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{"url":"https:\/\/www.airbestpractices.com\/system-assessments\/end-uses\/distillery-addresses-inappropriate-compressed-air-uses-saving-16600-ener?page=1","text":"Industrial Utility Efficiency\n\n# Distillery Addresses Inappropriate Compressed Air Uses Saving $16,600 in Energy Costs By addressing inappropriate uses of compressed air and making changes to the compressed air production side of their compressed air system, a distiller of fine alcohol products reduced its energy consumption by 30%, saving \\$16,600 per year in energy costs - with more potential savings possible.\n\n### Background\n\nThe facility has a fermenting, distilling, storage and distribution operations onsite, all of which use various amounts of 100 psi compressed air. The compressed air contacts various products at different stages of processing in the facility, so there is the need to produce the cleanest possible compressed air to maintain quality and product safety.\n\nThe compressed air system consisted of three water-cooled, oil-free air compressors of various ages, running in load unload mode. Air compressor sizes were 110 kW, 90 kW, and 160 kW.\u00a0 The operation typically used the 110 kW unit for base load with the 90 kW air compressor starting for peak duty. The 160 kW unit, being very old, was used for back-up.\n\nDrying is done using a heated blower style unit with dewpoint dependent switching to provide compressed air with - 40 oF pressure dewpoint. A refrigerated air dryer had been installed at one point for back-up duty, but was shut down due to maintenance issues.\n\nThe compressed air is directed throughout the plant through a system of steel piping. One medium sized 500-gallon storage receiver is located in the compressor room area as dry control storage, with various large receivers located out in the plant for peaking duty.\n\nThe compressed air is delivered to the multiple production areas through piping headers from which various branches are tapped to supply to each production operation.\u00a0 Installed data loggers showed minimal pressure loss across the piping system. Most of the pressure loss is across the drying and filtering system or across restrictor plates deliberately designed into the system.\n\n### System Baseline\n\nThe compressed air system electrical consumption was monitored as part of an extensive audit using amp loggers. Kilowatt readings using a hand held meter were taken for both the active air compressors to calibrate amps to power. System flow was recorded using the plant flow meter. Figure 2 shows the baseline during a two-week period.\n\n##### Figure 2: The distillery pays \\$66,420 per year for electricity to power the compressed air system. ##### Figure 3: The heated blower style dryer maintains a - 40 oF pressure dewpoint. The readings and observations during the measurement period showed the compressed air system was producing air at fair efficiency (21.7 kW\/100 cfm). The assessment found problems with the air dryer cooling purge flow, higher than needed discharge pressure, some small leakage and drainage, and some possible inappropriate uses, were causing higher than desired operating costs and occasional pressure issues. The study found that significant improvements were possible. ##### Figure 4: A typical baseline period shows the effects of dryer cooling purge. Click here to enlarge. ##### Figure 5: Devices like this compressed air-powered blowing horn were classed as possible inappropriate uses. ### Inappropriate Compressed Air Uses A survey of the demand side of the system was done including leakage. A total of 49 leakage points were found, estimated at 87 cfm. Various compressed air uses, which could be classed as inappropriate, were found including star seal blowing, bin vibrators, poorly adjusted dust collectors, compressed air vacuum, air horn blowers and fume blowing. Shown in the red line of Figure 4 is the shape of the compressed airflow demand curve during production over a one-week period at the end of the data logging. The pressure profile shows good pressure regulation when only one air compressor is required. Dips in pressure can be seen on occasion when two air compressors are required. The profile shows a somewhat flat pattern during production activities, with higher peak flows just after the air dryer heating cycle. This is caused by the dryer cooling cycle where a flow of compressed air is directed into the desiccant to remove the remaining heat. If this is not done, a dewpoint spike will develop when the dryer stitches sides. Additional peak flows are somewhat random, associated with clearing of alcohol lines after transfer of product. Compressed air is injected into stainless steel transfer lines to remove the remaining product before another ingredient is pumped. These dryer-associated peaks, with coincident line cleaning, required two air compressors on occasion to run to support plant pressure. Although the second unit barely loads, it and the dryer heater, contribute additional cost to peak electrical demand. Careful analysis of this profile showed that the air dryer was consuming higher than normal cooling purge, something that can easily be adjusted. It was thought that reduction of this would reduce the requirement for two air compressors, saving on peak demand. Savings could also be gained if leakage and inappropriate use could be reduced. Potentially inappropriate end uses found were as follows: \u2022 Air vacuum \u2013 A compressed air-powered drum vacuum was being used for cleanup purposes in an area where there is potential for explosion if spark-producing devices are used. The vacuum is created using a compressed air vortex, something that is safe, with few moving parts, but consuming about ten times the equivalent energy than a explosion-proof vacuum. ##### The distillery uses compressed air powered drum vacuums for cleanup, which is a inappropriate use of compressed air and not uncommon at many plants. \u2022 Dust collectors - Reverse-pulse compressed air cleaners are installed on various dust collectors in the facility. These cleaners appeared to be controlled with the process and turned off when not required, which is excellent practice. The pulse duty when these units operate was undesirable, however, the pulse duration appears too long and the pulse frequency too high. As such, it wastes compressed air. Observation of the pressure gauge on each filter manifold revealed the problem This gauge should drop only about 10 psi on a quick valve operation for about one tenth of a second. The actual pulses are more like three quarters of a second, pulling the pressure down to about 35 to 45 psi each time with a pulse spacing of only six seconds. This problem is very common with these types of filters. A fix is to install a receiver at each filter with fill control to filter out the compressed air pulses as shown in Figure 7. \u2022 Bin vibrators \u2013 Compressed air-powered vibrators were installed on the bottom of various bins to promote the flow of grain. These units consume six to 10 times the equivalent power of electric units. ##### Shown are bin vibrators used in the distillery\u2019s process. \u2022 Star seal blowing - Various blowing nozzles have been installed to clean the internals of star seals located on the top of some cookers. The constantly rotating seals meter the amount of ingredients going to the cooker, but because the input grain is dry, and the star seal is wet due to rising steam, the product sticks to the seal material, eventually creating clogs. These operations appear to be controlled with the process and therefore will shut off when not required. These blowers were measured and found to consume a peak of 126 cfm and average of 72 cfm. ##### Star seals are used on the cooker as shown to meter the amount of ingredients going to the vessel. \u2022 Alcohol vapor blowing \u2013 A pipe with holes drilled in it has been installed over the area where the product barrels are unloaded. This appears to be an attempt to control fumes. This blowing was not in service the few times the Maturing area was visited but the operator there reported it is still used to provide \u201cfree air conditioning\u201d on hot days. It appears that a fan-powered fume control system has been installed. As such, the compressed air powered ventilation can likely be removed. \u2022 Air motor pumps \u2013 The distillery uses two small pumps driven by air motors. Air motors are used to ensure safety since explosions are possible if spark-producing devices are used. The use of air motors to do the job of electric pumps is energy intensive as air motor power costs about 10 times more than a direct drive electric motor. ##### Figure 6: Dust collector operation can be improved using this solution (Source: Compressed Air Challenge.) ### Air Compressor Control Savings The existing air compressors were running in load\/unload mode with coordinated cascaded pressure bands. With this type of operation there are periods of time where the air compressor is unloaded running, consuming power, but producing no compressed air. This area of inefficiency is identified in the red as shown in Figure 7, which is a profile of C3 air compressor amp logs sorted highest to lowest to form a histogram, also called a load duration chart. This area could be addressed by choosing an air compressor with a control mode that doesn\u2019t have unloaded run time (like VSD operation). Further to this, due to air compressor cycling, there is a period of time where the air compressor is in transition between loaded and unloaded, or reverse, indicated with the green highlight. This can be addressed by eliminating air compressor cycles using VSD mode. And, in addition, there is extra power consumed by the air compressor if it runs at pressure higher than required by the plant as indicated with the yellow portion of Figure 7. This can be addressed by reducing the air compressor discharge pressure by adjustment of the pressure bands. Use of a VSD air compressor would allow a constant, lower plant pressure, rather than operating with a saw toothed waveform using load\/unload control. ##### Figure 7: The load duration profile on the original air compressor power input shows areas of possible savings. ### Conclusion At the distiller, the following action items were taken: 1. Installation of a new 132 kW oil-free VSD air compressor. 2. Adjustments to the dryer cooling purge. 3. Adjustments\/repair of dust collectors. 4. Repaired leakage, where economical. It\u2019s estimated that 75% of repairs were completed. Savings for these measures, based on air compressor histogram readings, are 257,000 kWh resulting in reduced electrical costs of \\$16,600 for a 30% energy savings. The initial studies were made possible with the financial and resource support of the local power utility. The final project was also partially funding with incentives from the power utility. Further potential savings should be expected if this compressed air user follows up on additional leakage repair and addresses additional inappropriate uses of compressed air.\n\nFor more information contact Ron Marshall, Marshall Compressed Air Consulting, tel: 204-806-2085, email:\u00a0ronm@mts.net","date":"2022-01-16 18:45:13","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\": 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\": 0.6200346946716309, \"perplexity\": 3065.5072675290294}, \"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-2022-05\/segments\/1642320300010.26\/warc\/CC-MAIN-20220116180715-20220116210715-00215.warc.gz\"}"}
| null | null |
Helmut Newcake in Paris is a 100% gluten free bakery serving brunch and lunch and also selling retail grocery items. The bakery is small and quirky with the option to eat-in or take-out the pastries made with organic gluten free flour. For lunch, they have dishes of the day that range from choices like sauteed chicken and veggies to souffle st. jacques. They serve brunch on Sundays, by appointment. Their range of pastries is large and is always growing. Many of them are also dairy free.
This is the first (and possibly only) gluten free patisserie in Paris. I made a special journey here in June and I could have stayed the whole weekend. The cakes and tarts were to die for, I was amazed to find perfect choux pastry and beautiful cakes that I wish I could have taken home. They also offer a Sunday brunch menu, but it's quite expensive - I had it as it was a one-off treat and it was pretty sunstantial. Marie & Francois (the owners) were lovely and they have clearly worked incredibly hard to achieve their dream - I hope it lasts. My advice is go there hungry, wear stretchy pants and take whatever you can home too.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,508
|
{"url":"https:\/\/www.physicsforums.com\/threads\/including-several-tex-files-in-one-multiple-defined-labels.1002691\/","text":"# Including several tex files in one: multiple defined labels\n\nHi PF!\n\nI have a divided a main.tex into chapters, chapter_1.tex, chapter_2.tex.... Each chapter is included in the main.tex via \\include{chapter_1}. Each chapter also references equations. Is there a way to have each of the chapter_1.tex only reference that file's equations, sections, figures? As is, I have multiple defined labels.\n\nThanks so much!\n\nWrichik Basu\nGold Member\n2020 Award\nWhy don't you generate the pdf for each chapter separately, and then combine them into one file? This would be better than including all those chapters into the main .tex file. Adobe online, for instance, allows you to combine files for free.\n\nDrClaude\nMentor\nWhy not simply change the labels? Each file\/chapter could have its own label prefix to make sure there can be no multiply defined labels (it is easy to perform a search and replace in each file to include those prefixes).\n\njoshmccraney, Orodruin, pasmith and 2 others\npasmith\nHomework Helper\nWhy don't you generate the pdf for each chapter separately, and then combine them into one file? This would be better than including all those chapters into the main .tex file. Adobe online, for instance, allows you to combine files for free.\n\nThat breaks things if one chapter makes references to the content of another, or you have page references, or you want to generate a single bibliography for the entire work.\n\nWrichik Basu, joshmccraney and FactChecker\nOrodruin\nStaff Emeritus","date":"2021-05-10 19:57:36","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\": 0.91082364320755, \"perplexity\": 2630.1743918141196}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-21\/segments\/1620243991759.1\/warc\/CC-MAIN-20210510174005-20210510204005-00122.warc.gz\"}"}
| null | null |
{"url":"http:\/\/mathematica.stackexchange.com\/questions\/7149\/how-to-keep-some-of-the-results-of-the-nestlist","text":"# How to keep some of the results of the NestList\n\nI'm using some iterative arithmetics to calculate wave propagation with the help of NestList. I have to use a small step size for iteration to guarantee the accuracy, which lead to too much data (e.g, 10^6 points) for further plotting, and the results consumes too much memories. Is there any clever way to cope with it? I didn't intend to use loops.\n\nThis is a simplified example:\n\nNestList[Sin, 1, 10^6]\n\n\nOne of my idea is to use Sow and Reap. But I've no idea to integrate them with NestList or Nest\n\nAny idea???\n\n-\nYou've tried NestList[Nest[f, #, k] &, start, m]? \u2013\u00a0J. M. Jun 21 '12 at 5:18\nVery good idea. \u2013\u00a0yulinlinyu Jun 21 '12 at 5:29\n@J.M. Surely deserves to be an answer ;-) \u2013\u00a0Vitaliy Kaurov Jun 21 '12 at 5:31\n\nDue to insistent public demand:\n\nIf, in a sequence of iterates $\\{x,f(x),f(f(x)),\\dots\\}$, one only needs every $k$-th iterate (say, for $k=3$, you want $\\{x,f(f(f(x))),f(f(f(f(f(f(x)))))),\\dots\\}$), then one can cleverly combine Nest[] and NestList[] like so:\n\nNestList[Nest[f, #, k] &, start, n]\n\n\nwhich yields a list containing the zeroth, $k$-th, $2k$-th, ... $nk$-th iterates.\n\n-\nFold[f[#1] &, x, Range[#]] & \/@ Range[0, 9, 3]\n(* or *)\nNest[f, x, #] & \/@ Range[0, 9, 3]\n(* both give: *)\n{x, f[f[f[x]]], f[f[f[f[f[f[x]]]]]], f[f[f[f[f[f[f[f[f[x]]]]]]]]]}\n\n\nEDIT: a variation on the second method:\n\nnestSkip[f_, x_, stepsize_Integer, numsteps_Integer] :=\nNest[f, x, stepsize #] & \/@ Range[0, numsteps]\n(* examples: *)\nnestSkip[g, y, 2, 2]\n(* ==> {y, g[g[y]], g[g[g[g[y]]]]} *)\nnestSkip[# + 5 &, 2, 3, 3]\n(* ==> {2, 17, 32, 47} *)\n\n-\nAnother very clever solution. \u2013\u00a0yulinlinyu Jun 21 '12 at 8:31\nThank you @yulinlinyu. \u2013\u00a0kglr Jun 21 '12 at 8:35\n\nJ. M.'s method is elegant but in some cases it is not optimal. This is because the inner Nest[f, #, k] & does not compile the same as an explicit series of function calls. In advantageous cases if we expand the inner operation in advance we can have very large performance gains.\n\njmNest[f_, k_, n_, start_] :=\nNestList[Nest[f, #, k] &, start, n]\n\nwNest[f_, k_, n_, start_] :=\nNestList[Evaluate @ Nest[f, #, k] &, start, n]\n\n\nA favorable test case:\n\njmNest[1 + # &, 200, 1*^5, 10] \/\/ RepeatedTiming \/\/ First\nwNest[1 + # &, 200, 1*^5, 10] \/\/ RepeatedTiming \/\/ First\n\n0.2407\n\n0.00191\n\n\nHere we get more than two orders of magnitude improvement, and it's easy to understand why as 200 applications of 1 + # & reduces to 200 + # &.\n\nIn a less trivial case such as three applications of Sin, which does not reduce to a simpler formula, we still have an improvement:\n\njmNest[Sin, 3, 1*^6, 10] \/\/ RepeatedTiming \/\/ First\nwNest[Sin, 3, 1*^6, 10] \/\/ RepeatedTiming \/\/ First\n\n0.08481\n\n0.05829\n\n\nThere are of course pathological cases as well where the symbolic expansion is far uglier than the sequential application:\n\njmNest[3 + # + Sqrt[# + 7] &, 15, 30, 10] \/\/ RepeatedTiming \/\/ First\nwNest[3 + # + Sqrt[# + 7] &, 15, 30, 10] \/\/ RepeatedTiming \/\/ First\n\n0.000460\n\n0.703\n\n\nI cannot think of a simple way to intelligently select between original and pre-expansion methods that does not introduce significant overhead. There is enough potential difference in speed that perhaps a ParallelTry is worth that overhead if unused cores are available for use.\n\n-\n\nCertainly not as elegant as JMs excellent solution, but it does use Sow and Reap as the OP requested and avoids repeated recalculation of intermediate results.\n\nClear[selectiveNestList];\nselectiveNestList[function_, step_, iterations_, initialValue_] :=\nReap[Nest[\nWith[{s = function[First@#]},\nIf[Mod[Last@#, step] === 0,\nSow[s]; {s, Last@# + 1}, {s, Last@# + 1}]] &, {initialValue,\n0}, iterations ]][[2, All, All, 1]] \/\/ First\n\n\nAnd in use:\n\nselectiveNestList[Cos, 2, 9, 4]\n\n\n{4, Cos[Cos[4]], Cos[Cos[Cos[Cos[4]]]],Cos[Cos[Cos[Cos[Cos[Cos[4]]]]]], Cos[Cos[Cos[Cos[Cos[Cos[Cos[Cos[4]]]]]]]]}\n\n-","date":"2016-07-23 15:08:14","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\": 0.19797031581401825, \"perplexity\": 5839.391990679413}, \"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-2016-30\/segments\/1469257823072.2\/warc\/CC-MAIN-20160723071023-00174-ip-10-185-27-174.ec2.internal.warc.gz\"}"}
| null | null |
All of the Particle devices use a boot loader. This small bit of code is the first thing that runs when the processor boots, and is responsible for initial processor setup, DFU mode, and loading system firmware.
## Automatic updating of the boot loader
In general you do not need to manage the boot loader directly. System firmware will update the boot loader itself when needed, and as you upgrade the system firmware.
## What version of the boot loader do I have?
Put the device in listening mode (blinking dark blue) by holding down the SETUP/MODE button and use the Particle CLI particle serial inspect command to find out what version of the boot loader is installed:
```
$ particle serial inspect
Platform: 6 - Photon
Modules
Bootloader module #0 - version 11, main location, 16384 bytes max size
Integrity: PASS
Address Range: PASS
Platform: PASS
Dependencies: PASS
System module #1 - version 108, main location, 262144 bytes max size
Integrity: PASS
Address Range: PASS
Platform: PASS
Dependencies: PASS
```
This shows that the Photon is running with system firmware 0.6.2 (version 108) and the boot loader from 0.6.2 (version 11). The mapping of these versions can be found on the [version mapping table page](https://github.com/spark/firmware/blob/develop/system/system-versions.md).
At the time of writing, the most common boot loader versions are:
- 7 (system firmware 0.4.9 - 0.6.0)
- 9 (0.6.1-rc, 0.6.1-final, 0.6.2-rc)
- 11 (0.6.2, 0.6.3, 0.6.4)
- 100 (0.7.0-rc.1)
- 101 (0.7.0-rc.5
## Boot loader downloads
You can download the v7 (0.4.9) bootloaders for the Photon, P1, and Electron on the [JTAG instructions page](https://docs.particle.io/faq/particle-tools/jtag/electron/#programming-the-boot-loader).
The boot loaders v11 (0.6.2) and v100 (0.7.0) can be found on the [0.7.0 release page](https://github.com/spark/firmware/releases/tag/v0.7.0-rc.2).
## Dim D7 (corrupted boot loader)
One of the things that can happen is that the boot loader sector (sector 0) becomes corrupted to erased. When this happens, when the processor starts the only thing that happens is the D7 LED will be on dimly. This is a side effect of being in JTAG/SWD mode.
The only thing you can do when this happens is reprogram the boot loader using JTAG/SWD. Since there is no code running at all, you can't use other modes like DFU mode and USB.
## Updating the bootloader in listening mode
Though somewhat counter-intuitive, you can update the boot loader when in listening mode (blinking blue), but not in DFU mode (blinking yellow).
The reason is that DFU mode runs from the boot loader flash memory sector. The STM32 processors used in the Particle devices run code directly from flash memory, not copied into RAM, so you can't reprogram the boot loader while running the boot loader.
Put the device in listening mode (blinking dark blue) by holding down the SETUP/MODE button.
Then you can use the CLI to flash in USB serial mode, for example:
```
$ particle flash --serial bootloader-photon.bin
```
Note that there are different boot loader binaries for each platform (Photon, P1, Electron, etc.).
This technique can be handy for manually downgrading the boot loader, but make sure you install the older version of system firmware first. The reason is that the device will reboot after flashing the boot loader by serial, and if you still have a newer version of system firmware, it will upgrade the boot loader again.
This can also be done OTA by specifying the device name or device ID instead of --serial, if the device is still able to get to breathing cyan or breathing magenta.
## Updating the boot loader by JTAG/SWD
If you have a corrupted or erased boot loader (Dim D7) or another problem that prevents getting into listening mode (blinking blue) and requires manually updating the boot loader, you'll need to use JTAG/SWD.
The [instructions for using JTAG/SWD mode are here](https://docs.particle.io/faq/particle-tools/jtag/electron/).
## Special considerations for downgrading from 0.7.0
System version 0.7.0 is a special case when it comes to the boot loader. While the 0.6.2 boot loader can safely run system firmware back to 0.4.9, this is not the case with the 0.7.0 boot loader (version 100).
When you downgrade from 0.7.0 to an earlier version like 0.6.2, you need to downgrade in a specific order, as described in the [release notes for 0.7.0](https://community.particle.io/t/particle-firmware-updates-thread/14378/49).
A much easier method, however, is to downgrade to a version that knows how to download the bootloader automatically:
These include:
- For Electron: [0.6.4](https://github.com/particle-iot/firmware/releases/tag/v0.6.4)
- For Photon/P1: [0.6.3](https://github.com/particle-iot/firmware/releases/tag/v0.6.3)
- For all: [0.5.5](https://github.com/particle-iot/firmware/releases/tag/v0.5.5)
### If you have upgraded to 0.7.0 and subsequently downgraded to 0.6.2 or earlier
If you previously upgraded to 0.7.0 and then downgraded, by particle update, or by the particle device doctor, to 0.6.2 without manually downgrading the boot loader, you'll have to upgrade to 0.7.0 again before downgrading again (sorry).
Note you only need to do this if you actually need 0.6.2. If you only need 0.6.x you should downgrade to 0.6.3 or 0.6.4 which know how to downgrade the bootloader automatically.
#### Photon
From the [release site](https://github.com/spark/firmware/releases/tag/v0.7.0-rc.3), download:
- [system-part1-0.7.0-rc.3-photon.bin](https://github.com/spark/firmware/releases/download/v0.7.0-rc.3/system-part1-0.7.0-rc.3-photon.bin)
- [system-part2-0.7.0-rc.3-photon.bin](https://github.com/spark/firmware/releases/download/v0.7.0-rc.3/system-part2-0.7.0-rc.3-photon.bin)
Then the Photon in DFU mode (blinking yellow) and apply the update:
```
particle flash --usb system-part1-0.7.0-rc.3-photon.bin
particle flash --usb system-part2-0.7.0-rc.3-photon.bin
```
#### P1
From the [release site](https://github.com/spark/firmware/releases/tag/v0.7.0-rc.3), download:
- [system-part1-0.7.0-rc.3-p1.bin](https://github.com/spark/firmware/releases/download/v0.7.0-rc.3/system-part1-0.7.0-rc.3-p1.bin)
- [system-part2-0.7.0-rc.3-p1.bin](https://github.com/spark/firmware/releases/download/v0.7.0-rc.3/system-part2-0.7.0-rc.3-p1.bin)
Then the P1 in DFU mode (blinking yellow) and apply the update:
```
particle flash --usb system-part1-0.7.0-rc.3-p1.bin
particle flash --usb system-part2-0.7.0-rc.3-p1.bin
```
#### Electron
From the [release site](https://github.com/spark/firmware/releases/tag/v0.7.0-rc.3), download:
- [system-part1-0.7.0-rc.3-electron.bin](https://github.com/spark/firmware/releases/download/v0.7.0-rc.3/system-part1-0.7.0-rc.3-electron.bin)
- [system-part2-0.7.0-rc.3-electron.bin](https://github.com/spark/firmware/releases/download/v0.7.0-rc.3/system-part2-0.7.0-rc.3-electron.bin)
- [system-part3-0.7.0-rc.3-electron.bin](https://github.com/spark/firmware/releases/download/v0.7.0-rc.3/system-part3-0.7.0-rc.3-electron.bin)
Then the Electron in DFU mode (blinking yellow) and apply the update:
```
particle flash --usb system-part1-0.7.0-rc.3-electron.bin
particle flash --usb system-part2-0.7.0-rc.3-electron.bin
particle flash --usb system-part3-0.7.0-rc.3-electron.bin
```
### Downgrading from 0.7.0 to 0.6.2
Note you only need to do this if you actually need 0.6.2. If you only need 0.6.x you should downgrade to 0.6.3 or 0.6.4 which know how to downgrade the bootloader automatically.
#### Photon
From the [release site](https://github.com/spark/firmware/releases/tag/v0.6.2), download:
- [system-part1-0.6.2-photon.bin](https://github.com/spark/firmware/releases/download/v0.6.2/system-part1-0.6.2-photon.bin)
- [system-part2-0.6.2-photon.bin](https://github.com/spark/firmware/releases/download/v0.6.2/system-part2-0.6.2-photon.bin)
- [bootloader-0.6.2-photon.bin](https://github.com/spark/firmware/releases/download/v0.7.0-rc.3/bootloader-0.6.2-photon.bin)
Put the device in listening mode (blinking blue) and apply the updates in this order:
```
particle flash --serial system-part2-0.6.2-photon.bin
particle flash --serial bootloader-0.6.2-photon.bin
particle flash --serial system-part1-0.6.2-photon.bin
```
Note: It must be done in this order (2, bootloader, 1). This update can also be done OTA. The bootloader can only be flashed serial or OTA, you cannot flash the bootloader in DFU mode (blinking yellow).
#### P1
From the [release site](https://github.com/spark/firmware/releases/tag/v0.6.2), download:
- [system-part1-0.6.2-p1.bin](https://github.com/spark/firmware/releases/download/v0.6.2/system-part1-0.6.2-p1.bin)
- [system-part2-0.6.2-p1.bin](https://github.com/spark/firmware/releases/download/v0.6.2/system-part2-0.6.2-p1.bin)
- [bootloader-0.6.2-p1.bin](https://github.com/spark/firmware/releases/download/v0.7.0-rc.3/bootloader-0.6.2-p1.bin)
Put the device in listening mode (blinking blue) and apply the updates in this order:
```
particle flash --serial system-part2-0.6.2-p1.bin
particle flash --serial bootloader-0.6.2-p1.bin
particle flash --serial system-part1-0.6.2-p1.bin
```
Note: It must be done in this order (2, bootloader, 1). This update can also be done OTA. The bootloader can only be flashed serial or OTA, you cannot flash the bootloader in DFU mode (blinking yellow).
#### Electron
From the [release site](https://github.com/spark/firmware/releases/tag/v0.6.2), download:
- [system-part1-0.6.2-electron.bin](https://github.com/spark/firmware/releases/download/v0.6.2/system-part1-0.6.2-electron.bin)
- [system-part2-0.6.2-electron.bin](https://github.com/spark/firmware/releases/download/v0.6.2/system-part2-0.6.2-electron.bin)
- [system-part3-0.6.2-electron.bin](https://github.com/spark/firmware/releases/download/v0.6.2/system-part3-0.6.2-electron.bin)
- [bootloader-0.6.2-electron.bin](https://github.com/spark/firmware/releases/download/v0.7.0-rc.3/bootloader-0.6.2-electron.bin)
Put the device in listening mode (blinking blue) and apply the updates in this order:
```
particle flash --serial system-part3-0.6.2-electron.bin
particle flash --serial system-part2-0.6.2-electron.bin
particle flash --serial system-part1-0.6.2-electron.bin
particle flash --serial bootloader-0.6.2-electron.bin
```
This update can also be done OTA. The bootloader can only be flashed serial or OTA, you cannot flash the bootloader in DFU mode (blinking yellow).
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,793
|
{"url":"https:\/\/ethos.bl.uk\/OrderDetails.do?uin=uk.bl.ethos.767930","text":"Use this URL to cite or link to this record in EThOS: https:\/\/ethos.bl.uk\/OrderDetails.do?uin=uk.bl.ethos.767930\nTitle: Conceptual design of a breed & burn molten salt reactor\nAuthor: Kasam, Alisha\nISNI:\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0000 0004 7651 6652\nAwarding Body: University of Cambridge\nCurrent Institution: University of Cambridge\nDate of Award: 2019\nAvailability of Full Text:\n Access from EThOS: Full text unavailable from EThOS. Please try the link below. Access from Institution:\nAbstract:\nA breed-and-burn molten salt reactor (BBMSR) concept is proposed to address the Generation IV fuel cycle sustainability objective in a once-through cycle with low enrichment and no reprocessing. The BBMSR uses separate fuel and coolant molten salts, with the fuel contained in assemblies of individual tubes that can be shuffled and reclad periodically to enable high burnup. In this dual-salt configuration, the BBMSR may overcome several limitations of previous breed-and-burn (B$\\&$B) designs to achieve high uranium utilisation with a simple, passively safe design. A central challenge in design of the BBMSR fuel is balancing the neutronic requirement of large fuel volume fraction for B$\\&$B mode with the thermal-hydraulic requirements for safe and economically competitive reactor operation. Natural convection of liquid fuel within the tubes aids heat transfer to the coolant, and a systematic approach is developed to efficiently model this complex effect. Computational fluid dynamics modelling is performed to characterise the unique physics of the system and produce a new heat transfer correlation, which is used alongside established correlations in a numerical model. A design framework is built around this numerical model to iteratively search for the limiting power density of a given fuel and channel geometry, applying several defined temperature and operational constraints. It is found that the trade-offs between power density, core pressure drop, and pumping power are lessened by directing the flow of coolant downwards through the channel. Fuel configurations that satisfy both neutronic and thermal-hydraulic objectives are identified for natural, 5$\\%$ enriched, and 20$\\%$ enriched uranium feed fuel. B$\\&$B operation is achievable in the natural and 5$\\%$ enriched versions, with power densities of 73 W\/cm$^3$ and 86 W\/cm$^3$, and theoretical uranium utilisations of 300 $\\mathrm{MWd\/kgU_{NAT}}$ and 25.5 $\\mathrm{MWd\/kgU_{NAT}}$, respectively. Using 20$\\%$ enriched feed fuel relaxes neutronic constraints so a wider range of fuel configurations can be considered, but there is a strong inverse correlation between power density and uranium utilisation. The fuel design study demonstrates the flexibility of the BBMSR concept to operate along a spectrum of modes ranging from high fuel utilisation at moderate power density using natural uranium feed fuel, to high power density and moderate utilisation using 20$\\%$ uranium enrichment.\nSupervisor: Shwageraus, Eugene Sponsor: University of Cambridge ; Ford Britain Trust\nQualification Name: Thesis (Ph.D.) Qualification Level: Doctoral\nEThOS ID: uk.bl.ethos.767930\u00a0 DOI:\nKeywords: nuclear ; molten salt ; mixed convection ; computational fluid dynamics ; breed & burn ; fast reactors ; fuel cycle ; Generation IV ; neutronic ; thermal-hydraulic ; power density ; uranium utilisation ; breeding ; natural uranium ; low-enriched ; fluoride salt ; chloride salt ; Serpent ; WIMS ; sodium fast reactor ; fast spectrum ; heat transfer correlation ; finite-difference model ; Monte Carlo ; OpenFOAM ; molten salt reactor ; MATLAB\nShare:","date":"2023-02-04 02:06:04","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\": 0.2844206988811493, \"perplexity\": 7221.537204668454}, \"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\/1674764500080.82\/warc\/CC-MAIN-20230204012622-20230204042622-00078.warc.gz\"}"}
| null | null |
\section{\label{sec:intro}Introduction}
One of the earliest recognitions of the importance of fluctuations in the energy dissipation rate in turbulence can be found in a footnote by Landau in the textbook on fluid mechanics \cite{Landau:1959}. The footnote explains that universal formulas for the small scales of structure functions do not exist because the energy dissipation rate will fluctuate on long time scales, and these fluctuations will be different in different flows. Frisch \cite{Frisch:1995} provides an extended discussion of the footnote. In the refined similarity theory by Kolmogorov~\cite{Kolmogorov:1962} and Obukhov~\cite{Obukhov:1962}, this insight on universality is extended to include fluctuations that result from the random character of the transfer of energy between scales, which is often called internal intermittency. Kolmogorov~\cite{Kolmogorov:1962} gives Landau credit for recognizing the importance of internal intermittency. However, this credit seems to be somewhat misplaced since the available published text by Landau observes only that large scale fluctuations in the energy dissipation will destroy universality of small scales \cite{Frisch:1995, Mouri:2006}. During the intensive effort to understand internal intermittency over the past 50 years, the direct application of Landau's insight about the importance of large scale fluctuations has often been obscured.
The refined similarity theory by Obukhov~\cite{Obukhov:1962} and Kolmogorov~\cite{Kolmogorov:1962} proposed that in the inertial range the moments of velocity differences between two points are universal functions when they are conditioned on the locally averaged value of the energy dissipation rate, $\varepsilon_r$, defined as the instantaneous energy dissipation rate averaged over a sphere of radius $r$. For simplicity we will consider the longitudinal component of the velocity differences, $\Delta_r u$. The conditional moments are
\begin{equation}
\label{eqn:Delta_r}
\langle(\Delta_r u)^p|\varepsilon_r \rangle = C_p(\varepsilon_r r)^{p/3},
\end{equation}
where $C_p$ are universal constants~\citep{Pope:2000}. Averaging this expression over a distribution of $\varepsilon_r$ yields
\begin{equation}
\langle(\Delta_ru)^p\rangle = C_p \langle\varepsilon_r^{p/3} \rangle r^{p/3} = C_p \frac{\langle\varepsilon_r^{p/3} \rangle}{\varepsilon^{p/3}} (\varepsilon r)^{p/3},
\label{refined_avg}
\end{equation}
where $\varepsilon=\langle \varepsilon_r \rangle$ is the mean energy dissipation rate. Since the moments of $\varepsilon_r$ depend on $r$, this means that the inertial range scaling law is modified by internal intermittency. Kolmogorov proposed that the fluctuations of $\varepsilon_r$ could be described with a power law scaling
\begin{equation}
\frac{\langle \varepsilon_r^p \rangle}{ \varepsilon ^p} \propto \left(\frac{L}{r}\right)^{\xi_p}.
\end{equation}
where $L$ is a length characterizing the energy input scale. In Kolmogorov (1962)\cite{Kolmogorov:1962}, a log-normal model was used to relate $\xi_p$ for all $p$ to $\xi_2=\mu$, which is commonly called the intermittency exponent. An extensive literature has explored the $r$ dependence of statistics of $\varepsilon_r$ in order to understand anomalous scaling exponents in the inertial range ~\cite{Sreenivasan:1997}.
However, the effects of fluctuations in the energy dissipation rate due to the large scales has been given much less attention, even though this is the direct application of Landau's original comment. Kolmogorov did state that the coefficients in the scaling law should not be universal, presumably because he recognized that large scale fluctuations would not be universal \cite{Kolmogorov:1962}.
Monin and Yaglom \cite{Monin:1971} provide a simple model at the beginning of their section titled ``Refined Treatment of the Local Structure of Turbulence, taking into account fluctuations in the dissipation rate''. An extended presentation of this model is in the textbook by Davidson~\cite{Davidson:2004}. They consider averaging together equal numbers of samples from two different turbulent states: state 1 with energy dissipation rate $\varepsilon_1=(1+\gamma)\langle \varepsilon \rangle$ and another state 2 with $\varepsilon_2=(1-\gamma)\langle \varepsilon \rangle$. Here $\langle \varepsilon \rangle$ is the mean energy dissipation rate and $\gamma$ is a measure of the difference in energy dissipation between the two states. Then equation~(\ref{refined_avg}) implies that a measured second order structure function in the inertial range averaged over equal contributions from each state would be
\begin{equation}
\langle(\Delta_ru)^2\rangle = \frac{C_2}{2} \left[ (1+\gamma)^{2/3} + (1-\gamma)^{2/3} \right] (\varepsilon r)^{2/3}.
\label{MY_model}
\end{equation}
So the large scale fluctuations in the energy dissipation are predicted to change the coefficient of the inertial range scaling law without changing the power law scaling. In this model, $\gamma$ must be less than or equal to one, so the coefficient of the second order structure function can decrease to as low as $C_2/2^{1/3} \approx 0.794 \: C_2$ for the case $\gamma=1$ where there is no energy injection in state 2.
This model is easily extended to the case where samples are included from state 1 with probability $\beta$ and from state 2 with probability $1-\beta$. Now the energy dissipation rates are $\varepsilon_1=(1+(1-\beta)\gamma/\beta)\langle \varepsilon \rangle$ and
$\varepsilon_2=(1-\gamma)\langle \varepsilon \rangle$. For this extended model, the measured structure function of order $p$ would be
\begin{equation}
\langle(\Delta_ru)^p\rangle= \kappa(\beta,\gamma) \; C_p \left(\langle \varepsilon \rangle r \right)^{p/3}.
\label{MY_model_duty}
\end{equation}
where the correction factor of the coefficient is
\begin{equation}
\kappa(\beta,\gamma)=\left[ \beta \left(1+\frac{1-\beta}{\beta} \gamma \right)^{p/3} + (1-\beta)\left(1-\gamma\right)^{p/3}\right].
\label{Eq:correction_factor}
\end{equation}
In the limiting case $\gamma=1$ and $\beta \rightarrow 0$, the coefficient for $p=2$ goes to zero, and the coefficients for $p>3$ go to infinity, so the effects of large scale fluctuations on the small scale statistics can be very large. In this limiting case, the flow consists of brief pulses of large energy input between long periods of no energy input.
In both Monin and Yaglom \cite{Monin:1971} and Davidson \cite{Davidson:2004}, the presentation of the model in equation (\ref{MY_model}) is followed with the observation that in typical situations this effect is not large.
Figure~\ref{fig:contour} shows a contour plot of the correction factor for $p=2$ in equation~\ref{Eq:correction_factor} as a function of the fluctuations in the energy input, $\gamma$, and the fraction of the time spent in the high energy input state, or duty cycle, $\beta$. The observation that the correction is not large in most cases is justified since the correction is less than 2.4\% for half of the parameter space for $p=2$. However, the correction can be very large in some flows. There is always a divergence for $\gamma=$1 and $\beta \rightarrow $0, and for large $p$, the correction is larger. Although this two state model is a simple idealization, we will show that it provides a reasonably good description of some of our data.
\begin{figure}
\begin{center}
\includegraphics[width=3.3in]{second_order_model_contour_plot.eps}
\caption[The correction factor $ \kappa $ for $p=2$]{\label{fig:contour} Contour plot of the correction factor $ \kappa $ from equation~(\ref{Eq:correction_factor}) for $p=2$, showing the change in coefficients in the inertial range the scaling law as a function
of the amplitude of fluctuations in the energy input $\gamma$, and the time spent in the high energy input state, or duty cycle, $\beta$. }
\end{center}
\end{figure}
In real flows, the energy dissipation rate and $\varepsilon_r$ have continuous distributions. In the continuous case, equation~(\ref{refined_avg}) can be used to predict the behavior of structure functions, but there are now contributions to the distribution of $\varepsilon_r$ from both internal intermittency and fluctuations in the energy input. In particular, $\varepsilon_r$ for $r \ge L$ has a distribution which is determined not by cascade processes but by the mechanisms creating the turbulence. In cases where internal intermittency can be ignored, we can estimate the fluctuations in the energy input and predict the coefficients of scaling law.
If the mean square velocity, $3U^2=\langle u_i u_i \rangle$ and the integral length scale, $L$, are defined using ensemble averages, then they can be considered to be time dependent. In this case, $\varepsilon \propto U^3/L$ provides an estimate of the instantaneous energy dissipation rate. If time averages of this dissipation rate are then used in equation~(\ref{refined_avg}) we obtain
\begin{equation}
\langle(\Delta_r u)^p\rangle = C_p \frac{\langle(U^3/L)^{p/3} \rangle}{\langle U^3/L\rangle^{p/3}} (\varepsilon r)^{p/3}.
\label{refined_model_u}
\end{equation}
In our flow, where $L$ has a weak dependence on the variations in the energy input, this simplifies to
\begin{equation}
\langle(\Delta_ru)^p\rangle = C_p \frac{\langle U^p \rangle}{\langle U^3 \rangle^{p/3}} (\varepsilon r)^{p/3}.
\label{refined_model}
\end{equation}
If internal intermittency is also important, then the two effects may be combined as
\begin{equation}
\langle(\Delta_ru)^p\rangle = C'_p \frac{\langle(U^3/L)^{p/3} \rangle}{\langle U^3/L\rangle^{p/3}} \left(\frac{L}{r}\right)^{\xi_p} (\varepsilon r)^{p/3}.
\label{refined_intermittent}
\end{equation}
It is important to determine the size of the effects of fluctuations in the large scale energy input in real turbulent flows. Surprisingly, there are no published results that we know of that document a dependence of coefficients of inertial range scaling laws for structure functions on systematic changes in the large scales of the flow. A compilation of experimental~\cite{Sreenivasan:1995} and simulation~\cite{Donzis:2010} results have given credence to the notion that the second order coefficients are close enough to independent of the flow that they can be treated as universal constants. At least three experimental studies have explored fluctuations in the large scale energy input in detail. Praskovsky et al. \cite{Praskovsky:1993} study two high Reynolds number flows, a mixing layer and a return channel. They find a conditional dependence of the second order structure functions on the instantaneous velocity and connect this with spatial and temporal variability of the energy flux passing through the cascade. They emphasize that the conditional dependence they observe is not in violation of the assumptions of the refined Kolmogorov theory since changes in the energy flux should change the small scales.
Sreenivasan et al. \cite{Sreenivasan:1996} use measurements in the atmospheric boundary layer to demonstrate the conditional dependence of structure functions on the velocity. They identify this conditional dependence as a result of mixed averages over regions of the flow with different energy dissipation rate and show that when properly normalized by the instantaneous local energy dissipation rate that the conditional dependence is removed in agreement with Kolmogorov's refined similarity hypotheses.
More recently, Mouri et al. \cite{Mouri:2006} explored the effects of large scale fluctuations of the turbulence energy dissipation rate. They measure grid and boundary layer turbulence and clearly confirm that the large scale energy fluctuations exist and that they affect small-scale statistics. They explicitly state the the large scale fluctuations do not affect the power law scaling or the coefficients of second order structure functions in the inertial range.
There is another set of literature exploring time dependent energy input in turbulence that has identified the presence of response maxima when the energy input oscillates with a period on the order of the large eddy turn-over time. This effect was first predicted in a mean field theory~\cite{vanderHeydt:2003}. It has been explored in a variety of models, numerical simulations and experiments~\cite{Cadot:2003, vanderHeydt:2003, Kuczaj:2006, Bos:2007, Kuczaj:2008, Jin:2008}. However, this work has focused on modulation periods near the turn-over time and seems not to have considered the effects on structure functions, which are most prominent for long modulation periods.
In this paper we present a series of experimental measurements of the effects of time-dependent energy input on the small scales of turbulence. We focus on second order structure functions where the effects of internal intermittency are small. We find that the coefficient of the inertial range scaling law depends on the fluctuations in the large scale energy input and measure coefficients that are more than 20\% below the value for the continuously driven case.
\section{Experiment}
The turbulence is generated in an octagonal Plexiglas tank that is 1 x 1 x 1.5 m$^3$ filled with approximately 1100 $l$ of filtered and degassed water. Two identical octagonal grids oscillate in phase to generate the turbulence. The grids have 8 cm mesh size, 36\% solidity, and are evenly spaced from the top and bottom of the tank with a 56.2 cm spacing between grids and a 1 cm gap between the grids and the tank walls. The grid oscillation has 12 cm amplitude and is powered by an 11kW motor. In these experiments, the grids were oscillated with frequencies up to 4 Hz which allows Taylor Reynolds numbers up to $R_\lambda=271$. Details about the experimental setup are available in Blum et al.~\cite{Blum:2010}.
We use stereoscopic particle tracking using four cameras as shown in figure~\ref{fig:cameras}. The cameras are two Bassler A504K video cameras capable of 1280 x 1024 pixel resolution at 480 frame per second, and two Mikrotron MC1362 cameras with the same pixel resolution and data rates, but with greater sensitivity. A 5 x 5 x 5 cm$^{3}$ detection volume at the center of the flow was illuminated with a pulsed 50 W Nd:YAG laser. A real-time image compression circuit with compression factors of 100 to 1000 enables us to acquire data continuously, which allows access to large data sets of particle trajectories~\cite{Chan:2007}.
\begin{figure}
\begin{center}
\includegraphics[width=4.5in]{Tank_Diagram8.eps}
\caption{\label{fig:cameras} Experimental setup. Four high speed cameras obtain stereoscopic images of a (5 cm)$^3$ volume at the center of the flow that is illumined by a pulsed Nd:YAG laser with 50 W average power. }
\end{center}
\end{figure}
Previous work with this experiment has shown that there are measurable fluctuations in the energy input even when the driving frequency of the oscillating grids is constant~\cite{Blum:2010}. Here we augment this effect by modulating the driving frequency of the oscillating grids.
For example, rather than driving the grids continuously at 3 Hz, we can drive it at 3 Hz for 15 s, and then halt for 15 s, and repeat. This produces a periodic time dependence in the energy input with a longer time scale than the grid oscillation period. Figure \ref{fig:cartoon} shows a schematic of the frequency modulation along with variable definitions. In this paper, we explore three different ways to augment the fluctuations in large scale energy input: (1) change $T$, the time to complete one modulation cycle (2) change the frequency modulation by holding $f_{high}$ constant and changing $f_{low}$ from 0 up to $f_{high}$, and (3) changing the duty cycle $t_{high}/T$. Figure~\ref{fig:cartoon_energy} shows a specific example of the time dependence of the mean square velocity, $\langle u_i u_i \rangle$, which is a measure of the energy in the large scales. The mean is obtained as a phase average over many cycles. It takes time for energy to propagate from the grid to the detection volume, so the energy lags several seconds after the grid frequency changes.
\begin{figure*}
\begin{center}
\includegraphics[width=5.7in]{demostration_of_grid_velocity_variation.eps}
\caption{\label{fig:cartoon} A sketch of the position and frequency of the oscillating grids as a function of time. $t_{high}$ is the time over which the grids oscillate at the higher frequency, $t_{low}$ is the time at lower frequency. $T$ is the cycle period, the time to complete one cycle of modulation from high to low frequency. $f_{high}$ is the high frequency of grids and $f_{low}$ is the low frequency. $\Delta f$ is the frequency differences between $f_{high}$ and $f_{low}$. }
\end{center}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[width=4.5in]{demostration_of_grid_energy_paper.eps}
\caption{\label{fig:cartoon_energy} Time dependence of the mean square velocity measured by phase averaging over many cycles for an experiment with $f_{high} = 3$ Hz, $f_{low} = 0$ Hz, Period $T =$ 24 s, and $50 \%$ duty cycle. Both the first and second cycle are phase averages over the whole experiments and hence identical.}
\end{center}
\end{figure*}
The inertia of the system used to drive the grids limited the rate at which the driving frequency could be changed. We were able to reduce the time required to stop or start to less than 1/3 of a second by minimizing the inertia in the experiment. The original version of this apparatus~\cite{Blum:2010} used a flywheel to improve symmetry between the up and down stroke of the oscillating grids. For this experiment we replaced the flywheel with a coupler. For the run with $f_{high} = 3$ Hz shown in figure~\ref{fig:cartoon_energy}, the start time is less than one oscillation and accounts for less than 3\% of the data. However, limitations from the inertia of the drive system did limit our experiments to periods of $T= 3$ s and greater, which resulted in the period of the modulation of the energy input always being longer than the large scale turn over time.
We conducted three sets of experiments to explore the effects of fluctuations of large scale energy input on small scales. Parameters for each of the experiments are given in table~\ref{table:LSI}. In the first set of experiments we made measurements with period $T$ of 3, 6, 12, 24, 48, and 384 seconds while always modulating the grid frequency with ($f_{high}$ - $f_{low}$) = (3 - 0) Hz, with a duty cycle of 50\%. We will refer to these experiments as ``varying the period''. In the second set of experiments, we held $f_{high} =$ 3 Hz and made measurements with$f_{low}$ of 3, 2, 1, and 0 Hz to get ($f_{high}$ - $f_{low}$) = (3 - 3), (3 - 2), (3 - 1), (3 - 0) Hz with $T=$ 30 s period and 50\% duty cycle. We will refer to these experiments as ``varying the amplitude''. In the third set of experiments we made measurements with duty cycles of 0\%, 25\%, 50\%, 75\% and 100\%, while always modulating the grid frequency with ($f_{high}$ - $f_{low}$) = (3 - 0) Hz and a period of $T = 30$ s. We will refer to these experiments as ``varying the duty cycle''. We also took data with continuous drive at grid frequencies ranging from 1 Hz to 4 Hz to vary Reynolds number as our control group to show that the effects we observe cannot be simply attributed to the changes in Reynolds number.
\begin{figure} [h]
\begin{center}
\includegraphics[width=2.8in]{energy_time.eps}
\caption{\label{fig:energytime} (a) Time dependence of the mean square velocity measured by phase averaging over many cycles. The motor is halted at t = 0, and turned back on after half a cycle period, $t=T/2$. Data is from the experiments with varying period. Symbols represent the cycle period, $T$ of $+$= 3 s, $\circ =$ 6 s, $\ast =$ 12 s, $\times =$ 24 s, $\Box =$ 48 s. The symbols $\circ$ and $+$ are only plotted every four data points, and other symbols show every data point. (b) The fluctuating energy versus $t/T$ with an additional data set $T =$ 384 s ($\diamond$). Here all data sets have symbols plotted for every other data point.}
\end{center}
\end{figure}
In figure \ref{fig:energytime}a we show the time dependence of the mean square velocity, $\langle u_i u_i \rangle$, for the set of experiments varying the period. Time zero is defined as the time when the energy input halts. For all of these experiments, the energy dissipates at approximately the same rate, so the decay curves nearly collapse. After half a period, the energy input resumes. For the experiments of longer period such as $T =$ 48 s, the energy has decayed to 10\% of its initial value after half a period. In figure \ref{fig:energytime}b, this data is shown with time normalized by the period. One additional data set with $T =$ 384 s is added in this plot. Only for this data set with a very long period does the fluid become approximately quiescent before the energy input is resumed.
\section{\label{sec: Results}Results}
\subsection{\label{sec:Coefficients_of_scaling_law}Coefficients of inertial range scaling law}
\subsubsection{Varying Period}
Figure~\ref{fig:structure_3rd} shows the third order structure functions of the experiments varying the period. The energy dissipation rate is determined from this data and the four-fifths law. When compensated by $\varepsilon r$, the inertial and dissipation ranges of these third order structure functions collapse fairly well, suggesting that the small scales of these turbulent flows are similar.
\begin{figure} [h!]
\begin{center}
\includegraphics[width=3in]{period_3rdstructure_function_r_eta.eps}
\caption{\label{fig:structure_3rd} Third order compensated structure functions for the experiments with varying period. Symbols represent the cycle period, $T$ of $+$= 3 s, $\circ =$ 6 s, $\ast =$ 12 s, $\times =$ 24 s, $\Box =$ 48 s, $\diamond =$ 384 s. Driving frequency modulation is ($f_{high}$ - $f_{low}$) = (3 - 0)Hz and the duty cycle is $50\%$. }
\end{center}
\end{figure}
However, the compensated second order structure functions shown in figure~\ref{fig:structure}a do not collapse well at all. The maximum of these compensated structure functions, which is an estimate of the coefficient in the inertial range scaling law, shows a 20\% decrease as the period increases. Increasing the fluctuations in the energy input does have a significant effect on the small scales of the flow. The shape of the second order structure functions shows little change, which is consistent with the idea that fluctuations in the energy input at large scales primarily change the coefficients in scaling laws while leaving the scaling exponents unchanged. Figure~\ref{fig:structure}b shows the second order structure functions scaled by the prediction of equation~(\ref{refined_model}). The good collapse of these curves after scaling indicates the effects of fluctuations in the energy input are largely captured by the refined model.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=3in]{period_structure_function_combined.eps}
\caption{\label{fig:structure} (a) Second order compensated structure functions for the experiments with varying period. Symbols are the same as \protect{figure~\ref{fig:structure_3rd}}. (b) Second order structure functions scaled by the ratio of moments of the energy dissipation rate predicted by the refined model in equation~\protect{\ref{refined_model}}. }
\end{center}
\end{figure}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=3in]{category_plot_period.eps}
\caption{\label{fig:category_period_gamma} Experimental measurements of inertial range scaling coefficient ($\bullet$) along with the prediction of the refined model ($\times$) for the experiments of varying period. The dotted line represents the prediction of the model by Monin and Yaglom. }
\end{center}
\end{figure}
Figure~\ref{fig:category_period_gamma} shows the measured coefficient of the inertial range scaling of the second order structure function, commonly labelled as Kolmogorov constant $C_2$. The decrease in the `constant' as the period increases is a clear indication that the previous assessment by Mouri et al.~\cite{Mouri:2006} and Praskovsky et al. \cite{Praskovsky:1993} that large scale fluctuations do not affect second order structure functions is only an approximation that is valid in cases where the fluctuations in the energy input are small. Figure~\ref{fig:category_period_gamma} also shows the prediction of our refined model from equation~(\ref{refined_model}) with the model value of $C_2=2.0$. The experimental measurement and the refined model are in fairly good agreement. There are many possible factors that contribute to the difference between the measurements and the model, including the difficulty in measuring scaling coefficients at modest Reynolds number and limitations of the estimate $\varepsilon \propto U^3/L$ in equation~(\ref{refined_model_u}). The dotted line is the prediction of the model by Monin and Yaglom. Our experimental measurements of the inertial range coefficient approaches this dotted line when the period is long as it should since in that case we are approaching the situation Monin and Yaglom consider where the energy input is constant in time for both the low frequency and high frequency state.
Measuring scaling coefficients from this data at modest Reynolds numbers has some difficulties. From the third order structure functions we extracted the energy dissipation rate by averaging the three bins at the maxima between $r/\eta=$15 and 68. For the second order structure functions, we used this same definition of the inertial range even though the peak of the second order compensated structure functions are at slightly larger $r$. This results in measured second order scaling coefficients being below the peak value. We tried using a different inertial range for the second order data. This makes small changes in the magnitude of the scaling coefficients, but has no effect on the conclusions we draw. Data at larger Reynolds numbers will be necessary to provide more precise quantitative measurements of how scaling coefficients depend on fluctuations in the energy input.
\begin{figure} [h!]
\begin{center}
\includegraphics[width=3in]{3Hz_2ndstructure_function_combined.eps}
\caption{\label{fig:structure_3Hz} (a) Second order compensated structure functions for the experiments with varying amplitude. (b) Second order structure functions scaled by the ratio of moments of the energy dissipation rate predicted by the refined model. Symbols represent different frequency modulations of ($f_{high}$ - $f_{low}$). $\times =$ (3 - 3) Hz, $\ast=$ (3 - 2) Hz, $\Box =$ (3 - 1) Hz, $\circ =$ (3 - 0) Hz. Cycle period $T$ is 30 s, and the duty cycle is $50\%$. }
\end{center}
\end{figure}
\subsubsection{Varying Amplitude}
Similar effects of the large scale energy fluctuations on small scales are also seen in the experiments where amplitude of the energy input is varied by changing the grid oscillation frequency. Figure \ref{fig:structure_3Hz}a shows the second order compensated structure functions for the data sets with varying amplitude. Similar to the experiments varying the period, the curves do not collapse, indicating that the coefficient of the scaling law depends on the large scales. Figure~\ref{fig:structure_3Hz}b shows the second order structure functions scaled by the prediction of equation~(\ref{refined_model}). The better collapse of these curves after scaling again indicates that the refined model is accurately describing the effects of fluctuating energy input.
\begin{figure}
\begin{center}
\includegraphics[width=3in]{category_plot_3Hz_gamma.eps}
\caption{\label{fig:category_3Hz_gamma}
Experimental measurements of the inertial range scaling coefficient ( $\bullet$), compared with predictions from the refined model ($\times$), and the Monin and Yaglom model ($\Box$) for the experiments of varying amplitude. The predictions of the refined model and the Monin and Yaglom model assume $C_2$ = 2. }
\end{center}
\end{figure}
Figure~\ref{fig:category_3Hz_gamma} shows the measured coefficient of the inertial range scaling along with predictions from the refined model and the Monin and Yaglom model. The main point is that increasing the amplitude of the fluctuations in the energy input systematically decreases the constant as predicted. Quantitatively, the refined model has coefficients larger than those measured meaning that it underestimates the effect of the large scale fluctuations. This deviation is likely due to the refined model using the time dependence of the rms velocity to estimate the fluctuations in the energy input, which does not capture all of the fluctuations. The Monin and Yaglom model works well for small amplitude of the energy input fluctuations, but for the largest fluctuation amplitude (3-0)Hz, it predicts a much larger effect of the large scale fluctuations than are observed experimentally. This is expected since this data is for period $T=30$ s, and there is not enough time for the energy to decay to the constant values assumed by the Monin and Yaglom model.
For the experiments varying the amplitude of the fluctuations in the energy input, we did not directly measure the phase averaged fluctuating velocity needed in the refined model. To make predictions with this model, we had to model the fluctuation velocity using the known values for continuous driving at different frequencies and the decay rate data in figure~\ref{fig:energytime}. The limitations of this model likely also contributes to the poorer agreement with the refined model in this case.
\begin{figure}
\begin{center}
\includegraphics[width=2.8in]{dutycycle_structure_function.eps}
\caption{\label{fig:structure_dutycycle} Second order compensated structure functions for the experiments with varying duty cycle. Symbols represent the duty cycle of $+$= 100\%, $\circ =$ 75\%, $\ast =$ 50\%, $\times =$ 25\%, $\Box =$ 48 s, $\diamond =$ 384 s. Driving frequency modulations is ($f_{high}$ - $f_{low}$) = (3 - 0)Hz and the period $T$ is $30$ s. }
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=2.8in]{category_plot_dutycycle.eps}
\caption{\label{fig:category_dutycycle_gamma} Experimental measurements of the inertial range scaling coefficient ($\bullet$), and the prediction of the Monin and Yaglom model ($\Box$) for the experiments of varying duty cycle. }
\end{center}
\end{figure}
\subsubsection{Varying Duty Cycle}
The set of experiments varying the duty cycle in figure~\ref{fig:structure_dutycycle} also shows that the compensated second order structure functions show strong dependence on fluctuations in the energy input. We show the measured inertial range scaling coefficient in figure~\ref{fig:category_dutycycle_gamma}. When the duty cycle is smaller, we observe a smaller coefficient. For 25\% duty cycle we see the smallest value of the inertial range scaling coefficient of 1.58. Note that 25\% duty cycle and 75\% duty cycle do not have the same coefficient. Because times with large energy input dominate the moments of the energy dissipation rate, the effects on the coefficient are largest for low duty cycle where bursts of large energy input are followed by a long quiescent period.
The predictions of the Monin and Yaglom model shown in figure~\ref{fig:category_dutycycle_gamma} are consistently below the measured coefficients. We expect that if the experiments were performed for larger period rather than $T=30$ s they would approach the Monin and Yaglom predictions.
\subsubsection{Varying Reynold's number}
The set of experiments varying Reynolds number for constant energy input in figure~\ref{fig:structure_Reynold} shows that the inertial range scaling coefficients for the second order structure functions do not have strong dependence on Reynolds number. We vary Reynolds number from $R_\lambda=$139 at 1 Hz continuous driving to $R_\lambda=$ 271 at 4 Hz continuous driving. The shape of the structure function changes at the lowest Reynolds number as expected, but after using the third order structure functions to determine the energy dissipation rate, the peak value remains relatively constant. This confirms that the variation we observe in the Kolmogorov constant is not simply the result of different effective Reynolds numbers in different experiments.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=3.2in]{Reynold_structure_function.eps}
\caption{\label{fig:structure_Reynold} Second order compensated structure functions for the experiments with varying Reynolds number. Symbols represent different Reynolds number, $R_\lambda$ of $\ast =$ 271, $\times =$ 250, $\Box =$ 237, $\diamond =$ 163. }
\end{center}
\end{figure}
\begin{table*}
\begin{center}
\begin{tabular}{|p{1.8cm} |p{0.9cm} |p{0.7cm} |p{1cm} |p{1cm}|p{1.1cm} |p{0.9cm}|l|p{1.4cm} |l|}
\hline
& $f_{high}$ (Hz)& $f_{low}$ (Hz) & $T$ (s) & Duty Cycle &U (cm/s) & $L$ (cm) & $\tau$ (s) & $\varepsilon$ (cm$^2$/s$^3$) & $R_\lambda$ \\ \hline
\multirow{2}{*}{Varying } &3 & 3 & 30 & 50$\%$ & 5.46 & 7.69 & 1.41 & 21.2 & 250 \\ \cline{2-10}
\multirow{2}{*}{amplitude} &3 & 2 & 30 & 50$\%$ & 4.72 & 7.48 & 1.58 & 14.1 & 230 \\ \cline{2-10}
&3 & 1 & 30 & 50$\%$ & 4.23 & 7.13 & 1.69 & 11 & 213 \\ \cline{2-10}
&3 & 0 & 30 & 50$\%$ & 4.21 & 7.16 & 1.7 & 10.4 & 212 \\ \hline
&3 & 0 & 3 & 50$\%$ & 4.44 & 8.28 & 1.86 & 10.6 & 235 \\ \cline{2-10}
\multirow{3}{*}{Varying}&3 & 0 & 6 & 50$\%$ & 4.71 & 8.51 & 1.81 & 12.3 & 245 \\ \cline{2-10}
\multirow{3}{*}{period}&3 & 0 & 12 & 50$\%$ & 4.54 & 8.44 & 1.86 & 11.1 & 240 \\ \cline{2-10}
&3 & 0 & 24 & 50$\%$ & 4.42 & 7.87 & 1.78 & 11 & 228 \\ \cline{2-10}
&3 & 0 & 48 & 50$\%$ & 4.07 & 6.48 & 1.59 & 10.4 & 198 \\ \cline{2-10}
&3 & 0 & 384 & 50$\%$ & 4.06 & 5.76 & 1.42 & 11.6 & 187 \\ \hline
\multirow{2}{*}{Varying} &3 & 0 & 30 & 100$\%$ & 5.46 & 7.69 & 1.41 & 21.2 & 250 \\ \cline{2-10}
\multirow{2}{*}{duty cycle}&3 & 0 & 30 & 75$\%$ & 4.92 & 7.58 & 1.54 & 15.7 & 236 \\ \cline{2-10}
&3 & 0 & 30 & 50$\%$ & 4.21 & 7.16 & 1.7 & 10.4 & 212 \\ \cline{2-10}
&3 & 0 & 30 & 25$\%$ & 3.27 & 6.86 & 2.1 & 5.1 & 183 \\ \hline
\multirow{2}{*}{Varying} &1 & N/A & N/A& 100$\%$ & 1.96 & 6.59 & 3.36 & 1.15 & 139 \\ \cline{2-10}
\multirow{2}{*}{Reynolds}&2 & N/A & N/A& 100$\%$ & 4.05 & 9.3 & 2.3 & 7.15 & 237 \\ \cline{2-10}
\multirow{2}{*}{Number}&3 & N/A & N/A& 100$\%$ & 5.46 & 7.69 & 1.41 & 21.2 & 250 \\ \cline{2-10}
&4 & N/A & N/A& 100$\%$ & 7.14 & 6.87 & 0.96 & 52.9 & 271 \\ \hline
\end{tabular}
\caption{ Experimental parameters and resulting statistics for different sets of experiments. Note that the case of $f_{high}$ = 3 Hz, $f_{low}$ = 3 Hz, Duty Cycle 50\% is the same data as the case of $f_{high}$ = 3 Hz, $f_{low}$ = 0 Hz, Duty Cycle 100\%.}
\label{table:LSI}
\end{center}
\end{table*}
\subsection{\label{sec: Conditional Structure Function} Conditional Structure Functions}
Previous work has used conditional structure functions to quantify the effects of the large scales on small scales in turbulent flows~\citep{Praskovsky:1993, Sreenivasan:1996, Sreenivasan:1998, Blum:2010, Blum:2011}. Velocity differences between two points separated by $\textit{\textbf{r}}$ are dominated by structures near scale $\textit{\textbf{r}}$ while velocity sums of two points are dominated by the large scales in the flow. So moments of velocity differences conditioned on sums provide a convenient way to observe the effects of the largest scales on other scales. We find that conditional structure functions provide a more sensitive measurement of the existence of fluctuations in the large scale energy input than the coefficients of inertial range structure functions. However, theoretical tools to predict the effects of large scale fluctuations on conditional structure functions are not available. In this section we present measured conditional structure functions as we systematically change the fluctuations in the energy input.
\begin{figure}
\begin{center}
\includegraphics[width=2.4in]{Run17-20_cond_sf_normalized_v2.eps}
\caption{\label{fig:3Hz_all} Eulerian second order conditional structure function versus large scale velocity for the experiments with varying amplitude. The frequencies modulated were ($f_{high}$ - $f_{low}$) = (a) (3 - 3) Hz (b) (3 - 2) Hz, (c) (3 - 1) Hz, (d) (3 - 0) Hz. Each curve represents the following separation distances $r/\eta$: $+$ = 2.67 to 5.33, $\circ$ = 5.33 to 10.67, $\ast$ = 10.67 to 21.33, $\times$ = 21.33 to 42.67, $\Box$ = 42.67 to 85.33, $\diamond$ = 85.33 to 170.67, $\vartriangle$ = 170.67 to 341.33, $\triangledown$ = 341.33 to 682.67.}
\end{center}
\end{figure}
\subsubsection{Varying amplitude}
Figure~\ref{fig:3Hz_all} shows conditional structure functions for the data sets varying the amplitude of the fluctuations in the energy input. We condition the structure function on the velocity component that is transverse to $\textit{\textbf{r}}$ denoted $\Sigma u_\perp$. In order to compare the conditional structure function for different length scales, we normalize the vertical axis by the unconditioned structure function. The horizontal axis is normalized by the characteristic velocity $U=(\langle u_i u_i/3 \rangle)^{1/2}$. In figure~\ref{fig:3Hz_all}a for constant driving of the oscillating grids, we see the results published by Blum et al.~\cite{Blum:2010} that the conditional structure functions for all length scales show a similar dependence on the large scale velocity. There is a slight dependence on length scale with the smallest length scales showing a stronger dependence on the large scale velocity. This small dependence on length scale remains unexplained since it is the opposite of the expectation that the small scales are approaching universality. The same effect is seen in DNS data in Ref.~\cite{Blum:2011}. However, in this paper we are focusing on fluctuations of the energy input and we will see that these produce much bigger effects than the small differences for different length scales.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=3.0in]{3Hz_different_grid_velocity.eps}
\caption{\label{fig:3Hz}(a) The velocity dependence of conditional second order structure functions of one separation distance $r/\eta =$ 10.67 to 21.33 for the experiments with varying amplitude. ($f_{high}$ - $f_{low}$) = (3 - 3) Hz ($+$), (3 - 2)Hz ($\circ $), (3 - 1) Hz ($\ast $), (3 - 0) Hz ($\times$) with 50\% duty cycle and $T =$ 30 s (b) The coefficient $b$ as a function of the separation distance for the experiments with varying amplitude. Symbols are the same as (a). }
\end{center}
\end{figure}
Figures~\ref{fig:3Hz_all}b-d show that increasing the fluctuations in the energy input produces a large increase in the dependence of the conditional structure functions on the large scale velocity. In each sub-figure, the curves for different length scales remain very similar, which confirms the fact observed earlier that fluctuating energy input does not change the length scale dependence. It primarily changes a pre-factor scaling the entire structure function. Note that figure~\ref{fig:3Hz_all}a still has dependence on the large scale velocity even though the oscillating grid is driven at a constant 3 Hz frequency. We interpret this as fluctuations in the energy input that remain even in the case of constant driving~\citep{Blum:2010}. To more directly compare the effects of changing the energy input fluctuations, we extract the curve for $r/\eta$ = 10.7 to 21.3 from figure~\ref{fig:3Hz_all}(a, b, c and d) and plot them on one graph as shown in figure \ref{fig:3Hz}a. In figure \ref{fig:3Hz_all} and figure \ref{fig:3Hz} the symmetry around zero large scale velocity is a result of conditioning on the transverse component of the large scale velocity for which $\Sigma u_\perp > 0$ is indistinguishable from $\Sigma u_\perp < 0$.
To quantify the observed dependence of the conditional structure function, we fit all the curves in figure \ref{fig:3Hz_all} to the functional form $au^4+bu^2+c$. Figure~\ref{fig:3Hz}b shows the fit coefficient $b$ as a function of the separation distance $r/\eta$. The coefficient $b$ measures the curvature of the conditional structure functions at the origin, and it captures the primary dependence on the large scale velocity. Measuring the coefficient of the second order term $b$ is also keeping with a previous study ~\citep{Sreenivasan:1998}. There is an increase by more than a factor of 5 in the curvature, $b$, as the fluctuations in the energy input increase from driving at 3 Hz continuously to alternating between 3 and 0 Hz. The degree to which all length scales show similar dependence on the large scales can also be evaluated from figure \ref{fig:3Hz}b. In section \ref{sec: Summary} we will show that changes in $b$ are closely related to the changes in the inertial range scaling coefficient that we presented in section \ref{sec:Coefficients_of_scaling_law}.
\subsubsection{\label{sec:Varying period} Varying period}
Figure \ref{fig:ontime}a shows the conditional second order structure functions for the experiments with varying period. When the period $T$ increases, there is a stronger dependence on large scale velocity. The two shortest periods $T =$ 3 s and 6 s have similar and relatively low curvatures. Increasing the period allows the turbulence to decay closer to quiescent before the energy input resumes, so the conditional dependence on the large scale velocity is stronger at longer periods. For the very long period, $T=384$ s, the conditional structure function has a different shape with a sharp minimum at the center of a region with less curvature. This is the result of the high energy state providing the samples with large velocity sum, while the low energy state provides only samples with velocity sum near zero. For this data at $T=384$ s there is also a much stronger dependence on the length scale as shown in figure \ref{fig:ontime}b.
\begin{figure}
\begin{center}
\includegraphics[width=2.8in]{Ontime_Period.eps}
\caption{\label{fig:ontime} (a) The velocity dependence of second order conditional structure functions of one separation distance $r/\eta =$ 10.67 to 21.33 for the experiments with varying period. Symbols represent the cycle period, $T$ of $+ =$ 3 s, $\circ =$ 6 s, $\ast =$ 12 s, $\times =$ 24 s, $\Box =$ 48 s, $\diamond =$ 384 s. Driving frequency modulations is ($f_{high}$ - $f_{low}$) = (3 - 0)Hz, and the duty cycle is $50\%$. (b) The coefficient $b$ as a function of separation distance for the experiments with varying period. Symbols are the same as (a). }
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=2.8in]{duty_cycle_variation_vertical.eps}
\caption{\label{fig:duty_cycle} (a) The velocity dependence of conditional second order structure functions of one separation distance $r/\eta =$ 10.67 to 21.33 for the experiments with varying duty cycle. Each curve shows the duty cycle of $+ =$ 100\% , $\circ = $ 75\% , $\ast = $ 50\%, $\times = $ 25\% with driving frequency modulations ($f_{high}$ - $f_{low}$) = (3 - 0)Hz and period $T = $30 s. (b) The coefficient $b$ as a function of separation distance for the experiments with varying duty cycle. Symbols are the same as (a).}
\end{center}
\end{figure}
\subsubsection{\label{sec:Varying Duty Cycle} Varying duty cycle}
Figure \ref{fig:duty_cycle}a shows the second order conditional structure functions for the experiments with varying duty cycle. It shows that reducing the duty cycle produces a large increase in the dependence of the conditional structure functions on the large scale velocity. The result is consistent with our previous findings that increasing the fluctuations of the large scale energy input increases the dependence of the second order conditional structure functions on the large scale velocity. Here all length scales show fairly similar dependence on the large scales as seen in figure \ref{fig:duty_cycle}b.
\begin{figure}
\begin{center}
\includegraphics[width=2.2in]{category_plot_cur_combined_2.eps}
\caption{\label{fig:cur} The relationship of the curvature $b$ of the conditional second order structure function with the coefficient of the inertial range scaling law. $\diamond$ is the parameterization 2(1-0.15$b$). $\bullet$ is the experimental measurements of the inertial range scaling coefficient. $\times$ is the refined model, and $\Box$ is the Monin and Yaglom model. }
\end{center}
\end{figure}
\subsection{\label{sec: Summary}Connecting Conditional Structure Functions and Coefficients of Inertial Range Scaling Law}
The curvature $b$ of the conditional structure functions increases as the fluctuations of the large scale energy input increases. This suggests that it might be possible to connect $b$ with changes in the coefficients of inertial range scaling law presented in section~\ref{sec:Coefficients_of_scaling_law}.
A simple linear parameterization $C_2=2(1- 0.15b)$ seems to match the measured scaling coefficients fairly well as shown in figure~\ref{fig:cur}. However, we do not have a solid theoretical foundation for choosing this functional form and the value of 0.15 is a rough fit. For weak fluctuations in the energy input, which includes most turbulent flows of interest, this parameterization seems to work fairly well. But for extreme cases it fails. At low duty cycles in figure~\ref{fig:cur}c, this parameterization is well above the measured coefficient. In the limit where one of the states is actually quiescent ($\gamma$ =1 in figure~\ref{fig:contour}), the curvature $b$ should go to infinity while the coefficient of the scaling law would not go negative. Conditional structure functions and coefficients of inertial range scaling law are both modified by fluctuations in the large scale energy input of turbulence. A more complete understanding of the relationship between these two could be very useful, since the effects of fluctuations in the large scale energy input are much easier to measure using conditional structure functions.
\section{Discussion}
In this paper we have focused on inertial range effects of fluctuations in the energy input because they are most easily measured with our apparatus. But it should be noted that the non-universality of the inertial range scaling coefficients implies non-universality of the functional form of structure functions in the dissipation range. Because the Kolmogorov scale depends on the energy flux, the functional form in the dissipation range will depend on the distribution of the energy flux which depends on the fluctuations of the energy input at large scales.
A problem facing research into the effects of large scale fluctuations on the small scales of turbulence is that the terminology that has accumulated over many years is not always as clear as it could be. The word `intermittency' appears to have entered the turbulence literature to describe the fluctuations between turbulent and non-turbulent fluid flowing past a point in a free shear flow. For example, the textbook by Hinze in 1959 uses `intermittent' only in this sense. The second edition of this textbook in 1975 introduces the use of a flatness factor to measure the `degree of intermittency' (p. 242), but even here, the goal is to quantify the fraction of the time that turbulence occurs. Over the decades a major change has occurred in how the word intermittency is used. In the parlance of a large part of the turbulence research community, intermittency has become associated with the rare events of large dissipation that are responsible for anomalous scaling~\citep{Sreenivasan:1997}. A good example of this usage is the book by Frisch ~\cite{Frisch:1995} which uses the word `intermittency' to refer to the fluctuations produced by uneven energy transfer through the cascade which we refer to above as internal intermittency. He briefly describes the turbulent to non-turbulent fluctuations seen in free shear flows with a footnote that says ``This phenomenon is known as `external intermittency'; its relation to the intermittency discussed in Chapter 8 is not clear''. In general use, the word `intermittency' has often taken on a connotation about large deviations from the mean that is entirely absent in the standard English definition of the word or in the traditional application of this word to turbulent flows. However, the old terminology is also still used. In the textbook by Pope (2000), the word intermittency is reserved for the turbulent to non-turbulent fluctuations in free shear flows while small scale effects are called `internal intermittency'. Other sources use the phrase `large-scale intermittency' to refer to the turbulent to non-turbulent fluctuations in free shear flows~\citep{Mi:2001}.
In this paper, we quantify the effects that fluctuations in the energy input at large scales have on the coefficients of inertial range power laws. The success of models based on the refined similarity hypotheses suggests we should use terminology that connects this phenomenon with the closely related phenomenon of internal intermittency that is already widely understood. However, the history of the terminology for these phenomena makes it difficult to find suitable terms. Davidson \cite{Davidson:2004} provides a clear description of the phenomenon of fluctuations at large scales and uses the phrases `integral-scale intermittency' and `large-scale intermittency' to refer to them in his section 6.5.1. We prefer this terminology, but the possibility of confusion with the older use of the phrase `large-scale intermittency' led us not to use this terminology in this paper.
One way to view the contributions from this and a previous sequence of papers~\citep{Blum:2010,Blum:2011} is that in quantifying the effects of large scale fluctuations on small scales, we find that large scale fluctuations which affect the entire cascade are a standard feature of turbulence and not a special feature of free shear flows or periodically modulated flows. Conditional structure functions are a sensitive way to quantify this dependence, and with them we find that the effects of large scale fluctuations can be detected in all flows except for a few special cases like turbulence behind a passive grid~\citep{Blum:2011}. This observation is in contrast to the usual assessment (see for example Praskovsky et al. \cite{Praskovsky:1993} and Mouri et al.~\cite{Mouri:2006}) where the large scale fluctuations are viewed as not affecting second order statistics except in free shear flows where conditional sampling of the turbulent regime can be used to restore the universal result.
Our interpretation is that, in general, turbulent flows have fluctuations in the large scale energy input. In many cases these are not large enough to have measurable effects on second order statistics, but by explicit control of the time dependence of the energy input we can make these effects big enough to produce a 20\% change in the Kolmogorov constant for the second order structure function. In other flows that appear to have constant energy input such as boundary layers, von-Karman flow between counter-rotating disks, etc., the strong inhomogeneity allows fluctuations at the large scales to intermittently transport fluid from different parts of the flow creating fluctuations in the energy input rate which should change the constants in inertial range scaling law in ways predicted by equation~(\ref{refined_avg}). The effects of turbulent to non-turbulent fluctuations in free shear flows are then seen to be a special case of this more general problem of transport in an inhomogeneous flow by the large scale fluctuations. To be sure it is an extreme case, where the entrained fluid has no vorticity and the viscous super-layer separating turbulent from non-turbulent fluid can be very thin. But the extreme case is smoothly connected to other flows where the large scale fluctuations entrain fluid with different turbulence characteristics. For example, experiments in a shearless mixing layer~\citep{Veeravalli:1989, Kang:2008} can continuously vary the turbulence on the two sides of the mixing layer from the extreme case of turbulent/non-turbulent to the case where the turbulence on both sides of the layer are the same.
In the future, we hope that the community can adopt some terminology that will allow us to talk more clearly about fluctuations at the large scales of turbulence. We have shown here that we can quantify and predict the the effects of large scale fluctuations using a refined similarity framework. These large scale fluctuations destroy universality in the Kolmorogov 1941 sense in exactly the way that Landau predicted, and they seem to naturally be called `large scale intermittency' since they are to the large scales what internal intermittency is to inertial and dissipation range scales. Furthermore they are the general case under which the traditional use of the phrase `large scale intermittency' can cleanly fall. We hope that further work on this topic will develop tools to more precisely quantify the fluctuations at large scales, and that this will lead to a consensus about the terminology to use in discussing these effects.
\section{\label{Conclusion}Conclusion}
Previous research has established that the small scales in turbulence are not entirely independent of the large scales~\citep{Kolmogorov:1962, Mouri:2006, Blum:2010}. Landau's footnote remark suggests that the fluctuations in the energy dissipation due to non-universal large scales will destroy the universality of small scales. Kolmogorov's paper on the refined similarity hypotheses~\citep{Kolmogorov:1962} identifies that the coefficients of inertial range scaling laws will not be universal. However, during the extensive effort to understand internal intermittency, the effects of fluctuations in the large scales have been largely ignored. The consensus in the literature has been that the coefficient of the inertial range scaling law for second order structure functions, known as the Kolmogorov constant $C_2$, is a universal constant~\citep{Praskovsky:1993,Sreenivasan:1995,Yeung:1997,Mouri:2006}.
In this paper, we systematically change the fluctuations in the energy input at the large scales and find that this leads to a decrease in the inertial range scaling coefficient that can be more than 20\%. An extension of the ideas in Kolmogorov's refined theory provides a model that successfully predicts these changes of the coefficients in inertial range scaling laws.
We also use structure functions conditioned on the velocity sum to measure the effect of fluctuations of large scale energy input on small scales. These conditional structure functions are able to identify the effects of fluctuations of the energy input even when the fluctuations are small. The curvature of the second order conditional structure functions appears to be determined by fluctuations in the energy input in a way similar to the changes in the Kolmogorov constant, but a quantitative understanding of this relationship is not available.
The turbulent flows that have been the focus of most laboratory and simulation work appear to have small enough fluctuations in the energy input that the effects on the second order Kolmogorov constant are usually negligible. However, in many geophysical flows such as turbulent clouds, the large scale fluctuations are a dominant feature of the flow. Our measurements show that fluctuations in the energy input at large scales can be determined by measuring the coefficients of inertial range scaling laws for conditional structure functions. This allows small scale measurements to provide a useful diagnostic of large scale dynamics. When it is possible to make direct measurements or predictions of the fluctuations in the large scale energy input, then the models we use here can provide prediction of the inertial range scaling coefficients from the properties of the large scales.
\noindent We would like to acknowledge support from the National Science Foundation under grant DMR-0547712 and DMR-1208990 and from COST Action MP0806. We thank Susantha Wijesinghe for his expertise on the real-time image compression circuit, and Greg Bewley, Eberhard Bodenschatz, Nick Ouellette, Arkady Tsinober, Zellman Warhaft and Haitao Xu for helpful conversations.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,880
|
USE $(database)
GO
SET ANSI_NULLS ON
GO
SET QUOTED_IDENTIFIER ON
GO
SET ANSI_PADDING ON
GO
print 'Creating Unique Keys on Bar'
--
ALTER TABLE [dbo].[Bar] ADD CONSTRAINT [UK_Bar] UNIQUE NONCLUSTERED
(
[Name] ASC
)WITH (PAD_INDEX = OFF, STATISTICS_NORECOMPUTE = OFF, SORT_IN_TEMPDB = OFF, IGNORE_DUP_KEY = OFF, ONLINE = OFF, ALLOW_ROW_LOCKS = ON, ALLOW_PAGE_LOCKS = ON) ON [PRIMARY]
GO
print 'Creating Unique Keys on BaseClass'
--
ALTER TABLE [dbo].[BaseClass] ADD CONSTRAINT [UK_BaseClass] UNIQUE NONCLUSTERED
(
[Guid] ASC
)WITH (PAD_INDEX = OFF, STATISTICS_NORECOMPUTE = OFF, SORT_IN_TEMPDB = OFF, IGNORE_DUP_KEY = OFF, ONLINE = OFF, ALLOW_ROW_LOCKS = ON, ALLOW_PAGE_LOCKS = ON) ON [PRIMARY]
GO
print 'Creating Unique Keys on BaseClassPartitioned'
--
ALTER TABLE [dbo].[BaseClassPartitioned] ADD CONSTRAINT [UK_BaseClassPartitioned] UNIQUE NONCLUSTERED
(
[Guid] ASC
)WITH (PAD_INDEX = OFF, STATISTICS_NORECOMPUTE = OFF, SORT_IN_TEMPDB = OFF, IGNORE_DUP_KEY = OFF, ONLINE = OFF, ALLOW_ROW_LOCKS = ON, ALLOW_PAGE_LOCKS = ON) ON [PRIMARY]
GO
print 'Creating Unique Keys on Foo'
--
ALTER TABLE [dbo].[Foo] ADD CONSTRAINT [UK_Foo] UNIQUE NONCLUSTERED
(
[Guid] ASC
)WITH (PAD_INDEX = OFF, STATISTICS_NORECOMPUTE = OFF, SORT_IN_TEMPDB = OFF, IGNORE_DUP_KEY = OFF, ONLINE = OFF, ALLOW_ROW_LOCKS = ON, ALLOW_PAGE_LOCKS = ON) ON [PRIMARY]
GO
print 'Creating Unique Keys on FooPartitioned'
--
ALTER TABLE [dbo].[FooPartitioned] ADD CONSTRAINT [UK_FooPartitioned] UNIQUE NONCLUSTERED
(
[Guid] ASC
)WITH (PAD_INDEX = OFF, STATISTICS_NORECOMPUTE = OFF, SORT_IN_TEMPDB = OFF, IGNORE_DUP_KEY = OFF, ONLINE = OFF, ALLOW_ROW_LOCKS = ON, ALLOW_PAGE_LOCKS = ON) ON [PRIMARY]
GO
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,491
|
Alto Jequitibá is een gemeente in de Braziliaanse deelstaat Minas Gerais. De gemeente telt 8.122 inwoners (schatting 2009).
