit{Prog}}{\mathit{Prog}'}$ reduces $\mathit{Prog}$ to $\mathit{Prog}'$.
The definition of the latter simply forms a method substitution $\mu$ from the top-level bindings
of $\mathit{Prog}$ and then reduces the main expression of $\mathit{Prog}$ under $\mu$ (rule \Rule{tl-prog}).
We defer the substitution of top-level--bound variables because they might be recursive.
The definition of the reduction relation for expressions extends over four rules.
Rule \Rule{tl-context} uses evaluation context $\REvCtxT$ to reduce inside an expression,
rule \Rule{tl-lambda} reduces function application in the usual way. Pattern matching
in rule \Rule{tl-case} assumes that the scrutinee is a constructor value $K\,\Multi[n]{V}$; the lookup
of a pattern clause matching $K$ yields at most one result as we assume that clauses
have distinct constructors. During a sequence of reduction steps, a variable bound by
$\mbox{\kw{let}}$ at the top-level might become a redex, as only $\lambda$-bound variables are substituted right away.
Thus, rule \Rule{tl-method} finds the value for the variable in the method substitution $\mu$.
\subsection{Example}
\label{sec:example-translation}
\boxfig{f:trans-example-src}{Example: FGG code (top) and its translation (middle)
with abbreviations (bottom)
}{
\vspace{-1ex}
\begin{flalign*}
\begin{array}{l@{}}
\mbox{\kw{type}}\ \Any\ \mbox{\kw{interface}}\ \{ \}\\
\mbox{\kw{type}}\ \Num\ \mbox{\kw{struct}}\ \{\, \val\ \IntT\, \}\\
\mbox{\kw{type}}\ \BoxT[\fgTyVar\, \Any]\ \mbox{\kw{struct}}\ \{\, \content\ \fgTyVar\, \}\\
\mbox{\kw{type}}\ \Eq[\fgTyVar\, \Any]\ \mbox{\kw{interface}}\ \{\, \eq(\that\,\fgTyVar)\,\bool\, \}\\
\mbox{\kw{func}}\ (\this \ \Num) \, \eq(\that\,\Num) \ \bool\ \{\,
\mbox{\kw{return}}\ \this.\val\,\mathtt{==}\,\that.\val \,\}
\\
\mbox{\kw{func}}\ (\this \ \BoxT[\fgTyVar\,\Eq[\fgTyVar]]) \,
\eq(\that\,\BoxT[\fgTyVar]) \ \bool\ \{\,
\mbox{\kw{return}}\ \this.\content.\eq(\that.\content) \,\}
\\
\mbox{\kw{func}}\ \main() \{\, \_ =
\BoxT[Num]\{\Num\{1\}\}.\eq(\BoxT[Num]\{\Num\{2\}\}) \,\}
\end{array}
&&%
\end{flalign*}
\hrule{}
\vspace{-1ex}
\begin{flalign*}
\begin{array}[t]{l@{~}r@{}l@{}}
\mbox{\kw{let}} &
X_{\eq,\Num} & {}=
\tllambda{\Quadr{\Void}{\This}{\Void}{\Tuple{\That}}} E_2 \,\mathtt{==}\, E_3
\\
& X_{\eq,\BoxT} & {}=
\tllambda{\Quadr{\Tuple{X_{\fgTyVar}}}{\This}{\Void}{\Tuple{\That}}} E_1
\\
\multicolumn{3}{l}{
\mbox{\kw{in}}~
X_{\eq,\BoxT}~\Quadr{\Tuple{V_3}}{\BoxVal{1}}{\Void}{\BoxVal{2}}
}
\end{array}
&&%
\end{flalign*}
\newcommand\Comment[1]{\multicolumn{2}{l@{}}{\small\color{GrayFgColor} \texttt{--}~#1}}
\par\noindent\hdashrule{\textwidth}{0.4pt}{1pt}
\begin{minipage}[t]{0.5\linewidth}
\vspace{-4ex}
\begin{flalign*}
\begin{array}[t]{r@{}l@{}}
\Comment{\textrm{translated body of~}\eq\textrm{~for~}\BoxT}
\\
E_1 & {}= \mbox{\kw{case}}\,V_1\,E_2\, \mbox{\kw{of}}\, \Pair{Y}{\Tuple{X_1}} \to
\\ & \qquad X_1~\Triple{Y}{\Void}{\Tuple{E_3}}
\\[\smallskipamount]
\Comment{\textrm{selectors for field content of~}\BoxT}
\\
E_2 & {}= \mbox{\kw{case}}\ \This\ \mbox{\kw{of}}\ \Tuple{X_1} \to X_1
\\
E_3 & {}= \mbox{\kw{case}}\ \That\ \mbox{\kw{of}}\ \Tuple{X_1} \to X_1
\end{array}
&&%
\end{flalign*}
\end{minipage}
\begin{minipage}[t]{0.5\linewidth}
\vspace{-4ex}
\begin{flalign*}
\begin{array}[t]{r@{}l@{}}
\Comment{\textrm{coercion}~\ensuremath{\subtypeOf{\fgTyVar}{\Eq[\fgTyVar]}}}
\\
V_1 & {}= \tllambda{Y} V_2\,(X_{\fgTyVar}\,Y)
\\[\smallskipamount]
\Comment{\textrm{identity coercion}~\ensuremath{\subtypeOf{\Eq[\fgTyVar]}{\Eq[\fgTyVar]}}}
\\
V_2 & {}= \tllambda{\Pair{Y}{\Tuple{X}}} \Pair{Y}{\Tuple{X}}
\\[\smallskipamount]
\Comment{\textrm{coercion}~\ensuremath{\subtypeOf{\Num}{\Eq[\Num]}}}
\\
V_3 & {}= \tllambda{X}\Pair{X}{\Tuple{\tllambda{\TripleY}{X_{\eq,\Num}\,\QuadrY{\Void}}}}
\end{array}
&&%
\end{flalign*}
\end{minipage}
}
We now give an example of the translation. The FGG code in the top part of \Cref{f:trans-example-src}
defines equality for numbers $\Num$ and for generic boxes $\BoxT[\fgTyVar\,\Any]$.
Interface $\Any$ defines no methods, it serves as an upper bound for otherwise unrestricted
type variables.
We take the liberty to assume a basic type $\IntT$ and an operator
$\mathtt{==}$ for equality.
Interface $\Eq[\fgTyVar]$ requires a method $\eq$
for comparing the receiver with a value of type $\fgTyVar$.
We provide implementations of $\eq$ for $\Num$ and $\BoxT[\fgTyVar]$. Comparing the content of
a box requires the F-bound $\Eq[\fgTyVar]$ \citep{Canning1989}.
The main function compares two boxes for equality.
\boxfig{f:trans-example}{Example: translation of the method declaration for $\BoxT$ and $\eq$}{
\begin{mathpar}
\inferrule*[rightstyle=\tiny\sc,leftstyle=\tiny\sc,right=method]{
\inferrule*[Right=call-iface]{
{\begin{array}[b]{@{}c@{}}
\inferrule*[Left=sub]{
\CircledText{1}\qquad
\inferrule*[Right=access]{
\ldots
}{
\tdExpTrans{\pair{\Delta}{\Gamma}}{\this.\content : \fgTyVar}{E_2}
}
}{
\tdExpTrans{\pair{\Delta}{\Gamma}}{\this.\content : \Eq[\fgTyVar]}{V_1\,E_2}
}
\end{array}
}
\quad
\inferrule*[Right=access]{
\ldots
}{
\tdExpTrans{\pair{\Delta}{\Gamma}}{\that.\content : \fgTyVar}{E_3}
}
\\
\methodSpecifications{\Eq[\fgTyVar]} = \eq(\that\,\fgTyVar)\bool
}{
\tdExpTrans{\pair{\Delta}{\Gamma}}{\this.\content.\eq(\that.\content) : \bool}{E_1}
}
\\
\Delta = \{ \fgTyVar : \Eq[\fgTyVar] \}\\
\Gamma = \{ \this : \BoxT[\fgTyVar], \that : \BoxT[\fgTyVar] \}
}{
\tdMethTrans{
\begin{array}[t]{@{}l@{}}
\mbox{\kw{func}}\ (\this \ \BoxT[\fgTyVar\,\Eq[\fgTyVar]]) \,
\eq(\that\,\BoxT[\fgTyVar]) \ \bool\ \{\\
\quad \mbox{\kw{return}}\ \this.\content.\eq(\that.\content) \\ \}
}
{X_{\eq,\BoxT} =
\tllambda{\Quadr{\Tuple{X_{\fgTyVar}}}{\This}{\Void}{\Tuple{\That}}} E_1
}
\end{array}
}
\end{mathpar}
\vspace{0.1cm}
\TransSectionCenter{
Subderivation~\CircledText{1}
}
\begin{mathpar}
\inferrule*[rightstyle=\sc\tiny,right=coerce-tyvar]{
\inferrule*[right=coerce-iface-iface]{
}{
\tdUpcast{\Delta}{\subtypeOf{\Eq[\fgTyVar]}{\Eq[\fgTyVar]}}{V_2}
}
\\
{
\begin{array}[b]{c}
(\fgTyVar : \Eq[\fgTyVar]) \in \Delta
\end{array}
}
}{
\tdUpcast{\Delta}{\subtypeOf{\fgTyVar}{\Eq[\fgTyVar]}}{V_1}
}
\end{mathpar}
} %
\boxfig{f:trans-example2}{Example: translation of the main function}{
\begin{mathpar}
\inferrule*[rightstyle=\sc\tiny,right=call-struct]{
\inferrule*[Right=methods-struct]{
\inferrule*[Right=type-inst-checked]{
\inferrule*[Right=coerce-struct-iface]{
\inferrule*[Right=methods-struct]{
\ldots
}{
\tdMethods{\eq(\That\,\Num)\,\bool}{\Void}{\EmptyFgTyEnv}{\Num}
}
}{
\tdUpcast{\EmptyFgTyEnv}{\subtypeOf{\Num}{\Eq[\Num]}}{V_3}
}
}{
\tdCheckSubst{\EmptyFgTyEnv}{\fgTyVar\,\Eq[\fgTyVar]}{\Num}{\LrAngle{\subst{\fgTyVar}{\Num}}}{\Tuple{V_3}}
}\\
\mbox{\kw{func}}\ (\this \ \BoxT[\fgTyVar\,\Eq[\fgTyVar]]) \, \eq(\that\,\BoxT[\fgTyVar]) \ \bool\,\ldots \in \fgDecls
}{
\tdMethods{\eq(\That\,\BoxT[\Num])\,\bool}{\Tuple{V_3}}{\EmptyFgTyEnv}{\BoxT[\Num]}
}
\\
\tdExpTransEmpty{\BoxT[\Num]\{\Num\{1\}\} : \BoxT[\Num]}{\Tuple{\Tuple{1}}}\\
\tdExpTransEmpty{\BoxT[\Num]\{\Num\{2\}\} : \BoxT[\Num]}{\Tuple{\Tuple{2}}}
}{
\begin{array}[t]{@{}l@{}}
\tdExpTransEmpty{\BoxT[\Num]\{\Num\{1\}\}.\eq(\BoxT[\Num]\{\Num\{2\}\}) : \bool\\ \qquad\qquad}{
X_{\eq,\BoxT}~\Quadr{\Tuple{V_3}}{\BoxVal{1}}{\Void}{\BoxVal{2}}
}
\end{array}
}
\end{mathpar}
} %
The middle part of the figure shows the translation of the FGG code, using abbreviations in the bottom part.
Variable $X_{\eq,\Num}$ holds the translation of the declaration of $\eq$ for $\Num$; it simply
compares $E_2$ (translation of $\this.val$) with $E_3$ (translation of $\that.\val$).
Remember that the translation of a method declaration takes a quadruple with coercions for the
bounded type parameters of the receiver, the receiver itself, coercions for the bounded type parameters of the
method, and the method arguments.
Here, $\Void$ is a tuple of size zero, corresponding to the non-existing type
parameters, $\Tuple{\That}$ denotes a tuple of size one, corresponding to the single argument
$\that$.
The translation of $\eq$ for $\BoxT$ is more involved. \Cref{f:trans-example} shows
its derivation.
We omit \enquote{obvious} premises
and some trivial details from the derivation trees.
Rule \Rule{call-iface} translates the body of the method. It coerces the
receiver to the interface type $\Eq[\fgTyVar]$ and then extracts the
method to be called via pattern matching, see $E_1$.
The construction of the coercion is done via
$\tdUpcast{\Delta}{\subtypeOf{\fgTyVar}{\Eq[\fgTyVar]}}{V_1}$,
see subderivation \CircledText{1}.
Coercion $V_1$ is slightly more complicated then necessary because the translation does not optimize
the identity coercion $V_2$.
Inside of $V_1$, we use $X_{\fgTyVar}$.
This variables denotes a coercion from $\fgTyVar$ to the representation of $\Eq[\fgTyVar]$;
it is bound by the $\lambda$-expression in the definition of $X_{\eq,\BoxT}$.
The translation of the main expression calls $X_{\eq,\BoxT}$ with appropriate
arguments, see \Cref{f:trans-example2} for the derivation.
The values $\Tuple{\Tuple{1}}$ and $\Tuple{\Tuple{2}}$ are nested tuples of size one,
representing numbers wrapped in $\Num$ and $\BoxT$ structs.
The method call of $\eq$ is translated by rule \Rule{call-struct},
relying on rule \Rule{methods-struct} to instantiate the type variable $\fgTyVar$
to $\Num$, as witnessed by the coercion $V_3$.
\subsection{Translation}
\label{sec:type-directed}
Before we dive into the technical details, we summarize our translation strategy.
\begin{description}
\item[Struct.] An FGG value at the type of a struct is represented in the TL as a
\emph{struct value}; that is, a tuple $\Tuple{\Multi[n]{V}}$ where $n$ is the number of
fields and $V_i$ represents the $i$-th field of the struct.
\item[Interface.] An FGG value at the type of an interface is represented in the TL as an
\emph{interface value}; that is a pair
$\Pair{V}{\mathcal{D}}$, where $V$ is a struct value realizing the interface
and $\mathcal D$ is a \emph{dictionary.}
\item[Dictionary.] A dictionary $\mathcal D$ for an interface with methods $\Multi{R}$
is a tuple $\Tuple{\Multi{V}}$ such that $V_i$ is a \emph{dictionary entry} for method $R_i$.
\item[Dictionary entry.] A dictionary entry for a method with signature
$R = m[\Multi{\fgTyVar\,\fgType_I}](\Multi{x\,\fgTypeAux})\fgTypeAux$
is a function accepting a triple:
(1) receiver, (2) tuple with coercions corresponding to the bounded type
parameters $\Multi{\fgTyVar\,\fgType_I}$ of the method,
(3) tuple for parameters $\Multi{x}$.