Aangrenzende gemeenten
De gemeente grenst aan Alto Caparaó, Caparaó, Luisburgo, Manhumirim en Iúna (ES).
Gemeente in Minas Gerais
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,033
|
Canada's Securities Regulators Seek Feedback on eXtensible Business Reporting Language ( XBRL )
Toronto – The Canadian Securities Administrators ("CSA") issued a notice to the Canadian market to seek feedback on a new business reporting tool called XBRL. CSA Notice 52-314, Securities Regulators Want Your Feedback on XBRL, provides information on XBRL, and links to an on-line survey to gather comments from the market about their level of awareness of XBRL and what the CSA's role should be in respect of XBRL.
"The CSA is committed to improving how information is collected and provided to investors, " says Jean St-Gelais, Chair of the CSA and President & Chief Executive Officer of the Autorité des marchés financiers (Québec). "This includes finding ways to use technology that could make it easier and more efficient for investors to receive, find, compare and analyze financial information. "
XBRL is a relatively new business reporting language that is emerging as an international standard for communicating business and financial data. The basic concept of XBRL is that it attaches standardized electronic "tags" to elements of information and these tags provide information about what the item represents. The notice provides a brief overview of XBRL, the benefits and costs associated with it, and current trends and developments around the world.
"There have been a number of significant developments with XBRL around the world, and we are interested in understanding the level of awareness of XBRL in the Canadian marketplace," said Jean St-Gelais.
To ensure that the survey reaches market participants, the CSA will email a copy of the Notice and survey to all public companies across Canada.
CSA Notice 52-314 Securities Regulators Want Your Feedback on XBRL, and the accompanying survey is available on several CSA members' websites.
The CSA is the council of the securities regulators of Canada's provinces and territories whose objectives are to improve, coordinate, and harmonize regulation of Canadian capital markets.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,576
|
These are the 20 health conditions that are considered to be so painful by the NHS that they can stop a person from performing their daily tasks.
Ranging from life-threatening conditions to chronic ailments, these health problems are widely experienced among the people of North East Lincolnshire, and everyone will know someone who suffers from at least one.
We cannot cure all of them, but with the help of medicine, plenty of rest, and support from loved ones, they are certainly manageable.
Below is the list of the 20 most painful health conditions according the NHS .
Shingles, also known as herpes zoster is an infection that causes a painful rash that typically appears as a rash or crop of blisters on one side of your body, often around the waistline.
The first signs of shingles can be a tingling or painful feeling in an area of skin, or a headache or feeling generally unwell. A rash will appear a few days later.
It can take up to 4 weeks for the rash to heal and your skin can be painful for weeks after the rash has gone, but it usually settles over time.
They are rare, but anyone can get them and they are most common in men and tend to start when a person is in their 30s or 40s.
Cluster headaches begin quickly and without warning. The pain is very severe and is often described as a sharp, burning or piercing sensation on one side of the head. It's often felt around the eye, temple and sometimes face. It tends to occur on the same side for each attack.
Frozen Shoulder is a condition that causes your shoulder to be painful and stiff, and without the correct treatment it can last for years.
In frozen shoulder, the joint becomes so tight and stiff that it's virtually impossible to carry out simple movements, such as raising your arm. Daily activities like taking off a T-shirt, lifting a kettle, putting on a coat or even combing your hair become an ordeal.
Broken bones are usually very painful.
A broken or cracked bone is known as a fracture, and can range from a hairline fracture to a severe breakage of the bone.
If the break is small, it's possible you might not feel any pain at all but, usually, a broken bone really hurts, especially when you try to move it. The pain is often described as feeling like a deep ache.
If you have a heart attack, you usually get a pain in the centre of your chest – often described as a sensation of heaviness, tightness or squeezing – that can be so bad it causes you to collapse.
The pain can feel like really bad indigestion, and sometimes spreads to your jaw, neck, back, arms or stomach.
If you suspect that you or someone else is having a heart attack, call for emergency help immediately.
A migraine typically feels like an intense headache on one side of the head. The pain is usually a moderate or severe throbbing sensation that gets worse when you move and prevents you from carrying out normal activities.
In some cases, the pain can occur on both sides of your head, and may affect your face or neck.
Migraines can cause vomiting and extreme sensitivity to light and sound. Sometimes, in cases of severe migraines, the best thing to do is find a dark, quiet place to lie down until the pain passes.
They can affect everybody and usually pass after a few hours.
Sciatica is the name given to an aching pain running down the leg. It's caused when the sciatic nerve – the longest nerve in the body, which stretches from your back to your feet – has been pinched or irritated by damage to the back.
Sciatica is different to general back pain. The pain of sciatica hardly affects your back at all – instead, it radiates out from your lower back, down the buttocks and into one or both of the legs, right down to the calf.
If you have the condition your symptoms may be worse when moving, sneezing or coughing.
Passing a kidney stone can produce a sudden, sharp, cramping pain in your lower back or the side of your abdomen, or occasionally in your groin. The pain may last for minutes or hours, with pain-free intervals in between.
Nobody knows exactly why we have an appendix, but removing it isn't harmful.
But too much pain after surgery is not a good thing, and you should never feel you have to "tough it out".
There are lots of effective painkillers on offer to keep your pain after surgery under control. In addition to making you more comfortable, well-controlled pain will help you get better faster and prevent long-term problems.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,339
|
// Copyright (c) 2012 The Chromium Authors. All rights reserved.
// Use of this source code is governed by a BSD-style license that can be
// found in the LICENSE file.
#include "ui/app_list/views/apps_grid_view.h"
#include <algorithm>
#include <set>
#include <string>
#include "base/guid.h"
#include "content/public/browser/web_contents.h"
#include "ui/app_list/app_list_constants.h"
#include "ui/app_list/app_list_folder_item.h"
#include "ui/app_list/app_list_item.h"
#include "ui/app_list/app_list_switches.h"
#include "ui/app_list/pagination_model.h"
#include "ui/app_list/views/app_list_drag_and_drop_host.h"
#include "ui/app_list/views/app_list_folder_view.h"
#include "ui/app_list/views/app_list_item_view.h"
#include "ui/app_list/views/apps_grid_view_delegate.h"
#include "ui/app_list/views/page_switcher.h"
#include "ui/app_list/views/pulsing_block_view.h"
#include "ui/app_list/views/top_icon_animation_view.h"
#include "ui/compositor/scoped_layer_animation_settings.h"
#include "ui/events/event.h"
#include "ui/gfx/animation/animation.h"
#include "ui/views/border.h"
#include "ui/views/controls/webview/webview.h"
#include "ui/views/view_model_utils.h"
#include "ui/views/widget/widget.h"
#if defined(USE_AURA)
#include "ui/aura/root_window.h"
#include "ui/aura/window.h"
#if defined(OS_WIN)
#include "ui/views/win/hwnd_util.h"
#endif // defined(OS_WIN)
#endif // defined(USE_AURA)
#if defined(OS_WIN)
#include "base/command_line.h"
#include "base/files/file_path.h"
#include "base/win/shortcut.h"
#include "ui/base/dragdrop/drag_utils.h"
#include "ui/base/dragdrop/drop_target_win.h"
#include "ui/base/dragdrop/os_exchange_data.h"
#include "ui/base/dragdrop/os_exchange_data_provider_win.h"
#endif
namespace app_list {
namespace {
// Distance a drag needs to be from the app grid to be considered 'outside', at
// which point we rearrange the apps to their pre-drag configuration, as a drop
// then would be canceled. We have a buffer to make it easier to drag apps to
// other pages.
const int kDragBufferPx = 20;
// Padding space in pixels for fixed layout.
const int kLeftRightPadding = 20;
const int kTopPadding = 1;
// Padding space in pixels between pages.
const int kPagePadding = 40;
// Preferred tile size when showing in fixed layout.
const int kPreferredTileWidth = 88;
const int kPreferredTileHeight = 98;
// Width in pixels of the area on the sides that triggers a page flip.
const int kPageFlipZoneSize = 40;
// Delay in milliseconds to do the page flip.
const int kPageFlipDelayInMs = 1000;
// How many pages on either side of the selected one we prerender.
const int kPrerenderPages = 1;
// The drag and drop proxy should get scaled by this factor.
const float kDragAndDropProxyScale = 1.5f;
// Delays in milliseconds to show folder dropping preview circle.
const int kFolderDroppingDelay = 250;
// Delays in milliseconds to show re-order preview.
const int kReorderDelay = 50;
// Delays in milliseconds to show folder item reparent UI.
const int kFolderItemReparentDealy = 50;
// Radius of the circle, in which if entered, show folder dropping preview
// UI.
const int kFolderDroppingCircleRadius = 15;
// Radius of the circle, in which if entered, show re-order preview.
const int kReorderDroppingCircleRadius = 30;
// Max items allowed in a folder.
size_t kMaxFolderItems = 16;
// RowMoveAnimationDelegate is used when moving an item into a different row.
// Before running the animation, the item's layer is re-created and kept in
// the original position, then the item is moved to just before its target
// position and opacity set to 0. When the animation runs, this delegate moves
// the layer and fades it out while fading in the item at the same time.
class RowMoveAnimationDelegate
: public views::BoundsAnimator::OwnedAnimationDelegate {
public:
RowMoveAnimationDelegate(views::View* view,
ui::Layer* layer,
const gfx::Rect& layer_target)
: view_(view),
layer_(layer),
layer_start_(layer ? layer->bounds() : gfx::Rect()),
layer_target_(layer_target) {
}
virtual ~RowMoveAnimationDelegate() {}
// gfx::AnimationDelegate overrides:
virtual void AnimationProgressed(const gfx::Animation* animation) OVERRIDE {
view_->layer()->SetOpacity(animation->GetCurrentValue());
view_->layer()->ScheduleDraw();
if (layer_) {
layer_->SetOpacity(1 - animation->GetCurrentValue());
layer_->SetBounds(animation->CurrentValueBetween(layer_start_,
layer_target_));
layer_->ScheduleDraw();
}
}
virtual void AnimationEnded(const gfx::Animation* animation) OVERRIDE {
view_->layer()->SetOpacity(1.0f);
view_->layer()->ScheduleDraw();
}
virtual void AnimationCanceled(const gfx::Animation* animation) OVERRIDE {
view_->layer()->SetOpacity(1.0f);
view_->layer()->ScheduleDraw();
}
private:
// The view that needs to be wrapped. Owned by views hierarchy.
views::View* view_;
scoped_ptr<ui::Layer> layer_;
const gfx::Rect layer_start_;
const gfx::Rect layer_target_;
DISALLOW_COPY_AND_ASSIGN(RowMoveAnimationDelegate);
};
// ItemRemoveAnimationDelegate is used to show animation for removing an item.
// This happens when user drags an item into a folder. The dragged item will
// be removed from the original list after it is dropped into the folder.
class ItemRemoveAnimationDelegate
: public views::BoundsAnimator::OwnedAnimationDelegate {
public:
explicit ItemRemoveAnimationDelegate(views::View* view)
: view_(view) {
}
virtual ~ItemRemoveAnimationDelegate() {
}
// gfx::AnimationDelegate overrides:
virtual void AnimationProgressed(const gfx::Animation* animation) OVERRIDE {
view_->layer()->SetOpacity(1 - animation->GetCurrentValue());
view_->layer()->ScheduleDraw();
}
private:
scoped_ptr<views::View> view_;
DISALLOW_COPY_AND_ASSIGN(ItemRemoveAnimationDelegate);
};
// Gets the distance between the centers of the |rect_1| and |rect_2|.
int GetDistanceBetweenRects(gfx::Rect rect_1,
gfx::Rect rect_2) {
return (rect_1.CenterPoint() - rect_2.CenterPoint()).Length();
}
// Returns true if the |item| is an folder item.
bool IsFolderItem(AppListItem* item) {
return (item->GetItemType() == AppListFolderItem::kItemType);
}
} // namespace
#if defined(OS_WIN)
// Interprets drag events sent from Windows via the drag/drop API and forwards
// them to AppsGridView.
// On Windows, in order to have the OS perform the drag properly we need to
// provide it with a shortcut file which may or may not exist at the time the
// drag is started. Therefore while waiting for that shortcut to be located we
// just do a regular "internal" drag and transition into the synchronous drag
// when the shortcut is found/created. Hence a synchronous drag is an optional
// phase of a regular drag and non-Windows platforms drags are equivalent to a
// Windows drag that never enters the synchronous drag phase.
class SynchronousDrag : public ui::DragSourceWin {
public:
SynchronousDrag(AppsGridView* grid_view,
AppListItemView* drag_view,
const gfx::Point& drag_view_offset)
: grid_view_(grid_view),
drag_view_(drag_view),
drag_view_offset_(drag_view_offset),
has_shortcut_path_(false),
running_(false),
canceled_(false) {}
void set_shortcut_path(const base::FilePath& shortcut_path) {
has_shortcut_path_ = true;
shortcut_path_ = shortcut_path;
}
bool CanRun() {
return has_shortcut_path_ && !running_;
}
void Run() {
DCHECK(CanRun());
running_ = true;
ui::OSExchangeData data;
SetupExchangeData(&data);
// Hide the dragged view because the OS is going to create its own.
const gfx::Size drag_view_size = drag_view_->size();
drag_view_->SetSize(gfx::Size(0, 0));
// Blocks until the drag is finished. Calls into the ui::DragSourceWin
// methods.
DWORD effects;
DoDragDrop(ui::OSExchangeDataProviderWin::GetIDataObject(data),
this, DROPEFFECT_MOVE | DROPEFFECT_LINK, &effects);
// Restore the dragged view to its original size.
drag_view_->SetSize(drag_view_size);
drag_view_->OnSyncDragEnd();
grid_view_->EndDrag(canceled_ || !IsCursorWithinGridView());
}
private:
// Overridden from ui::DragSourceWin.
virtual void OnDragSourceCancel() OVERRIDE {
canceled_ = true;
}
virtual void OnDragSourceDrop() OVERRIDE {
}
virtual void OnDragSourceMove() OVERRIDE {
grid_view_->UpdateDrag(AppsGridView::MOUSE, GetCursorInGridViewCoords());
}
void SetupExchangeData(ui::OSExchangeData* data) {
data->SetFilename(shortcut_path_);
gfx::ImageSkia image(drag_view_->GetDragImage());
gfx::Size image_size(image.size());
drag_utils::SetDragImageOnDataObject(
image,
image.size(),
drag_view_offset_ - drag_view_->GetDragImageOffset(),
data);
}
HWND GetGridViewHWND() {
return views::HWNDForView(grid_view_);
}
bool IsCursorWithinGridView() {
POINT p;
GetCursorPos(&p);
return GetGridViewHWND() == WindowFromPoint(p);
}
gfx::Point GetCursorInGridViewCoords() {
POINT p;
GetCursorPos(&p);
ScreenToClient(GetGridViewHWND(), &p);
gfx::Point grid_view_pt(p.x, p.y);
views::View::ConvertPointFromWidget(grid_view_, &grid_view_pt);
return grid_view_pt;
}
AppsGridView* grid_view_;
AppListItemView* drag_view_;
gfx::Point drag_view_offset_;
bool has_shortcut_path_;
base::FilePath shortcut_path_;
bool running_;
bool canceled_;
DISALLOW_COPY_AND_ASSIGN(SynchronousDrag);
};
#endif // defined(OS_WIN)
AppsGridView::AppsGridView(AppsGridViewDelegate* delegate,
PaginationModel* pagination_model,
content::WebContents* start_page_contents)
: model_(NULL),
item_list_(NULL),
delegate_(delegate),
pagination_model_(pagination_model),
page_switcher_view_(new PageSwitcher(pagination_model)),
start_page_view_(NULL),
cols_(0),
rows_per_page_(0),
selected_view_(NULL),
drag_view_(NULL),
drag_start_page_(-1),
drag_pointer_(NONE),
drop_attempt_(DROP_FOR_NONE),
drag_and_drop_host_(NULL),
forward_events_to_drag_and_drop_host_(false),
page_flip_target_(-1),
page_flip_delay_in_ms_(kPageFlipDelayInMs),
bounds_animator_(this),
is_root_level_(true),
activated_item_view_(NULL),
dragging_for_reparent_item_(false) {
SetPaintToLayer(true);
SetFillsBoundsOpaquely(false);
pagination_model_->AddObserver(this);
AddChildView(page_switcher_view_);
if (start_page_contents) {
start_page_view_ =
new views::WebView(start_page_contents->GetBrowserContext());
start_page_view_->SetWebContents(start_page_contents);
AddChildView(start_page_view_);
}
}
AppsGridView::~AppsGridView() {
// Coming here |drag_view_| should already be canceled since otherwise the
// drag would disappear after the app list got animated away and closed,
// which would look odd.
DCHECK(!drag_view_);
if (drag_view_)
EndDrag(true);
if (model_)
model_->RemoveObserver(this);
pagination_model_->RemoveObserver(this);
if (item_list_)
item_list_->RemoveObserver(this);
}
void AppsGridView::SetLayout(int icon_size, int cols, int rows_per_page) {
icon_size_.SetSize(icon_size, icon_size);
cols_ = cols;
rows_per_page_ = rows_per_page;
SetBorder(views::Border::CreateEmptyBorder(
kTopPadding, kLeftRightPadding, 0, kLeftRightPadding));
}
void AppsGridView::SetModel(AppListModel* model) {
if (model_)
model_->RemoveObserver(this);
model_ = model;
if (model_)
model_->AddObserver(this);
Update();
}
void AppsGridView::SetItemList(AppListItemList* item_list) {
if (item_list_)
item_list_->RemoveObserver(this);
item_list_ = item_list;
if (item_list_)
item_list_->AddObserver(this);
Update();
}
void AppsGridView::SetSelectedView(views::View* view) {
if (IsSelectedView(view) || IsDraggedView(view))
return;
Index index = GetIndexOfView(view);
if (IsValidIndex(index))
SetSelectedItemByIndex(index);
}
void AppsGridView::ClearSelectedView(views::View* view) {
if (view && IsSelectedView(view)) {
selected_view_->SchedulePaint();
selected_view_ = NULL;
}
}
void AppsGridView::ClearAnySelectedView() {
if (selected_view_) {
selected_view_->SchedulePaint();
selected_view_ = NULL;
}
}
bool AppsGridView::IsSelectedView(const views::View* view) const {
return selected_view_ == view;
}
void AppsGridView::EnsureViewVisible(const views::View* view) {
if (pagination_model_->has_transition())
return;
Index index = GetIndexOfView(view);
if (IsValidIndex(index))
pagination_model_->SelectPage(index.page, false);
}
void AppsGridView::InitiateDrag(AppListItemView* view,
Pointer pointer,
const ui::LocatedEvent& event) {
DCHECK(view);
if (drag_view_ || pulsing_blocks_model_.view_size())
return;
drag_view_ = view;
drag_view_offset_ = event.location();
drag_start_page_ = pagination_model_->selected_page();
ExtractDragLocation(event, &drag_start_grid_view_);
drag_view_start_ = gfx::Point(drag_view_->x(), drag_view_->y());
}
void AppsGridView::OnGotShortcutPath(const base::FilePath& path) {
#if defined(OS_WIN)
// Drag may have ended before we get the shortcut path.
if (!synchronous_drag_)
return;
// Setting the shortcut path here means the next time we hit UpdateDrag()
// we'll enter the synchronous drag.
// NOTE we don't Run() the drag here because that causes animations not to
// update for some reason.
synchronous_drag_->set_shortcut_path(path);
DCHECK(synchronous_drag_->CanRun());
#endif
}
void AppsGridView::StartSettingUpSynchronousDrag() {
#if defined(OS_WIN)
if (!delegate_)
return;
// Favor the drag and drop host over native win32 drag. For the Win8/ash
// launcher we want to have ashes drag and drop over win32's.
if (drag_and_drop_host_)
return;
delegate_->GetShortcutPathForApp(
drag_view_->item()->id(),
base::Bind(&AppsGridView::OnGotShortcutPath, base::Unretained(this)));
synchronous_drag_ = new SynchronousDrag(this, drag_view_, drag_view_offset_);
#endif
}
bool AppsGridView::RunSynchronousDrag() {
#if defined(OS_WIN)
if (synchronous_drag_ && synchronous_drag_->CanRun()) {
synchronous_drag_->Run();
synchronous_drag_ = NULL;
return true;
}
#endif
return false;
}
void AppsGridView::CleanUpSynchronousDrag() {
#if defined(OS_WIN)
synchronous_drag_ = NULL;
#endif
}
void AppsGridView::UpdateDragFromItem(Pointer pointer,
const ui::LocatedEvent& event) {
DCHECK(drag_view_);
if (!is_root_level_)
UpdateDragStateInsideFolder(pointer, event);
gfx::Point drag_point_in_grid_view;
ExtractDragLocation(event, &drag_point_in_grid_view);
UpdateDrag(pointer, drag_point_in_grid_view);
if (!dragging())
return;
// If a drag and drop host is provided, see if the drag operation needs to be
// forwarded.
gfx::Point location_in_screen = drag_point_in_grid_view;
views::View::ConvertPointToScreen(this, &location_in_screen);
DispatchDragEventToDragAndDropHost(location_in_screen);
if (drag_and_drop_host_)
drag_and_drop_host_->UpdateDragIconProxy(location_in_screen);
}
void AppsGridView::UpdateDrag(Pointer pointer, const gfx::Point& point) {
// EndDrag was called before if |drag_view_| is NULL.
if (!drag_view_)
return;
if (RunSynchronousDrag())
return;
gfx::Vector2d drag_vector(point - drag_start_grid_view_);
if (!dragging() && ExceededDragThreshold(drag_vector)) {
drag_pointer_ = pointer;
// Move the view to the front so that it appears on top of other views.
ReorderChildView(drag_view_, -1);
bounds_animator_.StopAnimatingView(drag_view_);
StartSettingUpSynchronousDrag();
if (!dragging_for_reparent_item_)
StartDragAndDropHostDrag(point);
}
if (drag_pointer_ != pointer)
return;
last_drag_point_ = point;
const Index last_drop_target = drop_target_;
DropAttempt last_drop_attempt = drop_attempt_;
CalculateDropTarget(last_drag_point_, false);
if (IsPointWithinDragBuffer(last_drag_point_))
MaybeStartPageFlipTimer(last_drag_point_);
else
StopPageFlipTimer();
gfx::Point page_switcher_point(last_drag_point_);
views::View::ConvertPointToTarget(this, page_switcher_view_,
&page_switcher_point);
page_switcher_view_->UpdateUIForDragPoint(page_switcher_point);
if (!EnableFolderDragDropUI()) {
if (last_drop_target != drop_target_)
AnimateToIdealBounds();
drag_view_->SetPosition(drag_view_start_ + drag_vector);
return;
}
// Update drag with folder UI enabled.
if (last_drop_target != drop_target_ ||
last_drop_attempt != drop_attempt_) {
if (drop_attempt_ == DROP_FOR_REORDER) {
folder_dropping_timer_.Stop();
reorder_timer_.Start(FROM_HERE,
base::TimeDelta::FromMilliseconds(kReorderDelay),
this, &AppsGridView::OnReorderTimer);
} else if (drop_attempt_ == DROP_FOR_FOLDER) {
reorder_timer_.Stop();
folder_dropping_timer_.Start(FROM_HERE,
base::TimeDelta::FromMilliseconds(kFolderDroppingDelay),
this, &AppsGridView::OnFolderDroppingTimer);
}
// Reset the previous drop target.
SetAsFolderDroppingTarget(last_drop_target, false);
}
drag_view_->SetPosition(drag_view_start_ + drag_vector);
}
void AppsGridView::EndDrag(bool cancel) {
// EndDrag was called before if |drag_view_| is NULL.
if (!drag_view_)
return;
// Coming here a drag and drop was in progress.
bool landed_in_drag_and_drop_host = forward_events_to_drag_and_drop_host_;
if (forward_events_to_drag_and_drop_host_) {
DCHECK(!IsDraggingForReparentInRootLevelGridView());
forward_events_to_drag_and_drop_host_ = false;
drag_and_drop_host_->EndDrag(cancel);
if (IsDraggingForReprentInHiddenGridView()) {
static_cast<AppListFolderView*>(parent())->
DispatchEndDragEventForReparent(true);
}
} else if (!cancel && dragging()) {
if (IsDraggingForReprentInHiddenGridView()) {
// Forward the EndDrag event to the root level grid view.
static_cast<AppListFolderView*>(parent())->
DispatchEndDragEventForReparent(false);
EndDragForReparentInHiddenFolderGridView();
return;
} else {
// Regular drag ending path, ie, not for reparenting.
CalculateDropTarget(last_drag_point_, true);
if (IsValidIndex(drop_target_)) {
if (!EnableFolderDragDropUI()) {
MoveItemInModel(drag_view_, drop_target_);
} else {
if (drop_attempt_ == DROP_FOR_REORDER)
MoveItemInModel(drag_view_, drop_target_);
else if (drop_attempt_ == DROP_FOR_FOLDER)
MoveItemToFolder(drag_view_, drop_target_);
}
}
}
}
if (drag_and_drop_host_) {
// If we had a drag and drop proxy icon, we delete it and make the real
// item visible again.
drag_and_drop_host_->DestroyDragIconProxy();
if (landed_in_drag_and_drop_host) {
// Move the item directly to the target location, avoiding the "zip back"
// animation if the user was pinning it to the shelf.
int i = drop_target_.slot;
gfx::Rect bounds = view_model_.ideal_bounds(i);
drag_view_->SetBoundsRect(bounds);
}
// Fade in slowly if it landed in the shelf.
SetViewHidden(drag_view_,
false /* show */,
!landed_in_drag_and_drop_host /* animate */);
}
// The drag can be ended after the synchronous drag is created but before it
// is Run().
CleanUpSynchronousDrag();
SetAsFolderDroppingTarget(drop_target_, false);
drop_attempt_ = DROP_FOR_NONE;
drag_pointer_ = NONE;
drop_target_ = Index();
drag_view_->OnDragEnded();
drag_view_ = NULL;
drag_start_grid_view_ = gfx::Point();
drag_start_page_ = -1;
drag_view_offset_ = gfx::Point();
if (IsDraggingForReprentInHiddenGridView())
dragging_for_reparent_item_ = false;
AnimateToIdealBounds();
StopPageFlipTimer();
// If user releases mouse inside a folder's grid view, burst the folder
// container ink bubble.
if (!is_root_level_ && !IsDraggingForReprentInHiddenGridView()) {
static_cast<AppListFolderView*>(parent())->
UpdateFolderViewBackground(false);
}
}
void AppsGridView::StopPageFlipTimer() {
page_flip_timer_.Stop();
page_flip_target_ = -1;
}
AppListItemView* AppsGridView::GetItemViewAt(int index) const {
DCHECK(index >= 0 && index < view_model_.view_size());
return static_cast<AppListItemView*>(view_model_.view_at(index));
}
void AppsGridView::SetTopItemViewsVisible(bool visible) {
int top_item_count = std::min(static_cast<int>(kNumFolderTopItems),
view_model_.view_size());
for (int i = 0; i < top_item_count; ++i)
GetItemViewAt(i)->SetVisible(visible);
}
void AppsGridView::ScheduleShowHideAnimation(bool show) {
// Stop any previous animation.
layer()->GetAnimator()->StopAnimating();
// Set initial state.
SetVisible(true);
layer()->SetOpacity(show ? 0.0f : 1.0f);
ui::ScopedLayerAnimationSettings animation(layer()->GetAnimator());
animation.AddObserver(this);
animation.SetTweenType(
show ? kFolderFadeInTweenType : kFolderFadeOutTweenType);
animation.SetTransitionDuration(base::TimeDelta::FromMilliseconds(
show ? kFolderTransitionInDurationMs : kFolderTransitionOutDurationMs));
layer()->SetOpacity(show ? 1.0f : 0.0f);
}
void AppsGridView::InitiateDragFromReparentItemInRootLevelGridView(
AppListItemView* original_drag_view,
const gfx::Rect& drag_view_rect,
const gfx::Point& drag_point) {
DCHECK(original_drag_view && !drag_view_);
DCHECK(!dragging_for_reparent_item_);
// Create a new AppListItemView to duplicate the original_drag_view in the
// folder's grid view.
AppListItemView* view = new AppListItemView(this, original_drag_view->item());
AddChildView(view);
drag_view_ = view;
drag_view_->SetPaintToLayer(true);
// Note: For testing purpose, SetFillsBoundsOpaquely can be set to true to
// show the gray background.
drag_view_->SetFillsBoundsOpaquely(false);
drag_view_->SetIconSize(icon_size_);
drag_view_->SetBoundsRect(drag_view_rect);
drag_view_->SetDragUIState(); // Hide the title of the drag_view_.
// Hide the drag_view_ for drag icon proxy.
SetViewHidden(drag_view_,
true /* hide */,
true /* no animate */);
// Add drag_view_ to the end of the view_model_.
view_model_.Add(drag_view_, view_model_.view_size());
drag_start_page_ = pagination_model_->selected_page();
drag_start_grid_view_ = drag_point;
drag_view_start_ = gfx::Point(drag_view_->x(), drag_view_->y());
// Set the flag in root level grid view.
dragging_for_reparent_item_ = true;
}
void AppsGridView::UpdateDragFromReparentItem(
Pointer pointer,
const ui::LocatedEvent& event) {
DCHECK(drag_view_);
DCHECK(IsDraggingForReparentInRootLevelGridView());
gfx::Point drag_point_in_grid_view;
ExtractDragLocation(event, &drag_point_in_grid_view);
UpdateDrag(pointer, drag_point_in_grid_view);
}
bool AppsGridView::IsDraggedView(const views::View* view) const {
return drag_view_ == view;
}
void AppsGridView::SetDragAndDropHostOfCurrentAppList(
ApplicationDragAndDropHost* drag_and_drop_host) {
drag_and_drop_host_ = drag_and_drop_host;
}
void AppsGridView::Prerender(int page_index) {
Layout();
int start = std::max(0, (page_index - kPrerenderPages) * tiles_per_page());
int end = std::min(view_model_.view_size(),
(page_index + 1 + kPrerenderPages) * tiles_per_page());
for (int i = start; i < end; i++) {
AppListItemView* v = static_cast<AppListItemView*>(view_model_.view_at(i));
v->Prerender();
}
}
gfx::Size AppsGridView::GetPreferredSize() {
const gfx::Insets insets(GetInsets());
const gfx::Size tile_size = gfx::Size(kPreferredTileWidth,
kPreferredTileHeight);
const int page_switcher_height =
page_switcher_view_->GetPreferredSize().height();
return gfx::Size(
tile_size.width() * cols_ + insets.width(),
tile_size.height() * rows_per_page_ +
page_switcher_height + insets.height());
}
bool AppsGridView::GetDropFormats(
int* formats,
std::set<OSExchangeData::CustomFormat>* custom_formats) {
// TODO(koz): Only accept a specific drag type for app shortcuts.
*formats = OSExchangeData::FILE_NAME;
return true;
}
bool AppsGridView::CanDrop(const OSExchangeData& data) {
return true;
}
int AppsGridView::OnDragUpdated(const ui::DropTargetEvent& event) {
return ui::DragDropTypes::DRAG_MOVE;
}
void AppsGridView::Layout() {
if (bounds_animator_.IsAnimating())
bounds_animator_.Cancel();
CalculateIdealBounds();
for (int i = 0; i < view_model_.view_size(); ++i) {
views::View* view = view_model_.view_at(i);
if (view != drag_view_)
view->SetBoundsRect(view_model_.ideal_bounds(i));
}
views::ViewModelUtils::SetViewBoundsToIdealBounds(pulsing_blocks_model_);
const int page_switcher_height =
page_switcher_view_->GetPreferredSize().height();
gfx::Rect rect(GetContentsBounds());
rect.set_y(rect.bottom() - page_switcher_height);
rect.set_height(page_switcher_height);
page_switcher_view_->SetBoundsRect(rect);
LayoutStartPage();
}
bool AppsGridView::OnKeyPressed(const ui::KeyEvent& event) {
bool handled = false;
if (selected_view_)
handled = selected_view_->OnKeyPressed(event);
if (!handled) {
const int forward_dir = base::i18n::IsRTL() ? -1 : 1;
switch (event.key_code()) {
case ui::VKEY_LEFT:
MoveSelected(0, -forward_dir, 0);
return true;
case ui::VKEY_RIGHT:
MoveSelected(0, forward_dir, 0);
return true;
case ui::VKEY_UP:
MoveSelected(0, 0, -1);
return true;
case ui::VKEY_DOWN:
MoveSelected(0, 0, 1);
return true;
case ui::VKEY_PRIOR: {
MoveSelected(-1, 0, 0);
return true;
}
case ui::VKEY_NEXT: {
MoveSelected(1, 0, 0);
return true;
}
default:
break;
}
}
return handled;
}
bool AppsGridView::OnKeyReleased(const ui::KeyEvent& event) {
bool handled = false;
if (selected_view_)
handled = selected_view_->OnKeyReleased(event);
return handled;
}
void AppsGridView::ViewHierarchyChanged(
const ViewHierarchyChangedDetails& details) {
if (!details.is_add && details.parent == this) {
if (selected_view_ == details.child)
selected_view_ = NULL;
if (drag_view_ == details.child)
EndDrag(true);
bounds_animator_.StopAnimatingView(details.child);
}
}
void AppsGridView::Update() {
DCHECK(!selected_view_ && !drag_view_);
view_model_.Clear();
if (!item_list_ || !item_list_->item_count())
return;
for (size_t i = 0; i < item_list_->item_count(); ++i) {
views::View* view = CreateViewForItemAtIndex(i);
view_model_.Add(view, i);
AddChildView(view);
}
UpdatePaging();
UpdatePulsingBlockViews();
Layout();
SchedulePaint();
}
void AppsGridView::UpdatePaging() {
int total_page = start_page_view_ ? 1 : 0;
if (view_model_.view_size() && tiles_per_page())
total_page += (view_model_.view_size() - 1) / tiles_per_page() + 1;
pagination_model_->SetTotalPages(total_page);
}
void AppsGridView::UpdatePulsingBlockViews() {
const int existing_items = item_list_ ? item_list_->item_count() : 0;
const int available_slots =
tiles_per_page() - existing_items % tiles_per_page();
const int desired = model_->status() == AppListModel::STATUS_SYNCING ?
available_slots : 0;
if (pulsing_blocks_model_.view_size() == desired)
return;
while (pulsing_blocks_model_.view_size() > desired) {
views::View* view = pulsing_blocks_model_.view_at(0);
pulsing_blocks_model_.Remove(0);
delete view;
}
while (pulsing_blocks_model_.view_size() < desired) {
views::View* view = new PulsingBlockView(
gfx::Size(kPreferredTileWidth, kPreferredTileHeight), true);
pulsing_blocks_model_.Add(view, 0);
AddChildView(view);
}
}
views::View* AppsGridView::CreateViewForItemAtIndex(size_t index) {
// The drag_view_ might be pending for deletion, therefore view_model_
// may have one more item than item_list_.
DCHECK_LE(index, item_list_->item_count());
AppListItemView* view = new AppListItemView(this,
item_list_->item_at(index));
view->SetIconSize(icon_size_);
#if defined(USE_AURA)
view->SetPaintToLayer(true);
view->SetFillsBoundsOpaquely(false);
#endif
return view;
}
AppsGridView::Index AppsGridView::GetIndexFromModelIndex(
int model_index) const {
int page = model_index / tiles_per_page();
if (start_page_view_)
++page;
return Index(page, model_index % tiles_per_page());
}
int AppsGridView::GetModelIndexFromIndex(const Index& index) const {
int model_index = index.page * tiles_per_page() + index.slot;
if (start_page_view_)
model_index -= tiles_per_page();
return model_index;
}
void AppsGridView::SetSelectedItemByIndex(const Index& index) {
if (GetIndexOfView(selected_view_) == index)
return;
views::View* new_selection = GetViewAtIndex(index);
if (!new_selection)
return; // Keep current selection.
if (selected_view_)
selected_view_->SchedulePaint();
EnsureViewVisible(new_selection);
selected_view_ = new_selection;
selected_view_->SchedulePaint();
selected_view_->NotifyAccessibilityEvent(
ui::AccessibilityTypes::EVENT_FOCUS, true);
}
bool AppsGridView::IsValidIndex(const Index& index) const {
const int item_page_start = start_page_view_ ? 1 : 0;
return index.page >= item_page_start &&
index.page < pagination_model_->total_pages() &&
index.slot >= 0 &&
index.slot < tiles_per_page() &&
GetModelIndexFromIndex(index) < view_model_.view_size();
}
AppsGridView::Index AppsGridView::GetIndexOfView(
const views::View* view) const {
const int model_index = view_model_.GetIndexOfView(view);
if (model_index == -1)
return Index();
return GetIndexFromModelIndex(model_index);
}
views::View* AppsGridView::GetViewAtIndex(const Index& index) const {
if (!IsValidIndex(index))
return NULL;
const int model_index = GetModelIndexFromIndex(index);
return view_model_.view_at(model_index);
}
void AppsGridView::MoveSelected(int page_delta,
int slot_x_delta,
int slot_y_delta) {
if (!selected_view_)
return SetSelectedItemByIndex(Index(pagination_model_->selected_page(), 0));
const Index& selected = GetIndexOfView(selected_view_);
int target_slot = selected.slot + slot_x_delta + slot_y_delta * cols_;
if (selected.slot % cols_ == 0 && slot_x_delta == -1) {
if (selected.page > 0) {
page_delta = -1;
target_slot = selected.slot + cols_ - 1;
} else {
target_slot = selected.slot;
}
}
if (selected.slot % cols_ == cols_ - 1 && slot_x_delta == 1) {
if (selected.page < pagination_model_->total_pages() - 1) {
page_delta = 1;
target_slot = selected.slot - cols_ + 1;
} else {
target_slot = selected.slot;
}
}
// Clamp the target slot to the last item if we are moving to the last page
// but our target slot is past the end of the item list.
if (page_delta &&
selected.page + page_delta == pagination_model_->total_pages() - 1) {
int last_item_slot = (view_model_.view_size() - 1) % tiles_per_page();
if (last_item_slot < target_slot) {
target_slot = last_item_slot;
}
}
int target_page = std::min(pagination_model_->total_pages() - 1,
std::max(selected.page + page_delta, 0));
SetSelectedItemByIndex(Index(target_page, target_slot));
}
void AppsGridView::CalculateIdealBounds() {
gfx::Rect rect(GetContentsBounds());
if (rect.IsEmpty())
return;
gfx::Size tile_size(kPreferredTileWidth, kPreferredTileHeight);
gfx::Rect grid_rect(gfx::Size(tile_size.width() * cols_,
tile_size.height() * rows_per_page_));
grid_rect.Intersect(rect);
// Page width including padding pixels. A tile.x + page_width means the same
// tile slot in the next page.
const int page_width = grid_rect.width() + kPagePadding;
// If there is a transition, calculates offset for current and target page.
const int current_page = pagination_model_->selected_page();
const PaginationModel::Transition& transition =
pagination_model_->transition();
const bool is_valid =
pagination_model_->is_valid_page(transition.target_page);
// Transition to right means negative offset.
const int dir = transition.target_page > current_page ? -1 : 1;
const int transition_offset = is_valid ?
transition.progress * page_width * dir : 0;
const int total_views =
view_model_.view_size() + pulsing_blocks_model_.view_size();
int slot_index = 0;
for (int i = 0; i < total_views; ++i) {
if (i < view_model_.view_size() && view_model_.view_at(i) == drag_view_) {
if (EnableFolderDragDropUI() && drop_attempt_ == DROP_FOR_FOLDER)
++slot_index;
continue;
}
Index view_index = GetIndexFromModelIndex(slot_index);
if (drop_target_ == view_index) {
if (EnableFolderDragDropUI() && drop_attempt_ == DROP_FOR_FOLDER) {
view_index = GetIndexFromModelIndex(slot_index);
} else if (!EnableFolderDragDropUI() ||
drop_attempt_ == DROP_FOR_REORDER) {
++slot_index;
view_index = GetIndexFromModelIndex(slot_index);
}
}
// Decides an x_offset for current item.
int x_offset = 0;
if (view_index.page < current_page)
x_offset = -page_width;
else if (view_index.page > current_page)
x_offset = page_width;
if (is_valid) {
if (view_index.page == current_page ||
view_index.page == transition.target_page) {
x_offset += transition_offset;
}
}
const int row = view_index.slot / cols_;
const int col = view_index.slot % cols_;
gfx::Rect tile_slot(
gfx::Point(grid_rect.x() + col * tile_size.width() + x_offset,
grid_rect.y() + row * tile_size.height()),
tile_size);
if (i < view_model_.view_size()) {
view_model_.set_ideal_bounds(i, tile_slot);
} else {
pulsing_blocks_model_.set_ideal_bounds(i - view_model_.view_size(),
tile_slot);
}
++slot_index;
}
}
void AppsGridView::AnimateToIdealBounds() {
const gfx::Rect visible_bounds(GetVisibleBounds());
CalculateIdealBounds();
for (int i = 0; i < view_model_.view_size(); ++i) {
views::View* view = view_model_.view_at(i);
if (view == drag_view_)
continue;
const gfx::Rect& target = view_model_.ideal_bounds(i);
if (bounds_animator_.GetTargetBounds(view) == target)
continue;
const gfx::Rect& current = view->bounds();
const bool current_visible = visible_bounds.Intersects(current);
const bool target_visible = visible_bounds.Intersects(target);
const bool visible = current_visible || target_visible;
const int y_diff = target.y() - current.y();
if (visible && y_diff && y_diff % kPreferredTileHeight == 0) {
AnimationBetweenRows(view,
current_visible,
current,
target_visible,
target);
} else {
bounds_animator_.AnimateViewTo(view, target);
}
}
}
void AppsGridView::AnimationBetweenRows(views::View* view,
bool animate_current,
const gfx::Rect& current,
bool animate_target,
const gfx::Rect& target) {
// Determine page of |current| and |target|. -1 means in the left invisible
// page, 0 is the center visible page and 1 means in the right invisible page.
const int current_page = current.x() < 0 ? -1 :
current.x() >= width() ? 1 : 0;
const int target_page = target.x() < 0 ? -1 :
target.x() >= width() ? 1 : 0;
const int dir = current_page < target_page ||
(current_page == target_page && current.y() < target.y()) ? 1 : -1;
#if defined(USE_AURA)
scoped_ptr<ui::Layer> layer;
if (animate_current) {
layer.reset(view->RecreateLayer());
layer->SuppressPaint();
view->SetFillsBoundsOpaquely(false);
view->layer()->SetOpacity(0.f);
}
gfx::Rect current_out(current);
current_out.Offset(dir * kPreferredTileWidth, 0);
#endif
gfx::Rect target_in(target);
if (animate_target)
target_in.Offset(-dir * kPreferredTileWidth, 0);
view->SetBoundsRect(target_in);
bounds_animator_.AnimateViewTo(view, target);
#if defined(USE_AURA)
bounds_animator_.SetAnimationDelegate(
view,
new RowMoveAnimationDelegate(view, layer.release(), current_out),
true);
#endif
}
void AppsGridView::ExtractDragLocation(const ui::LocatedEvent& event,
gfx::Point* drag_point) {
#if defined(USE_AURA) && !defined(OS_WIN)
// Use root location of |event| instead of location in |drag_view_|'s
// coordinates because |drag_view_| has a scale transform and location
// could have integer round error and causes jitter.
*drag_point = event.root_location();
// GetWidget() could be NULL for tests.
if (GetWidget()) {
aura::Window::ConvertPointToTarget(
GetWidget()->GetNativeWindow()->GetRootWindow(),
GetWidget()->GetNativeWindow(),
drag_point);
}
views::View::ConvertPointFromWidget(this, drag_point);
#else
// For non-aura, root location is not clearly defined but |drag_view_| does
// not have the scale transform. So no round error would be introduced and
// it's okay to use View::ConvertPointToTarget.
*drag_point = event.location();
views::View::ConvertPointToTarget(drag_view_, this, drag_point);
#endif
}
void AppsGridView::CalculateDropTarget(const gfx::Point& drag_point,
bool use_page_button_hovering) {
if (EnableFolderDragDropUI()) {
CalculateDropTargetWithFolderEnabled(drag_point, use_page_button_hovering);
return;
}
int current_page = pagination_model_->selected_page();
gfx::Point point(drag_point);
if (!IsPointWithinDragBuffer(drag_point)) {
point = drag_start_grid_view_;
current_page = drag_start_page_;
}
if (use_page_button_hovering &&
page_switcher_view_->bounds().Contains(point)) {
gfx::Point page_switcher_point(point);
views::View::ConvertPointToTarget(this, page_switcher_view_,
&page_switcher_point);
int page = page_switcher_view_->GetPageForPoint(page_switcher_point);
if (pagination_model_->is_valid_page(page)) {
drop_target_.page = page;
drop_target_.slot = tiles_per_page() - 1;
}
} else {
gfx::Rect bounds(GetContentsBounds());
const int drop_row = (point.y() - bounds.y()) / kPreferredTileHeight;
const int drop_col = std::min(cols_ - 1,
(point.x() - bounds.x()) / kPreferredTileWidth);
drop_target_.page = current_page;
drop_target_.slot = std::max(0, std::min(
tiles_per_page() - 1,
drop_row * cols_ + drop_col));
}
// Limits to the last possible slot on last page.
if (drop_target_.page == pagination_model_->total_pages() - 1) {
drop_target_.slot = std::min(
(view_model_.view_size() - 1) % tiles_per_page(),
drop_target_.slot);
}
}
void AppsGridView::CalculateDropTargetWithFolderEnabled(
const gfx::Point& drag_point,
bool use_page_button_hovering) {
gfx::Point point(drag_point);
if (!IsPointWithinDragBuffer(drag_point)) {
point = drag_start_grid_view_;
}
if (use_page_button_hovering &&
page_switcher_view_->bounds().Contains(point)) {
gfx::Point page_switcher_point(point);
views::View::ConvertPointToTarget(this, page_switcher_view_,
&page_switcher_point);
int page = page_switcher_view_->GetPageForPoint(page_switcher_point);
if (pagination_model_->is_valid_page(page))
drop_attempt_ = DROP_FOR_NONE;
} else {
DCHECK(drag_view_);
// Try to find the nearest target for folder dropping or re-ordering.
drop_target_ = GetNearestTileForDragView();
}
}
void AppsGridView::OnReorderTimer() {
if (drop_attempt_ == DROP_FOR_REORDER)
AnimateToIdealBounds();
}
void AppsGridView::OnFolderItemReparentTimer() {
DCHECK(!is_root_level_);
if (drag_out_of_folder_container_) {
static_cast<AppListFolderView*>(parent())->ReparentItem(
drag_view_, last_drag_point_);
// Set the flag in the folder's grid view.
dragging_for_reparent_item_ = true;
// Do not observe any data change since it is going to be hidden.
item_list_->RemoveObserver(this);
item_list_ = NULL;
}
}
void AppsGridView::OnFolderDroppingTimer() {
if (drop_attempt_ == DROP_FOR_FOLDER)
SetAsFolderDroppingTarget(drop_target_, true);
}
void AppsGridView::UpdateDragStateInsideFolder(
Pointer pointer,
const ui::LocatedEvent& event) {
if (IsDraggingForReprentInHiddenGridView()) {
// Dispatch drag event to root level grid view for re-parenting folder
// folder item purpose.
DispatchDragEventForReparent(pointer, event);
return;
}
// Regular drag and drop in a folder's grid view.
AppListFolderView* folder_view = static_cast<AppListFolderView*>(parent());
folder_view->UpdateFolderViewBackground(true);
// Calculate if the drag_view_ is dragged out of the folder's container
// ink bubble.
gfx::Rect bounds_to_folder_view = ConvertRectToParent(drag_view_->bounds());
gfx::Point pt = bounds_to_folder_view.CenterPoint();
bool is_item_dragged_out_of_folder =
folder_view->IsPointOutsideOfFolderBoundray(pt);
if (is_item_dragged_out_of_folder) {
if (!drag_out_of_folder_container_) {
folder_item_reparent_timer_.Start(FROM_HERE,
base::TimeDelta::FromMilliseconds(kFolderItemReparentDealy),
this, &AppsGridView::OnFolderItemReparentTimer);
drag_out_of_folder_container_ = true;
}
} else {
folder_item_reparent_timer_.Stop();
drag_out_of_folder_container_ = false;
}
}
bool AppsGridView::IsDraggingForReparentInRootLevelGridView() const {
return (is_root_level_ && dragging_for_reparent_item_);
}
bool AppsGridView::IsDraggingForReprentInHiddenGridView() const {
return (!is_root_level_ && dragging_for_reparent_item_);
}
gfx::Rect AppsGridView::GetTargetIconRectInFolder(
AppListItemView* drag_item_view,
AppListItemView* folder_item_view) {
gfx::Rect view_ideal_bounds = view_model_.ideal_bounds(
view_model_.GetIndexOfView(folder_item_view));
gfx::Rect icon_ideal_bounds =
folder_item_view->GetIconBoundsForTargetViewBounds(view_ideal_bounds);
AppListFolderItem* folder_item =
static_cast<AppListFolderItem*>(folder_item_view->item());
return folder_item->GetTargetIconRectInFolderForItem(
drag_item_view->item(), icon_ideal_bounds);
}
void AppsGridView::DispatchDragEventForReparent(
Pointer pointer,
const ui::LocatedEvent& event) {
static_cast<AppListFolderView*>(parent())->
DispatchDragEventForReparent(pointer, event);
}
void AppsGridView::EndDragFromReparentItemInRootLevel(
bool events_forwarded_to_drag_drop_host) {
// EndDrag was called before if |drag_view_| is NULL.
if (!drag_view_)
return;
DCHECK(IsDraggingForReparentInRootLevelGridView());
bool cancel_reparent = false;
scoped_ptr<AppListItemView> cached_drag_view;
if (!events_forwarded_to_drag_drop_host) {
CalculateDropTarget(last_drag_point_, true);
if (IsValidIndex(drop_target_)) {
if (drop_attempt_ == DROP_FOR_REORDER)
ReparentItemForReorder(drag_view_, drop_target_);
else if (drop_attempt_ == DROP_FOR_FOLDER)
ReparentItemToAnotherFolder(drag_view_, drop_target_);
else { // DROP_FOR_NONE_
cancel_reparent = true;
// Note(jennyz): cached_drag_view makes sure drag_view_ will be deleted
// after AnimateToIdealBounds() is called.
// There is a problem in layer() animation which cause DCHECK failure
// if a child view is deleted immediately before re-creating layer in
// layer animation. The layer tree seems marked dirty, and complaining
// when we try to re-create layer in AnimationBetweenRows when calling
// AnimateToIdealBounds.
cached_drag_view.reset(drag_view_);
}
}
if (!cancel_reparent) {
SetViewHidden(drag_view_,
false /* show */,
true /* no animate */);
}
}
// The drag can be ended after the synchronous drag is created but before it
// is Run().
CleanUpSynchronousDrag();
SetAsFolderDroppingTarget(drop_target_, false);
drop_attempt_ = DROP_FOR_NONE;
drag_pointer_ = NONE;
drop_target_ = Index();
if (!cancel_reparent)
drag_view_->OnDragEnded();
drag_view_ = NULL;
drag_start_grid_view_ = gfx::Point();
drag_start_page_ = -1;
drag_view_offset_ = gfx::Point();
dragging_for_reparent_item_ = false;
if (cancel_reparent)
CancelFolderItemReparent(cached_drag_view.get());
AnimateToIdealBounds();
StopPageFlipTimer();
}
void AppsGridView::EndDragForReparentInHiddenFolderGridView() {
if (drag_and_drop_host_) {
// If we had a drag and drop proxy icon, we delete it and make the real
// item visible again.
drag_and_drop_host_->DestroyDragIconProxy();
}
// The drag can be ended after the synchronous drag is created but before it
// is Run().
CleanUpSynchronousDrag();
SetAsFolderDroppingTarget(drop_target_, false);
drop_attempt_ = DROP_FOR_NONE;
drag_pointer_ = NONE;
drop_target_ = Index();
drag_view_->OnDragEnded();
drag_view_ = NULL;
drag_start_grid_view_ = gfx::Point();
drag_start_page_ = -1;
drag_view_offset_ = gfx::Point();
dragging_for_reparent_item_ = false;
}
void AppsGridView::OnFolderItemRemoved() {
DCHECK(!is_root_level_);
item_list_ = NULL;
}
void AppsGridView::StartDragAndDropHostDrag(const gfx::Point& grid_location) {
// When a drag and drop host is given, the item can be dragged out of the app
// list window. In that case a proxy widget needs to be used.
// Note: This code has very likely to be changed for Windows (non metro mode)
// when a |drag_and_drop_host_| gets implemented.
if (!drag_view_ || !drag_and_drop_host_)
return;
gfx::Point screen_location = grid_location;
views::View::ConvertPointToScreen(this, &screen_location);
// Determine the mouse offset to the center of the icon so that the drag and
// drop host follows this layer.
gfx::Vector2d delta = drag_view_offset_ -
drag_view_->GetLocalBounds().CenterPoint();
delta.set_y(delta.y() + drag_view_->title()->size().height() / 2);
// We have to hide the original item since the drag and drop host will do
// the OS dependent code to "lift off the dragged item".
DCHECK(!IsDraggingForReparentInRootLevelGridView());
drag_and_drop_host_->CreateDragIconProxy(screen_location,
drag_view_->item()->icon(),
drag_view_,
delta,
kDragAndDropProxyScale);
SetViewHidden(drag_view_,
true /* hide */,
true /* no animation */);
}
void AppsGridView::DispatchDragEventToDragAndDropHost(
const gfx::Point& location_in_screen_coordinates) {
if (!drag_view_ || !drag_and_drop_host_)
return;
if (GetLocalBounds().Contains(last_drag_point_)) {
// The event was issued inside the app menu and we should get all events.
if (forward_events_to_drag_and_drop_host_) {
// The DnD host was previously called and needs to be informed that the
// session returns to the owner.
forward_events_to_drag_and_drop_host_ = false;
drag_and_drop_host_->EndDrag(true);
}
} else {
// The event happened outside our app menu and we might need to dispatch.
if (forward_events_to_drag_and_drop_host_) {
// Dispatch since we have already started.
if (!drag_and_drop_host_->Drag(location_in_screen_coordinates)) {
// The host is not active any longer and we cancel the operation.
forward_events_to_drag_and_drop_host_ = false;
drag_and_drop_host_->EndDrag(true);
}
} else {
if (drag_and_drop_host_->StartDrag(drag_view_->item()->id(),
location_in_screen_coordinates)) {
// From now on we forward the drag events.
forward_events_to_drag_and_drop_host_ = true;
// Any flip operations are stopped.
StopPageFlipTimer();
}
}
}
}
void AppsGridView::MaybeStartPageFlipTimer(const gfx::Point& drag_point) {
if (!IsPointWithinDragBuffer(drag_point))
StopPageFlipTimer();
int new_page_flip_target = -1;
if (page_switcher_view_->bounds().Contains(drag_point)) {
gfx::Point page_switcher_point(drag_point);
views::View::ConvertPointToTarget(this, page_switcher_view_,
&page_switcher_point);
new_page_flip_target =
page_switcher_view_->GetPageForPoint(page_switcher_point);
}
// TODO(xiyuan): Fix this for RTL.
if (new_page_flip_target == -1 && drag_point.x() < kPageFlipZoneSize)
new_page_flip_target = pagination_model_->selected_page() - 1;
if (new_page_flip_target == -1 &&
drag_point.x() > width() - kPageFlipZoneSize) {
new_page_flip_target = pagination_model_->selected_page() + 1;
}
if (new_page_flip_target == page_flip_target_)
return;
StopPageFlipTimer();
if (pagination_model_->is_valid_page(new_page_flip_target)) {
page_flip_target_ = new_page_flip_target;
if (page_flip_target_ != pagination_model_->selected_page()) {
page_flip_timer_.Start(FROM_HERE,
base::TimeDelta::FromMilliseconds(page_flip_delay_in_ms_),
this, &AppsGridView::OnPageFlipTimer);
}
}
}
void AppsGridView::OnPageFlipTimer() {
DCHECK(pagination_model_->is_valid_page(page_flip_target_));
pagination_model_->SelectPage(page_flip_target_, true);
}
void AppsGridView::MoveItemInModel(views::View* item_view,
const Index& target) {
int current_model_index = view_model_.GetIndexOfView(item_view);
DCHECK_GE(current_model_index, 0);
int target_model_index = GetModelIndexFromIndex(target);
if (target_model_index == current_model_index)
return;
item_list_->RemoveObserver(this);
item_list_->MoveItem(current_model_index, target_model_index);
view_model_.Move(current_model_index, target_model_index);
item_list_->AddObserver(this);
if (pagination_model_->selected_page() != target.page)
pagination_model_->SelectPage(target.page, false);
}
void AppsGridView::MoveItemToFolder(views::View* item_view,
const Index& target) {
const std::string& source_item_id =
static_cast<AppListItemView*>(item_view)->item()->id();
AppListItemView* target_view =
static_cast<AppListItemView*>(GetViewAtSlotOnCurrentPage(target.slot));
const std::string& target_view_item_id = target_view->item()->id();
// Make change to data model.
item_list_->RemoveObserver(this);
std::string folder_item_id =
model_->MergeItems(target_view_item_id, source_item_id);
item_list_->AddObserver(this);
if (folder_item_id != target_view_item_id) {
// New folder was created, change the view model to replace the old target
// view with the new folder item view.
size_t folder_item_index;
if (item_list_->FindItemIndex(folder_item_id, &folder_item_index)) {
int target_view_index = view_model_.GetIndexOfView(target_view);
view_model_.Remove(target_view_index);
delete target_view;
views::View* target_folder_view =
CreateViewForItemAtIndex(folder_item_index);
view_model_.Add(target_folder_view, target_view_index);
AddChildView(target_folder_view);
} else {
LOG(ERROR) << "Folder no longer in item_list: " << folder_item_id;
}
}
// Fade out the drag_view_ and delete it when animation ends.
int drag_view_index = view_model_.GetIndexOfView(drag_view_);
view_model_.Remove(drag_view_index);
bounds_animator_.AnimateViewTo(drag_view_, drag_view_->bounds());
bounds_animator_.SetAnimationDelegate(
drag_view_, new ItemRemoveAnimationDelegate(drag_view_), true);
UpdatePaging();
}
void AppsGridView::ReparentItemForReorder(views::View* item_view,
const Index& target) {
item_list_->RemoveObserver(this);
model_->RemoveObserver(this);
AppListItem* reparent_item = static_cast<AppListItemView*>(item_view)->item();
DCHECK(reparent_item->IsInFolder());
AppListFolderItem* source_folder = static_cast<AppListFolderItem*>(
item_list_->FindItem(reparent_item->folder_id()));
// Move the item from its parent folder to top level item list.
// Must move to target_model_index, the location we expect the target item
// to be, not the item location we want to insert before.
int target_model_index = GetModelIndexFromIndex(target);
int current_model_index = view_model_.GetIndexOfView(item_view);
syncer::StringOrdinal target_position;
if (target_model_index < static_cast<int>(item_list_->item_count()))
target_position = item_list_->item_at(target_model_index)->position();
model_->MoveItemToFolderAt(reparent_item, "", target_position);
view_model_.Move(current_model_index, target_model_index);
if (source_folder->ChildItemCount() == 1)
RemoveLastItemFromReparentItemFolder(source_folder);
item_list_->AddObserver(this);
model_->AddObserver(this);
UpdatePaging();
}
void AppsGridView::ReparentItemToAnotherFolder(views::View* item_view,
const Index& target) {
DCHECK(IsDraggingForReparentInRootLevelGridView());
// Make change to data model.
item_list_->RemoveObserver(this);
AppListItem* reparent_item = static_cast<AppListItemView*>(item_view)->item();
DCHECK(reparent_item->IsInFolder());
AppListFolderItem* source_folder = static_cast<AppListFolderItem*>(
item_list_->FindItem(reparent_item->folder_id()));
AppListItemView* target_view =
static_cast<AppListItemView*>(GetViewAtSlotOnCurrentPage(target.slot));
AppListItem* target_item = target_view->item();
// Move item to the target folder.
const std::string& target_id_after_merge =
model_->MergeItems(target_item->id(), reparent_item->id());
if (target_id_after_merge != target_item->id()) {
// New folder was created, change the view model to replace the old target
// view with the new folder item view.
const std::string& new_folder_id = reparent_item->folder_id();
size_t new_folder_index;
if (item_list_->FindItemIndex(new_folder_id, &new_folder_index)) {
int target_view_index = view_model_.GetIndexOfView(target_view);
view_model_.Remove(target_view_index);
delete target_view;
views::View* new_folder_view =
CreateViewForItemAtIndex(new_folder_index);
view_model_.Add(new_folder_view, target_view_index);
AddChildView(new_folder_view);
} else {
LOG(ERROR) << "Folder no longer in item_list: " << new_folder_id;
}
}
if (source_folder->ChildItemCount() == 1)
RemoveLastItemFromReparentItemFolder(source_folder);
item_list_->AddObserver(this);
// Fade out the drag_view_ and delete it when animation ends.
int drag_view_index = view_model_.GetIndexOfView(drag_view_);
view_model_.Remove(drag_view_index);
bounds_animator_.AnimateViewTo(drag_view_, drag_view_->bounds());
bounds_animator_.SetAnimationDelegate(
drag_view_, new ItemRemoveAnimationDelegate(drag_view_), true);
UpdatePaging();
}
// After moving the re-parenting item out of the folder, if there is only 1 item
// left, remove the last item out of the folder, delete the folder and insert it
// to the data model at the same position. Make the same change to view_model_
// accordingly.
void AppsGridView::RemoveLastItemFromReparentItemFolder(
AppListFolderItem* source_folder) {
DCHECK(source_folder->ChildItemCount() == 1);
// Delete view associated with the folder item to be removed.
AppListItemView* folder_item_view = activated_item_view();
int folder_model_index = view_model_.GetIndexOfView(folder_item_view);
view_model_.Remove(folder_model_index);
delete folder_item_view;
// Now make the data change to remove the folder item in model.
AppListItem* last_item = source_folder->item_list()->item_at(0);
model_->MoveItemToFolderAt(last_item, "", source_folder->position());
// Create a new item view for the last item in folder.
size_t last_item_index;
item_list_->FindItemIndex(last_item->id(), &last_item_index);
views::View* last_item_view = CreateViewForItemAtIndex(last_item_index);
view_model_.Add(last_item_view, last_item_index);
AddChildView(last_item_view);
}
void AppsGridView::CancelFolderItemReparent(AppListItemView* drag_item_view) {
// The icon of the dragged item must target to its final ideal bounds after
// the animation completes.
CalculateIdealBounds();
// Remove drag_view_ from view_model_, it will be deleted after the animation.
int drag_view_index = view_model_.GetIndexOfView(drag_item_view);
view_model_.Remove(drag_view_index);
gfx::Rect target_icon_rect =
GetTargetIconRectInFolder(drag_item_view, activated_item_view_);
gfx::Rect drag_view_icon_to_grid =
drag_item_view->ConvertRectToParent(drag_item_view->GetIconBounds());
drag_view_icon_to_grid.ClampToCenteredSize(
gfx::Size(kPreferredIconDimension, kPreferredIconDimension));
TopIconAnimationView* icon_view = new TopIconAnimationView(
drag_item_view->item()->icon(),
target_icon_rect,
false); /* animate like closing folder */
AddChildView(icon_view);
icon_view->SetBoundsRect(drag_view_icon_to_grid);
icon_view->TransformView();
}
void AppsGridView::CancelContextMenusOnCurrentPage() {
int start = pagination_model_->selected_page() * tiles_per_page();
int end = std::min(view_model_.view_size(), start + tiles_per_page());
for (int i = start; i < end; ++i) {
AppListItemView* view =
static_cast<AppListItemView*>(view_model_.view_at(i));
view->CancelContextMenu();
}
}
bool AppsGridView::IsPointWithinDragBuffer(const gfx::Point& point) const {
gfx::Rect rect(GetLocalBounds());
rect.Inset(-kDragBufferPx, -kDragBufferPx, -kDragBufferPx, -kDragBufferPx);
return rect.Contains(point);
}
void AppsGridView::ButtonPressed(views::Button* sender,
const ui::Event& event) {
if (dragging())
return;
if (strcmp(sender->GetClassName(), AppListItemView::kViewClassName))
return;
if (delegate_) {
activated_item_view_ = static_cast<AppListItemView*>(sender);
delegate_->ActivateApp(static_cast<AppListItemView*>(sender)->item(),
event.flags());
}
}
void AppsGridView::LayoutStartPage() {
if (!start_page_view_)
return;
gfx::Rect start_page_bounds(GetLocalBounds());
start_page_bounds.set_height(start_page_bounds.height() -
page_switcher_view_->height());
const int page_width = width() + kPagePadding;
const int current_page = pagination_model_->selected_page();
if (current_page > 0)
start_page_bounds.Offset(-page_width, 0);
const PaginationModel::Transition& transition =
pagination_model_->transition();
if (current_page == 0 || transition.target_page == 0) {
const int dir = transition.target_page > current_page ? -1 : 1;
start_page_bounds.Offset(transition.progress * page_width * dir, 0);
}
start_page_view_->SetBoundsRect(start_page_bounds);
}
void AppsGridView::OnListItemAdded(size_t index, AppListItem* item) {
EndDrag(true);
views::View* view = CreateViewForItemAtIndex(index);
view_model_.Add(view, index);
AddChildView(view);
UpdatePaging();
UpdatePulsingBlockViews();
Layout();
SchedulePaint();
}
void AppsGridView::OnListItemRemoved(size_t index, AppListItem* item) {
EndDrag(true);
views::View* view = view_model_.view_at(index);
view_model_.Remove(index);
delete view;
UpdatePaging();
UpdatePulsingBlockViews();
Layout();
SchedulePaint();
}
void AppsGridView::OnListItemMoved(size_t from_index,
size_t to_index,
AppListItem* item) {
EndDrag(true);
view_model_.Move(from_index, to_index);
UpdatePaging();
AnimateToIdealBounds();
}
void AppsGridView::TotalPagesChanged() {
}
void AppsGridView::SelectedPageChanged(int old_selected, int new_selected) {
if (dragging()) {
CalculateDropTarget(last_drag_point_, true);
Layout();
MaybeStartPageFlipTimer(last_drag_point_);
} else {
ClearSelectedView(selected_view_);
Layout();
}
}
void AppsGridView::TransitionStarted() {
CancelContextMenusOnCurrentPage();
}
void AppsGridView::TransitionChanged() {
// Update layout for valid page transition only since over-scroll no longer
// animates app icons.
const PaginationModel::Transition& transition =
pagination_model_->transition();
if (pagination_model_->is_valid_page(transition.target_page))
Layout();
}
void AppsGridView::OnAppListModelStatusChanged() {
UpdatePulsingBlockViews();
Layout();
SchedulePaint();
}
void AppsGridView::SetViewHidden(views::View* view, bool hide, bool immediate) {
#if defined(USE_AURA)
ui::ScopedLayerAnimationSettings animator(view->layer()->GetAnimator());
animator.SetPreemptionStrategy(
immediate ? ui::LayerAnimator::IMMEDIATELY_SET_NEW_TARGET :
ui::LayerAnimator::BLEND_WITH_CURRENT_ANIMATION);
view->layer()->SetOpacity(hide ? 0 : 1);
#endif
}
void AppsGridView::OnImplicitAnimationsCompleted() {
if (layer()->opacity() == 0.0f)
SetVisible(false);
}
bool AppsGridView::EnableFolderDragDropUI() {
// Enable drag and drop folder UI only if it is at the app list root level
// and the switch is on and the target folder can still accept new items.
return switches::IsFolderUIEnabled() && is_root_level_ &&
CanDropIntoTarget(drop_target_);
}
bool AppsGridView::CanDropIntoTarget(const Index& drop_target) {
views::View* target_view = GetViewAtSlotOnCurrentPage(drop_target.slot);
if (!target_view)
return true;
AppListItem* target_item =
static_cast<AppListItemView*>(target_view)->item();
// Items can be dropped into non-folders (which have no children) or folders
// that have fewer than the max allowed items.
return target_item->ChildItemCount() < kMaxFolderItems;
}
// TODO(jennyz): Optimize the calculation for finding nearest tile.
AppsGridView::Index AppsGridView::GetNearestTileForDragView() {
Index nearest_tile;
nearest_tile.page = -1;
nearest_tile.slot = -1;
int d_min = -1;
// Calculate the top left tile |drag_view| intersects.
gfx::Point pt = drag_view_->bounds().origin();
CalculateNearestTileForVertex(pt, &nearest_tile, &d_min);
// Calculate the top right tile |drag_view| intersects.
pt = drag_view_->bounds().top_right();
CalculateNearestTileForVertex(pt, &nearest_tile, &d_min);
// Calculate the bottom left tile |drag_view| intersects.
pt = drag_view_->bounds().bottom_left();
CalculateNearestTileForVertex(pt, &nearest_tile, &d_min);
// Calculate the bottom right tile |drag_view| intersects.
pt = drag_view_->bounds().bottom_right();
CalculateNearestTileForVertex(pt, &nearest_tile, &d_min);
const int d_folder_dropping =
kFolderDroppingCircleRadius + kPreferredIconDimension / 2;
const int d_reorder =
kReorderDroppingCircleRadius + kPreferredIconDimension / 2;
// If user drags an item across pages to the last page, and targets it
// to the last empty slot on it, push the last item for re-ordering.
if (IsFirstEmptySlot(nearest_tile) && d_min < d_reorder) {
drop_attempt_ = DROP_FOR_REORDER;
nearest_tile.slot = nearest_tile.slot - 1;
return nearest_tile;
}
if (IsValidIndex(nearest_tile)) {
if (d_min < d_folder_dropping) {
views::View* target_view = GetViewAtSlotOnCurrentPage(nearest_tile.slot);
if (target_view &&
!IsFolderItem(static_cast<AppListItemView*>(drag_view_)->item())) {
// If a non-folder item is dragged to the target slot with an item
// sitting on it, attempt to drop the dragged item into the folder
// containing the item on nearest_tile.
drop_attempt_ = DROP_FOR_FOLDER;
return nearest_tile;
} else {
// If the target slot is blank, or the dragged item is a folder, attempt
// to re-order.
drop_attempt_ = DROP_FOR_REORDER;
return nearest_tile;
}
} else if (d_min < d_reorder) {
// Entering the re-order circle of the slot.
drop_attempt_ = DROP_FOR_REORDER;
return nearest_tile;
}
}
// If |drag_view| is not entering the re-order or fold dropping region of
// any items, cancel any previous re-order or folder dropping timer, and
// return itself.
drop_attempt_ = DROP_FOR_NONE;
reorder_timer_.Stop();
folder_dropping_timer_.Stop();
// When dragging for reparent a folder item, it should go back to its parent
// folder item if there is no drop target.
if (IsDraggingForReparentInRootLevelGridView()) {
DCHECK(activated_item_view_);
return GetIndexOfView(activated_item_view_);
}
return GetIndexOfView(drag_view_);
}
void AppsGridView::CalculateNearestTileForVertex(const gfx::Point& vertex,
Index* nearest_tile,
int* d_min) {
Index target_index;
gfx::Rect target_bounds = GetTileBoundsForPoint(vertex, &target_index);
if (target_bounds.IsEmpty() || target_index == *nearest_tile)
return;
int d_center = GetDistanceBetweenRects(drag_view_->bounds(), target_bounds);
if (*d_min < 0 || d_center < *d_min) {
*d_min = d_center;
*nearest_tile = target_index;
}
}
gfx::Rect AppsGridView::GetTileBoundsForPoint(const gfx::Point& point,
Index *tile_index) {
// Check if |point| is outside of contents bounds.
gfx::Rect bounds(GetContentsBounds());
if (!bounds.Contains(point))
return gfx::Rect();
// Calculate which tile |point| is enclosed in.
int x = point.x();
int y = point.y();
int col = (x - bounds.x()) / kPreferredTileWidth;
int row = (y - bounds.y()) / kPreferredTileHeight;
gfx::Rect tile_rect = GetTileBounds(row, col);
// Check if |point| is outside a valid item's tile.
Index index(pagination_model_->selected_page(), row * cols_ + col);
*tile_index = index;
return tile_rect;
}
gfx::Rect AppsGridView::GetTileBounds(int row, int col) const {
gfx::Rect bounds(GetContentsBounds());
gfx::Size tile_size(kPreferredTileWidth, kPreferredTileHeight);
gfx::Rect grid_rect(gfx::Size(tile_size.width() * cols_,
tile_size.height() * rows_per_page_));
grid_rect.Intersect(bounds);
gfx::Rect tile_rect(
gfx::Point(grid_rect.x() + col * tile_size.width(),
grid_rect.y() + row * tile_size.height()),
tile_size);
return tile_rect;
}
bool AppsGridView::IsFirstEmptySlot(const Index& index) const {
int last_possible_slot = (view_model_.view_size() - 1) % tiles_per_page();
return (index.page == pagination_model_->total_pages() - 1 &&
index.slot == last_possible_slot +1);
}
views::View* AppsGridView::GetViewAtSlotOnCurrentPage(int slot) {
if (slot < 0)
return NULL;
// Calculate the original bound of the tile at |index|.
int row = slot / cols_;
int col = slot % cols_;
gfx::Rect tile_rect = GetTileBounds(row, col);
for (int i = 0; i < view_model_.view_size(); ++i) {
views::View* view = view_model_.view_at(i);
if (view->bounds() == tile_rect)
return view;
}
return NULL;
}
void AppsGridView::SetAsFolderDroppingTarget(const Index& target_index,
bool is_target_folder) {
AppListItemView* target_view =
static_cast<AppListItemView*>(
GetViewAtSlotOnCurrentPage(target_index.slot));
if (target_view)
target_view->SetAsAttemptedFolderTarget(is_target_folder);
}
} // namespace app_list
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 932
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304-265-3482 baptist7@frontier.com Pastor Orville G. Wright.
Mount Carmel Baptist Church - PO Box 164 Beverly, WV 26253 Scott Lake Road, south of Beverly, 2 miles up from 219/250 304-637-9111 Pastor Rev.Leon W. Brown.
304-457-3247 Pastor Dr. Richard Kiley.
304-457-3206 philippibaptist@frontier.com Pastor Jonathan Villers.
304-842-3589 www.simpsoncreekbaptist.org Pastor Dr. C. Michael Hopkins.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,071
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{"url":"https:\/\/en.wikipedia.org\/wiki\/Gronwall%27s_inequality","text":"Gr\u00f6nwall's inequality\n\n(Redirected from Gronwall's inequality)\n\nIn mathematics, Gr\u00f6nwall's inequality (also called Gr\u00f6nwall's lemma or the Gr\u00f6nwall\u2013Bellman inequality) allows one to bound a function that is known to satisfy a certain differential or integral inequality by the solution of the corresponding differential or integral equation. There are two forms of the lemma, a differential form and an integral form. For the latter there are several variants.\n\nGr\u00f6nwall's inequality is an important tool to obtain various estimates in the theory of ordinary and stochastic differential equations. In particular, it provides a comparison theorem that can be used to prove uniqueness of a solution to the initial value problem; see the Picard\u2013Lindel\u00f6f theorem.\n\nIt is named for Thomas Hakon Gr\u00f6nwall (1877\u20131932). Gr\u00f6nwall is the Swedish spelling of his name, but he spelled his name as Gronwall in his scientific publications after emigrating to the United States.\n\nThe differential form was proven by Gr\u00f6nwall in 1919.[1] The integral form was proven by Richard Bellman in 1943.[2]\n\nA nonlinear generalization of the Gr\u00f6nwall\u2013Bellman inequality is known as Bihari\u2013LaSalle inequality. Other variants and generalizations can be found in Pachpatte, B.G. (1998).[3]\n\nDifferential form\n\nLet I denote an interval of the real line of the form [a,\u2009\u221e) or [a, b] or [a, b) with a < b. Let \u03b2 and u be real-valued continuous functions defined on I. If\u00a0u is differentiable in the interior Io of I (the interval I without the end points a and possibly b) and satisfies the differential inequality\n\n${\\displaystyle u'(t)\\leq \\beta (t)\\,u(t),\\qquad t\\in I^{\\circ },}$\n\nthen u is bounded by the solution of the corresponding differential equation y\u2009\u2032(t) = \u03b2(t)\u2009y(t):\n\n${\\displaystyle u(t)\\leq u(a)\\exp {\\biggl (}\\int _{a}^{t}\\beta (s)\\,\\mathrm {d} s{\\biggr )}}$\n\nfor all tI.\n\nRemark: There are no assumptions on the signs of the functions \u03b2 and\u00a0u.\n\nProof\n\nDefine the function\n\n${\\displaystyle v(t)=\\exp {\\biggl (}\\int _{a}^{t}\\beta (s)\\,\\mathrm {d} s{\\biggr )},\\qquad t\\in I.}$\n\nNote that v satisfies\n\n${\\displaystyle v'(t)=\\beta (t)\\,v(t),\\qquad t\\in I^{\\circ },}$\n\nwith v(a) = 1 and v(t) > 0 for all tI. By the quotient rule\n\n${\\displaystyle {\\frac {d}{dt}}{\\frac {u(t)}{v(t)}}={\\frac {u'(t)\\,v(t)-v'(t)\\,u(t)}{v^{2}(t)}}={\\frac {u'(t)\\,v(t)-\\beta (t)\\,v(t)\\,u(t)}{v^{2}(t)}}\\leq 0,\\qquad t\\in I^{\\circ },}$\n\nThus the derivative of the function ${\\displaystyle u(t)\/v(t)}$ is non-positive and the function is bounded above by its value at the initial point ${\\displaystyle a}$ of the interval ${\\displaystyle I}$:\n\n${\\displaystyle {\\frac {u(t)}{v(t)}}\\leq {\\frac {u(a)}{v(a)}}=u(a),\\qquad t\\in I,}$\n\nwhich is Gr\u00f6nwall's inequality.\n\nIntegral form for continuous functions\n\nLet I denote an interval of the real line of the form [a, \u221e) or [a, b] or [a, b) with a < b. Let \u03b1, \u03b2 and u be real-valued functions defined on\u00a0I. Assume that \u03b2 and u are continuous and that the negative part of \u03b1 is integrable on every closed and bounded subinterval of\u00a0I.\n\n\u2022 (a) If\u00a0\u03b2 is non-negative and if u satisfies the integral inequality\n${\\displaystyle u(t)\\leq \\alpha (t)+\\int _{a}^{t}\\beta (s)u(s)\\,\\mathrm {d} s,\\qquad \\forall t\\in I,}$\nthen\n${\\displaystyle u(t)\\leq \\alpha (t)+\\int _{a}^{t}\\alpha (s)\\beta (s)\\exp {\\biggl (}\\int _{s}^{t}\\beta (r)\\,\\mathrm {d} r{\\biggr )}\\mathrm {d} s,\\qquad t\\in I.}$\n\u2022 (b) If, in addition, the function \u03b1 is non-decreasing, then\n${\\displaystyle u(t)\\leq \\alpha (t)\\exp {\\biggl (}\\int _{a}^{t}\\beta (s)\\,\\mathrm {d} s{\\biggr )},\\qquad t\\in I.}$\n\nRemarks:\n\n\u2022 There are no assumptions on the signs of the functions \u03b1 and\u00a0u.\n\u2022 Compared to the differential form, differentiability of u is not needed for the integral form.\n\u2022 For a version of Gr\u00f6nwall's inequality which doesn't need continuity of \u03b2 and u, see the version in the next section.\n\nProof\n\n(a) Define\n\n${\\displaystyle v(s)=\\exp {\\biggl (}{-}\\int _{a}^{s}\\beta (r)\\,\\mathrm {d} r{\\biggr )}\\int _{a}^{s}\\beta (r)u(r)\\,\\mathrm {d} r,\\qquad s\\in I.}$\n\nUsing the product rule, the chain rule, the derivative of the exponential function and the fundamental theorem of calculus, we obtain for the derivative\n\n${\\displaystyle v'(s)={\\biggl (}\\underbrace {u(s)-\\int _{a}^{s}\\beta (r)u(r)\\,\\mathrm {d} r} _{\\leq \\,\\alpha (s)}{\\biggr )}\\beta (s)\\exp {\\biggl (}{-}\\int _{a}^{s}\\beta (r)\\mathrm {d} r{\\biggr )},\\qquad s\\in I,}$\n\nwhere we used the assumed integral inequality for the upper estimate. Since \u03b2 and the exponential are non-negative, this gives an upper estimate for the derivative of\u00a0v. Since v(a) = 0, integration of this inequality from a to t gives\n\n${\\displaystyle v(t)\\leq \\int _{a}^{t}\\alpha (s)\\beta (s)\\exp {\\biggl (}{-}\\int _{a}^{s}\\beta (r)\\,\\mathrm {d} r{\\biggr )}\\mathrm {d} s.}$\n\nUsing the definition of v(t) for the first step, and then this inequality and the functional equation of the exponential function, we obtain\n\n{\\displaystyle {\\begin{aligned}\\int _{a}^{t}\\beta (s)u(s)\\,\\mathrm {d} s&=\\exp {\\biggl (}\\int _{a}^{t}\\beta (r)\\,\\mathrm {d} r{\\biggr )}v(t)\\\\&\\leq \\int _{a}^{t}\\alpha (s)\\beta (s)\\exp {\\biggl (}\\underbrace {\\int _{a}^{t}\\beta (r)\\,\\mathrm {d} r-\\int _{a}^{s}\\beta (r)\\,\\mathrm {d} r} _{=\\,\\int _{s}^{t}\\beta (r)\\,\\mathrm {d} r}{\\biggr )}\\mathrm {d} s.\\end{aligned}}}\n\nSubstituting this result into the assumed integral inequality gives Gr\u00f6nwall's inequality.\n\n(b) If the function \u03b1 is non-decreasing, then part (a), the fact \u03b1(s) \u2264 \u03b1(t), and the fundamental theorem of calculus imply that\n\n{\\displaystyle {\\begin{aligned}u(t)&\\leq \\alpha (t)+{\\biggl (}{-}\\alpha (t)\\exp {\\biggl (}\\int _{s}^{t}\\beta (r)\\,\\mathrm {d} r{\\biggr )}{\\biggr )}{\\biggr |}_{s=a}^{s=t}\\\\&=\\alpha (t)\\exp {\\biggl (}\\int _{a}^{t}\\beta (r)\\,\\mathrm {d} r{\\biggr )},\\qquad t\\in I.\\end{aligned}}}\n\nIntegral form with locally finite measures\n\nLet I denote an interval of the real line of the form [a, \u221e) or [a, b] or [a, b) with a < b. Let \u03b1 and u be measurable functions defined on\u00a0I and let \u03bc be a non-negative measure on the Borel \u03c3-algebra of I satisfying \u03bc([a, t]) < \u221e for all tI (this is certainly satisfied when \u03bc is a locally finite measure). Assume that u is integrable with respect to \u03bc in the sense that\n\n${\\displaystyle \\int _{[a,t)}|u(s)|\\,\\mu (\\mathrm {d} s)<\\infty ,\\qquad t\\in I,}$\n\nand that u satisfies the integral inequality\n\n${\\displaystyle u(t)\\leq \\alpha (t)+\\int _{[a,t)}u(s)\\,\\mu (\\mathrm {d} s),\\qquad t\\in I.}$\n\n\u2022 the function \u03b1 is non-negative or\n\u2022 the function t\u03bc([a, t]) is continuous for tI and the function \u03b1 is integrable with respect to \u03bc in the sense that\n${\\displaystyle \\int _{[a,t)}|\\alpha (s)|\\,\\mu (\\mathrm {d} s)<\\infty ,\\qquad t\\in I,}$\n\nthen u satisfies Gr\u00f6nwall's inequality\n\n${\\displaystyle u(t)\\leq \\alpha (t)+\\int _{[a,t)}\\alpha (s)\\exp {\\bigl (}\\mu (I_{s,t}){\\bigr )}\\,\\mu (\\mathrm {d} s)}$\n\nfor all tI, where Is,t denotes to open interval (s, t).\n\nRemarks\n\n\u2022 There are no continuity assumptions on the functions \u03b1 and u.\n\u2022 The integral in Gr\u00f6nwall's inequality is allowed to give the value infinity.\n\u2022 If \u03b1 is the zero function and u is non-negative, then Gr\u00f6nwall's inequality implies that u is the zero function.\n\u2022 The integrability of u with respect to \u03bc is essential for the result. For a counterexample, let \u03bc denote Lebesgue measure on the unit interval [0,\u20091], define u(0) = 0 and u(t) = 1\/t for t (0, 1], and let \u03b1 be the zero function.\n\u2022 The version given in the textbook by S.\u00a0Ethier and T.\u00a0Kurtz.[4] makes the stronger assumptions that \u03b1 is a non-negative constant and u is bounded on bounded intervals, but doesn't assume that the measure \u03bc is locally finite. Compared to the one given below, their proof does not discuss the behaviour of the remainder Rn(t).\n\nSpecial cases\n\n\u2022 If the measure \u03bc has a density \u03b2 with respect to Lebesgue measure, then Gr\u00f6nwall's inequality can be rewritten as\n${\\displaystyle u(t)\\leq \\alpha (t)+\\int _{a}^{t}\\alpha (s)\\beta (s)\\exp {\\biggl (}\\int _{s}^{t}\\beta (r)\\,\\mathrm {d} r{\\biggr )}\\,\\mathrm {d} s,\\qquad t\\in I.}$\n\u2022 If the function \u03b1 is non-negative and the density \u03b2 of \u03bc is bounded by a constant c, then\n${\\displaystyle u(t)\\leq \\alpha (t)+c\\int _{a}^{t}\\alpha (s)\\exp {\\bigl (}c(t-s){\\bigr )}\\,\\mathrm {d} s,\\qquad t\\in I.}$\n\u2022 If, in addition, the non-negative function \u03b1 is non-decreasing, then\n${\\displaystyle u(t)\\leq \\alpha (t)+c\\alpha (t)\\int _{a}^{t}\\exp {\\bigl (}c(t-s){\\bigr )}\\,\\mathrm {d} s=\\alpha (t)\\exp(c(t-a)),\\qquad t\\in I.}$\n\nOutline of proof\n\nThe proof is divided into three steps. In idea is to substitute the assumed integral inequality into itself n times. This is done in Claim\u00a01 using mathematical induction. In Claim\u00a02 we rewrite the measure of a simplex in a convenient form, using the permutation invariance of product measures. In the third step we pass to the limit n to infinity to derive the desired variant of Gr\u00f6nwall's inequality.\n\nDetailed proof\n\nClaim 1: Iterating the inequality\n\nFor every natural number n including zero,\n\n${\\displaystyle u(t)\\leq \\alpha (t)+\\int _{[a,t)}\\alpha (s)\\sum _{k=0}^{n-1}\\mu ^{\\otimes k}(A_{k}(s,t))\\,\\mu (\\mathrm {d} s)+R_{n}(t)}$\n\nwith remainder\n\n${\\displaystyle R_{n}(t):=\\int _{[a,t)}u(s)\\mu ^{\\otimes n}(A_{n}(s,t))\\,\\mu (\\mathrm {d} s),\\qquad t\\in I,}$\n\nwhere\n\n${\\displaystyle A_{n}(s,t)=\\{(s_{1},\\ldots ,s_{n})\\in I_{s,t}^{n}\\mid s_{1}\n\nis an n-dimensional simplex and\n\n${\\displaystyle \\mu ^{\\otimes 0}(A_{0}(s,t)):=1.}$\n\nProof of Claim 1\n\nWe use mathematical induction. For n = 0 this is just the assumed integral inequality, because the empty sum is defined as zero.\n\nInduction step from n to n + 1: Inserting the assumed integral inequality for the function u into the remainder gives\n\n${\\displaystyle R_{n}(t)\\leq \\int _{[a,t)}\\alpha (s)\\mu ^{\\otimes n}(A_{n}(s,t))\\,\\mu (\\mathrm {d} s)+{\\tilde {R}}_{n}(t)}$\n\nwith\n\n${\\displaystyle {\\tilde {R}}_{n}(t):=\\int _{[a,t)}{\\biggl (}\\int _{[a,q)}u(s)\\,\\mu (\\mathrm {d} s){\\biggr )}\\mu ^{\\otimes n}(A_{n}(q,t))\\,\\mu (\\mathrm {d} q),\\qquad t\\in I.}$\n\nUsing the Fubini\u2013Tonelli theorem to interchange the two integrals, we obtain\n\n${\\displaystyle {\\tilde {R}}_{n}(t)=\\int _{[a,t)}u(s)\\underbrace {\\int _{(s,t)}\\mu ^{\\otimes n}(A_{n}(q,t))\\,\\mu (\\mathrm {d} q)} _{=\\,\\mu ^{\\otimes n+1}(A_{n+1}(s,t))}\\,\\mu (\\mathrm {d} s)=R_{n+1}(t),\\qquad t\\in I.}$\n\nHence Claim 1 is proved for n + 1.\n\nClaim 2: Measure of the simplex\n\nFor every natural number n including zero and all s < t in I\n\n${\\displaystyle \\mu ^{\\otimes n}(A_{n}(s,t))\\leq {\\frac {{\\bigl (}\\mu (I_{s,t}){\\bigr )}^{n}}{n!}}}$\n\nwith equality in case t\u03bc([a, t]) is continuous for tI.\n\nProof of Claim 2\n\nFor n = 0, the claim is true by our definitions. Therefore, consider n \u2265 1 in the following.\n\nLet Sn denote the set of all permutations of the indices in {1, 2, . . . , n}. For every permutation \u03c3Sn define\n\n${\\displaystyle A_{n,\\sigma }(s,t)=\\{(s_{1},\\ldots ,s_{n})\\in I_{s,t}^{n}\\mid s_{\\sigma (1)}\n\nThese sets are disjoint for different permutations and\n\n${\\displaystyle \\bigcup _{\\sigma \\in S_{n}}A_{n,\\sigma }(s,t)\\subset I_{s,t}^{n}.}$\n\nTherefore,\n\n${\\displaystyle \\sum _{\\sigma \\in S_{n}}\\mu ^{\\otimes n}(A_{n,\\sigma }(s,t))\\leq \\mu ^{\\otimes n}{\\bigl (}I_{s,t}^{n}{\\bigr )}={\\bigl (}\\mu (I_{s,t}){\\bigr )}^{n}.}$\n\nSince they all have the same measure with respect to the n-fold product of \u03bc, and since there are n! permutations in\u00a0Sn, the claimed inequality follows.\n\nAssume now that t\u03bc([a, t]) is continuous for tI. Then, for different indices i, j \u2208 {1, 2, . . . , n}, the set\n\n${\\displaystyle \\{(s_{1},\\ldots ,s_{n})\\in I_{s,t}^{n}\\mid s_{i}=s_{j}\\}}$\n\nis contained in a hyperplane, hence by an application of Fubini's theorem its measure with respect to the n-fold product of \u03bc is zero. Since\n\n${\\displaystyle I_{s,t}^{n}\\subset \\bigcup _{\\sigma \\in S_{n}}A_{n,\\sigma }(s,t)\\cup \\bigcup _{1\\leq i\n\nthe claimed equality follows.\n\nProof of Gr\u00f6nwall's inequality\n\nFor every natural number n, Claim\u00a02 implies for the remainder of Claim\u00a01 that\n\n${\\displaystyle |R_{n}(t)|\\leq {\\frac {{\\bigl (}\\mu (I_{a,t}){\\bigr )}^{n}}{n!}}\\int _{[a,t)}|u(s)|\\,\\mu (\\mathrm {d} s),\\qquad t\\in I.}$\n\nBy assumption we have \u03bc(Ia,t) < \u221e. Hence, the integrability assumption on u implies that\n\n${\\displaystyle \\lim _{n\\to \\infty }R_{n}(t)=0,\\qquad t\\in I.}$\n\nClaim\u00a02 and the series representation of the exponential function imply the estimate\n\n${\\displaystyle \\sum _{k=0}^{n-1}\\mu ^{\\otimes k}(A_{k}(s,t))\\leq \\sum _{k=0}^{n-1}{\\frac {{\\bigl (}\\mu (I_{s,t}){\\bigr )}^{k}}{k!}}\\leq \\exp {\\bigl (}\\mu (I_{s,t}){\\bigr )}}$\n\nfor all s < t in\u00a0I. If the function\u00a0\u03b1 is non-negative, then it suffices to insert these results into Claim\u00a01 to derive the above variant of Gr\u00f6nwall's inequality for the function\u00a0u.\n\nIn case t\u03bc([a, t]) is continuous for tI, Claim\u00a02 gives\n\n${\\displaystyle \\sum _{k=0}^{n-1}\\mu ^{\\otimes k}(A_{k}(s,t))=\\sum _{k=0}^{n-1}{\\frac {{\\bigl (}\\mu (I_{s,t}){\\bigr )}^{k}}{k!}}\\to \\exp {\\bigl (}\\mu (I_{s,t}){\\bigr )}\\qquad {\\text{as }}n\\to \\infty }$\n\nand the integrability of the function \u03b1 permits to use the dominated convergence theorem to derive Gr\u00f6nwall's inequality.\n\nReferences\n\n1. ^ Gronwall, Thomas H. (1919), \"Note on the derivatives with respect to a parameter of the solutions of a system of differential equations\", Ann. of Math., 20 (2): 292\u2013296, JFM\u00a047.0399.02, JSTOR\u00a01967124, MR\u00a01502565\n2. ^ Bellman, Richard (1943), \"The stability of solutions of linear differential equations\", Duke Math. J., 10 (4): 643\u2013647, doi:10.1215\/s0012-7094-43-01059-2, MR\u00a00009408, Zbl\u00a00061.18502\n3. ^ Pachpatte, B.G. (1998). Inequalities for differential and integral equations. San Diego: Academic Press. ISBN\u00a09780080534640.\n4. ^ Ethier, Steward N.; Kurtz, Thomas G. (1986), Markov Processes, Characterization and Convergence, New York: John Wiley & Sons, p.\u00a0498, ISBN\u00a00-471-08186-8, MR\u00a00838085, Zbl\u00a00592.60049","date":"2018-06-21 11:01:23","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 41, \"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\": 0.9820817112922668, \"perplexity\": 1688.026248732434}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-26\/segments\/1529267864139.22\/warc\/CC-MAIN-20180621094633-20180621114633-00249.warc.gz\"}"}
| null | null |
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La rivière Chaboillez est un affluent de la rivière La Reine, au Québec, au Canada, chevauchant les régions administratives de :
Nord-du-Québec : traversant le canton de Perron, dans la partie sud de la municipalité de Eeyou Istchee Baie-James ;
Abitibi-Témiscamingue : traversant le canton de Desméloizes, dans les municipalités de Normétal, Saint-Lambert et de La Reine, dans la municipalité régionale de comté (MRC) de Abitibi-Ouest.
La rivière Chaboillez coule surtout en zone forestière, sauf la zone inférieure qui traverse quelques zones agricoles. La foresterie constitue la principale activité économique de ce bassin versant ; l'agriculture, en second.
Le bassin versant de la rivière Chaboillez est surtout desservi par le chemin des et rang (sens est-ouest), la route du et rang Ouest, la route du et rang (sens est-ouest), ainsi que la route du et Rang (sens nord-sud).
Annuellement, la surface de la rivière est habituellement gelée de la mi-novembre à la mi-avril, toutefois la circulation sécuritaire sur la glace se fait généralement de la mi-décembre à la fin mars.
Géographie
La rivière Chaboillez prend sa source d'un ruisseau forestier à une altitude de dans la municipalité de Eeyou Istchee Baie-James.
Cette source est située à à l'est de la frontière de l'Ontario, à au nord-ouest du centre du village de Normétal, à au nord-est de l'embouchure de la rivière Chaboillez et à au nord-ouest du centre-ville de La Sarre.
Les principaux bassins versants voisins de la rivière Chaboillez sont :
côté nord : rivière Ménard, rivière Boivin, rivière Turgeon ;
côté est : rivière Des Méloizes, lac Macamic, rivière La Sarre ;
côté sud : rivière La Reine, lac Abitibi, rivière Des Méloizes ;
côté ouest : rivière La Reine, lac Abitibi, Boischere Creek.
À partir de sa source en zone forestière, la rivière Chaboillez coule sur environ selon les segments suivants :
vers le sud dans la municipalité de Eeyou Istchee Baie-James, jusqu'à la limite de la région administrative de l'Abitibi-Témiscamingue ;
vers le sud-est dans la municipalité de Saint-Lambert, jusqu'à la limite de Normétal ;
vers le sud-est dans Normétal, en recueillant un ruisseau (venant du nord-est), jusqu'à la municipalité de Saint-Lambert ;
vers le sud-est dans Saint-Lambert, jusqu'à la route des et rang (sens est-ouest) ;
vers le sud, jusqu'au cours d'eau Vachon (venant de l'est) ;
vers l'ouest, puis le sud, jusqu'à la route des et rang ouest ;
vers le sud, le sud-ouest, puis nord-ouest, jusqu'à la route des et rang (sens nord-sud) ;
vers le sud-ouest, puis l'ouest, en traversant à deux reprises la route du et rang (sens est-ouest), jusqu'à la limite de La Reine ;
vers le sud-ouest jusqu'à son embouchure.
L'embouchure de la rivière Chaboillez est localisé à :
à l'est de la frontière de l'Ontario ;
au sud du centre du village de Saint-Lambert ;
au nord-ouest du centre-ville de La Sarre ;
au nord de l'embouchure de la rivière La Reine ;
au nord-ouest du centre-ville de Rouyn-Noranda.
L'embouchure de la rivière Chaboillez est située dans un coude de rivière sur la rive nord-est de la rivière La Reine. De là, la rivière La Reine coule sur vers le sud, jusqu'à la rive nord du lac Abitibi. Puis, le courant traverse le lac Abitibi vers l'ouest sur jusqu'à son embouchure, en contournant cinq grandes presqu'îles s'avançant vers le nord et plusieurs îles.
À partir de l'embouchure du lac Abitibi, le courant emprunte le cours de la rivière Abitibi, puis de la rivière Moose pour aller se déverser sur la rive sud de la baie James.
Toponymie
Le mot Chaboillez constitue un patronyme de famille d'origine française.
Le toponyme « rivière Chaboillez » a été officialisé le à la Commission de toponymie du Québec, soit à la création de cette commission.
Notes et références
Annexes
Articles connexes
Rivière La Reine
Lac Abitibi
Rivière Abitibi
rivière Moose
Eeyou Istchee Baie-James
Normétal
Saint-Lambert
La Reine
Liste des cours d'eau du Québec
Liens externes
3Chaboillez
Abitibi-Ouest
Cours d'eau en Abitibi-Témiscamingue
Eeyou Istchee Baie-James
Cours d'eau au Nord-du-Québec
Projet:Croissant boréal
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"redpajama_set_name": "RedPajamaCommonCrawl"
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Irish or Anglo-Irish (2)
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Your search for Genre: "Prose" AND Literary Period: "Restoration" AND Metaphor Category: "Container" , "Architecture" returned 17 results(s) in 0.002 seconds
"For certain 'tis that Memory in Youth is infinitely more ready than in men of riper years, as appears from their different capacitys in learning of a Language; and then for Invention which always builds out of the Store-house of Memory, 'tis then most perfect and various when the Spirits are mos...
— Nourse, Timothy (c.1636–1699)
Date: 1690, 1694, 1695, 1700, 1706
"This is Memory, which is as it were the Store-house of our Ideas."
— Locke, John (1632-1704)
"For the narrow Mind of Man, not being capable of having many Ideas under View and Consideration at once, it was necessary to have a Repository, to lay up those Ideas, which at another time it might have use of."
"If therefore we will warily attend to the Motions of the Mind, and observe what Course it usually takes in its way to Knowledge, we shall, I think, find that the Mind having got any Idea, which it thinks it may have use of, either in Contemplation or Discourse; the first Thing it does, is to abs...
"The senses at first let in particular Ideas, and furnish the yet empty Cabinet: And the Mind by degrees growing familiar with some of them, they are lodged in the Memory, and Names got to them."
"Would the pictures coming into such a dark room but stay there, and lie so orderly as to be found upon occasion, it would very much resemble the understanding of a man, in reference to all objects of sight, and the ideas of them"
"Thirdly, Let us hence duly learn to prize and value our Souls; is the Body such a rare Piece, what this is the Soul? the Body is but the Husk or Shell, the Soul is the Kernel; the Body is but the Cask, the Soul the precious Liquor contained in it; the Body is but the Cabinet; the Soul the Jewel;...
— Ray [formerly Wray], John (1627–1705)
"St. Austin names Memory the Soul's Belly or Storehouse, or the Receptacle of the Mind, because it is appointed to receive and lay up as in a Treasury, those things that may be for our Benefit and Advantage."
— D'Assigny, Marius (1643-1717)
Date: Read 1680-1681, published 1705
"Memory then conceive to be nothing else but a Repository of Ideas formed partly by the Senses, but chiefly by the Soul it self: I say, partly by the Senses, because they are as it were the Collectors or Carriers of the Impressions made by Objects from without, delivering them to the Repository o...
— Hooke, Robert (1635-1703)
"They may perhaps be Monsters, and not Divinitys, or Sacred Truths, which are kept thus choicely, in some dark Corner of our Minds: The Specters may impose on us, whilst we refuse to turn 'em every way, and view their Shapes and Complexions in every light."
— Cooper, Anthony Ashley, third earl of Shaftesbury (1671-1713)
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\section{Introduction}
Extension of the electroweak symmetric sector of the standard model
(SM) to two or more Higgs doublets is a widespread curiosity, of which
two Higgs doublet models (2HDM) occupy the centre stage. Such models in
general suffer from the flavour changing neutral current (FCNC)
problem. A popular way of avoiding FCNC is to use some discrete
symmetry (or something that effectively leads to it), which restricts
the Yukawa interactions of the two doublets. Based on the nature of
such symmetry, four types of 2HDM are popular, namely, Type-I,
Type-II, Type-X (or lepton specific) and Type-Y (or flipped)
\cite{Gunion:1989we,Djouadi:2005gj,Branco:2011iw}. This
paper contains some observations related to Type-X 2HDM.
In this scenario, one scalar doublet in the flavour basis has Yukawa
couplings with quarks only, while the other one couples to leptons
alone (Yukawa coupling with neutrinos are neglected without affecting
other aspects of phenomenology). The physical states other than the
SM-like 125-GeV scalar, obtained on diagonalizing the mass matrices,
have very small coupling with quarks compared to those with leptons,
once all constraints including those from the Large Hadron Collider
(LHC) are taken into account. This considerably relaxes the lower
bounds on some of the physical masses. In particular, it has been
found \cite{Broggio:2014mna,Chun:2015hsa,Chun:2016hzs} that the
neutral pseudoscalar $A$ in Type-X 2HDM can be as
light at 40-60 GeV or even lighter in certain regions in the
parameter space, thanks to its generally low direct production rate at
the LHC and other colliders that have run so far.
\footnote{Such light pseudoscalars may also occur in further extensions of the
SM \cite{Bandyopadhyay:2015oga,Bandyopadhyay:2015tva, Goncalves:2016qhh}.}
And it is in part of these regions where the one-loop contribution induced by a light $A$
helps a good fit of the muon anomalous magnetic moment, especially for
high ($\ge 40$) values of $\tan\beta$, the ratio of the vacuum
expectation values of the two doublets~
\cite{Cheung:2001hz,Cheung:2003pw,Jegerlehner:2009ry,Lisanti:2009uy,Cao:2009as}.
It is therefore important not
only to look for LHC signals of this scenario \cite{Chun:2015hsa}, but also to {\em
actually reconstruct the mass of the light A}. We suggest a method of
doing precisely that.
The light pseudoscalar, for large $\tan\beta$, has a $\tau^+ \tau^-$
branching ratio close to unity, and a $\mu^+ \mu^-$ branching ratio on
the order of $0.35 \%$. Signals have been suggested in the multi-tau channels like $pp
\to HA \to \tau^+ \tau^- ~
\tau^+ \tau^-$ \cite{Su:2009fz,Kanemura:2011kx,Kanemura:2014bqa,Chun:2015hsa}.
However, the taus cannot be reconstructed in the
collinear approximation \cite{Rainwater:1998kj} since there are four neutrinos in the
final state. Besides, even if only one $A$ decays into a $\tau$-pair,
the visible $\tau$-decay product (like a $\tau$-induced jet) cannot be
treated in the collinear approximation at such low energies as that
possessed by the $\tau$ produced from an $A$ as light as 50 -- 60
GeV. Therefore, we cannot reliably obtain $m_A$ using the
$\tau$-pair(s). We find that the $\mu^+ \mu^-$ pair can come to one's
rescue here. With $pp \to hX \to AA
\to \tau^+ \tau^- ~ \mu^+ \mu^-$, one may reconstruct
$m_A$ from the muon pair, in association with a pair of tau-jets. We
show after a detailed simulation that such a strategy, combined
with that for suppressing SM backgrounds, isolates the signal events
carrying clear information on the pseudoscalar mass. It is thus
possible to achieve discovery-level statistical significance with an
integrated luminosity of about 100 $\rm{fb}^{-1}$ or less at 14 TeV.
In Section 2 we discuss the generic features of the
Type-X 2HDM with respect to the structure of the Yukawa and gauge couplings
of the physical scalars and point out how the parameter space of the model gets
constrained by the muon $g-2$ and precision observables. Section 3 is devoted
to the the LHC analysis of our signal that identifies the pseudoscalar resonance,
detailing the event selection criteria that helps in suppressing the backgrounds.
Section 4 includes a discussion of the results
in the context of the efficacy of the analysis scheme used for our signal.
We summarize and conclude in Section 5.
\section{The Type-X 2HDM Model and Constraints}
The Type-X 2HDM with $\Phi_{1,2}$ as the two doublets is characterised by the following
Yukawa structure:
\begin{equation}\label{eq:yukawa}
{\cal L}_Y= -Y^u\bar{ Q_L} \widetilde \Phi_2 u_R + Y^d \bar{ Q_L} \Phi_2 d_R+Y^e\bar{ l_L} \Phi_1 e_R + h.c.,
\end{equation}
where family indices are suppressed and $\widetilde \Phi_2=i\sigma_2\Phi_2^*$. This Yukawa
Lagrangian is the result of a $\mathbb{Z}_2$ symmetry~\cite{PhysRevD.15.1958}
which prevents tree level flavor changing neutral current.
Under $\mathbb{Z}_2$, the fields transform as $\Phi_2\rightarrow \Phi_2$ and $\Phi_1\rightarrow-\Phi_1$ combined with $e_R\rightarrow -e_R$ while
the other fermions are even under it.
Thus $\Phi_2$ couples only to the quarks whereas
$\Phi_1$ couples exclusively to the leptons.
The most general form of the scalar potential is
\begin{eqnarray}
\nonumber V_{\mathrm{2HDM}} &=& m_{11}^2\Phi_1^{\dagger}\Phi_1 + m_{22}^2\Phi_2^{\dagger}\Phi_2 -\Big[m_{12}^2\Phi_1^{\dagger}\Phi_2 + \mathrm{h.c.}\Big]
+\frac{1}{2}\lambda_1\left(\Phi_1^\dagger\Phi_1\right)^2+\frac{1}{2}\lambda_2\left(\Phi_2^\dagger\Phi_2\right)^2 \\
\nonumber && +\lambda_3\left(\Phi_1^\dagger\Phi_1\right)\left(\Phi_2^\dagger\Phi_2\right)+\lambda_4\left(\Phi_1^\dagger\Phi_2\right)\left(\Phi_2^\dagger\Phi_1\right)
+\Big\{ \frac{1}{2}\lambda_5\left(\Phi_1^\dagger\Phi_2\right)^2+\Big[\lambda_6\left(\Phi_1^\dagger\Phi_1\right) \\
&& +\lambda_7\left(\Phi_2^\dagger\Phi_2\right)\Big]\left(\Phi_1^\dagger\Phi_2\right) + \rm{h.c.}\Big\},
\label{eq:2hdmgen}
\end{eqnarray}
where all the couplings are assumed to be real.
The $\mathbb{Z}_2$ symmetry implies $\lambda_6=\lambda_7=0$. However, the term proportional to $m_{12}^2$,
which softly breaks $\mathbb{Z}_2$ can be non zero to keep the quartic coupling $\lambda_1$
below perturbativity limit \cite{ Gunion:1989we,Gunion:2002zf}.
Parameterizing the doublets as $\Phi_j=(\phi_j^+,(v_j+\phi^r_j+i\phi^i_j)/\sqrt{2})^T$, we obtain
the five physical massive states $A$, $h$, $H$, $H^{\pm}$ in terms of the two diagonalizing angles
$\alpha$ and $\beta$:
\begin{eqnarray}\label{}
A&=&-s_\beta \;\phi^i_1+c_\beta \;\phi^i_2,\quad H^+=-s_\beta\; \phi_1^+ +c_\beta\; \phi^+_2,\cr
h&=&-s_\alpha\; \phi^r_1+c_\alpha\; \phi^r_2,\quad~ H=c_\alpha\; \phi^r_1+s_\alpha\; \phi^r_2,
\end{eqnarray}
where $s_\alpha$ and $c_\beta$ stand for $\sin\alpha$ and $\cos\beta$, etc. The CP-even state
$h$ corresponds to the SM-like Higgs with mass $M_h=125$ GeV. Furthermore we look for the mass
hierarchy $M_A < M_h < M_H\simeq M_{H^\pm}$ which can be realised by setting $\lambda_4 + \lambda_5 \approx 0$.
The SM-like Higgs couples to the pseudoscalar with strength
$\lambda_{hAA} = -(\lambda_3 + \lambda_4 - \lambda_5)v$, where $v = \sqrt{{v_1}^2 + {v_2}^2} = 246$ GeV.
The Yukawa Lagrangian of Eq.(\ref{eq:yukawa}) can be rewritten in terms of the physical Higgs bosons,
$h, H, A$ and $H^\pm$:
\begin{eqnarray}
\nonumber \mathcal L_{\mathrm{Yukawa}}^{\mathrm{Physical}} &=&
-\sum_{f=u,d,\ell} \frac{m_f}{v}\left(\xi_h^f\overline{f}hf +
\xi_H^f\overline{f}Hf - i\xi_A^f\overline{f}\gamma_5Af \right) \\
&&-\left\{ \frac{\sqrt{2}V_{ud}}
{v}\overline{u}\left(m_{u}\xi_A^{u}P_L+m_{d}\xi_A^{d}P_R\right)H^{+}d +
\frac{\sqrt{2}m_l}{v}\xi_A^l\overline{v}_LH^{+}l_R + \mathrm{h.c.}\right\},
\label{eq:L2hdm}
\end{eqnarray}
where $f$ runs over all of the quarks and charged leptons, and $u$, $d$, and $l$ refer to the
up-type quarks, down-type quarks, and charged leptons, respectively.
The multiplicative factors of the
Yukawa couplings, {\it i.e.} $\xi_h^f$, $\xi_H^f$ and $\xi_A^f$ are given in Table \ref{Tab:YukawaFactors}. For $\sin(\beta-\alpha) \approx 1 $ the Yukawa coupling with the SM-like
Higgs $(h)$ are similar to that of the SM.
\begin{table}[t]
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|}
\hline
& $\xi_h^u$ & $\xi_h^d$ & $\xi_h^\ell$
& $\xi_H^u$ & $\xi_H^d$ & $\xi_H^\ell$
& $\xi_A^u$ & $\xi_A^d$ & $\xi_A^\ell$ \\ \hline
Type-X
& $c_\alpha/s_\beta$ & $c_\alpha/s_\beta$ & $-s_\alpha/c_\beta$
& $s_\alpha/s_\beta$ & $s_\alpha/s_\beta$ & $c_\alpha/c_\beta$
& $\cot\beta$ & $-\cot\beta$ & $\tan\beta$ \\
\hline
\end{tabular}
\end{center}
\caption{The multiplicative factors of Yukawa interactions in type X 2HDM}
\label{Tab:YukawaFactors}
\end{table}
In any type of the 2HDM, the couplings of scalars with a pair of gauge bosons are given by
\cite{Gunion:1989we,Djouadi:2005gj, Kanemura:2014dea}:
\begin{equation}
g_{hVV}=\mathrm{sin}(\beta-\alpha)g_{hVV}^{\mathrm{SM}},\,\,\,\,g_{HVV}=\mathrm{cos}(\beta-\alpha)g_{hVV}^{\mathrm{SM}},\,\,\,\,g_{AVV}=0,
\end{equation}
where $V$ = $Z,\,W^\pm$.
The couplings of $Z$ boson with the neutral scalars are,
\begin{align}
&hAZ_\mu:\,\frac{g_Z^{}}{2}\cos(\beta-\alpha)(p+p')_\mu,\quad
HAZ_\mu:\,-\frac{g_Z^{}}{2}\sin(\beta-\alpha)(p+p')_\mu,
\label{hhV}
\end{align}
where $p_\mu$ and $p'_\mu$ are
outgoing four-momenta of the first and the second scalars, respectively,
and $g_Z^{}=g_W^{}/\cos\theta_W^{}$.
For reasons already stated, we are concerned with the region corresponding to
$ \tan{\beta}\equiv v_2/v_1\gg 1$. This is because the contribution
to the muon {$g-2$} coming from the Barr-Zee \cite{PhysRevLett.65.2626} two-loop diagrams
can be substantial with a light pseudoscalar $A$ and
$\tau$ running in the loops. Constraints on 2HDM parameter space coming from $(g-2)_\mu$
have been analyzed in
Refs.~\cite{Dedes:2001nx,Cheung:2001hz,Krawczyk:2001pe,Krawczyk:2002df,Cheung:2003pw,Broggio:2014mna,Wang:2014sda,Chun:2015hsa,Ilisie:2015tra,Abe:2015oca,Han:2015yys,Cherchiglia:2016eui}
and it was shown
in the updated analysis ~\cite{Chun:2016hzs}
that light $A$ in Type-X 2HDM can explain $(g-2)_\mu$ at 2$\sigma$ while evading collider as well as
precision data constraints.
While it is true that in the Type-II 2HDM a light pseudoscalar can explain the $(g-2)_\mu$ anomaly,
there the lower bound on the charged Higgs mass is $M_{H^+}>580$ GeV
coming from the $B\to X_s\gamma$ measurement~\cite{Belle:2016ufb}. Such a heavy charged Higgs
is not compatible with the requirement of a light pseudoscalar~\cite{Broggio:2014mna}.
Similarly, in Type-I and Type-Y 2HDM, too, a very light pseudoscalar and its enhanced
coupling with the muons are not consistent, since that would also imply comparably
strong coupling to at least one type of quarks, leading to unacceptably large
$A$ production at hadron colliders.
Beside those models where the $A$ couples to muons proportionally to $\cot\beta$ cannot
explain $(g-2)_{\mu}$, since $\tan\beta \leq 1$ is disfavoured by a number of considerations.
It is only in the Type-X that a light $A$
can have enhanced coupling to the $\mu$ and the $\tau$, concomitantly suppressed
coupling to all quarks, and all phenomenological and other theoretical constraints
(vacuum stability, perturbativity etc.) duly satisfied \cite{Staub:2017ktc}. Keeping this in mind,
we proceed to find a strategy for reconstructing $M_A$ at the LHC.
\section{ Signal of a light $A$ : An analysis for the LHC}
The light pseudoscalar in Type-X 2HDM can be produced at the LHC via associated production along with
the SM Higgs and also via the decay of the SM like Higgs. The associated production is proportional to
$\cos^2(\beta-\alpha)$ and is suppressed for $(\beta-\alpha) \simeq \pi/2$,
leaving $h\to A\,A$ as the dominant production mode for the pseudoscalar.
The pseudoscalar is lepto-philic and almost exclusively
decays to $\tau$ lepton for large $\tan\beta$ with
a very small branching ratio to di-muon ($BR(A\to \mu\mu)\simeq (m_\mu/m_\tau)^2\simeq 0.35\%$).
This will lead to copious production of four-$\tau$ events ($AA \to \tau^+\tau^-\;\tau^+\tau^-$),
the characteristic Type-X signal which was analyzed in detail in
Refs.~\cite{Su:2009fz,Kanemura:2011kx,Kanemura:2014bqa,Chun:2015hsa}.
Since the decay of the $\tau$ involves neutrinos, full reconstruction of the four-$\tau$ system is
not possible which rules out any possibility of identifying a resonance peak. On the other hand if we
consider the decay $AA\to \mu^+\mu^-\;\tau^+\tau^-$ it is straightforward to identify the events owing to
clean di-moun invariant mass ($M_{\mu\mu}$) peak at $M_A$ which will be the `smoking gun' signal for a
light spin-0 resonance.
We show later that, in spite of the limited branching ratio for $A \to \mu^+\mu^-$, the $2\mu\;2\tau$ final
state can identify the $A$ peak well within the luminosity reach of the 14 TeV LHC.
\begin{table}[t]
\begin{center}
\begin{tabular}{|c|c|c|c|c|}\hline
Parameters & $M_{A}$ (GeV) & $\tan\beta$ & $\cos(\beta-\alpha)$ & $\lambda_{hAA}/v$ \\\hline\hline
BP1 & 50 & 60 & 0.03 & 0.02 \\\hline
BP2 & 60 & 60 & 0.03 & 0.03 \\\hline
\end{tabular}
\end{center}
\caption{Benchmark points for studying the discovery prospects of light pseudoscalar
in Type X 2HDM model at 14 TeV run of LHC. $\lambda_{hAA}$ is in units of $v = 246$ GeV.}
\label{tab:benchmark}
\end{table}
The signal we are exploring contains a pair of oppositely charged muons with exactly two
$\tau$--tagged jets produced via :
\begin{equation}
p\,p\to h\to A\,A\to \mu^+\mu^-\;\;\tau^+\tau^- \to \mu^+\mu^-\;\; {j_\tau}\,{j_\tau} +{\fmslash E_T},
\end{equation}
where $j_\tau$ is a $\tau$--tagged jet as a result of hadronic $\tau$-decay.
The NNLO cross section for the Higgs production via gluon fusion at 14
TeV LHC is $ 50.35$ pb \cite{higgs:xsection}.
The Type-X 2HDM model have been encoded using \texttt{FeynRules}~\cite{Christensen:2008py,Alloul:2013bka}
in order to generate the model files for implementation in $\texttt{MadGraph5\_aMC@NLO}$~\cite{Alwall:2011uj,Alwall:2014hca}
which was used for computing the required cross-sections and generating events for collider analyses.
We have chosen the benchmark points (BP) given in Table \ref{tab:benchmark} for our analysis.
As we have explained in the previous section, we want a light pseudoscalar which can explain the
muon $g-2$ anomaly at 2$\sigma$.
The benchmark points in the parameter space used here, corresponding to $M_A = 50, 60$ GeV, are
consistent with all phenomenological constraints. They also satisfy theoretical constraints
such as perturbativity and a stable electroweak vacuum \cite{Broggio:2014mna}.
The signal of a light $A$, which is our main focus here, does not depend on $M_H$ or $M_{H^\pm} $.
For both of our benchmark points, each of these masses is 200 GeV. For the chosen benchmark scenarios,
the branching ratio of Higgs to $AA$ is $BR(h\to A A)\simeq 15\%$ which is well below the upper limit of about
23$\%$ \cite{Aad:2015pla} on any non-standard decay branching ratio (BR) of the SM-like Higgs boson.
The choice of $\tan\beta$ ensures that the lepton universality bounds originating from
$Z$ and $\tau$ decays are satisfied \cite{Chun:2016hzs}.
\subsection{Backgrounds}
The major backgrounds to our signal process : $ \mu^+\mu^-\,j_\tau \,j_\tau$ come from the following channels
(A) $p p \to \mu^+ \mu^- + jets$, (B) $p p \to V V + jets (V = Z,W, \gamma^*) $ and (C) $p p \to t \bar t + jets$.
All the background events are generated with two additional partons and the events are matched up to
two jets using MLM matching scheme \cite{Mangano:2006rw,Hoche:2006ph} using the \emph{shower-kT} algorithm
with $p_T$ ordered showers. We use NNLO production cross section for $\mu^{\pm}\,\mu^{\mp}\,j\,j$
\cite{Catani:2009sm} and $ZZ$ \cite{Cascioli:2014yka}, whereas $t\,\bar t$ production cross section
is computed at N$^3$LO \cite{Muselli:2015kba}.
Apart from these three backgrounds there exist other SM processes like $VVV$,
$t\bar{t}V$ and $W^\pm Z$ which in principle could fake the proposed signal ($2\mu2\tau$)~\cite{CMS-PAS-HIG-16-036}.
However lower cross-section and the requirement of exactly two muons and two tau-tagged jets satisfying a tight
invariant mass window around the pseudoscalar mass effectively eliminates the contribution from these additional channels.
\subsection{Simulation and event selection}\label{sec:simulation}
\begin{figure}[t!]
\includegraphics[width = 7.5cm]{./M_taujets_MA_50.pdf}
\includegraphics[width = 7.5cm]{./M_taujets_MA_60.pdf}
\caption{The invariant mass of the 2 tau-tagged jets for $M_A = 50$ and $M_A = 60$.
The figures illustrate how the higher $p_T (j_\tau)$ threshold leads to more
precise reconstruction of the peak at $M_A$.}
\label{fig:IM_taujets}
\end{figure}
After generating both signal and background events with $\texttt{MadGraph5\_aMC@NLO}$, we have used
\texttt{PYTHIA6} \cite{Sjostrand:2006za} for the subsequent decay, showering and hadronization
of the parton level events. Decay of $\tau$ leptons is incorporated using \texttt{TAUOLA} \cite{Jadach:1993hs}
integrated in $\texttt{MadGraph5\_aMC@NLO}$. Both one- and three-prong $\tau$ decays have been
included in our analysis.
For event generation we have used the \texttt{NN23LO1} \cite{Ball:2014uwa} parton distribution
function and the default dynamic renormalisation and factorisation scales \cite{mad:scale} in $\texttt{MadGraph5\_aMC@NLO}$.
Finally, detector simulation was done using \texttt{Delphes3} \cite{deFavereau:2013fsa}.
Jets were reconstructed using the anti-kT algorithm \cite{Cacciari:2008gp} with $R = 0.4$. The $\tau$-tagging
efficiency and mistagging efficiencies of the light jets as $\tau$-jets are incorporated in \texttt{Delphes3}
as reported by the ATLAS collaboration~\cite{ATL-PHYS-PUB-2015-045}. We operate our simulation on
the Medium tag point for which the tagging efficiency of 1-prong (3-prong) $\tau$ decay is 70\% (60\%)
and the corresponding mistagging rate is 1\% (2\%).
The hadronic decays of the $\tau$ are associated with some missing transverse energy in the events.
For the signal events the $\tau$ leptons originate from the decay of a light
pseudoscalar ($A$) with mass 50 or 60 GeV. Hence, if the $p_T$ of the $\tau$-tagged jet has to be very
close to $m_A/2$, the corresponding missing energy in the final state is suppressed. The invariant
mass of the $\tau$-tagged jets will thus peak very close to the parent mass.
In Figure~\ref{fig:IM_taujets} we substantiate this claim by plotting the invariant mass of the
$j_\tau\,j_\tau$ system for two different jet $p_T$ thresholds. One can clearly see that for
$p_T(j_\tau) > 25$ GeV the invariant mass peaks at the parent pseudoscalar mass, whereas
$M(j_\tau j_\tau)$ is peaking at a lower value than $M_A$ for $p_T(j_\tau) > 20$ GeV. Also the
invariant mass peak is sharper for the higher $p_T$ threshold. The four-body invariant mass
$M_{2\mu 2j_\tau}$ also shows the same features and peaks close to $ M_h = 125 $ GeV as
depicted in Figure.~\ref{fig:IM_2mu2taujet}. It is evident that these variables can be very
efficient in minimizing the background events.
\begin{figure}[t]
\includegraphics[width = 7.5cm]{./M_2mu2taujets_MA_50.pdf}
\includegraphics[width = 7.5cm]{./M_2mu2taujets_MA_60.pdf}
\caption{The invariant mass of the 2$\mu$ and 2 tau-tagged jets for $M_A = 50$ and $M_A = 60$.
The figures illustrate how the higher $p_T (j_\tau)$ threshold leads to more
precise reconstruction of the peak at $M_h = 125$ GeV.}
\label{fig:IM_2mu2taujet}
\end{figure}
We use the following selection cuts to select our signal and reduce the accompanying backgrounds:
\begin{itemize}
\item {\bf Preselection Cuts:} We require the final state to have two oppositely charged muons
of mininum $p_T > 10$ GeV and $|\eta| < 2.5$. We also require two tau-tagged jets ($j_\tau$) of minimum $p_T$, $p_T(j_\tau) > 20, 25$ GeV within $|\eta| < 2.5$.
\item The invariant mass of the di-muon system ($M_{\mu\mu}$) satisfies the window, $$| M_{\mu\mu}~-~ M_A | < 7.5 \textrm{ GeV}. $$
\item The invariant mass of the two tau-tagged jets ($M_{j_\tau j_\tau}$) satisfies:
\begin{itemize}
\item for $p_T(j_\tau) > 20$ GeV : $ (M_A - 20) < M_{j_\tau j_\tau} < (M_A + 10) $ GeV
\item for $p_T(j_\tau) > 25$ GeV : $|M_{j_\tau j_\tau} - M_A | < 15$ GeV.
\end{itemize}
\item The invariant mass of two muons and two taujets ($M_{2\mu 2j_\tau}$)lies within the range :
\begin{itemize}
\item for $p_T(j_\tau) > 20$ GeV : $ (M_h - 20) < M_{2\mu 2j_\tau} < (M_h + 10)$ GeV.
\item for $p_T(j_\tau) > 20$ GeV : $|M_{2\mu 2j_\tau} - M_h| < 15$ GeV.
\end{itemize}
\end{itemize}
Notice that we have taken asymmetric cut-windows with respect to $M_A$
for $M_{j_\tau j_\tau}$ and $M_{2\mu 2j_\tau}$ for
$p_T(j_\tau) > 20$ GeV and symmetric ones for $p_T(j_\tau) > 25$ GeV. This has to do with the fact that
for lower $p_T(j_\tau)$ cut, the invariant mass peaks at a lower value compared to the parent mass.
\section{Results and Discussion}
In Table~\ref{Table:Cut-flow-pt-20}, we present
the cut flow for the signal and the various backgrounds for the benchmark points BP1 (BP2) where
the number of events are calculated at the integrated luminosity of 3000 $ \rm{fb}^{-1}$.
Note that some of the background events are estimated as upper bound (marked by an asterisk),
as the number of simulated events passing the cuts drop down to very small values at
some point in the cut flow table, even after simulating with $2\times 10^7$ events for
the background analysis. Since we adopt the Medium Tag point for tau-tagging, the mistagging rate
for a pair of light jets is $\sim ~10^{-4}$. This, along with a tight invariant mass window around $M_A$ helps
to get rid of a major fraction of the various background channels. Demanding that
$| M_{\mu\mu}~-~ M_A |<7.5$ GeV should take care of the $Z$ contribution in $p p \to \mu^+ \mu^- + jets $ and
$p p \to V V + jets $ . After the cut on $ M_{\mu\mu}$ only a feeble contribution from the photon
(and partly off-shell $Z$) continuum can contribute in the $p p \to \mu^+ \mu^- + jets $ channel.
\begin{table}[t]
\renewcommand{\arraystretch}{1.1}
\centering
\begin{tabular}{|c|c|c|c|c|}
\hline
Cuts & Signal & $p p \to \mu^+ \mu^-$ & $p p \to V V $ & $p p \to t \bar t $ \\
& & $+ jets$ & $+ jets $ & $+ jets$ \\
\hline\hline
\multicolumn{5}{|c|}{$p_T(j_\tau) > $ 20 GeV}\\
\hline\hline
Preselection & 858 (1480) & 41041 (41041) & 107890 (107890) & 14486 (14486) \\
\hline
$| M_{\mu\mu}~-~ M_A |<7.5$ GeV & 836 (1430) & 909 (779) & 1189 (1325) & 1637 (1697) \\
\hline
$M_{j_\tau j_\tau}>M_A - 20$ \& & 760 (1336) & 130 (390) & 307 (654) & 330 (419) \\
$M_{j_\tau j_\tau}<M_A+10$ GeV & & & & \\
\hline
$M_{2\mu 2j_\tau}>M_h-20$ \& & 698 (1283) & $<130$ ($<390)\ast$& 81 (109) & 65 (51) \\
$M_{2\mu 2j_\tau}<M_h+10$ GeV & & & & \\
\hline\hline
\multicolumn{5}{|c|}{$p_T(j_\tau) > $ 25 GeV}\\
\hline\hline
Preselection & 277 (493) & 28833 (28833) & 75209 (75209) & 11629 (11629) \\
\hline
$| M_{\mu\mu}~-~ M_A | < 7.5$ GeV & 269 (475) & 649 (390) & 794 (924) & 1324 (1396) \\
\hline
$| M(j_\tau j_\tau)-M_A |< 15$ GeV& 228 (420) & $<649$ (130) & 112 (416) & 182 (196) \\
\hline
$|M_{2\mu 2\,j_\tau} - M_h|<15$ GeV& 211 (410) &$<649\;(<130)\ast$ & 20 (15) & 27 (27) \\
\hline
\end{tabular}
\caption{Cut flow table for signal BP1(BP2) and different background processes with two different set of $p_T(j_\tau)$
cuts as described in Section~\ref{sec:simulation}. The number of events are computed with integrated luminosity
of 3000 $\rm{fb}^{-1}$. The number of background events also depends on benchmark points as $M_A$ changes. }
\label{Table:Cut-flow-pt-20}
\end{table}
We compute the statistical significance by using the formula,
$\mathcal{S} = \sqrt{2\left[(S+B)\textrm{ln}\left(1+\frac{S}{B}\right)-S\right]} $
where $S(B)$ are number of signal (background) events which survive the cuts.
In Figure~\ref{fig:significance} we have plotted the significance $\mathcal{S}$ as a function
of integrated luminosity for both the benchmark points where BP1(BP2) corresponds to
$M_A = 50(60)$ GeV. For BP1 it is possible to reach 5$\sigma$ sensitivity
at integrated luminosity of 70(400) $\rm{fb}^{-1}$ with $p_T({j_\tau}) > 20 (25) $ GeV.
For BP2 the 5$\sigma$ sensitivity is achievable at 40(125) $\rm{fb}^{-1}$ integrated luminosity.
Increasing the minimum $p_T(j_\tau)$ from 20 GeV to 25 GeV results in better invariant mass
peaks but provides fewer number of events which decreases the discovery prospect of the model.
However the luminosity requirement is well within the reach of high luminosity run at the LHC.\\
The benchmarks chosen in this work allowing for the $BR(h\to AA) \sim 15\%$ are
close to the borderline of the exclusion limit on $\sigma(h) \times BR(h\to AA) \times {BR(A \to \mu\mu)}^2$
\cite{CMS-PAS-HIG-15-011, Aggleton:2016tdd}, when this is translated for $A \to 2\mu 2\tau$.
However, one can still allow for a lower branching ratio for $h \to AA$, for example, close to 10$\%$, which keeps one well
within the exclusion limit, satisfying all the other constraints. This would entail the required luminosity
for a 5$\sigma$ discovery to be nearly double the values quoted above.
\begin{figure}[t]
\includegraphics[width = 7.5cm]{./Significance50.pdf}
\includegraphics[width = 7.5cm]{./Significance60.pdf}
\caption{Discovery potential of the light pseudoscalar decaying to di-muon and di-tau channel
using invariant mass variables for BP1(left panel) and BP2(right panel) at 14 TeV LHC. }
\label{fig:significance}
\end{figure}
If $A$ were a scalar instead of a pseudoscalar, then it would also have decays into
$W^\ast W^\ast$ and $Z^\ast Z^\ast$ competing with the $\mu^+\mu^-$ mode.
The non-observation of such final states, even with accumulating luminosity,
should act as a pointer to the $CP$-odd nature of $A$.
Secondly, the presence of such channels eats into the branching ratio of the A,
and suppresses the $\mu^+\mu^-$ channel rate, reducing it below detectability.
The fact that we can reconstruct the $A$ via the $\mu\mu$ peak (which is the main
point we make in this work) owes itself to the non-negligible branching ratio for
this mode, which would not have been possible if it were a scalar instead of a
pseudoscalar.
On the other hand, if A were a superposition of a scalar and a pseudoscalar field
(i.e. if CP were violated), then the taus coming from the decay of the other A
would consist of unequal admixtures of right-and left-polarised states (both for
$\tau^-$ and $\tau^+$). In principle, suitable triple products of vectors constructed
out of the tau-decay products would have asymmetric distributions if CP-violation
had taken place. However, the construction of such CP-asymmetric triple products
would have required us to reconstruct the taus fully. This would warrant the so-called
collinear approximation, where the $\tau$, the decay product jet and
a neutrino would all move along the same straight line. This approximation is
valid if the tau has an energy of at least about 40 GeV. In our case, for a
light (50 -- 60 GeV) A this energy is not possessed by the taus, and thus
their reconstruction is not reliable. Therefore, while one can distinguish a
pure pseudoscalar from a pure scalar in this channel, identifying a
CP-admixture is difficult.
It is possible to search for the heavy scalar $H$ and the pseudoscalar using the
$pp\to Z \to HA$ production channel. In principle, this
enables one to reconstruct the $H$ mass. However, this associated production rate
will be two orders of magnitude smaller than the rate for $pp \to h \to AA$, principally
due to the large Higgs production rate from gluon fusion. Nevertheless, if one notices a low-mass
$\mu^+\mu^-$ peak from $A$, one can look for a tau-pair peak simultaneously in such events. It
is relatively easy to reconstruct tau-leptons from the tau-jets in such a case, since these
taus are quite energetic and the collinear approximation~\cite{Rainwater:1998kj} will work
for them. Thus, in association with a light $A$ constructed in the way suggested
in our paper, a heavy $H$ can also be looked for, albeit at higher luminosity.
In addition, a light $A$ may of course be responsible for $4\tau$ final
states. Some channels leading to such a final state have been analyzed in \cite{Kanemura:2011kx,Kanemura:2014bqa,Chun:2015hsa}.
We observe that the $\geq 3\tau$ final state fares better in terms of the statistical
significance owing to the dominant branching ratio of $A \to \tau^+ \tau^-$ as compared to
the much smaller branching ratio of $A \to \mu^+ \mu^-$.
For instance, for a 5$\sigma$ discovery of $M_A = 60$ GeV with $M_H =200$ GeV,
the required luminosity is approximately 70 fb$^{-1}$ as against 218 fb$^{-1}$
for the $2\mu 2\tau$ final state.
However, the di-muon pair is a lot cleaner to reconstruct, and gives an accurate
handle on the mass determination for the parent pseudoscalar. Thus the $2\mu 2\tau$ state
is more informative when it comes to ``identifying'' the pseudoscalar.
\section{Summary and Conclusion}
While the Type-X 2HDM admits of a light pseudoscalar, the explicit reconstruction of its mass
is a challenging task. We propose to meet this challenge by making use of the small but
non-negligible branching ratio for $A\to \mu^+\mu^-$, especially in the region of the
parameter space, which best explains the muon anomalous magnetic moment. We have studied
the channel $pp\to h\to AA\to\mu^+\mu^-\,\tau^+\tau^-$, with the taus decaying into
a jet each. The $\mu^+\mu^-$ pair shows a conspicuous invariant mass peak at $M_A$.
Besides, an appropriate $p_T$- cut on the tau-tagged jets also creates a $j_\tau j_\tau$
mass distribution that has a peak in the neighbourhood of $M_A$. A proper window demanded
of the latter invariant mass helps the effective tagging and background reduction
for the $\mu^+\mu^-$ peak. We find that, for $M_A$ between 50 and 60 GeV, $M_A$ can
be reconstructed in this manner, with statistical significance of 4-5 $\sigma$
with an integrated luminosity not far exceeding 100 $\rm{fb}^{-1}$ in the 14 TeV run.
\section{Acknowledgements}\label{sec:Acknowledgements}
BM thanks Korea Institute for Advanced Study for hospitality while this project
was initiated. This work was partially supported by funding available from
the Department of Atomic Energy, Government of India, for the Regional Centre for
Accelerator-based Particle Physics (RECAPP), Harish-Chandra Research Institute.
SD thanks N. Chakrabarty for helpful discussions.
The authors acknowledge the use of the cluster computing setup available at
RECAPP and at the High Performance Computing facility of HRI.
\bibliographystyle{apsrev}
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In his 2012 book Time Cure, psychologist Philip Zimbardo introduced a groundbreaking therapeutic approach for PTSD sufferers, co-developed with Rosemary Sword. "Time Perspective Therapy" shifts mental focus from the past to the present, and from negative to positive events, helping the subject achieve a more balanced view of life. Featuring relatable real-life stories, this book describes how TPT can help anyone living with depression, anxiety or stress move beyond the negative experiences of the past--from toxic relationships to bullying--toward a more positive future.
Stanford University professor emeritus Philip G. Zimbardo is known for his landmark prison study, dramatized in the IFC film The Stanford Prison Experiment. His nonprofit Heroic Imagination Project teaches how to meet challenges with wise and effective action. He lives in San Francisco. Rosemary K.M. Sword is the author of numerous -TPT-related articles, coauthor of a popular PsychologyToday.com blog, and the developer of -TPT-based Aetas: Mind Balancing Apps for mobile devices. She lives in Makawao, Hawaii.
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@implementation MKTMockSettings
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{
if (_spiedObject)
[self letSpiedObjectHandleInvocation:invocation];
}
- (void)letSpiedObjectHandleInvocation:(NSInvocation *)invocation
{
[invocation setTarget:_spiedObject];
[invocation invoke];
}
@end
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{
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{"url":"https:\/\/courses.engr.illinois.edu\/ece110\/sp2020\/content\/courseNotes\/files\/?nodeMethod","text":"Node Method for Circuit Analysis\nsp2020\n\nECE 110\n\nCourse Notes\n\nExplore More\n\nLearn It!\n\nBridge topologies are common structures in circuits because they usually permit current to flow in either direction across the \"bridge\". This feature allows a motor (for example) connected across the bridge to be spun either clockwise or counterclockwise.\n\nFigure 1\n\nFig. 1: Resistor bridge circuit. Depending on the values of the resistors $R_1$ to $R_5$, the bridge current $I_5$ may be positive, zero or negative.\n\nThere are no pair of resistors in series or parallel, so we cannot use any shortcuts to help analyze the circuit. Instead, we have to use KVL, KCL and Ohm's law. On the rest of this page, we describe the node method, a systematic method to apply these fundamental circuit laws.\n\nSo far, we have always referred to voltage as a drop across an element. Now, we define a zero voltage level called ground, so that every node in a circuit can have a node voltage relative to that ground.\n\nFigure 2\n\nFig. 2: The symbol for ground. The ground symbol can be attached to any node in a circuit schematic to define the voltage at that node to be 0 V.\n\nOnce a ground symbol is attached to a circuit schematic, all other node voltages are then labeled relative to that zero level.\n\nFigure 3\n\nFig. 3: Node voltages. The ground symbol is attached to the node at the bottom, so the node voltage there is 0 V. The node voltage at the top is 15 V because the voltage drop across the voltage source $V_S$ is 15 V. The other two node voltages are unknown, so we label them $V_a$ and $V_b$.\n\nNow, the voltage drop across any resistor or the current through it can be expressed in terms of the node voltages.\n\nFigure 4\n\nFig. 4: KVL with node voltages. Equating total voltage rises and drops around the loop gives $V_a = V_5+V_b$.\n\nTherefore, we obtain the voltage drop across $R_5$ and the current through it by KVL and Ohm's law.\n\\begin{align}\nV_5 &= V_a-V_b\\\\\nI_5 &= \\frac{V_5}{R_5} = \\frac{V_a-V_b}{R_5} \\label{NDM-ER5}\n\\end{align}\nNotice that the arrow label associated with $I_5$ points from node voltage $V_a$ to $V_b$. Similarly, we express the other resistor currents in terms of the node voltages in Fig. 3.\n\\begin{align}\nI_1 &= \\frac{15-V_a}{R_1}\\\\\nI_2 &= \\frac{15-V_b}{R_2}\\\\\nI_3 &= \\frac{V_a-0}{R_3}\\\\\nI_4 &= \\frac{V_b-0}{R_4} \\label{NDM-ER4}\n\\end{align}\n\nThere are two unknown node voltages $V_a$ and $V_b$ in Fig. 4. The KCL equations at those two nodes are obtained by equating total currents flowing in with total currents flowing out.\n\\begin{align}\nI_1 &= I_3+I_5\\\\\nI_2+I_5 &= I_4\n\\end{align}\nSince all the currents flow through resistors, we can rewrite these equations in terms of the node voltages by using equations $\\eqref{NDM-ER5}$ to $\\eqref{NDM-ER4}$.\n\\begin{align}\n\\frac{15-V_a}{R_1} &= \\frac{V_a-0}{R_3}+\\frac{V_a-V_b}{R_5} \\label{NDM-KCA}\\\\\n\\frac{15-V_b}{R_2}+\\frac{V_a-V_b}{R_5} &= \\frac{V_b-0}{R_4} \\label{NDM-KCB}\n\\end{align}\nGiven resistor values $R_1$ to $R_5$, we can solve these two equations for the two unknown node voltages $V_a$ and $V_b$, from which we can determine any voltage drop or current in the circuit. The following table gives examples of calculating $V_a$ and $V_b$ (from equations $\\eqref{NDM-KCA}$ and $\\eqref{NDM-KCB}$) as well as the bridge current $I_5$ (from equation $\\eqref{NDM-ER5}$) given known values of $R_1$ to $R_5$.\n\nResistor Values Node Voltages Bridge Current\n\\begin{aligned} R_1&=R_2=R_3=R_5=5\\text{ k}\\Omega \\\\ R_4&=1\\text{ k}\\Omega \\end{aligned} \\begin{aligned} V_a&=6\\text{ V} \\\\ V_b&=3\\text{ V} \\end{aligned} \\begin{aligned} I_5 &=0.6\\text{ mA} \\end{aligned}\n\\begin{aligned} R_1&=R_2=R_3=R_5=5\\text{ k}\\Omega \\\\ R_4&=5\\text{ k}\\Omega \\end{aligned} \\begin{aligned} V_a&=7.5\\text{ V} \\\\ V_b&=7.5\\text{ V} \\end{aligned} \\begin{aligned} I_5 &=0\\text{ mA} \\end{aligned}\n\\begin{aligned} R_1&=R_2=R_3=R_5=5\\text{ k}\\Omega \\\\ R_4&=9\\text{ k}\\Omega \\end{aligned} \\begin{aligned} V_a&=8\\text{ V} \\\\ V_b&=9\\text{ V} \\end{aligned} \\begin{aligned} I_5 &=-0.2\\text{ mA} \\end{aligned}\nTable 1: Example node method KCL solutions for resistor bridge circuit.\n\nObserve that increasing the resistor value $R_4$ in the circuit in Fig. 1 changes the bridge current $I_5$ from positive to zero to negative.\n\nWhen there is just one voltage source in a circuit, you should attach the ground to one of the nodes connected to that voltage source, just as in Fig. 3. In this way, both nodes connected to the voltage source have node voltages that are known. But if there are two or more voltage sources not all connected to a common node, then at least one of them cannot be connected to the ground. We call these these ones floating voltage sources because their adjacent node voltages are unknown.\n\nFigure 5\n\nFig. 5: Voltage source bridge circuit. Two voltage sources $V_s$ and $V_x$ are not connected to a common node, so one of them becomes a floating voltage source. In this case, the ground is attached at the negative terminal of $V_s$, so $V_x$ is floating and the node voltages adjacent to it are unknown.\n\nThe problem with the floating voltage source $V_x$ is that the current $I_x$ cannot be expressed directly in terms of the node voltages. Thus, $I_x$ cannot be used in node method KCL equations to solve for the node voltages. We work around this problem by encapsulating the floating voltage source $V_x$ in a supernode. Then we can apply KCL to the supernode without involving $I_x$\n\nFigure 6\n\nFig. 6: Supernode encapsulating floating voltage source. The two unknown node voltages adjacent to the supernode can be labeled $V_a$ at the $-$ terminal of $V_x$ and $V_a+5$ at the $+$ terminal of $V_x$ because $V_x=5\\text{ V}$.\n\nAt the supernode, KCL gives $I_1+I_2 = I_3+I_4$, which can be rewritten in terms of node voltages as follows.\n\\begin{align}\n\\frac{15-V_a}{R_1} + \\frac{15-(V_a+5)}{R_2}&= \\frac{V_a-0}{R_3}+\\frac{(V_a+5)-0}{R_4} \\label{NDM-KCS}\n\\end{align}\nGiven resistor values $R_1$ to $R_4$, we can solve this equation for the unknown node voltage $V_a$. Knowing all the node voltages, we can determine any resistor's voltage drop or current. But to find the current $I_x$ through the floating voltage source, we need to apply KCL to one of the adjacent nodes (not the supernode). For example, KCL at the node labeled $V_a$ in Fig. 6 gives $I_1=I_3+I_x$. Therefore,\n\\begin{align}\nI_x &= I_1-I_3\\\\\n&= \\frac{15-V_a}{R_1} - \\frac{V_a-0}{R_3} \\label{NDM-IXE}\n\\end{align}\nThe following table gives examples of calculating $V_a$ (from equation $\\eqref{NDM-KCS}$) as well as the bridge current $I_x$ (from equation $\\eqref{NDM-IXE}$) given known values of $R_1$ to $R_4$.\n\nResistor Values Node Voltages Currents Bridge Current\n\\begin{aligned} R_1&=R_2=R_3=5\\text{ k}\\Omega \\\\ R_4&=1\\text{ k}\\Omega \\end{aligned} \\begin{aligned} V_a&=0\\text{ V} \\\\ V_a+5&=5\\text{ V} \\end{aligned} \\begin{aligned} I_1 &=3\\text{ mA} \\\\ I_3 &=0\\text{ mA} \\end{aligned} \\begin{aligned} I_x &=3\\text{ mA} \\end{aligned}\n\\begin{aligned} R_1&=R_2=R_3=5\\text{ k}\\Omega \\\\ R_4&=5\\text{ k}\\Omega \\end{aligned} \\begin{aligned} V_a&=5\\text{ V} \\\\ V_a+5&=10\\text{ V} \\end{aligned} \\begin{aligned} I_1 &=2\\text{ mA} \\\\ I_3 &=1\\text{ mA} \\end{aligned} \\begin{aligned} I_x &=1\\text{ mA} \\end{aligned}\n\\begin{aligned} R_1&=R_2=R_3=5\\text{ k}\\Omega \\\\ R_4&=9\\text{ k}\\Omega \\end{aligned} \\begin{aligned} V_a&=6.25\\text{ V} \\\\ V_a+5&=11.25\\text{ V} \\end{aligned} \\begin{aligned} I_1 &=1.75\\text{ mA} \\\\ I_3 &=1.25\\text{ mA} \\end{aligned} \\begin{aligned} I_x &=0.5\\text{ mA} \\end{aligned}\nTable 2: Example node method KCL solutions for voltage source bridge circuit.\n\nThe flowchart shown below efficiently solves a circuit with sources and resistors using the node method. Use this technique when you cannot simplify the circuit any further using circuit analysis shortcuts.\n\nFigure 7\n\nFig. 7: Flowchart for node method circuit analysis.\n\nAttach a ground to the node connected to as many voltage sources as possible. Use these voltage sources to label known node voltages.\n\nIf there are any floating voltages (ones connected to unknown node voltages), enclose each of them in a supernode.\n\nLabel all unknown node voltages using variables. Across a supernode, your variables should include the value of the enclosed floating voltage source.\n\nWrite a KCL equation at every supernode and at every other unknown node in terms of node voltages.\n\nDetermine the unknown node voltages by simultaneously solving the KCL equations.\n\nUse the node voltages to obtain desired quantities in the circuit.\n\nExplore More!","date":"2020-06-02 15:17:01","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\": 0.9986662864685059, \"perplexity\": 1681.7913930638595}, \"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-2020-24\/segments\/1590347425148.64\/warc\/CC-MAIN-20200602130925-20200602160925-00075.warc.gz\"}"}
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São Nicolau ist eine ehemalige Gemeinde (Freguesia) im Kreis (Concelho) von Mesão Frio im Norden Portugals. Die Gemeinde hatte 482 Einwohner auf einer Fläche von 0,45 km² (Stand 30. Juni 2011).
Die Gemeinde stellte eine der zwei Stadtgemeinden der Kleinstadt (Vila) Mesão Frio dar (Santa Cristina war die zweite). So befinden sich Sehenswürdigkeiten wie die romanischen Grabstätten (Arcas tumulares românicas), die namensgebende Gemeindekirche Igreja de São Nicolau und das historische Krankenhaus Hospital da Misericórdia hier.
Im Zuge der Gebietsreform vom 29. September 2013 wurden die Gemeinden São Nicolau, Vila Jusã und Santa Cristina zur neuen Gemeinde Mesão Frio (Santo André) zusammengeschlossen. São Nicolau ist Sitz dieser neu gebildeten Gemeinde.
Weblinks
Einzelnachweise
Ehemalige Freguesia in Portugal
Mesão Frio
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{"url":"https:\/\/iclr.cc\/virtual\/2021\/poster\/3036","text":"## Local Convergence Analysis of Gradient Descent Ascent with Finite Timescale Separation\n\n### Tanner Fiez \u00b7 Lillian J Ratliff\n\nKeywords: [ equilibrium ] [ gradient descent-ascent ] [ continuous games ] [ game theory ] [ theory ] [ convergence ] [ generative adversarial networks ]\n\nAbstract: We study the role that a finite timescale separation parameter $\\tau$ has on gradient descent-ascent in non-convex, non-concave zero-sum games where the learning rate of player 1 is denoted by $\\gamma_1$ and the learning rate of player 2 is defined to be $\\gamma_2=\\tau\\gamma_1$. We provide a non-asymptotic construction of the finite timescale separation parameter $\\tau^{\\ast}$ such that gradient descent-ascent locally converges to $x^{\\ast}$ for all $\\tau \\in (\\tau^{\\ast}, \\infty)$ if and only if it is a strict local minmax equilibrium. Moreover, we provide explicit local convergence rates given the finite timescale separation. The convergence results we present are complemented by a non-convergence result: given a critical point $x^{\\ast}$ that is not a strict local minmax equilibrium, we present a non-asymptotic construction of a finite timescale separation $\\tau_{0}$ such that gradient descent-ascent with timescale separation $\\tau\\in (\\tau_0, \\infty)$ does not converge to $x^{\\ast}$. Finally, we extend the results to gradient penalty regularization methods for generative adversarial networks and empirically demonstrate on CIFAR-10 and CelebA the significant impact timescale separation has on training performance.","date":"2021-09-19 01:27:41","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\": 0.8681188821792603, \"perplexity\": 550.359641713802}, \"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\/1631780056656.6\/warc\/CC-MAIN-20210919005057-20210919035057-00101.warc.gz\"}"}
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Rebecca Leah "Bekah" Brunstetter (born June 13, 1982) is an American writer. Her published plays include F*cking Art, which won top honors at the Samuel French Off-Off-Broadway Short Play Festival, I Used to Write on Walls, Oohrah!, Be a Good Little Widow, Going to a Place Where You Already Are, and The Cake, a play inspired by events leading to the US Supreme Court case Masterpiece Cakeshop v. Colorado Civil Rights Commission. She is a founding member of The Kilroys, which annually produces The Kilroys' List. Her television work includes writing for I Just Want My Pants Back, Underemployed, Switched at Birth, and American Gods, and both writing and producing on This Is Us.
Early life and education
Rebecca Leah Brunstetter was born on June 13, 1982, in Winston-Salem, North Carolina. She is the daughter of former North Carolina Senator Peter S. Brunstetter and Jodie Brunstetter. She was raised as the only daughter among three brothers in a conservative Christian home. Brunstetter wrote poems and short stories from a young age, and became involved with theater after moving from a private Christian middle school to Mount Tabor High School, a public school.
As a student at the University of North Carolina at Chapel Hill, Brunstetter initially pursued creative writing through poetry, but feedback from her creative writing professors convinced her to try playwriting. She wrote her first play as a first-year student and decided to pursue playwriting as a career. By the time Brunstetter graduated from UNC in 2004 with a Bachelor of Fine Arts degree, the UNC Theater Department had fully staged several of her plays. She continued to study playwriting for three years in the School of Drama at The New School, graduating with a Master of Fine Arts degree.
Career
Early plays
After graduating with her MFA, Brunstetter worked at a corporate job while continuing to write plays. I Used to Write on Walls, her play about three women who fall for a religious man who surfs and draws graffiti, premiered at the Gene Frankel Theatre Underground in New York in 2007, with Gwen Orel of Backstage calling it a "would-be feminist fable" that is "less convincing than cute". Duncan Pflaster of BroadwayWorld observed that the play seemed to "reinforce the stereotype that women need men to feel complete", but praised Brunstetter's writing and character development. Robert Hurwitt's review in the San Francisco Chronicle of the play's 2010 West Coast premiere also praised the quality of the dialogue, but called the lengthy play "a bit too much of one initially good thing".
In 2008, her play F*cking Art, about a cheerleader who visits her cancer-stricken classmate, was a winner at the 33rd Annual Samuel French Off-Off-Broadway Short Play Festival, and was subsequently published by Samuel French. The following year she was named Playwright in Residence at Ars Nova, and her play Oohrah!, a story about the family lives of people in a North Carolina town changing as veterans return home from Iraq, premiered off-Broadway at Stage 2 of the Atlantic Theater Company. In his review of Oohrah!, Charles Isherwood of The New York Times praised Brunstetter as an up-and-coming new playwright, but found the play "generally unconvincing". Joe Dziemianowicz of the New York Daily News assessed Oohrah! as "about as deep as your average sitcom", while drawing attention to the quality of Brunstetter's dialogue and writing of female characters.
Brunstetter's play Miss Lilly Gets Boned, a story about a religious woman whose disappointment in love causes her to plot revenge against a South African man who lost his wife in an elephant attack, premiered in 2010 at the Finborough Theatre. Writing for The Guardian, Michael Billington found the development of the main character to be unconvincing, but noted the high quality of the production and acting. In 2019, Rogue Machine Theatre produced the play's West Coast premiere in Los Angeles. Jeffrey Scott's BroadwayWorld review of the Los Angeles production praised the play, actors, and production quality, particularly the performance of Larisa Oleynik in the lead role, but also suggested that the story could be split into three one-act plays in future productions.
During her Ars Nova residency Brunstetter wrote a new play, titled Be a Good Little Widow, about the relationship between a woman and her husband's mother before and after the husband's death. Be a Good Little Widow premiered at Ars Nova in 2011, with a cast including Jill Eikenberry and Wrenn Schmidt. Writing for The New York Times, Adam Hetrick reviewed the play positively, praising Brunstetter for writing straightforward dialogue and genuinely emotional characters. A review by Chris Jones in the Chicago Tribune was less favorable to a 2011 Collaboraction staging of the play, calling Be a Good Little Widow "sincerely meant but structurally immature". Kathleen Foley later reviewed the Los Angeles premiere positively in the Los Angeles Times, observing that Brunstetter was adept at manipulating the audience's emotions to good effect.
Expanding into screenwriting
While working as a playwright, Brunstetter started a business writing audition monologues for actors and looked for other writing work to supplement her playwriting income. Her theater agent introduced her to a television agent in Los Angeles, and she was hired as a writer's assistant by MTV. After spending a season as an assistant on the short-lived MTV show I Just Want My Pants Back, she became a member of the writing staff for another MTV show, Underemployed, before moving to the ABC Family drama Switched at Birth, where she worked as a staff writer for three seasons.
Brunstetter continued her playwriting while working as a screenwriter, premiering her work Forgotten Corners of Your Dark, Dark Place, which starred actresses in wheelchairs, at the Theater Breaking Through Barriers' annual festival of new plays. Anita Gates of The New York Times praised the actresses' performances, but expressed concern that the play was unclear about whether or not it was mocking feminist self-examination groups. Brunstetter also collaborated with other Los Angeles-area writers to create The Kilroys' List, an annual list of plays by female and transgender playwrights modeled on the Black List but intended to promote gender equity. The list featured her own play The Oregon Trail, about a girl who withdraws from social life as she plays the video game The Oregon Trail. The play subsequently premiered at the Women's Voices Theater Festival, with Nelson Pressley of The Washington Post concluding that "even some over-explaining in the final steps doesn't erase the pleasure of this quest".
The Cake and American Gods
In 2015, Brunstetter began writing The Cake, a play about a baker who is asked to bake a cake for the wedding of her best friend's daughter but refuses because it is a same-sex wedding. The play was inspired by real-life events that eventually led to the Masterpiece Cakeshop v. Colorado Civil Rights Commission Supreme Court case, and by her father's opposition to same-sex marriage, a view with which she disagrees. The play premiered in Los Angeles, with Debra Jo Rupp in the starring role. Writing for the Los Angeles Times, Philip Brandes praised the play's narrative structure but noted that some of the dialogue "reads like a laundry list of liberal activist accusations". The play has been widely produced, including shows at the La Jolla Playhouse, Houston's Alley Theatre, and an Off-Broadway premiere at the Manhattan Theatre Club at New York City Center that Jesse Green of The New York Times described as "well-baked but not quite filling".
At the same time that she was writing The Cake, Brunstetter started work on the new Starz series American Gods, based on Neil Gaiman's novel of the same name. As part of the writing team for American Gods, Brunstetter helped develop the character of the goddess Easter. She was credited as a writer on the first-season finale, titled "Come to Jesus", which Oliver Sava singled out in Vulture as an exciting standout in an otherwise poorly-paced season.
This Is Us
Brunstetter also joined the NBC series This Is Us, first as a staff writer, then story editor, before becoming a supervising producer. She was nominated along with fellow producers for the Primetime Emmy Award for Outstanding Drama Series in 2017, 2018, and 2019. Her personal childhood bullying experience inspired a "heartbreaking" scene in This Is Us in which an overweight young girl is excluded from playing with the other girls. The show debuted in 2016, and received the highest ratings among new shows on American broadcast television in its first season. Brunstetter left the show after three seasons.
Going to a Place Where You Already Are and later work
In 2016, South Coast Repertory premiered Going to a Place Where You Already Are, a play the company commissioned from Brunstetter. The play follows a terminally ill woman who comes to believe in heaven, leading her to abandon further medical treatment against the wishes of her spouse, who does not believe in an afterlife. It is based in part on conversations Brunstetter had with her father's atheist parents about death and heaven. In the Los Angeles Times, Daryl Miller called Going to a Place Where You Already Are a "terrific new play", highlighting her simultaneously emotional and entertaining treatment of serious subjects. The play was also produced by the Boulder Ensemble Theater Company, where it was panned by Juliet Wittman in Westword as a saccharine remix of elements from This Is Us, and at Theater Alliance of Washington, DC, where John Stoltenberg of DC Metro Theater Arts called it "an extraordinary exploration of love in life and loss in death".
Brunstetter was one of several writers to receive an inaugural $5,000 Writers Alliance Grant from the Dramatists Guild Foundation in 2018, with Brunstetter's grant supporting a new commission from Theater Breaking Through Barriers. The resulting play, about a woman who borrows money from her politically conservative father under false pretenses to pay for an abortion, premiered the following year at Clurman Theatre in the Theatre Row Building under the title Public Servant. In The New York Times, Laura Collins-Hughes criticized Public Servant for its character and plot development, observing that the play's story seemed "grafted to fit" its politics. While lauding the inclusive casting of Public Servant, Deb Miller of DC Metro Theater Arts expressed disapproval of Brunstetter's "signature TV style", noting particularly her handling of the play's various dilemmas with "forced, overly sentimental, and unbelievably contrived" resolutions.
Collaborations
Brunstetter was hired in 2017 to adapt the bestselling self-help book The Secret for film. Her script adapts the book's ideas about the "law of attraction" into a story about the relationship between a widowed mother and a handyman who shares his thoughts on how the universe works. In 2019, singer Ingrid Michaelson announced that she and Brunstetter had been collaborating on an adaptation of The Notebook into a Broadway musical, with author Nicholas Sparks later confirming his involvement in the production.
With screenwriter Cinco Paul providing the songs, Brunstetter wrote the script for a musical comedy titled A.D. 16, about "a teenage Mary Magdalene in love with Jesus", that premiered in February 2022 at the Olney Theatre Center in Olney, Maryland. Peter Marks of The Washington Post praised the integration of modern music and themes with the Biblical setting, as well as the design and production of the show, concluding that A.D. 16 was "an occasion that merits its own hallelujah chorus". John Stoltenberg of DC Theater Arts also praised the musical, noting the effective movement between comedy and moral seriousness, and in particular the comedic portrayal of Jesus as a counterpoint to evangelical Christian ideas about masculinity. Rebecca Ritzel of Washington City Paper criticized the connection between the story and the music, saying that "the plot stops when the music starts", but favorably noted that both Paul and Brunstetter had drawn on their personal religious backgrounds to craft a "more subtle, and more empathetic" musical than similarly irreverent works like Godspell and Jesus Christ Superstar.
Personal life
Brunstetter married actor Morrison Keddie in 2016. In 2017, as a Valentine's Day present, Brunstetter wrote a short film script for Keddie based on a story about his uncle. The resulting film, titled Again, with Keddie portraying the lead role of a man who repeatedly watches the film Groundhog Day, was selected for the 2017 Tribeca Film Festival. Keddie has since provided the voice of George in the Barrington Stage Company production of The Cake. , Brunstetter resides in Los Angeles, California.
Works
I Used to Write on Walls, Samuel French, 2008,
F*cking Art, in Off Off Broadway Festival Plays, 33rd Series, Samuel French, 2008,
Oohrah!, Samuel French, 2010,
Be a Good Little Widow: A Funeral, Samuel French, 2011,
Going to a Place Where You Already Are, Samuel French, 2016,
The Cake, Samuel French, 2018,
References
Living people
American women dramatists and playwrights
American women screenwriters
Writers from Winston-Salem, North Carolina
The New School alumni
University of North Carolina at Chapel Hill alumni
21st-century American women writers
1982 births
21st-century American screenwriters
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Started up an amazing loot deal running right now.
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{
"redpajama_set_name": "RedPajamaC4"
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| 970
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Q: My array elements change value after I increment my counter I'm trying to read in a file to a character array, but whenever I increment my counter (for the array elements), my array elements seem to change value.
In the following code, I'm printing out the array values before the counter is incremented and after the counter is incremented (to find out the problem).
I'm not sure how to fix this (the code is in C)...
char fileContents[rowNum * colNum];
rewind(file);
int count = 0;
while((ch = fgetc(file)) != EOF){
if(ch != '\n'){
fileContents[count] = ch;
printf("%c ", fileContents[count]);
count++;
printf("%c %d\n", fileContents[count - 1], count - 1);
}
}
These are the file contents"
XXXX XXXX
XX XXX XX
X X X X X
XX XX XXX
XXXXXXXXX
XXXXXXXXX
XXXXXXXXX
XXXXXXXXX
A: Try
char fileContents[rowNum * colNum];
rewind(file);
int count = 0;
while((ch = fgetc(file)) != EOF){
if(ch != '\n'){
fileContents[count] = ch;
printf("%c ", fileContents[count]);
printf("%c %d\n", fileContents[count - 1], count);
count++;
}
}
I added the count++ statement after the printf because there is the possibility that you increase count and count = fileContents.size(); this implies fileContents[count] is out of bounds
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Birnau détient une église priorale de style rococo construite entre 1747 et 1750 par l'architecte autrichien Peter Thumb, pour l'abbaye impériale de Salem. Sa renommée résulte non seulement de son extraordinaire décoration intérieure mais aussi de sa situation, sur les hauteurs du lac de Constance.
Historique
Une première église de pèlerinage aurait été construite au . En 1741, devant l'afflux incessant des pèlerins, l'abbé de Salem décide de reconstruire l'édifice dans le style baroque et confie la mission d'établir les plans à l'architecte du Vorarlberg, Peter Thumb qui avait déjà beaucoup œuvré en Alsace et en Forêt-Noire.
La première pierre de la nouvelle église est posée le . Les travaux progressent très rapidement si bien que l'édifice peut être consacré le .
En 1804, l'abbaye de Salem est sécularisée et l'église de Birnau fermée au culte. Cloches et orgues sont vendues aux enchères.
L'édifice ne sera rendu au culte qu'en 1919. En effet, à cette date, l'église et les bâtiments annexes sont acquis par les moines de l'abbaye territoriale de Wettingen-Mehrerau (Autriche).
D'importants travaux de rénovation furent menés à bien au et en 1971, l'église priorale a été élevée au rang de basilique mineure par le Pape Paul VI.
Décoration et aménagements
L'église de Birnau compte parmi les plus beaux édifices de style rococo en Allemagne. Les fresques de plafond, dédiées à la vie de la Vierge Marie, et les riches stucs sont l'œuvre du peintre Gottfried Bernhard Göz, originaire de Moravie.
L'ensemble du mobilier est l'œuvre du sculpteur autrichien Joseph Anton Feuchtmayer. On lui doit notamment le très célèbre Honigschlecker ou putto au pot de miel, installé sur l'autel latéral de Saint-Bernard.
La tour de façade à bulbe abrite un ensemble de cinq cloches ainsi qu'une horloge du , toujours en fonction. L'orgue date de 1991.
Architecture baroque en Allemagne
Basilique en Allemagne
Abbaye fondée au XXe siècle
Église en Bade-Wurtemberg
Birnau
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// Licensed to the .NET Foundation under one or more agreements.
// The .NET Foundation licenses this file to you under the MIT license.
// See the LICENSE file in the project root for more information.
#nullable disable
using Microsoft.CodeAnalysis;
using System.Diagnostics;
namespace Microsoft.VisualStudio.LanguageServices.Implementation
{
internal sealed class MissingAnalyzerDependency
{
public MissingAnalyzerDependency(string analyzerPath, AssemblyIdentity dependencyIdentity)
{
Debug.Assert(analyzerPath != null);
Debug.Assert(dependencyIdentity != null);
AnalyzerPath = analyzerPath;
DependencyIdentity = dependencyIdentity;
}
public string AnalyzerPath { get; }
public AssemblyIdentity DependencyIdentity { get; }
}
}
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"redpajama_set_name": "RedPajamaGithub"
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\section{Introduction}
\setcounter{equation}{0}
\indent
In this brief note we consider three-dimensional incompressible Navier-Stokes equations in a domain $\Omega \subset {\mathbb{R}}^3$ :
\[
(NS)\left\{\begin{array}{ll}
v_{t}+(v \cdot \nabla) v =-\nabla p +\Delta v, \quad& (x,t) \in \Omega \times (0,T)\\
\nabla \cdot v=0,\qquad& (x,t) \in \Omega \times (0,T)\\
v(x,0)=v_0(x),\qquad & x \in \Omega
\end{array}\right.\]
where $v=(v_1, v_2, v_3) $ is the flow velocity and $p$ is the scalar pressure, respectively. The initial data $v_0$ satisfies
\[ \nabla \cdot v_0 =0.
\]
It is well known that the first equations of $(NS)$ can be rewritten as following equivalent form:
\begin{equation}\label{13}
v_t -v \times \omega =-\nabla \left( p+\frac{|v|^2}{2} \right)+\Delta v,
\end{equation}
where $\omega=\nabla \times v$ is the vorticity vector field. The global in time existence of a smooth solution to the system (NS) is one of the outstanding open problems in mathematical fluid mechanics.
On the other hand, the global in time existence of weak solution(Leray-Hopf weak solution) was proved first by Leray\cite{leray}.
There are numerous conditional regularity results of weak solutions by imposing the integrability conditions on the velocity or vorticity using scaling invariant function space for weak solutions to (NS) (see \cite{Bei1, Bei2, CKL, ESS, Giga, lady, ohyama, prodi, serrin} and references therein).
Besides the so-called Prodi-Serrin type regularity conditions, there are many studies on the geometric regularity conditions by imposing alignment of the direction of the vorticity (see \cite{Bei-Ber, Ber, Ber-Cor, Chae2, Const-Feff, Gruj1, Gruj2} and references therein).