\item[Coercion.] A structural subtype relation $\subtypeOf{\fgType}{\fgTypeAux}$ implies a \emph{coercion function}
to transform the target representation of an FGG value at type $\fgType$
into a representation at type $\fgTypeAux$.
\item[Bounded type parameter.] A bounded type parameter $\fgTyVar\,\fgType_I$
becomes a coercion parameter $X_{\fgTyVar}$ transforming
the type supplied for $\fgTyVar$ to its bound $\fgType_I$.
At instantiation sites, coercions need to be inserted.
\item[Method declaration.]
A method declaration
$\mbox{\kw{func}}\ (x\,t_S[\Multi{\fgTyVar\,\fgType_I}])\,m[\Multi{\fgTyVar'\,\fgType_I'}](\Multi{x\,\fgTypeAux})\,\fgTypeAux\,\{\mbox{\kw{return}}\,e\}$
is represented as a top-level function $\mT{m}{t_S}$ accepting a quadruple:
(1) tuple with coercions corresponding to the bounded type parameters $\Multi{\fgTyVar\,\fgType_I}$
of the receiver,
(2) receiver $x$,
(3) tuple with coercions corresponding to bounded type parameters $\Multi{\fgTyVar'\,\fgType_I'}$
of the method,
(4) tuple for parameters~$\Multi{x}$.
\end{description}
In essence, the above is a more detailed description of the translation scheme motivated in \Cref{sec:examples}.
The only difference is that dictionary entries and translations of methods are now represented as uncurried functions.
For example, instead of the curried representation in Figure~\ref{f:fgg-format3}
\renewcommand\subColored{yes}
\begin{lstlisting}[name=hsexample,escapechar=@,language=myhaskell,mathescape=true]
formatWithSep$_{\sub{Pair}}$ (toFormat$_{\sub{T}}$, toFormat$_{\sub{U}}$) this toFormat$_{\sub{S}}$ x = ...
toFormatWithSep$_{\sub{Pair}}$ (toFormat$_{\sub{T}}$, toFormat$_{\sub{U}}$) p =
(p, formatWithSep$_{\sub{Pair}}$ (toFormat$_{\sub{T}}$, toFormat$_{\sub{U}}$))
\end{lstlisting}
our actual translation scheme uses uncurried functions, as in the following code:
\begin{lstlisting}[name=hsexample,escapechar=@,language=myhaskell,mathescape=true]
formatWithSep$_{\sub{Pair}}$ ((toFormat$_{\sub{T}}$, toFormat$_{\sub{U}}$), this, toFormat$_{\sub{S}}$, x) = ...
toFormatWithSep$_{\sub{Pair}}$ (toFormat$_{\sub{T}}$, toFormat$_{\sub{U}}$) p =
(p, \(this,locals,arg) ->
formatWithSep$_{\sub{Pair}}$ ((toFormat$_{\sub{T}}$, toFormat$_{\sub{U}}$),locals,arg))
\end{lstlisting}
\renewcommand\subColored{no}
Using an uncurried representation instead of a curried representation is just a matter taste.
As we have carried out the semantic equivalence proof initially based on the uncurried representation,
we stick to the uncurried representation from now on.
\subsubsection{Conventions and Notations}
The translation relies on three total, injective functions with pairwise disjoint ranges
for mapping
FGG names to TL variables. The first function
$\expvarSet \to \TLVarSet$ translates a FGG variable $x$
to a TL variable $X$. To avoid clutter, we do not spell out
the translation function explicitly but use the abbreviation that a lowercase
$x$ always translates into its uppercase counterpart $X$.
The second function $\tyvarSet \to \TLVarSet$ translates
an FGG type variable $\fgTyVar$ into a TL variable, abbreviated
$\xTy{\fgTyVar}$. The third function $\methodNameSet \times \structNameSet \to \TLVarSet$
gives us the TL variable $\mT{m}{t_S}$ representing the translation
of a method $m$ for struct $t_S$.
Here is a summary of the shorthand notations for name translation functions, where
$\methodName{R}$ denotes the name part of method signature $R$.
\begin{mathpar}
x \leadsto X
\quad\qquad
\fgTyVar \leadsto \xTy{\fgTyVar}
\quad\qquad
\inferrule[]{
m = \methodName{R}
}{
\mbox{\kw{func}}\ (x \ {t_S}[\Multi{\fgTyVar\,\fgType_I}]) \, R \, \{ \mbox{\kw{return}}\ e \} \leadsto
\mT{m}{t_S}
}
\end{mathpar}
The notation for translating names slightly differs from the approach used in
the examples of \Cref{sec:examples}.
For instance, the coercion \texttt{toFormat$_{\sub{T}}$} from \Cref{f:fgg-format3}
is now named $X_{\sub{T}}$ and
\texttt{formatWithSep$_{\sub{Pair}}$} becomes
$X_{\texttt{formatWithSep},\texttt{Pair}}$.
The notation of the formal translation stresses that
$X_{\sub{T}}$ and $X_{\texttt{formatWithSep},\texttt{Pair}}$ are variables of the target
language.
An FGG type environment $\Delta$ as a mapping $\{\Multi{\fgTyVar : \fgType_{I}}\}$
from type variables $\fgTyVar_i$ to their upper bounds $\fgType_{Ii}$.
An FGG value environment $\Gamma$ is a mapping
$\{\Multi{x : \fgType}\}$ from FGG variables $x_i$ to their types $\fgType_i$.
An environment may contain at most one binding for a type variable or variable.
We write $\emptyset$ for the empty environment, $\Dom{\cdot}$ for the domain
of an environment, and $\cup$ for the disjoint union of two environments.
The notation $\distinct{\Multi{\mathfrak s}}$ asserts that $\Multi{\mathfrak{s}}$ is a sequence of disjoint items.
We let $[n]$ denote the set $\{1,\ldots,n\}$.
In the following, we assume that the declarations $\fgDecls$ of the FGG program being translated
are implicitly available in all rules. This avoids the need for
threading the declarations through all translation rules.
\subsubsection{Auxiliary Judgments}
\label{sec:auxiliary-judgments}
\boxfig{f:trans-auxiliaries}{Auxiliary judgments for the translation}{
\sTransSection{
\ruleform{
\tdCheckSubst{\Delta}{\Multi{\fgTyVar\,\fgType_I}}{\Multi{\fgTypeAux}}{\fgTySubst}{V}
}
}{Instantiation of bounded type parameters}
\begin{mathpar}
\inferrule[type-inst-checked]{
\fgTySubst = \Angle{\Multi[n]{\subst{\fgTyVar}{\fgTypeAux}}} \\
\tdUpcast{\Delta}{\subtypeOf{\fgTypeAux_i}{\fgTySubst \fgType_{Ii}}}{V_i} \\
\noteForall{i \in [n]}
}{
\tdCheckSubst{\Delta}{\Multi[n]{\fgTyVar\,\fgType_I}}
{\Multi[n]{\fgTypeAux}}{\fgTySubst}{\Tuple{\Multi[n]{V}}}
}
\end{mathpar}
\sTransSection{
\ruleform{
\tdMethods{R}{V}{\Delta}{\fgType_S} \qquad
\methodSpecifications{\fgType_I} = \{ \ForeachN{R} \}
}
}{
Method access
}
\begin{mathpar}
\inferrule[methods-struct]{
\mbox{\kw{func}}\ (x \ {t_S}[\Multi{\fgTyVar\,\fgType_I}]) \, R \, \{ \mbox{\kw{return}}\ e \} \in \fgDecls\\
\tdCheckSubst{\Delta}{\Multi{\fgTyVar\,\fgType_I}}{\Multi{\fgTypeAux}}{\fgTySubst}{V}
}{
\tdMethods{\fgTySubst R}{V}{\Delta}{t_S[\Multi{\fgTypeAux}]}
}
\inferrule[methods-iface]{
\mbox{\kw{type}}\ t_I[\Multi{\fgTyVar\,\fgType_I}] \ \mbox{\kw{interface}}\ \{ \ForeachN{R} \} \in \ForeachN{D}\\
\fgTySubst = \Angle{\Multi{\subst{\fgTyVar}{\fgTypeAux}}}
}
{
\methodSpecifications{t_I[\Multi{\fgTypeAux}]} = \{ \fgTySubst\ForeachN{R} \}
}
\end{mathpar}
}
\Cref{f:trans-auxiliaries} defines some auxiliary judgments.
The judgment $\tdCheckSubst{\Delta}{\Multi{\fgTyVar\,\fgType_I}}{\Multi{\fgTypeAux}}{\fgTySubst}{V}$,
defined by rule \Rule{type-inst-checked},
constructs a type substitution $\fgTySubst = \MultiSubst{\fgTyVar}{\fgTypeAux}$ and checks
that the $\Multi{\fgTypeAux}$ conform to their upper bounds $\Multi{\fgType_I}$.
Thus, it needs a type environment $\Delta$
to give meaning to free type variables and it returns a tuple
$\Tuple{\Multi[n]{V}}$ of $\lambda$-abstractions such that each $V_i$ coerces the actual type
argument to its upper bound. The relevant
premise for checking upper bounds is
$\tdUpcast{\Delta}{\subtypeOf{\fgTypeAux_i}{\fgTySubst \fgType_{Ii}}}{V_i}$, which
asserts that $\fgTypeAux_i$ is a structural subtype of $\fgTySubst \fgType_{Ii}$ giving raise
to a coercion function $V_i$.
The judgment will be defined and explained in the next subsection.
The lower part of \Cref{f:trans-auxiliaries} defines two judgments for looking up
methods defined for a struct or interface type.
Judgment $\tdMethods{R}{V}{\Delta}{\fgType_S}$
states that method signature $R$ is available for struct type $\fgType_S$ under
type environment $\Delta$, see rule \Rule{methods-struct}.
The value $V$ is a tuple of coercion functions resulting
from checking the bounds of the receiver's type parameters.
Judgment $\methodSpecifications{\fgType_I} = \{ \ForeachN{R} \}$
states that the set of method signatures
available for interface type $\fgType_I$ is $\{ \Multi{R} \}$, see rule \Rule{methods-iface}.
In this rule, we form the substitution
$\Angle{\Multi{\subst{\fgTyVar}{\fgTypeAux}}}$ by implicitly assuming that
$\Multi{\fgTyVar}$ and $\Multi{\fgTypeAux}$ have the same length. From now on, we
implicitly use the convention that two sequences forming a substitution are of the same length.
\subsubsection{Translation of Structural Subtyping}
\label{sec:transl-struct-subtyp}
\boxfig{f:upcast}{Translation of structural subtyping}{
\sTransSection{
\ruleform{\tdUpcast{\Delta}{\subtypeOf{\fgType}{\fgTypeAux}}{V}}
}{
Translation of structural subtyping
}
\begin{mathpar}
\inferrule[coerce-tyvar]{
Y~\textrm{fresh}\\
(\fgTyVar : \fgTypeAux_I) \in \Delta\\
\tdUpcast{\Delta}{\subtypeOf{\fgTypeAux_I}{\fgType}}{V}
}{
\tdUpcast{\Delta}{\subtypeOf{\fgTyVar}{\fgType}}
{\tllambda{Y} V\, (\xTy{\fgTyVar}\, Y)}
}
\inferrule[coerce-struct-iface]{
X,\MultiThree{Y}~\textrm{fresh}\\
\mbox{\kw{type}}\ t_I[\Multi{\fgTyVar\,\fgType_I}] \ \mbox{\kw{interface}}\ \{ \Foreach{R}{n} \} \in \ForeachN{D}\\
\fgTySubst = \Angle{\Multi{\subst{\fgTyVar}{\fgType}}}\\
\tdMethods{\fgTySubst R_i}{V_i}{\Delta}{\fgType_S}\\
m_i = \methodName{R_i}\\
U_i = \tllambda{\TripleY} X_{m_i,t_S}\,\QuadrY{V_i}
\quad\noteForall{i \in [n]}
}{
\tdUpcast{\Delta}{\subtypeOf{\fgType_S}{t_I[\Multi{\fgType}]}}
{\tllambda{X} \Tuple{X\Comma \Tuple{\Multi[n]{U}}} }
}
\inferrule[coerce-iface-iface]{
Y,\Foreach{X}{n}~\textrm{fresh}\\
\mapPerm : [q] \to [n]~\textrm{total}\\
\mbox{\kw{type}}\ t_I[\Multi{\fgTyVar\,\fgType_I}] \ \mbox{\kw{interface}}\ \{ \Foreach{R}{n} \} \in \ForeachN{D} \\
\mbox{\kw{type}}\ u_I[\Multi{\fgTyVarAux\,\fgTypeAux_I}] \ \mbox{\kw{interface}}\ \{ \Foreach{R'}{q} \} \in \ForeachN{D} \\
\Angle{\Multi{\subst{\fgTyVarAux}{\fgTypeAux}}} R'_i =
\Angle{\Multi{\subst{\fgTyVar}{\fgType}}} R_{\mapPerm(i)} \quad\noteForall{i \in [q]}
}{
\tdUpcast{\Delta}{\subtypeOf{t_I[\Multi{\fgType}]}{u_I[\Multi{\fgTypeAux}]}}
{ \tllambda{\Tuple{Y\Comma \Tuple{\Foreach{X}{n}}}}
\Tuple{Y\Comma \Tuple{X_{\mapPerm(1)}\Comma \ldots\Comma X_{\mapPerm(q)}}}
}
}
\end{mathpar}
}%
\Cref{f:upcast} defines the relation
$\tdUpcast{\Delta}{\subtypeOf{\fgType}{\fgTypeAux}}{V}$
for asserting that $\fgType$ is a structural subtype of $\fgTypeAux$, yielding
a coercion function $V$ to convert the target representations of $\fgType$ to $\fgTypeAux$.
Rule \Rule{coerce-tyvar} covers the case of a type variable $\fgTyVar$.
The premise states that type bound $(\fgTyVar : \fgTypeAux_I)$ exists in the environment.
By convention, $X_{\fgTyVar}$ is the corresponding coercion function.
We further find that $\tdUpcast{\Delta}{\subtypeOf{\fgTypeAux_I}{\fgTypeAux}}{V}$.
Hence, we obtain the coercion function for $\subtypeOf{\fgTyVar}{\fgTypeAux}$
by composition of coercion functions $V$ and $X_{\fgTyVar}$.
Rule \Rule{coerce-struct-iface} covers structs.
The premise $\tdMethods{\fgTySubst R_i}{V_i}{\Delta}{\fgType_S}$
asserts that each method with name $\methodName{R_i}$ of interface $t_I$ is defined for $\fgType_S$. Value
$V_i$ is a tuple with coercion parameters corresponding to the bounds of the receiver's type parameters.
Thus, $U_i = \tllambda{\TripleY} X_{m_i,t_S}\,\QuadrY{V_i}$
is the dictionary entry for the method:
a function accepting receiver $Y_1$, coercion parameters $Y_2$ corresponding to
bounded type parameters of the method,
and the arguments $Y_3$. As written earlier, dictionary entries and top-level functions
$X_{m_i,t_S}$ are uncurried. Thus, we need to deconstruct the argument triple $\TripleY$
and construct a new quadruple $\QuadrY{V}$ for calling $X_{m_i,t_S}$.
Rule \Rule{coerce-iface-iface} covers interfaces.
The idea is to build a dictionary for $u_I$ from
the methods in the dictionary for $t_I$.
As $t_I[\Multi{\fgType}]$ has to be a structural subtype of $u_I[\Multi{\fgTypeAux}]$,
the former must declare all methods of the latter.
Thus, the premise of the rule
requires the total function $\pi$ to be chosen
in such a way that the $i$-th method of $t_I$ is the same as the
$\pi(i)$-th method of $u_I$. Then we use pattern matching to deconstruct
the dictionary of $t_I$ as
$\Tuple{\Multi[n]{X}}$ and construct the wanted dictionary for $u_I$ as
$\Triple{X_{\pi(1)}}{\ldots}{X_{\pi(q)}}$.
\subsubsection{Translation of Expressions}
\label{sec:transl-expr}
\boxfig{f:trans-exp1}{Translation of expressions}{
\sTransSection{
\ruleform{\tdExpTrans{\pair{\Delta}{\Gamma}}{e : \fgType}{E}}
}{
Translating expressions
}
\begin{mathpar}
\inferrule[var]{
(x : \fgType) \in \Gamma
}{
\tdExpTrans{\pair{\Delta}{\Gamma}}{x : \fgType}{X}
}
\inferrule[struct]{
\fgTyOk{\Delta}{t_S[\Multi{\fgType}]}\\
\mbox{\kw{type}}\ t_S[\Multi{\fgTyVar\,\fgType_I}] \ \mbox{\kw{struct}}\ \{ \Foreach{f \ \fgTypeAux}{n} \} \in \ForeachN{D}\\\\
\tdExpTrans{\pair{\Delta}{\Gamma}}{e_i : \Angle{\Multi{\subst{\fgTyVar}{\fgType}}}\fgTypeAux_i}{E_i}
\quad\noteForall{i \in [n]}
}{
\tdExpTrans{\pair{\Delta}{\Gamma}}
{t_S[\Multi{\fgType}] \{ \Foreach{e}{n} \} : t_S[\Multi{\fgType}] }{\Tuple{\Foreach{E}{n}}}
}
\inferrule[access]{
\tdExpTrans{\pair{\Delta}{\Gamma}}{e : t_S[\Multi{\fgType}]}{E} \\
\mbox{\kw{type}}\ t_S[\Multi{\fgTyVar\,\fgType_I}] \ \mbox{\kw{struct}}\ \{ \Foreach{f \ \fgTypeAux}{n} \} \in \ForeachN{D}
}{
\tdExpTrans{\pair{\Delta}{\Gamma}}
{e.f_i : \Angle{\Multi{\subst{\fgTyVar}{\fgType}}} \fgTypeAux_i }
{ \mbox{\kw{case}}\ E\ \mbox{\kw{of}}\ \Tuple{\Foreach{X}{n}} \rightarrow X_i}
}
\inferrule[call-struct]{
\tdExpTrans{\pair{\Delta}{\Gamma}}{e : t_S[\Multi{\fgType}]}{E} \\
\tdMethods{m[\Multi{\fgTyVar'\,\fgType_I'}](\Foreach{x\,\fgTypeAux}{n})\fgTypeAux}{V}{\Delta}{t_S[\Multi{\fgType}]}\\
\tdCheckSubst{\Delta}{\Multi{\fgTyVar'\,\fgType_I'}}{\Multi{\fgType'}}{\fgTySubst}{V'}\\
\tdExpTrans{\pair{\Delta}{\Gamma}}{e_i: \fgTySubst \fgTypeAux_i}{E_i}
\quad\noteForall{i \in [n]}
}{
\tdExpTrans{\pair{\Delta}{\Gamma}}{e.m[\Multi{\fgType'}](\Foreach{e}{n}) : \fgTySubst \fgTypeAux}
{\mT{m}{t_S}~\Quadr{V}{E}{V'}{\Tuple{\Foreach{E}{n}}} }
}
\inferrule[call-iface]{
\tdExpTrans{\pair{\Delta}{\Gamma}}{e : \fgType_I}{E}\\
\methodSpecifications{\fgType_I} = \Foreach{R}{q}\\
R_j = m[\Multi{\fgTyVar'\,\fgType_I'}](\Foreach{x\,\fgTypeAux}{n})\fgTypeAux \quad(\textrm{for some}~j \in [q])\\
\tdCheckSubst{\Delta}{\Multi{\fgTyVar'\,\fgType_I'}}{\Multi{\fgType'}}{\fgTySubst}{V'}\\
\tdExpTrans{\pair{\Delta}{\Gamma}}{e_i : \fgTySubst \fgTypeAux_i}{E_i}
\quad\noteForall{i \in [n]}\\
Y, \Foreach{X}{q}\textrm{~fresh}
}{
\tdExpTrans{\pair{\Delta}{\Gamma}}
{e.m[\Multi{\fgType'}](\Foreach{e}{n}) : \fgTySubst\fgTypeAux}
{\mbox{\kw{case}}\ E \ \mbox{\kw{of}}\ \Tuple{Y\Comma \Tuple{\Foreach{X}{q}}} \to X_j\Triple{Y}{V'}{\Tuple{\Foreach{E}{n}}}}
}
\inferrule[sub]{
\tdExpTrans{\pair{\Delta}{\Gamma}}{e : \fgType}{E}\\
\tdUpcast{\Delta}{\subtypeOf{\fgType}{\fgTypeAux}}{V}
}{
\tdExpTrans{\pair{\Delta}{\Gamma}}{e : \fgTypeAux}{V \ E}
}
\end{mathpar}
}
\Cref{f:trans-exp1} defines the typing and translation relation for expressions.
The judgment $\tdExpTrans{\pair{\Delta}{\Gamma}}{e : \fgType}{E}$ states that
under type environment $\Delta$ and value environment $\Gamma$ the FGG expression $e$
has type $\fgType$ and translates to TL expression $E$.
Rule \Rule{var} retrieves the type
of FGG variable $x$ from the environment and translates $x$ to its TL counterpart $X$.
The context makes variable $X$ available, see \Cref{sec:transl-meth-progr}.
Rule \Rule{struct} typechecks and translates a struct literal $t_S[\Multi{\fgType}](\Multi{e})$.
Premise $\fgTyOk{\Delta}{t_S[\Multi{\fgType}]}$ checks that type $t_S[\Multi{\fgType}]$ is well-formed;
the definition of the judgment $\fgTyOk{\Delta}{\fgType}$ is given in \Cref{f:well-formedness} and will be explained
in the next subsection. Each argument $e_i$ translates to $E_i$, so the result is
$\Tuple{\Multi[n]{E}}$.
Rule \Rule{access} deals with field access $e.f_i$, where expression $e$ must have
struct type $t_S[\Multi{\fgType}]$ such that $t_S$ defines field $f_i$. Thus, $e$ translates to a tuple $E$,
from which we extract the $i$-th component via pattern matching.
Rule \Rule{call-struct} handles a method call $e.m[\Multi{\fgType'}](\Multi{e})$, where
receiver $e$ has struct type $t_S[\Multi{\fgType}]$ and translates to $E$.
The $V$ in the premise corresponds to a tuple of coercion functions that
result from checking the bounds of the receiver's type parameters, whereas
$V'$ is a tuple of coercion functions for the bounds of the type parameters of the method.
Argument $e_i$ translates to $E_i$.
According to our translation strategy, a method declaration for $m$ and $t_S$ is
represented as a top-level function $X_{m,t_S}$ accepting a quadruple:
coercions for the receiver's type parameters,
receiver,
coercions for the bounded type parameters local to the method, and method arguments. Thus,
the result of the translation is
$\mT{m}{t_S}\ \Quadr{V}{E}{V'}{\Tuple{\Multi{E}}}$.
Rule \Rule{call-iface} handles a method call $e.m[\Multi{\fgType'}](\Multi{e})$, where
receiver $e$ has interface type $\fgType_I$ and translates to $E$.
Similar to \Rule{call-struct}, $V'$ is a tuple of coercion functions that result from checking the bounds of the
type parameters local to the method.
Expressions $E_i$ are the translation of the arguments $e_i$.
Following our translation strategy, receiver $E$ is a pair where the first component
is a struct value and the second component is a
dictionary for the interface. Thus, we use pattern matching to extract the struct as $Y$
and the wanted method as $X_j$. This $X_j$ is a function accepting a triple:
receiver, coercions for bounded type parameters of the method, and method arguments.
Hence, the translation result is $X_j\,\Triple{Y}{V'}{\Tuple{\Multi{E}}}$.
The difference to rule \Rule{call-struct} is that there is no need to supply coercions for the bounded
type parameters of the receiver.
These coercions have already been supplied when building the dictionary,
see rule \Rule{coerce-struct-iface} of \Cref{f:upcast}.
The last rule \Rule{sub} is a subsumption rule allowing an expression $e$
with translation $E$ at type $\fgType$ to be assigned some supertype $\fgTypeAux$.
Subsumption between a type and some of its supertypes is implicit in FGG
via structural subtyping, but
our translation makes such conversions explicit.
Thus, premise
$\tdUpcast{\Delta}{\subtypeOf{\fgType}{\fgTypeAux}}{V}$
serves two purposes: it ensures that $\fgTypeAux$ is a supertype of $\fgType$
and it yields a coercion function $V$ from
$\fgType$ to $\fgTypeAux$. The translation
of $e$ at type $\fgTypeAux$ is then $V\,E$.
\subsubsection{Well-formedness}
\label{sec:well-form-judgm}
\boxfig{f:well-formedness}{Well-formedness}{
\sTransSection{
\ruleform{
\fgTyOk{\Delta}{\fgType}\qquad
\fgTyOk{\Delta}{\Multi{\fgType}}
}
}{
Well-formedness of types
}
\begin{mathpar}
\inferrule[ok-tyvar]{
(\fgTyVar : \fgType_I) \in \Delta
}{
\fgTyOk{\Delta}{\fgTyVar}
}
\inferrule[ok-tynamed]{
\fgTyOk{\Delta}{\Multi{\fgType}}\\
\mbox{\kw{type}}\,t[\Multi{\fgTyVar\,\fgType_I}]\ldots \in \fgDecls\\\\
\tdCheckSubst{\Delta}{\Multi{\fgTyVar\,\fgType_I}}{\Multi{\fgType}}{\fgTySubst}{V}
}{
\fgTyOk{\Delta}{t[\Multi{\fgType}]}
}
\inferrule[ok-many-ty]{
\fgTyOk{\Delta}{\fgType_i} \quad (\forall i \in [n])
}{
\fgTyOk{\Delta}{\Multi[n]{\fgType}}
}
\end{mathpar}
\sTransSection{
\ruleform{
\fgFormalsOk{\Delta}{\Multi{\fgTyVar\,\fgType_I}}\qquad
\fgTyOk{\Delta}{R}
}
}{
Well-formedness of type parameters and method signatures
}
\begin{mathpar}
\inferrule[ok-bounded-typarams]{
\Dom{\Delta} \cap \{ \Multi{\fgTyVar} \} = \emptyset\\
\distinct{\Multi{\fgTyVar}}\\\\
\fgTyOk{\Delta \cup \{ \Multi{\fgTyVar : \fgType_I} \}}{\Multi{\fgType_I}}
}{
\fgFormalsOk{\Delta}{\Multi{\fgTyVar\,\fgType_I}}
}
\inferrule[ok-msig]{
\fgFormalsOk{\Delta}{\Multi{\fgTyVar\,\fgType_I}}\\
\distinct{\Multi{x}}\\\\
\fgTyOk{\Delta \cup \{ \Multi{\fgTyVar : \fgType_I} \}}{\Multi{\fgTypeAux}\fgTypeAux}
}{
\fgTyOk{\Delta}{m[\Multi{\fgTyVar\,\fgType_I}](\Multi{x\,\fgTypeAux})\fgTypeAux}
}
\end{mathpar}
\sTransSection{
\ruleform{
\fgDeclOk{D}
}
}{
Well-formedness of declarations
}
\begin{mathpar}
\inferrule[ok-struct]{
t_S \textrm{~defined once in~}\fgDecls\\
\fgFormalsOk{\EmptyFgEnv}{\Multi{\fgTyVar\,\fgType_I}}\\
\fgTyOk{\{ \Multi{\fgTyVar : \fgType_I} \}}{\Multi{\fgTypeAux}}\\
\distinct{\Multi{f}}
}{
\fgDeclOk{
\mbox{\kw{type}}\ t_S[\Multi{\fgTyVar\,\fgType_I}]\
\mbox{\kw{struct}}\ \{ \overline{f \ \fgTypeAux} \}
}
}
\inferrule[ok-iface]{
t_I \textrm{~defined once in~}\fgDecls\\
\fgFormalsOk{\EmptyFgEnv}{\Multi{\fgTyVar\,\fgType_I}}\\
(\forall i \in [n])~\fgTyOk{ \{\Multi{\fgTyVar : \fgType_I} \} }{R_i}\\\\
\textrm{distinct method names in}~\Multi{R}
}{
\fgDeclOk{
\mbox{\kw{type}}\ t_I[\Multi{\fgTyVar\,\fgType_I}]\
\mbox{\kw{interface}}\ \{ \Multi[n]{R} \}
}
}
\inferrule[ok-method]{
\ForeachN{D}~\textrm{contains one}~\mbox{\kw{func}}\textrm{-declaration for}~t_S\textrm{~and~}m\\
\fgFormalsOk{\EmptyFgEnv}{\Multi{\fgTyVar\,\fgType_I}}\\
\fgTyOk{ \{ \Multi{\fgTyVar : \fgType_I} \} }{R}\\
(\mbox{\kw{type}}\ t_S[\Multi[n]{\fgTyVar\,\fgType_I'}]\,\mbox{\kw{struct}}\ldots) \in \fgDecls\\
\methodSpecifications{{\fgType_I'}_i} \subseteq \methodSpecifications{{\fgType_I}_i}\quad(\forall i \in [n])
}{
\fgDeclOk{
\mbox{\kw{func}}\ (x \ t_S[\Multi[n]{\fgTyVar\,\fgType_I}]) R \ \{ \mbox{\kw{return}}\ e \}
}
}
\end{mathpar}
}
\Cref{f:well-formedness} defines several well-formedness judgments.