Among the previous results, Chae\cite{Chae1} obtained local regularity criterion by imposing scaling invariant integrability conditions on $ v \times \frac{\omega}{|\omega|}$ or $\omega \times \frac{v}{|v|}$ which is a refinement of other Prodi-Serrin type condition on $v$ and $\omega$.
On the other hand, Lee\cite{lee} obtained regularity by assuming the smallness of the volume of the parallelepiped which is defined by the unit vectors $\frac{v}{|v|}$, $\frac{\omega}{|\omega|}$ and $\frac{\nabla \times \omega}{|\nabla \times \omega|}$.\\
We define nonlocal operator $\Lambda =(-\Delta)^{\frac12}$ as $\Lambda^{\beta} f= (-\Delta)^{\frac{\beta}{2}} f ={\mathcal{F}}^{-1}(|\xi|^{\beta} {\mathcal{F}}{f}(\xi))$ where ${\mathcal{F}}$ denotes a Fourier transform on ${ \mathbb{R} }^3$. We use a mixed type norms for $Q_{T}= { \mathbb{R} }^3 \times (0, T)$ :
\[
\| v\|_{ L^{\gamma, \alpha}_{x,t}(Q_{T})}:= \left\| \| v(\cdot, t)\|_{L^{\gamma}_{x}({ \mathbb{R} }^3) } \right\|_{L^{\alpha}_{t}(0, T)},\quad 1\leq \alpha, \gamma \leq \infty.
\]
We also denote $\{f\}_+(x):=\max \{ f(x),\, 0\}$. Also direction fields $\frac{\omega}{|\omega|}$ and $\frac{\Lambda^{\beta} v}{|\Lambda^{\beta}v|}$ are set to be zero when $\omega(x,t)=0$ and $\Lambda^{\beta}v(x,t)=0$, respectively.\\ First, we consider Prodi-Serrin type blow-up criterion in terms of some triple product, which improves the previous criterion of \cite{serrin}. We consider only $\Omega={ \mathbb{R} }^3$ case for simplicity.
\begin{theorem}\label{global}
Let $v$ be a local in time reguar solution of the Navier-Stokes equations (NS) in $Q_{T}:={ \mathbb{R} }^3 \times (0, T)$ with $v_0 \in H^{\frac{1}{2}}({ \mathbb{R} }^3)$. Then, we have,
\begin{description}
\item[(i)] if $v$, $\omega:=\nabla \times v$ and $\Lambda^{\beta} v$ satisfies that, for an absolute constant $\epsilon_0$ and some $\beta \in [1,\,2]$,
\begin{equation}\label{global-small-cond}
\left\|\left\{ \left( v \times \frac{\omega}{|\omega|} \right)\cdot \frac{\Lambda^{\beta} v}{|\Lambda^{\beta} v|}\right\}_{+}\right\|_{L^{3, \infty}_{x,t}(Q_{T})}\leq \epsilon_0,
\end{equation}
then a regular solution $v$ exists beyond $T$, that is, $v \in C([0,\, T+\delta) ; H^{\frac12}({ \mathbb{R} }^3))$ for some $\delta >0$.
\item[(ii)] $v$ blows up at $T_{*}$, which is a finite maximal time of local in time smooth solution to (NS), namely,
\[
\limsup_{ t \nearrow T_{*}} \| v(t) \|_{H^{m}} =\infty,\qquad \forall m \geq \frac12,
\]
if and only if for all
$\gamma \in (3, \infty]$ and $\alpha \in [2, \infty]$ with $3/\gamma +2/\alpha \leq 1$ and all $\beta \in [1.\, 2]$
\begin{equation}\label{global-v-cond}
\left\|\left\{\left( v \times \frac{\omega}{|\omega|} \right)\cdot \frac{\Lambda^{\beta} v}{|\Lambda^{\beta} v|}\right\}_{+} \right\|_{ L^{\gamma, \alpha}_{x,t}(Q_{T})}=\infty.
\end{equation}
\end{description}
\end{theorem}
\begin{remark}
From the standard local in time existence theory of Navier-Stokes equations, $v(t) \in H^{m}({ \mathbb{R} }^3)$ for any $m \in {\mathbb{N}}$ and $t \in (0,\, T_{*})$ where $T_{*}$ is a possible blow up time of local $H^{\frac12}$-solution. Therefore, any derivatives in Theorem \ref{global} are well-defined pointwise and $\Lambda^{\beta} v$ can be used as a test function.
\end{remark}
Since $\Lambda$ is a nonlocal operator, it does not seem easy to obtain local regularity criterion for Theorem \ref{global}. But for the case $\beta=2$, we can obtain a local regularity criterion for the triple product including $v$, $\omega$ and $-\Delta v = \nabla \times \omega$. \\
Our goal in this paper is to prove local regularity criterion by imposing integrability conditions on the triple product $\left( v \times \frac{\omega}{|\omega|}\right)\cdot \frac{\nabla \times \omega}{|\nabla \times \omega|}$ or $\left(\frac{v}{|v|} \times \omega \right)\cdot \frac{\nabla \times \omega}{|\nabla \times \omega|}$. This improves both of the results in \cite{Chae1, lee} as well as the previous Prodi-Serrin type conditions.
For the local analysis of a weak solution the notion of suitable weak solution is useful as shown in the partial regularity results (see \cite{CKN} and \cite{scheffer}).
Let $Q_{T}:= \Omega \times (0, T),$ For a point $z=(x,t)\in Q_{T}$, we denote
\[
B_{x,r} := \{ y \in {\mathbb{R}}^3 \, :\, |y-x|<r\},\quad Q_{z,r}=B_{x,r} \times (t-r^2, t).
\]
We also use the mixed space-time norms :
\[
\| v\|_{ L^{\gamma, \alpha}_{x,t}(Q_{z,r})}:= \left\| \| v(\cdot, t)\|_{L^{\gamma}_{x}(B_{x,r}) } \right\|_{L^{\alpha}_{t}(t-r^2, t)},\quad 1\leq \alpha, \gamma \leq \infty.
\]
We state the definition of a suitable weak solution to (NS) for local analysis.
\begin{defn}
A pair $(v,p)$ of measurable functions is a suitable weak solution of (NS) if the following conditions are satisfied :
\begin{description}
\item[(i)] $ v \in L^{\infty}(0, T; L^2(\Omega))\cap L^2(0, T; W^{1,2}(\Omega))$, $p \in L^{\frac32}(Q_T))$.
\item[(ii)] The pair $(v,p)$ satisfies (NS) in the sense of distribution.
\item[(iii)] The pair $(v,p)$ satisfies the local energy inequality,
\[
\int_{\Omega} |v(x,t)|^2 \phi(x,t) dx +2\int_0^t \int_{\Omega} |\nabla v(x, \tau)|^2 \phi (x, \tau) dx d\tau
\]
\[
\leq \int_0^t \int_{\Omega} \left( |v|^2 (\partial_t \phi +\Delta \phi)+(|v|^2 +2p) v \cdot \nabla \phi \right)dx d\tau
\]
for almost all $t \in (0, T)$ and all nonnegative scalar test function $\phi \in C_0^{\infty}(Q_{T})$.
\end{description}
\end{defn}
We say that a weak solution is regular at $z$, if $v$ is bounded in $Q_{z,r}$ for some $r>0$. This point $z$ is called a regular point.
Below we use extended definitions of the directional fields $v(x,t)/|v(x,t)|$, $\omega(x,t)/|\omega(x,t)|$ and $\nabla \times \omega(x,t)/|\nabla \times \omega(x,t)|$, which are set to zero whenever $v(x,t)=0$, $\omega(x,t)=0$ and $ \nabla \times \omega(x,t)=0$, respectively.
\begin{theorem}\label{thm1}
Let $z_0 =(x_0, t_0) \in Q_{T}$ with $\bar{Q}_{z_0,r} \subset Q_{T}$, and $(v,p)$ be a suitable weak solution of (NS) in $Q_{T}$ with the vorticity $ \omega =\nabla \times v$, where the derivatives are in the sense of distribution. Suppose $v$ and $\omega$ satisfy one of the following conditions :
\begin{description}
\item[(i)] There exists an absolute constant $\epsilon_0$ such that
\begin{equation}\label{CL-smallness-cond}
\left\|\left\{ \left( v \times \frac{\omega}{|\omega|} \right)\cdot \frac{\nabla \times \omega}{|\nabla \times \omega|}\right\}_{+}\right\|_{L^{3, \infty}_{x,t}(Q_{z_0,r})}\leq \epsilon_0.
\end{equation}
\item[(ii)] There exists $\gamma \in (3, \infty]$ and $\alpha \in [2, \infty]$ with $3/\gamma +2/\alpha \leq 1$ such that
\begin{equation}\label{CL-v-cond}
\left\{\left( v \times \frac{\omega}{|\omega|} \right)\cdot \frac{\nabla \times \omega}{|\nabla \times \omega|}\right\}_{+} \in L^{\gamma, \alpha}_{x,t}(Q_{z_0,r}).
\end{equation}
\item[(iii)] There exists $\gamma \in [2, \infty]$ and $\alpha \in [2, \infty]$ with $3/\gamma+2/\alpha \leq 2$ such that
\begin{equation}\label{CL-omega-cond}
\left\{\left( \frac{v}{|v|} \times \omega \right)\cdot \frac{\nabla \times \omega}{|\nabla \times \omega|} \right\}_{+}\in L^{\gamma, \alpha}_{x,t}(Q_{z_0,r}).
\end{equation}
Then, $z_0$ is a regular point.
\end{description}
\end{theorem}
\begin{remark} We note that there are many physical flows, including Beltrami flows (see \cite{Const-Maj}), for which the triple product vanishes. The above theorem says intuitively that even if the flow is far from the Beltrami flows,
if the projection of the vector $\nabla\times \omega $ on the plane spanned by $v$ and $\omega$ is ``controllable" in a local space-time region, then the flow is smooth in that region.
\end{remark}
\begin{remark}
In \cite{lee} it was proved that if there exists an absolute constant $\epsilon_0$ such that
\begin{equation}
\left\| \left( \frac{v}{|v|} \times \frac{\omega}{|\omega|} \right)\cdot \frac{\nabla \times \omega}{|\nabla \times \omega|}\right\|_{L^{\infty, \infty}_{x,t}(Q_{z_0,r})}\leq \epsilon_0,
\end{equation}
then $z_0$ is a regular point. As an easy consequence of (i) of Theorem \ref{thm1}, we can have, for $b>0$, that if there exists an absolute constant $\epsilon_0$ such that
\begin{equation}
\left\| |v|^{b}\left\{\left( \frac{v}{|v|} \times \frac{\omega}{|\omega|} \right)\cdot \frac{\nabla \times \omega}{|\nabla \times \omega|}\right\}_{+}\right\|_{L^{\frac{3}{b}, \infty}_{x,t}(Q_{z_0,r})}\leq \epsilon_0,
\end{equation}
then $z_0$ is a regular point. Hence, the result in \cite{lee} is a special case of Theorem \ref{thm1} as $b \rightarrow 0+$.
\end{remark}
\begin{remark}
Theorem \ref{thm1} (i) and (ii) can be considered as improvements of Theorem 1.1 (i) and (ii) in \cite{Chae1}. But Theorem \ref{thm1} (iii) can extend Theorem 1.1 (iii) of \cite{Chae1} only on the range $ \gamma \in [2,3]$ due to the technical difficulties. In order to extend Theorem 1.1 (iii) of \cite{Chae1} to the triple product on the range $\gamma \in (\frac32, 2) \cup (3, \infty]$, it seems necessary to develop different methods.
\end{remark}
\section{Proof of the Main Theorems}
First, we prove Theorem \ref{global} by using standard a priori estimates.\\
\begin{pfthm0}
Let $T_{*}$ be a maximal time of local existence of $H^{\frac{1}{2}}$ solution. Multiplying $\Lambda^{\beta} v$ on the both sides of \eqref{13} and integrating over ${ \mathbb{R} }^3$, we have, for $t<T_{*}$
\[
\frac12 \frac{d}{dt} \| \Lambda^{\frac{\beta}{2}} v \|_{L^2}^2+\| \nabla \Lambda^{\frac{\beta}{2}}v\|_{L^2}^2= \int_{{ \mathbb{R} }^3} ( v\times \omega)\cdot \Lambda^{\beta} v dx
\]
\[
\leq \int_{{ \mathbb{R} }^3} \left\{ \left(v \times \frac{\omega}{|\omega|} \right) \cdot \frac{\Lambda^{\beta} v}{|\Lambda^{\beta} v|} \right\}_{+}|\omega|\, |\Lambda^{\beta} v|\, dx
\]
\[
\leq \left\|\left\{ \left(v \times \frac{\omega}{|\omega|} \right) \cdot \frac{\Lambda^{\beta} v}{|\Lambda^{\beta} v|} \right\}_{+}\right\|_{L^{\gamma}} \| \omega \|_{L^p}\| \Lambda^{\beta} v\|_{L^q}:=I,
\]
where $p$ and $q$ safisfies $\frac{1}{p}+\frac{1}{q} =\frac{\gamma-1}{\gamma}$, $p \in [\frac{6}{5-\beta},\, \frac{6}{3-\beta}]$ and $q\in [\frac{6}{3+\beta}, \,\frac{6}{1+\beta}]$.\\
By the interpolation inequality, we have
\[
\| \omega \|_{L^p} \leq C \| \Lambda^{\frac{\beta}{2}} v \|_{L^2}^{\frac{3}{p}+\frac{\beta}{2}-\frac{3}{2}} \|\nabla \Lambda^{\frac{\beta}{2}}v\|_{L^2}^{\frac52-\frac{3}{p}-\frac{\beta}{2}}
\]
and
\[
\| \Lambda^{\beta} v \|_{L^q} \leq C\| \Lambda^{\frac{\beta}{2}} v \|_{L^2}^{\frac{3}{q}-\frac{\beta}{2}-\frac{1}{2}} \|\nabla \Lambda^{\frac{\beta}{2}}v\|_{L^2}^{\frac32-\frac{3}{q}+\frac{\beta}{2}} .
\]
Then we can estimate $I$ as
\[
I \leq C\left\|\left\{ \left(v \times \frac{\omega}{|\omega|} \right) \cdot \frac{\Lambda^{\beta} v}{|\Lambda^{\beta} v|} \right\}_{+}\right\|_{L^{\gamma}}\| \Lambda^{\frac{\beta}{2}} v\|_{L^2}^{\frac{\gamma-3}{\gamma}}\| \nabla \Lambda^{\frac{\beta}{2}} v\|_{L^2}^{\frac{\gamma+3}{\gamma}}.
\]
We first assume the condition (i) of Theorem \ref{global} holds true. Then we have
\[
\frac12 \frac{d}{dt} \| \Lambda^{\frac{\beta}{2}} v \|_{L^2}^2+\left[ 1-C \left\|\left\{ \left(v \times \frac{\omega}{|\omega|} \right) \cdot \frac{\Lambda^{\beta} v}{|\Lambda^{\beta} v|} \right\}_{+}\right\|_{L^{3}} \right]\| \nabla \Lambda^{\frac{\beta}{2}}v\|_{L^2}^2\leq 0
\]
If $\epsilon_0 < \frac{1}{C}$, then $ v \in L^{\infty} (0, T_{*}; H^{\frac{\beta}{2}}({ \mathbb{R} }^3))$. By the standard continuation argument, we have $v \in C((0, T_{*} +\delta); H^{\frac{1}{2}}({ \mathbb{R} }^3))$ for some $\delta>0$.\\
Next, we assume the condition (ii) of Theorem \ref{global} holds true. By Young's inequality, we have
\[
I \leq C\left\|\left\{ \left(v \times \frac{\omega}{|\omega|} \right) \cdot \frac{\Lambda^{\beta} v}{|\Lambda^{\beta} v|} \right\}_{+}\right\|_{L^{\gamma}}^{\frac{2\gamma}{\gamma-3}}\| \Lambda^{\frac{\beta}{2}} v\|_{L^2}^2 +\frac12 \| \nabla \Lambda^{\frac{\beta}{2}} v\|_{L^2}^2.
\]
Therefore, we obtain
\[
\frac{d}{dt} \| \Lambda^{\frac{\beta}{2}} v \|_{L^2}^2+\| \nabla \Lambda^{\frac{\beta}{2}}v\|_{L^2}^2 \leq C\left\|\left\{ \left(v \times \frac{\omega}{|\omega|} \right) \cdot \frac{\Lambda^{\beta} v}{|\Lambda^{\beta} v|} \right\}_{+}\right\|_{L^{\gamma}}^{\frac{2\gamma}{\gamma-3}}\| \Lambda^{\frac{\beta}{2}} v\|_{L^2}^2.
\]
By Gronwall's inequality, we have
\[
\sup_{t \in [0, T_{*})} \| \Lambda^{\frac{\beta}{2}} v (t)\|_{L^2}^2\leq \|v_0 \|_{H^{\frac{\beta}{2}}}^2 \exp \left[ C\left\|\left\{ \left(v \times \frac{\omega}{|\omega|} \right) \cdot \frac{\Lambda^{\beta} v}{|\Lambda^{\beta} v|} \right\}_{+}\right\|_{L^{\gamma, \frac{2\gamma}{\gamma-3}}_{x,t}(Q_{T_{*}})}^{ \frac{2\gamma}{\gamma-3}} \right].
\]
Note that $\left\|\left\{ \left(v \times \frac{\omega}{|\omega|} \right) \cdot \frac{\Lambda^{\beta} v}{|\Lambda^{\beta} v|} \right\}_{+}\right\|_{L^{\gamma, \frac{2\gamma}{\gamma-3}}_{x,t}(Q_{T_{*}})}^{ \frac{2\gamma}{\gamma-3}} < \infty$ due to \eqref{global-v-cond}. Hence, $v \in C((0, T_{*} +\delta); H^{\frac{1}{2}}({ \mathbb{R} }^3))$ for some $\delta>0$. This concludes the proof.
\end{pfthm0}
Before proceding our proof, we recall the notion of an epoch of possible irregularity of the suitable weak solution of the Navier-Stokes equations. It is well known that for weak solution there exists a closed set $E \subset I=[0, T]$ such that solutions are regular on $I \setminus E$ and $1/2$-dimensional Hausdorff measure of $E$ is zero. Moreover, $E$ can be written as $\displaystyle{I\setminus \left\{\bigcup_{i \in {\mathcal{I}}} (\alpha_i, \beta_i)\right\}}$ where ${\mathcal{I}}$ is at most countable and $(\alpha_i, \beta_i)$ are disjoint open intervals in $[0, T]$. As in \cite{Galdi}, we call $\beta_i$ as an epoch of possible irregularity. We recall the following Lemma proved by Neustupa and Penel\cite{neu-pen} on the epoch of possible irregularity for suitable weak solutions.
\begin{lemma}\label{lem1}
Let $z_0=(x_0, t_0) \in Q_{T}$. Suppose $v$ is a suitable weak solution of the Navier-Stokes equations in $Q_{T}$ and $t_0$ be an epoch of possible irregularity. Then there exist positive numbers $\tau$, $r_1$ and $r_2$ with $r_1 < r_2$ such that the follwings are satisfied :
\begin{description}
\item[(a)] $\tau$ is sufficiently small so that $t_0$ is only one epoch of possible irregularity in time interval $[t_0-\tau, t_0]$.
\item[(b)] The closure $B_{x_0, r_2} \times (t_0-\tau, t_0)$ is contained in $Q_{T}$, i.e., $\bar{B}-{x_0, r-2} \times [t_0-\tau, t_0] \subset Q_{T}$.
\item[(c)] $((\bar{B}_{x_0, r_2} \setminus B_{x_0, r_1}) \times [t_0-\tau, t_0])\cap {\mathcal{S}}=\phi$, where ${\mathcal{S}}$ is the set of possible singular points of $v$.
\item[(d)] $v$, $v_t$, and $p$ are, together with all their space derivatives, continuous on $(\bar{B}_{x_0, r_2} \setminus B_{x_0, r_1}) \times [t_0-\tau, t_0].$
\end{description}
\end{lemma}
\begin{pfthm1}
First, we assume that $t_0$ is an epoch of possible irregularity for $v$ in $Q_{z_0, r}$. Suppose that $0<r_1<r_2<r$ and $r^2 <\tau$ are the positive numbers in Lemma \ref{lem1}. For simplicity, we denote $B_1 =B_{x_0, r_1}$ and $B_{2}=B_{x_0, r_2}$. We choose cut-off function $\varphi \in C_0^{\infty}(B_2)$ such that $\varphi=1$ on $B_1$ and set $u=\varphi v -V$ where $V \in C_0^2 (B_2 \setminus \bar{B}_1)$ satisfies $\mbox{div }V= ( v\cdot \nabla)\varphi$. We note that $(v \cdot \nabla)\varphi$ satisfied the compatibility condition :
\[
\int_{B_2 \setminus \bar{B}_1} (v\cdot \nabla)\varphi dx = \int_{\partial B_2}\varphi v \cdot n_2 dS -\int_{\partial B_1} v \cdot n_1 dS=0,
\]
where $n_i$ is a unit outward normal vector to the sphere $\partial B_i$.
Using Bogovski$\rm{\breve{i}}$'s Theorem(see \cite{Bogo} or \cite[Theorem III.3.1]{Galdi}), we can prove that there exists at least one $V$ satisfying above properties. Then, by a straightforward calculation, $u$ satisfies
\begin{equation}\label{eq-main}
u_t -\varphi v \times \omega +\nabla \left( \varphi \left( p +\frac{|v|^2}{2}\right)\right) -\Delta u =h , \qquad \mbox{div }u=0,
\end{equation}
where $h$ satisfies
\[
h=-\frac{\partial V}{\partial t} +\left( p +\frac{|v|^2}{2} \right) \nabla \varphi -v \Delta \varphi -2 (\nabla \varphi \cdot \nabla)v +\Delta V.
\]
We note that $h(\cdot, t)$ is sufficiently smooth and supported in the region $(\bar{B}_2\setminus B_1)$.
Multiplying $-\Delta u$ on the both sides of \eqref{eq-main} and integrating, we have
\[
\frac12\frac{d}{dt} \| \nabla u \|_{L^2(B_2)}^2 + \| \Delta u \|_{L^2(B_2)}^2\]
\[ = \int_{B_2} v \times (\varphi \omega ) \cdot (\nabla \times \nabla \times (\varphi v)) dx -\int_{B_2} v \times (\varphi \omega) \cdot \nabla\times( \nabla \times V) dx -\int_{B_2} \Delta u \cdot h dx
\]
\[
\leq \int_{B_2} v \times (\varphi \omega ) \cdot (\nabla \times \nabla \times (\varphi v)) dx + C\| v\|_{L^2}^2 \| \varphi \omega \|_{L^2}^2\]
\[ +C\| D^2 V \|_{L^{\infty}(B_2)}^2 +\frac{1}{8} \| \Delta u \|_{L^2(B_2)}^2 +C\| h \|_{L^2(B_2)}^2
\]
\[
\leq C \int_{B_2} |v|^2 \, |\varphi \omega|\, |\nabla^2 \varphi| dx +C \int_{B_2} |v| \, |\varphi \omega|\, |\nabla \varphi|\, |\nabla v | dx+ \int_{B_2} \varphi^2\left((v \times \omega) \cdot ( \nabla \times \omega)\right)_{+} dx
\]
\[
+ C\| v\|_{L^2}^2 \| \varphi \omega \|_{L^2}^2
+C\| D^2 V \|_{L^{\infty}(B_2)}^2 +\frac{1}{8} \| \Delta u \|_{L^2(B_2)}^2 +C\| h \|_{L^2(B_2)}^2
\]
\[
:= I_1+I_2+I_3+ C\| v\|_{L^2}^2 \| \varphi \omega \|_{L^2}^2
+C\| D^2 V \|_{L^{\infty}(B_2)}^2 +\frac{1}{8} \| \Delta u \|_{L^2(B_2)}^2 +C\| h \|_{L^2(B_2)}^2.
\]
$I_1$ and $I_2$ can be easily estimated as follows :
\[
I_1 \leq C \int_{B_2} |v|^2 \, |\nabla \times u -\nabla \varphi \times v +\nabla \times V|\, |\nabla^2 \varphi| dx
\]
\[
\leq C\|v \|_{L^3(B_2\setminus B_1)}^2 \| \nabla u \|_{L^2(B_2)}^{\frac12}\| \Delta u \|_{L^2(B_2)}^{\frac12}+ C (\| v \|_{L^3(B_2\setminus B_1)}^3+1)
\]
\[
\leq C\| v\|_{L^3(B_2\setminus B_1)}^2 \| \nabla u \|_{L^2(B_2)}^2 + \frac18 \| \Delta u \|_{L^2(B_2)}^2 +C (\| v \|_{L^3(B_2\setminus B_1)}^3+1),
\]
and
\[
I_2 \leq C \int_{B_2} |v| \, |\nabla \times u -\nabla \varphi \times v +\nabla \times V|\, |\nabla \varphi|\, |\nabla v| dx
\]
\[
\leq C(\| v\|_{L^3(B_2\setminus B_1)}\| \nabla u\|_{L^6(B_2)}+\| v\|_{L^4(B_2\setminus B_1)}^2 +\| v\|_{L^2(B_2\setminus B_1)}) \| \nabla v \|_{L^2(B_2)}
\]
\[
\leq \frac18 \| \Delta u \|_{L^2(B_2)}^2 +C(\| v\|_{L^3(B_2\setminus B_1)}^2+1) \| \nabla v \|_{L^2(B_2)}^2.
\]
Here, we note that
\[
\| v\|_{L^3(B_2\setminus B_1)} \leq C,
\]
for some constant $C$ and all $t \in [t_0-r_2^2, t_0]$ due to the choice of $r_1$ and $r_2$ in Lemma \ref{lem1}.\\
Let us set $\kappa := \left\{\left( v \times \frac{\omega}{|\omega|} \right)\cdot \frac{\nabla \times \omega}{|\nabla \times \omega|}\right\}_{+}$. Then $I_3$ can be estimated as
\[
I_3 \leq \int_{B_2} \kappa \, |\varphi \omega|\, |\varphi \nabla \times \omega|\, dx
\]
\[
\leq \int_{B_2} \kappa\, | \nabla \times u-\nabla \varphi \times v+\nabla \times V|\, |\Delta u -\nabla \varphi \times \omega-\Delta \varphi v +\Delta V|\, dx
\]
\[
\leq C\int_{B_2} \kappa\, |\nabla u |\, |\Delta u| dx +C\int_{B_2} \kappa\, |g_1 |\, |\Delta u| dx+C\int_{B_2} \kappa\, |\nabla u |\, |g_2| dx
\]
\[
:=I_3^1+I_3^2+I_3^3,
\]
where we set $g_1=\nabla \varphi \times v-\nabla \times V$ and $g_2 =\nabla \varphi \times \omega+\Delta \varphi v -\Delta V$. Since $g_1$ and $g_2$ are smooth functions supported on $(B_2 \setminus \bar{B}_1) \times (t_0-\tau, t_0]$, we estimate
\[
I_3^2 ,\, I_3^3 \leq C \| v\|_{L^2} \| g_1\|_{L^{\infty}} \| \Delta u \|_{L^2}+ C \| v\|_{L^2} \| g_2 \|_{L^{\infty}} \| \nabla u \|_{L^2} \leq C \| v\|_{L^2}^2 + C\| \nabla u \|_{L^2}^2 +\frac18 \| \Delta u \|_{L^2}^2.
\]
We first assume the condition of Theorem \ref{thm1} holds true. In this case, we estimate
\begin{equation}
I_3^1 \leq C \| \kappa \|_{L^3(B_2)} \| \nabla u \|_{L^6 (B_2)} \| \Delta u \|_{L^2(B_2)} \leq C_1 \epsilon_0 \| \Delta u \|_{L^2(B_2)}^2
\end{equation}
Combining all the estimates $I_1$, $I_2$, $I_3^1$, $I_3^2$ and $I_3^3$, we have
\begin{eqnarray}
\frac{d}{dt} \| \nabla u \|_{L^2(B_2)}^2 + \| \Delta u \|_{L^2(B_2)}^2 & \leq& 2C_1 \epsilon_0 \|\Delta u \|_{L^2(B_2)}^2\nonumber\\
&&+ C(\| \nabla u \|_{L^2(B_2)}^2+\| \nabla v \|_{L^2(B_2)}^2+ \| h \|_{L^2}^2+1) \label{CL-21}
\end{eqnarray}
for $t \in (t_0-r_2^2, t_0]$, and an absolute constant $C_1$. If $C_1 \epsilon_0 <\frac12$, then integrating \eqref{CL-21} in time over $[t_0-r_2^2, t_0]$, we can obtain $\nabla u \in L^{2, \infty}_{x,t}(Q_{z_0, r_2})$, and therefore $\nabla v \in L^{2, \infty}_{x,t}(Q_{z_0, r_1})$. Applying Corollary 2.1 in \cite{Chae1}, we conclude that $z_0$ is a regular point.\\
Next, we assume that the condition (ii) of Theorem \ref{thm1} holds true, and estimate
\begin{eqnarray}
I_3^1 &\leq& C \| \kappa \|_{L^{\gamma}(B_2)} \| \nabla u \|_{L^{\frac{2\gamma}{\gamma-2}}} \| \Delta u \|_{L^2} \nonumber\\
&\leq & C \| \kappa \|_{L^{\gamma}(B_2)} \| \nabla u \|_{L^2}^{\frac{\gamma-3}{\gamma}} \| \Delta u \|_{L^2}^{\frac{\gamma+3}{\gamma}} \nonumber \\
& \leq & C \| \kappa \|_{L^{\gamma}(B_2)}^{\frac{2\gamma}{\gamma-3}} \| \nabla u \|_{L^2}^2 +\frac18 \| \Delta u \|_{L^2}^2,
\end{eqnarray}
where we used the interpolation inequality,
\[
\| \nabla u \|_{L^{\frac{2\gamma}{\gamma-2}}} \leq C \| \nabla u \|_{L^2}^{1-\frac{3}{\gamma}} \| \Delta u \|_{L^2}^{\frac{3}{\gamma}},
\]
for $ \gamma \in (3, \infty]$. Since $\kappa \in L^{\gamma, \alpha}_{x,t}(Q_{z_0,r_2})$ with $3/\gamma+2/\alpha \leq 1$ and $\gamma>3$, we have
\[
\| \kappa \|_{L^{\gamma, \frac{2\gamma}{\gamma-3}}_{x,t}(Q_{z_0, r_2})}^{\frac{2\gamma}{\gamma-3}} \leq \| \kappa \|_{L^{\gamma, \alpha}_{x,t}(Q_{z_0, r_2})}^{\frac{2\gamma}{\gamma-3}} r_2^{\frac{2\gamma}{\gamma-3}(1-\frac{3}{\gamma}-\frac{2}{\alpha})}<\infty.
\]
Similarly to the previous case, we conclude that $z_0$ is a regular point for $v$ by Gronwall's inequality.\\
Let us set $\eta := \left\{\left(\frac{v}{|v|} \times \omega \right)\cdot \frac{\nabla \times \omega}{|\nabla \times \omega|}\right\}_{+}$. Then $I_3$ can be estimated as
\[
I_3 \leq \int_{B_2} \eta \, |\varphi v|\, |\varphi \nabla \times \omega|\, dx
\]
\[
\leq \int_{B_2} \eta\, | u+ V|\, |\Delta u -\nabla \varphi \times \omega-\Delta \varphi v +\Delta V|\, dx
\]
\[
\leq C\int_{B_2} \eta\, | u |\, |\Delta u| dx +C\int_{B_2} \eta \, |V |\, |\Delta u| dx+C\int_{B_2} \eta \, | u |\, |g_2| dx
\]
\[
:=J_3^1+J_3^2+J_3^3,
\]
where we set $g_2 =\nabla \varphi \times \omega+\Delta \varphi v -\Delta V$. Since $V$ and $g_2$ are smooth functions supported on $(B_2 \setminus \bar{B}_1) \times (t_0-\tau, t_0]$, we estimate
\[
J_3^2 + J_3^3 \leq C \| \nabla v\|_{L^2}^2 + C\| u \|_{L^2}^2+\frac18 \| \Delta u \|_{L^2(B_2)}^2.
\]
Now we assume (iii) of Theorem \ref{thm1} holds true, then we estimate $J_3^1$ as
\begin{eqnarray*}
J_3^1 &\leq& C \|\eta \|_{L^{\gamma}(B_2)} \| u \|_{L^{\frac{2\gamma}{\gamma-2}}} \| \Delta u \|_{L^2}
\\
&\leq& \left\{ \begin{array}{ll} C\|\eta \|_{L^{\gamma}(B_2)} \| u \|_{L^6}^{\frac{2\gamma-3}{\gamma} } \| \Delta u \|_{L^2}^{\frac{3}{\gamma}} \qquad &\mbox{if } 2 \leq \gamma \leq 3\\ C\|\eta \|_{L^{\gamma}(B_2)} \| u \|_{L^2}^{\frac{\gamma-3}{\gamma} }\| \nabla u \|_{L^2}^{\frac{3}{\gamma} } \| \Delta u \|_{L^2} \qquad &\mbox{if } \gamma >3 \end{array}\right.\\
&\leq& \left\{ \begin{array}{ll} C\|\eta \|_{L^{\gamma}(B_2)}^{\frac{2\gamma}{2\gamma-3}} \|\nabla u \|_{L^2}^2+\frac18 \| \Delta u \|_{L^2}^{2} \qquad &\mbox{if } 2 \leq \gamma \leq 3\\ C\|\eta \|_{L^{\gamma}(B_2)}^2 \| \nabla u \|_{L^2}^{\frac{6}{\gamma} } +\frac18\| \Delta u \|_{L^2}^2 \qquad &\mbox{if } \gamma >3 \end{array}\right.
\end{eqnarray*}
Since $\eta \in L^{\gamma, \alpha}_{x,t}(Q_{z_0,r_2})$ with $3/\gamma+2/\alpha \leq 2$, $\gamma\geq 2$ and $\alpha \geq 2$, we have
\[
\| \eta \|_{L^{\gamma, \frac{2\gamma}{2\gamma-3}}_{x,t}(Q_{z_0, r_2})}^{\frac{2\gamma}{2\gamma-3}} \leq \| \eta \|_{L^{\gamma, \alpha}_{x,t}(Q_{z_0, r_2})}^{\frac{2\gamma}{2\gamma-3}} r_2^{\frac{2\gamma}{2\gamma-3}(2-\frac{3}{\gamma}-\frac{2}{\alpha})}<\infty,
\]
and
\[
\| \eta \|_{L^{\gamma, 2}_{x,t}(Q_{z_0, r_2})}^2 \leq \| \eta \|_{L^{\gamma, \alpha}_{x,t}(Q_{z_0, r_2})}^2r_2^{\frac{2(\alpha-2)}{\alpha}}< \infty.
\]
Similarly to the previous case, we conclude that $z_0$ is a regular point for $v$ by Gronwall's inequality.\\
Next, we suppose that $t_0$ is a singular time which is not an epoch of possible irregularity. Then there exists a time $t^{*}$ in $(t_0-r^2, t_0)$ and $0<\tilde{r}_1<\tilde{r}_2<r$ such that $v$ is regular on $B_{x_0, \tilde{r}_2} \setminus B_{x_0, \tilde{r}_1} \times [t^{*}, t_0]$. Assume that $v$ is not regular on $B_{x_0, \tilde{r}_1} \times [t^{*}, t_0]$, then there exists $s \in (t^{*}, t_0]$ such that the suitable weak solution is regular on $B_{x_0, \tilde{r}_1} \times [t^{*},s)$ and singularity occurs at $(y,s) \in B_{x_0, \tilde{r}_1}\times \{s\}$. Then we take a local neighborhood of $(y,s)$ contained in $B_{x_0, \tilde{r}_2} \times [t^{*},s).$ Hence we can show $(y,s)$ is a regular point by the repetition of the above argument as in the case of the epoch of possible irregularity. It gives a contradiction to the assumption that $(y,s)$ is a singular point and hence $v$ is regular on $B_{x_0, \tilde{r}_1} \times [t^{*}, t_0]$. This completes the proof.
\end{pfthm1}
\section*{Acknowledgments}
This work was partially supported by NRF grants no. 2016R1A2B3011647.
|
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Hans Karl August Simon von Euler-Chelpin (ur. 15 lutego 1873 w Augsburgu, zm. 6 listopada 1964 w Sztokholmie) – szwedzki chemik organik, biochemik, pochodzenia niemieckiego, laureat Nagrody Nobla z chemii w 1929 roku.
Był synem oficera wojsk Królestwa Bawarii, większość dzieciństwa spędził w Wasserburgu am Inn. Uczęszczał do szkół w Monachium, Würzburgu i Ulm, następnie, w latach 1891–1893 studiował malarstwo w Akademii Sztuk Pięknych w Monachium pod kierunkiem Ludwiga Schmida-Reutte oraz Franza von Lenbacha. Chęć głębszego zrozumienia kolorów skłoniła go do podjęcia studiów naukowych.
W latach 1893–1895 studiował chemię na Uniwersytecie Humboldtów w Berlinie. Jego wykładowcą chemii był m.in. Hermann Emil Fischer, a fizyki Emil Warburg i Max Planck. W latach 1896–1897 pracował pod kierownictwem Walthera Nernsta na Uniwersytecie w Getyndze, a w 1897 roku został asystentem Svante Arrheniusa w Królewskim Instytucie Technicznym w Sztokholmie. W 1900 roku przeniósł się na Uniwersytet w Sztokholmie. W latach 1906–1941 był tam profesorem chemii ogólnej i organicznej, a od 1929 roku kierował nowo utworzonym Instytutem Biochemii przy tej uczelni.
Prowadził badania nad enzymami, witaminami i fermentacją cukrów. Wyjaśnił udział enzymów w fermentacji cukrów, za co został uhonorowany Nagrodą Nobla w 1929 roku, wspólnie z uczonym brytyjskim sir Arthurem Hardenem.
Jego syn Ulf von Euler (1905-1983) otrzymał medyczną Nagrodę Nobla w 1970 roku.
Przypisy
Bibliografia
Absolwenci Uniwersytetu Humboldtów w Berlinie
Ludzie urodzeni w Augsburgu
Ludzie związani ze Sztokholmem
Nobliści – chemia
Szwedzcy chemicy
Urodzeni w 1873
Wykładowcy uczelni w Szwecji
Zmarli w 1964
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dystopia, feminist story, kim liggett, review, the grace year, YA
Review: The Grace Year by Kim Liggett
07:00 Jo No comments
Published: 10th October 2019 | Publisher: Del Rey | Cover Designer: | Source: Publisher via NetGalley
Kim Liggett's Website
No one speaks of the grace year.
It's forbidden.
We're told we have the power to lure grown men from their beds, make boys lose their minds, and drive the wives mad with jealousy. That's why we're banished for our sixteenth year, to release our magic into the wild before we're allowed to return to civilization.
But I don't feel powerful.
I don't feel magical.
Tierney James lives in an isolated village where girls are banished at sixteen to the northern forest to brave the wilderness - and each other - for a year. They must rid themselves of their dangerous magic before returning purified and ready to marry - if they're lucky.
It is forbidden to speak of the grace year, but even so every girl knows that the coming year will change them - if they survive it...
The Grace Year is The Handmaid's Tale meets Lord of the Flies - a page-turning feminist dystopia about a young woman trapped in an oppressive society, fighting to take control of her own life. From Goodreads.
Book Depository | Wordery | Goodreads
I received this eProof for free from Del Rey via NetGalley for the purposes of providing an honest review.
Rep: Side lesbian character.
Trigger/Content Warnings: This book features pregnancy, birth, bullying, death, ogling, discussion of sex, fade-to-black sex scene, sex shaming language, predators, sexual assault, forced prostitution, rape culture, violence against women, execution by hanging, forced watching of execution, discussion of skinning girls alive, an axe wound, suicide, cannibalism, transphobia: girl grabbed by the crotch to "check" she's not "really" a boy, homophobia: same sex attraction leads to violent punishment, acting on same sex attraction is punishable by death, and purposeful public outing of lesbian character.
View/Hide Trigger & Content Warnings
Marketed as The Handmaid's Tale meets Lord of the Flies, The Grace Year by Kim Liggett sounded absolutely right up my street! And while it has an incredible premise, and started off really strong, I ultimately found it quite disappointing and forgetable.
Tierney lives in this incredibly patriachal society, where men oppress women for the sake of control and because of fear. I woman is to be seen and not heard, be a wife, and produce babies - preferably sons. During their sixteenth year, all girls are sent off to an isolated island, and locked inside an infenced encampment, for their grace year. They spend their whole year there to rid themselves of their magic. Because all girls have magic: it makes the men crazy with lust and makes wives jealous. The day before the start of their grace year, the women are all lined up, after all the men have discussed and decided which girls will become the wives the eligible men. There are always more grace year girls than there are eligible men. The men give veils to their chosen intended, and that's that. The women have no say. They're then sent off on their grace year with the understanding that when they come back - if they come back, because never does everyone make it home after their grace year - those with veins will become wives, those without will become indentured labourers. This is how it's always been. Once their in their encampment, they are left to their own devices for a full year, to rid themselves of magic, and survive. But with the girls slowly starting having mental issues, and with the poachers - the sons of the women who have been forced into prostitution in the Outskirts as punishment, or sisters sent their if their older sister does not return from their grace year, dead or alive - circling outside the fence, waiting for anyone to leave, so they can skin them alive, (and sell their body parts back to the town, who will consume them to bring back their youth and vitality), 33 girls quickly becomes fewer and fewer.
As I said, The Grace Year started really strong. The town the women live in is horrifying. Women literally have no rights at all. Everything they have is given to them by the men, and the men can just as easily take it away again. They have all the power, and they'll use it. Sick of your wife who isn't getting pregnant? Say you've seen her using magic, and she'll be executed. Then you can find yourself another from this year's grace year girls. It's disgusting, and just so, so awful. I was raging, and I loved it, certain this was going to be a book where, as it's a standalone, we'd see women take back their power. What I read, during the grace year, was women turning against women, because why not take some power when you finally have some freedom?
But also, I kind of just expected more. There was no background for the world building. Why do girls have magic? What actually happens if they don't release it? When was the grace year first thought up? When was it decided that the way the people live now is in the "best interests" for everyone? Why is this world the way it is? I have absolutely no idea, because we're not told anything. And was just really slow. Sometimes it would feel like things were going somewhere, but then you'd discover that actually, months had gone by. I get it, they're spending a whole year there, and there will be times when nothing much of note is going to happen, but to skip months in a few lines? And then having months go by where we see not a huge deal happening at all, really? And when things did happen, they never quite made it to the level of horrifying I was expecting. There was this sense, throughout, that things could get really bad, and at times they do, but it's softened. A lot of it we don't actually see - things are always happening where Tierney is not, and maybe Liggett didn't want the readers having to read such terrible things, or she simply didn't want to have to write them, but I mean we see a an execution at the beginning, and there are other seriously awful - and probably very triggering things, especially when it comes to the trans/homophobia - she does write, I just didn't understand why we didn't see it all. Don't give us this horrific world, and then shy away from the realities of it. I'm not satying I want to see terrible things happen to the girls, but these things aren't just terrible, they also hugely affect the plot, and we just don't see them.
And the way things ended was just so disappointing. I was thinking, "Really, this is the way the story is going to go? This is the path Tierney is taking? After everything? Really?" I got the importance of the very end, but this is a standalone story. Nothing is coming after. And this is what we get? It was such a let down. I just couldn't believe this was the whole story. For me, it just wasn't enough. There was no real closure for me. I just didn't see the point of the story, of reading it, if that was the ending I was getting. And it's also really forgetable; at the time of writing this, I actually forgot what I read. I knew I had a review to write, but couldn't for the life of me remember for what book. I had to look it up. It was that ineffectual for me.
So yeah, this story just wasn't for me, sadly. But a lot of people have really loved it, so do read some other reviews before deciding whether or not to read it.
Thank you to Del Rey via NetGalley for the eProof.
What do you expect from a feminist dystopia? What are some of your favourite feminist dystopias? Will you be picking up The Grace Year? Let me know in the comments!
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\section{\label{intro}Introduction}
Progress in solving string theory in various backgrounds can be
done by considering sectors of the theory that decouple from
the rest of the degrees of freedom in a suitable limit. Such
decoupled sectors are characterized by having an altogether
different asymptotic symmetry compared to that of the parent
string theory. A well known example to such a truncation is
the BMN \cite{Berenstein:2002jq} sector of string theory in
AdS$_5\times S^5$. Once a consistent sector is found, a
complete worldsheet theory with the appropriate symmetries can
be written down without further reference of the parent theory.
Non-relativistic string theory \cite{jGom_Oog} (see also
\cite{DanGuiKru}) in flat space is another consistent
sector of string theory, whose world-sheet conformal field
theory description has the appropriate Galilean symmetry
\cite{Brugues:2004an}. Non-relativistic superstrings and
non-relativistic superbranes \cite{JGom_Kam_Town,Garcia:2002fa}
are obtained as a certain decoupling limit of the full relativistic
theory. The basic idea behind the decoupling limit is to take a
particular non-relativistic limit in such a way that the light
states satisfy a Galilean-invariant dispersion relation, while the
rest decouple. For the case of strings, this can be accomplished by
considering wound strings in the presence of a background $B$-field
and tuning the $B$-field so that the energy coming from the
$B$-field cancels the tension of the string. In flat space,
once kappa symmetry and diffeomorphism invariance are fixed,
non-relativistic strings are described by a free field theory
in flat space. In AdS$_5\times S^5$ \cite{GGK_AdS5XS5}, the
world-sheet theory reduces to a supersymmetric free field theory
in AdS$_2$.
It is an interesting question whether similar non-relativistic
string actions can be constructed in an expanding spacetime
and if so, whether non-relativistic strings could play a
cosmological role in the form of cosmic strings. The study
of cosmic strings\footnote{The dynamics of cosmic strings can be
described by considering perturbations around a static solitonic
string solution. Keeping all orders in the perturbations results
in a relativistic effective string action, while keeping only up
to quadratic order gives rise to the non-relativistic string action
we will consider in this paper \cite{JGom_Kam_Town}.} has been
catalysed in the past few years mainly due to theoretical
motivations, in particular
the realisation that they are generic in Supersymmetric Grand
Unified Theory (SUSY GUT) models \cite{Jeannerot} and brane
inflation \cite{BMNQRZ,SarTye}. The latter possibility is of
particular significance as it provides a potential observational
window to superstring physics \cite{PolchStab,PolchIntro}. Further,
the fact that the Planck satellite and laser interferometers such
as LISA and LIGO may be able to probe a significant part of cosmic
string tensions relevant to these models \cite{DamVil}, opens
the possibility of detecting cosmic strings in the foreseeable
future.
One can think of situations in which ordinary cosmic strings
could behave non-relativistically. Network simulations in
a matter or radiation dominated universe \cite{All_Shell}
suggest that, at late times, string segments move relatively
slowly and coherently on the largest scales, but also show
evidence that small-scale-structure \cite{Mart_Shell_sss,Polch_Rocha}
which is largely responsible for damping energy from the network
through the formation of minuscule loops, remains relativistic
as Hubble damping is inefficient at scales much smaller than the
horizon \cite{book}. However, the situation is different for
strings in de Sitter spacetime, where Hubble damping can be very
efficient rendering the strings essentially non-relativistic.
This may be relevant for late time cosmology as observations
\cite{Perlmutter,WMAP3} suggest that the universe is already
entering a de Sitter phase. Further, non-relativistic string
networks have been considered as Solid Dark Matter (SDM)
\cite{BuchSper,BatCarChMo} and more recently \cite{Alexand}
as an alternative explanation of galactic rotation curves.
It would thus be desirable to have an effective diffeomorphism
invariant action\footnote{Note that Ref.~\cite{BuchSper} considers
an action applicable to a `continuous medium' with internal
structure, which is invariant under limited reparametrisations
preserving the worldlines of the constituent particles. Here
we consider a \emph{diffeomorphism invariant} non-relativistic
action.} describing the dynamics of non-relativistic strings
in a cosmological context. On the other hand, the fact that one
can construct a consistent worldsheet theory of non-relativistic
strings at quantum level (in flat space) also motivates the
study of \emph{fundamental} non-relativistic strings in an
expanding spacetime. In this paper, we point out that a
non-relativistic diffeomorphism invariant action can be obtained
in the case of a Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW)
spacetime as a limit of the relativistic Nambu-Goto action, and
study the dynamics and cosmological evolution of non-relativistic
strings.
The structure of the paper is as follows. In section
\ref{rel_string} we review some basic results about the
Nambu-Goto action and the dynamics of relativistic strings
in an expanding universe, which will be useful for comparison
to the non-relativistic case. In section \ref{non_rel_lim} we
obtain a non-relativistic diffeomorphism invariant worldsheet
action by taking a particular limit of the Nambu-Goto action
in expanding spacetime. We move on in section \ref{NR_dynamics}
to study non-relativistic string dynamics as described by this
action. The physical interpretation of non-relativistic strings
as well as their possible coupling to cosmology-in particular
the effective equation of state of an `ideal gas' of
non-interacting, non-relativistic strings-are discussed in
section \ref{physical}. The effect of string interactions is
left for section \ref{VOS}, where macroscopic models for the
cosmological evolution of both relativistic and non-relativistic
string networks are discussed. In section \ref{RelVnonRel} we
solve numerically the non-relativistic network model for a wide
range of parameters and compare the results to those of the
relativistic model. In section \ref{discuss} we discuss possible
applications of non-relativistic strings in condensed matter and
cosmological contexts. Finally, we have three appendices which
describe an alternative derivation of our non-relativistic string
action as a semiclassical expansion \cite{semiclass} around the
vacuum solution (appendix \ref{semicl}), the Hamiltonian formulation
of relativistic and non-relativistic string dynamics (appendix
\ref{hamiltonian}) and the construction of a spacetime energy-momentum
tensor for the non-relativistic string (appendix \ref{NR_EM_tensor}).
\section{\label{rel_string} Relativistic String in Expanding Spacetime}\setcounter{equation}{0}
Let us first consider a string moving in a $D+1$ dimensional
spacetime with metric $G_{MN}\, \,(M,N=0,1,2,\ldots,D)$. Its
world history is described by a two-dimensional spacetime surface,
the string worldsheet $x^M=x^M(\sigma^i)$, $i=0,1$.
The dynamics is governed by the Nambu-Goto action
\begin{equation}\label{nambu}
S_{NG}=-T_R \! \int \! \sqrt{-\gamma}\, d^2\sigma \, ,
\end{equation}
where $T_R$ is the string tension and $\gamma$ is the determinant
of the pullback of the background metric on the worldsheet,
$\gamma_{ij}=G_{MN}(x)\partial_i x^M \partial_j x^N$.
The equations of motion for the fields $x^M$ obtained from this
action are given by
\begin{equation}\label{eom}
\nabla^2 x^M + \Gamma^M_{N\Lambda} \gamma^{ij}
\partial_i x^N \partial_j x^\Lambda = 0 \, ,
\end{equation}
where $\Gamma^M_{N\Lambda}$ is the ($D+1$)-dimensional Christoffel
symbol and $\nabla^2 x^M$ the covariant Laplacian of the worldsheet
fields $x^M$.
By varying the action with respect to the background metric $G_{MN}$
we obtain a spacetime energy-momentum tensor
\begin{equation}\label{emt}
T^{MN}(y^\Lambda)=\frac{1}{\sqrt{-G}}\,T_R \! \int \! d^2\sigma
\sqrt{-\gamma} \gamma^{ij}\partial_i x^M
\partial_j x^N \, \delta^{(D+1)}(y^\Lambda-x^\Lambda(\sigma^i))\,.
\end{equation}
The rigid symmetries of (\ref{nambu}) are given by the Killing
vectors of $G_{MN}$. The Nambu-Goto action is also invariant
under 2D diffeomosrphisms of the worldsheet coordinates $\sigma^i$.
One can use this freedom to fix the gauge by imposing two conditions
on $x^M(\sigma^i)$.
Now consider string propagation in an expanding Universe
described by a flat FLRW metric
\begin{equation}\label{FLRW_metric}
G_{MN} = a(x^0)^2 \eta_{MN}
\end{equation}
in conformal time $x^0\!\equiv\!\eta$.
A convenient gauge choice in this case is the \emph{transverse temporal
gauge} given by:
\begin{equation}\label{trans_temp}
\left\{
\begin{array}{l}
\dot x x^\prime = 0 \\
\tau=x^0
\end{array}
\right.
\end{equation}
The equations of motion (\ref{eom}) become~\cite{Tur_Bhatt}:
\begin{eqnarray}
\left\{
\begin{array}{l}
\dot\epsilon=-2\frac{\dot a}{a} \epsilon \dot{\bf x}^2
\label{eom_eps} \\
\ddot {\bf x} + 2\frac{\dot a}{a} \left( 1 - \dot{\bf x}^2 \right)
\dot {\bf x} = {\left(\frac{{\bf x}^{\prime}}{\epsilon}\right)}^{\prime}
\epsilon^{-1} \label{eom_x}
\end{array}
\right.
\end{eqnarray}
where
\begin{equation}\label{epsilon}
\epsilon = \frac{ {-x^\prime}^2 }{ \sqrt{-\gamma} } = \left(
\frac{ {{\bf x}^\prime }^2 } { 1-\dot{\bf x}^2 } \right)^{1/2}\, .
\end{equation}
The variable $\epsilon$ is related to the canonical momentum
associated to the field $x^0(\tau)$. Indeed, in the transverse
gauge $\gamma_{01}\!\equiv\!\gamma_{\tau\sigma}\!=\!0$, we have:
\begin{equation}\label{p_0_Lagr}
p_0=-\frac{T_R}{2} \sqrt{-\gamma} \gamma^{ij} \frac{\partial \gamma_{ij}}
{\partial_\tau x^0}= - T_R a(x^0)^2 \epsilon \dot x^0 \, ,
\end{equation}
which, after imposing the temporal gauge condition $\dot x^0=1$,
becomes $-T_R a(x^0)^2 \epsilon$.
In the transverse temporal gauge, the energy-momentum tensor (\ref{emt})
of a relativistic string in a FLRW background is \cite{book}:
\begin{equation}\label{emt_trans_temp}
T^{MN}(\eta,y^I)=\frac{1}{a(\eta)^{D+1}}\,T_R \! \int \! d\sigma
\left( \epsilon \dot x^M \dot x^N -\epsilon^{-1} x^{\prime M}
x^{\prime N} \right) \, \delta^{(D)}(y^I-x^I(\sigma,\eta))\, ,
\end{equation}
where, having integrated out $\delta(\eta-x^0(\tau))$, $I$ runs from
$1$ to $D$.
To construct the string energy one projects $T^{MN}$ on a spatial
hypersurface $\eta\!=\!{\rm const}$, with induced metric $h$ and normal
covectors $n_M\!=\!\left(a(\eta),{\bf 0}\right)$, integrating over the
$D$ spatial coordinates:
\begin{eqnarray}\label{energy}
E(\eta) &=& - \! \int \! \sqrt{h} n_M n_N T^{MN} d^D{\bf y} \nonumber \\
&=& \! \int \! \sqrt{h} T^0_{\; 0} d^D{\bf y} \, .
\end{eqnarray}
Thus, due to the foliation, the energy can be constructed from the
$00$ component of the energy momentum tensor. Since, $\sqrt{h}=
a(\eta)^D$, equation (\ref{energy}) becomes:
\begin{equation}\label{string_energy}
E(\eta) = \! \int \! T^0_{\;0} a(\eta)^D d^D{\bf y}
= a(\eta) T_R \! \int \! \epsilon \, d\sigma \, .
\end{equation}
Therefore, the energy of the relativistic string is simply the
tension times the {\it physical} string length, taking into account
relativistic length contraction.
\section{\label{non_rel_lim}Non-Relativistic Limit of Nambu-Goto
Action in FLRW}\setcounter{equation}{0}
Now consider a string, charged under a background antisymmetric
2-tensor field $B_{MN}$, propagating in $D+1$ FLRW spacetime.
The string couples to $B$ through a topological Wess-Zumino term,
so that the total action reads:
\begin{equation}\label{total_action}
S=S_{NG}+S_{WZ}=-T_R \! \int \! \sqrt{-\gamma}\, d^2\sigma
+ q \! \int \! B^* \, ,
\end{equation}
where $q$ is the string charge and $B^*$ the pullback of $B$
on the worldsheet. We consider a relativistic string aligned in
the $x^0$, $x^1$ directions, its transverse coordinates being
$X^a$ with $a=2,3,...D$.
The non-relativistic limit \cite{jGom_Oog, JGom_Kam_Town, DanGuiKru}
of this string consists of rescaling the longitudinal coordinates
\begin{equation}\label{rescale}
x^\mu \rightarrow \omega x^\mu\,, \quad \mu=0,1
\end{equation}
and taking the limit $\omega \rightarrow \infty$. This yields a
divergent term, coming from $S_{NG}$, which (in some geometries)
can be canceled by an appropriate choice of a closed $B_{MN}$.
If we assume that the string is wrapped on a spatial circle
\begin{equation}\label{circle}
x^1 \sim x^1 + 2\pi R
\end{equation}
then the chosen $B_{01}$ cannot be set to zero by a gauge
transformation.
The above procedure generally works in flat spacetime (in fact
one only needs the longitudinal part of the metric be flat
\cite{JGom_Kam_Town}), but for a curved background it is not
guaranteed that there is a choice of closed $B$ which cancels
the diverging piece of the action. Non-relativistic
superstring actions have been obtained in the case of
AdS$_5\times S^5$ \cite{GGK_AdS5XS5}.
We will now see that the non-relativistic limit can also be taken
in the case of a FLRW background. We write the Lagrangian density
of the Nambu-Goto piece as:
\begin{eqnarray}
{\cal{L}}_{NG} &=& -T_R \sqrt{-{\rm det}[g_{ij}+G_{ab}(\eta)\partial_i X^a
\partial_j X^b]} \nonumber \\
&=&-T_R \sqrt{-{\rm det}g_{ij}}\sqrt{{\rm det}[\delta^i_j+g^{ik}
G_{ab}(\eta)\partial_k X^a \partial_j X^b]} \label{L_NG} \, ,
\end{eqnarray}
where $g_{ij}=G_{\mu\nu}\partial_i x^\mu \partial_j x^\nu$ and $G_{ab}(\eta)=
a(\eta)^2 \delta_{ab}$.
Then, assuming a power law expansion $a(\eta)\!=\!\eta^{\alpha/2}$
(for example $\alpha=2$ resp. $4$ in radiation resp. matter dominated
era) we obtain the non-relativistic limit of $S_{NG}$ by the rescaling
(\ref{rescale}), which implies
\begin{equation}
a(\eta) \rightarrow \omega^{\alpha/2} a(\eta)\label{a_rescale} \, .
\end{equation}
Expanding the Lagrangian density in powers of the parameter $\omega$
we then obtain:
\begin{equation}\label{L_NG_expand}
{\cal{L}}_{NG} = -T_R \omega^\alpha \left\{ \omega^2 \sqrt{-{\rm det}g}
+ \frac{1}{2}\sqrt{-{\rm det}g} g^{ij}G_{ab}(\eta)\partial_i X^a
\partial_j X^b + {\cal O}\left(\frac{1}{\omega^2}\right)\right\} \, .
\end{equation}
We can then rescale the string tension by
\begin{equation}\label{T_rescale}
T_R\, \omega^\alpha \rightarrow T_0
\end{equation}
and take the limit $\omega \rightarrow \infty$, yielding a finite
and a divergent piece:
\begin{eqnarray}
{\cal L}_{\rm reg}&=&-\frac{T_0}{2} \sqrt{-{\rm det}g} g^{ij}
G_{ab}(\eta)\partial_i X^a \partial_j X^b \label{L_reg} \\
{\cal L}_{\rm div}&=&-T_0\omega^2\sqrt{-{\rm det}g} =-T_0\omega^2
a(\eta)^2 \sqrt{-{\rm det}(\eta_{\mu\nu}\partial_i x^\mu
\partial_j x^\nu)} \label{L_div} \, .
\end{eqnarray}
The divergent piece can be canceled by choosing an appropriate closed
$B_{\mu\nu}$. Indeed if we choose\footnote{The chosen $B_{\mu\nu}$
is closed. Working in zweibeins $e^\mu$ we have: ${\rm d}B =
\frac{1}{2}{\rm d}[a(x^0)^2\epsilon_{\mu\nu}e^\mu \wedge
e^\nu] = {\rm d} [ a(x^0)^2 e^0 \wedge e^1 ] = 2a \dot a \, e^0
\wedge e^0 \wedge e^1 + a^2 ({\rm d}e^0 \wedge e^1 + e^0 \wedge
{\rm d}e^1 ) = a^2 (-w^{01} \wedge e^1 \wedge e^1 ) - a^2 e^0
\wedge w^{10} \wedge e^0 = 0$ where we have used ${\rm d}e +
w\wedge e=0$, Cartan's structure equation with zero torsion.}
$B_{\mu\nu}=a(\eta)^2 \epsilon_{\mu\nu}$ the Wess-Zumino part
of the Lagrangian becomes:
\begin{equation}\label{L_WZ}
\frac{1}{2} \omega^2 (q\omega^\alpha) a(\eta)^2 \epsilon_{\mu\nu}
\epsilon^{ij} \partial_i x^\mu \partial_j x^\nu .
\end{equation}
This term precisely cancels the divergent piece (\ref{L_div}) if one
tunes the rescaled charge $(q\omega^\alpha)$ with the string tension
$T_0$. We are thus left with the Non-Relativistic string action:
\begin{equation}\label{S_NR}
S_{NR}=-\frac{T_0}{2} \! \int \! \sqrt{-{\rm det}g}
g^{ij}G_{ab}(\eta)\partial_i X^a \partial_j X^b d^2\sigma \, .
\end{equation}
This action can also be derived by a `semiclassical approximation'
\cite{semiclass}
from the classical solution:
\begin{equation}
x_0^M = \left\{
\begin{array}{cll}
\tau &,& M=0 \\
\lambda\sigma &,& M=1 \\
0 &,& M=a\in (2,\ldots,D)
\end{array}
\right.
\end{equation}
(see Appendix~\ref{semicl} for details).
\section{\label{NR_dynamics}Non-Relativistic String Dynamics}\setcounter{equation}{0}
The action (\ref{S_NR}) is characterised by 2D diffeomorphism invariance
with respect to the worldsheet coordinates $\sigma^i$ and global Galilei
invariance (modulo time translations due to time dependence of the
metric) with respect to the transverse spacetime coordinates $X^a$.
The canonical variables satisfy two primary constraints:
\begin{eqnarray}
&& p_\mu \epsilon^{\mu\rho} \eta_{\rho\nu} x^{\prime\nu} + \frac{1}{2}
\left(\frac{P_aP_b}{T_0}G^{ab}(x^0) + T_0 X^{\prime a}X^{\prime b}
G_{ab}(x^0) \right) = 0 \label{energy_constr} \\
&& p_\mu x^{\prime\mu} + P_a X^{\prime a} = 0 \, , \label{pxp_constr}
\end{eqnarray}
where $p_\mu$, $P_a$ are the canonical momenta corresponding to $x^\mu$
and $X^a$.
Varying the action with respect to the transverse and longitudinal fields
$X^a$ and $x^\mu$ respectively, one obtains the equations of motion:
\begin{eqnarray}
\begin{array}{l}
\partial_i\left(\sqrt{-g}g^{ij}\partial_j X^a\right)+\sqrt{-g}g^{ij}
\Gamma^a_{bc} \partial_i X^b \partial_j X^c + \sqrt{-g}g^{ij}
\partial_i X^b \partial_j x^0 \, (\partial_0 G_{bc}) \, G^{ca} = 0
\end{array}
\nonumber \\
&& \label{eom_trans} \\
\begin{array}{rl}
\partial_i\left[\sqrt{-g}\partial_kx^\nu\eta_{\mu\nu} a(x^0)^2 \left(
g^{ik}g^{mn}-2g^{im}g^{kn} \right) \right. \partial_m & \left. \!\!
X^a \, \partial_n X^b \, G_{ab}(x^0) \right] \\
& = \sqrt{-g} g^{mn} \partial_m X^a \partial_n X^b \frac{\partial G_{ab}(x^0)}
{\partial x^\mu}
\end{array}
\nonumber \\
&& \label{eom_long}
\end{eqnarray}
subject to the boundary condition (\ref{circle}).
For the metric (\ref{FLRW_metric}) the Christoffel symbols
$\Gamma^a_{bc}$ vanish and the transverse equations of motion
(\ref{eom_trans}) relate the covariant divergence of the
transverse fields $X^a$ to the time derivative of the transverse
metric $G_{ab}$. We can use the 2D reparametrisation invariance
of the action to fix the gauge. For our discussion it will be
convenient to work in the \emph{static gauge}
\begin{equation}\label{nonrel_gauge}
x^0-\tau=0 \, , \quad x^1-\lambda\sigma=0 \, ,
\end{equation}
identifying worldsheet and background times, while allowing for
multiple windings of the non-relativistic string. Indeed, defining
$\sigma\in [0,2\pi)$, the periodicity condition (\ref{nonrel_gauge})
requires that
\begin{equation}\label{lambda}
\lambda=nR \, ,
\end{equation}
where $n$ is the string winding number.
After fixing the gauge, the physical degrees of freedom of the
non-relativistic string are the transverse coordinates $X^a$
and the corresponding momenta $P_a$. The equation of motion
(\ref{eom_trans}) becomes:
\begin{equation}\label{eom_gauge}
\ddot X^a = -2\frac{\dot a}{a} \dot X^a + \lambda^{-2}
X^{\prime\prime a} \, ,
\end{equation}
which is the wave equation with a cosmological damping term
$-2\frac{\dot a}{a} \dot X^a$. This equation (for $\lambda\!=\!1$)
has been used by Vilenkin \cite{Vil_CS} to describe small
perturbations around a straight cosmic string, and was obtained
by taking the limit $\dot X^2\!\ll\! 1$, $X^{\prime 2}\!\ll\! 1$
of the relativistic equations of motion in the static gauge.
Here, there is also a winding number $\lambda$. One might be
tempted to say that, for $\dot a/a=0$, equation (\ref{eom_gauge})
implies a wave propagation velocity
\begin{equation}\label{v_0}
v_0^2=\lambda^{-2} \, .
\end{equation}
However, one should remember that the physical coordinates are not
$\sigma,\tau$ but rather $x^1\!=\!\lambda \sigma,\, x^0=\tau$, so
rewriting (\ref{eom_gauge}) in terms of the physical variables
we get (in the case $\dot a/a=0$)
\begin{equation}\label{eom_phys}
\partial_{x^0}^2 X^a = \partial_{x^1}^2 X^a \, ,
\end{equation}
which describes a wave propagating at the velocity of light. The
non-relativistic string allows the propagation of waves along the
longitudinal directions with the speed of light. However, the
transverse velocities are not restricted, in contrast to the case
of the relativistic string.
An `energy'
\begin{equation}\label{P_0}
{\cal P}_0=\frac{1}{2\lambda} \! \int \! d\sigma \left(\frac{P_aP_b}{T_0}
G^{ab}(x^0) + T_0 X^{\prime a}X^{\prime b} G_{ab}(x^0) \right)
\end{equation}
can be obtained from the constraint (\ref{energy_constr}), which
in the gauge (\ref{nonrel_gauge}) becomes:
\begin{equation}\label{P_0_gauge}
{\cal P}_0 = \frac{1}{2} \! \int \! d\sigma \left( \lambda T_0
\dot X^a \dot X^b + \lambda^{-1} T_0 X^{\prime a}X^{\prime b}
\right) G_{ab}(x^0) \, .
\end{equation}
This can be interpreted as the sum of the kinetic and potential
energies of transverse excitations along the string. The actual
string energy, obtained by integrating the projection of the
energy-momentum tensor on a constant $x^0$ hypersurface is (see
appendix \ref{NR_EM_tensor}):
\begin{equation}\label{E_nonrel}
E(x^0)=a(x^0)^{-1}\,{\cal P}_0 \, .
\end{equation}
Since $x^0$-translation is not an isometry of (\ref{FLRW_metric})
the Lagrangian is not time-translationally invariant and $p_0$ is not
conserved. In fact, its time evolution can be found from the
longitudinal equations of motion (\ref{eom_long}).
In the gauge (\ref{nonrel_gauge}) the $\mu=0$ component of
(\ref{eom_long}) becomes:
\begin{equation}\label{p_0_dot}
\frac{1}{2} \left(\dot X^a \dot X_a + \lambda^{-2} X^{\prime a}
X^{\prime}_a \right)\dot{} = \lambda^{-2}\left(X^{\prime a}
\dot X_a\right)^\prime + \frac{\dot a}{a} \left( \lambda^{-2}
X^{ \prime a} X^{\prime}_a - \dot X^a \dot X_a \right) \, .
\end{equation}
Integrating we obtain:
\begin{equation}\label{P_0_dot}
\dot{\cal P}_0 = a \dot a \lambda T_0 \!\int\! d\sigma \left(
\lambda^{-2} X^{ \prime a} X^{\prime b} - \dot X^a \dot X^b
\right)\delta_{ab} \, ,
\end{equation}
where the boundary term gives no contribution due to the
periodicity condition (\ref{circle}).
Similarly, from the constraint (\ref{pxp_constr}) we define the
momentum ${\cal P}_1$ along the string
\begin{equation}\label{P_1}
{\cal P}_1=-\frac{1}{\lambda} \! \int \! P_a X^{\prime a} d
\sigma = - T_0 \! \int \! \dot X_a X^{\prime a} d\sigma
\end{equation}
in the gauge (\ref{nonrel_gauge}). Translational invariance
then dictates that ${\cal P}_1$ is conserved, as can be easily
verified using the equations of motion.
\section{\label{physical}Physical Interpretation and Cosmology}\setcounter{equation}{0}
\subsection{The NR Particle vs NR String Limit}
The non-relativistic limit is generally understood as a low velocity
limit, which can be formally obtained by sending the speed of light
$c$ to infinity. This procedure works, at least in the case of the
point particle, although there are some conceptual issues involved
when taking limits of dimensionful constants like $c$
\cite{Duff,DufOkVen}. A safer route is to keep $c$ constant and
rescale the time coordinate by a dimensionless parameter, say $\omega$,
taking the limit $\omega\rightarrow\infty$. One can thus obtain a
reparametrisation invariant, non-relativistic action for the point
particle. The naive application of this to the case of the string
fails\footnote{The string obtained in this limit has a fixed length
and no physical oscillations (see Ref.~\cite{Yastremiz}).} but this
problem was solved with the realisation \cite{jGom_Oog} (see also
\cite{DanGuiKru}) that in order to obtain a Galilei invariant string
action one has to rescale both longitudinal coordinates, not the time
coordinate only. In a sense, one can speak of a non-relativistic
`particle' limit, obtained by taking $v\ll 1$ and a non-relativistic
limit for extended objects for which one has to rescale all worldvolume
coordinates, as we did in section \ref{non_rel_lim} for the case of
the string.
The rescaling of the longitudinal string direction corresponds
to the assumption $(\partial y/ \partial x)^2\ll 1$, which one makes when
deriving the wave equation by applying Newton's 2nd law on an
infinitesimal string segment. In the rescaling prescription
we followed, the waves move along the string at the speed of
light as the string tension equals the mass per unit length.
One usually thinks of non-relativistic strings as `violin-type'
having a small tension compared to their mass per unit length
and thus a subluminal `sound speed' along the string. In this
sense, the strings we consider here are `hybrid', having a
relativistic speed of propagation along the string, but transverse
Galilei invariance. However, it is precisely this hybrid action
(in flat space) which arises in the simplest Lorentz invariant field
theories when one studies the low-energy dynamics of domain wall
solutions. Strings with subluminal propagation speeds (which would
correspond to a differentiation between the string mass per unit
length and the tension) can arise in more complicated models, which
allow for spontaneously broken longitudinal Lorentz invariance
through a current generation mechanism on the string
worldsheet\footnote{See Ref.~\cite{BlPil_Redi} for a discussion
of the relation between strings with broken longitudinal
Lorentz invariance and Kaluza-Klein strings in one dimension
higher.} \cite{Witten, Carter_cwc}. To obtain such string actions
as a non-relativistic limit of the Nambu-Goto action, one would
have to rescale each of the longitudinal coordinates by a
different factor and take both factors to infinity while keeping
their ratio constant.
Note that, in order to ensure that the antisymmetric field $B$ used
to cancel the divergent piece of the action cannot be gauged away,
the non-relativistic string had to wind a compact dimension, say
$x^1\sim x^1 + 2\pi R$. In fact, the divergent piece of the action
is a total derivative with respect to the worldsheet coordinates
\cite{JGom_Kam_Town}, so if the action (\ref{S_NR}) was to be
interpreted as an effective non-relativistic action, one could
simply drop this term without requiring that the string is wound.
However, if the non-relativistic string is to be interpreted as a
fundamental object, consistency requires a non-trivial winding. In
this case, there are two distinct scales, namely $T_0$ of dimension
mass-squared, which appears in the action (\ref{S_NR}), and the mass
scale $m=2\pi n R T_0$, related to the geometry (through the
compactification radius $R$) and the string winding number $n$.
In fact, when quantising the non-relativistic string \cite{jGom_Oog},
one encounters again the necessity of winding, as the mass $m$ is
needed to define the energy states of the non-relativistic string
spectrum. In the flat case there are no physical states with
zero winding number \cite{jGom_Oog}.
Also note that in deriving the non-relativistic string action
(\ref{S_NR}) we have defined the tension $T_0$ by a rescaling
of the relativistic string tension $T_R$, appearing in the
Nambu-Goto action (see equation (\ref{T_rescale})):
\begin{equation}\label{T0_TR}
T_0 = \omega^\alpha T_R \, ,
\end{equation}
where the expansion exponent $\alpha$ is positive and $\omega$ is
taken to infinity. Interpreting $T_0$ as the physically relevant
quantity which is to be kept constant, the relativistic tension
$T_R$ goes to zero as $\omega$ tends to infinity.
Finally, we comment on the stability of the non-relativistic string.
A closed non-relativistic string is more stable to breakage than
its relativistic counterpart\footnote{We thank F. Passerini for
discussions of this point.}. This is a consequence of the winding,
which only allows a discrete number of potential `splitting points'
along the string. From an astrophysical perspective, ordinary
cosmic string loops decay through gravitational radiation, which
mainly couples to the kinetic energy of the fluctuations. In
particular, the power in gravitational radiation scales with the
sixth power of the root mean square (rms) velocity (see for example
\cite{book, vosk}). Thus, if such non-relativistic strings were
to play an astrophysical role, their decay rate would be power-law
suppressed. For long stings, the main energy-loss mechanism is
through string intercommutation, which removes string length from
the long string network. This is also expected to be suppressed
for non-relativistic strings as the interaction rates are
proportional to the string velocities. We shall now consider
the possibility of coupling non-relativistic strings to cosmology.
\subsection{Coupling to Gravity and Cosmology}
The non-relativistic action we have analysed describes the dynamics
of the independent degrees of freedom of the non-relativistic string,
namely the transverse excitations. In obtaining this action we have
introduced a closed $B$ field, which cancels the divergent piece
corresponding to the rest energy of the string. Alternatively, if
we are not interested in quantisation, we can simply drop the
divergent part of the action-without introducing the $B$ field-because
it is a total derivative (cf the case of the point-particle).
Here, we will follow the latter approach. The energy-momentum
tensor of the non-relativistic string (Appendix \ref{NR_EM_tensor})
therefore describes the energy of the transverse excitations but does
not include a contribution from the rest mass of the string. However,
when one couples non-relativistic matter to General Relativity
it is necessary to include the rest mass $m_0 c^2$ in the
energy-momentum tensor, which gives the main contribution to
the $T^{00}$ part while kinetic contributions are subdominant.
Following this logic we will add the rest mass of the string to
the $T^{00}$ part of the energy-momentum tensor of Appendix
\ref{NR_EM_tensor}, which can then be coupled to Einstein's
equations. From now on we work in $D=3$ spatial dimensions.
Consider a cosmological setup where the cosmic fluid has a
component due to a gas of non-interacting, non-relativistic strings.
To obtain the energy density of the string fluid, one has to sum the
contributions of all string segments in the network and, as we
discussed, it is the rest energy of the segments which will give the
dominant contribution. This is in analogy to a gas of non-relativistic
particles (dust), where the dominant contribution to the energy
density is $\rho \equiv T^0_{\; 0} = m_0 n + {\cal O}(v^2)$, where
$m_0$ is the particle rest mass, $n$ the rest frame number density
and $v$ the rms particle velocity. The off-diagonal terms of the
energy-momentum tensor of the particle fluid average out to zero
by summing over all particles with random velocities in all
directions, whereas the $T^i_{\; i}\equiv -p$ components are
proportional to the kinetic energy density, which for
non-relativistic particles is negligible so that $p\ll \rho$.
In the case of a `string gas' one can obtain an effective
energy-momentum tensor in an analogous manner, by approximating
the string network as a collection of straight string segments
moving with average velocity $v$, and averaging over string
orientations and directions of motion. Let us first consider
the relativistic case. The effective energy-momentum tensor
can be constructed by considering a straight string oriented
in the $\hat {\bf z}$ direction say, and Lorentz-boosting its
energy-momentum tensor in the $\pm \hat {\bf x}$ and $\pm \hat
{\bf y}$ directions, then averaging and repeating the same
procedure for strings oriented in the $\hat {\bf x}$ and $\hat
{\bf y}$ directions \cite{Kolb_Turn}. The result is:
\begin{equation}\label{T_fin}
\langle T^\mu_{\; \nu} \rangle = \frac{\mu}{3L^2}
\left(
\begin{array}{cccc}
3 \gamma^2 & 0 & 0 & 0 \\
0 & (1-v^2\gamma^2) & 0 & 0 \\
0 & 0 & (1-v^2\gamma^2) & 0 \\
0 & 0 & 0 & (1-v^2\gamma^2) \\
\end{array}
\right) \, ,
\end{equation}
where $\mu$ is the string tension, $L$ the average separation between
nearby strings in the network and $\gamma=(1-v^2)^{-1/2}$ a Lorentz
factor corresponding to $v$. From (\ref{T_fin}) the equation of
state can be read:
\begin{equation}\label{eos}
-p \equiv \langle T^i_{\; i} \rangle = \frac{1}{3} ( \gamma^{-2}
- v^2 ) \langle T^0_{\; 0} \rangle = \frac{1}{3} (1-2v^2) \langle
T^0_{\; 0} \rangle \Rightarrow p=-\frac{1}{3} (1-2v^2) \rho \, .
\end{equation}
A similar procedure can be followed for non-relativistic strings,
which are generally expected to have much smaller string velocities.
Indeed, for relativistic strings the constraint $\dot x^2+x^{\prime
2}\equiv 0$ in the conformal gauge imposes that critical points on
the string move with the speed of light, but for non-relativistic
strings the physical string velocities can take any value. One can
thus obtain the equation of state for such a non-relativistic string
gas by using Lorentz boosts with $\gamma=1$ or, alternatively, by
performing transverse Galilean boosts instead. The result is again
$p=-\frac{1}{3}(1-2v^2) \rho$, but with the difference that one can
safely assume $v\ll 1$, unlike the relativistic network case, where
the strings oscillate relativistically at small scales, while there
is no known mechanism which is efficient enough to damp these
excitations. Indeed, Hubble damping is inefficient at scales much smaller
than the horizon, and for large scales, of order the string correlation
length, numerical simulations (see for example \cite{All_Shell})
demonstrate that string segments move more slowly and coherently, but at
speeds large enough to produce significant deviations from the
equation of state $w\equiv p/\rho=-1/3$.
Note that one can apply an analogous procedure for strings which
have a tension $T$ smaller than their mass per unit length $\mu$
($T<\mu$). In this case the resulting equation of state is:
\begin{equation}\label{eos_Tmu}
p=-\frac{1}{3} [ T/\mu(1-v^2) - v^2 ] \rho = -\frac{1}{3} [
v_0^2 - (1+v_0^2) v^2 ] \rho \, ,
\end{equation}
where we have defined the `sound speed' along the string $v_0 =
\sqrt{T/\mu}$. Equation (\ref{eos_Tmu}) can in general lead to both
positive or negative equation of state with $p>-\rho/3$. This is in
contrast to vacuum (non-interacting) cosmic strings with $\mu=T$, where
the rms speed does not exceed $1/\sqrt{2}$ so the equation of state is
nonpositive (\ref{eos}) with $p \ge -\rho/3$. However, this is to be
expected because in the limit $T\rightarrow 0$ the `string' describes a
line-like structure of dust particles with $0<p\ll \rho$. In fact,
taking $v_0\rightarrow 0$ in equation (\ref{eos_Tmu}) gives $p=\rho
v^2/3$, or, in terms of the kinetic energy density $\rho_k$,
\begin{equation}\label{kin}
p=\frac{2}{3} \rho_k \, ,
\end{equation}
which is precisely the equation of state for a gas of
non-relativistic particles, following from ordinary kinetic theory
considerations. In connection to the discussion of the previous
sections, obtaining this kind of non-relativistic string from the
Nambu-Goto action involves a rescaling of the longitudinal directions
by different factors, the ratio of which determines the propagation
speed $v_0$.
Finally, note that this discussion only applies to a `perfect' gas
of non-interacting strings. String intercommutations typically result
in the removal of energy from the network in the form of closed
string loops, significantly altering the above picture. Thus, a
\emph{frustrated} string network, with $w\simeq -1/3$, $\rho \propto
a^{-2}$ eventually dominates over matter or radiation, but turning
on string interactions will result to a different equation of state.
For abelian string networks, where interactions are efficient,
the resulting scaling law is $\rho \propto t^{-2}$, where $t$ is
cosmic time, which scales like radiation in the radiation era and
like matter in the matter era. The cosmological evolution of
non-relativistic string networks, including the possible effects of
string intercommutation will be discussed in the next section.
\section{\label{VOS} Velocity Dependent One-Scale (VOS) Models}\setcounter{equation}{0}
In this section we discuss analytic models for the evolution of
macroscopic variables describing the large-scale properties of
a string network. We will first review results for relativistic
strings and then construct a macroscopic evolution model for
non-relativistic strings, based on the action (\ref{S_NR}).
To set up the physical picture we briefly summarise Kibble's
one-scale model \cite{Kibble}, which captures the basic qualitative
features of network evolution.
Monte-Carlo simulations of cosmic string formation suggest that
to a good approximation the strings have the shapes of random walks
at the time of formation \cite{Vach_Vil}. Such `Brownian' strings
can be described by a characteristic length $L$, which determines
both the typical radius of curvature of strings and the typical
distance between nearby string segments in the network. On average
there is a string segment of length $L$ in each volume $L^3$ and
thus the density of the cosmic string network at formation is
\begin{equation}\label{rho}
\rho=\frac{\mu L}{L^3}=\frac{\mu}{L^2} \,,
\end{equation}
where $\mu$ is the string mass per unit length, which for
relativistic strings is equal the `tension' $T_R$ appearing
in the Lagrangian. Assuming that the strings are simply stretched
by the cosmological expansion we have $\rho \propto a(t)^{-2}$.
This decays slower than both matter ($\propto a^{-3}$) and radiation
($\propto a^{-4}$) energy densities and so such non-interacting
strings would soon dominate the universe.
Now consider the effect of string interactions. As the network
evolves, the strings collide or curl back on themselves creating
small loops, which oscillate and radiatively decay. Via these
interactions enough energy is lost from the network to ensure that
string domination does not actually take place. Each string segment
travels on average a distance $L$ before encountering another nearby
segment in a volume $L^3$. Assuming relativistic motion ($v\approx 1$)
and that the produced loops have an average size $L$, the
corresponding energy loss is given by $\dot\rho_{\rm{loops}}\approx
L^{-4} \mu L$. The energy loss rate equation is therefore
\begin{equation}\label{rholoss}
\dot\rho\approx -2\frac{\dot a}{a}\rho - \frac{\rho}{L}\,.
\end{equation}
Equation (\ref{rholoss}) has an attractor `scaling' solution in
which the characteristic length $L$ stays constant relative to
the horizon $d_H\sim t$ \cite{Kibble}. The approach of string
networks to a scaling regime has been verified by high-resolution
simulations \cite{BenBouch, All_Shell}.