The judgments $\fgTyOk{\Delta}{\fgType}$ and $\fgTyOk{\Delta}{\Multi{\fgType}}$
assert that a single type and multiple types, respectively, are well-formed
under type environment $\Delta$. A type variable is well-formed if it's contained
in $\Delta$ (rule \Rule{ok-tyvar}). A named type $t[\Multi{\fgType}]$ is well-formed
if its type arguments are well-formed and if they are subtypes of the upper bounds
in the definition of $t$. The latter is checked by the premise
$\tdCheckSubst{\Delta}{\Multi{\fgTyVar\,\fgType_I}}{\Multi{\fgType}}{\fgTySubst}{V}$
of rule \Rule{ok-tynamed}. As discussed earlier, the $V$
is a tuple of coercion functions. We ignore these coercion functions as they do not have an operational
meaning. The type substitution $\fgTySubst$ is also ignored.
Thus, our translation demonstrates that bounds in struct and interface declarations could be eliminated from FGG.
We stick with them to stay close to the original presentation of FGG.
Judgment $\fgTyOk{\Delta}{\Multi{\fgTyVar\,\fgType_I}}$ asserts that bounded type parameters
$\Multi{\fgTyVar\,\fgType_I}$ are well-formed
under type environment $\Delta$ (rule \Rule{ok-bounded-typarams}).
Judgment $\fgTyOk{\Delta}{R}$
ensures that a method signature
is well-formed (rule \Rule{ok-msig}). To form the combined environment
$\Delta \cup \{\Multi{\fgTyVar\,\fgType_I}\}$ in the premise requires disjointness of the type variables in
$\Delta$ and $\Multi{\fgTyVar}$.
This can always be achieved by $\alpha$-renaming the
type variables bound by $R$.
Judgment $\fgDeclOk{D}$ validates declaration $D$.
A struct declaration is well-formed if it is defined only once
(restriction \Restriction{fgg-unique-structs}{} in \Cref{sec:dynamic-semantics}), if
all field names are distinct (restriction \FGGUniqueFields{}),
and if the field types are well-formed.
An interface declaration is well-formed if it is defined only once, if all its method signatures are well-formed, and if
all methods have distinct names.
A method declaration for $t_S$ and $m$ is well-formed if there is no other declaration
for $t_S$ and $m$ (restriction \FGGUniqueReceiver{}), if the method signature is well-formed,
and if each bound $\fgType_{Ii}$ of the method declaration is a structural subtype of the
corresponding bound $\fgType_{Ii}'$ in the declaration of $t_S$.
In FGG, this boils down to checking that the methods
of $\fgType_{Ii}'$ are a subset of the methods of $\fgType_{Ii}$.
The well-formedness conditions for method declarations
do not impose restrictions on the method body. We will deal with this in the upcoming translation
rule for methods.
\subsubsection{Translation of Methods and Programs}
\label{sec:transl-meth-progr}
\boxfig{f:trans-meth-prog}{Translation of methods and programs}{
\sTransSection{
\ruleform{\tdMethTrans{\mbox{\kw{func}}\ (x \ t_S[\Multi{\fgTyVar\,\fgType_I}])\,R\,\{\mbox{\kw{return}}\,e\}}{X = V} }
}{
Translating method declarations
}
\begin{mathpar}
\inferrule[method]{
\Delta = \{\Multi{\fgTyVar\,\fgType_I}, \Multi{\fgTyVarAux\,\fgTypeAux_I}\}\\
\Gamma = \{ x : t_S[\Multi{\fgTyVar}], \Multi{x : \fgTypeAux} \}\\
x \notin \{\Multi{x}\}\\
\tdExpTrans
{\pair{\Delta}{\Gamma}}
{e : \fgTypeAux}{E}
\\
V = \lambda \Quadr{\Tuple{\Multi{\xTy{\fgTyVar_i}}}}
{X}
{\Tuple{\Multi{\xTy{\fgTyVarAux_i}}}}
{\Tuple{\Multi{X}}}
. E
}{
\tdMethTrans
{\mbox{\kw{func}}\ (x \ t_S[\Multi{\fgTyVar\,\fgType_I}]) \,
m[\Multi{\fgTyVarAux\,\fgTypeAux_I}](\Multi{x \ \fgTypeAux})\, \fgTypeAux \ \{ \mbox{\kw{return}}\ e \}}
{\mT{m}{t_S} = V}
}
\end{mathpar}
\sTransSection{
\ruleform{\tdProgTrans{P}{\mathit{Prog}}}
}{
Translating programs
}
\begin{mathpar}
\inferrule[prog]{
\Multi{D} \textrm{~implicitly available in all subderivations}\\
\tdExpTrans{\pair{\emptyset}{\emptyset}}{e : \fgType}{E} \\\\
\fgDeclOk{D_i} \\
(\textrm{for all}~D_i \in \Multi{D})\\\\
\tdMethTrans{D_i}{X_i = V_i}\\
(\textrm{for all}~D_i = \mbox{\kw{func}}\,\ldots \in \Multi{D})\\
}
{ \turnsG{\mathsf{prog}}
{\ForeachN{D} \ \mbox{\kw{func}}\ \mbox{\mathem main} () \{ \_ = e \}} \leadsto
\mbox{\kw{let}}\ \ForeachN{X_i = V_i}\ \mbox{\kw{in}}\ E
}
\end{mathpar}
} %
\Cref{f:trans-meth-prog} defines the translation for method declarations and programs.
Rule \Rule{method} deals with method declarations
$\mbox{\kw{func}}\ (x \ t_S[\Multi{\fgTyVar\,\fgType_I}]) \, m[\Multi{\fgTyVar'\,\fgType_I'}](\Multi{x \ \fgTypeAux}) \, \fgTypeAux \ \{ \mbox{\kw{return}}\ e \}$.
The translation of such a declaration is the binding $\mT{m}{t_S} = V$.
According to our translation
strategy, $V$ must be a function accepting a quadruple:
coercions $\Tuple{\Multi{X_{\fgTyVar_i}}}$ for the bounded type parameters of the receiver,
receiver $X$ corresponding to $x$,
coercions $\Tuple{\Multi{X_{\fgTyVarAux_i}}}$ for the bounded type parameters local to the method,
and finally method arguments $\Multi{X}$ corresponding to $\Multi{x}$. Binding
all these variables with a $\lambda$ makes them available in the translated body $E$.
Judgment $\tdProgTrans{P}{\mathit{Prog}}$ denotes the translation of an FGG program $P$ to a TL program $\mathit{Prog}$.
Rule \Rule{prog}
typechecks the main expression $e$ under empty environments against some type
$\fgType$ to get its translation $E$. Next, the rule requires all struct or interface declarations
to be well-formed. Finally, it translates each method declarations to a binding $X_i = V_i$.
The resulting TL program is then
$\mbox{\kw{let}}\ \ForeachN{X_i = V_i}\ \mbox{\kw{in}}\ E$.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,845
|
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{
"redpajama_set_name": "RedPajamaC4"
}
| 7,104
|
This Letter to Debt Collector may be used by anybody who has been alleged to owe a debt. It may be sent to a debt collection agency, lender, creditor, or any other person or organisation that is alleging that a debt exists.
The letter may be used to dispute the debt entirely, or to assert that the debt is actually less than has been alleged. It may be used to ask for further information or documentation in relation to the alleged debt. It may be used to notify the recipient of the letter that the sender of the letter is experiencing financial hardship. It may also be used to propose a settlement, including an offer to pay a reduced amount, and a proposal regarding a payment schedule. The letter may even be used to notify the recipient that the sender is experiencing harassment from the debt collector or the debt collector's employees.
A person who is alleged to owe a debt can use this letter to get on the front foot with a debt collector, and to take control of any alleged debt.
This letter may be used to respond to a debt collection agency, lender, creditor, or any other person or organisation that is alleging that a debt exists.
If the debt collector does not have many contact details for the sender of the letter, then the sender may want to take care when preparing the letter, not to provide personal information that will help the debt collector to pursue the sender more aggressively. For example, if the debt collector does not know the sender's phone number, then the sender may want to be careful not to enter a phone number when preparing the letter.
Enter relevant information in the letter when prompted. If the sender wants to provide any further documentation with the letter (such as a medical report to verify that the sender is experiencing financial hardship), that documentation can be attached once the letter has been completed.
The letter and any relevant attachments may then be sent to the debt collector.
Importantly, this letter is not designed for use in response to formal court documents. For example, if the debt collector has commenced legal proceedings and has served formal court documents in relation to those proceedings, then it may be necessary to obtain assistance from a lawyer in order to deal with those court documents and to follow court procedures in order to respond.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 623
|
LCMT2 () – білок, який кодується однойменним геном, розташованим у людей на короткому плечі 15-ї хромосоми. Довжина поліпептидного ланцюга білка становить 686 амінокислот, а молекулярна маса — 75 602.
Кодований геном білок за функціями належить до трансфераз, метилтрансфераз.
Задіяний у такому біологічному процесі, як процесинг тРНК.
Білок має сайт для зв'язування з S-аденозил-L-метіоніном.
Література
Примітки
Див. також
Хромосома 15
Трансферази
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,091
|
Q: How to select all text in TextBoxs using Caliburn.Micro? In my WPF application I use Caliburn.Micro to bind between View and ViewModel.
In my View I have StackPanel with Text box
<StackPanel>
<TextBox x:Name="Ctx_A" />
<TextBox x:Name="Ctx_B" />
<TextBox x:Name="Ctx_C" />
<TextBox x:Name="Ctx_D" />
</StackPanel>
Now I want to select all text after focus on TextBox. In my ViewModel I write method TextBoxGotFocus
public void TextBoxGotFocus(object sender)
{
var tb = sender as TextBox;
if (tb != null)
{
tb.SelectAll();
}
}
Next I go back to my View and write something like that:
<StackPanel>
<TextBox x:Name="Ctx_A" cal:Message.Attach=" [Event GotFocus]=[Action TextBoxGotFocus($source)]"/>
<TextBox x:Name="Ctx_B" cal:Message.Attach=" [Event GotFocus]=[Action TextBoxGotFocus($source)]"/>
<TextBox x:Name="Ctx_C" cal:Message.Attach=" [Event GotFocus]=[Action TextBoxGotFocus($source)]"/>
<TextBox x:Name="Ctx_D" cal:Message.Attach=" [Event GotFocus]=[Action TextBoxGotFocus($source)]"/>
</StackPanel>
Is there any possible to attach this method to StackPanel instead TextBox?
A: I would remove the TextBoxGotFocus() method from your view model, because it's blurring the lines between your view and your view model. View models shouldn't be using any WPF controls in them. Instead, the easiest way I can think of to do this is to create a class that inherits from TextBox, and then attach an event to that:
public class AutofocusTextBox : TextBox
{
public AutofocusTextBox()
{
GotFocus += (sender, e) => SelectAll();
}
}
Then in your XAML, you can just use it like this:
<local:AutofocusTextBox ... />
That will keep your XAML a lot cleaner
A: All Credit goes to Dr. WPF... Source
in the abcView.xaml
<TextBox Text="{Binding Amount}"
MinWidth="100"
cal:Message.Attach="[Event GotFocus] = [Action GotFocusMethod($source)];
[Event PreviewMouseLeftButtonDown] = [Action SelectivelyIgnoreMouseButton($source, $eventArgs)]" />
in the abcViewModel.cs
// called by ui... used to highlight a textbox on focus
public void GotFocusMethod(object source)
{
var based = source as TextBox;
based.SelectAll();
} // close gotfocusmethod
// called by ui... used to highlight a textbox on focus
public void SelectivelyIgnoreMouseButton(object sender, MouseButtonEventArgs eve)
{
TextBox tb = (sender as TextBox);
if (tb != null)
{
if (!tb.IsKeyboardFocusWithin)
{
eve.Handled = true;
tb.Focus();
}
}
} // close selectivelyignoremousebutton()
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,884
|
\section{introduction}
\label{sec:introduction}
After the first data release of the cosmic microwave background (CMB)
temperature fluctuations observed by the Wilkinson Microwave
Anisotropy Probe (WMAP), anomalous signatures in the CMB on large
angular scales, so called ``large-angle anomalies''
have been claimed by many authors
\citep{RalstonJain:04,deOliveiraetal:04,Hansenetal:04,Hajianetal:05,
Moffat:05,LandMagueijo:06,Bernuietal:06,Copietal:07,Eriksenetal:07,
Monteserinetal:08,Samaletal:09}
and also confirmed recently by the Planck mission \citep{PlanckXXIII:13}.
So far, the origin has been veiled
in mystery. They may be due to
(a) difference in a priori significance and a posteriori significance
\citep{AurichLustig:10,PontzenPeiris:10,Efstathiouetal:10,Bennettetal:11}
(b) incomplete subtraction of
foreground emissions
\citep{Abramoetal:06,Cruzetal:11,Hansenetal:12}
(c) contribution from large scale structures via the integrated Sachs-Wolfe effect
\citep{InoueSilk:06,InoueSilk:07,Rassatetal:07,Afsholdietal:09,FP:10,Rassatetal:13,
TomitaInoue:08, SakaiInoue:08,Inoue:12,PlanckXXIII:13}, or kinetic Sunyaev-Zel'dovich
effect \citep{Peiris:10},
(d) possible systematics from instruments \citep{Hansonetal:10}
(e) incomplete treatment of masking \citep{Kimetal:12}, or
(f) extensions of inflationary models
\citep{Aurichetal:07,Gumrukcuogluetal:07,Rodrigues:08,Bernuietal:08,
Cruzetal:08,Fialkov:10,ZhengBunn:10,Liuetal:13}.
In real observation, we cannot observe the whole sky with a
sufficiently high signal-to-noise (S/N) ratio. A conservative
approach is to mask out the region (e.g., the Zone of Avoidance),
where the S/N ratio is low and just ignore the data on it.
For estimating the power spectrum of fluctuations in the sky, we can
use deconvolution techniques \citep[e.g.][]{Hivon:2002} assuming an
isotropy prior on the power spectrum. However, for
estimating the density field itself, we need to
develop methods that can reconstruct the phases (if expressed in
complex numbers) as well as the amplitudes of missing fluctuations.
In order to reconstruct missing data on the masked region, we need to
find the inverse of the masking operator.
However, in general, the masking matrix is singular
thus not invertible. We need to assume a certain prior information about the
underlying data such as isotropy and smoothness
\citep{Abrialetal:08,Kimetal:12,BucherLouis:12,Starcketal:13} or their
derivatives \citep{Inoueetal:08} in order to regularize the inverse operator.
Since the result depends on the choice of the prior, the robustness of each reconstruction method
should be mutually checked.
In this paper, we propose a new method for regularizing the inverse of
masking operators using the iterative spherical harmonic expansion
(IHE). As our aim is to probe the origin of the large-angle CMB
anomalies, we ignore the effect of noise in the signal in the
following, which is a good approximation for large-angle fluctuations.
This paper is organized as follows.
In section \ref{sec:iteration}, we describe the formulation of our
reconstruction method using the iterative spherical harmonic expansion
and
our simulations for random Gaussian fluctuations.
In section \ref{sec:result}, we show the
accuracy of the reconstruction, which depends on the pixelization of the
sky, the underlying power spectrum, and the number of iteration.
And then we compare the accuracy of
our method to that of the brute-force and the singular value decomposition (SVD)
method. In section \ref{sec:summary}, we give our conclusion.
\section{Iterative harmonic expansion}
\label{sec:iteration}
\subsection{Theory}
\label{ssec:theory}
\begin{figure*}[tp]
\begin{tabular}{ccc}
\includegraphics[width=0.32\linewidth]{map_masked.eps}&
\includegraphics[width=0.32\linewidth]{map_naive.eps} &
\includegraphics[width=0.32\linewidth]{map_diff_naive.eps} \\
(a) Original map with mask &
(c) Pseudo-$a_{\ell m}$ reconstruction &
(e) Pseudo-$a_{\ell m}$ residual \\
\includegraphics[width=0.32\linewidth]{map_smoothed.eps} &
\includegraphics[width=0.32\linewidth]{map_rec.eps} &
\includegraphics[width=0.32\linewidth]{map_diff_rec.eps} \\
(b) Original smoothed map without mask &
(d) IHE reconstruction &
(f) IHE residual \\
\end{tabular}
\caption{Plots of an original and reconstructed maps.
(a) Original simulated map with $\ell_{\rm cut}=30$ and a sky cut
$|b|<20^\circ$, which is used as an input.
(b) Original simulated map smoothed with $\ell_{\rm cut}=7$ on a full sky (no mask).
(c) Reconstructed map with $\ell_{\max}=7$ using pseudo-$a_{i}$.
(d) Reconstructed map with $\ell_{\max}=7$ using IHE with $N_{\rm ite}=10$ iterations.
(e) Residual map after subtraction:(b)-(c).
(f) Residual map after subtraction:(d)-(b).
Color scale (-1.87 to 1.55) is the same for all the figures.
\label{fig:example}
}
\end{figure*}
Suppose a density field $\delta(\uv{\gamma})$ on a
unit sphere where $\uv{\gamma}$ represents a unit vector that specifies
the angular position in the sky. The observed field is given by
\begin{align}
\delta_{\rm obs}(\uv{\gamma})=W(\uv{\gamma})\delta(\uv{\gamma}),
\end{align}
where $W=0$ in the masked region and $W=1$ elsewhere.
Then we can obtain the pseudo harmonic coefficient $\tilde{a}_i$
by simply integrating over the whole sky,
\begin{align}
\label{eq:pseudo-init}
\tilde{a}_{i}
&=
\int d\Omega~ \delta(\uv{\gamma}) W(\uv{\gamma}) Y^*_{i}(\uv{\gamma})
\end{align}
where $W_{ij}=\int d\Omega~ W Y_{j} Y_{i}^*$ is
the $(i,j)$ component of the mode coupling matrix
due to incomplete sky coverage, $Y_i(\uv{\gamma})$ is a
spherical harmonic,
and $d\Omega$ is the surface element on a unit sphere.
For simplicity, we use the single subscript index
$i = \ell^2 + \ell + m + 1$ where $\ell$ and $m$ represent a
multipole moment and an azimuthal number, respectively.
In terms of true harmonic coefficients to be reconstructed $a_{j}^{\rm
true}$, Eq.~(\ref{eq:pseudo-init}) can be written as
\begin{align}
\label{eq:pseudo-wlm}
\tilde{a}_{i} = \sum_j^{\infty} ~ a_{j}^{\rm true} W_{ij}.
\end{align}
The mode coupling matrix $\bvec{W}$ is explicitly given as
\cite{Hivon:2002}
\begin{align}
\label{eq:wij-pixel}
W_{i_1i_2}
&=
\sum_{i_3}w_{i_3}
\int d\Omega~ Y_{i_3}^{*} Y_{i_2} Y_{i_1} \nonumber \\
&=
\sum_{i_3}w_{i_3}
(-1)^{m_2}
\left[
\frac{(2\ell_1+1)(2\ell_2+1)(2\ell_3+1)}{4\pi}
\right]^{1/2} \nonumber \\
& \hspace{0.5cm}\times
\left(
\begin{array}{ccc}
\ell_1&\ell_2&\ell_3 \\
0 &0 &0
\end{array}
\right)
\left(
\begin{array}{ccc}
\ell_1&\ell_2&\ell_3\\
m_1 &-m_2 &m_3
\end{array}
\right),
\end{align}
where $w_{i}=\int d\Omega~ W Y^*_{i}$ and $(:::)$ denotes the Wigner 3$j$-symbols.
Note that the subscript $i_j$ depends only on $\ell_j$ and $m_j$,
i.e. $i_j=\ell_j^2+\ell_j+m_j+1$.
As shown in Eq.~(\ref{eq:pseudo-wlm}), $a_{i}^{\rm true}$'s can be
obtained by inverting the mode coupling matrix $\bvec{W}$. However, in
general, it can be singular and not invertible.
In order to regularize the inversion, we propose the
following iterative scheme.
The iteration process starts from
\begin{align}
\label{eq:iteration3}
\tilde{a}_{i}^{(1)}
&=
\int d\Omega~ \delta(\uv{\gamma}) W(\uv{\gamma}) Y_{i}^{*}(\uv{\gamma}),
\end{align}
which we call ``the pseudo-$a_i$'s''.
For $m>1$, the $m$-th set of $\tilde{a}_{i}$'s is constructed iteratively
from two maps;
the original observed map at the unmasked region and the map reconstructed from
the inverse transform of the $(m-1)$-th $\tilde{a}_{i}$'s at the masked
region,
\begin{align}
\label{eq:iteration1}
\tilde{a}_{i}^{(m)}
&=
\int d\Omega~ \left[
\delta(\uv{\gamma}) W(\uv{\gamma})
+ \tilde{\delta}^{(m-1)}(\uv{\gamma}) S(\uv{\gamma})
\right] Y_{i}^{*}(\uv{\gamma}),
\end{align}
where
\begin{align}
\label{eq:iteration2}
\tilde{\delta}^{(m-1)}(\uv{\gamma})
&=
\left\{
\begin{array}{ll}
\displaystyle{\sum_i^{\infty}} ~ \tilde{a}_{i}^{(m-1)}Y_{i}(\uv{\gamma}) & m\ge 2, \\
0 & m= 1, \\
\end{array}
\right.
\end{align}
and $S=1-W$. In what follows, we assume that fluctuations whose
angular scales are larger than that of the masked region do not
significantly correlate with fluctuations
smaller than the masked region. In that case, the summation in
Eq.~(\ref{eq:pseudo-wlm}) and Eq.~(\ref{eq:iteration2}) can be
truncated at a certain multipole $i_{\rm max}$ as long as we concern
large-angle fluctuations corresponding to multipoles
$i\ll i_{\rm max}$. In what follows, we sum up $a_i$'s up to the multipole
$i_{\rm imax}$, and we omit the summation symbol when no confusion arises.
Substituting Eq.~(\ref{eq:iteration2}) into
(\ref{eq:iteration1}) recursively, we obtain the general formula of the $m$-th
iterated harmonic coefficients as a series of $S_{ij}$,
\begin{equation}
\label{eq:neuman_series}
\tilde{a}_{i}^{(m)}
=
\tilde{a}_{j}^{(1)}
(
\delta^{K}_{ij}+S_{ij}+S_{ik}S_{kj}+\cdots+[S^{m-1}]_{ij}
),
\end{equation}
where $\delta^{K}_{ij}$ is the Kronecker delta, and
$S_{ij}=\delta^{K}_{ij}-W_{ij}$.
The map between $m$-th iterated harmonic coefficient
$\tilde{a}_{i}^{(m)}$ is equivalent to the product of $\tilde{a}_{i}^{(1)}$'s
and the Taylor expansion of the inverse of the mode coupling matrix $\bvec{W}$
truncated at the $(m-1)$-th order,
\begin{equation}
\label{eq:direct_inv}
\tilde{a}_{i}^{(m)}
\simeq
\tilde{a}_{j}^{(1)} [ (\delta^{K}-S)^{-1} ]_{ij}
=
\tilde{a}_{j}^{(1)} [W^{-1}]_{ij},
\end{equation}
Thus $\tilde{a}_{i}^{(m)}$'s approximately represent
fluctuations in which the mode coupling matrix $\bvec{W}$ is
deconvolved. The number of iteration
$N_{\rm ite}$ should be evaluated using Monte-Carlo
simulations, which is discussed later in Sec.\ref{sec:result}.
We can regard the above process as a mapping
from a observable to the estimator,
\begin{equation}
F_{\rm IHE}: \tilde{a}_i \rightarrow \tilde{a}^{(m)}.
\label{eq:mapping}
\end{equation}
As we will see in Sec.~\ref{sec:result}, the accuracy of the IHE method
depends on the maximum multipole $\ell_{\rm max}$ to be reconstructed,
the spectrum of the underlying
field $\delta$, the mask geometry, and the number of iteration
$N_{\rm ite}$.
Thus we can write
$F_{\rm IHE}=F_{\rm IHE}(\ell_{\rm max}, \delta, \bvec{W},N_{\rm ite})$.
Note that the computation time for estimating $\tilde{a}^{(N_{\rm ite})}$ can be significantly reduced
if we use Eq.~(\ref{eq:iteration1}) instead of
Eq.~(\ref{eq:neuman_series}).
The reason is as follows. As the rank of the matrix
$S_{ij}$ is of the order $\ell_{\rm max}^2$, it costs
$N_{\rm pix}\ell_{\rm max}^4$ by Eq.~(\ref{eq:wij-pixel}), and
matrix algebra of Eq.~(\ref{eq:neuman_series}) costs
$\sim N_{\rm ite} \ell_{\rm max}^6$ computations.
If one uses Eq.~(\ref{eq:iteration1}), it would be of the order
$N_{\rm ite} N_{\rm pix}\ell_{\rm max}^2$, where $N_{\rm ite}$ is the
number of iteration, and $N_{\rm pix}$ is the number of pixels which tile
the sky, and $\ell_{\rm max}$ is the maximum multipole to be reconstructed.
For example,
for given $\ell_{\rm max}=10$, $N_{\rm pix}=12\times 1024^2$ pixels
and 5 time iteration, the computation time will be reduced by a factor
of $\sim 25$.
\subsection{Simulation}
\label{ssec:data}
In order to assess the accuracy of reconstruction, we generate 1000
random isotropic Gaussian density fields in the sky. We use the code
{\it synfast}, which is publicly available as a package in Healpix
\footnote{http://healpix.jpl.nasa.gov/} \cite{Gorski:2005} for
generating random Gaussian maps. First, we use the
Harrison-Zel'dovich spectrum as the input power spectrum. It gives an
angular power spectrum $C_{\ell} \propto l^{n}$ where $n=-2$ on large
angular scales, in the Einstein de-Sitter universe which correspond to
the Sachs-Wolfe plateau of the CMB power spectrum. As a simple model
of the zone of avoidance, we consider an azimuthally symmetric mask
$W(\uv{\gamma})=0$ at the Galactic latitude $b$ smaller than 20 degrees,
i.e. $|b|<20^\circ$ and $W=1$ otherwise. We use pixels that are
sufficiently smaller than the size of the mask and the fluctuation
scales to be reconstructed in order to reduce the errors due to the
pixelization effect (see Sec. \ref{ssec:pixel}). We adopt the Healpix
resolution $N_{\textrm{side}}= 512$ (the total number of pixels on the
whole sky is $N_{\textrm{pix}} = 12\times
N_{\textrm{side}}^2=3145728$). The input power spectrum should be
truncated at sufficiently small scales $\ell_{\rm cut}$ compared to
the scale to be reconstructed $\ell_{\rm max}$, i.e. $\ell_{\rm max}
\ll \ell_{\rm cut}$. The cut-off scale $\ell_{\rm cut}$ is inferred
from the detector resolution scale in CMB observations or the shot
noise dominant scales in galaxy surveys. In our simulated maps, we
set $\ell_{\rm cut}=30$, which is sufficiently smaller than the
claimed anomalous multipoles in the CMB. We find that the choice of
$\ell_{\rm cut}$ is not sensitive to our result.
\section{Result}
\label{sec:result}
In this section, we first explore the pixelization effect, which may
affect the accuracy of reconstruction in Sec.~\ref{ssec:pixel} and
then we compare our reconstruction method with other inversion methods
using the brute-force or the singular value decomposition (SVD) in
Sec.~\ref{ssec:svd}. Finally, we discuss the condition of the original
maps that can be reconstructed by using the IHE in
Sec.\ref{ssec:dependence}.
\subsection{Pixelization effect}
\label{ssec:pixel}
\begin{figure}[tp]
\includegraphics[width=\linewidth]{plot2.eps}
\caption{
The accuracy of map reconstruction averaged over 1000
realizations of Gaussian maps as a function of
the iteration number
for a mask $|b|<20^\circ$ with $\ell_{\rm max}=7$ for various
pixel resolution $N_{\textrm{side}}$.
Both solid and dashed lines are obtained by using
Eqs.~(\ref{eq:iteration1}) and (\ref{eq:iteration2}).
Dashed lines represent values for which approximated
$S_{ij}$'s for pixel-based masks are used. Solid lines show
values for which the
exact expression of Eq.~(\ref{eq:wij-nopixel}) is used.
Dotted lines indicate values obtained by using
the direct inversion of the mode coupling matrix $\bvec{W}$,
the second equation in Eq.~(\ref{eq:direct_inv}).
In the bottom panel, we show the relative differences
of the accuracy in percentage.
$N_{\rm side}=1024$ provides accuracy better
than a few percent. $N_{\rm ite}\sim 10$ with $N_{\rm side}=512$ can achieve
sub-percent accuracy.