Equation (\ref{rholoss}) was derived on physical grounds and it only
captures the basic processes involved in string evolution, namely
the stretching and intercommuting of strings. It does not take into
account other effects like the redshifting of string velocities due
to Hubble expansion. In fact, it neglects completely the evolution
of string velocities, making the crude approximation that they
remain constant throughout cosmic history. However, we can construct
a more accurate Velocity-dependent One-Scale (VOS) model, based on
the Nambu-Goto action (\ref{nambu}).
\subsection{\label{rel_VOS}Relativistic Strings}
The relativistic VOS model \cite{vos,vosk} extends Kibble's
one-scale model, abandoning the constant string velocity
approximation and introducing an extra variable, the rms velocity
of string segments, whose dynamics is governed-as we will see-by
a macroscopic version of the relativistic equations of motion
(\ref{eom_x}). Although the simple one-scale model captures most
of the qualitative features of macroscopic string evolution, this
correction is crucial for quantitative modelling. Indeed, the
average string velocity enters linearly in the loop production
term, which provides the main energy loss mechanism of the string
network, and so the evolution of string velocities significantly
affects the string energy density. The resulting VOS model is
still very simple depending on only one free parameter\footnote{
Strictly speaking there are two parameters in the VOS model, the
loop production efficiency $\tilde c$ and the momentum parameter
$k$. For the second parameter, however, there exists a
physically motivated ansatz (\ref{kans_R}), which expresses it
in terms of the rms velocity $v(t)$. Once this choice is made,
one is only left with the freedom of tuning $\tilde c$ when
trying to fit numerical simulations.} but, remarkably, it has
been shown to accurately fit numerical simulation data throughout
cosmic history \cite{vostests}. We briefly sketch how the model
is constructed from the microscopic equations of section
\ref{rel_string}. This will be useful for comparison to the
non-relativistic case.
Consider the relativistic string energy defined in section
\ref{rel_string} (equation (\ref{string_energy})):
\[
E(\eta) = a(\eta) T_R \! \int \! \epsilon \, d\sigma
\]
and take the first derivative with respect to conformal time $\eta$.
Using the equation of motion (\ref{eom_eps}) for $\epsilon$, one
finds
\begin{equation}\label{E_dot_rel}
\dot E = \frac{\dot a}{a} \left( 1 - 2 v^2 \right) E \, ,
\end{equation}
where $v^2\!=\! \int \! \epsilon \dot{\bf x}^2 \, d\sigma /
\! \int \! \epsilon \, d\sigma\! \equiv\! \langle \dot {\bf x}^2
\rangle$ is the worldsheet average of the square of transverse
velocities. For a network of strings the energy density $\rho$
is related to the total string energy by $E\propto \rho a(\eta)^3$.
Therefore:
\begin{equation}\label{rho_dot}
\frac{\dot \rho}{\rho} = \frac{\dot E}{E} - 3\frac{\dot a}{a}
= -2 \frac{\dot a}{a} \left(1+v^2\right) \, .
\end{equation}
To this we add a phenomenological term \cite{Kibble, book} describing
the production of loops when strings collide and curl back on
themselves. The resulting network density evolution equation is:
\begin{equation}\label{rho_dot_full}
\dot\rho = -2 \frac{\dot a}{a} \left(1+v^2\right) \rho - \tilde c
\frac{v \rho}{L} \, ,
\end{equation}
where $\tilde c$ is the loop production efficiency related to the
integral of an appropriate loop production function over all
relevant loop sizes \cite{book}. This is treated as a free
parameter which can be determined by comparison to numerical
simulations.
In the VOS model, the rms velocity $v$ appearing in equation
(\ref{rho_dot_full}) is promoted to a dynamical variable whose
evolution is given by a macroscopic version of the Nambu-Goto
equation of motion (\ref{eom_x}). This equation can be obtained
by differentiating $v^2$ and eliminating $\ddot {\bf x}$ using
the equation of motion. This introduces the second spatial
derivative ${\bf x}^{\prime\prime}$ which corresponds to string
curvature and can be expressed in terms of the mean curvature
radius of the network. Differentiating $v^2$ and using equation
(\ref{eom_eps}) we find:
\begin{equation}\label{v_square_dot}
2v \dot v = \langle \dot {\bf x}^2 \rangle \dot{} = 2\langle
\dot{\bf x}\cdot \ddot{\bf x} \rangle - 2\frac{\dot a}{a}
\left( \langle \dot{\bf x}^2 \rangle^2 -\langle \dot{\bf x}^4
\rangle \right) \, .
\end{equation}
The second term is of purely statistical nature and has the effect
of `renormalising' the coefficient of the $\frac{\dot a}{a} v^4$
term which will find later. It has been demonstrated numerically
\cite{vos} to have small magnitude and thus can be neglected.
Keeping only the first term and using the equation of motion for
${\bf x}$ we find:
\begin{equation}\label{v_vdot}
v\dot v = \frac{\int\! \dot{\bf x} \cdot {\bf x}^{\prime\prime}
\epsilon^{-1} \, d\sigma}{\int\!\epsilon\, d\sigma} + \frac{\int\!
\dot{\bf x} \cdot {\bf x}^\prime (\epsilon^{-1})^\prime \,
d\sigma}{\int\!\epsilon \, d\sigma} - 2 \frac{\dot a}{a} \left(
\langle \dot{\bf x}^2 \rangle - \langle \dot{\bf x}^4 \rangle
\right) \, .
\end{equation}
The second term vanishes due to the gauge condition $\dot{\bf x}
\cdot{\bf x}^\prime = 0$. Further, within our approximations
$\langle \dot{\bf x}^4 \rangle\simeq \langle \dot{\bf x}^2
\rangle^2$ so the third term becomes $2\frac{\dot a}{a} v^2
(1-v^2)$. For the first term we need to express ${\bf
x}^{\prime\prime}$ in terms of the local curvature vector.
We define
\begin{equation}\label{ds}
ds = \sqrt{ {\bf x}^{\prime2} } d\sigma = \sqrt{1-\dot{\bf x}^2}
\epsilon d\sigma
\end{equation}
and the physical (local) radius of curvature by
\begin{equation}\label{R}
\frac{d^2 {\bf x}}{ds^2}=\frac{a(\eta)}{{\cal R}} \hat{\bf u} \, ,
\end{equation}
where $\hat{\bf u}$ is a unit vector. Then:
\begin{equation}\label{x_pp}
{\bf x}^{\prime\prime} = \frac{d^2{\bf x}}{d\sigma^2}
= {\bf x}^{\prime2} \frac{d^2 {\bf x}}{ds^2} + {\bf x}^\prime
\frac{d \sqrt{{\bf x}^{\prime2}}}{ds}
\end{equation}
Due to the constraint $\dot{\bf x} \cdot{\bf x}^\prime = 0$ the
second term vanishes on dotting with $\dot{\bf x}$ so we have:
\begin{equation}\label{xdot_x_pp}
\int\! \dot{\bf x} \cdot {\bf x}^{\prime\prime} \epsilon^{-1} \,
d\sigma =
\int\! \dot{\bf x}
\cdot \frac{d^2 {\bf x}}{ds^2} ( 1 - \dot{\bf x}^2 ) \epsilon \,
d\sigma = a(\eta) \langle (\dot{\bf x} \cdot \hat{\bf u} ) ( 1 -
\dot{\bf x}^2 ) / {\cal R} \rangle \int\! \epsilon \, d\sigma \, .
\end{equation}
We define the momentum parameter $k$ \cite{vosk} by the equation:
\begin{equation}\label{k}
\langle (\dot{\bf x} \cdot \hat{\bf u} ) ( 1 - \dot{\bf x}^2 )
/ {\cal R} \rangle = \frac{kv}{{\cal R}} (1-v^2) \, ,
\end{equation}
where ${\cal R}$ is now the average string radius of curvature,
numerically close to the correlation length $L$ for Brownian networks
\cite{book, vos, Aust_Cop_Kib}. With this definition, equation
(\ref{v_vdot}) becomes:
\begin{equation}\label{dv_dtau}
\dot v = \frac{a(\eta)}{{\cal R}}k(1-v^2) - 2\frac{\dot a}{a}v(1-v^2) \, .
\end{equation}
Changing to cosmic time $t$, with $dt = a d\eta$ and
$\dot{}=a\frac{d}{dt}$ we finally obtain:
\begin{equation}\label{dv_dt}
\frac{dv}{dt} = (1-v^2) \left( \frac{k}{{\cal R}} - 2Hv \right) \, ,
\end{equation}
where $H=a^{-1} \frac{da}{dt}$ is the Hubble parameter. Note that,
since
\[
v^2=\langle \dot{\bf x}^2 \rangle=\left\langle \left(
\frac{d{\bf x}} {d\eta} \right)^2 \right\rangle = \left\langle
\left( a \frac{d{\bf x}}{dt} \right)^2 \right\rangle \,
\]
and the physical coordinates ${\bf x}_{\rm phys}$ are given in
terms of the comoving ones ${\bf x}$ by ${\bf x}_{\rm phys} = a
{\bf x}$, the rms velocity $v$ has the interpretation of physical
\emph{peculiar} velocity of string segments. Equation (\ref{dv_dt})
has therefore a clear physical meaning: the rms peculiar velocities
of string segments are produced by string curvature and damped by
cosmological expansion.
The momentum parameter $k$ is a measure of the angle between the
curvature vector and the velocity of string segments and thus it is
related to the smoothness of the strings. As $v$ increases towards
relativistic values the accumulation of small-scale structure renders
the strings wiggly. Velocities become uncorrelated to curvature and
$k$ decreases. In particular it can be shown analytically that for
flat space, where $v^2=1/2$, the momentum parameter vanishes for a wide
range of known solutions \cite{vos, Carl_thesis}.
An accurate ansatz for the momentum parameter $k$ for relativistic
strings has been proposed in \cite{vosk}
\begin{equation}\label{kans_R}
k = k(v) = \frac{2\sqrt{2}}{\pi}\frac{1-8 v^6}{1+8 v^6} \, ,
\end{equation}
satisfying $k(1/\sqrt{2})=0$.
Note that the fact that $v=1/\sqrt{2}$ in flat spacetime, can be
shown analytically for closed loops only, but for long strings it
is observed in numerical simulations \cite{book}. For expanding
or contracting spacetimes, $v$ is less or greater than $1/\sqrt{2}$
respectively. Hence for an expanding universe, string velocities
are subject to the constraint:
\begin{equation}\label{vconstr}
v^2 \le \frac{1}{2} \, .
\end{equation}
In a matter or radiation dominated universe, Hubble expansion
is too weak to significantly reduce string velocities, which
remain close to $1/2$ at short scales \cite{book}. This
limitation does not apply to non-relativistic strings.
\subsection{\label{NRVOS}Non-Relativistic Strings}
For the non-relativistic string the energy of the excitations is
given by (see Appendix \ref{NR_EM_tensor}):
\begin{equation}\label{E_exc}
E_{\rm exc}=a(\eta) \frac{1}{2} \! \int \! d\sigma \left(
\mu \dot{\bf X}^2 + \mu \lambda^{-2} {\bf X}^{\prime 2}
\right)=a(\eta)^{-1}\,{\cal P}_0 \, ,
\end{equation}
where ${\bf X}$ are the \emph{transverse} string coordinates and
we have defined the tension $\mu\!=\!\lambda T_0$. To that we must
add the string mass
\begin{equation}\label{E_0}
E_0 = a(\eta) \mu \! \int \! d\sigma \, ,
\end{equation}
so that the total energy is:
\begin{equation}\label{E_tot}
E=E_0 + E_{\rm exc}=a(\eta) \mu \! \int \! d\sigma + a(\eta)^{-1}
\,{\cal P}_0 \, .
\end{equation}
Then, differentiating with respect to conformal time ( $\dot {}\
\! = \frac{d}{d\eta}$), we have:
\begin{eqnarray}
\dot E &=& \frac{\dot a}{a} E_0 + (a^{-1}{\cal P}_0)\dot{}
= \frac{\dot a}{a} E_0 - \frac{\dot a}{a} E_{\rm exc} + a^{-1}
\,\dot{\cal P}_0 \nonumber \\
&=& \frac{\dot a}{a} \left(1 + \frac{1}{2} W^2 -
\frac{3}{2} V^2 \right) E_0 \label{E0dot_E0} \, ,
\end{eqnarray}
where we have used equations (\ref{P_0_gauge}), (\ref{p_0_dot})
and defined the rms quantities:
\begin{equation}\label{V}
V^2= \frac{\int\! d\sigma \dot {\bf X}^2}{\int \! d\sigma} \equiv
\langle \dot {\bf X}^2 \rangle \, ,
\end{equation}
and
\begin{equation}\label{W}
W^2 = \frac{\int\! d\sigma \lambda^{-2} {\bf X}^{\prime 2}}{\int
\! d\sigma} \equiv \langle \lambda^{-2} {\bf X}^{\prime 2}\rangle
= \langle (\partial_{x^1}{\bf X})^2
\rangle \, ,
\end{equation}
corresponding to the average velocity of string segments and the
average magnitude of string tangent vectors. The latter quantity
parametrises small-scale perturbations on the string, $W=0$
corresponding to strings which are straight at scales smaller
than the correlation length\footnote{With this interpretation,
one expects that $W$ should have the effect of reducing the
effective radius of curvature of the network. As we will see
later, this is indeed the case.}. Thus, the term $W^2/2$ in
equation (\ref{E0dot_E0}) corresponds to the average elastic
energy due to short-scale string deformations. In the
non-relativistic limit one has $W^2\ll 1$.
Defining the energy density $\rho \propto E a^{-3}$, and using
\begin{equation}\label{E_dot_E0}
\frac{\dot E}{E_0} \simeq \frac{\dot E}{E} = \frac{\dot\rho}{\rho}
+ 3 \frac{\dot a}{a} \, ,
\end{equation}
we find
\begin{equation}\label{rho_dot_NR}
\dot\rho = -\frac{\dot a}{a} \left( 2 - \frac{1}{2} W^2
+ \frac{3}{2} V^2 \right) \rho - \tilde c V \frac{\rho}{L} \, ,
\end{equation}
where we have included a phenomenological loop production term,
as in the relativistic case.
From (\ref{V}) we have
\begin{equation}\label{v_square_dot_NR}
2 V \dot V = \langle \dot {\bf X}^2 \rangle^{\cdot} =
2 \langle \dot{\bf X}\cdot \ddot{\bf X} \rangle - 2\frac{\dot
a}{a}\left( \langle \dot{\bf X}^2 \rangle^2 -\langle \dot{\bf X}^4
\rangle \right)
\end{equation}
as before. We neglect the statistical terms and using the
non-relativistic equation of motion (\ref{eom_gauge}) we find:
\begin{equation}\label{vv_dot_NR}
V\dot V = \frac{\int\!\dot{\bf X} \cdot \ddot{\bf X} \, d\sigma}
{\int \! \, d\sigma} = \frac{\int\! \lambda^{-2} \dot{\bf X}
\cdot {\bf X}^{\prime\prime}\, d\sigma}{\int \! \, d\sigma}
- 2\frac{\dot a}{a} V^2
\end{equation}
In order to express ${\bf X}^{\prime\prime}$ in terms of the
string curvature vector we define:
\begin{equation}\label{ds_NR}
ds = \sqrt{ 1 + (\partial_{x^1} {\bf X})^2 } dx^1=
\lambda \sqrt{ 1 + \lambda^{-2} {\bf X}^{\prime2} } d\sigma
\end{equation}
and the physical radius of curvature:
\begin{equation}\label{R_NS}
\frac{d^2 {\bf Y}}{ds^2}=\frac{a(\eta)}{{\cal R}}\hat{\bf u}\,,
\end{equation}
where we have introduced the 3-vector ${\bf Y}\!=\!\left(x^1,{\bf
X}\right)$ and a unit 3-vector $\hat{\bf u}$. Now:
\begin{equation}\label{X_pp_NR}
{\bf X}^{\prime\prime} = \frac{d^2{\bf X}}{d\sigma^2} =
\lambda^2 \left(1+\lambda^{-2}{\bf X}^{\prime2}\right) \frac{d^2
{\bf X}}{ds^2} + \lambda {\bf X}^\prime \, \frac{d \sqrt{1+
\lambda^{-2}{\bf X}^{\prime2}}}{ds}
\end{equation}
In this case, the second term will not cancel on dotting with $\dot
{\bf X}$, because $\dot {\bf X} \cdot {\bf X}^\prime \ne 0$ for the
non-relativistic string. Instead we have two terms:
\begin{equation}\label{Xdot_X_pp_NR}
\lambda^{-2} \int\! \dot{\bf X} \cdot {\bf X}^{\prime\prime} \,
d\sigma = \int\! \dot{\bf X} \cdot \frac{d^2 {\bf X}}{ds^2}
\left(1 + \lambda^{-2}{\bf X}^{\prime2}\right) \, d\sigma +
\lambda^{-2} \int\! \dot{\bf X} \cdot{\bf X}^\prime \left(\ln
\sqrt{1 + \lambda^{-2}{\bf X}^{\prime2}} \right)^\prime
d\sigma \, .
\end{equation}
For the first term we note that, since $\dot{\bf X}$ is normal
to $(x^1,{\bf 0})$ in Cartesian coordinates,
\begin{equation}\label{Xdot_curv}
\dot{\bf X} \cdot \frac{d^2 {\bf X}}{ds^2} = \dot{\bf X} \cdot
\frac{d^2{\bf Y}}{ds^2}
\end{equation}
and so we can use equation (\ref{R_NS}) to write:
\begin{equation}\label{int_Xdot_curv}
\int\! \dot{\bf X} \cdot \frac{d^2 {\bf X}}{ds^2} \left(1
+ \lambda^{-2}{\bf X}^{\prime2}\right) \, d\sigma =
a(\eta) \frac{k V}{{\cal R}} (1 + W^2) \!\int\! d\sigma\, .
\end{equation}
Here, in analogy to the relativistic case, we have defined a
momentum parameter $k$ by:
\begin{equation}\label{kdef_NR}
\left\langle \left(1 + \lambda^{-2}{\bf X}^{\prime2}\right)
(\dot{\bf X} \cdot \hat{\bf u}) /{\cal R} \right\rangle =
\frac{k V}{{\cal R}} (1 + W^2) \, .
\end{equation}
For the second term in (\ref{Xdot_X_pp_NR}) we have:
\begin{eqnarray}
&& \lambda^{-2} \int\! \dot{\bf X} \cdot{\bf X}^\prime \left(\ln\sqrt{1
+ \lambda^{-2}{\bf X}^{\prime2}} \right)^\prime d\sigma
= \lambda^{-2} \int\! \dot{\bf X} \cdot{\bf X}^\prime \frac{{\bf X}^\prime
\cdot {\bf X}^{\prime\prime}\lambda^{-2}}{1+ \lambda^{-2}{\bf
X}^{\prime2}} \, d\sigma
\nonumber \\
&& \ \ = \lambda^{-2} \int\! \left(\dot{\bf X} \cdot{\bf X}^\prime \right)
\left({\bf X}^\prime \cdot \hat {\bf u}\right) \frac{a(\eta)}{{\cal R}}
\, d\sigma + \lambda^{-3} \int\! \left(\dot{\bf X} \cdot{\bf X}^\prime
\right){\bf X}^{\prime2} \frac{{\bf X}^\prime \cdot {\bf X}^{\prime
\prime}\lambda^{-2}}{\left(1+ \lambda^{-2}{\bf X}^{\prime2}\right)^2} \,
d\sigma \nonumber \\
&& \ \ = a(\eta) \frac{k^\prime V W^2}{{\cal R}} \!\int\! d\sigma
+ {\cal O}(VW^4) \, ,
\label{Xdot_Xp_NR}
\end{eqnarray}
where we have used equation (\ref{X_pp_NR}) and defined the
parameter $k^\prime$ by:
\begin{equation}\label{k_prime}
\left\langle \lambda^{-2} \left(\dot{\bf X} \cdot{\bf X}^\prime
\right) \left({\bf X}^\prime \cdot \hat {\bf u}\right)/{\cal R}
\right \rangle = \frac{k^\prime V W^2}{\cal R}
\end{equation}
Putting all the terms together, equation (\ref{vv_dot_NR}) can be
rewritten (in terms of cosmic time $t$) as:
\begin{equation}\label{dV_dt}
\frac{dV}{dt}=\frac{1}{{\cal R}} \left( k + k^{\prime\prime} W^2
\right) - 2HV \, ,
\end{equation}
with
\begin{equation}\label{kpp}
k^{\prime\prime}\equiv k + k^{\prime}\, .
\end{equation}
Equations (\ref{rho_dot_NR}), (\ref{dV_dt}) form the
Non-Relativistic Velocity dependent One-Scale (NRVOS) model.
In principle one should consider $W$ as a third dynamical variable
and try to derive an evolution equation, as in the case of $V$.
As a first approximation we will assume that time variations in $W$
do not have a significant impact, $W$ remaining always small, and
we will treat it as a constant parameter. This approximation will
be tested in the next section, where we will solve the NRVOS
equations numerically, for different choices of the $W$ parameter.
Finally, one comment is in order regarding the magnitude of the
parameter $k^\prime$. From its definition in equation (\ref{Xdot_Xp_NR})
one expects $k^\prime\ll k$. Indeed, $k^\prime$ measures the
average value of $(\dot{\bf X}\cdot{\bf X}^\prime)({\bf X}^\prime \cdot
\hat{\bf u})$ the first factor of which contains uncorrelated vectors,
while for the second factor string tangents will generally be normal to
the local curvature vector. On the other hand $k$ corresponds to the
average value of $\dot{\bf X} \cdot \hat {\bf u}$ and these two vectors
are correlated, at least for smooth strings/small excitation velocities.
Given that the $k^{\prime\prime}$ term in (\ref{dV_dt}) is already
suppressed by a factor ${\cal O}(W^2)$ it is a good approximation to
set $k^{\prime\prime}\simeq k$. Then, $W$ has the effect of
`renormalising' the effective radius of curvature ${\cal R}
\rightarrow {\cal R}/(1+W^2)$ (or equivalently the momentum
parameter $k\rightarrow k(1+W^2)$), as may be expected from its
interpretation as a short-scale structure parameter.
\section{\label{RelVnonRel}Relativistic vs Non-Relativistic Network Evolution}\setcounter{equation}{0}
In this section we solve numerically the NRVOS equations for a
non-relativistic string network and compare to the relativistic
case. The naive expectation is that non-relativistic networks
are denser than their relativistic counterparts because the small
string velocities reduce the effect of the loop production term.
Physically, the transverse excitations on strings are non-relativistic
so fewer loops are produced per unit time due to string
self-intersections. Long string segment interactions are also
suppressed due to the low collision rate corresponding to
small velocities.
To close the NRVOS equations we need to specify an ansatz for the
non-relativistic momentum parameter $k$. For a velocity dependent
model like the one we have developed, it is not consistent to treat
$k$ as a constant parameter. Further, in the relativistic case, its
dependence on the rms velocity $v$ (equation (\ref{kans_R})) is
important in determining the scaling values of the network variables.
The functional dependence of the momentum parameter on $v$ can be
obtained by considering `curvature' and `bulk' contributions to string
velocities, as explained in Ref.~\cite{vosk}. Following the discussion
in that reference we take:
\begin{equation}\label{k_NR}
k(v) = k_0 (1-v^2) \, ,
\end{equation}
where $k_0$ is a constant. This has the same functional dependence
as the low-velocity limit of $k(v)$ in Ref.~\cite{vosk}, but here
we have left the overall normalisation $k_0$ as a free parameter.
This reflects the fact that the non-relativistic string limit is
not merely a low-velocity one. There is a difference between
slowly moving, straight, relativistic strings and wiggly,
non-relativistic strings. The defining property of the non-relativistic
string is that its transverse excitations be Galilei, as opposed to
Lorentz, invariant. The difference between relativistic and
non-relativistic strings is in the transverse perturbations.
In an effective description, non-relativistic strings can be
thought of as having a short wavelength cut-off on the string
excitations. As a result, arbitrarily small-wavelength relativistic
perturbations are not excited and this translates into a reduced
curvature parameter normalisation $k_0$. The string can be
thought of as a massive rigid rod, but with tension $T$ equal to its
mass per unit length $\mu$ \footnote{Relativistic invariance in the
longitudinal directions implies that the waves along the string travel
at the speed of light $c$ \cite{JGom_Kam_Town}. Note the difference
to the other notion of non-relativistic string with $T<\mu$ and
longitudinal speed $v<c$.}. In analogy to the
relativistic case, where the overall normalisation was determined
by comparison to a known analytic solution \cite{vosk}, $k_0$ can
be obtained in the non-relativistic case by comparison to a given
model of non-relativistic string. In the general discussion below
we will simply treat it as a free parameter and examine its effect
on the network evolution.
Equations (\ref{rho_dot_NR}), (\ref{dV_dt}) and (\ref{k_NR}) have
been solved numerically for a range of parameters $k_0$ and
$W$. This was done by rewriting equation (\ref{rho_dot_NR})
in terms of the correlation length $L=\sqrt{\mu/\rho}$ and then
introducing a function $\gamma(t)=L/t$. Under the assumption
$L\simeq {\cal R}$, the resulting equation for $\gamma(t)$ together
with (\ref{dV_dt}) form a non-autonomous system of coupled ODE's,
which can be integrated numerically. During matter or radiation
domination, this system has an attractor solution in which both
$\gamma(t)$ and $v(t)$ tend to constant values (scaling). Here,
we present numerical results for a radiation dominated universe.
In Fig.~\ref{fig_comparison} we plot the evolution of the string
energy density and rms velocity for both a non-relativistic
and a relativistic string network, that is, the solution of
equations (\ref{rho_dot_NR}), (\ref{dV_dt}), (\ref{k_NR})
in the former case and (\ref{rho_dot_full}), (\ref{dv_dt}),
(\ref{kans_R}) in the latter. To highlight the effect of
non-relativistic velocities, we have chosen a value of the
parameter $k_0$ which gives a scaling value of $V\simeq 0.1$ and taken
$W<V$. We have also assumed that both networks have the same loop
production efficiency parameter $\tilde c$ and chosen the value
$\tilde c=0.23$, suggested by relativistic network simulations
\cite{vos}. As expected, the non-relativistic network has a much
higher scaling string density compared to the non-relativistic one.
Of course, non-interacting strings ($\tilde c=0$) do not converge
to a scaling solution.
\begin{figure}
\begin{center}
\includegraphics[width=2.7in,keepaspectratio]{rhoR.eps}
\includegraphics[width=2.7in,keepaspectratio]{vR.eps}
\includegraphics[width=2.7in,keepaspectratio]{rhoNR_v01.eps}
\includegraphics[width=2.7in,keepaspectratio]{vNR_v01.eps}
\end{center}
\caption{\label{fig_comparison} Relativistic versus
non-relativistic network evolution. Non-relativistic networks
evolve to slower and much denser scaling configurations than
their relativistic counterparts. Here, we have plotted the
evolution of the normalised string density and rms velocity
for a relativistic network and a non-relativistic one with
$V\simeq 0.1$.}
\end{figure}
We now explore the dependence of non-relativistic string evolution
on the parameters $k_0$ and $W$. In Fig.~\ref{fig_NR_v0} we plot
the normalised string density $\rho t^2/\mu=\gamma^{-2}$ and the
rms string velocity $V$ as functions of cosmic time $t$ for different
choices of $k_0$ producing string velocities $0<V<1$. We have assumed
a constant value of $W<V$, but below we will consider the effect of
varying $W$ also, allowing for $W>V$. It is apparent from
Fig.~\ref{fig_NR_v0} that the rms string velocities are controlled
by the parameter $k_0$. Reducing $k_0$ leads to smaller $V$, which
in turn implies a higher string density, due the reduced energy loss
term. The fact that the scaling value of the rms velocity is not
universal for non-relativistic strings, but instead depends on the
parameter $k_0$, is not surprising. In the relativistic case, there
is a distinct upper speed limit $c=1$ and the relativistic constraint
implies that the rms velocities are smaller than, but not far off,
$1/\sqrt{2}$ (see for example Ref.~\cite{book}). On the other hand,
in any non-relativistic theory velocities are unbounded.
\begin{figure}
\begin{center}
\includegraphics[width=2.7in,keepaspectratio]{rhoNR_v05.eps}
\includegraphics[width=2.7in,keepaspectratio]{vNR_v05.eps}
\includegraphics[width=2.7in,keepaspectratio]{rhoNR_v01.eps}
\includegraphics[width=2.7in,keepaspectratio]{vNR_v01.eps}
\includegraphics[width=2.7in,keepaspectratio]{rhoNR_v005.eps}
\includegraphics[width=2.7in,keepaspectratio]{vNR_v005.eps}
\end{center}
\caption{\label{fig_NR_v0} Evolution of normalised string density
and rms velocity for a non-relativistic network with constant $W<V$
for different choices of the parameter $k_0$. Reducing the value
of $k_0$ results to lower rms string velocities, which in turns
implies a slower rate of string interactions. This results in a
dramatic enhancement of string network density.}
\end{figure}
We then consider the impact of varying the parameter $W$. Looking
at the first term of equation (\ref{rho_dot_NR}), which describes
dilution due to cosmic expansion, one observes that $W^2$ and $V^2$
appear with opposite signs, so a large $W$ could counterbalance
(or even reverse) the effect of $V$ on this term. However, if
both $V,W\ll 1$ they play no significant role in that term. Thus,
one only needs to check the case $W>V$ when $V,W$ are not negligible.
Fig.~\ref{fig_W} shows the time evolution of $\rho$ for a choice of
$k_0$ leading to $V\simeq 0.1$, for the cases $W=0, 0.1, 0.5$.
The first two figures show identical evolutions, even though
in the second one $W\simeq V$. In the third case, however, where
$W^2=25 V^2$ the effect of $W$ counterbalances that of $V$ in the
dilution term of (\ref{rho_dot_NR}), resulting in an appreciable
reduction of the string scaling density, at the $10\%$ level.
Since the most important impact of string velocities is through
the loop production term of (\ref{rho_dot_NR}), the basic prediction
of the model, which is a dramatic enhancement of the string scaling
density (Fig. \ref{fig_comparison}), remains robust.
\begin{figure}
\begin{center}
\includegraphics[width=2.7in,keepaspectratio]{rhoNR_v01_w0.eps}
\includegraphics[width=2.7in,keepaspectratio]{vNR_v01_w0.eps}
\includegraphics[width=2.7in,keepaspectratio]{rhoNR_v01_w01.eps}
\includegraphics[width=2.7in,keepaspectratio]{vNR_v01_w01.eps}
\includegraphics[width=2.7in,keepaspectratio]{rhoNR_v01_w05.eps}
\includegraphics[width=2.7in,keepaspectratio]{vNR_v01_w05.eps}
\end{center}
\caption{\label{fig_W} Dependence of normalised string density
on the parameter $W$ for a network with $V\simeq 0.1$.
The plots correspond to $W=0, 0.1$ and $0.5$ respectively.
Increasing $W$ does not significantly alter the scaling density
until $W$ becomes greater than $V$. For $W=0.5=5V$, the scaling
is reduced by $10\%$, so it remains two orders of magnitude greater
than that of relativistic strings.}
\end{figure}
\section{\label{discuss}Discussion}\setcounter{equation}{0}
So far we have studied the dynamics and macroscopic evolution of
non-relativistic strings in some generality, without discussing
any specific setup in which they could be relevant. However,
non-relativistic string-like objects arise in several contexts
and have been considered before in the literature.
For example, Ref.~\cite{Mart_Moor_Shell} studied non-relativistic
vortex-strings with motivations from both cosmology \cite{book} and
condensed matter physics \cite{Schwarz85, Schwarz88}. In that
reference, the non-relativistic limit was taken at the level of the
equations of motion by requiring small string velocities $\dot X^2\ll
1$. Here, we have taken the non-relativistic limit at the level of
the string action but this involved a rescaling procedure which
corresponds to having both $\dot X^2\ll 1$ and $(\partial X/\partial \zeta)^2
\ll 1$, where $\zeta$ is the physical length along the string.
The non-relativistic evolution model we have developed in section
\ref{NRVOS} can be applied to the condensed matter context considered
in \cite{Mart_Moor_Shell} by introducing a friction term relevant
to that situation. Adding this term and setting $\dot a/a=0$
equation (\ref{rho_dot_NR}), expressed in terms of the correlation
length, reads:
\begin{equation}\label{}
2\frac{dL}{dt}=\tilde c V + \frac{L}{\ell_d} V^2\, ,
\end{equation}
as in Ref.~\cite{Mart_Moor_Shell}, where $\ell_d$ is the relevant
damping length scale. The velocity evolution equation is also modified
by the addition of a friction term $-\ell_d/L$, again as in
Ref.~\cite{Mart_Moor_Shell}. The system has a solution with $L\propto
t^{1/2}$, which is actually observed experimentally for defects in
condensed matter systems and liquid crystals \cite{Mermin, SalVol, ChDTY}.
In cosmology, slowly-evolving string networks have been invoked
in order to obtain a negative equation of state \cite{SperPen}.
Bucher \& Spergel \cite{BuchSper} have proposed a Solid Dark Matter
(SDM) model, which could be realised in terms of a frustrated string
or domain wall \cite{BatBuchSper} network. Rigidity and stability in
this scenario have been studied in Ref.~\cite{BatCarChMo}. More recently,
a string network of the SDM kind was revived \cite{Alexand} in an
attempt to explain the flat rotational curves and the Tully-Fisher
relation observed in galaxies, which were the main motivation for
the development of MOND\footnote{For a recent review on the MOND
scenario see Ref.~\cite{MOND}.} theories. The fundamental difficulty
\cite{BatCarChMo} with the SDM scenario is to explain how an essentially
non-relativistic network can naturally arise from an initial tangle
of (relativistic) Nambu-Goto strings like the ones believed to be
produced in cosmological phase transitions. Indeed, Hubble damping
is inefficient at subhorizon scales \cite{book} and there is no known
mechanism efficient enough to damp the relativistic short-scale excitations
on strings. These affect the equation of state through the velocity
dependent term in equation (\ref{eos}), leading to $w>-1/3$\footnote{
One could argue that the velocity which enters the equation of state
is the coherent string velocity at the scale of the string
correlation length rather than the rms short-scale velocity. While
it is true that the coherent velocities are typically smaller,
numerical simulations \cite{All_Shell} suggest $v_{\rm coh}\simeq 0.15$
so one still expects significant departure from $w=-1/3$. Furthermore,
small-scale structure has the effect of `renormalising' the string
mass per unit length \cite{book, Mart_Shell_sss} and string tension
so that equation (\ref{eos_Tmu}) should be used instead of (\ref{eos}).
This also increases the value of $w$.}. Further, numerical
evidence is now accumulating supporting that scaling behaviour in
field theory strings and domain walls is rather generic \cite{walls},
so that frustrated networks seem hard to obtain. On the other hand
the analysis we did in section \ref{RelVnonRel} points towards a SDM
picture for non-relativistic strings, where the above problems are
not present. Here, string velocities can be arbitrarily small and,
as we saw in section \ref{RelVnonRel}, network densities are
dramatically enhanced so that strings could even dominate the
universe before scaling is reached.
Note that the procedure for obtaining the non-relativistic string
action (\ref{S_NR}) required at least one of the spatial directions
to be compact. If the action (\ref{S_NR}) is to be treated as a
classical effective action this global property can be ignored, but
if it is taken to describe a fundamental object, then the winding
around a compact dimension is required at quantum level. The fact
that a consistent non-relativistic string theory based on the action
(\ref{S_NR}) can be constructed \cite{jGom_Oog} allows one to take
the view that there is a fundamental winding string obeying this
action. Then, a cosmological setup like that of sections \ref{NRVOS}
and \ref{RelVnonRel} can still be considered as long as the
compactification radius is larger than the horizon. This possibility
of having a universe with non-trivial topology is not observationally
excluded. Cosmological observations constrain the local geometry
as described by the metric to be nearly flat \cite{WMAP3}, but the
global topology of spatial hypersurfaces need not be that of the
covering space. Indeed, topological identifications under
freely-acting subgroups of the isometry group are allowed, and
the WMAP sky maps appear to be compatible with finite flat topologies
with fundamental domain significantly greater than the distance to
the decoupling surface \cite{Luminet} (see also \cite{Cornish}).
One can therefore imagine a situation where fundamental
non-relativistic strings are wound around 1-cycles in a
non-simply-connected universe, in a setup analogous to that
of the Brandenberger-Vafa scenario \cite{BrandVaf}. If the
compactification radius is larger than the horizon, as required
by cosmological observations, a network of such wound strings
behaves like an open string network. An analogous situation
occurs in ordinary cosmic string simulations, where the
network evolves in a periodic box and there is a class of long
strings (determined mainly by initial conditions) which wind around
the box. As the universe expands these strings tend to straighten
out and behave essentially non-relativistically \cite{Paul_private}.
These strings are usually discarded as artifacts of the periodicity
of the box, but in a universe of compact topology, such
configurations can play a physical role.
Finally, in theories with compact extra dimensions one has the
possibility of non-relativistic strings winding 1-cycles in the
internal space. Analogous (but relativistic) objects have been
considered in the context of brane inflation
\cite{BarnBCS, MatsNecl, cycloops}, which are topologically trapped
and behave like monopoles. Although the copious production of such
objects in the early universe is inconsistent with the existence of
an early radiation era, there are regions in parameter space where
they are allowed and in some cases can provide candidates for dark
matter. The situation of non-relativistic strings wrapping an internal
dimension is qualitatively similar, but the corresponding energy
spectrum is different than in the relativistic case.
The outstanding question arising from the above is to what extent
such non-relativistic strings are `natural' or `generic' objects
in cosmology. Even though non-relativistic strings exist in some
part of the moduli space of string theory, there is at present no
mechanism which produces them in a cosmological setup. Nevertheless,
it is clear that the non-relativistic string action and the VOS model
developed here are applicable at least as effective descriptions of
cosmic- and vortex-strings in certain situations. Indeed, the action
we have considered is the only sensible non-relativistic limit, having
$T=\mu$, of the standard Nambu-Goto action, and is precisely the action
one obtains when considering the low energy dynamics of topological
defects in field theory. The macroscopic NRVOS model based on this
action, provides a semi-analytic tool for the study of the cosmological
evolution of non-relativistic strings. Possible situations of
cosmological interest involving non-relativistic strings include strings
in de Sitter space, Solid Dark Matter, wound strings, etc, as discussed
above. Further, in a condensed matter application we have noted that
our model reproduces the correct scaling law, as experimentally observed.
It would be interesting to go one step further and perform numerical
simulations of string network evolution based on the non-relativistic
string action presented here. The comparison of macroscopic string
evolution and small-scale structure to the relativistic case could
provide an independent means of probing the effect of small-scale
structure on string networks, which is an area of current interest
and active research.
\begin{acknowledgments}
We are grateful to Carlos Martins for reading the manuscript
and making valuable comments. It is also a pleasure to thank
Roberto Emparan, Jaume Garriga, Gary Gibbons, Jaume Gomis, Paul
Shellard and Paul Townsend for discussions. This work has been
supported in part by the European EC-RTN project MRTN-CT-2004-005104,
European Network on Random Geometry (ENRAGE) project
MRTN-CT-2004-005616, MCYT FPA 2004-04582-C02-01 and
CIRIT GC 2005SGR-00564. We would like to thank the Galileo Galilei
Institute for Theoretical Physics for its hospitality and INFN
for partial support during the completion of this work.
\end{acknowledgments}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,119
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Q: How make dates list from date ranges with additional data on row in Google Sheets? Issue:
Make table with fields "Member, Date, Rate" from table with fields "Member, Rate, From Date, To Date".
The result table must be fully automatically generated. (Using formulas: Arrayformula, Query and more)
Example:
Left side source table, right side result table.
A: use:
=ARRAYFORMULA(QUERY(SPLIT(FLATTEN(IF((A2:A5<>"")*(DAYS(D2:D5; C2:C5)>=SEQUENCE(1; 1000; 0));
A2:A5&"×"&C2:C5+SEQUENCE(1; 1000; 0)&"×"&B2:B5; )); "×"); "where Col2 is not null"))
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 655
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Here is the long-awaited webinar presentation from last Friday (December 21, 2012). John Paul demonstrates the Atlas Line live across a variety of charts, explaining how this indicator can be used to identify trends. By knowing the direction of the market in advance, you can position trades long or short ahead of time. Entry prices are provided to you automatically by the indicator, letting you know exactly when to enter. The included live training will show you how to manage the trade including the three different stop strategies, including the proprietary Pullbacks and Strength trades. You can also get an idea of how to set up your charts correctly.
To find out more about this Atlas Line, visit the official page.
The January Effect – Was John Paul's Crystal Ball Correct?
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,371
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package com.linkedin.pinot.broker.requesthandler;
import com.linkedin.pinot.common.request.FilterOperator;
import com.linkedin.pinot.common.utils.request.FilterQueryTree;
import java.util.ArrayList;
import java.util.Iterator;
import java.util.List;
/**
* Optimizer that flattens nested logical operators of the same kind. For example, AND( a, AND (b, c)) is the same as
* AND(a, b, c).
*/
public class FlattenNestedPredicatesFilterQueryTreeOptimizer extends FilterQueryTreeOptimizer {
public static int MAX_OPTIMIZING_DEPTH = 5;
@Override
public FilterQueryTree optimize(FilterQueryOptimizerRequest request) {
FilterQueryTree filterQueryTree = request.getFilterQueryTree();
flatten(filterQueryTree, null, MAX_OPTIMIZING_DEPTH);
return filterQueryTree;
}
/**
* Flatten the operators if parent and child have the same AND or OR operator.
* (e.g. AND( a, AND (b, c)) is the same as AND(a, b, c). This helps when we re-order
* operators for performance.
*
* It does so by looking at the operator of the 'parent' and 'node'. If they are same, and
* collapsible, then all the children of 'node' are moved one level up to be siblings of
* 'node', rendering 'node' childless. 'node' is then removed from 'parent's children list.
*
* @param node The node whose children are to be moved up one level if criteria is satisfied.
* @param parent Node's parent who will inherit node's children if criteria is satisfied.
* @param maxDepth is the maximum depth to which we recurse
*/
private void flatten(FilterQueryTree node, FilterQueryTree parent, int maxDepth) {
if (node == null || node.getChildren() == null || maxDepth == 0) {
return;
}
// Flatten all the children first.
List<FilterQueryTree> toFlatten = new ArrayList<>(node.getChildren().size());
for (FilterQueryTree child : node.getChildren()) {
if (child.getChildren() != null && !child.getChildren().isEmpty()) {
toFlatten.add(child);
}
}
for (FilterQueryTree child : toFlatten) {
flatten(child, node, maxDepth - 1);
}
if (parent == null) {
return;
}
if (node.getOperator() == parent.getOperator()
&& (node.getOperator() == FilterOperator.OR || node.getOperator() == FilterOperator.AND)) {
// Move all of 'node's children one level up. If 'node' has no children left, remove it from parent's list.
List<FilterQueryTree> children = node.getChildren();
Iterator<FilterQueryTree> it = children.iterator();
while (it.hasNext()) {
parent.getChildren().add(it.next());
it.remove();
}
// 'node' is now childless
// Remove this node from its parent's list.
parent.getChildren().remove(node);
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,484
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\section*{Introduction}
The development and maintenance of tissues requires close coordination of biochemical and mechanical signaling~\cite{lecuit2011force,heisenberg2013forces,hannezo2019mechanochemical}. There is, for instance, mounting evidence for the key role played by tissue material properties and their regulation during embryonic development~\cite{petridou2019tissue}. Tissues must be able to adjust their mechanical properties in response to internal and external stimuli. In particular, epithelial tissues, which line all cavities in the body and demarcate organs, must sustain substantial mechanical stresses while also supporting numerous biological processes such as selective diffusion and absorption/secretion~\cite{ross2006histology}. In homeostasis, the tissue must maintain its shape and resist deformation while remaining flexible. The tissue must also be able to regenerate and repair itself, often with fast turnover, e.g., in gut epithelia~\cite{krndija2019active}. Furthermore, in morphogenesis, the tissue must take up a specific shape and function~\cite{wolpert2015principles}. During metastasis, however, the shape is lost and cancer cells invade surrounding healthy tissue~\cite{weinberg2013biology}. All of these processes require that cells be able to move, often over distances much larger than the cell size. During cell migration, however, the tissue must maintain its integrity. It is, therefore, not surprising that tissues exhibit rich viscoelastic behavior~\cite{forgacs1998viscoelastic}. Unlike passive viscoelastic materials, a tissue can actively tune its rheological response, making the study of its rheology not only important for understanding biological functions but also an interesting problem from the perspective of the physics of active matter systems~\cite{marchetti2013hydrodynamics}.
Collective cell migration has been extensively studied in biology~\cite{friedl2009collective} and biophysics~\cite{alert2020physical}. In vitro studies of confluent cell monolayers~\cite{poujade2007collective,trepat2009physical,tambe2011collective,brugues2014forces,etournay2015interplay} focused on the physical aspects of force generation and transmission and showed that cell migration is an inherently collective phenomenon. Some aspects of collective cell migration are remarkably similar to slow dynamics of structural glasses~\cite{angelini2011glass,park2015unjamming,bi2015density, bi2016motility,atia2018geometric,sussman2018anomalous,czajkowski2019glassy}. This suggests that many of the observed behaviors share common underlying mechanisms and can be understood, at least at mesoscales (i.e., distances beyond several cell diameters), using physics of dense active systems~\cite{henkes2020dense}. A particularly intriguing observation is that tuning cell density~\cite{szabo2006phase,angelini2011glass,sadati2013collective}, strength of cell-cell and cell-substrate interactions~\cite{garcia2015physics}, or cell shape parameters~\cite{bi2016motility,merkel2018geometrically} can cause the collective migration to stop. In other words, the tissue undergoes fluid to solid transition. Signatures of such behavior have been reported in several in vitro~\cite{park2015unjamming,mitchel2020primary} and developmental systems~\cite{benazeraf2010random,lawton2013regulated,mongera2018fluid}. This suggests that important aspects of morphogenetic development might rely on tissue's ability to undergo phase transitions~\cite{petridou2019tissue}.
How a tissue responds to external and internal mechanical stresses will depend on its rheological (i.e., material) properties. While there have been numerous studies focusing on the rheology of a single cell~\cite{desprat2006microplates,salbreux2012actin,berthoumieux2014active}, much less is known about tissue rheology, particularly during development. In order to develop a comprehensive understanding of tissue mechanics, such insight is key. Though single cell measurements are valuable, the mechanics of a tissue can be drastically different from that of its constituent cells. The stiffness of cell monolayers, for example, is orders of magnitude higher than the stiffness of constituent cells, while the time dependent mechanical behaviors of monolayers in response to deformation vary depending on the magnitude of loading~\cite{harris2012characterizing}. Embryonic cell aggregates have been shown to behave elastically (i.e., solid-like) at short time scales but flow like fluids at long time scales, which facilitates both the robustness needed to maintain integrity and the flexibility to morph during development~\cite{forgacs1998viscoelastic}. Experiments have characterized the mechanical behaviors of tissues at various loading conditions, which led to a phenomenological description that models the relaxation properties of epithelial monolayers based on fractional calculus~\cite{bonfanti2020unified}. Notably, a recent particle-based model that includes cell division and apoptosis provided a plausible microscopic model for nonlinear rheological response~\cite{matoz2017nonlinear}. Particle-based models are, however, unable to capture geometric aspects such as cell shape. It is, therefore, necessary to investigate rheological response in geometric models.
The vertex model~(VM)~\cite{nagai2001dynamic,farhadifar2007influence,fletcher2014vertex} and more recent, closely related self-propelled Voronoi models~(SPV)~\cite{bi2016motility,barton2017active} have played an important role in modeling mechanics of epithelial tissues since they account for the shapes of individual cells and provide a link to cellular processes, such as cell-cell adhesion, cell motility, and mitosis~\cite{fletcher2014vertex}. These geometric models are also able to capture the solid to fluid transition and demonstrate rich and unusual nonlinear mechanical behavior~\cite{moshe2018geometric,popovic2020inferring}. While the mechanical properties of the VM and SPV have been extensively studied, most works to date focused on the long-time behavior, e.g., by studying the quasistatic shear modulus~\cite{bi2015density}, effective diffusion constant of cells that is related to the viscosity of tissues~\cite{bi2016motility}, and correlations between a structural property called ``softness'' and the likelihood of topological rearrangements of cells~\cite{tah2021quantifying}. The rheological properties of the VM that cover a broad range of time scales, however, have not yet been systematically explored. In this paper, we present a detailed study of the rheology of the VM by studying its response to applied oscillatory shear and bulk deformations of small amplitude, i.e., in the linear response regime. We measured the response stresses and used them to compute the storage and loss moduli in both the solid and fluid phases. We show that the dynamical response of the VM can be fitted to standard spring-dashpot viscoelastic models over seven decades in the driving frequency and that the solid-fluid transition can be tuned by applying pre-deformation to the system. Thus we argue that the VM makes a suitable basis for studies of tissue dynamics beyond the quasistatic limit.
\section*{\label{results}Results}
\noindent{\bf Vertex model dynamics.} In the VM, the state of an epithelial tissue is approximated as a polygonal tiling of the plane. The degrees of freedom are vertices, i.e., meeting points of three or more cell-cell junctions. In the simplest formulation, junctions are assumed to be straight lines. The energy of the VM is a quadratic function of cell areas and perimeters~\cite{farhadifar2007influence}, i.e.,
\begin{equation}
E=\sum_{C}\left[\frac{K_C}{2}\left(A_C-A_{C0}\right)^2+\frac{\Gamma_C}{2}\left(P_C-P_{C0}\right)^2\right], \label{eq:vm_energy}
\end{equation}
where $K_C$ and $\Gamma_C$ are the area and perimeter elastic moduli. $A_C$ and $A_{C0}$ are the actual and preferred areas of cell $C$, respectively. Similarly, $P_C$ and $P_{C0}$ are, respectively, the actual and preferred perimeters of the same cell. In this work, we assumed $K_C$, $\Gamma_C$, $A_{C0}$, and $P_{C0}$ to be identical for all cells (i.e., $K_C\equiv K,\Gamma_C\equiv\Gamma,A_{C0}\equiv A_0,P_{C0}\equiv P_0$). Furthermore, we fixed the values of $K$ and $A_0$, and measured the energy in units of $KA_0^2$, stresses in units of $KA_0$, and lengths in units of $A_0^{1/2}$. Since the ratio between the perimeter and area elastic moduli does not qualitatively change the behavior of the VM~\cite{farhadifar2007influence,bi2015density}, we fixed that ratio to $\Gamma/(KA_0)\approx0.289$. The only variable parameter in simulations was the preferred cell perimeter $P_0$, which sets the dimensionless cell-shape parameter, defined as the ratio $p_0=P_0/\sqrt{A_0}$.
The cell-shape parameter, $p_0$, plays a central role in determining whether the system behaves as a fluid or solid~\cite{bi2015density}. Bi, \emph{et al}.~\cite{bi2015density} argued that the rigidity transition occurs at $p_0=p_c\approx3.812$ for a random polygonal tiling. The transition point is, however, at $p_c=\sqrt{8\sqrt{3}}\approx3.722$ for a regular hexagonal tiling~\cite{staple2010mechanics}. In the fluid phase, the energy barrier for neighbor exchanges vanishes and cells can flow past each other~\cite{bi2014energy}. As $p_0$ is reduced below $p_c$, the energy barrier becomes finite, neighbor exchanges cease and the system becomes solid. While the transition point for hexagonal tilings can be understood in terms of the mechanical stability and the excess perimeter~\cite{moshe2018geometric}, the mechanism that leads to a larger value for random tilings is more subtle and not fully understood ~\cite{yan2019multicellular}.
\begin{figure*}[htbp]
\centering
\includegraphics[width=0.95\textwidth]{Fig1.pdf}
\caption{Loss and storage shear moduli in the solid (top row) and fluid phase (bottom row). An overlay of the representative reference (grey) and sheared (yellow) configurations in (a)~the solid and (b)~the fluid phase. The magnitude of the shear is highly exaggerated for demonstration purposes. (c-d) show the representative storage ($G^\prime$) and loss ($G^{\prime\prime}$) shear moduli as functions of the shearing frequency, $\omega_0$, for different values of the cell-shape parameter, $p_0$, where we removed the dissipative contribution to $G^{\prime\prime}$ due to the velocity field resulting from the affine part of the deformation that dominates at large frequencies (see text).
Dashed curves are the fits based on (c)~the Standard Linear Solid (SLS) model in the solid phase and (d)~the Burgers model in the fluid phase. (e-f) The collapse of the moduli curves for different values of $p_0$ for (e)~the solid phase and (f)~the fluid phase. The insets show the representation of (e)~the SLS model and (f)~the Burgers model in terms of the springs and dashpots. The majority of the data corresponds to the system of nearly square shape with $N_x=15$ cells in the horizontal direction, and we also show examples of larger systems with $N_x=37$ and $N_x=51$ cells in the horizontal direction.}
\label{fig:simpleShear_rheology}
\end{figure*}
For definiteness, in all simulations we always started from a regular hexagonal tiling in a nearly square box subject to periodic boundary conditions, where the initial cell areas $A_C$ matched the preferred areas $A_0$ (see Methods). For the solid phase with $p_0\lesssim3.722$, a hexagonal tiling is the ground state of the energy in Eq.~(\ref{eq:vm_energy})~\cite{staple2010mechanics} and it was directly used to investigate rheological properties. Note that there was some residual hydrostatic stress (due to the mismatch of cell perimeters $P_C$ from the preferred values $P_0$), which could be eliminated by the appropriate relaxation of the simulation box. This hydrostatic stress, however, does not qualitatively affect the rheological behavior of the system (see Supplementary Information, Sec.~\ref{SI:hydrostatic_stress} for further discussion). For the fluid phase with $p_0\gtrsim3.722$, the hexagonal tiling corresponds to a saddle point of the energy in Eq.~(\ref{eq:vm_energy})~\cite{staple2010mechanics}. Thus a small random perturbation was applied to each vertex by a displacement drawn from the Gaussian distribution with zero mean and standard deviation $1.5\times10^{-4}\sqrt{A_0}$ (see Methods) and then the system was relaxed using the FIRE algorithm~\cite{bitzek2006structural} to reach a local energy minimum. Note that the energy landscape in the fluid phase has many local minima and a large number of soft modes (see Supporting Information, Sec.~\ref{SI:dynamical_matrix}). Thus we repeated simulations to investigate rheological properties for multiple configurations corresponding to different local energy minima (see Methods).
In order to probe the dynamical response of the VM, we need to specify the microscopic equations of motion for vertices. Assuming the low Reynolds number limit, which is applicable to most cellular systems due to their slow speed, inertial effects can be neglected~\cite{purcell1977life}. The equations of motion are then a force balance between friction and elastic forces due to deformations of cell shapes, i.e.,
\begin{equation}\label{Eq:overdamped_dynamics}
\gamma\dot{\mathbf{r}}_i=\mathbf{F}_i.
\end{equation}
Here, $\mathbf{r}_i$ is the position vector of vertex $i$ in a laboratory frame of reference, $\mathbf{F}_i=-\nabla_{\mathbf{r}_i}E$ is the mechanical force on vertex $i$ due to deformation of cells surrounding it, $\gamma$ is the friction coefficient, and dot denotes the time derivative. In simulations we fixed the value of $\gamma$, which sets the unit of time as $\gamma/\left(KA_0\right)$.
We note that a precise model for dissipation in epithelial tissues is at present not known. While having a mechanism of dissipating the external energy input is central in this study, its precise microscopic details are, however, not important. We therefore assume that each vertex experiences dissipative drag proportional to its instantaneous velocity, which is a common assumption in discrete models of soft materials~\cite{frenkel2001understanding}. Furthermore, we neglected thermal fluctuations of vertices and hence omit the stochastic term in Eq.~(\ref{Eq:overdamped_dynamics}). This is a reasonable assumption since typical energy scales in tissues significantly exceed the thermal energy, $k_BT$, at room temperature $T$, where $k_B$ is the Boltzmann constant. It is important, however, to note that in epithelia there are other sources of stochasticity (e.g., fluctuations of the number of force-generating molecular motors) that are important for tissue scale behaviors~\cite{curran2017myosin}. Here, we do not consider such effects but point out that they could be directly included in the model as additional forces in Eq.~(\ref{Eq:overdamped_dynamics}).
\noindent{\bf Response to a shear deformation.} The hexagonal ground state in the solid phase and states corresponding to local energy minima in the fluid phase were then used to investigate the rheological behavior by applying an oscillatory affine shear deformation (Fig. \ref{fig:simpleShear_rheology}a,b). At each time step, we first applied the affine shear deformation to the simulation box and all vertices that was followed by internal relaxation of vertices according to Eq.~(\ref{Eq:overdamped_dynamics}) (see Methods). The affine shear deformation can be described by a deformation gradient tensor defined as $\hat{\boldsymbol{F}}=\partial\mathbf{x}/\partial\mathbf{X}_0$, where the mapping $\mathbf{x}=\mathbf{x}\left(\mathbf{X}_0,t\right)$ maps the reference configuration $\mathbf{X}_0$ to a spatial configuration $\mathbf{x}$ at time $t$. For simple shear, the deformation gradient tensor is $\hat{\boldsymbol{F}}=\big(\begin{smallmatrix}1 & \epsilon(t)\\ 0 & 1\end{smallmatrix}\big)$, where $\epsilon\left(t\right)=\epsilon_0\sin\left(\omega_0 t\right)$. Sufficiently small amplitude $\epsilon_0=10^{-7}\ll1$ was used to probe the linear response properties.
The response stress tensor, $\hat{\boldsymbol{\sigma}}_C\left(t\right)$, for each cell $C$ was computed using the formalism introduced in \cite{nestor2018relating} (see Supplementary Information, Sec.~\ref{SI:stress}). The average stress tensor $\hat{\boldsymbol{\sigma}}\left(t\right)=\sum_C w_C\hat{\boldsymbol{\sigma}}_C\left(t\right)$, with $w_C = A_C/\sum_C A_C$, was used as a measure for the response of the system. The dynamic shear modulus $G^*\left(\omega_0\right)=\tilde{\tau}(\omega_0)/\tilde{\epsilon}(\omega_0)$ was then calculated at a given frequency $\omega_0$ of applied shear strain, where $\tilde{\tau}(\omega)$ and $\tilde{\epsilon}(\omega)$ are the Fourier transforms of the response shear stress $\tau(t)=\hat{\sigma}_{xy}\left(t\right)$ and the applied strain $\epsilon(t)$, respectively (see Methods). We ensured that the simulations were sufficiently long for a steady state to be reached (see Methods and Supplementary Information, Sec.~\ref{SI:steady_state}). The real part of the dynamic shear modulus, $G^\prime=\text{Re}\left(G^*\right)$, is the storage shear modulus and the imaginary part, $G^{\prime\prime}=\text{Im}\left(G^*\right)$, is the loss shear modulus. The storage shear modulus corresponds to the in-phase response and measures the elastic (i.e., reversible) response of the system, while the loss shear modulus corresponds to the out-of-phase response and measures the irreversible dissipation~\cite{larson1999structure} (see also Supplementary Information, Sec.~\ref{SI:rheology}). Note that the stress contribution due to the friction with the substrate resulting from the part of the velocity field due to the externally imposed affine deformation was removed in the analysis. This is because at high frequencies, the stresses are dominated by the friction due to external driving that masks the internal material response that we are interested in exploring (see Fig.~\ref{fig:moduli_affineVelocity} and Supplementary Information, Sec.~\ref{SI:stress}).
For systems under an oscillatory simple shear, storage and loss shear moduli were obtained for different values of $p_0$ and different system sizes in the solid and the fluid phases with frequencies $\omega_0$ of the applied shear strain spanning over seven orders of magnitude, as shown in Fig.~\ref{fig:simpleShear_rheology}c,d. Most simulations were performed for systems with nearly square shapes with $N_x=15$ cells in the horizontal direction. We repeated several simulations for systems with $N_x=37$ and $N_x=51$, which showed that the finite size effects are negligible (Fig.~\ref{fig:simpleShear_rheology}c-f).
In the solid phase there are two different regimes (see Fig.~\ref{fig:simpleShear_rheology}c). At low frequencies, $\omega_0$, the storage shear modulus $G^{\prime}$ has a constant value, while the loss shear modulus scales as $G^{\prime\prime}\propto\omega_0$. At high frequencies the storage shear modulus $G^{\prime}$ has a higher constant value, while the loss shear modulus scales as $G^{\prime\prime}\propto\omega_0^{-1}$. Such rheological behavior is characteristic for the Standard Linear Solid (SLS) model~\cite{larson1999structure}. Storage and loss shear moduli for the SLS model are~\cite{larson1999structure}, respectively,
\begin{subequations}
\label{G_SLS}
\begin{align}
G_\text{SLS}^\prime\left(\omega_0\right)&=\frac{E_2+\frac{\eta_1^2}{E_1^2}\omega_0^2\left(E_1+E_2\right)}{1+\frac{\eta_1^2}{E_1^2}\omega_0^2},\label{eq:Gprime}\\
G_\text{SLS}^{\prime\prime}\left(\omega_0\right)&=\frac{\omega_0\eta_1}{1+\frac{\eta_1^2}{E_1^2}\omega_0^2}\label{eq:Gprimeprime},
\end{align}
\end{subequations}
where we used the representation of the SLS model (Fig.~\ref{fig:simpleShear_rheology}e, inset) that consists of a spring with elastic constant $E_2$ connected in parallel with a Maxwell element, which comprises a spring with elastic constant $E_1$ and a dashpot with viscosity $\eta_1$ connected in series.
The above expressions in Eqs.~(\ref{G_SLS}) were used to fit the storage and loss shear moduli obtained from simulations. The fitted curves, represented with dashed lines in Fig.~\ref{fig:simpleShear_rheology}c, show an excellent match with the simulation data, indicating that the SLS model is indeed appropriate to describe the shear rheology in the solid phase. This was also confirmed in Fig.~\ref{fig:simpleShear_rheology}e, where we collapsed the storage and loss shear moduli for different values of the shape parameter, $p_0$, by rescaling the moduli and frequencies with the fitted values of spring and dashpot constants.
As the value of the $p_0$ increases, we observe that the storage shear modulus reduces at all frequencies and that the loss shear modulus reduces at high frequencies. Furthermore the crossover between the two regimes shifts towards lower frequencies (Fig.~\ref{fig:simpleShear_rheology}c). This is because the elastic constants $E_1$ and $E_2$ decrease linearly with increasing $p_0$ and they become zero exactly at the solid-fluid transition with $p_0=p_c\approx 3.722$ (Fig.~\ref{fig:SimpleShear_constant}a).
The dashpot constant $\eta_1$ is nearly independent of $p_0$ and scales with the friction parameter $\gamma$, which is the only source of dissipation in the VM. The crossover between the two regimes for both the storage and loss shear moduli corresponds to a characteristic time scale, $\eta_1/E_1$, which diverges as $p_0$ approaches the solid-fluid transition due to the vanishing elastic constant (Fig.~\ref{fig:SimpleShear_constant}c). Note that the values of the elastic constants $E_1$ and $E_2$ can be estimated analytically. In the quasistatic limit ($\omega_0\rightarrow 0$), the external driving is sufficiently slow that the system can relax internally. In this limit, Murisic, \emph{et al.}~showed that the storage shear modulus is
\begin{equation}\label{Eq:Gquasi}
G^{\prime}(\omega_0\rightarrow 0)=E_2=\frac{1}{2}KA_0\big(1-[\alpha(p_0,\Gamma/KA_0)]^2\big),
\end{equation}
where $\alpha(p_0,\Gamma/KA_0)$ is a scaling factor chosen such that the hydrostatic stress vanishes once the system box size is rescaled from $L$ to $\alpha L$~\cite{murisic2015discrete}. In the high frequency limit ($\omega_0\rightarrow \infty$), on the other hand, the system follows the externally imposed affine deformation and has no time for internal relaxation. Thus, by considering the energy cost for a hexagonal tiling under affine deformation, we obtained the storage shear modulus (see Supplemental Information, Sec.~\ref{SI:tuning_phase_transition})
\begin{equation}
G^{\prime}(\omega_0\rightarrow \infty)=E_1+E_2=3\sqrt{3}\Gamma\left(1-\frac{p_0}{p_c}\right).\label{Eq:Gaffine}
\end{equation}
The above Eqs.\ (\ref{Eq:Gquasi}) and (\ref{Eq:Gaffine}) were used to extract the values of elastic constants $E_1$ and $E_2$, which showed excellent agreement with the fitted values from simulations (Fig.~\ref{fig:SimpleShear_constant}a).
\begin{figure}[tbp]
\centering
\includegraphics[width=0.98\columnwidth]{Fig2.pdf}
\caption{(a-b) Fitted values of spring-dashpot models for the system under simple shear. (a)~Elastic constants as a function of target cell-shape parameter, $p_0$. In the solid phase (i.e., for $p_0 < p_c\approx3.722$), fitted values of the spring constants show excellent match with the analytical predictions obtained from Eqs.~(\ref{Eq:Gquasi}) and (\ref{Eq:Gaffine}) (dashed lines). Inset shows the spring constants near the critical point. (b)~Dashpot viscosity constants as a function of the target cell-shape parameter, $p_0$. (c-d) Characteristic time scales in (c)~the solid and (d)~fluid phase obtained from the fitted values of the elastic constant and the dashpot viscosity. The normalization factor $t^*=\gamma/(K A_0)$ sets the unit of time. For the fluid phase (i.e., for $p_0 > p_c\approx3.722$), errorbars correspond to the standard deviation for simulations that were repeated for configurations that correspond to different local energy minima.}
\label{fig:SimpleShear_constant}
\end{figure}
In the fluid phase, the storage and loss shear moduli show a markedly different behavior (Fig.~\ref{fig:simpleShear_rheology}d). There are three different regimes with two crossover frequencies, which correspond to two characteristic time scales. At low frequencies, $\omega_0$, the storage shear modulus $G^\prime\propto \omega_0^2$ and the loss shear modulus $G^{\prime\prime}\left(\omega_0\right)\propto \omega_0$. The storage modulus approaches $0$ for $\omega_0\to0$, which indicates that the system is indeed a fluid. At high frequencies the storage shear modulus has a constant value, while the loss shear modulus scales as $G^{\prime\prime}\left(\omega_0\right)\propto \omega_0^{-1}$. To capture this behavior we used the Burgers model, which consists of two Maxwell models connected in parallel (Fig.~\ref{fig:simpleShear_rheology}f, inset), to fit the shear moduli measured in the simulations. The storage and loss shear moduli for a Burgers model are~\cite{larson1999structure}, respectively,
\begin{subequations}
\begin{align}
G_\text{Burg}^\prime\left(\omega_0\right)&=\frac{p_{1} q_{1} \omega_0^{2}-q_{2} \omega_0^{2}\left(1-p_{2} \omega_0^{2}\right)}{p_{1}^{2} \omega_0^{2}+\left(1-p_{2} \omega_0^{2}\right)^{2}},\label{Gp_liq}\\
G_\text{Burg}^{\prime\prime}\left(\omega_0\right)&=\frac{p_{1} q_{2} \omega_0^{3}+q_{1} \omega_0\left(1-p_{2} \omega_0^{2}\right)}{p_{1}^{2} \omega_0^{2}+\left(1-p_{2} \omega_0^{2}\right)^{2}},\label{Gpp_liq}
\end{align}
\end{subequations}
where $p_{1}=\eta_{1}/E_{1}+\eta_{2}/E_{2}$, $p_2=\eta_1\eta_2/(E_1E_2)$, $q_1=\eta_1+\eta_2$, $q_2=\eta_1\eta_2(E_1+E_2)/(E_1E_2)$. The dashed curves in Fig.~\ref{fig:simpleShear_rheology}d show fits of the storage and loss shear moduli for a range of values of $p_0$, which show good agreement with simulations. Unlike for the solid phase, it is not possible to collapse the data for storage and loss shear moduli onto single universal curves because the fluid phase is characterized by two independent timescales $\eta_1/E_1$ and $\eta_2/E_2$. Thus we show two different collapses for the storage and loss shear moduli in the low frequency range (Fig.~\ref{fig:simpleShear_rheology}f) and in the high frequency range (Fig.~\ref{fig:rescale_fluid_highF} in the Supplementary Information, Sec.~\ref{SI:collapse_fluid_high_freq}).
As the value of the $p_0$ decreases, we observe that both the storage and loss shear moduli reduce at intermediate and high frequencies, but they increase at low frequencies (Fig.~\ref{fig:simpleShear_rheology}d). We also observe that the first crossover shifts towards lower frequencies, while the second crossover remains at approximately the same frequency. This is because the elastic constants $E_1$ and $E_2$ decrease linearly toward zero as $p_0$ approaches the solid-fluid transition at $p_c\approx3.722$ (Fig.~\ref{fig:SimpleShear_constant}a). The dashpot constant $\eta_2$ also decreases linearly toward zero, while the dashpot constant $\eta_1$ increases but remains finite as $p_0$ approaches the solid-fluid transition (Fig.~\ref{fig:SimpleShear_constant}b). As a consequence, one of the characteristic time scales $\eta_1/E_1$ diverges, while the second time scale $\eta_2/E_2$ remains finite as as $p_0$ approaches the solid-fluid transition (Fig.~\ref{fig:SimpleShear_constant}d). The diverging characteristic time scale captures the macroscopic behavior of the system, while the second time scale ($\sim \gamma/KA_0$) captures the microscopic details of the VM.
Finally, we note that the values of the spring and dashpot constants are somewhat sensitive to the local energy minimum configuration used to probe the response in the fluid phase. The errorbars in Fig.~\ref{fig:SimpleShear_constant} show standard deviation for different configurations that were obtained by using the same magnitude of the initial perturbation (see Methods). In Fig.~\ref{fig:initial_kick} in the Supplementary Information, Sec.~\ref{Sup:initial_perturbation}, we show how the values of the spring and dashpot constants are affected when configurations were obtained by using different magnitudes of the initial perturbation.
\begin{figure*}[htbp]
\centering
\includegraphics[width=0.95\textwidth]{Fig3.pdf}
\caption{Loss and storage bulk moduli in the solid (top row) and fluid phase (bottom row). An overlay of the representative reference (grey) and biaxially deformed (yellow) configurations in (a)~the solid and (b)~the fluid phase. The magnitude of the bulk deformation is highly exaggerated for demonstration purposes. (c-d) show the representative storage ($B^\prime$) and loss ($B^{\prime\prime}$) bulk moduli as functions of the deformation frequency, $\omega_0$, for different values of the cell-shape parameter, $p_0$, where we removed the dissipative contribution to $B^{\prime\prime}$ due to the velocity field resulting from the affine part of the deformation that dominates at large frequencies (see text). For the solid phase in (c), the loss bulk modulus $B^{\prime\prime}\equiv0$. For the fluid phase in (d), dashed curves are the fits based on the Standard Linear Solid (SLS) model. (e-f) The collapse of the moduli curves for different values of $p_0$ for (e)~the solid phase and (f)~the fluid phase. The insets show the representation of (e)~the spring model and (f)~the SLS model in terms of the springs and dashpots.}
\label{biaxial_rheology}
\end{figure*}
\noindent{\bf Response to bulk deformations.} We further studied the bulk rheological properties of the system by applying an oscillatory biaxial deformation (Fig.~\ref{biaxial_rheology}a,b) described by the deformation gradient $\hat{\boldsymbol{F}}=\big(\begin{smallmatrix}1+\epsilon(t)&0\\0&1+\epsilon(t)\end{smallmatrix}\big)$,
where $\epsilon(t)=\epsilon_0\sin\left(\omega_0 t\right)$. We applied a sufficiently small amplitude $\epsilon_0=10^{-7}\ll1$ to probe the linear response properties characterized by the average normal stress $\sigma(t)=\frac{1}{2}\left[\hat \sigma_{xx}(t)+\hat \sigma_{yy}(t)\right]$, where we again removed the stress contribution due to the friction with the substrate resulting from the part of the velocity field due to the externally imposed affine deformation. As in the simple shear test, we then computed the dynamic bulk modulus as $B^*(\omega_0)=\tilde \sigma (\omega_0)/\tilde \epsilon(\omega_0)$ from which we obtained the storage bulk modulus $B^\prime=\text{Re}\left(B^*\right)$ and the loss bulk modulus $B^{\prime\prime}=\text{Im}\left(B^*\right)$ (see Fig.~\ref{biaxial_rheology}c,d).
In the solid phase, the storage bulk modulus is independent of the driving frequency and the loss bulk modulus is zero. Thus the response of the system can be captured by a single spring $E_\text{solid}$ (Fig.~\ref{biaxial_rheology}e, inset). This is because the hexagonal tiling is stable to biaxial deformation in the solid phase and thus there is no internal relaxation. The measured value of the storage bulk modulus matches the analytical prediction
\begin{equation}
B_\text{theory}=2KA_0+\sqrt[\leftroot{-1}\uproot{0}4]{12}\Gamma p_0
\label{eq:bulk_solid}
\end{equation}
by Staple, \emph{et al.}~\cite{staple2010mechanics},
where the hexagonal tiling is assumed to undergo affine deformation under biaxial deformation. Storage bulk moduli, normalized by $B_\text{theory}$, for different values of $p_0$ all collapse to 1 (Fig.~\ref{biaxial_rheology}e).
In the fluid phase, the bulk response behavior of the system can be described by the SLS model (Fig.~\ref{biaxial_rheology}f, inset). While it might appear counter-intuitive to model a fluid with the SLS model, this is a direct consequence of the fact that in the fluid state, the bulk modulus is finite but the shear modulus vanishes, i.e., the fluid flows in response to shear but resists bulk deformation. The fitted storage and loss bulk moduli for the SLS model [see Eq.~(\ref{G_SLS})] show an excellent match with the simulation data (Fig.~\ref{biaxial_rheology}d). This was also confirmed in Fig.~\ref{biaxial_rheology}f, where we collapsed the storage and loss bulk moduli for different values of $p_0$.
\begin{figure}[bp]
\centering
\includegraphics[width=0.98\columnwidth]{Fig4.pdf}
\caption{Fitted values of spring-dashpot models for the system under bulk deformation as a function of the target cell-shape parameter, $p_0$. (a)~In the solid phase ($p_0 < p_c\approx3.722$), the bulk storage modulus $E_{\text{solid}}$ agrees with the analytical prediction $B_{\text{theory}}$ in Eq.~(\ref{eq:bulk_solid}). At the solid-fluid transition point ($p_0=p_c\approx 3.722$), it continuously changes to the high frequency limit of the bulk storage modulus, i.e., $B^{\prime}(\omega_0\rightarrow \infty)=E_1+E_2$, of the fluid phase. The low frequency limit of the bulk storage modulus is $B^{\prime}(\omega_0\rightarrow 0)=E_1$ in the fluid phase. (b)~Dashpot viscosity constant as a function of $p_0$. For the fluid phase ($p_0 > p_c\approx3.722$), errorbars correspond to the standard deviation for simulations that were repeated for configurations that correspond to different local energy minima. }
\label{biaxial_constant}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=0.95\textwidth]{Fig5.pdf}
\caption{Tuning the solid to fluid transition by applying uniaxial deformation. (a)~The solid-fluid transition boundary in the $a-p_0$ plane, where $a$ measures the amount of uniaxial deformation described by the deformation gradient $\hat{\boldsymbol{F}}=\big(\begin{smallmatrix}a & 0 \\ 0 & 1\end{smallmatrix}\big)$. Blue line shows the analytical prediction from Eq.~(\ref{Eq:phaseBoundary}), which matches the stability analysis with the Hessian matrix (red dots). (b,c)~The fitted values of the (b)~spring and (c)~dashpot constants for the SLS model in the solid phase and the Burgers model in the fluid phase when the system is under uniaxial compression ($a=0.95$), no pre-deformation ($a=1$), and under uniaxial tension ($a=1.05$). }
\label{transition_uniaxial}
\end{figure*}
The fitted values of spring elastic and dashpot viscosity constants for different values of $p_0$ are plotted in Fig.~\ref{biaxial_constant}. In the fluid phase, the storage bulk modulus in the high frequency limit ($B^{\prime}(\omega_0\rightarrow \infty)=E_1+E_2$) continuously increases from the value for the solid phase $B_{\text{theory}}$ in Eq.~(\ref{eq:bulk_solid}) as the system transitions from solid to fluid (Fig.~\ref{biaxial_constant}a). The storage bulk modulus in the quasistatic limit ($B^{\prime}(\omega_0\rightarrow 0)=E_2$) emerges at the transition point with a finite value and increases as $p_0$ increases from $p_c$ (Fig.~\ref{biaxial_constant}a). Fig.~\ref{biaxial_constant}b shows that the dashpot constant diverges as the $p_0$ decreases toward $p_c$. Thus, the characteristic time scale $\eta_1/E_1$ also diverges, but for a different reason than for the shear deformation, where the spring constant is vanishing (see Fig.~\ref{fig:SimpleShear_constant}). Finally, we note that, unlike for the response to shear, the values of the spring and dashpot constants for bulk deformation are not sensitive to the local energy minimum configuration used to probe the response in the fluid phase, which is reflected in very small errorbars in Fig.~\ref{biaxial_constant}. This is because the bulk moduli are dominated by the changes in cell areas.
{\bf Response to a shear deformation of a uniaxially pre-deformed system.} The solid-fluid transition for the regular hexagonal tiling occurs when $p_0\approx3.722$, above which the hexagonal tiling is unstable. This is consistent with the vanishing of the affine shear modulus in Eq.~(\ref{Eq:Gaffine}) at the transition point. If the regular hexagonal tiling is compressed or stretched uniaxially by a factor $a$, which is described by the deformation gradient $\hat{\boldsymbol{F}}=\big(\begin{smallmatrix}a & 0 \\ 0 & 1\end{smallmatrix}\big)$, then the high frequency limit of the storage shear modulus that is dominated by affine deformation becomes (see Supplemental Information, Sec.~\ref{SI:tuning_phase_transition}),
\begin{equation}
\begin{split}
G^{\prime}_\text{affine}\left(a\right)=&\frac{2\sqrt{2}\Gamma}{3^{7/4}a}\left(1+\frac{1}{\left(1+3a^2\right)^\frac{3}{2}}\right)\\
& \times \left(-3p_0+\sqrt[4]{192}\left(1+\sqrt{1+3a^2}\right)\right).
\end{split}
\end{equation}
By setting the affine shear modulus to 0, we obtained the solid-fluid transition boundary in the $a-p_0$ plane as
\begin{equation}\label{Eq:phaseBoundary}
a\left(p_0\right) = \sqrt{\frac{\sqrt{3}p_0^2}{8}-\frac{p_0}{\sqrt[4]{12}}}.
\end{equation}
The above analytical prediction for the phase boundary (Fig.~\ref{transition_uniaxial}a, blue line) shows an excellent agreement with the stability analysis in terms of the eigenvalues of the Hessian matrix of the energy function (Fig.~\ref{transition_uniaxial}a, red dots). A given configuration is stable if all eigenvalues of the Hessian matrix are positive and the loss of mechanical stability occurs when the lowest eigenvalue becomes $0$. For a given $p_0$, the value of the lowest eigenvalue reduces with decreasing $a$, i.e., as the magnitude of compression is increased. Thus, the compression (stretching) shifts the solid-fluid transition towards the lower (higher) values of $p_0$ (see Fig.~\ref{transition_uniaxial}a).
We also probed the response to oscillatory shear applied to uniaxially compressed and stretched systems. This analysis was done on both the uniaxially deformed hexagonal tiling in the solid phase as well as a system in the fluid phase obtained by relaxing the unstable uniaxially deformed hexagonal tiling after an initial random perturbation (see Methods). The response to the shear deformation is qualitatively similar and can still be described by the SLS model in the solid phase and the Burgers model in the fluid phase. Fig.~\ref{transition_uniaxial}b,c shows fitted values of the parameters for spring-dashpot models when the system is under uniaxial compression ($a=0.95$), no pre-deformation ($a=1$, i.e., same as Fig.~\ref{fig:SimpleShear_constant}a,b), and uniaxial tension ($a=1.05$). In both the solid and fluid phases, all spring elastic constants decrease to $0$ as $p_0$ approaches the critical value predicted by Eq.~(\ref{Eq:phaseBoundary}). The dashpot viscosity remains constant in the solid phase. Once the system enters the fluid phase as $p_0$ increases, a new second dashpot constant emerges and increases from $0$, while the value of the first dashpot constant decreases. As in the simple shear case, we note that the values of the spring and dashpot constants are somewhat sensitive to the local energy minimum configuration used to probe the response in the fluid phase. The errorbars in Fig.~\ref{fig:SimpleShear_constant} show standard deviation for configurations that were obtained by using different random initial perturbation (see Methods).
Finally, we note that besides uniaxial deformation, the solid-fluid transition point can be tuned by other modes of deformation (see Fig.~\ref{fig:PhaseDiagram} and Supplementary Information, Sec.~\ref{SI:tuning_phase_transition}).
\section*{Discussion}
We have performed a detailed analysis of the rheological properties of the Vertex Model subject to small-amplitude oscillatory deformations over seven orders of magnitude in the driving frequency. Our analysis shows that the VM exhibits non-trivial viscoelastic behavior that can be tuned by a single dimensionless geometric parameter - the shape parameter, $p_0$. In order to characterize the response, we constructed constitutive rheological models that use combinations of linear springs and dashpots connected in series and in parallel. These models allowed us to match the shear response of the VM to that of the Standard Linear Solid model in the solid phase and the Burgers model in the fluid phase. In the low-frequency, i.e., quasistatic regime, our results are fully consistent with many previous studies \cite{staple2010mechanics,murisic2015discrete,bi2015density,bi2016motility}. Our work, however, provides insights into the time-dependent response of the VM over a broad range of driving frequencies, which is important if one is to develop full understanding of the rheological properties of the VM and how those inform our understanding of the rheology of epithelial tissues.
We also showed that the critical value for the solid-fluid transition can be tuned by applying pre-deformation. Interestingly, under uniaxial and biaxial (i.e., isotropic) compression the solid to fluid transition shifts to lower values of $p_0$, leading to the non-intuitive prediction that one can fluidize the system by compressing it. This is, however, not surprising since the transition is driven by a geometric parameter that is inversely proportional to the square root of the cell's native area. Compressing the system reduces its area and, hence, effectively increases $p_0$. It is, however, important to note that this is just a property of the VM and it does not necessarily imply that actual epithelial tissue would behave in the same way. Cells are able to adjust their mechanical properties in response to applied stresses and it would be too simplistic to assume that compression would directly lead to changes in the native area. In fact, experiments on human bronchial epithelial cells show that applying apical-to-basal compression, which effectively expands the tissue laterally (i.e., corresponds to stretching in our model) fluidizes the tissue~\cite{park2015unjamming}.
Furthermore, the transition from solid phase to fluid phase is accompanied by the emergence of a large number of soft modes. As we have already alluded, it has recently been argued that these soft modes lead to a nonlinear response distinct from that obtained in classical models of elasticity~\cite{moshe2018geometric}. Approximately half of the eigenmodes are zero modes (see Fig.~\ref{fig:DOS} in the Supplementary Information, Sec.~\ref{SI:dynamical_matrix}). While the analysis of soft modes in the VM is an interesting problem~\cite{yan2019multicellular}, it is beyond the scope of this work.
Other models in this class have intriguing non-trivial mechanical properties, such as the existence of topologically protected modes~\cite{kane2014topological,lubensky2015phonons,paulose2015topological,huber2016topological,rocklin2017transformable,mao2018maxwell}.
We note that while we studied the dynamical response over a wide range of frequencies, our work focused on the behavior in the linear response regime, where there are effectively no plastic events, i.e., while being allowed, T1 transformations typically did not occur during the process of probing the rheology. A full understanding of the VM rheology would also need to allow for cell rearrangements. This is, however, a very challenging problem and first steps in addressing it have only recently been made~\cite{popovic2020inferring}.