\label{fig:pixel}
}
\end{figure}
We need to assess the accuracy of the mode coupling matrix $\bvec{W}$
since it may be close to being singular. In other words, the
difference between $W(\uv{\gamma})$ and $W(\uv{\gamma}_i)$ should be carefully
checked where $\uv{\gamma}$ and $\uv{\gamma}_i$ are unpixelized and pixelized
positions on a unit sphere, respectively.
In order to do so, we need to estimate uncertainties due to
pixelization of the sky as it could be potential sources of errors.
For an azimuthally symmetric mask $W=0$ for $|b|\le b_0$, we do not
need pixelization, since $W_{ij}$ can be analytically calculated as
\begin{equation}
\label{eq:wij-nopixel}
W_{ij}
=
2 \pi \left(
1 - \int_{\pi/2-b_0}^{\pi/2+b_0}
Y_{i}(\theta,0)
Y_{j}(\theta,0)
\sin\theta
d\theta
\right),
\end{equation}
where $\theta$ is the polar angle, and $\theta = \pi/2 - b$. We
compare the eigenvalues of $\bs{W}$ obtained by using
Eq.~(\ref{eq:wij-pixel}) and Eq.~(\ref{eq:wij-nopixel}) to study the
accuracy of the calculated matrix. The difference depends on the
resolution of pixels $N_{\rm side}$. We adopt $N_{\rm side}=16, 128,
512, 1024$ and find that the differences in the eigenvalues are less
than $0.01 \%$ for $N_{\rm side}\geq 512$. In order to evaluate the
accuracy of map reconstruction, we use the L$^2$ norm
of the fractional difference between the reconstructed map and the original map,
\begin{align}
\label{eq:l2norm_l}
D^2_{\ell}
\equiv
\frac{\sum_{i}^{N{\rm pix}} [\delta_{\rm rec}(\uv{\gamma}_i;\ell)-
\delta_{\rm true}(\uv{\gamma}_i;\ell)]^2}
{\sum_{i}^{N{\rm pix}} \delta_{\rm true}^2(\uv{\gamma}_i;\ell)}
\end{align}
where $\delta_{\rm rec}$ and $\delta_{\rm true}$ describe the reconstructed and
the original density fluctuations contributed from the $\ell$-th
multipole mode,
\begin{align}
\label{eq:delta_l}
\delta(\uv{\gamma}; \ell)
\equiv
\sum_{m=-\ell}^{\ell} a_{\ell m} Y_{\ell m}(\uv{\gamma}),
\end{align}
respectively.
An overall fractional difference $D^2$ is defined by substituting
$\delta(\uv{\gamma})$ for $\delta(\uv{\gamma},\ell)$ in Eq.~(\ref{eq:l2norm_l})
where $\delta(\uv{\gamma})$ is a sum of all the multipole contributions,
i.e. $\delta(\uv{\gamma})=\sum_{\ell} \delta(\uv{\gamma};\ell)$.
In Fig.~\ref{fig:pixel}, we can see the difference between a pixelized
mask and an unpixelized ``smooth'' mask given by
Eq.~(\ref{eq:wij-nopixel}). Using Healpix, we pixelize the mask with
different resolution $N_{\rm side}$ where the total number of pixel on
the sky is defined as $N_{\rm pix}=12\times N_{\rm side}^2$. Before
looking into the pixelization effect, we first study the overall
behaviour of reconstruction accuracy as a function of the number of
iteration. As we increase the number of iteration, the accuracy of
reconstruction gradually improves at $N_{\rm ite} < 10$
then it begins to degrade. Finally, the $D^2$ accuracy
converges to the one obtained by using the brute-force inversion
(dotted lines in the Fig.~\ref{fig:pixel} ). As we can see in
the bottom panel of Fig.~\ref{fig:pixel}, the differences
in $D^2$ between the pixelized and
unpixelized $W_{ij}$ are significant for $N_{\rm side}=16$ and $128$,
while they are less than 1\% at the $D^2$ minimum for
$N_{\rm side}\geq 512$. We then conclude that the pixelization effect
can be ignored when we use sufficiently fine pixels with $N_{\rm side}\geq 512$.
\subsection{Comparison with $W^{-1}$ and SVD}
\label{ssec:svd}
\begin{figure}[tp]
\begin{tabular}{l}
\hspace{1cm}(a) $C_\ell \propto \ell^{ 0}$ \\
\includegraphics[width=\linewidth]{plot.power-0.0.eps}\\
\hspace{1cm}(b) $C_\ell \propto \ell^{-2}$ \\
\includegraphics[width=\linewidth]{plot.power-2.0.eps}\\
\hspace{1cm}(c) $C_\ell \propto \ell^{-4}$ \\
\includegraphics[width=\linewidth]{plot.power-4.0.eps}\\
\end{tabular}
\caption{
In the right panels, we plot $D^2$ accuracy as a function of
iteration number. Three different rows show the different
underlying density fluctuations that have power-law spectrum of
$C_{\ell} \propto \ell^{0}$, $\ell^{-2}$ and $\ell^{-4}$ from
top to bottom.
For a red ($n=0$) spectrum, we do not see any improvement.
However, for a flat ($n=-2$) spectrum, there
exists an optimal number of iteration around $N_{\rm ite}=10$
that minimizes $D^2$. The optimal $N_{\rm ite}$ depends on the
$\ell_{\rm max}$. For a blue ($n=-4$) spectrum, $D^2$
converges to a few percent accuracy after sufficient number of
iteration.
For all cases, after a sufficient number of iteration,
$D^2$ converges to the value which is obtained by
the direct inversion.
In the left panels, we show the result for the SVD method for the
comparison as a function of eigenvalue threshold $\lambda_k$.
\label{fig:svd}
}
\end{figure}
\begin{figure*}
\begin{tabular}{lll}
~~~~ (a) $C_{\ell}\propto \ell^{0}$ &
~~~~ (b) $C_{\ell}\propto \ell^{-2}$ &
~~~~ (c) $C_{\ell}\propto \ell^{-4}$ \\
\includegraphics[width=0.33\linewidth]{SVD_LmaxDeptIndivL_power-0.0.eps}&
\includegraphics[width=0.33\linewidth]{SVD_LmaxDeptIndivL_power-2.0.eps}&
\includegraphics[width=0.33\linewidth]{SVD_LmaxDeptIndivL_power-4.0.eps}\\
\end{tabular}
\caption{
The IHE reconstruction accuracy $D_{\ell}^2$
for each multipole component.
From right to left, the indices of underlying power spectrum are
$n=0, -2$ and $-4$, respectively. From top to bottom, the components of
multipoles $\ell=1$ to $4$ are shown. We reconstruct
the masked map with maximum multipoles of $\ell_{\rm max}=3,5,7$ and 9.
The error bars represent
1 $\sigma$ standard deviation calculated from 1000 random realizations.
\label{fig:l2norm_indivL}
}
\end{figure*}
Using Eq.~(\ref{eq:direct_inv}), we can compare our
method to the direct brute-force inversion of the matrix $\bvec{W}$
and the singular value decomposition\citep{Efstathiou:2004}.
If the mode coupling matrix $\bvec{W}$ is invertible, we obtain the
unique solution of the underlying density fluctuation within the mask.
However, if the matrix $\bvec{W}$ contains some eigenvalues which are close to
zero, where the matrix is close to being singular, the inversion
causes big errors. In such a case, we can remove the singularity by
replacing the small eigenvalues with zero, so called the SVD method.
More specifically, the matrix $\bvec{W}$ can be decomposed into
three matrices as,
\begin{equation}
\bvec{W}
=
\bvec{U} ~\bvec{\Sigma} ~\bvec{V}^{\dagger},
\label{eq:svd}
\end{equation}
where $\bvec{V}$ and $\bvec{U}$ are $i_{\rm max}\times i_{\rm max}$
unitary matrices and a superscript $\dagger$ denotes the Hermitian conjugate.
$\bvec{\Sigma}$ is the diagonal matrix which consists of the eigenvalues
of $\bvec{W}$, or called as the singular values.
The order of the eigenvalues in $\Sigma$ is arbitrary but
they are arranged in a descending order so that the decomposition is
determined uniquely. Let the $k-$th eigenvalue be $\lambda_k$,
and the eigenvalues smaller than $\lambda_k$ are set to zero.
Then the pseudo inversion of $W$ is written in terms of
$\Sigma^{+}$, the rank-$k$ diagonal matrix that consists of the
reciprocal of the non-zero eigenvalues that are equal to or larger than
$\lambda_k$, as
\begin{equation}
\bvec{W}^{+}
=
\bvec{V} ~\bvec{\Sigma}^{+} ~\bvec{U}^{\dagger},
\label{eq:svd_inv}
\end{equation}
which gives
\begin{equation}
a^{\rm est}_i
=
\tilde{a}_j W^{+}_{ij}.
\end{equation}
The choice of threshold $\lambda_k$ is not trivial, and should be
carefully determined {\it a priori} because the mapping $F_{\rm SVD}$
depends on various factors including the threshold $\lambda_k$: i.e.
$F_{\rm SVD}=F_{\rm SVD}(\ell_{\rm max}, \delta, {\boldsymbol W}, \lambda_k)$.
In this context, the {\it singularity} can only be defined
in terms of $\lambda_k$.
More specifically, the mapping $F_{\rm SVD}$ is contaminated by the
eigenvalues which are close to being {\it singular} if
$\partial D^2 / \partial \lambda_k < 0$. In such cases,
the threshold of eigenvalues should be increased
to implement more accurate reconstruction.
The brute-force inversion corresponds to
$F_{\rm inv}=F_{\rm SVD}(\ell_{\rm max}, \delta, {\boldsymbol W}, \lambda_k=0)$.
In the left panels of Fig.~\ref{fig:svd}, $D^2$ is shown as a
function of the minimum non-zero eigenvalue $\lambda_k$. The solid
lines show the different maximum multipole to be reconstructed,
$\ell_{\rm max}=3,5,7,9$ from dark to light color. The different
three rows show the different power of the input power spectra which
will be discussed in detail in the next section. Let us focus on the
middle panel. For $\ell_{\rm max}=3$ and $5$, $D^2$ increases
monotonically as we increase the threshold. It means that the
$F_{\rm SVD}$ is not contaminated by singular eigenvalues
as $\partial D^2/\partial \lambda_k > 0$.
On the other hand, for $\ell_{\rm max} = 7$ and $9$, $D^2$ has a
minimum around $\lambda_k\sim 0.1$. Thus the eigenvalues
smaller than $\sim 0.1$ may be the contaminant of the mapping $F$
and we can better
estimate the original density fluctuations when we limit the
eigenvalues as $\lambda < 0.1$. In the right panel of
Fig.~\ref{fig:svd}, we show $D^2$ for our IHE method as a
function of the number of iteration. As in Fig.~\ref{fig:pixel},
$D^2$ has the minimum around $N_{\rm ite}\simeq 10$ depending on the
maximum multipole. After a sufficient number of iteration, $D^2$
values converge to those obtained by using the brute-force inversion,
which are shown as the horizontal dashed lines in Fig.\ref{fig:pixel}.
Our IHE method with a certain finite number of iteration is always better than
the SVD method. However, in practice, we should know a priori the
optimal number of iteration. The number depends on the mask
geometry, the maximum
multipole to be reconstructed, and the underlying spectrum of
the density fluctuation. We can estimate the optimal iteration number
by carrying out Monte-Carlo simulations. In the next section we will see how the
result changes for different types of underlying power spectrum.
\subsection{Dependence on underlying power spectrum}
\label{ssec:dependence}
Reconstruction accuracy depends on the underlying power spectrum
$C_\ell\propto \ell^{n}$ of
the fluctuation as each harmonic mode is not independent in the masked
incomplete sky even the underlying fluctuation is Gaussian.
We consider three power law indices, $n=0, -2$ and $-4$, which
correspond to the following three cases.
The projected two dimensional galaxy or dark matter distribution is
approximated as $C_{\ell}^g \propto \ell^0$
\cite[e.g.][]{Frithetal:05} on large scales, and
the ordinary Sachs-Wolfe spectrum is approximated as
$C_{\ell}^{\rm SW}\propto \ell^{-2}$ \citep{SachsWolfe:67}.
On very large-angular scales,
the integrated Sachs-Wolfe effect, which gives
$C_{\ell}^{\rm iSW}\propto \ell^{-4}$ \cite[e.g.][]{Cooray:02}
dominates the CMB in the standard $\Lambda$CDM scenario.
Given the typical scale of mask $\theta_{\rm M}$, it is impossible to
reconstruct the fluctuation whose scale is smaller than
$\ell_{\rm M} \ge 180/\theta_{\rm M}$.
Because of the mode coupling, fluctuations with angular scales
corresponding to $\ell_{\rm M}$ are strongly affected by
fluctuations with smaller angular sizes.
If the spectral index is negative, the amplitude of a smaller scale fluctuation
is weak and does not strongly disturb the large scale fluctuations.
Thus the deconvolution mapping $F$ is less affected by singularities.
However, if the slope of the spectrum is flat or positive,
large scale modes are highly contaminated by the noisy small scale
fluctuations.
In Fig.~\ref{fig:svd}, the top and bottom panels show
$D^2$ accuracy for $n=0$ and $n=-4$. For $n=0$ case, the SVD has a minimum at
$\lambda_k\sim 0.5$ that is larger than the case of $n=-2$. It
means that
$F_{\rm SVD}(n=0)$ is more affected by singularities than $F_{\rm SVD}(n=-2)$.
The IHE result in the right panels
shows $N_{\rm ite}=1$ gives the best accuracy and it gradually
degrades to the value given by the brute-force method.
On the other hand, $F_{\rm SVD}(n=-4)$ is not affected by singularities
because it always shows
$\partial D^2 / \partial \lambda_k < 0$ for the SVD method.
In Fig.~\ref{fig:l2norm_indivL}, we also show the reconstruction accuracy
for each multipole component, $D^2_{\ell}$. The behaviour
is similar to the one described in Fig.~\ref{fig:svd}.
The reconstruction accuracy is always better for low multipoles
and that the optimal number of
iteration depends also on the multipole component we are going to
reconstruct.
\section{summary}
\label{sec:summary}
We have developed a new method based on the iterative harmonic
expansions (IHE) for reconstructing the missing fluctuations
in the masked region in the sky.
Reconstructing the data at a masked region is
known as an inverse problem. Our method is equivalent to the brute-force
inversion in the limit that the iteration number goes to infinity.
However, in some cases, the finite truncation of the
iteration gives a better estimate of the underlying fluctuation.
The reconstruction accuracy depends on the geometrical shape of the
mask, the maximum multipole mode to be included in the IHE analysis,
the tilt of the underlying power spectrum, and the multipole component
of the map to be reconstructed.
As an example, we have applied the method to reconstruct the missing data
on an azimuthally symmetric mask. We have considered three
types of gaussian fluctuations with the power-law indices $n=0, -2$ and
$-4$, which correspond to the galaxy power spectrum,
the ordinary Sachs-Wolfe spectrum, and the integrated Sachs-Wolfe spectrum,
respectively.
For $n=-2$ case, we have found that there exists an optimal finite number of
iteration that makes the reconstruction more accurate
than the SVD method or the brute-force method.
For $n=0$ case, the pseudo-$a_{\ell m}$ is the
best estimator for the projected density fluctuations.
For $n=-4$ case, the brute-force inversion method gives the best accuracy.
In that case, the IHE method can help to reduce the computation time of
inversion.
It would be interesting to see how the significance of the
large-angle CMB anomaly changes when we use different methods of
map reconstruction. The IHE method may help to unveil the origin of
the CMB anomaly. We will explore this problem
in our future work.
\begin{acknowledgments}
We thank Masahiro Takada, Eiichiro Komatsu, Issha Kayo, Takahiro
Nishimichi for useful discussions.
AN is supported in part by the Grant-in-Aid for the Scientific
Research Fund (No. 23340061), by JSPS Core-to-Core Program
`` International Research Network for Dark Energy'', by World Premier
International Research Center Initiative (WPI Initiative), MEXT, Japan
and by the FIRST program ``Subaru Measurements of Images and Redshifts
(SuMIRe)'', CSTP, Japan.
\end{acknowledgments}
\input{ms.bbl}
\end{document}
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| 1,683
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Employing a plumber is an essential choice you need to make as a property holder on the grounds that there are things that you have to put into consideration. The duties of the plumber will involve repairing your water heater, repairing your plumbing system among various other responsibilities. You will be offered low quality services on the off chance that you enlist a plumber that isn't qualified, the plumber may even wind up damaging your property instead. This can bring about a ton of costs when you need to do repairs for the spots the unqualified plumber damaged. There are tips that you can utilize when you are searching for a plumber to hire.
Knowing the references of the plumber is essential, you can ask for the plumber to give you references of the customers he has offered services to before. When you get the contacts of the customers that the plumber has served before, you can solicit the kind of services they got from the plumber. If you will get good work done, is the sort of consolation you will search for in the event that you have a ton of work to be done.
You should enlist a plumber that has insurance, the insurance he has ought to be of the correct level with the goal that he will be covered if there is any occurrence of any mishap.
How long the plumbing organization has been in operation should be put into thought, if the organization has been in operation for a drawn out stretch of time then it has workers that are well skilled. Businesses that have been in operation for quite a while implies that they are effective, if customers continue returning to the business as a result of the quality services they are offered ,the business will stay in operation for a long time.
Once you have distinguished a couple of plumbing organizations that you will have the capacity to work with, guarantee that you think about the costs of the services. In order to know how much cash you will spend on the plumbing work, you have to know the quotes of the expenses from various plumbing organizations you had selected. Costs of paying workers and call out expenses will be incorporated into the quotes you will get for the plumbing work. The plumbing organization you pick should charge you cost that will fit your budgetary plan. You can consider hiring a plumber that offers 24 hour services, this is vital on the grounds that you can have the capacity to get in touch with them whenever when you have an emergency.
|
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| 3,160
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Murphy, Himes hold roundtable to hear stories of support for the Affordable Care Act
Danbury News Times
NORWALK — Renee Bea of Darien has watched her 7-year-old daughter struggle with migraines. Bea said she's grateful for the protections in the federal Affordable Care Act for people with pre-existing conditions, but is concerned that recent efforts to reduce or invalidate the law could jeopardize those.
"It just floors me that at seven, she has a pre-existing condition," she said on Thursday in Norwalk. "She might be denied insurance in the future."
Bea was one of more than 30 people in attendance at a "Share Your Story" roundtable organized by U.S. Sen. Chris Murphy, D-Connecticut. Murphy launched the campaign in May to rally support for the Affordable Care Act to allow residents to share their health care stories, particularly what would happen if the Affordable Care Act was repealed. Murphy emphasized the particular need to raise support for the law because in 2018, the Texas Attorney General and 17 others sued the federal government over the Affordable Care Act and a federal judge ruled that the law was unconstitutional. The U.S. Fifth Circuit Court of Appeals will hear arguments in this lawsuit in July.
Murphy said if the ruling is upheld, the results would be disastrous for the state.
"To Connecticut, that is a humanitarian disaster," he said.
U.S. Rep. Jim Himes, D-Conn, who was also in attendance, said that there have been considerable gains made in coverage and protection through the Affordable Care Act.
"Were the Texas case to succeed, (the lawsuit) puts us back in that world," he said.
Joan Coprio, of Stamford, said she was diagnosed with stage 3 Hodgkin's lymphoma and had to do multiple cycles of six day inpatient chemotherapy.
"If I did not have insurance, it would have been unaffordable," she said.
Coprio said while she had gotten insurance through her employer, it was nice to know that health insurance plans were available through the Affordable Care Act in case she had lost her job during that time period.
Alyssa MacKenzie of New Canaan said she was thankful and privileged to be covered on her father's insurance because of her complications with lupus. She estimated that some of her medications can cost up to $30,000 a month.
"It wouldn't just be me," she said, stating that other members of her family also have the autoimmune disease and could be affected by changes to the Affordable Care Act.
Suzanne Curto, director of behavioral health at Norwalk Community Health Center, where the event was held, said that their patients would be extremely affected by changes to the act, since many are covered through the expanded Medicaid program.
"We are servicing clients who are most in need," she said. "Imagine dealing with multiple medical issues and poverty and homelessness."
Both Murphy and Himes said they would take the stories shared with them back to Washington, D.C., to work to preserve and enhance the Affordable Care Act.
"We continue to work really hard on means to make the Affordable Care Act better," Murphy said.
|
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| 267
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<?xml version="1.0" encoding="utf-8"?>
<menu xmlns:android="http://schemas.android.com/apk/res/android" >
<item
android:id="@+id/page"
android:actionProviderClass="net.meiolania.apps.habrahabr.ui.PageActionProvider"
android:showAsAction="always"
android:title="@string/count_pages"/>
</menu>
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| 8,552
|
Q: .htaccess RewriteRule url to new Bonjour,
I try to redirect urls of a website
Example :
www.exemple.net/?p=2
to
www.exemple.net/index-2.html
www.exemple.net/?p=35
to
www.exemple.net/index-35.html
etc...
So i add this lines to my .htacess :
RewriteCond %{QUERY_STRING} ^(?)p=(.*)$ [NC]
RewriteRule .* /index-%1.html [L,R=301]
But i'm redirected to http://www.exemple.net/index-2.html?p=2
My .htaccess :
RewriteEngine On
RewriteCond %{HTTP_HOST} ^exemple.net$
RewriteRule ^(.*) http://www.exemple.net/$1 [QSA,L,R=301]
RewriteCond %{QUERY_STRING} ^(?)p=(.*)$ [NC]
RewriteRule .* /index-%1.html [L,R=301]
RewriteBase /
RewriteCond %{REQUEST_FILENAME} !-f
RewriteCond %{REQUEST_FILENAME} !-d
RewriteRule . /index.php [L]
Can you help me please ?
A: You need to add a question mark at the end of the destination to discard the query string:
RewriteRule .* /index-%1.html? [L,R=301]
Alternatively, use the QSD flag:
RewriteRule .* /index-%1.html [QSD,L,R=301]
A: Try :
#Redirecting /?p=2
RewriteCond %{THE_REQUEST} ^[A-Z]{3,}\s/\?p=([^&\s]+) [NC]
RewriteRule ^ /index-%1.html? [NE,NC,R,L]
#Redirecting /transfer?p=2
RewriteCond %{THE_REQUEST} ^[A-Z]{3,}\s/transfer\?p=([^&\s]+) [NC]
RewriteRule ^ /transfer-%1.html? [NE,NC,R,L]
#Redirecting /blog?p=2
RewriteCond %{THE_REQUEST} ^[A-Z]{3,}\s/blog\?p=([^&\s]+) [NC]
RewriteRule ^ /blog-%1.html? [NE,NC,R,L]
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 439
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Le Richard J. Daley Center également connu comme Daley Plaza est un gratte-ciel situé à Chicago, dans l'État de l'Illinois aux États-Unis. L'immeuble a été nommé d'après Richard J. Daley, qui fut maire de Chicago de 1955 à 1976 et est le premier centre civique de la Ville de Chicago. À proximité se trouve l'hôtel de ville de Chicago (Chicago City Hall) et le James R. Thompson Center, qui abrite certains bureaux du gouvernement de l'État de l'Illinois. Plusieurs sculptures monumentale en acier se trouvent près du bâtiment, dont le Chicago Picasso de Pablo Picasso et le Flamingo d'Alexander Calder.
Situé à l'angle de Randolph Street et Washington Street entre Dearborn Street et Clark Street, le Richard J. Daley Center est considéré comme l'un des immeubles les plus innovants du Mouvement moderne de l'architecture de Chicago. Il culmine à 198 mètres et possède trente et un étages.
Le bâtiment a été conçu dans le style international par l'architecte Jacques Brownson du cabinet C. F. Murphy Associates et achevé en 1965. À l'époque il était le plus haut bâtiment à Chicago, mais n'a occupé ce titre que pendant quatre ans jusqu'à ce que le 875 North Michigan Avenue (ex John Hancock Center) soit achevé.
D'abord connu sous le nom de Civic Center of Chicago, le bâtiment a été rebaptisé pour le maire Daley le , sept jours après sa mort.
Articles connexes
Liste des plus hautes constructions de Chicago
Architecture à Chicago
Gratte-ciel à Chicago
Richard
Richard
Architecture moderniste aux États-Unis
|
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| 3,898
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PLEIN AIR PAINTOUT: Take a peek at what 22 artists are coming up with during the public observation days of the Santa Fe Springs Plein Air Paintout. They will be painting the North Florida Springs through Sunday, when a reception will be held from 5 to 7 p.m. at Lanza Gallery & Art Supplies, 23645 W. U.S. 27 in High Springs. Friday, you can watch the artistry at Gilchrist Blue Springs State Park, 7450 NE 60th St. in High Springs; Saturday, head out to Downtown High Springs. Some of the locations are free, while others have entrance fees. Pick up your map to find out where the artists will be and all of the information throughout the paintout at Lanza Gallery & Art Supplies. Light refreshments will be served during the reception. All of the finalized art also will be on display and for sale. The exhibit will run Sunday through May 18. For more information, call 474-9922.
HOME & GARDEN SHOW: The 20th annual North Central Florida Home & Garden Show will be held Saturday and Sunday at the Stephen C. O'Connell Center. The yearly event will feature booths, seminars and experts from all over North Central Florida with a focus on everything house-related. If you're buying a new home, building a new one or planning to remodel part — or all — of your current residence, you'll find helpful tips, tricks, items and businesses to help ease the process. Some of the popular topics covered during the event will include home repair, landscaping, home security, home decor and electronics. The event will be held from 10 a.m. to 5 p.m. Saturday and 11 a.m. to 4 p.m. Sunday. The Stephen C. O'Connell Center is located at 250 Gale Lemerand Drive. For more information, check out the Home & Garden Show Program in Friday's Gainesville Sun.
CHAT WITH AUTHOR ELIZABETH BERG: Pick the mind of author Elizabeth Berg this Saturday during the current installment of the Alachua County Library District 2019 Author Series. Berg will discuss her latest book, "Night of Miracles," a feel-good novel about surprising friendships, community and how small acts of kindness can change a life. Released in November, "Night of Miracles" is a companion novel to Berg's 2017 book, "The Story of Arthur Truluv." Berg's first novel, "Durable Goods," won the American Library Association's Best Book of the Year award. Her novel "Open House" was an Oprah's Book Club selection. Berg worked as a nurse for 10 years before winning a magazine essay contest and embarking on her writing career. She will answer questions and be available for book signings during the event. Copies of her books will be for sale, and library copies may be reserved at aclib.us/author-series. The event begins at 2:30 p.m. at Headquarters Library, 401 E. University Ave. For more information, visit aclib.us.
CRUISE-IN: The Gainesville Street Rods Monthly Cruise-In, which raises funds to support Stop Children's Cancer, will be held from 5:30 to 8:30 p.m. Saturday at the Springhill Publix, which is located at Northwest 39th Avenue and Interstate 75. The car show is free to spectators and $5 for anyone who wants to register their vehicle as part of the show. Registration is open to all makes of legal, street-driven vehicles. There also will be a $100 jackpot drawing for registered vehicles and a trophy award for the "Young Cruisers of the Month." The show will host a DJ and a 50/50 drawing to benefit Stop Children's Cancer. For more information, call 658-1477.
OLD FASHIONED RIVER PARTY: Celebrate Manatee Springs with live music and a fish fry from 10 a.m. to 6 p.m. Saturday in the picnic area of Manatee Springs State Park. The 27th annual Old Fashioned River Party will showcase more than a dozen acoustic acts, including fiddlers, banjo players, guitarists, singer-songwriters, old-time musicians and yodelers. Noted local conservationist Whitey Markle will do a set of his songs in the afternoon. A fish fry will be served from 1 to 2 p.m., and the food is included in the park entrance fee of $6 per vehicle. The event is intended to raise awareness of the springs and to help protect them for future generations. Manatee Springs State Park is located at 11650 NW 115th St. in Chiefland. For more information, call 493-6072.