Regardless of whether cells in an epithelial tissue are arrested or able to move, the rheological response of the tissue is viscoelastic with multiple time scales~\cite{bonfanti2020unified}. This response arises as a result of the complex material properties of individual cells combined with four basic cellular behaviors: movement, shape change, division, and differentiation. The tissue not only has a non-trivial rheological response but is also able to tune it. There is growing evidence that this ability of biological systems to tune their rheology, and in particular, transition between solid-like and fluid-like behaviors, plays a key role during morphogenesis \cite{petridou2019tissue}. How such cellular processes are regulated and coordinated to form complex morphological structures is only partly understood. It is, however, clear that the process involves mechano-chemical feedback between mechanical stresses and the expression of genes that control the force-generating molecular machinery in the cell. Any models that aim to describe morphological processes, therefore, need to include coupling between biochemical processes and mechanical response. The base mechanical model, however, must be able to capture the underlying viscoelastic nature of tissues. Our work provides evidence that the vertex model, a commonly used model to study mechanics of epithelial tissues, has interesting non-trivial rheological behavior. This, combined with its ability to capture both fluid- and solid-like behavior by tuning a single geometric parameter shows it to be an excellent base model to build more complex descriptions of real tissues.
\section*{Methods}
Most simulations were performed for systems with nearly square shapes with $N_x=15$ cells in horizontal direction (i.e., $N=240$ cells in total) subject to periodic boundary conditions. The shape of the simulation box was chosen to be as near to a square as allowed by the geometry of a hexagon and the size of the box was such that it accommodated $N$ cells of area $A_C$ that matched the preferred areas $A_0$. Simulations of larger system sizes ($N_x=37$, $51$, i.e., $N=1406$, $2652$ cells, respectively) were performed for a subset of values of $p_0$ to explore the finite size effects. No quantitative differences between the system with $N=240$ cells and larger systems were observed. Since the ratio between the perimeter and elastic moduli does not qualitatively change the behavior of the VM~\cite{bi2015density,farhadifar2007influence}, we used $\Gamma/(KA_0)\approx0.289$ for all simulations.
We applied an oscillatory affine deformation of frequency $\omega_0$ either as simple shear described by the deformation gradient $\hat{\boldsymbol{F}}=\big(\begin{smallmatrix}1&\epsilon(t)\\0&1\end{smallmatrix}\big)$ or biaxial deformation described by $\hat{\boldsymbol{F}}=\big(\begin{smallmatrix}1+\epsilon(t)&0\\0&1+\epsilon(t)\end{smallmatrix}\big)$, where $\epsilon(t)=\epsilon_0\sin(\omega_0 t)$. In all simulations, we used a small magnitude of deformation, i.e., $\epsilon_0=10^{-7}$, so that we probed the linear response and the measured moduli were independent of the magnitude of the deformation. In every time step, the system evolved according to the overdamped dynamics in Eq.~(\ref{Eq:overdamped_dynamics}) after the affine deformation was applied. Equations of motion were integrated using the first order Euler method with the time step, $\Delta t \approx 0.00866\gamma/(KA_0)$. Measurements of the response stresses for each cell and the entire system were taken 25 times within each cycle of oscillatory deformation by using Eq.~(\ref{eq:stress_tensor}) in the Supporting Information, Sec.~\ref{SI:stress}.
To ensure that we were probing the steady state, we performed the following analysis. For example, in the case of shear deformation, the shear stress signal $\tau(t)$ was divided into blocks of length $T=3T_0$, each containing 3 cycles of the time period $T_0=2 \pi/\omega_0$ of the driving shear deformation. Within each block $n$, we performed the Fourier transform of $\tau(t)$ and obtained $\tilde{\tau}_n(\omega)$ as
\begin{equation}
\tilde{\tau}_n(\omega)=\frac{1}{T}\int_{(n-1)T}^{nT}\tau(t)e^{i\omega t}dt,
\end{equation}
where $n$ is a positive integer. Similar Fourier transform analysis was performed for the strain, $\epsilon(t)$. The length of the simulation was chosen such that it contained a sufficient number of blocks in order for the $\tilde{\tau}_n(\omega_0)$ to reach a steady state value $\tilde{\tau}(\omega_0)$. The obtained steady state value of $\tilde{\tau}(\omega_0)$ was used to calculate the storage and loss shear moduli as described in the main text. Analogous procedure was applied to the normal stress, $\sigma(t)$, in the case of the bulk deformation. Please refer to the Supplementary Information, Sec.~\ref{SI:steady_state} for a representative example of the steady state analysis.
In the solid phase, we performed rheological tests on a hexagonal tiling. In the fluid phase, however, the hexagonal tiling is unstable. Instead, the hexagonal tiling was randomly perturbed and then relaxed to a nearby local stable state using the FIRE optimizer~\cite{bitzek2006structural}. The local energy minimum was determined with the relative accuracy of $10^{-12}$. A random perturbation was applied to each vertex $i$, i.e., each vertex was displaced from its original position in the hexagonal tiling by a vector $\delta \mathbf{r}_i = \delta x_i \mathbf{e}_x + \delta y_i \mathbf{e}_y$, where $x_i$ and $y_i$ were Gaussian random variables with zero mean and standard deviation $1.5\times10^{-4} \sqrt{A_0}$.
The rheology of a local stable state was then probed following the same procedure as in the solid case.
During the energy minimization and oscillatory deformations, T1 transitions were allowed but were not common. T1 transitions were implemented following the procedure introduced by Spencer, \emph{et al.}~\cite{spencer2017vertex}.
\section*{Acknowledgements}
This research was primarily supported by NSF through the Princeton
University's Materials Research Science and Engineering Center
DMR-2011750 and by the Project X Innovation Research Grant from the Princeton School of Engineering and Applied Science. RS acknowledges support by the UK BBSRC (Award
BB/N009789/1). This project was initiated during the KITP program ``Symmetry, Thermodynamics and Topology in Active Matter'' (ACTIVE20), and it is supported in part by the National Science Foundation under Grant No.\ NSF PHY-1748958. We would like to acknowledge useful discussions with Mikko Haataja.
\section*{Data availability} The data supporting the results and findings of this study is available from the corresponding authors upon reasonable request.
\section*{Code availability} Simulation and analysis codes used in this study are available from the corresponding authors upon reasonable request.
\section*{Contributions} ST, RS, and AK conceived the study and designed the project. ST and NS performed numerical simulations and analysed the data. RS developed the Vertex Model code used in numerical simulations. ST, NS, RS, and AK wrote the paper.
\section*{Competing interests}
The authors declare no competing interests.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 5,277
|
Peter Stebbings es un actor canadiense, más conocido por haber interpretado a Kevin Sharp en la serie Madison y a Paul Deeds en Traders.
Biografía
Peter sale con la actriz Charlotte Sullivan, la pareja se casó y en el 2015 se anunció que estaban esperando a su primer bebé juntos.
Carrera
En 1990 interpretó a Godwin en un episodio de la serie 21 Jump Street protagonizada por Johnny Depp.
En 1994 se unió al elenco recurrente de la serie Madison donde dio vida a Kevin Sharpe hasta 1998.
En 1997 se unió al elenco de la serie Traders donde interpretó a Paul Deeds, un banquero inversionista hasta el 2000.
En 1998 apareció en la serie The Outer Limits donde interpretó a Seth Todtman durante el episodio "Final Exam", más tarde en el 2000 apareció de nuevo en la serie durante el episodio "Revival" donde dio vida a Luke.
En el 2002 se unió al elenco de la serie Jeremiah donde interpretó a Markus Alexander el líder de Thunder Mountain quien busca formar nuevas alianzas con los sobrevivientes.
En el 2010 se unió al elenco recurrente de la serie The Murdoch Mysteries donde interpreta a James Pendrick.
En el 2011 se unió al elenco de la serie The Listener donde interpretó al abogado Alvin Klein, hasta el 2014. Ese mismo año apareció en la película Immortals donde interpretó a Helios, el dios del sol y un soldado ateniense.
En el 2013 apareció en la última temporada de la serie The Borgias, donde interpretó al Cardenal DeLuca.
En el 2014 se unió a la serie Crossbones donde interpretó a James Balfour, hasta el final de la serie ese mismo año luego de que fuera cancelada al finalizar su primera temporada debido al bajo rating.
Filmografía
Series de televisión
Películas
Director, escritor y productor
Teatro
Premios y nominaciones
Referencias
Enlaces externos
Peter Stebbings on TV.com
Peter Stebbings - Zimbio
Actores de Vancouver
Actores de televisión de Canadá
Nacidos en Vancouver
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,945
|
namespace hector_nav_msgs
{
template <class ContainerAllocator>
struct GetRecoveryInfoResponse_
{
typedef GetRecoveryInfoResponse_<ContainerAllocator> Type;
GetRecoveryInfoResponse_()
: trajectory_radius_entry_pose_to_req_pose()
, radius_entry_pose()
, req_pose() {
}
GetRecoveryInfoResponse_(const ContainerAllocator& _alloc)
: trajectory_radius_entry_pose_to_req_pose(_alloc)
, radius_entry_pose(_alloc)
, req_pose(_alloc) {
}
typedef ::nav_msgs::Path_<ContainerAllocator> _trajectory_radius_entry_pose_to_req_pose_type;
_trajectory_radius_entry_pose_to_req_pose_type trajectory_radius_entry_pose_to_req_pose;
typedef ::geometry_msgs::PoseStamped_<ContainerAllocator> _radius_entry_pose_type;
_radius_entry_pose_type radius_entry_pose;
typedef ::geometry_msgs::PoseStamped_<ContainerAllocator> _req_pose_type;
_req_pose_type req_pose;
typedef boost::shared_ptr< ::hector_nav_msgs::GetRecoveryInfoResponse_<ContainerAllocator> > Ptr;
typedef boost::shared_ptr< ::hector_nav_msgs::GetRecoveryInfoResponse_<ContainerAllocator> const> ConstPtr;
}; // struct GetRecoveryInfoResponse_
typedef ::hector_nav_msgs::GetRecoveryInfoResponse_<std::allocator<void> > GetRecoveryInfoResponse;
typedef boost::shared_ptr< ::hector_nav_msgs::GetRecoveryInfoResponse > GetRecoveryInfoResponsePtr;
typedef boost::shared_ptr< ::hector_nav_msgs::GetRecoveryInfoResponse const> GetRecoveryInfoResponseConstPtr;
// constants requiring out of line definition
template<typename ContainerAllocator>
std::ostream& operator<<(std::ostream& s, const ::hector_nav_msgs::GetRecoveryInfoResponse_<ContainerAllocator> & v)
{
ros::message_operations::Printer< ::hector_nav_msgs::GetRecoveryInfoResponse_<ContainerAllocator> >::stream(s, "", v);
return s;
}
} // namespace hector_nav_msgs
namespace ros
{
namespace message_traits
{
// BOOLTRAITS {'IsFixedSize': False, 'IsMessage': True, 'HasHeader': False}
// {'nav_msgs': ['/opt/ros/indigo/share/nav_msgs/cmake/../msg'], 'geometry_msgs': ['/opt/ros/indigo/share/geometry_msgs/cmake/../msg'], 'actionlib_msgs': ['/opt/ros/indigo/share/actionlib_msgs/cmake/../msg'], 'std_msgs': ['/opt/ros/indigo/share/std_msgs/cmake/../msg']}
// !!!!!!!!!!! ['__class__', '__delattr__', '__dict__', '__doc__', '__eq__', '__format__', '__getattribute__', '__hash__', '__init__', '__module__', '__ne__', '__new__', '__reduce__', '__reduce_ex__', '__repr__', '__setattr__', '__sizeof__', '__str__', '__subclasshook__', '__weakref__', '_parsed_fields', 'constants', 'fields', 'full_name', 'has_header', 'header_present', 'names', 'package', 'parsed_fields', 'short_name', 'text', 'types']
template <class ContainerAllocator>
struct IsFixedSize< ::hector_nav_msgs::GetRecoveryInfoResponse_<ContainerAllocator> >
: FalseType
{ };
template <class ContainerAllocator>
struct IsFixedSize< ::hector_nav_msgs::GetRecoveryInfoResponse_<ContainerAllocator> const>
: FalseType
{ };
template <class ContainerAllocator>
struct IsMessage< ::hector_nav_msgs::GetRecoveryInfoResponse_<ContainerAllocator> >
: TrueType
{ };
template <class ContainerAllocator>
struct IsMessage< ::hector_nav_msgs::GetRecoveryInfoResponse_<ContainerAllocator> const>
: TrueType
{ };
template <class ContainerAllocator>
struct HasHeader< ::hector_nav_msgs::GetRecoveryInfoResponse_<ContainerAllocator> >
: FalseType
{ };
template <class ContainerAllocator>
struct HasHeader< ::hector_nav_msgs::GetRecoveryInfoResponse_<ContainerAllocator> const>
: FalseType
{ };
template<class ContainerAllocator>
struct MD5Sum< ::hector_nav_msgs::GetRecoveryInfoResponse_<ContainerAllocator> >
{
static const char* value()
{
return "a93581be8e34e3c09aeafc6b9b990ad5";
}
static const char* value(const ::hector_nav_msgs::GetRecoveryInfoResponse_<ContainerAllocator>&) { return value(); }
static const uint64_t static_value1 = 0xa93581be8e34e3c0ULL;
static const uint64_t static_value2 = 0x9aeafc6b9b990ad5ULL;
};
template<class ContainerAllocator>
struct DataType< ::hector_nav_msgs::GetRecoveryInfoResponse_<ContainerAllocator> >
{
static const char* value()
{
return "hector_nav_msgs/GetRecoveryInfoResponse";
}
static const char* value(const ::hector_nav_msgs::GetRecoveryInfoResponse_<ContainerAllocator>&) { return value(); }
};
template<class ContainerAllocator>
struct Definition< ::hector_nav_msgs::GetRecoveryInfoResponse_<ContainerAllocator> >
{
static const char* value()
{
return "nav_msgs/Path trajectory_radius_entry_pose_to_req_pose\n\
geometry_msgs/PoseStamped radius_entry_pose\n\
geometry_msgs/PoseStamped req_pose\n\
\n\
\n\
\n\
================================================================================\n\
MSG: nav_msgs/Path\n\
#An array of poses that represents a Path for a robot to follow\n\
Header header\n\
geometry_msgs/PoseStamped[] poses\n\
\n\
================================================================================\n\
MSG: std_msgs/Header\n\
# Standard metadata for higher-level stamped data types.\n\
# This is generally used to communicate timestamped data \n\
# in a particular coordinate frame.\n\
# \n\
# sequence ID: consecutively increasing ID \n\
uint32 seq\n\
#Two-integer timestamp that is expressed as:\n\
# * stamp.sec: seconds (stamp_secs) since epoch (in Python the variable is called 'secs')\n\
# * stamp.nsec: nanoseconds since stamp_secs (in Python the variable is called 'nsecs')\n\
# time-handling sugar is provided by the client library\n\
time stamp\n\
#Frame this data is associated with\n\
# 0: no frame\n\
# 1: global frame\n\
string frame_id\n\
\n\
================================================================================\n\
MSG: geometry_msgs/PoseStamped\n\
# A Pose with reference coordinate frame and timestamp\n\
Header header\n\
Pose pose\n\
\n\
================================================================================\n\
MSG: geometry_msgs/Pose\n\
# A representation of pose in free space, composed of postion and orientation. \n\
Point position\n\
Quaternion orientation\n\
\n\
================================================================================\n\
MSG: geometry_msgs/Point\n\
# This contains the position of a point in free space\n\
float64 x\n\
float64 y\n\
float64 z\n\
\n\
================================================================================\n\
MSG: geometry_msgs/Quaternion\n\
# This represents an orientation in free space in quaternion form.\n\
\n\
float64 x\n\
float64 y\n\
float64 z\n\
float64 w\n\
";
}
static const char* value(const ::hector_nav_msgs::GetRecoveryInfoResponse_<ContainerAllocator>&) { return value(); }
};
} // namespace message_traits
} // namespace ros
namespace ros
{
namespace serialization
{
template<class ContainerAllocator> struct Serializer< ::hector_nav_msgs::GetRecoveryInfoResponse_<ContainerAllocator> >
{
template<typename Stream, typename T> inline static void allInOne(Stream& stream, T m)
{
stream.next(m.trajectory_radius_entry_pose_to_req_pose);
stream.next(m.radius_entry_pose);
stream.next(m.req_pose);
}
ROS_DECLARE_ALLINONE_SERIALIZER;
}; // struct GetRecoveryInfoResponse_
} // namespace serialization
} // namespace ros
namespace ros
{
namespace message_operations
{
template<class ContainerAllocator>
struct Printer< ::hector_nav_msgs::GetRecoveryInfoResponse_<ContainerAllocator> >
{
template<typename Stream> static void stream(Stream& s, const std::string& indent, const ::hector_nav_msgs::GetRecoveryInfoResponse_<ContainerAllocator>& v)
{
s << indent << "trajectory_radius_entry_pose_to_req_pose: ";
s << std::endl;
Printer< ::nav_msgs::Path_<ContainerAllocator> >::stream(s, indent + " ", v.trajectory_radius_entry_pose_to_req_pose);
s << indent << "radius_entry_pose: ";
s << std::endl;
Printer< ::geometry_msgs::PoseStamped_<ContainerAllocator> >::stream(s, indent + " ", v.radius_entry_pose);
s << indent << "req_pose: ";
s << std::endl;
Printer< ::geometry_msgs::PoseStamped_<ContainerAllocator> >::stream(s, indent + " ", v.req_pose);
}
};
} // namespace message_operations
} // namespace ros
#endif // HECTOR_NAV_MSGS_MESSAGE_GETRECOVERYINFORESPONSE_H
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,006
|
Alison Skipworth est une actrice britannique, née Alison Mary Elliott Margaret Groom à Londres (Angleterre, Royaume-Uni) le , morte à Los Angeles (Californie, États-Unis) le .
Biographie
Ayant épousé Frank Skipworth en 1882, elle sera connue, tout au long de sa carrière, sous le nom d'Alison Skipworth. Au cinéma, elle participe exclusivement à des films américains, s'étant établie aux États-Unis vers 1900, et débute dans quatre courts métrages muets en 1912. Après deux autres films muets en 1920 (, adaptation de la pièce éponyme de Rachel Crothers, qu'elle avait créée l'année précédente — 1919 — à Broadway) et 1921, elle apparaît surtout durant la période du parlant, dans 48 films, de 1930 à 1938.
Très active au théâtre, Alison Skipworth débute à Londres en 1894, et à Broadway (New York) en 1899. Elle joue régulièrement sur les scènes new-yorkaises jusqu'en 1930, puis après s'être retirée du cinéma, de 1938 à 1942 (année où elle met un terme définitif à sa carrière), dans des pièces principalement, mais aussi dans deux comédies musicales. Parmi ses partenaires à Broadway, citons-en trois : Lionel Atwill (L'Élévation de Henri Bernstein, en 1917), qu'elle retrouvera au cinéma dans deux de ses films les plus connus, Le Cantique des cantiques (1933) et La Femme et le Pantin (1935), tous deux avec également Marlene Dietrich ; Basil Rathbone (trois pièces, entre 1923 et 1926, dont La Grande Duchesse et le garçon d'étage d'Alfred Savoir et Le Cygne de Ferenc Molnár) ; et Miriam Hopkins (The Garden of Eden, en 1927), aux côtés de laquelle elle tournera ensuite dans le film Becky Sharp (1935).
Filmographie complète
1912 : A Mardi Gras Mix Up, The Pilgrimage, Into the Jungle et A Political Kidnapping, courts métrages, réalisateur(s) non-connu(s)
1920 : de John Stuart Robertson
1921 : Handcuffs or Kisses de George Archainbaud
1930 : Strictly Unconventional de David Burton
1930 : Raffles de George Fitzmaurice et Harry d'Abbadie d'Arrast
1930 : Outward Bound de Robert Milton
1930 : Du Barry, Woman of Passion de Sam Taylor
1930 : L'Amant de minuit (Oh, for a Man!) de Hamilton MacFadden : Laura
1931 : The Virtuous Husband de Vin Moore
1931 : L'Ange de la nuit (Night Angel) d'Edmund Goulding
1931 : Devotion de Robert Milton
1931 : d'Alfred E. Green
1931 : Cette nuit ou jamais (Tonight or Never) de Mervyn LeRoy
1932 : The Unexpected Father de Thornton Freeland
1932 : Si j'avais un million (If I had a Million) de James Cruze
1932 : High Pressure de Mervyn LeRoy
1932 : Sinners in the Sun d'Alexander Hall
1932 : Nuit après nuit (Night after Night) d'Archie Mayo
1932 : Madame Racketeer de Harry Wagstaff Gribble et Alexander Hall
1933 : Tonight Is Ours de Stuart Walker
1933 : Le Cantique des cantiques (The Song of Songs) de Rouben Mamoulian
1933 : He learned about Women de Lloyd Corrigan
1933 : A Lady's Profession de Norman Z. McLeod
1933 : Le Club de minuit (Midnight Club) d'Alexander Hall et
1933 : Alice au pays des merveilles (Alice in Wonderland) de Norman Z. McLeod
1933 : Tillie et Gus (Tillie and Gus) de Francis Martin
1934 : Six d'une sorte (Poker Party) de Leo McCarey
1934 : Coming-Out Party de John G. Blystone
1934 : Wharf Angel de William Cameron Menzies et
1934 : Shoot the Works de Wesley Ruggles
1934 : Une femme diabolique (The Notorious Sophie Lang) de Ralph Murphy
1934 : Le capitaine déteste la mer (The Captain Hates the Sea) de Lewis Milestone
1934 : Here is my Heart de Frank Tuttle
1935 : L'Intruse (Dangerous) d'Alfred E. Green
1935 : Un drame au casino (The Casino Murder Case) d'Edwin L. Marin
1935 : Doubting Thomas de David Butler
1935 : Becky Sharp de Rouben Mamoulian et Lowell Sherman
1935 : Shanghai de James Flood
1935 : Hitch Hike Lady d'Aubrey Scotto
1935 : La Femme et le Pantin (The Devil is a Woman) de Josef von Sternberg
1935 : Une femme dans la rue (The Girl from Avenue) d'Alfred E. Green
1936 : Une princesse à bord (The Princess Comes Across) de William K. Howard
1936 : L'Enchanteresse (The Gorgeous Hussy) de Clarence Brown
1936 : Two in a Crowd d'Alfred E. Green
1936 : Satan Met a Lady de William Dieterle
1936 : White Hunter d'Irving Cummings
1937 : Stolen Holiday de Michael Curtiz
1937 : Two Wise Maids de Phil Rosen
1938 : de Bernard Vorhaus
1938 : Drôle d'équipe (Wide Open Faces) de Kurt Neumann
1938 : Ladies in Distress de Gus Meins
Théâtre (à Broadway)
Pièces, sauf mention contraire
1899-1900 : The Manoeuvres of Jane de Henry Arthur Jones, avec Ferdinand Gottschalk
1900 : The Ambassador de John Oliver Hobbs
1900 : The Interrupted Honeymoon de F. Kinsey Peile
1900 : The Man of Forty de Walter Frith
1901-1902 : The Way of the World de Clyde Fitch
1903 : Mme Flirt (The Frisky Mrs. Johnson) de Paul Gavault et Georges Berr, adaptation et mise en scène de Clyde Fitch, avec Ferdinand Gottschalk
1903 : Captain Dieppe d'Anthony Hope et Harrison Rhodes, avec Charles Lane
1904 : Man proposes d'Ernest Denny
1905 : Friquet de Pierre Berton
1905 : Fritz in Tammany Hall, comédie musicale, musique de Jean Schwartz, lyrics de William Jerome, livret de John J. McNally
1906 : Cymbeline de William Shakespeare
1910-1911 : Suzanne de Frantz Fonson et Fernand Wicheler, adaptation de C. Haddon Chambers, avec Billie Burke
1913 : The Old Firm de Harry et Edward Paulton
1913-1914 : The Marriage Game d'Anne Crawford Flexner, avec Charles Trowbridge
1916-1917 : Major Pendennis, adaptation par Langdon Mitchell du roman L'Histoire de Pendennis (The History of Pendennis) de William Makepeace Thackeray, avec Walter Kingsford, Leonard Willey
1917 : L'Élévation (Elevation) de Henri Bernstein, avec Lionel Atwill
1918 : The Woman of the Index de Lillian Trimble Bradley et George Broadhurst
1918 : Betty at Bay de Jessie Porter
1919 : 39 East de Rachel Crothers, avec Luis Alberni, Blanche Friderici, Henry Hull
1921-1922 : Lilies of the Field de William J. Hurlbut, avec Mary Philips, Cora Witherspoon
1922 : The Torch Bearers de George Kelly, avec Mary Boland
1923-1924 : Le Cygne (the Swan) de Ferenc Molnár, adaptation de Melville Baker, avec Philip Merivale, Basil Rathbone
1925 : Avril enchanté (The Enchanted April) de Kane Campbell, adaptation du roman éponyme d'Elizabeth von Arnim, avec Elisabeth Risdon
1925 : La Grande Duchesse et le Garçon d'étage (The Grand Duchess and the Waiter) d'Alfred Savoir, adaptation mise en scène par Frank Reicher, avec Elsie Ferguson, Basil Rathbone, Frederick Worlock
1926 : Port O' London de George W. Oliver, avec Walter Kingsford, Basil Rathbone
1926 : Ashes of Love de la comtesse Vera Cathcart, avec Lumsden Hare
1926 : Buy, buy, Baby de Russell Medcraft et Norma Mitchell, avec Laura Hope Crews, Thurston Hall, Verree Teasdale
1926-1927 : New York Exchange de Peter Glenny
1927 : Jenny de Corning White, avec Edward Arnold
1927 : The Garden of Eden d'Avery Hopwood, d'après Rudolf Bernauer et Rudolph Oesterreicher, avec Miriam Hopkins, Douglass Montgomery, Ivan F. Simpson
1927 : Spellbound de Frank Vosper
1927-1928 : Los Angeles de Max Marcin et Donald Ogden Stewart, produite par George M. Cohan, avec Jack La Rue, Helen Vinson
1928 : Mrs. Dane's Defense de Henry Arthur Jones, avec Stanley Logan, Robert Warwick
1928 : Say when, comédie musicale, musique et lyrics de divers, livret de Calvin Brown, d'après la pièce Love in a Mist de Gilbert Emery et Amelie Rivers, avec Gene Raymond
1928-1929 : Angela de Fanny Todd Mitchell, d'après A Royal Family de Robert Marshall, avec Eric Blore, Jeanette MacDonald
1929 : Cafe de Danse, adaptation de Leontrovitch Mitchell et Clarke Silvernail, mise en scène par (et avec) Gregory Ratoff
1929 : Button, Button de Maurice Clark, mise en scène par Maurice Clark et Henry C. Potter, avec Ann Shoemaker
1929 : A Primer for Lovers de William J. Hurlbut, avec Gavin Muir, Robert Warwick
1930 : Marseilles de Sidney Howard, d'après la trilogie marseillaise de Marcel Pagnol, avec Guy Kibbee
1938 : 30 Days Hath September d'Irving Gaumont et Jack Sobell
1939-1940 : When we are married de John Boynton Priestley, avec Sally O'Neil, Tom Powers, Philip Tonge
1941 : First Stop to Heaven de Norman Rosten, avec Eduard Franz
1942 : Lily of the Valley de (et mise en scène par) Ben Hecht, avec Joseph Pevney
Notes et références
Liens externes
Actrice britannique de cinéma
Actrice britannique de théâtre
Naissance à Londres
Naissance en juillet 1863
Décès en juillet 1952
Décès à 88 ans
Décès à Los Angeles
Personnalité inhumée au cimetière de Kensico
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Mika's Fortune
Written by Faith Emiry
Cover designed by Ivy Wong
Copyright 2018 Ripple Foundation
All rights reserved. No part of this publication may
be produced in whole or in part, by any means
without permission of the publisher of this book.
For information, please email info@ripplefoundation.ca
ISBN: 978-1-927864-05-0
Published by Ripple Foundation
Visit our website ripplefoundation.ca
First e-book edition 2015
About the Author
Faith Emiry was born and raised in rural Northern Ontario where her family operates a dairy farm and a strawberry patch. Her interests include social justice, photography, dodgeball, public speaking, and travel. Faith has been writing stories for her younger sister for a number of years. Her stories are most often lively, funny adventures based on everyday life. "Mika's Fortune" was inspired by Faith's annoyance in finding disappointing fortunes inside fortune cookies. When she is not reading or writing, Faith can be found at 4H events or spending time with her beloved cousin, Shasta, to whom this book is dedicated.
At Ripple Foundation we believe that creativity, education, and imagination are key components to success. We are dedicated to fostering these areas in the lives of youth across Canada. Our mandate is to raise funds and develop and facilitate initiatives that raise awareness and promote creativity and education in youth across the country.
Since 2012, Ripple Foundation and its group of dedicated volunteers have run the Kids Write for Kids program, a national, annual creative writing challenge for children in grades 4-8. In 2018, Ripple Foundation launched the Write It Workshop Series, a creative writing program designed to inspire young authors. The series focuses on improving the writing skills of children and youth and increasing their confidence in writing.
Visit ripplefoundation.ca for more information.
The net proceeds from the sale of all books are donated to the winners' charity of choice. To view all titles written by winners of Kids Write 4 Kids, visit ripplefoundation.ca/books.
To find out when our next Kids Write 4 Kids creative challenge will take place, visit ripplefoundation.ca/contest.
Join us on social media to spread the word and inspire young writers in your community.
Facebook | Twitter | Instagram | YouTube | Wave Blog
Mika's Fortune
Mika loved cookies. Ginger snaps, sugar, chocolate chip, oatmeal... she loved them all, but her favourite kind of cookie was... fortune cookie. That's why when her mom brought a big box of fortune cookies home from the factory, Mika didn't hesitate to grab a handful from out of the box.
Mika was just about to open up her cookie when her mom smiled and said, "Ah, ah, ah. The factory is working on something new, Mika. These cookies could be incredibly dangerous." She took the cookies from Mika and returned them to the box.
The next day, just after her mom had left, temptation was too much for Mika. She ran to the kitchen where she had last seen the cookies. However, the cookies weren't there.
_"She probably hid them."_ Mika thought to herself as she began to look around the kitchen. Sure enough, her mom had hid them – on the top shelf in her room – just out of Mika's reach. This didn't stop Mika though. She climbed up the shelf to grab a cookie.
_"I'll just take one,"_ she decided. She examined the cookie. _"I wonder why she said they're dangerous. They look the same."_ She removed the fortune and nibbled on the edge. _"They taste the same,"_ she thought.
When she had finished the cookie she looked down and read the fortune. _**You will win a contest**_ it said. Mika smiled. She had never gotten a fortune like this one before.
"Too bad the fortunes are fake," she said to herself.
A few minutes later the phone rang. *ring, ring, ring* Mika ran to answer it. "Hello?" "Congratulations!! You've won!!" The voice on the other side shouted.
"I've won – what?"
"Why, a lifetime supply of chocolate!" The man on the other side stated as if it was obvious. "A lifetime supply of chocolate! Is this a joke?"
"No, no. We will be sending it as soon as possible." "As soon as possible?" She asked still confused.
"As soon as possible. Congratulations." She heard a click on the other side of the line. She hung up the phone stunned. Then she remembered the cookie.
_**You will win a contest.**_
"It must be a coincidence." She thought to herself. _"....Maybe I'll try one more... Just to be sure."_ She returned to her mother's room and climbed the shelf once again to get another fortune cookie.
After taking the fortune out and gobbling down the delicious cookie, Mika began to read.
_**It will snow tonight.**_
"Yeah right. We live in Hawaii, and it's summer. I guess it was just a coincidence." She laughed.
Yet, later that night as Mika sat eating dinner with her mother, something caught her eye. Mika ignored it, until a minute later when her mother gasped.
"Is That... _Snow_?" Her mother asked. "It's summer, and we live in... _Hawaii_." Mika looked outside. Her eyes grew very big. Fluttering down from the sky, there were many white crystal-like features. There was snow.
_"They work,"_ Mika muttered to herself, stunned. _"The fortune cookies come true!"_
The next morning, Mika didn't need a cookie. Instead she went outside to play in the fluffy snow. She made snowmen, castles and snow angels and by the end of the day, it had been the most exciting day of her life. Mika decided to take one more cookie.
_"It won't do any harm, all the fortunes make people happy."_ She thought back on all the good fortunes that had come true already. She returned to the box, taking another cookie, cracking it open and reading the fortune.
_**You will find yourself allergic to candy.**_ Mika was horror-struck.
_"Allergic to candy? That's the worst thing that could possibly happen to me. What will I do?"_ Mika panicked. She couldn't tell her mom. She had disobeyed her, but she didn't know what to do herself. _"Why did I look at those cookies?"_ she muttered.
All of a sudden, her mother returned home.
"Mika?" she called. "Where are you?" She entered her room to find Mika sitting on the floor beside the box of fortune cookies. "WHAT DID YOU DO?" Her eyes were horror-struck. Just then, everything clicked. She put it all together.
"The snow, the chocolate... You've been eating these cookies this whole time?" Mika's mother asked. Mika's head hung low in shame. Her mother glanced at the fortune on the floor beside her. _"Oh, Mika,"_ she sighed.
"Can we fix it mommy?" Mika asked, worried.
"I might be able to find a way," her mother responded. "Come with me."
Mika didn't know where they were going, but she didn't care, as long as this was fixed. When they arrived, Mika realized they were at the fortune cookie factory where her mother worked. 'We need to make our own cookie," her mother told her, "to undo your fortune." She got her keys from her pocket and unlocked the door.
Into the factory they went, passing many big machines until they finally reached a big door. "This is it" her mother said. She typed a code in on a screen that was on the door. A fingerprint scanner appeared and she pressed her fingerprint onto it. It made a series of beeps, then the doors slid open. Inside there was an enormous machine. Mika thought the machine was even bigger than any dinosaur she'd seen in stories. Her mother approached a screen and spoke.
"Now we need to type in the fortune we want. We'll need to be very detailed, or it could react in a way we don't want it to."
Mika nodded her head and thought. "What about, _you can eat candy again_?" She asked. Her mother shook her head. "Even more detailed than that, Mika. Let's try... _all the past fortunes you have opened will be forgotten and erased from the past_." Mika's eyes grew very wide. "No mommy, that means that my chocolate will disappear, and so will the snow!"
Her mother sighed and replied "Mika, we have to do this if you want to get rid of the last fortune... you know what, what if we change it just a little bit... to... _all the past fortunes you have ever opened will be forgotten by everyone but yourself and your family, and will become erased from the past?_ "
Mika thought for a moment "So I'll still remember everything... and so will you?"
"Yes sweetie".
"Then I guess that would work... if it's the only way." Her mother gently smiled and began to type the words into the screen.
The machine grew very loud. *Click, clump, swoop, squish, **boom**. Click, clump, swoop, squish, **boom**...* The noise continued for eight long minutes, then came to a quick stop. One small fortune cookie fell into the bottom cubby of the machine.
"Here you go." Her mother passed the cookie to Mika. Nervously, and hoping with all her heart it would work, Mika opened the package. She took the fortune from the cookie, ate the cookie, and read the fortune aloud. _**"All the past fortunes you have ever opened will be forgotten by everyone but you and your family, and will be erased from the past." She looked back at her mother. "Did it work?"**_ She asked hopefully.
"Let's find out," she said. They walked outside. The snow was gone. They walked home. The trucks of chocolate were gone. Her mother smiled. "It worked." She went inside. "Wait here, Mika." Her mother went inside, only to return back in a few minutes. "Here." She passed Mika a piece of candy and Mika nervously popped it in her mouth. She chewed it and swallowed. No allergic reaction.
_"IT WORKED!!!"_ Mika shouted joyfully, jumping up and down. When she was finally calmed down she looked back to her mother. "I am sorry I ate the cookies. You told me not to, and I did anyway."
Her mother smiled. "Thank-you for apologizing. I'm sorry too. I shouldn't have brought home a big box of dangerous cookies when I knew you liked them so much." Mika hugged her mother.
She would never disobey her mother again... unless she did.
# Table of Contents
1. Title Page
2. About the Author
3. About Kids Write 4 Kids
4. Kids Write 4 Kids eBooks
5. Mika's Fortune
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"Don't Toss Us Away" is a song written by Bryan MacLean and recorded by country rock band Lone Justice in 1985 on their self-titled debut album. In 1988, the song was recorded by American country music singer Patty Loveless, who released the song as the second single from her album Honky Tonk Angel, in February 1989. Loveless' version reached the number five position on the Billboard Hot Country Singles chart in May 1989.
Background
Loveless and her brother, Roger Ramey, heard the song while driving with Tony Brown. All of them agreed it would be a good song for her to record, as the ballad fit her vocal style quite well. Maria McKee, songwriter Bryan MacLean's sister, told Loveless at the studio that she was singing/recording the song the way her brother had always intended.
Chart positions
The song charted for 17 weeks on the Billboard Hot Country Singles chart, reaching number 5 during the week of May 6, 1989.
Year-end charts
References
1985 songs
1989 singles
Patty Loveless songs
Song recordings produced by Tony Brown (record producer)
Songs written by Bryan MacLean
Country ballads
MCA Nashville Records singles
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TUTOR PERINI CORP TPC S
November 15, 2019 - 8:34am EST by
Novana
Price: 18.32 EPS 1.5 2.0
Shares Out. (in M): 50 P/E 12.2 9.2
Market Cap (in $M): 916 P/FCF 75 0
Net Debt (in $M): 629 EBIT 161 200
TEV ($): 1,545 TEV/EBIT 9.6 7.7
Borrow Cost: General Collateral
EQT Corporation EQT 09/11/2020
PACIFIC GAS & ELECTRIC CO PCG 06/29/2020
Summary investment thesis
Following a near doubling of the stock in the last 2 months, also fuelled by a short squeeze, we believe TPC represents a very attractive opportunity to short a structural cash burner with potential 100% downside in a bear case scenario and very little upside in a base case scenario. We believe the short squeeze was triggered by a very superficial reading of the company's latest set of quarterly results. In Q3-19, TPC reported their best ever quarterly free flow, equal to $200m. As we will explain below, we believe this seemingly positive development is nothing but a smokescreen to cover up deep fundamental issues with this business. We think there is a high likelihood TPC will need to resort to a rescue capital increase to salvage the situation, leading to substantial downside to current share price.
Summary TPC history and business description
TPC today is one of the largest contractors in the US, ranked 10th overall:
TPC has a long and interesting history. Perini Corp. has been providing construction services for over 100 years. In 1999, Ronald Tutor, the CEO and main shareholder of Tutor-Saliba, a California based contractor, became the Chairman and CEO of Perini. Perini Corp. ended up acquiring Tutor-Saliba via a merger in 2008, creating what is now Tutor Perini ("TPC"). Ron Tutor, going for 80, is still the CEO of the company. TPC core business was general construction and building services. In particular, TPC targeted 2 key markets: large building projects in the leisure / gambling / health sectors, especially in Nevada (think of Caesars Palace and MGM) and some civil projects in the Northeast, primarily in Boston and NYC. The former was the core business of Perini, the latter was the business of Ron Tutor, Tutor-Saliba.
Under the stewardship of Ron Tutor, the business grew dramatically, especially in its core building segment. In the first decade since Ron Tutor became CEO, TPC sales grew from little over $600m in 1999 to over $5bn by 2009. The building segment alone grew to over $5bn in sales by 2008. Growth was fuelled primarily by large projects in Las Vegas where TPC had a stronghold. TPC had a strong reputation for efficient operator. In 2005 Forbes named Perini (now TPC) "One of the Best Managed Companies in America," ranking it No. 1 in the construction industry. By 2008, this business was on fire, the Hospitality and Gaming segment alone contributed $3.7bn in sales. TPC was involved in the construction services of the MGM, MIRAGE CityCenter, Cosmopolitan Resort and Casino, Wynn Encore Hotel and Planet Hollywood's Westgate Tower. In 2008, the building sector alone contributed $152m in EBIT, the highest ever recorded for TPC in that segment. TPC was the established leader in Building entertainment:
Things started to deteriorate quickly though. The GFC hit and large projects in Vegas were abandoned. The building segment imploded with sales going from $5.1bn in 2008 to $1.5bn by 2012, with obvious impact on margins:
To compensate for losses in the Building segment, Ron Tutor, right after Perini merged with his own Tutor-Saliba company, embarked on an acquisition spree, focusing on the segment he was familiar with: Civil. Ron Tutor acquired several businesses in the Civil space, such as Lunda Construction, Becho and Frontier-Kempter in the Civil space and Fisk Electric and Five Star Electric in the Specialty Contractor space. These transactions helped the Civil segment and the Specialty Contractor segment grow sales between from $0.5bn in 2008 to $2.6bn by 2012. In our view, this is where TPC problems began. TPC and his ambitious CEO simply overstretched themselves.
It's important to note that the Civil segment is a much trickier one than building. Projects are typically longer and more complicated. Cost overruns are common and for this reason, typical contractors try to limit their exposure by having variable pricing or cost-plus type of contracts, rather than fixed price contracts. Furthermore, large Civil projects are often subject to many variation orders during the construction period, leading to tricky contractual resolutions. Finally, while the building segment is predominantly dealing with private clients, the Civil segment is predominantly dealing with public entities, local municipalities or the federal government. Competition is somewhat lower as bidders require scale and expertise. Because of this, margins could in theory be better than in Building, were contractor earn LSD margins. Other civil competitors such as Fluor, KBR, Parsons, Granite, Skanska, Dragados, Kiewit Corporation etc. can earn mid-to-high single margins on larger project if managed well. Tutor Perini is still a smaller competitor, ranked 10th in the US but 2nd overall in the transportation (roads, bridges, tunnels) segment. It is therefore extremely puzzling to see TPC generating double digit EBIT margins in its Civil segment when all peers make mid-to-low single digit margins.
The Civil segment has been the real driver of performance for TPC in recent years, representing now c. ¾ of the EBIT generated by the company before central costs and over half of its backlog. Civil revenues went from less than $400m in 2009 to nearly $2bn in 2015. Run Tutor was absolutely determined to have the business grow back to the heyday of the Las Vegas boom through aggressive growth in Civil. He nearly succeeded as by 2016, TPC sales were back to $5bn from little over $3bn in 2010.
The growth in Civil though came at significant risks for TPC. Large construction projects are complicated. Conservative players will try to minimise risks structuring contracts where excess costs will be borne by the customer, or at least shared. Not so Tutor Perini – the proportion of contracts under fixed prices increased dramatically over time and it represented 80% of sales in 2018:
Peers such as Granite, Fluor, Skanska typically keep this percentage between 30% and 60%. TPC has effectively gone for broke, betting the house on potentially lucrative contracts that could become huge liabilities if mismanaged. Think about a $4bn contract where TPC aims to make 10% margin. A 20% cost overrun will crush margins to -10%. This is exactly what TPC has been doing since 2009.
Aggressive Accounting issues
The problem with construction companies is that revenue recognition is subject to management own estimates. "Because control transfers over time, revenue is recognized to the extent of progress towards completion of the performance obligations. The selection of the method to measure progress towards completion requires judgment and is based on the nature of the products or services provided… Due to the nature of the work required to be performed on many of the Company's performance obligations, estimating total revenue and cost at completion is complex, subject to many variables and requires significant judgment". This significant issue is compounded by the fact that larger Civil projects are naturally subject to order variations and changes to the original contract. The contractor continues to incur in costs and recognises revenues and margins associated with these costs, not knowing know though whether the client will ever pay for these.
TPC clearly highlights this problem amongst the risk factors in the annual report:
"If we are unable to accurately estimate contract risks, revenue or costs, the timing of new awards, or the pace of project execution, we may incur a loss or achieve lower than anticipated profit"
"Our contracts require us to perform extra, or change order, work which can result in disputes or claims and adversely affect our working capital, profits and cash flows"
"Our actual results could differ from the assumptions and estimates used to prepare our financial statements."
Put it very simply, TPC highlights 3 main risks with this business:
TPC may sign contracts that are loss making
Because these contracts are loss making, TPC will end up burning cash
TPC may recognise profits on such contracts when in reality they are loss making
This is the crux of our thesis: TPC has signed loss making contracts since 2010, burning cash throughout, while telling the market they have been generating healthy margins.
This problem expresses itself at the balance sheet level under "Costs and estimated earnings in excess of billings", otherwise known as Unbilled Receivables. These represent the receivables that the client owns the contractor for revenue recognised by TPC where costs have been incurred on a cash level but the client couldn't be billed. The 2 larger categories of unbilled receivables are "claims" and "unapproved change orders". Claims occur when there is a dispute regarding both a change in the scope of work and the price associated with that change. Unapproved change orders occur when a change in the scope of work results in additional work being performed before the parties have agreed on the corresponding change in the contract price. Over the last 9 years, since TPC focused its prospects on the Civil segment, unbilled receivables grew exponentially from little over $100m in 2010 to nearly $1.2bn in 2019.
Particularly worrying was the growth in claims, growing from close to zero in 2010 to over $700m in 2019:
Claims are the most troublesome item on balance sheet because it represents revenues (and margins) that have been recognised in the P&L that are highly uncertain in terms of collectability. The clients dispute both the change in the scope ("I never told you to add the extra ditch to the foundation of that bridge...") and the price associated with that change ("you cannot possibly charge me $1m for that extra ditch"). Over the last 9 years, TPC recognised c. $1bn in revenue that not only wasn't paid for by the customer, it wasn't even billed.
The issue becomes apparent when we analyse cash flow. The gap between reported EBIT and free cash flow since 2009 is striking:
The change in fortunes at TPC since the company decided to aggressively move into the Civil segment is remarkable. Between 2003 and 2010, when TPC was involved in Las Vegas type of construction projects, they generated c. $500m in free cash flow, which is over 50% of the EBIT generated in the period notwithstanding the negative FCF reported during the GFC in 2009. Vice versa, between 2011 and 2019, TPC generated c. $1.6bn in cumulative EBIT and negative cumulative free cash flow. Over the last decade, TPC did not generate a single dollar in cumulative free cash flow.
The above suggests that TPC continued to recognise P&L profits on contracts that were essentially unprofitable. The huge balance of claims will never be entirely paid off as some of these receivables are not going to be recognised by clients.
Several large impairments made in recent years suggest the above interpretation of financials is correct. Following the numerous acquisitions discussed above, TPC had nearly $900m of Goodwill on balance sheet by 2011. Over the years, the company was forced to take several impairments to these acquisitions, the latest in Q2-19 in the Civil sector. Following these impairments, Goodwill today is only $205m. There have been in the last years impairments totalling nearly $700m suggesting the true underlying value of the acquired business is significantly lower than previously expected. Still, TPC didn't write off any receivable nor restated previously recognised revenue. We believe the day of reckoning is approaching fast.
Stretched balance sheet
The inability to generate cash discussed above, led the company to incur in substantial amounts of debt to fund operations. Large contractors should ideally be debt free, given the capital-intensive nature of their business. Because TPC couldn't generate any cash, net debt went from $450m in 2011 to nearly $800m, bringing total leverage to very uncomfortable levels.
We actually believe the situation is much worse than it looks. The company obviously reports only debt balance at quarter end. Intra-quarter, we think the company is continuously drawing on its revolver to fund day to day operations. This shouldn't happen at a well-run, large scale contractor. The average cost of debt for TPC should be c. 6%, give or take:
However, if we calculate the actual interest cost on a quarterly basis, we notice that the implied interest rate paid by TPC on the average debt balance of the period is much higher than this:
The only way to explain this is to assume heavy intra-quarter revolver borrowings. Doing a simple back of the envelope calculations, we can imply from the above that on average, TPC carries at least $300m of debt on balance sheet more than it reports at quarters end. Average cost of debt is c. 200bps higher than it should be, or c. $15m a year. The revolver costs c. 4% a year. 4% of $300m is $12m. Tutor Perini is financially much more stressed than it would like you to believe.
Chronic inability to collect
Since 2010, when Ron Tutor embarked on an aggressive growth strategy in the Civil segment, TPC failed to collect unbilled receivables. Put it differently, TPC continued to generate positive EBIT on paper but negative free cash flow. Over the years, this became an important focus for management for 2 reasons:
Notwithstanding record backlog and good growth in earnings, the stock wasn't going anywhere as investors realised that TPC was unable to collect on its unbilled receivables
TPC was starved for cash and needed cash quickly. Collection of old receivables would be an obvious place to start in order to generate cash
What we try to illustrate below is management chronic inability to collect receivables notwithstanding their bombastic claims to the contrary. Ron Tutor promised time and time again he'll tackle the issue, but his track record should really make investors very sceptic.
Unbilled receivables increased from $139m in Q4-10 to $905m in Q4-15. At Q4-15 results, in February 2016, Ronald Tutor finally had to concede to the market that the company had an issue with collections and that the unbilled receivable balance needed to be resolved. As far as I can tell, it's the first time he tackled the issue publicly. This is what he said:
"The increase in the cost in excess account has certainly gotten our attention and we are working hard to drive it down. Note that just three years ago the cost in excess balance was about half of what it currently is. We expect to reach that level by no later than the fourth quarter of 2017".
Quarter by quarter, management kept repeating this mantra. Just 3 months later, at Q1-16 results, the CFO said "in late February, we communicated our plan to reduce unbilled costs, that's the cost and estimated earnings in excess of billings reflected on our balance sheet from $905 million at the end of 2015, to about half that amount by the end of 2017, so over a two-year period. To accomplish this, we have been intensely focused on resolving numerous claims and unapproved change orders, as well as billing and collecting other unbilled amounts…We're starting to see considerable traction in our efforts as we have made very good progress in the first quarter of 2016".
A quarter later (Q2-16), same promises were reiterated: "We expect to make further progress in reducing our debt level as we collect substantial cash that we are owed throughout the balance of this year and beyond in accordance with our previously stated goal".
Even as the company was clearly not delivering on its stated plan, it continued to boast confidence in these targets. A quarter later (Q3-16) the CEO stated on the call " ...this is Ron Tutor. I'm more confident today than the last time we spoke that we will collect the dollars, and we are currently making great strides…". I won't bore you with comments from every single quarter, suffice to say that the company missed its targets miserably.
Unbilled receivables balance was $905m in 2015. The CEO vowed to bring it back to 2012 levels ($465m) by Q4-2017. Instead, unbilled receivables increased a further $28m by Q4-17 to $933m, missing management target by nearly half a billion dollars.
Having missed its 2017 targets, management then focused on 2018 and 2019. On its Q4-17 call, the CFO stated the following: "we remain dedicated to significantly reducing our unbilled costs. Based on how negotiations are developing and expected to further progress, we anticipate more progress in resolving and reducing certain of our larger unbilled cost issues in 2018 and beyond…our operating cash flow in 2018 is expected to exceed net income…Deleveraging, in other words, debt reduction, remains a top priority in terms of capital allocation. We plan to utilize as much operating cash we generate in 2018 as possible for this purpose". The annual guidance given at Q4-17 results implied net income at the mid-point of c. $100m. Operating cash flow should have been higher. It ended being $21m, some 80% lower. Furthermore, TPC was supposed to pay down debt during 2018. Instead, net debt increased from $543m to $645m. Once again, management failed to deliver on its promises.
We illustrated the above to give some background to Q3-19 results, where management showed positive cash generation in the quarter and made new promises about cash collections in 2020. Given management track record, we urge investors to treat their promises with extreme caution.
Enter Q3-19 results
Expectations were high going into the quarter. During TPC's Q2-19 call, the CFO promised strong cash generation in H2: "We expect more substantial reductions in unbilled during the second half of this year and into 2020, as we continue to focus our efforts on negotiating, litigating and settling the various claims". The stock rallied over 50% from Q2 ahead of Q3, in expectation of a cash flow relief. Note that the stock is heavily shorted (c. 23% S.I.) and forensic accountant specialists like Gradient have a well-known negative thesis on the name, so modest positive news can produce outsized short squeezes. Q3 results involved a reduction in EPS guidance from $1.60-1.80 to $1.40-1.55 but also the best quarter ever in the company's history in terms of cash flow. TPC generated $200m in FCF in the quarter. The surprise sent the shares 18% higher on the day and nearly 100% up from Q2. We think the reaction gives us a very interesting entry point on the short and believe the market has not looked properly at the drivers of cash generation.
The CEO was quick to tout the strong cash flow generation as a victory for him on the collection front, vindicating his earlier claims. The CEO said "The Company generated a new quarterly record $222.9 million of operating cash for the third quarter…The strong operating cash flow was driven by collections associated with certain dispute resolutions, as well as from the Company's continued focus on improved working capital management… Our strong cash flow was driven by significant collections associated with several settlements". The perception that TPC got a handle on collections sent the shares higher. Furthermore, the CEO indicated that 9 individual disputes (claims) totalling $257m will be resolved (either settled or via legal means) in Q1-2020. This further positive development reinforced the perception that collections are a problem of the past and significant cash flow will be generated in the near term. Even bears Gradient effectively threw in the towel on the short and upgraded the stock following these comments: "we note that a portion of our concern is scheduled to be resolved via litigation early next year".
In our view, the market was blinded by the strong cash generation in the quarter, ignoring the real sources of cash. Contrary to what management would like you to believe, settlements of claims had very little to do with cash generation this past quarter. A per table below, the claims balance continued to climb higher each quarter this year:
Rather, what is driving cash flow, is the increase in "Billings in excess of costs and estimated earnings", otherwise known as Deferred Revenues
Understanding movements in Deferred Revenues
Billings in excess of costs and estimated earnings, or deferred revenues, is defined as the excess of contract billings to date over the amount of contract costs and profits (or contract revenue) recognized to date. In other words, it represents the cash amount clients advanced to TPC for large project that wasn't spent yet. It's a sort of advance payment, typically 5-10% of the contract value, that clients pay contractors to help them managing their working capital. In a normally managed business, one would expect this balance to remain stable over time, at least as a proportion of revenues. There is no logical reason to explain why clients would decide to prepay larger proportion of projects over time. However, this is exactly what happened to TPC clients in recent years:
Until 2017, deferred revenue balance was in the $300-400m range, or 6-8% of sales. Over the last 2-3 years, the balance jumped to over $800m, or 18% of sales, nearly 3x larger than the "normal" level. This was a huge source of cash flow for TPC in recent periods. Since Q1-17, TPC raised nearly $500m in cash from changes in deferred revenue: $126m in 2017, $116m in 2018 and $246m in the 9 months to September 2019. It's fair to say that without this huge increase in deferred, TPC would probably be bust by now.
What is driving this huge increase in Deferred? This is probably the most interesting and misunderstood element of TPC business. We think that TPC is desperate for cash, they cannot collect receivables and the only way to fund themselves is to ask clients to do it for them. We think TPC is signing as many projects as they can because every time a new project is signed, the client will have to put down some 5-10% of the total project cost up front, thereby advancing working capital to an otherwise stressed balance sheet. We think TPC is effectively committing themselves to a huge amount of liabilities (remember – at least 80% of contracts are fixed price, if there is a cost overrun or the contract was bid overly aggressively, TPC is on the hook for the losses) in order to get some cash in the door. This is also very clear just by looking at the backlog. Backlog ballooned to $11bn from $6bn 3 years ago while sales are still lower than they were then.
TPC would justify this by saying that a large backlog means higher revenues in the future. While this may well still happen in the future, the huge increase in backlog didn't bring any revenue growth over the last 4 years:
Let's think about this for a second. Over the last 4 years, revenues went nowhere but backlog went up by over $4bn. We also know that over the same period, deferred revenues increased by c. $0.5bn. A typical contract has c. 10% down payment / advance payment. Signing incremental $4-5bn of backlog should lead to $400-500m of cash inflow from deferred. This is exactly what TPC did – they signed as many contracts as they could in order to get hold of the cash.
How will the story unfold?
We think that TPC is playing a dangerous game that could lead to its own demise. Signing contracts to get advance payments is great, but what TPC is doing is effectively signing liabilities. They will need to deliver and perform on these contracts, or otherwise be subject to financial penalties. We believe there are 2 fundamental issues TPC will need to deal with:
While we have no proof of this, we suspect these contracts carry a huge amount of risk to TPC and many of them are likely to be unprofitable. We base this hypothesis on the following 4 considerations:
Past contracts were clearly unprofitable at a cash flow level, so there is no reason to suspect these are going to be any better
The proportion of fixed price contracts increased to 80% from less than 50% in 2016, suggesting TPC is willing to take on a lot more risk
TPC needs to sign as many contracts as they can for working capital reasons, as explained above. It is likely that in order to win these competitive bids, they had to bid rather aggressively
TPC financial profile is questionable. Its bond yields 7%. All else being equal, clients would rather advance the cash to a financially sound competitor. TPC may have had to underbid in order to win a tender
Even assuming the contract signed are profitable, TPC will need to unwind the positive working capital effect from advances from customers. Unless backlog continues to increase and they continue to lock themselves in ever increasing larger future liabilities, the deferred revenue balance will inevitably shrink. These contracts will eventually need to be worked on. This is going to be a huge use of cash for TPC going forward
It's very hard to predict how and when TPC will fold but we think the issues described above are huge and there are no easy fixes. We believe the only way to restore a healthy balance sheet would be to raise a substantial amount of fresh equity in a highly dilutive capital increase.
We have no smoking gun. We can only highlight several yellow flags that taken together paint a picture of a company with a business vastly underperforming reported financials while carrying a substantially stressed balance sheet and no easy way out of this mess.
We could be wrong but the risk-reward appears very attractive. Even assuming TPC will be able to collect overdue receivables and continue to fund its operations, current valuation doesn't leave much room to the upside. Over the last decade, TPC generated on average c. $220-260m in EBITDA. Even assuming these numbers are real (and we think they are not given aggressive accounting issues), the current share price implies a valuation multiple of 6-7x EV/EBITDA, which is at the high end of where peers are trading (GVA is on 6x, FLR on 5x, BBY LN on 6.5x). If, on the other hand, our bear case plays out, the equity could be worthless.
I and/or others I advise hold a material investment in the issuer's securities.
|
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| 7,429
|
CCR Files EEOC Discrimination Charges for Three African-American Applicants to the NYFD
Judge Throws Out Charges in "Los Angeles Eight" Case
Los Angeles - In a decision received today, Los Angeles Immigration Judge Bruce J. Einhorn ordered an end to deportation proceedings against Khader Hamide and Michel Shehadeh, members of the "Los...
CCR joins NLG in Seeking Reparations for Victims of Tulsa Race Riots
CCR along with the National Lawyer's Guild, the National Conference of Black Lawyers, and the National Association for the Advancement of Colored People filed an amicus brief in the case Alexander v...
CCR Visits Cincinnati to Meet with Boycott Leaders
The Center for Constitutional Rights visited Cincinnati to release its report on the findings of the hearing convened by CCR at the University of Cincinnati in May, 2003. The visit included meetings...
Docket: CCR Files Amicus Brief in Support of Voting Rights for Felons
Federal Appeals Court Finds for Minority Public School Teachers Who Say NYC Board of Education Discriminated Against Them
BLACK FIREFIGHTERS FORMALLY BECOME PARTIES IN DOJ LAWSUIT CHARGING NYC AND FDNY WITH DISCRIMINATORY HIRING PRACTICES
On July 17, 2007, the Center for Constitutional Rights (CCR) formally filed to intervene on behalf of the Vulcan Society in the Department of Justice's lawsuit against the City of New York. That suit...
CCR Condemns Murder of 18-Year-Old Khiel Coppin at Hands of NYPD
November 13, 2007, New York, NY – The Center for Constitutional Rights (CCR), which brought the historic NYPD racial profiling case Daniels v. City of New York in 1999 in the wake of the Amadou...
City Data Released Today on Fire Department Tests and Applicants of Color Does Not Tell Whole Story, Say CCR Attorneys
November 27, 2007, New York – Today, the City of New York released preliminary data from its January 2007 firefighter written exam and claimed that the percentage of Black and Latino firefighters who...
CCR Charges NYPD With Racial Profiling in Federal Lawsuit
New York, NY, January 31, 2008 – Today, in federal court in Manhattan, the Center for Constitutional Rights (CCR) is filing a companion case to its ground breaking racial profiling case, Daniels et...
|
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| 1,606
|
Harold William Crommelin (9 April 1903 – 20 May 1998) was an Australian businessman and politician who was a Liberal Party member of the Legislative Assembly of Western Australia from 1956 to 1968, representing the seat of Claremont.
Early life and career
Crommelin was born in Perth to Annie Florence (née Loton) and Reginald Crommelin. His maternal grandfather was Sir William Loton, a member of parliament and Mayor of Perth. Crommelin attended Toowoomba Grammar School (in Queensland) and Hale School, and after leaving school worked as a farmer, eventually purchasing his own property near Pingelly. He later worked as a car salesman and accounting clerk, eventually becoming a partner in a clothing firm. During World War II, his company made greatcoats for the army. Crommelin was elected to the South Ward of the Claremont Municipal Council in 1953, and would serve as a councillor until 1963.
At the 1956 state election, Crommelin won the seat of Claremont, defeating two other Liberal candidates. One of those was the incumbent member, Charles North, who had held the seat for 32 years and was also a former Speaker of the Legislative Assembly. Crommelin was re-elected unopposed at the 1959 election and was subsequently made deputy chairman of committees in the Legislative Assembly. He held that position until his retirement at the 1968 election, which coincided with an electoral redistribution that abolished the seat of Claremont. Crommelin died in Perth in May 1998, aged 95. At the time of his death, only one other MP in Western Australia (Sir Leslie Diver) had lived to a greater age. Crommelin had married Peggy Noreen Taylor in 1926, with whom he had two sons.
References
1903 births
1998 deaths
Liberal Party of Australia members of the Parliament of Western Australia
Members of the Western Australian Legislative Assembly
People educated at Toowoomba Grammar School
People educated at Hale School
Politicians from Perth, Western Australia
Western Australian local councillors
|
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{"url":"https:\/\/en.wikisource.org\/wiki\/Page:LorentzGravitation1916.djvu\/28","text":"Page:LorentzGravitation1916.djvu\/28\n\ndirection of one of the coordinates e. g. of ${\\displaystyle x_{e}}$ over the distance ${\\displaystyle dx_{e}}$. We had then to keep in mind that for the two sides the values of ${\\displaystyle u_{b}}$, which have opposite signs, are a little different; and it was precisely this difference that was of importance. In the calculation of the integral\n\n ${\\displaystyle \\int u_{b}\\sum (c){\\frac {\\partial \\pi {}_{ba}}{\\partial x_{c}}}\\mathrm {x} _{c}d\\sigma }$ (39)\n\nhowever it may be neglected. Hence, when we express the components ${\\displaystyle u_{b}}$ in terms of the quantities ${\\displaystyle \\psi _{ab}}$, we may give to these latter the values which they have at the point ${\\displaystyle P}$.\n\nLet us consider two sides situated at the ends of the edges ${\\displaystyle dx_{e}}$ and whose magnitude we may therefore express in ${\\displaystyle x}$-units ${\\displaystyle dx_{j}dx_{k}dx_{l}}$ if ${\\displaystyle j,k,l}$ are the numbers which are left of 1, 2, 3, 4 when the number ${\\displaystyle e}$ is omitted. For the part contributed to (38) by the side ${\\displaystyle \\Sigma _{2}}$ we found in \u00a7 26\n\n${\\displaystyle \\psi {}_{be}dx_{j}dx_{k}dx_{l}}$\n\nWe now find for the part of (39) due to the two sides\n\n${\\displaystyle \\psi {}_{be}\\sum (c){\\frac {\\partial \\pi {}_{ba}}{\\partial x_{c}}}\\left[\\int \\limits _{2}\\mathrm {x} _{c}d\\sigma -\\int \\limits _{1}\\mathrm {x} _{c}d\\sigma \\right]}$\n\nwhere the first integral relates to ${\\displaystyle \\Sigma _{2}}$ and the second to ${\\displaystyle \\Sigma _{1}}$. It is clear that but one value of ${\\displaystyle c}$, viz. ${\\displaystyle e}$ has to be considered. As everywhere in ${\\displaystyle \\Sigma _{1}:\\mathrm {x} _{c}=0}$ and everywhere in ${\\displaystyle \\Sigma _{2}:\\mathrm {x} _{c}=dx_{e}}$ it is further evident that the above expression becomes\n\n${\\displaystyle \\psi {}_{eb}{\\frac {\\partial \\pi {}_{ba}}{\\partial x_{c}}}dW}$\n\nThis is one part contributed to the expression (36). A second part, the origin of which will be immediately understood, is found by interchanging ${\\displaystyle b}$ and ${\\displaystyle e}$. With a view to (37) and because of\n\n${\\displaystyle \\psi {}_{eb}=-\\psi {}_{be}}$\n\nwe have for each term of (36) another by which it is cancelled. This is what had to be proved.\n\n\u00a7 31. Now that we have shown that equation (32) holds for each element ${\\displaystyle \\left(dx_{1},\\dots dx_{4}\\right)}$ we may conclude by the considerations of \u00a7 21 that this is equally true for any arbitrarily chosen magnitude and shape of the extension ${\\displaystyle \\Omega }$. In particular the equation may be applied to an element ${\\displaystyle \\left(dx'_{1},\\dots dx'_{4}\\right)}$ and by considerations exactly similar to","date":"2017-03-27 09:11:08","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 28, \"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\": 0.8396167159080505, \"perplexity\": 189.2920204447221}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-13\/segments\/1490218189466.30\/warc\/CC-MAIN-20170322212949-00607-ip-10-233-31-227.ec2.internal.warc.gz\"}"}
| null | null |
Q: Interesting JavaScript inheritance pattern I have recently watched a video where Douglas Crockford was explaining inheritance patterns of Javascript. The video itself is pretty old - it was filmed 6 years ago - but still useful. In that video he showed one inheritance pattern he kinda invented (although I am not sure who the author is). This is the code using his approach:
// imitation of new operator
function objectConstructor(obj, initializer, methods) {
// create prototype
var func, prototype = Object.create(obj && obj.prototype);
// add methods to the prototype
if(methods) Object.keys(methods).forEach(function(key) {
prototype[key] = methods[key];
});
// function that will create objects with prototype defined above
func = function() {
var that = Object.create(prototype);
if(typeof initializer === 'function') initializer.apply(that, arguments);
return that;
}
func.prototype = prototype;
prototype.constructor = func;
return func;
}
var person = objectConstructor(Object, function(name) {
this.name = name;
}, {
showName: function() {
console.log(this.name);
}
});
var employee = objectConstructor(person, function(name, profession) {
this.name = name;
this.profession = profession;
}, {
showProfession: function() {
console.log(this.profession);
}
});
var employeeInfo = employee('Mike', 'Driver');
employeeInfo.showName(); // Mike
employeeInfo.showProfession(); // Driver
Unfortanately, he didn't show the invocation. So, this part
var employeeInfo = employee('Mike', 'Driver');
employeeInfo.showName();
employeeInfo.showProfession();
is mine. It generally works, but it turns out that I repeat this.name = name; for both "classes" - person and employee. I played around but I didn't manage to make it work properly without that repetition. Seems I cannot get name because such a property isn't contained in the prototypal chain for employee. I didn't succeed either in mixing in stuff like person.call(this, arguments). So, apart from whether it is cool/nice/smart/sensible etc. or not in 2017, how could I remove this.name = name; from employee and get the same result? Or everything is ok and this approach doesn't suppose it?
A: Since the func constructor completely disregards this, passing any context to it via call or apply will not work. Creating a way to copy over the super class' properties after creating an object is one of the ways you could accomplish your task.
// imitation of new operator
function objectConstructor(obj, initializer, methods) {
// create prototype
var func, prototype = Object.create(obj && obj.prototype);
// add methods to the prototype
if(methods) Object.keys(methods).forEach(function(key) {
prototype[key] = methods[key];
});
// function that will create objects with prototype defined above
func = function() {
var that = Object.create(prototype);
if(typeof initializer === 'function') initializer.apply(that, arguments);
return that;
}
func.prototype = prototype;
prototype.constructor = func;
return func;
}
function copyProperties(source, target) {
for (var prop in source) {
if (source.hasOwnProperty(prop)) {
target[prop] = source[prop];
}
}
}
var person = objectConstructor(Object, function(name) {
this.name = name;
}, {
showName: function() {
console.log(this.name);
}
});
var employee = objectConstructor(person, function(name, profession) {
copyProperties(person.apply(null, arguments), this);
this.profession = profession;
}, {
showProfession: function() {
console.log(this.profession);
}
});
var employeeInfo = employee('Mike', 'Driver');
employeeInfo.showName(); // Mike
employeeInfo.showProfession(); // Driver
A: Here is your snippet with 2 small modifications so that you can do a super(name) type of call.
I've placed comments were I've made the modifications.. with prefix keith:
// imitation of new operator
function objectConstructor(obj, initializer, methods) {
// create prototype
var func, prototype = Object.create(obj && obj.prototype);
// add methods to the prototype
if(methods) Object.keys(methods).forEach(function(key) {
prototype[key] = methods[key];
});
// function that will create objects with prototype defined above
func = function() {
var that = Object.create(prototype);
if(typeof initializer === 'function') initializer.apply(that, arguments);
return that;
}
func.prototype = prototype;
//keith: store the initialization in constructor,
//keith: as func is already creating the object..
prototype.constructor = initializer;
return func;
}
var person = objectConstructor(Object, function(name) {
this.name = name;
}, {
showName: function() {
console.log(this.name);
}
});
var employee = objectConstructor(person, function(name, profession) {
//keith: call our super person(name)
person.prototype.constructor.call(this, name);
this.profession = profession;
}, {
showProfession: function() {
console.log(this.profession);
}
});
var employeeInfo = employee('Mike', 'Driver');
employeeInfo.showName(); // Mike
employeeInfo.showProfession(); // Driver
|
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| 7,670
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<h1 class="title">{{site.contact_title}}</h1>
<p class="subtitle">{{site.contact_description}}</p>
<div class="contact-icons">
{% if site.email != null %}
<a class="social-link" aria-label="My E-Mail" href="mailto:{{site.email}}">
<div class="social">
<svg class="social-svg" viewBox="0 0 48 48">
<use x="12" y="12" width="24" height="24" viewBox="0 0 48 48" xlink:href="{{ "/assets/img/solid.svg#envelope" | prepend: site.baseurl }}"></use>
</svg>
</div>
</a>
{% endif %}
{% if site.github_username != null %}
<a class="social-link" aria-label="My GitHub" target="_blank" rel="noreferrer"
href="https://github.com/{{site.github_username}}">
<div class="social">
<svg class="social-svg" viewBox="0 0 48 48">
<use x="12" y="12" width="24" height="24" viewBox="0 0 48 48" xlink:href="{{ "/assets/img/brands.svg#github" | prepend: site.baseurl }}"></use>
</svg>
</div>
</a>
{% endif %}
{% if site.codepen_username != null %}
<a class="social-link black" aria-label="My Codepen" target="_blank" rel="noreferrer"
href="https://codepen.io/{{site.codepen_username}}">
<div class="social">
<svg class="social-svg" viewBox="0 0 48 48">
<use x="12" y="12" width="24" height="24" viewBox="0 0 48 48" xlink:href="{{ "/assets/img/brands.svg#codepen" | prepend: site.baseurl }}"></use>
</svg>
</div>
</a>
{% endif %}
{% if site.dev_username != null %}
<a class="social-link black" aria-label="My DEV" target="_blank" rel="noreferrer"
href="https://dev.to/{{site.dev_username}}">
<div class="social">
<svg class="social-svg" viewBox="0 0 48 48">
<use x="12" y="12" width="24" height="24" viewBox="0 0 48 48" xlink:href="{{ "/assets/img/brands.svg#dev" | prepend: site.baseurl }}"></use>
</svg>
</div>
</a>
{% endif %}
{% if site.linkedin_username != null %}
<a class="social-link linkedin" aria-label="My LinkedIn" target="_blank" rel="noreferrer"
href="https://www.linkedin.com/in/{{site.linkedin_username}}">
<div class="social">
<svg class="social-svg" viewBox="0 0 48 48">
<use x="12" y="12" width="24" height="24" viewBox="0 0 48 48" xlink:href="{{ "/assets/img/brands.svg#linkedin-in" | prepend: site.baseurl }}"></use>
</svg>
</div>
</a>
{% endif %}
{% if site.twitter_username != null %}
<a class="social-link twitter" aria-label="My Twitter" target="_blank" rel="noreferrer"
href="https://twitter.com/{{site.twitter_username}}">
<div class="social">
<svg class="social-svg" viewBox="0 0 48 48">
<use x="12" y="12" width="24" height="24" viewBox="0 0 48 48" xlink:href="{{ "/assets/img/brands.svg#twitter" | prepend: site.baseurl }}"></use>
</svg>
</div>
</a>
{% endif %}
</div>
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| 4,119
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const handlers = {}
$(() => {
const app = Sammy('#root', function () {
this.use('Handlebars', 'hbs');
// home page routes
this.get('/index.html', handlers.getHome);
this.get('/', handlers.getHome);
this.get('#/home', handlers.getHome);
// user routes
this.get('#/register', handlers.getRegister);
this.get('#/login', handlers.getLogin);
this.post('#/register', handlers.registerUser);
this.post('#/login', handlers.loginUser);
this.get('#/logout', handlers.logoutUser);
// ADD YOUR ROUTES HERE
this.get('#/allSongs', handlers.getAllSongs);
this.get('#/createSong', handlers.getCreateSong);
this.get('#/mySongs', handlers.getMySongs);
this.get('#/removeSong/:id', handlers.removeSong);
this.get('#/removeSongMy/:id', handlers.removeSongMy);
this.get('#/likeSong/:id', handlers.likeSong);
this.get('#/listenSong/:id', handlers.listenSong);
this.get('#/listenSongMy/:id', handlers.listenSongMy);
this.post('#/createSongPost', handlers.postSong);
});
app.run('#/home');
});
|
{
"redpajama_set_name": "RedPajamaGithub"
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| 3,522
|
{"url":"http:\/\/mathhelpforum.com\/calculus\/133953-sparametric-equations.html","text":"# Math Help - sparametric equations\n\n1. ## sparametric equations\n\nHi:\nI have done part a for this parametric equation question. Before I continue to part b could somebody please check I have succeeded with part a.\nthank you\n\n2. You have done part (a) correctly. For part (b), you are given that the slope of the normal to the curve is -1\/2, which means that the slope of the tangent (which is equal to the value of the derivative) at that point is 2.\n\nTherefore, solve for t:\n\n$\\frac{2(6e^{4t} - 1)}{18e^{2t} - 1} = 2$\n\n3. here it is. Seems a little long winded. Have I done correct and is this the best way?\nthanks","date":"2015-07-06 14:33:38","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 1, \"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\": 0.7638067007064819, \"perplexity\": 372.3592369075233}, \"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-2015-27\/segments\/1435375098464.55\/warc\/CC-MAIN-20150627031818-00065-ip-10-179-60-89.ec2.internal.warc.gz\"}"}
| null | null |
\section{Introduction}
Researchers studying racial disparities often do not have self-reported demographic data readily available, and must rely on proxies such as name and location to predict race (e.g. \textcite{zhang_assessing_2018,fiscella_use_2006}).
Similary, researchers studying racial discrimination are often interested in racial \textit{signal} encoded in names \parencite{bertrand_are_2004,kang_whitened_2016}.
In hiring discrimination studies, for example, the recruiter's perception of a candidate's race (as proxied by name) becomes important, in which case it's useful to estimate the likelihood of name belonging to a particular race \parencite{parasurama_who_2020}.
Over the years, new methods and models have been proposed to incrementally improve the accuracy of race prediction models \parencite{fiscella_use_2006,imai_improving_2016,ambekar_name-ethnicity_2009,sood_predicting_2018,xie_rethnicity_2021}.
This paper contributes to this line of research by presenting a transformer-based race and ethnicity prediction model, which, to the best of my knowledge, achieves state-of-the-art results in predictive accuracy.
\section{Data}
The race prediction model uses the U.S. Florida voter registration dataset\footnote{https://dataverse.harvard.edu/dataset.xhtml?persistentId=doi:10.7910/DVN/UBIG3F} collected by \textcite{sood_predicting_2018}.
In total, there are 13,089,545 names across 5 race categories.
Table \ref{tab:race_counts} reports the counts and label for each race category.
Note that \texttt{unknown}, \texttt{other}, and \texttt{multiracial} categories are dropped, because there are too few examples.
\begin{table}[H]
\centering
\input{tables/race_counts.tex}
\caption{Florida voter registration data race label and counts}
\label{tab:race_counts}
\end{table}
\section{Models}
The main point of departure from \textcite{sood_predicting_2018} is that I replace their LSTM classifier with transformer-based models.
\subsection{Fine-tuning a pre-trained BERT model}
As a baseline, I start with a pretrained BERT\footnote{https://huggingface.co/bert-base-uncased} model for sequence classification \parencite{vaswani_attention_2017,devlin_bert_2019}.
Although BERT classification typically operates at the sentence level with sequences of tokens, it can also be used at the token level with the sequences of word pieces (i.e. sequences of characters).
First, in the preprocessing step, the name is lowercased, and the first name and the last name are concatenated by an underscore.
For example \texttt{George Smith} becomes \texttt{george\_smith}.
Then the normalized name gets tokenized into words or word pieces depending on whether the name is in the model's vocabulary.
For example, \texttt{george\_smith} gets tokenizen into \texttt{{[[CLS], george, \_, smith, [SEP]]}}.
Here, \texttt{george} and \texttt{smith} become distinct tokens because both names are part of the vocabulary.
If a name is not in the vocabulary, it gets tokenized into word pieces.
For example, \texttt{satoshi} and \texttt{nakamoto} are not in the vocabulary, therefore \texttt{satoshi\_nakamoto} gets tokenized into \texttt{[[CLS], sato, \#\#shi, \_, nak, \#\#amo, \#\#to, [SEP]]}
For the training step, following \textcite{sun_how_2019}, I use the following hyperparameters: \texttt{N\_EPOCHS=4, BATCH\_SIZE\_PER\_GPU=128, LEARNING\_RATE=2e-5, WEIGHT\_DECAY=2e-5}.
See the \href{https://wandb.ai/parasu/raceBERT-public/runs/39jr1hrc/overview}{wandb project page} for a complete list of hyperparameters and configs.
Table \ref{tab:pretrained_race_model_performance} reports the performance of the pre-trained model on a hold-out test set, and Table \ref{tab:pretrained_race_model_performance_comparison} compares raceBERT's performance to ethnicolr's performance (Note that ethnicolr does not have \texttt{aian} as a category).
raceBERT achieves better performance across all race categories compared to ethnicolr.
On average, there is a 4.4\% improvement in the f1-score, with the highest performance improvement coming from non-white names, with improvements as much as 18\% for black names.
\begin{table}[H]
\centering
\input{tables/race_performance.tex}
\caption{raceBERT hold-out performence metrics (pre-trained language model)}
\label{tab:pretrained_race_model_performance}
\end{table}
\begin{table}[H]
\centering
\input{tables/race_performance_without_aian.tex}
\caption{raceBERT hold-out performence improvement (pre-trained language model)}
\label{tab:pretrained_race_model_performance_comparison}
\end{table}
I repeat the training process with the Wikipedia ethnicity dataset \parencite{ambekar_name-ethnicity_2009,sood_predicting_2018} and report the results in Appendix A.
As with the race model, raceBERT achieves an overall 5\% improvement over ethnicolr.
\subsection{Training a model from scratch}
Although the pre-trained BERT model achieves significant performance improvements over ethnicolr, a few limitations remain.
First, with a vocabulary size of 30,000, the model is needlessly large for the task at hand.
Most of the tokens in the vocabulary will never be used.
Second, many commonly occurring names are in the vocabulary, which raises the question of whether the model is simply memorizing names rather than learning from character sequences of names, in which case the generalizability of the model will suffer.
To overcome these issues, I train a roBERTa model from scratch with a much smaller vocabulary size of 500 \parencite{liu_roberta_2019}.
In the preprocessing step, the first name is lowercased, and the last name is uppercased, which are then concatenated by a space.
For example \texttt{George Smith} becomes \texttt{george SMITH}.
In theory, the mixed lower/upper case will make it easier for the model to discriminate between first and last names.
Using the transformed names, I train a Byte Pair Encoding tokenizer with a max vocabulary size of 500 to learn the most commonly occurring character sequences, which is then used to tokenize all names.
For example, \texttt{george SMITH} gets tokenized into \texttt{[[CLS], ge, or, ge, SMITH, [SEP]]}.
Likewise, \texttt{satoshi NAKAMOTO} gets tokenized into \texttt{[[CLS], sa, t, o, sh, i, N, A, K, AM, O, T, O, [SEP]]}.
Next, I train a masked language model to learn the character sequences using the roBERTa masked language architecture \texttt{(ATTENTION\_HEADS=12, HIDDEN\_LAYERS=6)}.
Using the weights from the language model, I initialize a roBERTa sequence classification model and train using the following hyperparameters: \texttt{N\_EPOCHS=4, BATCH\_SIZE\_PER\_GPU=128. LEARNING\_RATE=2e-5, WEIGHT\_DECAY=2e-5}.
All other configs can be found at the \href{https://wandb.ai/parasu/raceBERT-public/runs/39jr1hrc/overview}{wandb project page}.
Table \ref{race_model_performance} reports the performance of the model on a hold-out test set, and \autoref{race_model_performance_without_aian} reports the performance improvement compared to ethnicolr.
The model's performance metrics are almost the same as the pre-trained BERT model, but with a much smaller vocabulary and (theoretically) greater generalizability.
As such, I use this as the default model for the python package.
\begin{table}[H]
\centering
\input{tables/race_performance_bert_scratch.tex}
\caption{raceBERT hold-out performence metrics}
\label{race_model_performance}
\end{table}
\begin{table}[H]
\centering
\input{tables/race_bert_scratch_without_aian.tex}
\caption{raceBERT hold-out performence improvements}
\label{race_model_performance_without_aian}
\end{table}
\subsection{Code, Configs, and Resources}
All of the training code, as well as the raceBERT python package code, is on \href{https://github.com/parasurama/raceBERT}{Github}\footnote{https://github.com/parasurama/raceBERT}.
The trained models are uploaded to the \href{https://huggingface.co/pparasurama/raceBERT}{huggingface hub}\footnote{https://huggingface.co/pparasurama/raceBERT}.
All training configurations, hyperparameters, and learning curves can be found at the \href{https://wandb.ai/parasu/raceBERT-public}{wandb project page}\footnote{https://wandb.ai/parasu/raceBERT-public}.
\section{Conclusion \& Limitations}
This paper presents a new transformer-based model for predicting race from names and demonstrates the performance improvements over existing state-of-the-art models.
One limitation of this model is that it's trained on the Florida voter registration dataset, which is not necessarily representative of the U.S. population.
As such, the accuracy scores may vary when used on datasets from U.S. regions with different demographic distributions.
Another limitation -- one that's shared by all race prediction models -- is that the model is not 100\% accurate.
While this is admissable when studying racial disparities in the aggregate, or when estimating the likelihood of a name belonging to a particular race, it's not advisable to use this model in applications where it is important to know individual race.
\pagebreak
\printbibliography
\pagebreak
\section*{Appendix}
|
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}
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Category: Bills (continued)
90 Players in 90 Days: Buffalo Bills LB Lorenzo Alexander
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Are you Human? 12+48=?
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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The Horseback Ride to Tocori Waterfalls is a unique tour that combines natural beauty, nature, and local culture into one unique experience. In one action-packed tour, you will horseback ride on a country farm, walk through the jungle, observe and swim beneath a waterfall, and then spend time at the family farm, enjoying a traditional Costa Rican meal!
The day starts when you are picked up from your local area hotel. Those staying in Manuel Antonio will drive for about 20 minutes to the farm after picking up other adventurers. Guests staying in Jaco, Uvita, or Dominical will have a longer drive.
When you arrive at Rancho Don Gilberto, you will meet your guide. He will fit you with a helmet, provide a brief safety and instructional speech on horseback riding, and then you will be matched with your horse.
You will ride along a trail that eventually cuts through fields and forest for about an hour. Then you will walk along a trail towards the waterfalls. Along the way, your guide will tell you about the plants, animals, and history of the area.
There are four waterfalls at the Tocori Waterfalls, 1 requires almost no walking, but if you can walk about 1/4 mile, you will discover the rest. Don't forget your camera for the stunning views! And don't forget your bathing suit so you can swim below the falls!
After taking in the scenery and cooling off in the stream, you will ride back to the private ranch for a delicious Costa Rican style meal. Casados, BBQ, and fresh local fruits will be served. You will also have time to see and learn about indigenous herbs and plants.