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{"url":"http:\/\/en.wikipedia.org\/wiki\/Slip_(vehicle_dynamics)","text":"# Slip (vehicle dynamics)\n\nIn (automotive) vehicle dynamics, slip is the relative motion between a tire and the road surface it is moving on. This slip can be generated either by the tire's rotational speed being greater or less than the free-rolling speed (usually described as percent slip), or by the tire's plane of rotation being at an angle to its direction of motion (referred to as slip angle)[1].\n\nIn rail vehicle dynamics, this overall slip of the wheel relative to the rail is called creepage. It is distinguished from the local sliding velocity of surface particles of wheel and rail, which is called micro-slip.\n\nThe slip is generally given as a percentage of the difference between the surface speed of the wheel compared to the speed between axis and road surface, as:\n\n$slip=\\frac{\\omega r - v}{v},$\n\nwhere $\\omega$ is rotational speed of the wheel, $r$ is wheel radius and $v$ is vehicle speed. Meaning a positive slip means the wheels are spinning and negative that they are skidding. Locked brakes, $\\omega r=0$, means that $slip$ is -100% and spinning on the spot, $v=0$ and $\\omega r$$0$, means that $slip=$\u221e.","date":"2014-07-29 14:16:52","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 10, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5408310294151306, \"perplexity\": 969.4868449206936}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-23\/segments\/1406510267330.29\/warc\/CC-MAIN-20140728011747-00434-ip-10-146-231-18.ec2.internal.warc.gz\"}"}
| null | null |
{"url":"https:\/\/zbmath.org\/?q=an%3A1415.11170","text":"# zbMATH \u2014 the first resource for mathematics\n\nThe Eisenstein cocycle and Gross\u2019s tower of fields conjecture. (English. French summary) Zbl\u00a01415.11170\nSummary: This paper is an announcement of the following result, whose proof will be forthcoming. Let $$F$$ be a totally real number field, and let $$F \\subset K \\subset L$$ be a tower of fields with $$L\/F$$ a finite abelian extension. Let $$I$$ denote the kernel of the natural projection from $$\\mathbb {Z}[\\mathrm{Gal}(L\/F)]$$ to $$\\mathbb {Z}[\\mathrm{Gal}(K\/F)]$$. Let $$\\Theta \\in \\mathbb {Z}[\\mathrm{Gal}(L\/F)]$$ denote the Stickelberger element encoding the special values at zero of the partial zeta functions of $$L\/F$$, taken relative to sets $$S$$ and $$T$$ in the usual way. Let $$r$$ denote the number of places in $$S$$ that split completely in $$K$$. We show that $$\\Theta \\in I^{r}$$, unless $$K$$ is totally real in which case we obtain $$\\Theta \\in I^{r-1}$$ and $$2\\Theta \\in I^r$$. This proves a conjecture of Gross up to the factor of 2 in the case that $$K$$ is totally real and $$\\#S \\neq r$$. In this article we sketch the proof in the case that $$K$$ is totally complex.\n\n##### MSC:\n 11R42 Zeta functions and $$L$$-functions of number fields 11R80 Totally real fields\nFull Text:\n##### References:\n [1] Cassou-Nogu\u00e8s, P, Valeurs aux entiers n\u00e9gatifs des fonctions zeta et fonctions zeta $$p$$-adiques, Invent. Math., 51, 29-59, (1979) \u00b7 Zbl\u00a00408.12015 [2] Charollois, P; Dasgupta, S, Integral Eisenstein cocycles on $${ GL}_n$$, I: sczech\u2019s cocycle and $$p$$-adic $$L$$- functions of totally real fields, Camb. J. Math., 2, 49-90, (2014) \u00b7 Zbl\u00a01353.11074 [3] Charollois, P; Dasgupta, S; Greenberg, M, Integral Eisenstein cocycles on $${ GL}_n$$, II: shintani\u2019s method, Commentarii Mathematici Helvetici, 90, 435-477, (2015) \u00b7 Zbl\u00a01326.11072 [4] Dasgupta, S., Spie\u00df, M.: The Eisenstein cocycle, partial zeta values, and Gross-Stark units. Preprint, available at http:\/\/arxiv.org\/abs\/1411.4025 \u00b7 Zbl\u00a00809.11029 [5] Deligne, P; Ribet, K, Values of abelian $$L$$-functions at negative integers over totally real fields, Invent. Math., 59, 227-286, (1980) \u00b7 Zbl\u00a00434.12009 [6] Diaz y Diaz, F; Friedman, E, Signed fundamental domains for totally real number fields, Proc. Lond. Math. Soc., 104, 965-988, (2014) \u00b7 Zbl\u00a01325.11117 [7] Greither, C; Popescu, C, An equivariant main conjecture in Iwasawa theory and applications, J. Algebr. Geom., 24, 629-692, (2015) \u00b7 Zbl\u00a01330.11070 [8] Gross, BH, On the values of abelian $$L$$, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 35, 177-197, (1988) \u00b7 Zbl\u00a00681.12005 [9] Hill, R, Shintani cocycles on $$\\textbf{GL}_n$$, Bull. L.M.S., 39, 993-1004, (2007) \u00b7 Zbl\u00a01192.11030 [10] Hu, S., Solomon, D.: Properties of higher-dimensional Shintani generating functions and cocycles on $$\\textbf{PGL}_3(\\textbf{Q})$$. Proc. L.M.S. 82, 64-88 (2001) \u00b7 Zbl\u00a01045.11035 [11] Sczech, R, Eisenstein group cocycle for $${ GL}_n$$ and values of $$L$$-functions, Invent. Math., 113, 581-616, (1993) \u00b7 Zbl\u00a00809.11029 [12] Shintani, T, On the evaluation of zeta functions of totally real fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 23, 393-417, (1976) \u00b7 Zbl\u00a00349.12007 [13] Spie\u00df, M, On special zeros of $$p$$-adic $$L$$-functions of Hilbert modular forms, Invent. Math., 196, 69-138, (2014) \u00b7 Zbl\u00a01392.11027 [14] Spie\u00df, M, Shintani cocycles and the order of vanishing of $$p$$-adic Hecke $$L$$-series at $$s=0$$, Math. Ann., 359, 239-265, (2014) \u00b7 Zbl\u00a01307.11125 [15] Steele, A, Adic shintani cocycles, Math. Res. Lett., 21, 403-422, (2014) \u00b7 Zbl\u00a01317.11116 [16] Stevens, G.: Eisenstein, The, measure and real quadratic fields. Th\u00e9orie des nombres, pp. 887-927 (Qu\u00e9bec:de Gruyter, Berlin (1987)) (1989) \u00b7 Zbl\u00a00408.12015\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":"2021-09-20 00:37:24","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\": 0.9020698666572571, \"perplexity\": 1774.4071979523537}, \"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-2021-39\/segments\/1631780056902.22\/warc\/CC-MAIN-20210919220343-20210920010343-00615.warc.gz\"}"}
| null | null |
import math
import numpy as np
import h5py
import matplotlib.pyplot as plt
import tensorflow as tf
from tensorflow.python.framework import ops
def load_dataset():
train_dataset = h5py.File('datasets/train_signs.h5', "r")
train_set_x_orig = np.array(train_dataset["train_set_x"][:]) # your train set features
train_set_y_orig = np.array(train_dataset["train_set_y"][:]) # your train set labels
test_dataset = h5py.File('datasets/test_signs.h5', "r")
test_set_x_orig = np.array(test_dataset["test_set_x"][:]) # your test set features
test_set_y_orig = np.array(test_dataset["test_set_y"][:]) # your test set labels
classes = np.array(test_dataset["list_classes"][:]) # the list of classes
train_set_y_orig = train_set_y_orig.reshape((1, train_set_y_orig.shape[0]))
test_set_y_orig = test_set_y_orig.reshape((1, test_set_y_orig.shape[0]))
return train_set_x_orig, train_set_y_orig, test_set_x_orig, test_set_y_orig, classes
def random_mini_batches(X, Y, mini_batch_size=64, seed=0):
m = X.shape[1] # number of training examples
mini_batches = []
np.random.seed(seed)
permutation = list(np.random.permutation(m))
shuffled_X = X[:, permutation]
shuffled_Y = Y[:, permutation].reshape((Y.shape[0], m))
num_complete_minibatches = math.floor(
m / mini_batch_size) # number of mini batches of size mini_batch_size in your partitionning
for k in range(0, num_complete_minibatches):
mini_batch_X = shuffled_X[:, k * mini_batch_size: k * mini_batch_size + mini_batch_size]
mini_batch_Y = shuffled_Y[:, k * mini_batch_size: k * mini_batch_size + mini_batch_size]
mini_batch = (mini_batch_X, mini_batch_Y)
mini_batches.append(mini_batch)
# Handling the end case (last mini-batch < mini_batch_size)
if m % mini_batch_size != 0:
mini_batch_X = shuffled_X[:, num_complete_minibatches * mini_batch_size: m]
mini_batch_Y = shuffled_Y[:, num_complete_minibatches * mini_batch_size: m]
mini_batch = (mini_batch_X, mini_batch_Y)
mini_batches.append(mini_batch)
return mini_batches
def convert_to_one_hot(Y, C):
Y = np.eye(C)[Y.reshape(-1)].T
return Y
def create_placeholders(n_x, n_y):
X = tf.placeholder(dtype=tf.float32, shape=(n_x, None))
Y = tf.placeholder(dtype=tf.float32, shape=(n_y, None))
return X, Y
def initialize_parameters():
W1 = tf.get_variable("W1", [25, 12288], initializer=tf.contrib.layers.xavier_initializer())
b1 = tf.get_variable("b1", [25, 1], initializer=tf.zeros_initializer())
W2 = tf.get_variable("W2", [12, 25], initializer=tf.contrib.layers.xavier_initializer())
b2 = tf.get_variable("b2", [12, 1], initializer=tf.zeros_initializer())
W3 = tf.get_variable("W3", [6, 12], initializer=tf.contrib.layers.xavier_initializer())
b3 = tf.get_variable("b3", [6, 1], initializer=tf.zeros_initializer())
parameters = {"W1": W1,
"b1": b1,
"W2": W2,
"b2": b2,
"W3": W3,
"b3": b3}
return parameters
def forward_propagation(X, parameters):
# Retrieve the parameters from the dictionary "parameters"
W1 = parameters['W1']
b1 = parameters['b1']
W2 = parameters['W2']
b2 = parameters['b2']
W3 = parameters['W3']
b3 = parameters['b3']
Z1 = tf.add(tf.matmul(W1, X), b1)
A1 = tf.nn.relu(Z1)
Z2 = tf.add(tf.matmul(W2, A1), b2)
A2 = tf.nn.relu(Z2)
Z3 = tf.add(tf.matmul(W3, A2), b3)
return Z3
def compute_cost(Z3, Y):
logits = tf.transpose(Z3)
labels = tf.transpose(Y)
cost = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits(logits=logits, labels=labels))
return cost
def model(X_train, Y_train, X_test, Y_test, learning_rate=0.0001,
num_epochs=2000, minibatch_size=32, print_cost=True):
ops.reset_default_graph()
seed = 3
(n_x, m) = X_train.shape
n_y = Y_train.shape[0]
costs = []
X, Y = create_placeholders(n_x, n_y)
parameters = initialize_parameters()
Z3 = forward_propagation(X, parameters)
cost = compute_cost(Z3, Y)
optimizer = tf.train.AdamOptimizer(learning_rate=learning_rate).minimize(cost)
init = tf.global_variables_initializer()
saver = tf.train.Saver()
with tf.Session() as sess:
sess.run(init)
for epoch in range(num_epochs):
epoch_cost = 0.
num_minibatches = int(m / minibatch_size)
seed = seed + 1
minibatches = random_mini_batches(X_train, Y_train, minibatch_size, seed)
for minibatch in minibatches:
(minibatch_X, minibatch_Y) = minibatch
_, minibatch_cost = sess.run([optimizer, cost], feed_dict={X: minibatch_X, Y: minibatch_Y})
epoch_cost += minibatch_cost / num_minibatches
# Print the cost every epoch
if print_cost is True and epoch % 100 == 0:
print("Cost after epoch %i: %f" % (epoch, epoch_cost))
if print_cost is True and epoch % 5 == 0:
costs.append(epoch_cost)
saver.save(sess, "model/recogn.ckpt")
# plot the cost
plt.plot(np.squeeze(costs))
plt.ylabel('cost')
plt.xlabel('iterations (per tens)')
plt.title("Learning rate =" + str(learning_rate))
plt.show()
parameters = sess.run(parameters)
print("Parameters have been trained!")
correct_prediction = tf.equal(tf.argmax(Z3), tf.argmax(Y))
accuracy = tf.reduce_mean(tf.cast(correct_prediction, "float"))
print("Train Accuracy:", accuracy.eval({X: X_train, Y: Y_train}))
print("Test Accuracy:", accuracy.eval({X: X_test, Y: Y_test}))
return parameters
X_train_orig, Y_train_orig, X_test_orig, Y_test_orig, classes = load_dataset()
# Flatten the training and test images
X_train_flatten = X_train_orig.reshape(X_train_orig.shape[0], -1).T
X_test_flatten = X_test_orig.reshape(X_test_orig.shape[0], -1).T
# Normalize image vectors
X_train = X_train_flatten / 255.
X_test = X_test_flatten / 255.
# Convert training and test labels to one hot matrices
Y_train = convert_to_one_hot(Y_train_orig, 6)
Y_test = convert_to_one_hot(Y_test_orig, 6)
parameters = model(X_train, Y_train, X_test, Y_test)
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,689
|
Samsung Galaxy J2 Pro is now official in the Philippines; Features a 5-inch SAMOLED display
By Bryan February 17, 2018
The Samsung Galaxy J2 Pro is officially available in the Philippines. The phones features a 5-inch Super AMOLED display, a quad-core Qualcomm processor, and a Php7,490 price tag.
Samsung Galaxy J2 Pro Philippines
The Samsung Galaxy J2 Pro is now officially available in the Philippines. Priced at Php7,490, the smartphone features a 5-inch Super AMOLED qHD display with a resolution of 960 x 540.
Samsung teases Galaxy S9 camera features with videos
Samsung Galaxy J7+ and Galaxy J7 Pro now come in pink
Samsung Galaxy A8 (2018) and Galaxy A8+ (2018) now official in the Philippines
Under the hood, the smartphone is powered by a 1.4GHz quad-core Qualcomm Snapdragon 425 processor, paired with 1.5GB of RAM and 16GB of storage. It has a dedicated expansion microSD card slot, an FM radio, and a 2,600 mAh battery.
As for the cameras, the Galaxy J2 Pro packs an 8MP rear sensor with a f/2.2 auto-focus lens, and an LED flash. The front camera uses a 5MP sensor and another LED flash.
The Samsung Galaxy J2 Pro is now available with a suggested retail price of Php7,490. It is now available in Samsung Galaxy Experience Stores and authorized dealers nationwide.
Samsung Galaxy J2 Pro specs:
5-inch qHD Super AMOLED Display (960 x 540; 220 ppi)
1.4 GHz Quad-Core Qualcomm Snapdragon 425 Processor
1.5 GB RAM
16GB Storage +microSD up to 256GB (Dedicated Slot)
8MP Main Camera, f/2.2, AF, LED Flash
5MP Front Camera, f/2.2, LED Flash
Wi-Fi b/g/n
Android 7.1 Nougat, TouchWiz
Energizer smartphones on sale at PC Express
First official Xiaomi Mi Store in the Philippines is now open
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,910
|
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