Following lunch, you will be returned to your hotel. The tour is expected to last around 4 – 5 hours including transfer times from Manuel Antonio and longer for those staying in Dominical, Uvita, or Jaco. Of course, this is subject to change due to the nature of the tour. Riders should not weigh over 200 pounds. Participants should be able to ride at an easy pace for around two hours. Your adventure includes bilingual guide, lunch, drinks, and transportation.
Cancellation Policies Please send cancellations by e-mail to our main office in Quepos, Costa Rica. Unfortunately, there will be no refund on cancellations 7 days before the confirmed pick up time for any tour. For group tours, there are no refunds on cancellations made 7 days before the date of the tour. In case of a cancellation, Iguana Tours will credit the agency the total amount prepaid when following our cancellation policies. If there are any questions in regard to these policies, please call our Director of Operations directly.
Latitude: N 9°25'45.71". Longitude: W 85°50'23.36"
A fun day with friendly local guides, well trained horses, beautiful scenery and excellent home-cooked meal.
There were only my husband and I and one other tourist on this trail ride. The pictures and videos I brought home made my fellow equestrian friends very jealous !
This tour really gave you a sense of CR, the jungle, the way of life, how animals are kept and how they run this ranch. Our guide brought his son along. The guide was very informative and made us feel very safe, as we were in the deep jungle for the ride. I loved the overwhelming sounds of the jungle while we quietly rode thru the jungle. So amazing ! I have been riding horses all of my life and have been on trail rides all over the USA, but this ride tops my list !! We also loved the home cooked, local food, coffee and fruits we were served after the trail ride. What a great memory !!
The guides on this tour were friendly but we all felt like they were rushing us through the tour. We also didn't ride the horses for very long, probably only 30 minutes total. Also, the waterfall they mention swimming in is literally a water hole at the bottom of a waterfall. Definitely not big enough for the whole tour of us to swim in. The horses are well trained as well as the guides. The home cooked Costa Rican meal at the end was also very good!
Seemed a bit unorganized, it was disconcerting to have no guide in front to lead the way. Guides were knowledgeable, but not very personable.
This tour was well organized and exceeded expectations. It was a safe adventure with my 11-year old child. Lunch afterwards was a treat!
good choice for outfitter. Guide was an old timer with good stories and experience. beautiful waterfalls.
This was a but less "guided" once we got on the horses than expected but as a result was rather exciting and we juts loved it. Fantastic ride, beautiful waterfall and short walk and plenty of time to relax at the destination. Wish we'd have taken our swimsuits. Worrying when the guide was bucked off his horse but it all ended well. He had a great perspective on history, local life, politics, economy - we learned a lot.
The ride back to the house was so fun through the water.. we could have rode it longer.
We were the first to be picked up and the last to be dropped off so we spent more time in the van than anyone else. Didn't appreciate that part. But I enjoyed the ride once we got off the road and into the trails and forest. Lots of butterflies, including the giant blue morpho! The small waterfall and swimming hole were refreshing after the hot ride--felt good to cool off!
the saddle is not anything special. may tell clients to ware more padded or horse ride appropriate clothes. they say come ready to go swimming in waterfall. that's ok but kinda a hassle to get wet and then ride again.
Now during the rainy season, the trails are muddy and slippery which scared the crap out of us lol but still made for a good time. Gilberto was a fun guide and the cooked food after the trail was great!
The tour guides seemed disorganized and confused, taking forever to get us on our horses and on the trail. This was also much more strenuous than expected - one on our party had to get off her horse and walk because her horse would not walk but wanted to trot the whole way. I would not want to do this again.
Horse's were great. Inexperienced at riding made the ride seem longer.
This Guide Roberto was accommodating, patient and very attentive to his clients. It was an excellent experience.
AMAZING- Gilberto our guide was awesome- really nice guy who loves horses. We especially liked the history of the indigenous people he told us about and all the history of the surrounding area, the waterfall was beautiful and refreshing. The lunch at the end of the ride was delicious.
I would just try to schedule this in the am during rainy season.
This tour needs to have limitations....we and 2 other families that were at Si Como No all had the same experience. The tour was fine, but the horseback riding was rough. We were sore for DAYS after - miserabley sore!! Maybe you should just tell people if it's been awhile since they've been on a horse, then this is not the tour for them. Thank goodnedss this tour was our last one - it ruined my last day in CR.
Did not go on this tour because there is a weight limit of 230lbs and I weigh 250. So they adjusted and took us to the mangrove tour. That was okay.
Our 2 guides were very nice and the older 'cowboy' guide man definitely shared his life experiences with us and made the tour interesting. Had us smelling and tasting different vegetation and telling us how he lives off the land and only uses herbs for medicine...very nice tour.
Two hours on a four-wheel-drive horse is just too long. I was bruised and sore for days. The waterfall is beautiful, but not worth the hour on horseback to get there. Lunch afterward was very good.
Fun tour! I just don't think I'm that great at horseback riding - I got very sore.
Although Alberto is a very smart man and was good to point out plant life and inform us of the healing properties of the plants....he was a very serious man and a little difficult to talk to. Our lunch that was prepared out at the ranch was very nice and I would have like to see more of the ranch and learn more about what they have there, but it was almost like it wasn't allowed....too bad.
The reason for the lower logistics rating was we were all picked up from our hotels then went to a central office and had to change vans. This van had no air condition for the long ride to the ranch. The tour and lunch were great. The main man that was a guide was excellent. Told us about what we were seeing. The woman was not very helpful.
Overall, we had a good time. The waterfall and the surrounding area were quite beautiful, but the bulk of the ride is through residential streets where it is super hot and dusty and not very scenic. Would have preferred a ride that spent more time in the forested area. Lunch was great. The main guide was great. The other guides were more horse-people than people-people, but that was fine. I would choose a different tour next time.
The tour was good but the guide was not very friendly. As people who do not ride horses everyday, I wish that the guide would have been more helpful in helping us control the horses so that our rides were more smooth.
The good news was that we were the only participants that morning and got a private tour. However, half of the horseback ride was along a road, and although it wasn't a very busy road, it did make me a little nervous. My horse must have picked up on my nervousness and reacted a little scared and was a little skittish. So, the entire time we were on horseback I felt tense. I did loosen up on the way back though, so that was more enjoyable. The tour guide was nice, but not very talkative. I think the experience was ok. Not sure I would do it again.
Given the length of the horseback ride and the rocky path, this tour would be better for someone with good experience on horses. The waterfall was small and thus somewhat disappointing.
If you enjoy horseback riding, this is the tour. You get to ride in different terrains and locations before reaching the waterfalls: Regular street, unpaved street, muddy and steep paths and even crossing and walking in the river! Overall a great experience especially for children.
The trip to the falls was beautiful, but I have decided horseback riding is not my cup of tea. The guide was great and the lunch we were served was great.
|
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{"url":"http:\/\/mathhelpforum.com\/calculus\/168705-integral-refresher-question.html","text":"# Math Help - Integral Refresher Question\n\n1. ## Integral Refresher Question\n\nHi, I'm having a little bit of trouble with a integration question, here is the text from the question.\n\nFind the antiderivitive of the given functions:\n\n$F(x) = x(1-x^2)^7$\n\nBasically I just have no idea how to expand $(1 - x^2)^7$\n\n2. Originally Posted by Crell\nHi, I'm having a little bit of trouble with a integration question, here is the text from the question.\n\nFind the antiderivitive of the given functions:\n\n$F(x) = x(1-x^2)^7$\n\nBasically I just have no idea how to expand $(1 - x^2)^7$\nDont expand!\n\nUse substition.\n\nu=1-x^2\n\n3. Well, you could expand it using the \"binomial theorem\": $(a+ b)^n= \\sum_{i=0}^n \\begin{pmatrix}n \\\\ i\\end{pmatrix}a^ib^{n- i}$\nFor $(1- x^2)^7= 1- 7x^2+ 21x^4+ \\cdot\\cdot\\cdot- 7x^{12}+ x^{14}$\n\nBut you don't want to do that. Instead, make the substituition $u= 1- x^2$ so that $(1- x^2)^7= u^7$ and $du= -2 xdx$.\n\n4. Originally Posted by HallsofIvy\n$du= -2 xdx$.\nThank you much, this is the part I was missing.\n\n$\n\n\\int x(1-x^2)^7dx\n\n$\n\nAside:\n\nLet $u = 1-x^2$\n\nDerive this equation\n\n$\\frac{du}{dx} = -2x$\n\nSolve for $dx$\n\n$dx = \\frac{du}{-2x}$\n\nManipulate the first equation to make it appear like $-2x(u)^7dx$ so that when you substitute dx, the x is eliminated.\n\nTherefore:\n\n$-\\frac{1}{2} \\int -2x(u)^7dx$\n\nSubstitute $dx = \\frac{du}{-2x}$ into first equation\n$-\\frac{1}{2} \\int \\frac {-2x(u)^7du}{-2x}$\n\n$-\\frac{1}{2} \\int {(u)^7du}$\n\nIntegrate, sub in u and simplfy.\n\n$- \\frac{1}{2} [\\frac {(u)^8}{8}]$\n\n$-\\frac{1}{2} [\\frac {(1-x^2)^8}{8}]$\n\n$-\\frac {(1-x^2)^8}{16}$\n\n5. Did you differentiate that answer to cheek it for yourself?\n\n6. Have now, thanks.\n\n7. This is a perfect place to use my favourite rule of integration:\n\n$\\displaystyle\\int{f'(x)f(x)^n} dx=\\frac{1}{n+1}f(x)^{n+1}+c$\n\nSo let $f(x)=(1-x^2)$\n$f'(x)=-2x$\n$n=7$\n\n$f'(x)f(x)^n= -2x(1-x^2)^7$\nYou'll notice that it isn't quite a match to what you are trying to integrate. However, if you divide by $-2$, it is exactly what you are trying to integrate. Therefore, we can apply the rule as normal, but we must remember to divide our final answer by -2.\n\n$\\displaystyle\\int x(1-x^2)^7 dx$\n\n$=\\displaystyle\\frac{-1}{2}\\times\\frac{(1-x^2)^8}{8} + C$\n\n$=\\displaystyle -\\frac{(1-x^2)^8}{16} + C$\n\nI think in this case, this method is much nicer than substitution - if you spot the rule, it will take a few lines of working to obtain the full solution with no messy work required at all.\n\n8. In speaking of the binomial theorem, you can find the anti-derivative\nwithout performing an expansion, but in somewhat undesirable forms:\n\n$\\displaystyle \\sum_{k=0}^{\\infty}\\frac{(-1)^kx^{2k+2}}{2k+2}\\binom{7}{k}$ or $\\displaystyle \\sum_{k=0}^{7}\\frac{(-1)^k(x)^{2k+2}}{2k+2}\\binom{7}{k}$.\n\nPlus the constant of integration. The u-sub does the job far better.\n\nSpoiler:\n1.\n\n$\\displaystyle x(1-x^2)^7 = x\\left(\\sum_{k=0}^{\\infty}\\binom{7}{k}(-1)^kx^{2k}\\right) = \\sum_{k=0}^{\\infty}\\binom{7}{k}(-1)^{k}x^{2k+1}.$\n\n\\begin{aligned}\\displaystyle \\therefore \\int x(1-x^2)^7\\;{dx} & = \\int \\left(\\sum_{k=0}^{\\infty}\\binom{7}{k}(-1)^{k}x^{2k+1}\\right)\\;{dx} \\\\& = \\sum_{k=0}^{\\infty} \\left(\\int \\binom{7}{k}(-1)^{k}x^{2k+1}\\;{dx}\\right) \\\\& = \\sum_{k=0}^{\\infty}\\frac{(-1)^kx^{2k+2}}{2k+2}\\binom{7}{k}+C.\\end{aligned}\n\n2.\n\n\\displaystyle \\begin{aligned}x(1-x^2)^7 & = x\\left(\\sum_{k=0}^{7}\\binom{7}{k}(-1)^kx^{2k}\\right) \\\\& = \\sum_{k=0}^{7}\\binom{7}{k}(-1)^kx^{2k+1}.\\end{aligned}\n\n\\displaystyle \\begin{aligned} \\therefore \\int x(1-x^2)^7\\;{dx} & = \\int \\left(\\sum_{k=0}^{7}\\binom{7}{k}(-1)^kx^{2k+1}\\right)\\;{dx} \\\\& = \\sum_{k=0}^{7} \\left(\\int \\binom{7}{k}(-1)^kx^{2k+1}\\;{dx}\\right) \\\\& = \\sum_{k=0}^{7}\\frac{(-1)^k(x)^{2k+2}}{2k+2}\\binom{7}{k}+C.\\end{aligned}","date":"2014-07-23 00:04:24","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\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 38, \"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\": 0.9775760769844055, \"perplexity\": 840.7455154302772}, \"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\/1405997869778.45\/warc\/CC-MAIN-20140722025749-00093-ip-10-33-131-23.ec2.internal.warc.gz\"}"}
| null | null |
Q: RESOLVIDO – Pré-definir o tamanho do eixo y no Chart.Js Edit: Pessoal, acho que já resolvi. Em options é possível colocar esse código:
scales: {
y: {
min: 0,
max: 120,
ticks: {
stepSize: 20
}
}
}
Com ele consigo predefinir o tamanho do eixo y, e também colocar os múltiplos (nesse caso optei por múltiplos de 20).
A dúvida era:
Estou fazendo um gráfico com Chart.Js e estou precisando fazer uma mini gambirra para uma das linhas da Meta aparecer bonitinha, ao invés de colocar o yMin e yMax iguais, estou colocando o yMax com 0.01 a mais que o yMin, pois percebi que quando o valor da linha é maior que um valor redondo (60, 80, 100 etc.) automaticamente sobe +20 na altura do y.
Eu gostaria que por padrão o topo do y já ficasse com um valor um pouco maior que o maior valor da Meta, pois caso o topo fique igual ao valor da Meta, fica meio feio, vejam:
Meu código (eu ainda vou simplificá-lo):
const ctx = document.getElementById('myChart').getContext('2d');
const myChart = new Chart(ctx, {
type: 'bar',
data: {
labels: data_final, //['Red', 'Blue', 'Yellow', 'Green', 'Purple', 'Orange'],
datasets: [{
label: '# of Votes',
data: qtde_produzida, //[12, 19, 3, 5, 2, 3],
backgroundColor: [
'rgba(255, 99, 132, 0.2)',
'rgba(54, 162, 235, 0.2)',
'rgba(255, 206, 86, 0.2)',
'rgba(75, 192, 192, 0.2)',
'rgba(153, 102, 255, 0.2)',
'rgba(255, 159, 64, 0.2)'
],
borderColor: [
'rgba(255, 99, 132, 1)',
'rgba(54, 162, 235, 1)',
'rgba(255, 206, 86, 1)',
'rgba(75, 192, 192, 1)',
'rgba(153, 102, 255, 1)',
'rgba(255, 159, 64, 1)'
],
borderWidth: 2
}]
},
options: {
scales: {
y: {
beginAtZero: true
}
},
plugins: {
annotation: {
annotations: {
line1: {
type: 'line',
mode: 'horizontal',
scaleID: 'y-axis-0',
yMin: meta[0],
yMax: meta[0],
borderColor: 'rgba(255, 99, 132, 1)',
borderWidth: 3,
label: {
content: "Meta: " + meta[0],
enabled: true,
backgroundColor: 'rgba(255, 99, 132, 1)',
}
},
line2: {
type: 'line',
mode: 'horizontal',
scaleID: 'y-axis-0',
yMin: meta[1],
yMax: meta[1],
borderColor: 'rgba(54, 162, 235, 1)',
borderWidth: 3,
label: {
content: "Meta: " + meta[1],
enabled: true,
backgroundColor: 'rgba(54, 162, 235, 1)',
}
}
}
}
}
}
});
Obrigado.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,156
|
M98 (NGC 4192) e спирална галактика, разположена по посока на съзвездието Косите на Вероника.
Открита е от Пиер Мешен през 1781. М98 е част от галактичния свръхкуп в Дева
Ъгловите ̀и размери са 9′.8 × 2′.8. Видимата ̀и звездна величина е +11.0, а разстоянието до нея е 60 млн. св.г..
Външни препратки
Spiral Galaxy M98 @ SEDS Messier pages
WIKISKY.ORG: SDSS image, M98
Вижте също
Списък на обектите на Месие
Бележки
98
Галактики
Астрономически обекти, открити през 1781 година
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 7,465
|
Kessleria saxifragae es una especie de polilla del género Kessleria, familia Yponomeutidae.
Fue descrita científicamente por Stainton en 1868.
Referencias
Enlaces externos
Kessleria catalogueoflife.org
saxifragae
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,783
|
Q: Nesting multiple same type class inside one parent I want to create classes like below:
.colored{
&{
.red{background-color:red;}
.blue{background-color:blue;}
.gray{background-color:gray;}
}
}
and it is not working but if I write like this:
.colored{
&.red{
background-color:red;
}
&.blue{
background-color:blue;
}
&.gray{
background-color:gray;
}
}
then it works. Are there any reasons why the first version does not work as expected?
A: Reason:
It is because of the way nesting works (and is supposed to work) in Less. When you nest a selector under another selector, the inner selector is considered as applicable for an element which is a child of the element represented by the outer selector.
The below is code
.colored{
&{
.red{background-color:red;}
.blue{background-color:blue;}
.gray{background-color:gray;}
}
}
is equivalent to the following (because & will be replaced by the parent selector which is .colored)
.colored{
.red{background-color:red;}
.blue{background-color:blue;}
.gray{background-color:gray;}
}
Now as I mentioned earlier, this would compile to .colored .red, .colored .blue, .colored .gray (note the space between the selectors).
On the other hand when you insert the & immediately before .red, .blue, .gray, it means that the parent selector and the nested selector apply to the same element (and not a child). The selectors that it would output is .colored.red, .colored.blue, .colored.gray (note that there is no space).
Solution:
This is the same code as in your question and it is the recommended solution. I am posting it again in the answer only to make it complete.
.colored{
&.red{
background-color:red;
}
&.blue{
background-color:blue;
}
&.gray{
background-color:gray;
}
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,337
|
Q: malloc : Anonymous mapping and magic area I was just fiddling around memory mappings and wanted to view user space virtual memory region mappings. Wrote some lines like
char *ptr = NULL;
printf("Allocating 300KB\n");
ptr = malloc (300*1024);
printf("Allocated at %p.. sleeping\n", ptr);
sleep (30);
free (ptr);
printf("Freed... sleeping\n");
sleep (30);
On running the program, pmap on the pid shows allocated region as:
00007f73b1e57000 316K rw--- [ anon ]
while program o/p says:
Allocated at 0x7f73b1e57010.. sleeping
Is this 16KB extra allocation for what we call magic region on allocation?
In the kernel, the corresponding vm_area_struct will hold ranges visible to program or the entire range from starting of magic region?
A: The difference is not 16KB, it is 16 bytes. Which most probably corresponds to the header that malloc has to allocate before your memory block so as to link blocks together, etc.
A: To start with any OS which has a Memory management unit manages all it's memory (heap, code space, stacks I/O memory) using the MMU, all memory exists in a virtual space and page tables are used to translate the virtual addresses into physical addresses, the mapping to physical memory is dependent on the OS
malloc will return a pointer to heap memory using sbrk call which in turn will increase the heap size, the MMU when this memory is accessed will then allocates actual physical page and maps to the virtual address.
According to pmap manual page the output shows and not the allocated memory block size from malloc but the virtual mapping size.
"Virtual Mapping Size (Kbytes)
The virtual size in kilobytes of each mapping."
For a quick experiment to check if the block size of the memory returned from malloc should be equal to the output from pmap.
To prove the point I did a quick test using this code
int main(int argc, char **argv)
{
char *timeBuf = (char *)malloc(100);
printf("allocated address is %p\n",timeBuf);
int i;
for(i =0 ;i < atoi(argv[1]);i++)
{
}
return 0;
}
the pmap output is:
`0000000001338000 132K rw--- [ anon ]`
The returned pointer from malloc:
allocated address is 0x1338010
I think the 16 bytes is kept by malloc for book keeping in it's headers as mentioned in previous answers. As you can se the allocated memory in program is just 100 bytes but the pmap virtual memory size is 132K
So to answer your question in short no this is not related to the magic area.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,528
|
ADK STAB AO-40 is an effective antioxidant for long term thermal stability in ABS and rubbers. High performance antioxidant for engineering plastics and elastomers.
ADK STAB AO-50 is a highly effective and versatile, low melting antioxidant for all plastics and elastomers. ADK STAB AO-50 shows good compatibility with most common polymers and is a good substitute for BHT. General purpose antioxidant recommended for all plastics and rubbers.
ADK STAB AO-60 has a high molecular weight and is a highly effective thermal and process stabiliser with low volatility. ADK STAB AO-60 shows synergistic activity with phosphite process stabilisers and thioethers for long term thermal stability. General purpose antioxidant suitable for most plastics, elastomers and synthetic fibers.
ADK STAB AO-80 is a high molecular weight type highly effective hindered phenolic antioxidant that has low volatility and excellent gas fading resistance. ADK STAB AO-80 also shows excellent synergism with phosphites and high molecular weight thioethers for long term thermal aging and is suitable for most plastics, elastomers and synthetic fibers.
CCP STAB 390E is an excellent antioxidant with the advantage of a pumpable liquid. Offers low volatility, low fogging and favorable compatibility, particularly recommended for polyols, polyurethanes and textiles. Excellent fit for processes requiring liquids, emulsions or suspensions.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,764
|
# MRS MOREAU'S WARBLER
_How Birds Got their Names_
STEPHEN MOSS
For Suzanne: after whom, one day,
I hope to name a new species of warbler.
# CONTENTS
1. Title Page
2. Dedication
3. List of Illustrations
4. Introduction
5. Prologue
6. [1. SOUND AND ECHOES
_The Origins 0f Bird Names_ ](9781783350926_chapter_01.html#Chapter1)
7. 1: The Cuckoo's Calling
8. 2: Trade Routes and Translations
9. 3: Invasions and Conquests
10. 4: The Nature of Birdsong
11. 5: The Sound Approach
12. [2. INVASION AND CHANGE
_The Beginnings of English_ ](9781783350926_chapter_02.html#Chapter2)
13. 1: The Ravens' Lament
14. 2: Red Tails and White Arses
15. 3: Sex, Chaucer and Blackbirds
16. 4: Fifty Shades of Green?
17. [3. HISTORY AND SCIENCE
_The Birth of Ornithology_ ](9781783350926_chapter_03.html#Chapter3)
18. 1: Dirty Underwear
19. 2: Folk and Fowls
20. 3: Pioneers and Puffins
21. 4: A Little Latin
22. 5: A Correspondence Course
23. [4. TAMING NATURE
_The Organisation of Bird Names_ ](9781783350926_chapter_04.html#Chapter4)
24. 1: A Man of Kent
25. 2: Flaming Galahs and Fairy-Wrens
26. 3: The Nature Poet
27. 4: The Military Man
28. [5. EPONYMS AND EXPLORATION
_Bird Names go Global_ ](9781783350926_chapter_05.html#Chapter5)
29. 1: The Museum Man
30. 2: Eponymous Birds
31. 3: Into the North
32. 4: Scotland's Forgotten Genius
33. 5: Exploration and Empire
34. [6. TWENTIETH-CENTURY FLOCKS
_The Names we use Today_ ](9781783350926_chapter_06.html#Chapter6)
35. 1: Redbreasts and Hedge Sparrows
36. 2: Tit-Willow and Willow Tit
37. 3: Reed Warblers and Roasted Larks
38. 4: Canada Geese and Crossbills
39. 5: Eiderdowns, Cranes and Kites
40. 6: Hobbies and Spies
41. [7. TOMORROW NEVER KNOWS
_The Future of Bird Names_ ](9781783350926_chapter_07.html#Chapter7)
42. 1: Bird Names at a Crossroads
43. 2: Titmice and Ring-Doves
44. 3: Politics and Political Correctness
45. 4: Splitting Species
46. 5: New Birds, New Names
47. Epilogue
48. Acknowledgements
49. Appendix
50. Index
51. About the Author
52. Also by the Author
53. Copyright
# ILLUSTRATIONS
1 Cuckoo – _Cuculus canorus_
2 Redstart – _Phoenicurus phoenicurus_
3 Puffin – _Fratercula arctica_
4 Dartford Warbler – _Sylvia undata_
5 Leach's Petrel – _Oceanodroma leuchorhoa_
6 Robin – _Erithacus rubecula_
7 Kestrel – _Falco tinnunculus_
# INTRODUCTION
And out of the ground the Lord God formed every beast of the field, and every fowl of the air; and brought them unto Adam to see what he would call them: and whatsoever Adam called every living creature, that was the name thereof. And Adam gave names to all cattle, and to the fowl of the air...
Genesis, 2:19-20
Swallow and starling, puffin and peregrine, blue tit and blackcap. We use these names so often that few of us ever pause to wonder about their origins. What do they mean? Where did they come from? And – Old Testament mythology aside – who originally created them?
Sometimes it's easy to assume that we know what a bird's name means, and often that assumption is quite correct. Treecreepers creep around trees, whitethroats have a white throat, and cuckoos do indeed call out their name.
The origin of other names can seem obvious, but may not be quite as straightforward as first appears. Even the simplest of English bird names, 'blackbird', turns out to be more complicated than you might imagine. There is also a whole range of folk names, from 'scribble lark' to 'sea swallow' and 'flop wing' to 'furze wren', each of which has its own tale to tell about our language, history and culture.i
Ornithologists have often been rather dismissive of 'folk names', as though they are somehow inferior to the official, authorised ones. Yet, as the French scholar Michel Desfayes points out in his monumental two-volume work on the origins of European bird names, it is purely a matter of chance that, while some folk names remained localised, others were adopted as the name we still use today.1
*
Another pressing question is, _when_ were birds given their names? Broadly speaking, it is reasonable to assume that most common and familiar birds were named a long time ago, by ordinary people – hence the term 'folk' names – while scarce and unfamiliar birds were named much more recently, by professional ornithologists.
Another general rule is that most early names were based on some obvious feature of the bird itself: its sound, colour or pattern, shape or size, habits or behaviour. Some of our longest-standing names reflect this, such as cuckoo and chiffchaff, blackcap and whitethroat, woodpecker and great tit.
Once the professionals got involved, from the seventeenth century onwards, names began to be based on more arcane aspects of birds' lives, such as where they live or the locality where they were found. These include habitat-based names such as reed, sedge and willow warblers, along with place-based names such as Dartford warbler and Manx shearwater. Many compound names, such as black-tailed and bar-tailed godwits, and pink-footed and white-fronted goose, also arose during this period, to help tell similar species apart.
The final category of bird names – most of which also originated fairly recently, during the eighteenth and nineteenth centuries – is in many ways the most beguiling. These are the species called after people, such as Montagu's harrier, Bewick's swan, Cetti's warbler and Leach's petrel.
The stories behind these birds, and the people after whom they were named, are told in Chapters 4 and 5. They include the country parson Gilbert White, author of _The Natural History of Selborne_ ; James Clark Ross, a young midshipman who shot his eponymous gull on a failed expedition to reach the fabled North-West Passage; and the disgraced military officer George Montagu who, following a midlife crisis, fled to Devon with his mistress, where he pursued the study of birds for the rest of his days.
*
According to the opening book of the Old Testament, once Adam had been created, almost the very first thing he did was to give names to the birds. As one commentator has shrewdly pointed out, this means that – more dubious claimants aside – taxonomists can justifiably claim to be the world's oldest profession.2
Since then, names have always fascinated us, yet they can also frustrate us. In _Romeo and Juliet_ , Shakespeare's lovelorn heroine laments,
What's in a name? That which we call a rose
By any other name would smell as sweet.
Superficially at least, the Bard makes a valid point. As philosophers have long argued, the name we give to a person, place or object often has little or no connection with its sense and meaning: if we called a rose something completely different, it would still be the same flower.
But is that always the case? After all, names are not always random or meaningless labels, unconnected with the object to which they are attached. More than any other words, names carry with them the baggage of their etymological history: a history that, once we begin to investigate more deeply, reveals unexpected origins, and often yields a profound association between the name and the object that bears it. That's certainly true of onomatopoeic names, which derive from the sound the bird makes, and also of many names based on a bird's colour, pattern, habits and habitat.
At other times, though, a bird's name can cause confusion and misunderstanding. Some lead us down a blind alley, as in hedge sparrow – long used for the dunnock – which is not a sparrow at all, but an accentor. Other misleading names include stone curlew, a bird only distantly related to the true curlews; and bearded tit, which is neither bearded (it sports magnificent 'moustaches'), nor a tit.
In an ideal world, the names we give to birds would all make perfect sense. But in the real – and far more fascinating – world, they do not. This is for one simple reason: they were not handed down to mankind since time immemorial, as depicted in the Book of Genesis. Instead, they were coined by a whole range of different people, over many thousands of years, from the prehistoric era to the present day.
For this pressing urge to name the things we see around us dates back to our earliest ancestors. Initially, at the dawn of human civilisation, it would have been for purely practical reasons. Our hunter-gatherer ancestors would have soon realised that they needed to give names to the various wild creatures they came across, so they could easily distinguish between those that might be good to eat, and those that might kill and eat them.
As the evolutionary biologist Carol Kaesuk Yoon has pointed out, the ability to name things – and then recall what they were named at a later date – would have been essential for survival: 'Anyone living in the wild who could not reliably order, name, communicate about, and remember which organisms were which – who could not do good caveman taxonomyii – would most likely have led a considerably tougher and possibly shorter existence.'
From roughly ten thousand years ago, the coming of agriculture brought a new dimension to the naming of living things. Those first farmers needed to know when a particular wild flower would come into bloom, or at what time of year a migratory bird would depart and return. Understanding the timing of these events allowed them to chart the changing of the seasons, and know when to plant and harvest their precious crops.
As a result of these primordial needs, human beings evolved to notice the plants and animals around them, perceive their similarities and differences, and give them names based on those characteristics. Indeed, had our ancestors not learned to read the natural world, and shape that world around their needs, it is unlikely that human society and culture would have made such rapid and spectacular progress.
So although in the modern world we no longer need to learn names to know which creatures are good to eat – and which might in turn eat us – there can be no doubt that we live with the legacy of that early impulse. Having evolved as a hunter-gatherer, we continue to use many of those skills and techniques, even today. For what is birding, if not a sublimated form of hunting?
*
Names – and in particular the names of the other living things around us – help us make sense of the world. But do they do more than that? Do they also affect the way we perceive the very objects to which we give those names? And if, as we can probably agree, they do, then is this a positive or negative thing?
A strong case can be made for the idea that when we know the names of living creatures, it helps us appreciate the diversity of the natural world, and treat other species better. The Indian entrepreneur Aishwarya Shiva Pareek goes a step further, making an explicit connection between naming and being human: 'This is the main objective of human life... to give unique identity to unknown things in our native languages and to categorize them... Without us these things are nameless...'3
But this anthropocentric world-view has its dangers. It raises the valid concern that by naming living creatures, and bringing them under our own sphere of control, we may somehow diminish them. As the author Joanne Harris perceptively notes, 'A named thing is a tamed thing.'4 When we give a wild creature a name, are we not perhaps extending mankind's sovereignty over other species, in an act that goes right back to Adam's naming of 'every fowl of the air' in the first book of the Bible?
So on the one hand, it is clear that names enable us to better know, understand and appreciate the natural world. Yet on the other hand, they can create an artificial barrier between the rest of nature and humankind. 'Names are masks', argues the American novelist Matthew Woodring Stover, 'they get in the way':5 naming can reinforce the growing gulf between humanity and other living things.
The author John Fowles had no doubt on which side of the argument he belonged: 'Even the simplest knowledge of the names and habits of flowers or trees... removes us a step from total reality towards anthropocentrism; that is, it acts mentally as an equivalent of the camera viewfinder. Already it destroys or curtails certain possibilities of seeing, apprehending and experiencing.'6
Fowles's belief that by naming other species we create a distance between them and us, as when we look at the world through the lens of a camera, is a potentially seductive idea.iii Indeed, this creative tension – whether names bring us closer to the natural world or distance us from it – reverberates through this book. But although I can sympathise with Fowles's point, I must come down firmly on the side of the namers.
I believe that by giving linguistic labels to the multifarious wonders of life around us – by watching, seeing, focusing on and separating one organism from another, closely related species – we are then better able to understand and appreciate the natural world in all its glorious variety and confusion.
*
Sometimes, of course, the origin of a bird's name is simply lost. We can only guess at the meaning of the names we call many of our commonest and most familiar birds: swan, goose, sparrow and starling. As the man who spent more time studying the origins of bird names than virtually anyone, the late Professor W. B. Lockwood, pointed out, 'There is good reason to believe that in a number of cases answers may for ever elude us.'7
What we _do_ know is that the process of naming birds was, like so many other aspects of our language, strongly influenced by major events in our own history. This began with the initial shift from a nomadic, hunter-gathering existence to the beginnings of settled agriculture on the fertile river plains of present-day Iraq, more than ten thousand years ago. It continued via the emergence of the ancestral language of so many modern tongues, Proto-Indo-European, on the steppes of central Eurasia, about three thousand years before the birth of Christ. And it developed and changed as a result of the successive invasions and conquests of our own islands, and our later expansion and empire-building, both of which helped define the nature of the English language spoken not just by 65 million Britons, but also as a _lingua franca_ around the rest of the world.
The story of how birds got their names takes us on a journey through the major events in our language, history and culture. We shall discover how a small band of Anglo-Saxon invaders began the process of giving English names to birds; how the Norman Conquest led to a linguistic and cultural divide between lords and servants, still reflected in many modern bird names; and how writers like Chaucer and Shakespeare made their own important contributions to our knowledge and understanding of what our birds were called.
Yet, as we'll also find, despite radical changes in our language a surprising number of names dating back well before 1066 are still in use today, including yellowhammer, redstart and wheatear, all of whose real meanings are very different from their apparent ones. The persistence of these ancient names (all at least a thousand years old, and probably far older) reflects the extraordinary tenacity of names of any kind – whether of birds, people or places – to persist in the language long after other words from that time have been lost.
Some of our bird names are even older than the Anglo-Saxon era. These include gull (from Cornish), auk (Old Norse), ptarmigan and capercaillie (Scottish Gaelic), rook, crow and raven (West Germanic) and goose. The last is possibly the oldest of all the names we still use today, and may go all the way back to the language spoken on the steppes of eastern Europe and western Asia more than five thousand years ago.
As already noted, though, not all bird names are quite so ancient. From the seventeenth century onwards, as more and more species were discovered, a cohort of professional ornithologists – men such as William Turner and John Ray, Thomas Pennant and William MacGillivray – devised new names and attempted to codify and standardise those already in use. Some new names were created from scratch, while others were based on ones already long in existence, with many folk names ultimately gaining formal status as the 'official name' for the species.iv
Meanwhile, the Ages of Exploration and Empire saw a vast increase in the number of species discovered around the world, many of which were given their new name by intrepid Britons as they explored the far reaches of the globe. From the yellow-bellied sapsucker of North America to the locust finch of Africa, and the many-coloured rush-tyrant of Patagonia to the short-billed leaftosser of the Amazonian rainforest, the world's ten thousand and more different species of bird now sport a mind-boggling variety of common names.
Back home in Britain, by the start of the twentieth century the vast majority of birds had been given the names we use today. Even so, there have been a number of changes during living memory, such as the switch from 'redbreast' to robin, and 'hedge sparrow' to dunnock.
But throughout this period, the wishes of tidy-minded scientists have often been trumped by what Lockwood calls 'ordinary users of the language... [who] do not necessarily feel bound by the prescriptions of the ornithologists, indeed... will generally not even be aware of them.'8 So however much the bird books insist on the official name dunnock, many people still choose to call the little bird foraging unobtrusively around the base of their shrubbery a hedge sparrow.
So what of the future? As we shall see in the final chapter of this book, a radical change in the way scientists classify species is already leading to an explosion in new names, even as the birds themselves are threatened with extinction. Yet despite the pressures of globalisation, and the resulting homogenisation of the English language, most bird names are still proving remarkably resistant to change.
So next time you hear the croaking call of the raven, remember that the name we use for this huge and fearsome corvid is not all that different from what our prehistoric ancestors might have called it, as they stared up into a cold, grey sky and watched these huge black birds passing overhead. For me, that revelation is, in equal measure, both astonishing and comforting.
*
How did I come to write this book? It began with the influence of my late mother, Kay Moss, who in spite of her rather limited formal education passed on to me her deep and abiding love of the English language, and also encouraged me in my lifelong passion for birds. Together, these have made me endlessly curious about the origin of bird names.
I can still recall sitting in my grammar-school playground some time during the mid-1970s with my friend Daniel,v testing each other on the scientific names of British birds. In those days I certainly knew my _Anthus pratensis_ from my _Prunella modularis_ , and my _Crex crex_ from my _Coccothraustes coccothraustes_ , even if I struggle to remember some of them now.vi
In the early 1980s, when I was studying English at Cambridge, I made a special study of the bird poetry of John Clare (see Chapter 4). Later on, as I pursued a career as a writer and TV producer, I began to take a closer interest in the cultural side of our relationship with birds. This culminated in the BBC 4 television series and accompanying book _Birds Britannia_.9 Subtitled 'How the British Fell in Love with Birds', this examined the profound and longstanding connection between the British and our birdlife, expressed through both popular and high culture.
While making that series I interviewed my friend and fellow birder David Lindo (aka 'The Urban Birder'). Like me, David acquired his fascination with birds at a very early age, and in a similar suburban setting (he in Wembley, me in Shepperton), during the late 1960s and early 1970s.
Like most young birders in those days, David knew no one else who shared his interest, and so resorted to making up his own names for the species he saw. Sparrows were 'baby birds', starlings 'mummy birds' and blackbirds 'daddy birds'. We may smile, but that early desire to name and categorise shows that we have an instinct to give names to the living things we see around us, even in early childhood.
When it comes to naming birds there is also – and I may be touching on a controversial subject here – some difference between the sexes. Broadly speaking, most male birders have an urge to put a name to every bird they see or hear, often interrupting ordinary day-to-day conversations to do so (in what the TV presenter and keen birder Mike Dilger calls 'birding Tourette's'). This can result in a perhaps unhealthy obsession with keeping lists: of birds seen in your garden, on your local patch, in your home city or county, in the UK and ultimately around the world.vii
Women, on the other hand, often take a more holistic (and perhaps less stressful) approach – preferring to take a deeper interest in what the bird is doing, and why, rather than always needing to label it. Of course, not all men are obsessive listers and not all women are fascinated by bird behaviour, but there is more than a grain of truth in this distinction.
I hope that _Mrs Moreau's Warbler_ will appeal to both groups equally. Anyone interested in detail can find out how many of our birds got their names; while those who prefer the big-picture view can better understand the sweep of history and how it shaped the names we call our birds today.
And if you still prefer to give your own names to the birds, then may I refer you to the performance-poet A. F. Harrold,10 whose splendid verse 'Among The Ornithologists' mixes wonder, imagination and confusion in equal measure to produce a cornucopia of evocative names. These beguile and inspire us – as all good bird names should:
Like the Fool at Court I can see the truth, speak a true name:
This one I'll call the _Fifth Day of Christmas Bird_ for its eye's gold ring,
Here's the _Nervous Bugger_ who's always a step ahead, twittering,
I'll call this one the _Golden Glimpse_ as I miss it sitting still again,
But here's the _Puffed-Up Lover Bird_ , strutting grey and wooing.
A stately _Snaked-neck Bird_ makes its slow way along the stream.
A _Single Drop of Blood in the Darkest Night Bird_ paddles out of a dream
And under the river bank, and as I wonder what it's doing
I see the _Surprising Single Snowfall In The Night Bird_ , twig in beak,
build an unruly, unshapely, unhandsome home of a nest
and think it's doing fine. And look! _A Blue Sphere With A Yellow Vest_
cocks a momentary eye at me, but then declines to speak.
For all I know it's just named me inside its tiny brain
Or left me unlabelled, unpinned down, free to be anything I claim.viii
Most of all, this book is a tribute to the pioneering and far-sighted men and women who named our birds. Many of these people are anonymous: our distant ancestors, whose curiosity about the natural world led them to try to create order by giving names to the creatures they saw. Others are long dead, but not forgotten: their names live on in the plethora of eponymous bird names, mostly coined during the eighteenth and nineteenth centuries, but some – such as Mrs Moreau's warbler – devised more recently.
It is these heroes and heroines who are the centre of this book; they, and the myriad variety of more than ten thousand different kinds of birds, in every corner of the globe, which bear the names they bestowed on them.
Stephen Moss
Mark, Somerset
May 2017
#### Notes
1 Michel Desfayes, _A Thesaurus of Bird Names_ , Musée cantonal d'histoire naturelle (Paris, 1998).
2 John Wright, _The Naming of the Shrew_ (London, 2014).
3 <http://aishwaryashivapareek.com/post/104398138332/does-a-cat-know-he-is-a-cat-does-a-dog-know-he-is>
4 Joanne Harris, _Runemarks_ (London, 2007).
5 Matthew Woodring Stover, _Caine's Law_ (London, 2012).
6 John Fowles, _The Tree_. This has recently been republished by Little Toller Books (Dorset, 2016), with a perceptive introduction by the author William Fiennes.
7 W. B. Lockwood, _The Oxford Book of British Bird Names_ (London, 1984).
8 ibid.
9 BBC 4 (2009) and (London, 2011).
10 A. F. Harrold, 'Among The Ornithologists', in Of Birds & Bees (Reading, 2008).
i Referring to yellowhammer, common tern, lapwing and Dartford warbler respectively.
ii Carol Kaesuk Yoon, _Naming Nature_ (New York and London, 2009). 'Good caveman taxonomy' applied to plants, too. Knowing which plant was good to eat, and which might be poisonous, would also have been vital. Later, this working knowledge of plant names, and their various therapeutic uses, would develop into the earliest form of medicine.
iii In his Introduction to a short work, _Animal. Vegetable. Mineral._ (London, 2016), the nature writer Tim Dee has expanded on this theme: 'Go to your window in the morning, open your curtains and think how not one blackbird you might see knows that it is a blackbird; not one tree cares that it is an oak, an ash, or a lime. Not one; and yet the blackbird lives as a blackbird not as a blackcap; the ash is an ash and not an alder. We are right to tell the difference because difference tells.'
iv It's important to note that, as Lockwood points out, only when an ornithologist has specifically coined a name can we date its creation precisely. In other cases, even though we may be able to discover the first recorded mention of the name in print (for example, by looking it up in the _Oxford English Dictionary_ ), we have no idea how far back the usage of the name may go.
v Now Professor Daniel Osorio of Sussex University, one of the world experts on the way birds and other organisms perceive colour, and still a dear friend.
vi Meadow pipit, dunnock, corncrake and hawfinch respectively.
vii In case you're wondering, I keep all of those lists, which currently (spring 2017) stand at 84, 97, 214, 374 and 2,627 species respectively.
viii I guess that the birds are, respectively, blackbird, pied wagtail, goldcrest, wood pigeon, mute swan, moorhen, coot and blue tit – but you may prefer your own versions!
# PROLOGUE
_Mrs Moreau's Warbler_
Winifred's warbler ( _Scepomycter winifredae_ ), also known as Mrs Moreau's warbler, is a species of bird in the Cisticolidae family... endemic to montane forest in the Uluguru Mountains in Tanzania. It is threatened by habitat loss.
WIKIPEDIA ENTRY: 'Mrs Moreau's Warbler'
When I think back to the year 1970, lists of names often come to mind. John, Paul, George and Ringo, whose band, the Beatles, broke up in April of that year. Lovell, Haise and Swigert, who in that same month, against all the odds, guided their stricken spacecraft Apollo 13 back to Earth. Jairzinho, Tostão, Rivellino and the incomparable Pelé, Brazil's formidable forward line, who thrashed Italy 4-1 to win the World Cup, thus forever defining football as 'the beautiful game'.
All these people – and their incredible achievements – made a lasting impression on me. But there was one other name that would shape my life even more profoundly: that belonging to the wife of a now long-forgotten ornithologist.
I was ten years old, and had been obsessed with birds for as long as I could remember. To encourage my interest, for four shillings a week (the pre-decimal equivalent of 20p), my mother subscribed to a weekly 'partwork' of magazines, with the beguiling title _Birds of the World._
Every Saturday morning, I would wait eagerly for the paperboy to drop the latest issue through our letterbox, and then spend the rest of the day absorbed in its contents – the full-colour photographs, the text packed with fascinating facts about the world's birds and their extraordinary lifestyles.
Even in nine large-format volumes, _Birds of the World_ could only cover a fraction of the 8,600 or so different kinds of bird known to exist at that time. But in a concession to completeness, its editor John Gooders had decided to include a full list of every single species. So it was that, some time in late 1970, on page 2,110 of Volume VII, part 3, I came across the name of the bird that gave this book its title: Mrs Moreau's warbler.
Something about the strangeness of the name struck me, even then. I already knew – or could guess – that birds could be called after their colour or their size, their habits or their habitat, the sound they made, or the place where they came from. Some, I also realised, were named after people: even at this early stage in my ornithological education I had heard of Leach's petrel, Montagu's harrier and Bewick's swan.
But ' _Mrs_ Moreau's warbler'? How on earth had this species acquired such an unusual name? A clue lay in the words in italics beneath: _Scepomycter winifredae._ Even at this early age, I was able to deduce that the bird had been named after a woman called Winifred Moreau.
Nowadays, of course, I can simply Google the name and click on the brief but informative Wikipedia entry. But no such easy shortcuts to knowledge were available back in the dark ages of my childhood. And my mum was calling me downstairs for tea. So I put down the magazine and, for the moment at least, forgot all about Mrs Moreau's warbler.
Yet as the years went by, and my interest in bird names grew, my thoughts kept returning to this obscure little bird, the woman after whom it was named, and her husband, one of the greatest ornithologists of the twentieth century.
*
Reginald Ernest Moreau – known to his friends and colleagues simply as 'Reg' – was born in 1897. The Moreausi were a typically respectable, middle-class family, living an unremarkable existence in the Surrey town of Kingston-upon-Thames.
Then one day, when Reg was about ten years old, their quiet, comfortable lives were shattered. Returning home from work, his stockbroker father was struck by the open door of a passing train. Although Mr Moreau senior survived the accident, he became a manic-depressive and was never able to work again. As a result of their straitened circumstances, the family moved out of town to a more modest property in rural Surrey. There, during long bicycle trips around the local countryside, Reg developed his lifelong interest in birds.
In 1914, the year the First World War broke out, the seventeen-year-old Reg left school and took an exam to enter the Civil Service. He just managed to scrape through, in ninety-ninth place out of a hundred, and ended up in the Army Audit Office in Aldershot. Then, however, he fell ill with rheumatoid arthritis. The family doctor prescribed a complete change, and Reg applied for a posting abroad, to Egypt's capital Cairo.
He took to colonial life immediately, as his son David recalled many years later:
Once in Egypt, he began to behave like the Indiana Jones character that he had clearly always wanted to be. Adopting a bush hat, khaki shirts and shorts... he began making long journeys by ancient car, rail and on foot into the surrounding desert. He took to flies, protesting camels, leather water bottles and Bedouin as if Kingston-on-Thames [ _sic_ ] had never existed.1
Reg Moreau spent much of the next thirty years or so living and working in Africa. He became an expert in the study of bird migration: the epic, twice-yearly journeys made by hundreds of millions of birds, as they travel between the northern latitudes of Eurasia and the vast continent of Africa.
In his final years, by then living in the quiet Oxfordshire village of Berrick Salome, he brought together his lifetime's work into a book, _The Palearctic-African Bird Migration Systems_. This was published in 1972, but sadly Reg did not live to see it in print, having died, aged seventy-three, on 30 May 1970.
Despite the less-than-snappy title, the book was a masterpiece, distilling decades of hard-won knowledge and experience into clear, precise prose. Even now, almost fifty years after it was published, it is full of insights into the incredible journeys made by migrating birds.
As Reg Moreau lay on his deathbed, in the spring of 1970, he had time to write a short page of acknowledgements, which began with heartfelt thanks to his wife Winifred: 'This book would never have been written but for the devotion of my darling diminutive wife, known to generations of ornithologists as Winnie.'
A touching tribute, certainly. Yet Winnie Moreau contributed far more to their relationship than simple devotion. She was also a leading ornithologist in her own right, and an equal partner with Reg in their field trips and discussions; so much so that perhaps, in a less chauvinistic era, she might have been given a joint credit for the book.
Winnie and Reg first met on a fine spring day in the early 1920s, in a chance encounter that would radically shape the course of their lives. At the time, she was picking wild flowers and he was watching migrant birds. But this meeting did not take place on some windswept English headland, but under clear blue skies near the port city of Alexandria, where Winnie – a vicar's daughter from Cumberland – was working as a nanny.
More than forty years later, in 1966, Reg recalled that first meeting:
Here one March afternoon, where the steppe was still bright with flowers and was twinkling with short-toed larks and wheatears, I came across a small person picking scarlet ranunculuses... She was knowledgeable in birds. Improbably we met twice more, for an hour or two, before she returned to England. We were married in Cumberland in June 1924.
After the wedding, they returned to Egypt. Four years later, they moved to Amani, a hill station in the scenically beautiful and biologically fascinating Usambara Mountains of north-east Tanganyika (now Tanzania), where Reg had taken up a new post in the accounts department of a biological research station.
But while auditing may have been his profession, his main passion – shared by his wife – was ornithology. Fired up by their new and exotic surroundings, Reg and Winnie embarked on a long-term study of the birds around their new home. As well as the long-distance migrants that would form the subject of his book, they also focused on the sedentary 'Eastern Arc endemics': a unique group of very localised species, found nowhere else in the world but here.
In 1938, a year before the outbreak of the Second World War, Reg and Winnie embarked on an expedition to the Uluguru mountain range, several days' journey south of the Usambaras. There, high in the montane forest, they discovered an obscure and endangered songbird which, in a perhaps surprising act of marital devotion, he named _Scepomycter winifredae_ – Mrs Moreau's warbler.
I say surprising, because in the few rather grainy, black-and-white photographs of him that survive, the short, stout, bald and bespectacled Reg bears more than a passing resemblance to Captain Mainwaring from _Dad's Army_. But beneath that stern-looking exterior he was a sociable and fun-loving man. And he clearly had a romantic streak, as the naming of this obscure little bird after his wife proves.
When Reg Moreau died in 1970 his obituaries were uniformly warm and positive. He was remembered as 'a squat, square figure [with]... a rugged face, a heavy square jaw, thick glasses, and just a fringe of curly hair which he brushed upwards'. His rather unusual dress sense was also mentioned: '[He was] adorned frequently in the summer with a transparent green eyeshade, and more often than not, if the weather was warm, with huge knees and strong shoes protruding from a pair of shorts.'
But most of all, Reg Moreau was regarded a key influence on both professional and amateur ornithologists. As my friend and mentor James Ferguson-Lees recalled just before his death, he was always keen to share his vast knowledge and experience, yet also prepared to listen to other people's thoughts and opinions. 'Reg was a remarkable man – a great enthusiast about birds and bird migration – like a God to us youngsters!'ii
Winnie, though, remained tantalisingly vague, the dutiful wife hovering in the background. Although six years older than Reg, she survived for another eleven years, dying in 1981, in her ninetieth year.
Now, almost forty years later, she is finally being recognised as an equal partner in Reg's life and work, not simply his willing and devoted assistant.2 The American academic Nancy J. Jacobs has discovered that in his writings on new birds discovered in the Usambaras, Reg always used the first person plural, to highlight that these had been jointly found and named by him and Winnie.iii As Reg himself wrote: 'The frequent use of the pronoun "we"... is a natural result of our close collaboration.'3
Amidst their busy lives, Reg and Winnie also found time to raise two children: a daughter, Prinia – named after a family of African songbirds – and a son, David, who later made a career for himself as an author of rather racy novels, mostly set in expatriate circles in Tanzania.iv
David Moreau – who narrowly escaped being christened 'Buphagus' after the scientific name for the oxpeckers – depicted his parents as a loving but rather unpredictable couple. He claimed that Reg once warned Prinia to 'cover your ears. There's going to be a loud bang', just before he shot and wounded a leopard hiding beneath her bed.
Reg delighted in reciting saucy limericks to his dinner guests, while Winnie frequently cared for abandoned baby birds, tucking them into a sock, which she then placed inside her bra. Indeed, she once did so while entertaining the visiting provincial governor. Such recollections suggest that Reg and Winnie Moreau's long and happy marriage and family life were enlivened by a great sense of fun.v
The only photograph I can find of Reg and Winnie together comes from late in his life, long after they had returned to England. They stand side-by-side in front of a brick fireplace: he wearing a jacket, tie and jumper, she looking rather smarter, in a neat two-piece outfit. Both are smiling, as well they might, given their many achievements: not least the discovery of the warbler that bears Winifred's name.
*
In January 2017, almost half a century after I first read about Mrs Moreau's warbler, I finally travelled to the Uluguru Mountains in eastern Tanzania, on a quest to see this bird for myself. For the story of that journey – and whether or not I succeeded – you will have to wait until the end of this book...
#### Notes
1 David Moreau, _More Wrestling than Dancing_ (London, 1990).
2 Nancy J. Jacobs, 'The Intimate Politics of Ornithology in Colonial Africa', _Journal of the Society for Comparative Study of Society and History_ , 2006.
3 W. L. Sclater and Reginald Moreau, _Taxonomic and Field Notes on Some Birds of North-Eastern Tanganyikan Territory_ , Ibis 2: 487–522 (1932).
i The rather exotic family name came from a French ancestor who had moved to London to sell books.
ii James Ferguson-Lees was one of the most influential birdwatchers and ornithologists of the second half of the twentieth century. He was a successful author, editor, conservationist and dedicated field birder, who influenced his own and subsequent generations. It was a privilege and a pleasure to get to know him in his later years, until his death, just after his eighty-eighth birthday, in January 2017.
iii The only other female ornithologist to rival Winifred Moreau is Maria Koepcke. Born Maria von Mikulicz-Radecki in Leipzig, Germany, in 1924, she and her husband Hans pioneered ornithology in Peru, before her untimely death in an air crash on Christmas Eve, 1971. She has two species of bird named after her: Koepcke's hermit (a type of hummingbird) and Koepcke's screech-owl.
iv I later discovered that Reg himself had also written a collection of short stories under the barely concealed pseudonym 'E. R. Morrough' – because, working for the Civil Service, he was not permitted to publish under his own name.
v _More Wrestling than Dancing_ , the memoir by Reg's son David, contains many more wonderful anecdotes and descriptions of family life with the Moreaus.
# SOUND AND ECHOES
_The Origins of Bird Names_
Names turned over by time, like the plough turning the soil. Bringing up the new while the old were buried in the mud.
Joe Abercrombie, _The Heroes_
## _1: The Cuckoo's Calling_
The sound, as it percolates into my consciousness with the full force of an early-morning espresso, is quite unmistakable. Two notes float across the fresh spring landscape, hanging momentarily in the warm, still air, before fading away. Way out of sight, in the far distance, a second bird echoes with another round of notes, followed by a third, this time almost beyond the horizon.
'Cuck-ooo, Cuck-ooo, Cuck-ooo...'
The spring call of the male cuckoo.i The very name encapsulates its sound, and is so familiar that, even if you have never caught a glimpse of the bird itself, you are instantly aware of its identity. Despite the cuckoo's recent decline, it remains the classic harbinger of spring; even today, a letter to _The Times_ newspaper traditionally marks the first sighting of the bird each year.
In the West Yorkshire village of Marsden, local people still celebrate the cuckoo's annual return towards the end of April with the 'Cuckoo Day Festival'. There is a craft fair, a village procession and that staple ritual of English village life: a maypole around which Morris dancers, complete with white handkerchiefs, perform their terpsichorean displays.
Along with other 'cuckoo fairs' that used to take place up and down the country, the Marsden festival was once a key event in the rural calendar. It marked the shift from winter into spring, with all the hope the new season brings. Traditionally, villagers also took part in the ritual of 'penning the cuckoo': building a wall in order to capture the returning bird, and so supposedly prolong the summer. In rural Shropshire, as soon as the first cuckoo was heard each year, farm labourers would down tools and drink beer for the rest of the day.
So why, of all our spring migrants, was the cuckoo's return so widely marked and celebrated? After all, it is not a showy bird: even where cuckoos are common, in the far north of Scotland, they are still more often heard than seen. The reason for the cuckoo's fame is, of course, its distinctive and inimitable sound. As the Victorian clergyman-naturalist, the Revd C. A. Johns, pointed out, the cuckoo's call is closer to the human voice than that of any other bird. This, surely, explains why it has been so important to rural communities, for whom it was the unmistakable signal that winter was finally over, and spring was here to stay.
The cuckoo's sound appears in the very first entry of the _Oxford Book of English Verse_. It is the subject of a poem created by an anonymous scribe some time during the mid-thirteenth century, and widely regarded as the earliest verse written in something clearly recognisable as English:
Sumer is icumen in,
Lhude sing cuccu!ii
Surprisingly, perhaps, this is the very first recorded use of the word 'cuckoo' in written English. That's because its origins lie across the Channel: it came into our language from the Old French word _cucu_ , which derives from the Latin _cuculus_ , still used in the cuckoo's scientific name. Both of these are, of course, also onomatopoeic.
Before this time, people would have used a very different name: 'yek', which came from the Old English 'geac'. This is similar to the names for the cuckoo in today's Scandinavian languages (such as the Swedish _gök_ ), indicating its ancient Germanic lineage.
The old name remained remarkably resistant to the more obvious charms of the new one. Cuckoo did not gain the upper hand until quite late on, as can be seen in the writings of Randle Holme, who in 1688 stated: 'The Cuckow is in some parts of England called a Gouke.' Incredibly, in some parts of northern England and Scotland the word has survived right up to the present day: the wildlife sound recordist Geoff Sample remembers growing up in Northumberland during the 1960s and hearing people being called 'a daft old gowk'.iii
*
The cuckoo – or rather the geac – first appears in written Old English in the earliest dictionary of our language, the _Corpus Glossary_ , which dates back to AD 725. It can also be found in a contemporary poetic tribute to the monk Guthlac of Crowland (later canonised as Saint Guthlac), who lived from 673 to 714.
For much of his life, Guthlac lived as a hermit on a small island in the Lincolnshire Fens. When he first arrived in this watery wonderland at the start of spring, it's hardly surprising that one of the first birds he encountered was the cuckoo:
Bright was the glorious plain and his new home;
sweet the birds' song; earth blossomed forth;
Cuckoos heralded the year.1
This early reference to the species – which in the original is referred to by its Anglo-Saxon name 'geac' – is unusual: according to the great ornithologist and broadcaster James Fisher the cuckoo is one of just sixteen species of bird recorded in Anglo-Saxon literature.iv
Yet it's only by pure chance that these particular names lived on to the present day, while others did not. As Fisher points out, the entire surviving corpus of Old English writings totals less than a quarter of a million words. So doubtless many other birds were named in written works that sadly perished from fires, flood or simple neglect.
But we do have one vitally important manuscript from this period. Dating from the final decades of the first millennium – somewhere between AD 960 and 990 – the _Exeter Book_ is the largest collection of extant Old English writings, and one of the oldest surviving books of poetry in the world.v
*
On a fine spring afternoon, I was briefly tempted to join the sun-seekers lounging on the grass on Exeter's Cathedral Green. But instead I headed indoors, to the red sandstone library and archive, tucked out of sight around the corner of the cathedral. As I entered, a charm of goldfinches flew overhead, delivering their light, tinkling songs – a good omen, I hoped.
I had come, along with a handful of other curious visitors, on the one day each month when the _Exeter Book_ is on display to the public. We were shown round by Stuart, one of those people whose deep historical knowledge is matched by an engaging ability to deliver fascinating facts.
As Stuart pointed out, this stout volume has had its ups and downs in the millennium or more since it first arrived here. It was, at some stage, used as a chopping board for cutting manuscripts (and still shows the stains from glue pots on some of its pages), and probably lay on a dusty bookshelf for most of its long lifetime. Indeed, the _Exeter Book_ was only truly appreciated when, some time during the seventeenth or eighteenth centuries, these ancient manuscripts began to be valued once again.
The reason the book was overlooked was simple: hardly anyone could read or understand its contents. That was because less than a couple of centuries after it had been produced, the English language had changed out of all recognition.
Soon after the Norman Conquest, Anglo-Saxon began to be neglected as a written language. Even a few decades after they were transcribed, therefore, the poems contained in the _Exeter Book_ would have been incomprehensible to any but the most determined scholar. So in many ways it is incredible that it has survived at all.
Stuart beckoned us forward, so we could examine the volume more closely. To my surprise, the first impression was not of poetry, but of densely written, evenly spaced prose. As he explained, that is because sheepskin parchment was so expensive that the scribe could not afford to waste space by writing in short lines, so he filled each page all the way up to the margins. The yellowish sheets are etched with words written in dark-brown ink, made from a mixture of oak galls, gum to make it sticky, and either vinegar or urine as a preservative. This unpromising recipe worked: after more than a millennium the book still looks clean and fresh, and the script has hardly faded at all.
The _Exeter Book_ contains roughly forty poems – and almost a hundred verse riddles – composed many centuries earlier, and handed down through the generations by word-of-mouth. Amongst the riddles is a verse devoted to a very familiar bird:
In former days my mother and father
forsook me for dead, for the fullness of life
was not yet within me. But a kinswomen
graciously fitted me out in soft garments,
as kind to me as to her own children,
tended and took me under her wing;
until under shelter, unlike her kin,
I matured as a mighty bird (as was my fate).
My guardian then fed me until I could fly
and wander more widely on my
excursions; she had the less of her own
sons and daughters by what she did thus.2
This is, of course, the cuckoo. Whoever wrote this riddle was clearly aware of this bird's unusual habit of laying its eggs in the nests of other species, and fooling them into raising its young, at the expense of their own offspring.
Fascinating though this and the other riddles are, they were not what I had come to see. I wanted to read (or, given my lack of fluency in Anglo-Saxon, gaze at) a much longer work: the 124-line autobiographical verse known as _The Seafarer_.
Written by an anonymous mariner, some time towards the end of the seventh century, this haunting and evocative poem wonderfully captures the hardship of life on the high seas. More importantly, for anyone searching for the origins of English bird names, _The Seafarer_ is an ornithological goldmine:
There I heard nothing but the roar of the sea,
of the ice-cold wave, and sometimes the song of the wild swan;
I had for my amusement the cry of the gannet
and the sound of the whale instead of the laughter of men,
the sea-mew singing instead of the drinking of mead.
Storms beat on the rocky cliffs, where the tern, ice on its wings, gave answer;
Very often the dewy-winged eagle screamed...3
In an earlier translation, James Fisher chose different identities for some of the wild creatures in the poem, suggesting that the 'whale' could have been a flock of whimbrels (a smaller cousin of the curlew), and that the 'sea-mew' (a kind of gull) was the kittiwake. His translation runs as follows:
There heard I naught but seething sea,
Ice-cold wave, awhile a song of swan.
There came to charm me gannet's pother
And whimbrels' trills for the laughter of men,
Kittiwake singing instead of mead.
Storms there the stacks thrashed, there answered them the tern
With icy feathers; full oft the erne wailed round
Spray-feathered...4
Fisher speculated that _The Seafarer_ would have been written around the year AD 685, at Bass Rock, a vast and noisy seabird colony just off the east coast of Scotland. He suggested that the (whooper) swans would have been heading north, back to their breeding grounds in Iceland; while the whimbrels would have just arrived back from Africa, en route to Shetland or Scandinavia. As Fisher pointed out, this could only have occurred during a brief window at the height of spring migration – in his view, the week from 20 to 27 April.
The language in which _The Seafarer_ was written is not easy for the modern reader to comprehend, but even in the original West Saxon (a dialect of Old English) we can recognise some species, including 'ganot' (the gannet, our largest seabird), and 'stearn' (the tern, one of our smallest).
Both 'earn' (erne, or white-tailed eagle) and 'mæw' (mew, a kind of gull) are of very ancient origin, almost certainly predating Old English. They were ultimately supplanted by 'eagle', from Norman French, and 'gull' – which, perhaps uniquely amongst modern English bird names, comes from one of the south Celtic languages, probably Cornish.vi Yet they have endured as folk names right up to the present day.vii
Variations on the word 'mew' – including 'maw', 'maa' and 'ma' – are still heard to describe common or herring gulls in the Lowland Scots dialect. The word also survives in the North American name for the common gull ('mew gull'), and in a more ancient form in the name fulmar, from the Old Norse, which means 'foul gull', because of the bird's habit of spitting smelly, sticky oil on any intruders that come too near its nest.
*
The continued existence of ancient names such as gowk, mew and erne, along with many other names from the same period, is not merely a quaint historical footnote in our story. Instead, it goes to the very heart of the way we use language.
We live in an age of globalisation; as a result, our language is being pulled in two different and conflicting directions. One trend sees English becoming simpler, as different dialects merge and disappear under the onslaught of the mass media and the Internet. Yet at the same time, it is becoming more rich and varied, through its longstanding habit of borrowing words from other tongues. In the linguist David Crystal's memorable phrase, English is still 'a vacuum-cleaner of a language, sucking in words from any other language that its speakers come into contact with...'5
Yet one key area of language – the names we use for birds – goes against both these trends, by staying more or less the same. Some, indeed perhaps the majority, of the names we use every day have remained virtually unchanged over centuries, and in some cases for millennia. This is all the more surprising, given the extraordinary shifts that have occurred in the English language during the past 1,500 years.
*
If we try to read poems such as _The Seafarer_ and _Beowulf_ in their original Old English, they appear utterly impenetrable. Even the Middle English used by Chaucer and the _Gawain_ poet can at first be hard to understand, though on a closer look (or better still, when read out loud) it does become more or less comprehensible.
For most of us, the first easily recognisable works, written in what we now call Early Modern English, appeared during the late sixteenth and early seventeenth centuries. The poetry of Edmund Spenser and John Donne, the poems and plays of William Shakespeare, and the majestic King James Bible, are often regarded as the zenith of our literary achievement, and are also the earliest still readable examples of the global language now spoken by millions of people around the world.
Given these dramatic changes, it is little short of astonishing that so many bird names with Anglo-Saxon origins have lasted to the present day – albeit often in a rather different form from the original. This is one of the most intriguing aspects of the story of our bird names, and one to which I shall return many times, for it tells us much about the crucial importance of the natural world in our society, history and culture.
But before I do, we need to go even further back in time. For although many of the names we use for birds today have changed, or been lost and forgotten along the way, a handful go back well before the beginnings of English: to the very dawn of human civilisation, roughly 3,000 years before the birth of Christ.
Their origins lie very far from here: with a small group of early farmers living thousands of miles to the east of Britain, on the vast open grasslands of central Eurasia – the place we now know as the Russian steppes.
## _2: Trade Routes and Translations_
Try to imagine, if you can, the day-to-day existence of those first farmers on the steppes of central Eurasia, so distant from us in space and time. In the words of the seventeenth-century philosopher Thomas Hobbes, we can surely guess that their lives would have been 'solitary, poor, nasty, brutish, and short'.
We can picture them spending long, hard days cultivating the steppe grasslands, planting and harvesting their meagre crops, and caring for their precious livestock. They would also have needed to cope with the vagaries of weather and climate, which could so easily mean the difference between success and failure and, ultimately, survival and death.
For these early farmers, life had changed little for several millennia, ever since their own ancestors had first renounced the nomadic, hunter-gatherer lifestyle in favour of agriculture, which required a permanent, settled home. Life in one place may have been easier, in some ways, yet it would still have been very tough and unrelenting.
But then, roughly 5,000 years ago, the world began to change. Two developments – one cultural, the other technological – dramatically improved the lives of these ancient people.
The first was the domestication of the horse, arguably the most important wild creature ever to be subjugated for human use. The second, which followed soon afterwards, was the invention of the spoked wheel. For the first time in human history, this simple breakthrough allowed people to build fast, light and manoeuvrable vehicles. These could in turn be pulled over longer and longer distances by the newly tamed horse. The eventual dominance of vehicular transport over our lives had begun.
The newly developed wagons and carts, pulled by horses, made life much easier, allowing heavy items such as firewood and crops to be carried on short journeys from woods and fields to villages and homes. But more importantly for our story, they also opened up the possibility of moving goods and people over far longer distances.
Thanks to these long-forgotten people, the greatest change in human history was set in motion: the beginning of trade between different groups, communities and, ultimately, nations. At first, they would have simply bartered their produce with their immediate neighbours, perhaps exchanging a bushel of wheat for a couple of chickens. But over time, the bleak, hostile and treeless steppe where they lived turned into a thriving trade corridor, which would eventually stretch for thousands of miles, to and from Europe in the west and Asia in the east.
Opening up this transcontinental route had another, even more profound, effect on later civilisations. As these ancient steppe-dwellers gradually migrated westwards and eastwards, their language – originally spoken by only a handful of people in this remote and landlocked location – began to spread across a vast swathe of Europe and Asia. In the process, it changed and developed into a huge range of new tongues, including Latin, Welsh, French, German, Hindi, Swedish, Spanish, Greek and English.
At first sight, these languages do not appear to have all that much in common. They do of course share some common terms, borrowed from one another relatively recently: English in particular has proved adept at appropriating words as varied as chutney and bungalow (from Hindi), schadenfreude and kitsch (from German) and coracle and corgi (from Welsh). But we are far more aware of the differences in vocabulary, word and sentence structure between one language and another, than any similarities.
Yet as linguists first discovered back in the eighteenth century, many of these differences are in fact superficial, and even apparently dissimilar languages may be related. And just as the similarities in facial appearance between two people are often because they share a common ancestor, languages too have a 'family tree'.
So, while it may come as a surprise to anyone who has struggled with a phrasebook while attempting to make themselves understood abroad, all these languages, and many more, are ultimately descended from a single tongue. Known by linguists as 'Proto-Indo-European' or PIE, this was first spoken on those windswept central Eurasian grasslands, roughly three thousand years before the birth of Christ. Extraordinary though it may seem, the languages that descend from it are still spoken by roughly half the world's population – almost four billion people. And as David Anthony points out in his book, _The Horse, the Wheel and Language_ ,6 this means that the languages we speak today are almost entirely the result of those two developments that give his book its intriguing title.
We have no written records of the actual words those people used to speak to one another as they went about their day-to-day lives. Yet by comparing words still used in one modern language with their equivalents in another, linguists have been able to painstakingly reconstruct some of their lost vocabulary.
Amongst those words, there are a tiny number that, amazingly, have lasted – albeit in different forms in various modern languages – all the way down to the present day. These include the name of a species of bird that would have been very familiar indeed to our distant ancestors: the goose – or, as linguists now believe it would have been originally called, _ghans_.
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Long before the domestication of the horse, the invention of the wheel, or even the earliest agriculture, prehistoric peoples right across Europe and Asia would have been aware of the twice-yearly migration of geese.
Looking up each autumn, they would have seen straggling, V-shaped skeins of birds arriving from the north, silhouettes etched against the grey skies as the land echoed with their distinctive, honking calls. They also would have noted the date when the flocks headed back north towards their breeding grounds in spring.viii
During the winter months, when vast flocks of geese fed on grasslands and wetlands, they would no doubt have used whatever primitive weapons they had – rocks, stones and perhaps flint spears – to try to kill the plump, tasty birds, so they could supplement their meagre diet.
It would only have been a matter of time before it occurred to more intelligent individuals that, rather than spending time and effort trying to hunt and kill geese, there might be an easier way to ensure a regular, reliable and year-round supply of eggs, flesh and feathers. So it was that, almost 5,000 years ago, the greylag goose became only the third (or possibly fourth) species of bird – after the chicken, duck and perhaps the pigeon – to be domesticated.
The central importance of geese to our ancestors' lives meant that these birds would have been given a vernacular name far earlier than more obscure, less useful species. That is no surprise. But what is truly extraordinary is that this name has lasted – in different forms in different languages – all the way down to the present – especially given the ways languages have evolved, and vocabulary has changed, over thousands of years.
Take a look at the modern name for goose in both German and Dutch: _gans_. At first sight this does not appear very similar to the word we use in English; but think of the name we give to a male goose, 'gander', and the connection becomes clearer. Likewise, the Spanish name, _ánsar_ , may not appear to have much in common with 'goose'. But it is remarkably similar to the scientific name of the greylag goose, _Anser anser_ and, via _gans_ , to goose. So even if bird names in different languages may not appear to be related, a closer look reveals that they often are.
The point of this exercise in linguistic archaeology is this: because these European languages began to diverge from one another roughly 5,000 years ago, we can show that the precursor of these related words for goose in use today must have already been in existence at that time. And that means it must go all the way back to the Proto-Indo-European spoken by those early traders, on the Central Asian steppes.
Thus, of all our bird names, 'goose' can justifiably claim to be the oldest.
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Other names we still use today go almost as far back in time; again, we can demonstrate this by looking at another crucial period: the Early Iron Age. Lasting from roughly 1000 to 500 BC, this period saw the first widespread use of iron and steel, smelted from iron ore, to make tools and weapons.
This major technological breakthrough coincided with – and also triggered – a series of important social and cultural changes. These included more advanced agriculture, the first major religious written texts (including the early books of the Old Testament) and, most importantly for our story, the development of the earliest written languages, through the invention of abstract alphabetic characters.
The first alphabets arose in the Middle East, later spreading westwards into Europe, where the Greeks developed the form that would become the ancestor of all European alphabets. In north-west Europe, another ancestral tongue had not yet been written down, but was spoken across a wide geographical area. Proto-Germanic, as it was later called, eventually split into two forms. One branch, to the north, evolved into the various Scandinavian languages such as Danish, Swedish and Norwegian, while the other developed into modern German and Dutch and – following successive invasions into Britain from continental Europe – English.
Although English has since diverged markedly from these continental tongues, we can still identify many words that share a common origin, and therefore must date back to this distant time. Prominent amongst these are some of our best-known bird names, including swallow and swan.
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The swallow and the swan are two birds that, like the goose, would have been very familiar to our ancestors right across northern Europe.
Like the cuckoo, the swallow is one of the classic signs of the coming of spring. A long-distance migrant, it spends about half the year raising a family in our rural barns and outbuildings, before returning south to Africa each autumn to spend the winter there, hunting for insects amongst the vast gatherings of game animals on the grassy savannah.
'Swan' could refer to one of three closely related species: the resident mute swan, with its black-and-orange bill, or the black-and-yellow-billed Bewick's and whooper swans. These are both winter visitors to Britain and north-west Europe, and like the geese they fly south and west in autumn and head back north and east in spring.
The English word swan is linguistically almost identical to the German _schwan_ and the Dutch _zwaan_ , the differences simply being the result of the standard shifts in pronunciation and spelling between the three languages. Likewise, swallow is _Schwalbe_ in German and _zwaluw_ in Dutch. That these birds have virtually the same name in all three modern European tongues is clear evidence that they share a common origin in the language known as West Germanic, which was spoken around the time of Christ's birth.
But that's not the whole story. For the names of both species can also be found in Old Norse, as _svanr_ and _svala_.ix Because, like West Germanic, Old Norse is also derived from Proto-Germanic, we know these names must go back even further, to at least 500 BC.x
Simply knowing that these names have a common origin in the ancestral language of northern Europe still leaves one crucial thing unexplained: how did they end up being used here in Britain? As with so many aspects of our culture, they did so via a series of dramatic events: a series of invasions that brought people – and their languages – from mainland Europe to our island home.
## _3: Invasions and Conquests_
The first great historical invasion of our isles is, as every schoolchild knows, the conquest of the Ancient Britons – led by Queen Boadicea (also known as Boudicca) – by the Roman Empire. Yet despite ruling much of Britain for close to half a millennium, following Julius Caesar's arrival in 55 BC, the Romans never quite managed to fully subsume this outlying land and its recalcitrant people into their mighty empire. This was never more apparent than in the stubborn resistance amongst ordinary folk to speaking the language of their conquerors.
Although Latin was widely spoken amongst the Romans, and continued to be used as the language of scholarship long after they left, the Ancient Britons managed to keep hold of their own languages for the whole of the Roman occupation. This was very different from the situation in Gaul (modern-day France), where Latin rapidly replaced the indigenous language, driving it to outlying lands such as Brittany. This explains why the modern French language is so closely related to Latin.
Ironically, it was only when the Romans finally departed – more than four centuries after their initial invasion – that the various native tongues finally began to decline. The cause was the arrival of a new group of invaders, this time from the near continent.
They were a motley bunch: variously known as the Angles, Saxons and Jutes, and hailing from Denmark, southern Sweden, the Low Countries and north-west Germany. They succeeded by taking advantage of the social chaos left by the decline of the Roman Empire, and the continued warring between the various groups of Britons left behind.
Having crossed the North Sea to land on the east coast, they eventually extended their influence throughout much of the area we now call England. Here, the existing Romano-British population intermingled and interbred with the newcomers. In the outlying parts of the British Isles – present-day Ireland, Wales and Scotland, the Isle of Man and Cornwall – which the invaders did not manage to reach, those peoples, often erroneously lumped together as Celts,xi retained their separate identity. They also continued to speak their own languages, the precursors of modern Irish and Scottish Gaelic, and Welsh.
But the conquest of these isles by those invaders from the east was not as brutal, or as sudden, as we might imagine. It took place over several hundred years, from the middle of the fifth century to the end of the seventh. So most historians, rather than seeing this as a single, momentous event, now regard it as a more gradual, measured process: not so much an invasion as a migration.
Of all the many lasting influences these newcomers had on their new home, by far the most important and enduring was their language. Known as Anglo-Saxon or Old English, this ancient tongue marked the birth of what is now spoken as a first or second language by more than two billion people, all over the world.
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This eventful period in our history also saw the first appearance of a significant number of English bird names, many of which – including rook and raven, sparrow and wheatear, gannet and crow – we still use today. And even though some species have since been given a more modern name, other old names still managed to cling on until relatively recently. These include 'erne', meaning sea eagle, and 'ruddock', for robin.
It is important to remember that all these names would have been part of an almost exclusively _spoken_ language, rather than a written one. Centuries before the invention of the printing press, written works were rare indeed, and the vast majority of the population was functionally illiterate. As a result, the oral tradition thrived, with stories and poems – such as _Beowulf_ – passed down the ages from one generation to another with remarkable fidelity. So it is not surprising that the names given to birds also arose in a purely oral setting, being coined by ordinary people to describe the creatures they saw every day as they toiled in the fields and forests.
Many of these early names are onomatopoeic: they imitate or echo the sounds made by the birds themselves. There are two good reasons for this: one cultural and one practical.
From a cultural point of view, there is growing evidence that we possess a 'music instinct': the ability to make sense of what we hear in the world around us, and the urge to imitate it ourselves.xii What could be more natural than a human being, having heard a bird sing, trying to mimic it? Surely one reason why so many ancient bird names are based on sound could be that our distant ancestors learned to sing by listening to birds. If so, that would make song the earliest art form – well before the emergence of cave paintings.
Another reason is more pragmatic. In an age long before the invention of optical aids such as binoculars and telescopes, which allow us to see feather-by-feather detail, visual features were far less important in identifying birds. By far the easiest way to tell one species apart from another, similar-looking one would have been by listening to the sound it made. If our ancestors then wanted to remember what a bird was called – perhaps because it was particularly good to eat or, like the cuckoo, marked the changing of the seasons – then the logical next step would be to turn this sound into the bird's name.
But this process wasn't as straightforward as simply repeating the sound; first this had to be transliterated into human speech. And as we shall now discover, this is not quite as simple as it might appear.
## _4: The Nature of Birdsong_
At this stage in our story, we need to make a brief digression. Let's start with the reason birds make sounds in the first place.
The primary way birds communicate with one another can be divided into songs and calls. The purpose of song is to defend a territory and attract a mate, while the various calls perform specific functions such as warning against predators, begging for food, or simply keeping in touch with other birds in the same flock.
Sound is not the only way birds communicate, of course. Many species use their brightly coloured plumage and visual displays to do so. These include the extraordinary courtship dances of the multi-coloured birds of paradise, the strutting parade of the male peacock and, closer to home, the display of the black grouse – these are just three of the best known examples among many in the bird world.
But communicating by sound has three major advantages over vision. First, it is more consistent, working in poor light or even total darkness, or when the bird is hidden in a woodland, hedgerow or dense reed bed. Sound also carries further than vision: the bittern's low, booming call can be heard several kilometres away. And sound has another major advantage: when a bird is calling or singing it does not always need to show itself, meaning that it can hide from predators, whereas during a visual performance it makes itself vulnerable to attack.
During the breeding season, male birds – and in the northern hemisphere these are usually the only ones that sing – need to defend a territory against their rivals. At the same time, they must attract and keep a female, otherwise all their efforts will have been in vain. That is why on a fine spring day, from long before dawn until after dusk, a songbird will sing his heart out, at a time when he could be doing all kinds of other essential tasks, such as building a nest or finding food.
Few other kinds of behaviour in nature are quite so persistent; and none perform two such critically important functions. The performance-poet A. F. Harrold summed up this dual purpose with admirable clarity and brevity in his verse, 'Dawn Chorus':
From hedgerow, telephone wire,
aerial and tree
sings out a double-edged request
_fuck off or fuck me_.7
But it's not just _why_ birds sing that is important; we also need to understand _how_ they do so. The way they form sounds is fundamentally different to the way we do, because of their very different anatomy.
Human beings make sounds by using our lungs to pump air through our larynx and vocal cords, which fine-tune pitch and tone. We then use our lips and tongue to articulate these sounds to make specific words and phrases.
When a bird sings or calls, it uses an organ called a syrinx.xiii This is the avian equivalent of our larynx, but with one crucial difference. The human larynx is situated at the top of the trachea (or windpipe), but a bird's syrinx is much lower down, at the junction of the two bronchi, the passages that carry air in and out of the lungs. This means that the bird can mix two sources of sound, simultaneously producing two different songs at the same time – in what the ornithologist C. H. Greenewalt dubbed the 'two-voice' phenomenon.xiv
That is perhaps why we feel so inadequate when we hear a master songster like the nightingale or song thrush. We admire birds partly because we find them so difficult to imitate – with the possible exception of a handful of species that make far simpler sounds, such as the cuckoo. And when we try to represent their sounds in our own language, for example to form the names of birds, we struggle to do so, with different people hearing each sound – and then trying to vocalise it – in their own individual manner.
There is also variation in the way people speak any language over time, as we have seen, and so the way we use bird sounds to form names has also varied considerably. Today, when we hear a wood pigeon make its monotonous yet strangely soothing sound, we represent it with the word 'coo'. But according to the linguist W. B. Lockwood our ancestors heard exactly the same sound quite differently, representing it as 'doove', from which we get the modern name 'dove'.xv Although it may not be immediately obvious, this is just one example of how the call of a bird can end up as its name, through the power of onomatopoeia.xvi
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There are many others. Take the crow family. Globally there are about 120 different species of crow, only eight of which live in Britain. Four of these are mainly black – the carrion crow, jackdaw, rook and raven – while the other four are more striking and varied in appearance: the chough, with its bright red bill and feet, the grey-and-black hooded crow, the black-and-white magpie and the multi-coloured jay.
At first sight – or perhaps I should say first hearing – the only onomatopoeic name appears to be jackdaw, whose name mimics the 'chack, chack' sound the birds make as flocks fly overhead to roost at dusk on a cold winter's day, looking like scraps of black bin-bags caught by the wind.xvii Yet the other three mainly black species, the raven, rook and carrion crow, are also named after their distinctive sounds.
Each name reflects a version of their harsh cries: just try saying them out loud in the tone of the bird and that becomes far clearer. Given the familiarity of these species, which thrived alongside the early settlers as they ploughed the earth to grow crops, and their superficially similar, mainly black plumage, it is not surprising that they were called after their sound rather than their appearance.
The names raven, rook and crow can all be found in Old English,xviii which in turn, as we have seen, derived from earlier Germanic languages, the ancestors of modern-day German, Dutch and Scandinavian tongues as well as English. So we might reasonably expect the names we use today for these members of the crow family to be found in other northern European tongues – and we'd be absolutely right. A quick glance at the Scandinavian and Dutch languages soon confirms the links between these birds' names, and their common origin in the sounds made by each species. Rook is _råka_ in Swedish, _råge_ in Danish and _roek_ in Dutch, while the crow is _kråka_ , _krage_ and _kraai_ , and the raven is _korp_ , _ravn_ and _raaf_. And we know that because they are so similar in all these languages, they must be very ancient indeed – going back for thousands of years.
Imagine those early hunters, clad in animal skins and carrying primitive spears, glancing up as a raven passed overhead. They would have heard that deep, penetrating cry: a sound so resonant you can feel it passing into the core of your body. Is it too fanciful to assume that one man, inspired by this extraordinary sound, was tempted to imitate the calling bird, and was then copied in turn by his companions? From there it is but a short step to the bird's call becoming its name, and then persisting – with minor changes – to this very day..
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But what of the chough, another member of the crow family? Unlike the other 'black' crows, choughs are easy to distinguish, with their glossy blue-black plumage, comically red legs and a long, crimson, de-curved bill, which they poke into the short turf on clifftops to find their invertebrate food.
Take a walk along a Welsh coastal headland, and you may hear the chough's cries being swept away by the fierce wind: a strong, resonant 'chow, chow' sound. How 'chow' became 'chough' is due to one of the English language's most troublesome suffixes. In the English language the suffix 'ough' can be pronounced in at least ten, and arguably twelve, different ways: as in the words cough, rough, plough, through, though, thought, thorough, hough (an alternative spelling of 'hock'), slough (pronounced 'slew' in American English, and meaning a marshy lake), lough (a word used in Ireland, also for a lake, or loch),xix hiccough and Middlesbrough.
Given this profusion of different ways of pronouncing those four letters, which so confuses the poor learner of English (whether a native child or foreign adult), it is reasonable to surmise that the name of the chough was originally pronounced 'chow' (to rhyme with plough). Some time later, it must have changed to 'chuff' (to rhyme with rough), the pronunciation we still use today.xx
## _5: The Sound Approach_
Neither the cuckoo nor those various kinds of crow could be said to have a tuneful voice. Indeed, paradoxically, it is the very simplicity of their sounds that explains why they were originally adopted as the bird's name. Birds with complex, varied songs, such as the blackbird, robin and nightingale, are rarely given onomatopoeic names; those that have simple, repetitive and above all memorable songs, like the cuckoo and chiffchaff, are.
But for many other groups and species of bird, the link between the sound and the name is not so clear. Who would have thought, for instance, that the names rail, crake, kite, smew, bittern and knot all have an onomatopoeic origin? In each case the link between name and sound has become corrupted and changed over time, so that the original connection is not always evident.
With other names, that link with the bird's sound is still there, but may take a little delving to uncover it. Nightjar is, like many English bird names, an amalgamation of two words: the first being obvious, as these curious birds are indeed nocturnal, the second less so. 'Jar' is in fact a corruption of the word 'churr', representing the bird's weird rattling call, which echoes across moors and heaths at dusk on spring and summer evenings, and which to the untrained ear sounds more mechanical than avian in origin.
Before so many of our heaths and commons were destroyed by the onset of modern agriculture, the nightjar would have been a far more familiar bird than it is today. Hence it has a plethora of now obsolete folk names, many of which confuse the bird (deliberately, perhaps, because of its nocturnal habits) with another creature of the night, such as 'churn owl', 'goat owl' and, my favourite, 'fern owl'.xxi
Another name for the nightjar, which was still included in the very first bird book I ever owned, _The Observer's Book of Birds_ , is 'goatsucker'. This curious name derives from the notion that nightjars were supposed to feed under the cover of darkness on the milk of goats. Like so many other old wives' tales, there is not a shred of evidence for this; however, given that these mysterious birds may have been attracted to paddocks containing domestic livestock because of the concentration of insects found there at dusk, it is perhaps just about understandable.xxii
Getting back to onomatopoeic names, I can't resist including a name that my two younger sons still find hilarious, even in their teenage years: hoopoe. The hoopoe – pronounced 'hoo-poo', which explains my boys' amusement – is one of Europe's most striking and unmistakable birds: a boldly patterned black, white and pinkish-orange bird with a prominent crest and appallingly insanitary nesting habits. My friend Marek Borkowski, who lives in the middle of the Biebrza Marshes in Poland, has hoopoes nesting in his garden, and tells me that on hot summer days the stench from their nest inside a tree-hole is almost unbearable.
In fact, though, the name 'hoopoe' derives from the bird's call, a pair of echoing, staccato notes, which carries over a surprisingly long distance. It's not a sound we hear very often in Britain, where the species is a scarce visitor and very occasional breeder, but if you visit a patch of rough farmland in southern or eastern Europe during the spring or early summer you have a good chance of hearing it.
The sound-based origin of the hoopoe's name becomes clearer when we discover that its scientific name is _Upupa epops_ , which is doubly onomatopoeic. Richard Holme, writing in the late seventeenth century, referred to 'A Upupa... [which] is in our country speech called a Whoophoo, or Whopee, or Hoopoe, and Howpe'.xxiii
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The birds that make the most complex sounds are, as you might expect, songbirds: the various species and families that make up roughly half of the world's 10,700 or so bird species. But because their songs are so elaborate, they do not often lend themselves to onomatopoeic names.
The exceptions are those whose songs or calls are suitably simple and memorable, such as the metronomic, constantly repeated two-note song of the chiffchaff. Heard on a fine spring day, the chiffchaff is far easier to recognise and remember than the more complex song of its cousin the willow warbler, which pours out a silvery series of notes descending the scale with a rather wistful, plaintive tone.
But how we translate even simple birdsongs into names varies across different languages. And just as French cockerels say 'cocorico', Dutch ones go 'kukeleku' and Chinese say 'goh-geh-goh-goh' (whereas as we British know, they are actually saying 'cock-a-doodle-doo'), so other nations disagree about exactly what sound the chiffchaffxxiv is making. Germans call the species _Zilpzalp_ , the Dutch _tjiftjaf_ , while the Finns (whose Finno-Ugric language bears no resemblance to other major European tongues, apart from Hungarian and Estonian) prefer the rather splendid _tiltaltti_.
Other bird names based on sound include, appropriately, 'chat', as in those two charismatic little birds of moor and heath, the stonechat and whinchat. Listening to a stonechat's call, which sounds like two pebbles being knocked together, we might understandably conclude that this is the origin of its name. But as is so often the case with bird names, things are not quite so straightforward.
In fact, the name 'stone chack' was originally given to the wheatear, the stonechat's larger cousin, because of that bird's habit of perching on prominent stones in its moorland breeding territory, while uttering a lip-smacking call. Only as recently as the late eighteenth century was it applied to the stonechat – first as 'stone chatterer', then 'stone chatt', and finally as the name we use today.
This may appear rather messy and confusing, but that is the nature of bird names. Most were not decided by an elite group of experts, but emerged organically when ordinary folk, living in different parts of the country, chose their own names for the birds they came across. And thank goodness they did, for otherwise we might have to rely on professional ornithologists to name our birds, which would no doubt have produced far less varied and imaginative results.xxv
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To discover other names based on the calls of songbirds takes a little more digging, as over time the original sound has often been obscured by shifts in spelling and pronunciation.
It may not be immediately obvious, but the name 'finch' is another example of onomatopoeia. It comes from the commonest member of the family in Britain – our third most numerous breeding species after the wren and robin – the chaffinch.
Looking at a male chaffinch, with his splendid pink breast, dove-grey head and white flashes on his wings, you might assume that such a bird would have been named after its colourful and striking appearance. Yet the word 'finch' actually derives from the Old English 'finc', from the bird's rather monotonous call, usually represented today as 'pink'. Perhaps because the chaffinch is so ubiquitous, this name was later applied to other species such as the goldfinch and greenfinch, and thus to the family as a whole. While the goldfinch and greenfinch clearly took the prefix of their names from their appearance, the 'chaff' part of the chaffinch's name comes from its preference for feeding on grain amongst the chaff produced by the threshing process.
The original sound made by the chaffinch has now been largely lost in the English version of the name. But it is far more apparent in the modern Dutch _vink_ , the German _fink_ and the various Scandinavian languages ( _fink_ or _finke_ ), suggesting that the original name is even older than we might think, going back well before the birth of Christ. And we can still detect it in several English folk names for the species, all of which are more obviously based on its sound, such as 'pink', 'chink', 'twink', 'tink' and 'spink'.xxvi
One characteristic of the chaffinch is that different birds in different parts of the country have distinct local accents. Thus whereas those around my home in Somerset end their song with a fairly standard flourish, on a visit to the Scottish city of Dundee I discovered, to my amusement, that they finish with what sounds remarkably like 'ginger-beer' – leading local children to dub the chaffinch the 'ginger-beer bird'.
The chaffinch's propensity to vary its song from region to region is not a new discovery. Writing in 1600, in his translation of an older French text, 'practitioner in physicke' Richard Surflet observed: 'The spinke is a very beautifull and melodious birde, but all spinkes haue not one and the same tunes.'
*
Not all birds called after the sound they make have onomatopoeic names. Warblers do not all warble, but the name is apt enough to have been used for two totally unrelated families, one in the Old World (Sylviidae), and one in the New World (Parulidae).xxvii
Given the importance of song when we try to identify these often skulking birds, it is perhaps surprising that, apart from the chiffchaff, only two European warblers have been named after their sound. The best known of these is the grasshopper warbler, an elusive streaky brown bird that announces its presence via its reeling song, which sounds like a cross between an insect and an angler letting out a fishing-reel at speed.xxviii The other is the melodious warbler, a large yellow-and-green species found in western Europe, which regularly turns up in southern Britain on autumn migration. It does indeed have an attractive song, although to my ears it is no more tuneful than, say, the blackcap or garden warbler.xxix
The song thrush, too, is named after its persistent and repetitive melody, which can be heard in our parks, gardens and hedgerows from January through to June. Yet it is not universally popular: although many people (including myself) love the song thrush's chatty tone, others (including my wife Suzanne) find it rather tedious. As with any form of music, an appreciation of birdsong is clearly a matter of personal taste.
*
Of all the birds named after their sound few have a greater claim to the title of the world's greatest songster than the nightingale. The male's extraordinary outpouring of notes and phrases, emerging from the densest thicket at full volume on a spring evening, really has to be heard to be believed.
Although it may not be immediately obvious, the nightingale's name is also a reference to its sound: the 'gale' element derives from a Germanic word meaning 'songstress'. This is also found in the modern German, Dutch and Scandinavian names for the bird, _Nachtigall_ , _nachtegaal_ and _nattergal_ – all of which mean 'night singer' – and all of which are, as can be seen by their similarity to one another, very ancient indeed.
The nightingale has always been widely celebrated for its song, by poets, writers and musicians going all the way back to the Ancient Greeks and Romans, and reaching its zenith in the works of the nineteenth-century Romantic Poets such as John Keats.
Two very different poets – John Clare and T. S. Eliot – even attempted to reproduce the specific sounds made by the bird. In 'The Progress of Rhyme', Clare deploys a series of ever more bizarre and eccentric phrases:
'Chew-chew chew-chew,' and higher still:
'Cheer-cheer cheer-cheer,' more loud and shrill
'Cheer-up cheer-up cheer-up,' and dropt
Low: 'tweet tweet jug jug jug,' and stopt
One moment just to drink the sound
Her music made, and then a round
Of stranger witching notes was heard:
'Wew-wew wew-wew, chur-chur chur-chur,
Woo-it woo-it': could this be her?xxx
In T. S. Eliot's masterpiece _The Waste Land_ , the poet also uses onomatopoeia to convey the nightingale's song:
...yet there the nightingale
Filled all the desert with inviolable voice
And still she cried, and still the world pursues,
'Jug Jug' to dirty ears.
In some ways, though, our long and fervent admiration of the nightingale's song strikes me as rather odd. For when people hear one singing for the very first time, they are sometimes shocked. The nightingale's curious outpouring of grunts, tweets and whistles can take a while to get used to – especially if you are used to the more tuneful, sedate, and above all predictable songs of the song thrush, blackbird and robin.
Especially on first hearing, listening to a nightingale is a bit like trying to appreciate modern jazz: you have to relax, forget your preconceptions and allow the sound to wash over you, rather than trying to follow individual melodies. After a while you get used to the bird's improvisational technique, and can at last begin to enjoy what you are hearing.xxxi Then, as has happened ever since human beings first listened to this small, brown and rather unprepossessing-looking bird, you can simply admire one of the most extraordinary of all the world's natural sounds.
*
Both the nightingale and the cuckoo, the bird with whose sound we began our story, are long-distance migrants: travelling back and forth each spring and autumn between their African winter quarters and their breeding grounds in Britain and Europe.
In the past few years, many of the mysteries surrounding these incredible global journeys have been solved. One major breakthrough is that scientists are now able to place tiny, ultra-lightweight transmitters on the birds before they leave our shores, which have allowed us to track their movements in forensic detail.xxxii
When the British Trust for Ornithology decided to promote their cuckoo-tracking scheme by giving names to the individual birds being followed, they named one bird after the lead character in a children's book by John Miles. This literary cuckoo was called 'Gowk', after the folk name for the species that goes all the way back to Old English.
Sadly, after a promising start, in which Gowk safely crossed the Channel in mid-July, the tag transmitted only a few low-quality signals, which then disappeared. The conclusion was inescapable: that this particular cuckoo had failed to make it to Africa, and died en route. But his name lives on; and in doing so reflects the extraordinary persistence of bird names in our language – especially those derived from the sound the bird makes.
For me, it also shows why we should continue to cherish these ancient names: both those that are still in everyday use, like cuckoo, and those, like gowk, that survive only as folk names in certain parts of the country. The wonderful variety of names we give to birds is a reflection of the crucial importance of nature, in both our language and in our lives.
When we delve into the origins of a bird's name, and discover how it first came into being, we discover something vital about ourselves, about our history, and most of all about our relationship with the natural world. At a time when so many species of birds are under threat, we should cherish that deep and lasting connection: not only for what it tells us about our past, but also how it can inform our future, allowing us to better appreciate our interdependence with global biodiversity.
Which brings me back to the cuckoo – or, as I should perhaps say, the gowk. For had it not been for one cataclysmic event, almost a thousand years ago, this is the name we would still be using for this annual harbinger of spring. That event, which took place on a fine autumn day in the year 1066, would forever alter the course of our nation's history and culture. It would also change the very language we speak – including, of course, many of the names we give to birds.
In the next chapter, I shall explore the profound consequences of the final and most momentous invasion of our island nation: the Norman Conquest.
#### Notes
1 From _The Exeter Book, an Anthology of Anglo-Saxon Poetry_ , Israel Gollancz (ed.) (London, 1893).
2 From translation by Kevin Crossley-Holland (Enitharmon Press, 2008).
3 From W. S. Mackie (ed.), _The Exeter Book, Part II: Poems IX-XXXII_ (London, 1934).
4 James Fisher, _The Shell Bird Book_ (London, 1966).
5 See David Crystal, 'English as a Classical Language, <https://archive.org/stream/Omnibus42/09%20Crystal%20English%20as%20a%20Classical%20Language_djvu.txt>
6 David Anthony, _The Horse, the Wheel and Language_ (Princeton, 2007)
7 A. F. Harrold, _Of Birds & Bees_ (Reading, 2008).
i Throughout this chapter, I have used the word 'call' to describe the spring sound of birds such as the cuckoo, hoopoe and crows, which are not usually thought of as 'songbirds'. However, as the wildlife sound recordist Geoff Sample has pointed out ( _in litt._ ), these 'calls' have exactly the same function as song: to defend a territory against rival males, while at the same time attracting females.
ii Summer is coming in
The cuckoo sings loudly!
iii As he points out, this is tautological, as 'gowk' is also a dialect word meaning a foolish person.
iv In _The Shell Bird Book_ (1966), in my view the most readable yet scholarly history of Britain's birds ever written. The other 15 species are: robin, crane, (white-tailed) sea eagle, crow, wood pigeon, nightingale, swallow, chaffinch, raven, whooper swan, gannet, whimbrel, kittiwake, tern and quail. All would have been named by the year AD 700.
v Donated to Exeter Cathedral some time during the mid-eleventh century by its first bishop, Leofric, it remains there to this day, protected by his ominous warning: 'If anyone should take it away from thence, let him lie under eternal malediction'.
vi Earlier versions in these languages include the Welsh 'gwˆ ylan', Cornish 'guilan' and Breton 'goelann'. 'Puffin' may also be of Cornish origin.
vii Although it is no longer in general use, 'erne' regularly features as a crossword clue – the answer usually being 'sea eagle', but occasionally just 'eagle' or 'seabird'.
viii In many cultures, the date of the autumn arrival of geese would be used to predict the weather for the season to come – the belief was that an early arrival date meant a hard winter, and a late arrival a mild one. In reality the arrival date of migratory wildfowl has no link with the weather in the coming winter, and is purely a result of immediate weather conditions at the time of travel.
ix Their equivalents in modern Scandinavian languages are _svan/svane_ and _svala/svale_ – again, clearly related to our modern English names.
x As to the actual meaning of these ancient names, Lockwood suggests that swallow derives from a word meaning 'cleft stick'– a reference to the bird's long, forked tail; while 'swan' may come from a word meaning 'noise', which he speculates may refer to the sound made by the mute swan's wings as they fly overhead. I must say I am not entirely convinced.
xi The idea that these people were a single, homogenous group known as 'Celts' is an eighteenth-century invention; in reality they were a motley group of different tribes with little in common with one another.
xii See, for example, Philip Ball, _The Music Instinct_ (2010), and also the relatively new science of 'Biomusicology', a term coined by the veteran Swedish musicologist Nils L. Wallin in 1991, which looks at the connections between the sounds made by birds and other wild creatures and the music made by humans.
xiii The syrinx has recently been found to have evolved far earlier than we thought: evidence of its existence has been found in the fossilised skeleton of a duck-like bird, _Vegavis iaai_ , that lived more than 66 million years ago, in the age of the non-avian dinosaurs.
xiv The wildlife sound recordist Chris Watson once allowed me to listen through headphones to a blackbird singing. Using a parabolic reflector to magnify the sound enabled me to hear a whole series of normally inaudible high-pitched notes, uttered simultaneously with the deeper ones that we usually hear.
xv The _OED_ suggests a different etymology, linking 'dove' with a now lost Old English word meaning 'to dive' or 'to dip'; I have to say I side with Lockwood here.
xvi Incidentally, the name 'turtle dove' comes from the bird's soft, repetitive call, usually written as 'tur-tur-tur'.
xvii The use of the prefix 'Jack' may of course simply be a nickname, as in Robin redbreast or Jenny wren. 'Jack' often signifies male birds, but given that, according to the _OED_ , this is particularly used for birds of prey, in which the males are significantly smaller than the females, it may also imply smallness (as in 'jack curlew' for whimbrel, or jack snipe). In the case of the jackdaw, it could be all three at once: signifying a nickname, the bird's smaller size and its onomatopoeic call!
xviii As _hraefn_ , _hroc_ and _crawe_ respectively. The modern Icelandic word for the raven, _hrafn_ (which is also used as a Christian name) is almost identical to the one our Anglo-Saxon ancestors would have used. Incidentally, in both Old English and Old Norse the 'f' sound would have been pronounced as a 'v', making _hraefn_ sound even more similar to the modern 'raven'.
xix Surprisingly there is no link to the Gaelic word 'loch' – 'lough' in fact comes from Middle English.
xx The linguist David Crystal ( _in litt_ ) confirms this, pointing out that such transitions between the 'ow' and 'uff' sounds are reasonably frequent in English.
xxi The nineteenth-century poet John Clare wrote one of his most evocative sonnets, entitled 'The Fern Owl's Nest', on the nightjar – see Chapter 4.
xxii A much better diet-based name, invented by the ornithologist and sometime poet Mike Toms, is 'moth-gobbler'.
xxiii The Dutch have an even greater liking for coining names based on bird sounds than we do. Their name for the hoopoe – _hop_ – is commendably succinct, as is _oehoe_ for the eagle owl, an incongruously short name for such a huge and impressive bird. The Germans go one better in the brevity stakes: their name for the eagle owl is simply _uhu_ , while the French prefer the non-onomatopoeic (and rather pretentious) _grand-duc d'Europe_.
xxiv Oddly, despite the ubiquity of the chiffchaff's song, some languages have chosen names entirely unrelated to its sound. The Scandinavian languages all use the habitat-based name _gransanger_ , which translates as 'spruce warbler'; the French choose a motion-based name, _pouillot véloce_ , or 'speedy warbler'; and the Spanish plump for the species' ubiquity, _mosquitero común_.
xxv The exceptions to this are those species named after people (see Chapter 5).
xxvi This last name is still widely used in Scandinavia, and would have been brought here by the Viking invaders, over one thousand years ago. In Britain, Spink is a locally common surname, especially in Yorkshire and Norfolk, where the Vikings would have first landed, and is thought to have originated as a nickname for someone who chattered like a finch.
xxvii Worldwide, there are almost 400 different warbler species, including willow, sedge and reed warblers in Europe, and yellow, blackpoll and magnolia warblers in North America.
xxviii The grasshopper warbler belongs to the genus Locustella, named from the species' sonic similarity to a grasshopper or cricket.
xxix The melodious warbler's scientific name, _Hippolais polyglotta_ , is a further nod to its vocal talents.
xxx Oddly, Clare – who as an expert field-naturalist would surely have known that only the male bird sings – followed poetic convention by depicting the bird as a female in his sonnet 'The Nightingale'.
xxxi Unlike, it must be said, modern jazz.
xxxii For both the nightingale and the cuckoo, this new information cannot come a moment too soon. Both species are suffering steep declines in numbers, and both are in serious danger of disappearing from large swathes of their former haunts during the next decade or so.
# INVASION AND CHANGE
_The Beginnings of English_
We need words to name and designate things. But we only have a static language with which to express ourselves.
Piet Mondrian
## _1: The Ravens' Lament_
Sunday, 15 October 1066 dawned bright, clear and cold across the rolling Sussex landscape. Soon after sunrise, the autumn mists began to melt away, revealing a scene of utter devastation. Six thousand men – two-thirds of them English, the rest Norman – lay dead.
The only movement came from the hordes of glossy, blue-black ravens descending on the stiffening corpses, plucking out their eyes and stabbing at their open wounds to feed on the exposed flesh. The only sound to pierce the deathly silence was the occasional deep, hoarse cry, as one raven pushed a rival away from its own gruesome plunder.
Later that day, the victorious Normans and defeated English returned to the battlefield to claim their dead. Two monks from Waltham Abbey, which had been re-endowed and rebuilt by King Harold Godwinson just six years earlier, began the grisly task of looking for the body of their deposed ruler amongst the accumulated piles of human remains.
Finally, after hours of searching, they came across what they believed to be his corpse. According to one contemporary source, rather than being slain by the proverbial arrow through his eye, as famously depicted in the Bayeux Tapestry, Harold had been brutally hacked to death by four Norman knights:
The first, cleaving his breast through the shield with his point, drenched the earth with a gushing torrent of blood; the second smote off his head below the protection of the helmet and the third pierced the inwards of his belly with his lance; the fourth hewed off his thigh and bore away the severed limb: the ground held the body thus destroyed.i
Harold's body had been so badly mutilated they had to summon Harold's first wife, Edith the Fair (known also, because of her grace and beauty, as Edith Swan-neck) to confirm his identity. Her reaction to seeing her former lover's corpse in this terrible state is not recorded.
His mother Gytha, stricken with grief after losing three of her sons in the battle, requested that Harold's body be returned, allegedly offering his own weight in gold in exchange. Initially William of Normandy, leader of the invading Norman forces, refused, curtly adding: 'Harold mounted guard on the coast while he was alive; he may continue his guard now he is dead.'
But eventually William did relent, and Harold's body – or what was left of it after the ravages inflicted by the Norman army and the ravens – was taken back to Waltham Abbey for a Christian burial. Meanwhile William the Bastard, Duke of Normandy, rode to London to claim the English throne.
On Christmas Day 1066, after several months of skirmishes and political wrangling, he was finally crowned in Westminster Abbey as King William I. The Norman Conquest had well and truly begun.
*
Historians love to dwell on what they call 'counterfactuals' – speculating on what might have happened had the outcome of a particular historical event been different from what actually transpired. Of all the many alternative scenarios, one of the most intriguing is to consider the history and development of the English language had Harold, rather than William, triumphed at Hastings.
One thing can be said for certain: English would be far less varied, in both syntax and vocabulary, than the language we speak today. Modern English benefits from 'hybrid vigour': the amalgamation of the Norman French spoken by the invaders, and the Old English spoken by the defeated Anglo-Saxons. As the journalist and literary critic Allan Massie points out:
If you were to begin by asking, in Monty Python style, 'What have the Normans ever done for us?' you might first reply that the most enduring consequence of the Conquest is the richness of the English language, with its Anglo-Saxon base and Franco-Latin superstructure.1
Thanks to its mongrel origins, modern English is a fabulously varied and flexible language: not hidebound by complex and unnecessary grammatical rules, and containing a wealth of alternative words for each object or concept – well over twice as many as other languages.
But this linguistic transformation did not happen overnight. At first, just as had occurred between the Roman invaders and the Ancient Britons, the invading Normans and defeated English kept themselves to themselves. Socially – and more importantly for our story, linguistically – the two groups lived separate lives, fuelled by mutual resentment and suspicion.
English remained purely a spoken language, 'an uncultivated tongue', fit only for labourers, servants and peasants. Norman French, on the other hand, enjoyed a far more elevated status. It was spoken by the nobles, but importantly it was also a written language, used in legal documents, and in the popular genre known as 'Romance' literature. Meanwhile, a third language, Latin (at this point still a spoken as well as a written language), was primarily used in the religious and educational spheres. The end result was a kind of linguistic and social apartheid, with English firmly at the bottom of the pecking order.
The clearest indication of this is the language spoken by kings. Many 'English' monarchs of this period not only spoke French, but also spent most of their time on the other side of the Channel, where they still controlled vast areas of land. Henry II was away from England for almost two-thirds of his reign, while his son Richard the Lionheart (more accurately known as Richard Coeur de Lion) never actually learned to speak the language of his new realm of England.
The fact that royalty – and by extension, the nobility – spoke French, while the labourers continued to speak English, is reflected in the very different words we still use today for farm animals and for the meat they produce. If you walked into a restaurant and ordered a 'cow steak' you would get some pretty funny looks; as you would if you asked for a pig pie, sheep shank or deer casserole.
This is because, while we use names derived from Old English for the creatures themselves – cow, pig, sheep and deer – we call their meat by French names: beef, pork, mutton and venison. This is a direct consequence of the relative social status of two groups of people in the post-conquest world: the English peasants, who tended the animals, and the French nobles, who ate their meat.
*
Not surprisingly the new arrivals – and their new and unfamiliar language – also influenced bird names. In Old English, the bird we know as the kingfisher was called an _isen_ or _isern_ , from a word meaning iron-coloured – i.e. blue – which survives in several modern European names for the species, including the German _Eisvogel_ and Dutch _ijsvogel_.
From roughly the year 1000 the name 'fisher' first appears (as _fiscere_ ) in Old English. Sometime in the fourteenth or early fifteenth centuries the compound name 'king's fisher' emerged, probably as a direct translation of the French _roi pêcheur_. This in turn may be linked to the Fisher King, the mythical figure of the Grail legend, the last in the long line of those charged with safeguarding the holy relic. The new name soon gained dominance over any older ones.
Occasionally, instead of the old Anglo-Saxon name giving way to the new Norman one, they both survived. We still use the names 'dove' and 'pigeon': 'dove' mainly for smaller members of the family Columbidae, and 'pigeon' for the larger ones. But their origins are very different. As we have seen, dove probably derives from an Old English word based on the bird's sound, while pigeon comes from the Old French word _pijon_ ,ii and does not appear in English until the late fourteenth century.
Not surprisingly, given the influence of the conquerors, several other bird names we still use today derive directly from Norman French. These include mallard and wigeon, pheasant and partridge, kestrel and merlin, eagle and peregrine.iii At first sight these names – and indeed the birds themselves – do not appear to have much in common with one another; so we might easily assume that their common French origin is simply a linguistic accident.
But take a closer look. All these species are either wildfowl or gamebirds, which would have been pursued for food and sport; or raptors, used for falconry and hunting. In exactly the same way that beef, pork and venison have French names because they were too expensive for the commoners to eat, so the invading Norman aristocracy gave names to all these birds, as these were the ones they encountered most often in their day-to-day lives. The new names rapidly displaced the Old English ones that had been used until then, which have long since fallen into disuse.
So even at this early stage in our society, the differences between the elite nobility and the labouring classes that would come to define English society were already beginning to show.
You might reasonably assume that as Old English gradually merged with Norman French to create Middle, and later Modern, English, then the now mostly incomprehensible Anglo-Saxon names would vanish too, to be replaced by those with a Latin origin. But as we shall see, the opposite proved to be the case. A surprisingly large proportion of the names we still use today – including redstart, yellowhammer, fieldfare, lapwing and wheatear – have their origins in the pre-Conquest tongue.
Yet if you assume you know what these names actually mean, you may need to think again. For in that change from Old to Middle, and later to Modern English, something very strange happened – something that reveals that names, and other proper nouns, behave in a significantly different way from other words in our language.
## _2: Red Tails and White Arses_
What is the commonest surname in Britain, and indeed throughout much of the English-speaking world? It is, of course, Smith. More than half a million people in the United Kingdom, and over two million in the United States, are called Smith, which is also the commonest surname in Australia and the second commonest in Canada.iv
The name Smith means 'one who works with metal' – and dates all the way back to Anglo-Saxon times, when men were often named after the job they did. It is not the only profession-based surname still in widespread use today: others include Cooper (meaning barrel-maker), Mason (as in stonemason), Miller (of grain), Turner (of wood) and Taylor (of clothes), all of which feature in the list of the top hundred commonest surnames in England and Wales.
Given that none of these professions is widely practised today (though some of us like to think of ourselves as 'wordsmiths'), their survival is clearly down to the longstanding custom of giving children the same surname as their father, however irrelevant the original meaning of that name may now be; a practice that began far back in the thirteenth and fourteenth centuries.
Many common place names in Britain also contain elements that reveal their pre-Norman origins. So we still use the suffix '-by', an Old Norse word meaning 'settlement', as in Selby and Whitby, '-ham', the Old English for farm or homestead, and the ubiquitous '-chester', which originally came into our language from Latin, indicating a Roman fort. As with surnames, once a name of a particular place has become established in common use, it proves remarkably resistant to change.
You might not be surprised to learn that the names we use for birds are no exception to this rule. But to discover their original meaning requires a degree of linguistic detective work, because they are effectively 'in disguise'. Over the centuries many have changed into a completely different word, by means of a process that linguists call 'folk [or false] etymology'.
Take two well-known British birds: the redstart, a close relative of the robin, and the yellowhammer, a canary-coloured member of the bunting family. We use these names so frequently that we no longer even notice just how peculiar they are. Yet if we think about them, they make absolutely no sense at all: redstarts are not especially jumpy, and nor do yellowhammers have a particularly percussive call.
These names, along with many others, including fieldfare, lapwing and wheatear, are classic examples of the peculiar propensity for archaic words to survive in names far longer than they would in other aspects of our language. But as their original meaning became less and less clear, the Anglo-Saxon names were eventually transformed into more familiar words. These usually have little or no connection with the original meaning, and so can mislead the unwary.
Thus the Old English word _steort_ , meaning tail, transmuted into 'start' (hence redstart, and its scarcer cousin, the black redstart). Likewise, the 'hammer' in the name of our most colourful bunting is nothing to do with tools, but is a corruption of _Ammer_ , the word still used for bunting in German today.v
Yet even by the late eighteenth century, the name yellowhammer, while widely used as a folk name, was still not fully established as the official name for this species. The naturalist Thomas Pennant preferred the alternative 'yellow bunting', in a tidy-minded attempt to bring the yellowhammer in line with its cousins the reed, corn and snow buntings. But less than a century later, the Victorian ornithologist William Yarrell changed the name back again, adding a helpful explanation of its origin for his readers: 'I have ventured to restore to this bird what I believe to have been its first English name, Yellow Ammer. The word Ammer is a well-known German term for Bunting.'
Other familiar birds provide further examples of how folk etymology can mislead us. Around my Somerset home, the first sign that winter is on its way comes with the arrival each autumn of large flocks of redwings, fieldfares and lapwings, refugees from the north. These birds travel here to enjoy the benefits of our relatively mild winter climate and the plentiful food this brings.
Their names, at least at first glance, appear to make perfect sense. The redwing is a small, dark thrush from Iceland, sporting a rufous patch on its flanks. The fieldfare, its larger, more colourful cousin, arrives each November from its breeding grounds in Scandinavia and northern Russia, greedily feasting on berries in hedgerows and probing for worms as it 'fares over' muddy fields. And the lapwing, also known as the peewit from its distinctive piping call, has a very distinctive flight action; as flocks pass overhead on a fine winter's day, the alternating flashes of dark upperwing and white underwing appear to 'lap' through the sky like swimmers in a calm blue sea.
And yet of this trio of winter visitors to my own corner of the West Country, only the redwing's name actually means what it suggests: a bird with a red coloration on (or near) its wing. The other two have far more complex origins, and very different meanings from the obvious ones.
The linguist W. B. Lockwood made short work of the _OED_ 's more prosaic interpretation of the fieldfare's name: 'The name is clearly corrupt; the explanation that the meaning is somehow fieldfarer is just the obvious guess – and quite as obviously improbable... for dozens of species fare over fields.'2
He had a point. As an alternative, Lockwood proposed that the name of what Chaucer called 'the frosty feldefare' comes from a long-lost Old English phrase meaning 'grey piglet', a reference to the bird's colour and its harsh, grunting call. Only once Old English gave way to the more modern, French-influenced language, and this meaning had become obsolete – just like the suffixes 'start' and 'ammer' – did this ungainly thrush gain its misleading modern name.vi
To most people, the meaning of the name 'lapwing' appears equally obvious. Yet, once again, Lockwood disagreed with the easy explanation. Digging down into Old English, he discovered a reference from before the Norman Conquest to the bird as the 'hléapewince'. He believed that the name refers to the prominent tuft of feathers on top of the bird's head, and translated it as 'movable crest'.vii
We can trace the gradual changes in the bird's name through time: from 'hléapewince' (first noted in AD 1050), through 'lhapwynche' (1340), 'lappewinke' (1390), 'lapwyng' (1430) and finally the one we use today, lapwing (1591). Soon afterwards, in 1604, that name appeared in the climactic final act of Shakespeare's _Hamlet_ , in which Horatio taunts the hurried departure of Osric by comparing him with a newly hatched chick:
*
This Lapwing runs away with the shell on his head.
This casual insult reveals that Shakespeare had more ornithological expertise than we might give him credit for: he clearly knew that lapwing chicks are precocial, meaning that they are able to leave the nest almost immediately after they are born.
The best known of what my old French teacher Mr Schrecker used to call 'faux amis' (false friends, because they mislead the unwary) is the name 'wheatear'.
Wheatears are showy members of the chat family – cousins of the robin and nightingale – with roughly two-dozen representatives, mainly found in the arid, stony deserts of North Africa and the Middle East. Just one species (officially known as the northern wheatear to distinguish it from its heat-loving southern cousins) has managed to extend its range northwards into the temperate regions of Europe, including Britain.
Wheatears are one of the earliest spring migrants to arrive back from Africa, turning up in fields of short-cropped grass from the middle of March onwards. Even though this kind of grass and wheat are not the same plant, people have tended to assume that the bird's name must be somehow connected with our most widespread arable crop.
But this perky little bird has nothing whatsoever to do with ears of wheat. The Anglo-Saxon name, which unfortunately has not survived in print, was probably 'wheteres'. The final 's' on this singular noun provides a crucial clue to its real meaning: 'white-arse'. This is a reference to the wheatear's most prominent feature, its bright white rump, which is revealed as soon as the bird takes to the wing and flits away from you. To confirm this, we need look no further than two dialect names for the species: 'white rump', from Northumberland, and the blunter 'white ass' from Cornwall.
By the seventeenth century the origin of the wheatear's name had already been long forgotten, as can be seen from two contemporary accounts. In _Worthies of England_ , published posthumously in 1662, the historian Thomas Fuller pronounced with great authority that: 'It [the wheatear] is so called, because fattest when wheat is ripe... whereon it feeds.'3
The other comes from the English poet John Taylor who, in August 1653, set out on what he called 'The certain travailes of an uncertain journey', 'for no other intent or purpose, but to pleasure himself, and to please his friends in the first place'. Taylor's perambulations coincided with the start of the autumn bird migration season, during which he came across 'rare Birds I never saw before', adding in a dreadful example of doggerel:
Th' are called wheat-ears, less than lark or sparrow,
Well roasted, in the mouth they taste like marrow.
Having never actually eaten a wheatear, well-roasted or not, I cannot attest to Taylor's culinary tastes, but I do know that his ornithological knowledge was severely limited, as this later couplet reveals:
The name of wheat-ears, on them is ycleap'd,
Because they come when wheat is yearly reap'd.4
We can only hope that the rest of his account was more accurate than this entirely false piece of speculation (and rather bad poetry).
But what is puzzling is how the English language could have changed so radically that these contemporaries of Marvell and Milton could no long decode the names that had been coined a thousand years earlier, back in Anglo-Saxon times. _When_ this rift in understanding happened is easy to answer – some time during the thirteenth century. _How_ , and especially _why_ it did so, is a little trickier. And the explanation helps to solve a puzzle that may have already occurred to you: why is a blackbird called a blackbird, when so many other birds are also black?
## _3: Sex, Chaucer and Blackbirds_
The way English evolved from an obscure and rather inflexible Germanic tongue into the rich, fluid, complex language we all speak today comes down, as with so many things, to sex.
Fortunately for the future of the English language, the initial stand-off between Normans and Saxons could not last for ever. The mutual attraction between noble French lords and comely English maidens (and perhaps also between aristocratic French ladies and muscular sons of the soil) inevitably led to social and sexual interactions between the two groups. Soon afterwards, these turned into more formal and permanent liaisons. Thus by the late twelfth century one chronicler could observe: 'Now that the English and Normans have been dwelling together, marrying and giving in marriage, the two nations have become so mixed that it is scarcely possible today... to tell who is English, who of Norman race.'5
Then, as the English monarchs withdrew from their possessions in France following successive military defeats, something rather odd happened. In other countries where two or more languages are habitually spoken, such as Switzerland and Belgium, they usually remain clearly separate, each used by a different community. Alternatively, as in Scotland and Ireland where English displaced Gaelic, the conquerors' language eventually triumphs at the expense of the original one.
But in England, the competing tongues of what had started off as Old English and Norman French underwent a kind of mutually agreed merger, creating a completely new language with features from both its parents: what we now call Middle English.
This new language was in many ways more complex than those it replaced, with a far more extensive vocabulary, thanks to the borrowing of 'loan words' from Norman French and Latin. Some of these displaced the Old English word entirely, but more often than not the old and new words co-existed alongside one another, endowing our language with a plethora of synonyms. Thus today we can choose between words of both Germanic and Romance origin, which mean more or less the same thing: for example, 'kingly' and 'royal', 'pretty' and 'beautiful', or 'wed' and 'marry'.
But crucially, this new language was also more straightforward than its parent tongues, in several important ways. Technical aspects such as inflexions (sets of endings added to words to indicate grammatical case, number and gender) were either lost or greatly simplified.viii It is not unreasonable to suggest that this, along with the rich extra vocabulary provided by merging the two different languages, was one important reason for the eventual adoption of English as a global _lingua franca_.
Of course, as we have seen in the previous chapter, this process had actually begun long before the Norman Conquest, with the Viking invasions of northern and eastern England, from the late eighth century until around the time of the Norman invasion, leading to many of the Scandinavian invaders settling on this side of the North Sea. Not only did Old English borrow many loan-words from Old Norse, but the shifting grammar also began at this point, with the system of grammatical inflexions being streamlined. 'In order to facilitate communication', notes Professor Simon Horobin from the University of Oxford, 'the two groups of speakers must have placed less stress on the inflexional endings; as a consequence, the Old English system of inflexions began to break down.'6
But the most revolutionary change was the almost total disappearance of grammatical gender. Old English, like modern German, had three different genders – masculine, feminine and neuter – but during the transformation into Middle English these distinctions gradually disappeared. Interestingly, this process appears to have begun in the north of England, under the influence of the Vikings; gender disappeared there some time during the twelfth century, earlier than it vanished in the south.
The impact of this cannot be overstated. As we British discover to our cost when we come to learn a foreign language, what must seem perfectly natural to generations of French, Spanish and German children – the use of gender to qualify nouns, as in 'le chat', 'el perro' and 'das Auto' – is a real struggle for native English speakers, for whom gender in language effectively disappeared almost a thousand years ago.
By the time of Geoffrey Chaucer, who was writing in the closing years of the fourteenth century, the competing claims of Old English and Norman French were over, and Middle English was firmly established – a language that, with a little effort on the part of the reader, can still be understood today.
*
Although Chaucer is rightly celebrated for his classic works, most notably _The Canterbury Tales_ , we should not forget that he also contributed to the slow but steady growth in the understanding of Britain's birdlife. The ornithologist and broadcaster James Fisher described him as an 'ornithological hero', and while this may be a slight overstatement, it does have some validity. As Fisher points out, Chaucer not only knew the names of more than forty species of birds, he also added several others to the embryonic list of birds seen in Britain, which by the time of his death in 1400 had reached the landmark 100 species.
One of Chaucer's best-known works, 'The Parlement of Foules', features more than thirty British birds, which have gathered together to choose their mates. These range in size from the robin to the swan, and include resident species like the lark and the lapwing, and migrants such as the turtle dove, cuckoo, nightingale and swallow, along with more exotic visitors (perhaps commoner in those days), including the stork and the crane.
But one common and widespread bird is missing from this otherwise comprehensive catalogue of species: the blackbird. Its absence conceals a fascinating story of one of the most profound changes of all: the switch from 'fowl', the standard word for all birds in Chaucer's time, to the one we use today: 'bird'.
*
Bird names don't come much more basic than that of the most familiar member of the thrush family: the blackbird. It's a bird, and it's black. End of.
Except that, when you start to think about it, lots of other birds are black, too. Crows, rooks, ravens and jackdaws would also have been common, widespread and very familiar to our rural ancestors. So out of all these 'black birds', why did they choose just one species for this epithet?
This is only confusing because we are looking at the name from the wrong angle. For the key word here is not 'black', but 'bird'. This goes back to the Anglo-Saxon word _brid_ , which comes from the same root as the words 'breed' and 'brood'. It had a very different meaning from the modern word 'bird', and referred purely to baby birds or fledglings, as the _OED_ explains in its primary definition of the word:
Bird, _n_. The general name for the young of the feathered tribes; a young bird; a chicken, eaglet, etc.; a nestling. _The only sense in Old English; found in literature down to 1600_. [My italics]
So what were adult birds called? Until long after the Norman Conquest they were known as fowls, from the Old English 'fugol' (or in _Beowulf_ , 'fugle').ix
But gradually, around the time Chaucer was writing (towards the end of the fourteenth century), the original meaning of 'bird' was starting to change. Indeed, Chaucer himself occasionally uses the word in its modern sense: in his poem 'The Legend of Good Women', written around the year 1385, he makes a passing reference to 'whanne the brid began to synge...'
The title 'Parlement of Foules' is one of the last recorded examples of fowl being used to refer to all shapes and sizes of bird, large and small. About this time, 'bird' started to be used to denote the smaller species – those that today we call songbirds, although the name 'foules' persisted for larger birds.
So whereas today the meaning of the spoken phrase 'there's a black bird' could be unclear, referring perhaps to a crow or raven rather than a blackbird, in those days such ambiguity was not a problem, as those larger species would have been known as 'foules'.x So the name 'blackbird' made perfect sense.
The use of 'fowl' to mean all birds persisted well into the seventeenth and eighteenth centuries, as in the _King James Bible_ of 1611, which in the opening chapter of the Book of Genesis refers to 'every fowl of the air'. But the new distinction between the two words was becoming more established. In Dr Johnson's epic _Dictionary of the English Language_ , first published in 1755, the great lexicographer defines the word 'bird' thus:
A general term for the feathered kind; a fowl. In common talk, fowl is used for the larger, and bird for the smaller kind of feathered animals.
This distinction can still be found in some parts of Scotland, where larger birds are known as 'fowls' and smaller ones are 'birds' – as in 'muir [moor] fowl', a common Scots term for red grouse. But elsewhere the use of 'fowl' is today mostly confined to domestic birds such as chickens, or other groups of larger birds. These include 'wildfowl' and 'waterfowl' (ducks, geese and swans), and the names guineafowl and peafowl (the correct term for the species we usually call the peacock).xi
*
Given the very specialised original meaning of the word 'bird', it is perhaps not surprising that the first recorded use of the name 'blackbird' is as recent as 1350, as 'blacbrid'.
So what would the blackbird have been called before this time? A clue lies in the name we still use today for its upland counterpart, the ring ouzel. Alternatively spelled 'ousel' or 'wosel', this derives from the same root as the modern German word for the blackbird, _Amsel_. Also known as the mountain (or moor, fell or hill) blackbird, the male ring ouzel can be told apart from its commoner cousin by the distinctive white band across its upper breast – hence the name ring ouzel.
A further clue to the blackbird's old name comes in Act 3 of Shakespeare's _A Midsummer Night's Dream_ , when Bottom refers to 'the Woosell cocke, so blacke of hew, with orange-tawny bill...' This must be the blackbird, as ring ouzels have pale edges to their feathers and a pale lemon-yellow bill, whereas male blackbirds are indeed all black, and their bill is a deep orange-yellow shade. Even though the name 'blackbird' has been around for more than 600 years, in some parts of northern England and the Midlands blackbirds were still widely referred to as 'ouzels' well into the twentieth century, in another example of the remarkable persistence of folk names. The Nottinghamshire-born author D. H. Lawrence certainly used the name 'black ousel', having learned it from his mother, who would often lapse into archaic words and phrases from her childhood.xii
Not surprisingly, given our fondness for this common and familiar species, the name 'blackbird' has spread around the world. Today it is used for several close relatives of our species, such as the grey-winged, Indian, Tibetan and white-collared blackbirds of Asia, and also for a group of entirely unrelated birds found in the Americas, the family Icteridae, which includes the colourful New World orioles and oropendolas as well as the crow-like grackles, parasitic cowbirds and colourful meadowlarks.
The New World use of the name 'blackbird' dates back to the earliest settlers in North America. In 1602, the Norfolk clergyman turned adventurer John Brereton wrote of the birds he encountered in northern Virginia: 'We saw in the country... Doves, Sea-pies [Oystercatchers], Blacke-birds with carnation wings.'7
This refers to the most abundant North American bird, the red-winged blackbird, which does indeed superficially resemble our own species – until it takes flight, to reveal bright crimson epaulettes on each shoulder.
In this strange new world, it is perhaps no surprise that when homesick settlers such as Brereton encountered a new species of bird, they looked for any superficial resemblance to a more familiar species back home, and named it accordingly – something we shall investigate further in Chapter 4.
## _4: Fifty Shades of Green?_
The blackbird is just one of over 130 species on the official 'British List'xiii (totalling just over 600 species in all) whose names feature at least one colour.
Not surprisingly, given their prominence in the plumage of so many birds, red and black are the top two colours, with thirty-two and thirty species respectively. Bird names we have already come across featuring these colours include redstart, red grouse and red-wing, along with blackcap, blackbird and – with two for the price of one – black redstart.
Next on the list comes yellow, with fourteen species, including yellowhammer and yellow wagtail, then white, with thirteen species, including whitethroat (and of course lesser whitethroat). Grey is surprisingly high on the list, in fifth place, with twelve species, including grey heron, phalarope, wagtail and plover, while gold/golden and green have nine and eight species respectively, including several 'golden' plovers, golden eagle, goldfinch, greenfinch and green woodpecker.
One colour that comes surprisingly low on the list, with just seven species, is blue. Yet while blue may be a ubiquitous colour in the human world, when it comes to the plumage of birds it is fairly rare. Of common British birds only the kingfisher is predominantly blue, and is named after its feeding habits rather than appearance; the blue tit, on the other hand, has a prominent blue crown and tail, but is in fact mainly yellow and green.xiv
*
It may seem obvious to name a bird after its colour, but it is a system not without pitfalls, and colour-based names can sometimes confuse the unwary. For instance, female and young male blackbirds are brown, while the female blackcap has a chestnut-coloured crown. Grey wagtails are indeed grey, but they also show prominent flashes of lemon yellow, and so are often mistaken for their cousin, the yellow wagtail.xv
Later on, as ornithologists began to explore further afield, they soon found that, as they discovered more species, they needed more and more subtle ways of telling them apart. And so a second tranche of colour-based names began to emerge. These go beyond the usual reds, blacks, whites, greens, blues and yellows to encompass more complex and subtle shades. The best known of these is 'pied', as in pied flycatcher and pied wagtail. Others include roseate (tern), snowy (owl), tawny (owl and pipit), dun (meaning brown, as in dunnock and dunlin), buff (buff-bellied pipit and buff-breasted sandpiper), coal (tit), rose (-ringed parakeet) and ruddy (duck).
Heading north, the globe-trotting ornithologists came across two Arctic species of seabird, whose pallid plumages led to them being dubbed ivory and glaucous gulls – the latter from the Greek _glaukós_ (via Latin _glaucus_ ), and meaning pale bluish-grey or green.
As they explored other bird-rich continents such as Africa, Asia and South America, they discovered various reddish-yellow species, including an antshrike, babbler, owl, wren and whistling duck, all of which were given the Latin-derived epithet 'fulvous'. Darker, more reddish-brown ducks, hawks, partridges and pygmy owls were described as 'ferruginous', from the Latin for iron-coloured or rusty; while a tiger heron and imperial pigeon, whose colour can best be described, in the wise words of the _OED_ , as 'of a colour tending to reddish; somewhat rufous...' were named 'rufescent'.
Within a particular family, too, many different subtleties of colour and shade are needed in order to name a host of similar-looking species. Take the Old World Warblers: a family comprising roughly 280 species found in Europe, Asia and Africa (including, of course, Mrs Moreau's eponymous bird). These birds are famously tricky to identify in the field, mainly because very few (with the exceptions of the blackcap and whitethroat – both named after their most obvious plumage feature) are easy to tell apart from their cousins. Indeed, most appear to be basically brown, green or yellowish in shade: what birders often contemptuously dismiss as 'LBJs' or 'little brown jobs'.
If we just look at two colours, yellow and green, there is a plethora of subtly different ways of naming a species after its shade. Some are obvious, such as yellow-breasted and lemon-throated. Others are far more refined, among them the descriptors olivaceous, sulphur-bellied and icterine – the last deriving from the Greek _ikteros_ , meaning jaundiced, from the erroneous belief that a sighting of a yellow bird was supposed to cure this medical condition.
In the Americas there is another family of superficially similar-looking birds, known as wood-warblers, which like the American blackbirds were named by homesick settlers after familiar species they recalled from back in Britain. This family includes even more birds whose names feature a green, gold or yellow hue, such as black-throated green, citrine (meaning lemon-coloured), golden-cheeked, green-tailed, grey-and-gold, olive-capped, yellow-rumped and a dozen different species of yellowthroat.xvi
There are some species whose colours and shades are so subtle they can only be described with what the ornithologist and author Jeremy Mynott called 'a nice note of ruminative hesitation'.8 These include _greenish_ warbler, _reddish_ egret, _yellowish_ flycatcher and the rather sad-sounding _greyish_ mourner, a South American flycatcher.
But of all the names of birds on the British list named after colours and shades, the most fascinating story of all is the supposed origin of a name given to two rare vagrants, a shrike and a wheatear: each of which rejoices in the name Isabelline. The story of how they acquired this unusual epithet takes us to the next stage in the story of how birds got their names: the beginnings of modern ornithology.
#### Notes
1 Allan Massie, _Daily Telegraph_ , 12 October 2012. <http://www.telegraph.co.uk/history/9606163/In-everything-we-say-there-is-an-echo-of-1066.html>
2 Lockwood, _op. cit._
3 Thomas Fuller, _The Historie of the Worthies of England_ (London, 1662).
4 Charles Hindley (ed.), _The Works of John Taylor: The Water-Poet_ (London, 1872).
5 Charles Johnson (ed. and trans.), _Dialogus de Scaccario_ (1177) (London, 1950).
6 Simon Horobin, _How English became English_ (Oxford, 2016).
7 John Brereton, _Briefe and True Relation of the Discoverie of the North Part of Virginia in 1602_ (1602), Wisconsin Historical Society Digital Library and Archives.
8 Jeremy Mynott, _Birdscapes_ (Princeton, 2009).
i As described by William of Jumièges, writing just four years after the event, in 1070.
ii Which ultimately derives from the Latin _pipion_ , meaning 'young bird', a word that also comes from its sound (from 'pipiare', meaning 'to cheep').
iii Peregrine is from a Latin word meaning 'coming from foreign parts'. This first appeared in its Latin form _peregrinus_ around the year 1250, when the writer Albertus Magnus noted that young birds caught on migration proved better for falconry than those taken straight from the nest. It has since gained the more general meaning of 'wandering', as in the word peregrination, which originally referred to a lifelong spiritual journey or pilgrimage, but now usually refers to any kind of meandering voyage.
iv It is also a very common name in German ( _Schmidt_ or _Schmitt_ ) and Dutch ( _Smid_ or _Smidt_ ), and has direct parallels in Romance languages, such as the Italian _Ferrero_ (meaning ironworker) and the French _Fabre_. The pseudonym 'John Smith' is also the one most frequently adopted by British men who do not wish to reveal their true identity for personal or nefarious reasons.
v Confusingly, though, in German the yellowhammer is the _Goldammer_ , while several English folk names also prefer gold to yellow, as in 'golden amber' and 'gladdie' or 'go-laddie', now obsolete West Country names which probably derived from the phrase 'gold laddie'.
vi The first written reference to 'fieldfare' – with the modern (and etymologically incorrect) spelling – appears in John Florio's pioneering dictionary, _Worlde of Wordes_ , in 1598.
vii The _OED_ dissents from this view, however, maintaining that the original name was a combination of two words meaning 'to leap' and 'to totter, waver or wink', and so does indeed refer to the bird's flight, in which the alternating dark upperwing and white underwings look rather like a winking eye.
viii Modern English does still uses some inflexions, for example the possessive – 'the girl's book' – and to indicate the difference between singular and plural – 'boy' and 'boys'.
ix Closely related to the modern Dutch and German words for bird, _Vogel_.
x In practice, as David Crystal has pointed out, we can still tell the difference, as 'black bird' is stressed equally on both syllables, whereas in 'blackbird' only the first syllable is stressed.
xi It also survives in a handful of ancient folk names, such as 'rain fowl', for the green woodpecker, whose ringing call is supposed to herald a change in the weather; and 'garefowl', for the now extinct great auk.
xii Another, unrelated species also retained the name 'ouzel' until at least the middle of the nineteenth century, as this line from Charles Kingsley reveals: 'The startled water-ousel, with his white breast, flitted a few yards.' The mention of the white breast immediately gives away the bird's identity: the author of _Westward Ho!_ and _The Water-Babies_ is of course referring to the dipper.
xiii The official British List of birds accepted as having been seen in a truly wild state in Great Britain (England, Scotland and Wales and associated waters) has been maintained by the British Ornithologists' Union (BOU) since 1879. The current figures were taken in spring 2017, but do change as new birds are added and/or their names are revised.
xiv The remaining colours that appear in the names of birds on the British List are brown and purple, with four each, including brown shrike and purple heron; and finally pink, with just one: pink-footed goose. Incidentally, the reason no British bird has the name 'orange' is that this colour did not enter the English language until the late Middle Ages, by which time most common birds – including the robin redbreast – had already been named.
xv Any 'yellow' wagtail seen in winter – or on a fast-flowing river at any time of year – is definitely a grey wagtail, as yellow wagtails are spring and summer visitors to Britain, usually found in wet meadows.
xvi Although most New World warblers are named after the predominant shades of yellow or green, there are also cerulean, a deep shade of blue, and plumbeous, meaning lead-coloured.
# HISTORY AND SCIENCE
_The Birth of Ornithology_
'What's the use of their having names,' the Gnat said, 'if they won't answer to them?'
'No use to them,' said Alice; 'but it's useful to the people that name them, I suppose. If not, why do they have names at all?'
Lewis Carroll, _Through the Looking-Glass_
## _1: Dirty Underwear_
12 August 1566 was an auspicious day for the Spanish royal family. It saw the birth of Princess Isabella Clara Eugenia, a little girl whose royal pedigree was second to none. She was not only the daughter of the mighty King Philip II of Spain, but also the granddaughter of the two most powerful rulers in the Renaissance world: the Holy Roman Emperor Charles V and the French king Henry II.
Princess Isabella's arrival was a triumph of hope over experience. Her 21-year-old mother, Elisabeth of Valois, had previously miscarried twin girls, and had also survived a bout of smallpox, which nearly killed her. So Princess Isabella's father greeted his daughter's safe delivery with more than the usual relief and joy.i
During her long and eventful life, Isabella did her best to live up to her distinguished royal antecedents. She was a strong and determined woman, and more than held her own in this male-dominated world. Indeed, her stubbornness is at the root of a curious legend: one that links her with the names of several species of bird and mammal.
The adjective 'Isabelline' is found in the English name of three species of bird: a shrike, a wheatear and a bush-hen (a type of rail from the Indonesian province of Sulawesi), and in the scientific names of several others. It has also been used to describe a distinctively pale form of the Himalayan brown bear, _Ursus arctos isabellinus_ , and several breeds of horses and dogs.
All these creatures have one thing in common: their predominant coloration is a dirty greyish-brown, as though they need a good wash. Given the legend of how this very unusual shade got its name, this is rather apt.
*
From infancy, Isabella was expected to marry for strategic and political reasons, rather than for love. At just two years old, she was betrothed to her cousin, the future Holy Roman Emperor Rudolf II. But when she reached maturity, it became clear that the eccentric, depressed and probably homosexual Rudolf was never going to honour his original agreement to wed her.
In 1599 – at the relatively advanced age of thirty-two – she finally married Rudolf's younger brother, Archduke Albert of Austria. A later portrait, by the artist Peter Paul Rubens, shows the couple posing side-by-side, for, despite the somewhat inauspicious start to their married life, they remained devoted to one another until Albert's death in 1621.
Back in 1598, a year before the couple wed, the dying Philip II had installed Albert and Isabella as joint rulers of the Netherlands. Saying that they were in charge of this far-flung corner of the vast Spanish Empire was one thing; maintaining control over it proved quite another. To combat the growing threat from Spain, the English and Dutch joined forces to defend the city of Ostend, and Archduke Albert was sent to win it back.
In a gesture of marital devotion and self-sacrifice, his loving wife Isabella vowed not to change her undergarments until the siege was over. But unfortunately for her, Albert and especially the beleaguered people of Ostend, there was no quick resolution to this bloody conflict. The siege lasted another three long years, during which more than 100,000 people perished, in what was described by one contemporary as 'a long carnival of death'.ii
After the eventual Spanish victory, in the autumn of 1604, Isabella was finally permitted to remove her underwear. By then, it had turned a rather unpleasant shade of greyish-brown, duly dubbed 'Isabelline', after the defiant monarch.
*
There is only one teensy little problem with this rather repugnant, yet strangely touching, tale: the word 'Isabella' had been used to describe exactly the same colour several years before the siege ever took place. It first surfaces in 1600, in an inventory of royal garments, amongst whose pages we find 'one rounde gowne of Isabella-colour satten... set with silver spangles'.1
Confronted with this troublesome fact, proponents of the legend quickly switch their attention to another famous queen: Isabella of Castile, who reigned jointly with her husband Ferdinand of Aragon during the latter years of the fifteenth century. Conveniently, Ferdinand also took part in a siege – of the Moorish stronghold of Granada. Although this conflict lasted less than just nine months, from April 1491 to January 1492, there would still have been plenty of time for his spouse's underwear to turn that rather dingy shade of greyish-brown.
I hate to contradict such beguiling tales, but both these stories may have arisen as a convenient way to explain an etymological coincidence. It appears that 'Isabelline' is more likely to be a corruption of the Italian word _zibellino_. This name was given to a pelt of an animal such as a marten or sable, worn as a fashion accessory by wealthy women during the sixteenth century. It may originally derive from an Arabic word meaning 'lion' – and therefore mean 'lion-coloured'.iii
The theory goes that when this term was first heard in England, it somehow became associated with Queen Isabella, and the word changed as a result. I must say, though, I still prefer the royal underwear theory.
## _2: Folk and Fowls_
Whatever the truth about its origins, the use of the name 'Isabelline' marks the beginning of an era that saw an explosion in the number of species given English names.
According to James Fisher, from 1460 to 1776 the 'British List' doubled from roughly 114 species at the start of this period, to 228 species by the end. Given that about 250 or so species of breeding, migrant and wintering birds are found in Britain today, we can see that by the late eighteenth century the vast majority of common birds had been given English names – or, to put it more precisely, _official_ English names. Most familiar species already had a wealth of folk names: some widely used throughout Britain, others confined to particular localities or regions.
Folk names are just that – everyday names coined by everyday folk. Many, though quaint and wonderfully descriptive, have long since vanished through disuse. Others, though – including robin, dunnock and wren – ultimately became the accepted name for the species.
*
As we have already seen, colour and shade feature in many early bird names, often dating back to well before the Norman Conquest. But there's a problem with naming birds after an aspect of their appearance. And that is that, unlike sounds, colours are not unique to a single species. So it's not surprising that the same name was often used to describe several different, and often unrelated, species.
Take the name 'blackcap'. Today this refers solely to the greyish-brown warbler _Sylvia atricapilla_ , whose males sport a prominent black crown. But over time the name has also been used for other species of bird with black on their heads, including marsh, coal and great tits, reed bunting, and even black-headed gull. One name for several species.
The opposite was also true. The proliferation of folk names up and down the country meant that a single species was often called completely different things in different places, a recipe for muddle and misunderstanding.
So in parts of Scotland and northern England the yellowhammer (which, as we have already seen, might be called the 'golden amber', 'gladdie' or 'go-laddie') was known as the 'yoldring'.iv Even more confusingly, in other parts of Britain it was called the 'goldfinch', the name normally given to the small, tuneful member of the finch family with a red face and yellow flashes on its wings.
In turn, the goldfinch also had a wealth of alternative names (as you might expect for such a common and colourful bird). These included 'red cap', from its crimson face patch; 'thistle finch', an allusion to its diet of thistle seeds; and the sartorially inventive 'proud tailor', referring to its smart plumage. More obscure names included 'sheriff's man', after the black and gold livery worn by sheriffs, and 'King Harry'. As Lockwood suggested, the latter was probably a cheeky reference to King Henry VIII: 'that monarch's ostentatious attire being linked with the ornate plumage of the bird'.2
Sadly, the vast majority of these alternative folk names are now long forgotten. However, despite the fact that modern English is becoming more homogenous, due in part to the influence of television and the Internet, a handful of folk names have managed to survive to the present-day. Across much of Britain people still call lapwings 'peewits' (or in Lincolnshire, 'pyewipes'), after their plaintive, penetrating call; while the green woodpecker is still occasionally known as the 'yaffle', in reference to the laughing sound it makes as it flies away from you.v
But the drive towards coining official names would inevitably lead to a simplification in the plethora of different folk names for the same species, as well as a reduction in the potentially confusing use of the same name for different species. This coincided with a much deeper understanding of Britain's birds, our knowledge of which would increase dramatically during the period in question, from 1500 to 1750.
*
The impulse behind the giving of standard names to birds was the great outpouring of learning taking place at this time: from the sixteenth-century Renaissance, through the seventeenth-century Age of Reason, to the eighteenth-century Age of Enlightenment, during which, as the historian Roy Porter has noted, 'the key... concept was Nature'.3
This period also saw great advances in the way we perceived, catalogued and categorised what Shakespeare called 'nature's mysteries', with the seismic shift from what historian Keith Thomas termed 'the cruelty of indifference' to a far more modern and enlightened approach to the natural world.4
Until this time, mankind's dominion over nature was more or less absolute. Indeed, as this verse from the _Book of Genesis_ shows, it had been sanctioned at the highest level of authority, the Old Testament: 'And God blessed them, and God said unto them, be fruitful, and multiply, and replenish the earth, and subdue it: and have dominion over the fish of the sea, and over the fowl of the air, and over every living thing that moveth upon the earth.'
We humans have never quite managed to shake off the notion that we can and should exploit, without compunction or consequence, the Earth's natural resources. But during this period from 1500 to 1750 we did at least begin to move away from a primarily exploitative relationship between human beings and wild animals, and towards a more inclusive, benevolent approach – what the environmental historian Dr Rob Lambert calls the shift 'from use to delight'.
One consequence of this change in attitudes was that we began to appreciate the natural world for its own sake, rather than simply as a resource for us to exploit. Very gradually – and through intermediate stages such as the Victorian mania for shooting, collecting and stuffing specimens of wild birds – this would ultimately lead to the rise of popular, nature-based hobbies including, of course, birdwatching.5
For the very first time in history, we were finally beginning to regard other creatures with a combination of empathy, understanding and curiosity – not all that different from the outlook millions of us have towards nature today. Birds, being more ubiquitous and visible than any other creatures, were at the forefront of this new approach. And this led to a slew of new, and now very familiar, names.
## _3: Pioneers and Puffins_
The movement towards standardising the official names of birds (along with many other species of animals and plants) was begun by a cohort of people who, for the first time, called themselves 'naturalists'.vi Amongst these pioneering men (and a few exceptional women like the seventeenth-century lepidopterist Eleanor Glanville) was the herbalist, natural historian and theologian William Turner (1509/10-68).
For an academic scholar and scientist, Turner led a turbulent and often dangerous life. He held controversial, reformist views, at a time when every change to the holder of the English throne dictated the prevailing religion. Thus in 1538, towards the end of the reign of Henry VIII, he was imprisoned for preaching without a licence. When he was released he left England to travel around Europe, where his radical opinions found a more receptive audience. After Henry's death in 1547, and the accession of the more sympathetic King Edward VI, Turner returned home to become Dean of Wells in Somerset.
But unfortunately for both Turner and his brand of radical Protestantism, Edward's reign was soon cut short by chronic ill health. When the young king died in 1553, aged just 15, his Catholic half-sister Mary succeeded him. Turner's life was now in mortal danger, and he fled abroad once again. He escaped just in time, narrowly avoiding the dreadful fate of his contemporaries Nicholas Ridley and Hugh Latimer, both of whom were burned at the stake by Queen Mary for heresy.
After Mary's half-sister Elizabeth acceded to the throne in 1558, and the Catholic faith gave way to Protestant beliefs for the final time in this unsettled period in our history, Turner could at last return permanently to England.
I include these biographical details as we cannot separate the religious crusader from the pioneering naturalist; indeed, in his fiery personality the two vocations fused, as the editor and translator of his ornithological writings noted:
'It must be understood that, his scientific work apart, nearly the whole of Turner's life was spent in religious controversy...'6
Perhaps it was this love of argument and desire to go against the grain that allowed him to make such advances in our ornithological knowledge. For Turner was in every sense a true 'Renaissance Man', in the modern meaning of the phrase: both as a polymath, straddling different fields of learning, and also in his constant striving to advance the boundaries of human knowledge about the world – in his case, the natural world.
William Turner certainly did that, taking a particular interest in the classification and naming of birds. _Avium Praecipuarum_ , written in Latin and published in 1544, was the first printed book entirely devoted to ornithology. As its full title suggests,vii he relied heavily on the classical works of Pliny and Aristotle, who had been the last people to attempt to classify birds, between 1,500 and 1,900 years earlier. Incredibly, in all that time virtually no progress had been made in our ornithological knowledge.
Turner set out to systematically record the names of every known bird in Greek, Latin, English and German – a daunting task, given that hardly any prior research had been done on the naming of species. He obtained much of his information by talking to bird-catchers and wildfowlers, for whom a working knowledge of the range of names used for different kinds of birds was essential.
James Fisher notes that Turner listed more than a hundred different kinds of British bird, of which no fewer than fifteen were named for the first time. These included the hen harrier, woodlark and brambling – the latter with the unusual spelling of 'bramlyng'. According to Lockwood, this is a corruption of 'brandling', and refers to the brindled plumage of this handsome finch, rather than having anything to do with the blackberry-producing plant. If so, then the modern name 'brambling' is yet another of those 'faux amis': unwary linguistic traps set to confuse us.
Turner also first coined the name 'creeper' for the species we today call the treecreeper.viii This minuscule bird, often seen moving jerkily around a gnarled tree trunk as it searches for tiny insects, already had a wide range of folk names, including 'tree climber', 'tree clipper', 'tree speeler' (from a Scottish word meaning climber), 'bark creeper' and 'bark runner'. The most evocative example, perhaps, was the Somerset name 'tree mouse'. But a word of caution: the suffix 'mouse', as used in the old name for tits, 'titmouse', is thought to derive from the Old English word 'mose', simply meaning 'small creature', so does not necessarily refer to the treecreeper's rather rodent-like habits.
Turner also either found or coined the name 'nut jobber' for the nuthatch, deriving from the now obsolete word 'job', meaning to peck or jab at something. The modern suffix, 'hatch', derives from the word 'hack', from the bird's habit of jamming a nut into a crevice in the bark of a tree, and then pecking at it to break off morsels to eat.
Turner was not simply naming the birds; he was also trying to distinguish genuine accounts of their behaviour from purely spurious and inaccurate ones, many of which had gained currency through continued repetition down the centuries. Given how little was known about Britain's birdlife at this time, it is remarkable how accurate many of Turner's observations were. He wrote that 'Cranes... breed in England in marshy places', and noted that bitterns 'can easily be driven into nets by the use of a stalking horse'.
Not surprisingly, Turner did repeat several long-held errors and misinterpretations. He confused the osprey with the sea eagle, repeated the hoary old myth that barnacle geese hatch from goose barnacles, and claimed that the heron 'screams while it couples and (they say) emits blood from its eyes'. He also thought – along, to be fair, with many others – that male and female hen harriers were separate species, a confusion not cleared up until more than 250 years later, by George Montagu (see Chapter 4).
But given the virtually blank slate from which he was working, William Turner did remarkably well to begin the long process of unravelling the complexities of the multiple names given to Britain's birds. He died peacefully in London in 1568, secure in the knowledge that not only had his once radical religious beliefs now entered the mainstream of English society, but he had also made substantial advances in our knowledge and understanding of English bird names.
*
During the following hundred years or so, progress in the naming of birds almost ground to a halt – perhaps because Englishmen and women had other things on their minds during these turbulent times. First, there was the transition from the last of the Tudors, Elizabeth, to the Stuart kingship of James I and his unfortunate son Charles I. Then came the period leading up to and after the English Civil War during the 1640s and 1650s, during which one king was beheaded and another deposed, and Britain was briefly run as a republic by Lord Protector Oliver Cromwell. As the title of a book by historian Christopher Hill puts it, this really was a time when 'the world turned upside down'.
Not until the decade following the Restoration of the English monarchy, under Charles II in 1660, was any real progress made towards giving official, widely accepted names to the remainder of Britain's birds. The impetus for this came from two men from very different social classes, who nevertheless shared a passion for all things ornithological: John Ray and Francis Willughby.
John Ray (1627–1705) came from humble beginnings. He was a village blacksmith's son from Essex, who has been described as 'solitary, modest, principled, persistent... The last of the heroes whose work gradually shifted the study of plants away from superstition and towards science'.ix
Ray's life changed dramatically at the age of sixteen, when he won a bursary to study at Cambridge. He remained at Cambridge as a fellow, and later taught Francis Willughby (1635–72), a handsome young country gentleman whose family seat, Middleton Hall in Warwickshire, is now the site of an RSPB nature reserve. Despite their very different backgrounds, the two men soon discovered that they shared a mutual interest – the new science of ornithology – following which they became firm friends and trusted colleagues.
After touring Britain and Europe to watch and study wildlife, they returned home with ambitious plans to publish a masterwork summarising their observations and conclusions. But in 1672, at the age of just thirty-six, Willughby caught pleurisy and died. Grief-stricken at his friend's passing, Ray wrote that Willughby's death was 'to the infinite and unspeakable loss and grief of myself, his friends, and all good men'. In tribute, he pledged to publish their joint work posthumously, and soon afterwards did so, first in Latin (1676) and then in English (1678).
Willughby's _Ornithology_ , as it is usually called,x was the first volume devoted entirely to birds to be written in English. As a result, it had an enormous influence on later ornithologists and writers. Ray's modesty in ascribing authorship solely to his late friend is typical of the man hailed by James Fisher as 'the greatest of all field naturalists'.
Some bird names appear for the very first time in Willughby's _Ornithology_ , yet we know the species themselves were recorded far earlier. One example is the name crossbill, whose name refers to the way the upper mandible of the beak crosses over the lower one.
Crossbills are neither exclusively resident nor migratory, but nomadic in their habits, wandering long distances after the end of the breeding season in search of the conifers on whose cones they feed. In some years, large flocks of crossbills arrive in Britain from continental Europe, turning up in locations where they have not previously been seen. When they do, they can be easily identified, thanks to that unique physiological feature commemorated in their name.
So although the name 'crossbill' does not appear in written form until Willughby's _Ornithology_ , we know that they occurred in Britain more than 400 years earlier – all because of a sharp-eyed Benedictine monk and historian named Matthew Paris.
Sometime in the autumn of 1251, Paris observed a flock of unfamiliar birds feeding in the grounds of his monastery at St Albans in Hertfordshire: 'About the fruit season there appeared, in the orchards chiefly, some remarkable birds, which had never been seen in England, somewhat larger than larks...' This could have referred to any number of fruit-eating species, including redwings and fieldfares; but the clincher to the species' identity comes in the next line: 'The beaks of these birds were crossed, so that by this means they opened the fruit as if with pincers or a knife.'
This observation is accompanied by a line drawing in the book's margin, which clearly shows a crossbill feeding on a large seed, using the specialised bill unique to this group of birds. This is also the first known reference to its unusual migratory habits, and reminds us that long before the great explosion in learning from the sixteenth to the eighteenth centuries, led by men like Ray and Willughby, perceptive individual observers like Matthew Paris had been gradually adding to our accumulated knowledge of Britain's birds.
Ray and Willughby finally brought to an end the obsession with classical observers such as Aristotle and Pliny who, although they made some remarkable observations for their time, had also made serious mistakes. Aristotle, for instance, had believed that the redstart turned into the robin during the winter months, and that swallows hibernated under water (a belief that endured surprisingly late, well into the eighteenth century, and was even given credence by Gilbert White).
By discarding many of the classical world's false assumptions, which had so hindered the progress of modern ornithology, Ray and Willughby opened the door to a new, more rigorous and scientific, approach to bird study.
One of their simplest yet most revolutionary methods was to classify birds first according to habitat (Land and Water Birds), and then further subdivide these into smaller categories, such as birds that swim, and those that wade; or birds with hooked bills, and those without. In so doing, Ray and Willughby managed to produce a classification that, with a few exceptions, looks remarkably like the one we still use today.
They also sorted out various areas of confusion. As we have seen, until this time, the same name might be used for two completely different and unrelated species. One such was the word 'shoveler'. We might reasonably assume that any reference to a shoveler (or its variants, 'shoveller' and 'shovelard') would refer to the familiar and colourful duck of that name, which uses its specialised, shovel-shaped beak to filter tiny items of food from the surface of the water. Yet until the late seventeenth century the name applied to a very different species: the bird we now call a spoonbill. As its name suggests, the spoonbill also has a peculiarly spatula-shaped bill, which like the shoveler it uses when feeding.
This long-legged waterbird was once common in England, found in the vast, soggy wetland known as the Great Fen that covered much of East Anglia. But some time during this period, as the fens were drained for farming and settlement, and the birds were hunted for food, the spoonbill disappeared as a British breeding species.
As late as 1796, one observer, Captain J. G. Stedman, could still write that 'the shoveler, or spoon-bill... is about the size of a goose'. But by then, as the species became less and less familiar, the name shoveler had been transferred to the large, colourful duck with a similarly shaped bill. The first reference to this comes from John Ray, in 1674. Four years later, in Willughby's _Ornithology_ , Ray confirmed the new name spoonbill (from an older term, 'spoon-billed heron'), with the explanation: 'The Bill... is of the likeness of a Spoon...' – which indeed it is.
The confusion between the names spoonbill and shoveler is far from the only one to emerge from this key period in our history, when the names given to different species were in constant flux. Contestants in pub quizzes are easily misled by the answer to the apparently simple question: which British seabird has the scientific name _Puffinus puffinus_? To most people's surprise, it is not the puffin, but the Manx shearwater.
The origin of the name 'puffin' is a mystery; it has been suggested that it might derive from a Cornish word (perhaps via another Celtic language, Breton), but this cannot be confirmed. What we do know is that it originally referred to the young Manx shearwater.
Like puffins, shearwaters nest in underground burrows on remote offshore islands such as Lundy and Skomer.xi For centuries, sailors would stop off at their breeding colonies to harvest the plump chicks, whose bodies have a very high fat content. These would then be salted for food to sustain the mariners on their long sea voyages, when fresh meat would be scarce.
Later this was turned into a thriving and profitable trade, no doubt helped by the convenient classification of shearwaters as fish, which could therefore be eaten during the period of Lent, when meat was forbidden. As Thomas Moffett observed in the late sixteenth century, this custom was sanctioned at the highest level of the Catholic Church: 'Puffins, whom I may call the feathered fishes, are accounted even by the holy fatherhood of Cardinals to be no flesh but rather fish.'7
At this time the bird we now call the puffin was known as the 'sea parrot', because of its large, colourful beak. But some time during the seventeenth or eighteenth centuries, confusion arose between the two species. This presumably occurred because both nest in burrows, and the young superficially resemble one another, being grey, plump and rather fluffy (and in the case of the puffin chick, lacking the distinctive bill it develops in adulthood). So the common name transferred from one species to the other; yet the Manx shearwater retains the scientific name _Puffinus puffinus_ to this day.
The puffin's generic name, _Fratercula_ , comes from the Latin word meaning 'friar', which one commentator has suggested arose 'perhaps as a reference to the bird's habit, when rising from the sea, of clasping its feet as though in prayer'.8 The name was coined by the Swiss ornithologist Conrad Gessner. Writing to his English friend John Caius,xii he joked that 'If you imagine that this bird was white, and that you then put on a black cloak with a cowl, you could give this bird the name of "little friar of the sea" ( _Fratercula arctica_ ).' The name stuck.
The origin of the name of another seabird, the storm petrel, has given rise to another confusing myth. W. B. Lockwood noted that when feeding, these tiny birds (barely larger than a house martin) tap the water with their feet as they fly low over the surface of the sea. He suggested that this 'pitter-patter' action led to the name, though the _OED_ again demurs, suggesting that although the origins are now long lost, it may come from the sounds these bird make while mating, or even their smell.
The explorer and naturalist William Dampier, writing in 1703, went a step further, forging an entirely spurious link with the New Testament account of St Peter miraculously walking on the waves: 'As they fly... they pat the Water alternately with their Feet, as if they walkt upon it; tho' still upon the Wing. And from hence the Seamen give them the name of Petrels, in allusion to St Peter's walking upon the Lake of Gennesareth.'9
This story has taken root in many other European cultures, as can be seen from the folk names _Petersvogel_ (German), _Søren-Peder_ (Norwegian), and _ave de San Pedro_ (Spanish), all of which assume the same entirely spurious Biblical connection.
But perhaps the most intriguing mystery surrounds the name 'scoter'. Scoters are a group of sea ducks found across the higher latitudes of the northern hemisphere. All have one thing in common: they are predominantly black in colour. It is this characteristic that might explain the origin of their peculiar name, which like so many others was first noted by John Ray in the late seventeenth century, when he referred to 'the black Diver or Scoter: _Anas niger_ '.
The theory goes like this. In other European languages, and in the English folk name 'black diver', the scoter is named after its appearance. Thus in German, _Russente_ translates as 'soot duck', while the Dutch _zwarte zee-eend_ means 'black sea duck'. It is hardly far-fetched to suggest that this species was originally known in English as a 'sooter', and that sometime during this period the name was mistranscribed as scoter. Sadly we shall never know if this is correct: the word 'sooter' used in reference to the bird's name has never been found.
In his book _Lapwings, Loons and Lousy Jacks_10 Ray Reedman offers an alternative explanation: that 'scoter' may derive from the phrase 'sea-coot'. This name has in the past been used for several birds with a predominantly black plumage, including the cormorant, guillemot, scoter and American coot, but its supposed link with the name scoter is pure conjecture.
*
Ray and Willughby were not only well travelled, but well-read too, so were keenly aware of the work of earlier writers. These included the sixteenth-century Swiss ornithologist Conrad Gessner, brilliantly described by Anna Pavord as 'a one-man search engine, a sixteenth-century Google'.11
One of Ray's rare errors came about through a mistranslation of Gessner's name for the species we know today as the waxwing. Gessner had named this bird the Bohemian jay, because of its superficial resemblance to that species, and also its irregular autumnal wanderings south and west from its Scandinavian breeding grounds in search of the berries on which it feeds. Unfortunately, Ray mistranslated 'Bohemian jay' as 'Bohemian chatterer', which, as Lockwood notes, 'is especially unfortunate seeing that the waxwing is a very silent bird' (though it does sometimes call in flight, making a tinkling call rather like a 1980s Trimphone).
A now obsolete name for the waxwing, the 'silk-tail', refers to the bird's smooth, silky plumage, and also finds an echo in its current scientific name, _Bombycilla garrulus_ – which translates as 'noisy silk-tail' (another confusing reference to the bird's sound). The modern name 'waxwing' was not coined until 1817, which strikes me as surprisingly late in the day, given that those strange red markings on the bird's wings really do look like the wax once used to seal up envelopes.
Another misleading name, this time chosen by Willughby, for one of our most enigmatic birds of prey, is honey buzzard. He chose the name when he discovered combs of wasps in the bird's nest, and a century later Thomas Pennant adopted it as the official English name for the species.
But both Willughby and Pennant were mistaken: honey buzzards do not feed on the sweet and sticky honey; what they are actually after are the juicy grubs hidden away in the cells of the comb. The species' scientific name – _apivorus_ , meaning 'bee-eating' – is much closer to the mark, as is the Dutch name _wespendief_ (meaning 'wasp thief'), a reference to the way the bird steals the comb away from those pesky insects, in order to consume the grubs at its leisure.xiii
Overall, though, despite these few errors, Ray and Willughby set the course for the major advances in the standardisation of bird names, which would continue apace in the coming century.
## _4: A Little Latin_
Meanwhile, a second linguistic revolution was occurring – not in English names, but in Latin ones.xiv And while this may at first appear to be an arcane and scholarly by-way in our story of the origin of English bird names, nothing could be further from the truth.
For like other great sea-changes in history – from the invention of printing to the advent of the Internet – the system known by the tongue-twisting phrase 'binomial nomenclature' changed the world, by allowing knowledge and understanding of living things to progress internationally without being held back by linguistic complexity and confusion, as had been the case until then.
This approach, which would eventually create the system of classification still used by scientists around the world today, was pioneered by a Swedish scientist widely known by the Latinised version of his name, Carl Linnaeus.
Linnaeus has been memorably described as 'Sweden's most important contribution to world culture until Abba'.12 Born in a small southern Swedish town in 1707, he came from relatively humble beginnings: his father was a Lutheran pastor of peasant stock. But he went on to become one of the most famous men of his time, acclaimed as the 'father of modern taxonomy'. The Swiss-born French philosopher Jean-Jacques Rousseau put it more simply: 'tell him I know of no greater man on Earth'.
Later scientists and writers – along with amateur naturalists and birders – also owe Linnaeus an enormous debt of gratitude. As one saying goes: 'God created the world, Linnaeus put it in order.'xv
So how did Linnaeus achieve such universal admiration and praise? He did so by simplifying the existing, and ludicrously cumbersome, method of naming species, and introducing a system we still use today to classify every single one of the world's millions of living organisms.
Before Linnaeus, plants and animals, including birds, were usually classified using an ornate and increasingly unwieldy system, which formed the Latin name from complex descriptions, sometimes many words long. Thus the common plant hoary plantain, whose current Latin name is _Plantago media_ , was lumbered with the ludicrous epithet _Plantago foliis ovato-lanceolatus pubescentibus, spica cylindrica, scapo tereti_ (which roughly translates as 'plantain with pubescent ovate-lanceolated leaves, a cylindrical spike and a terete scape').
Nor did birds escape this fate: the shoveler, now _Anas clypeata_ , laboured under the seven-word phrase _Anas platyrhynchos altera sive clypeata Germanis dicta_ , which the ornithologist Professor Tim Birkhead translates as 'another duck with a broad bill, or, according to the Germans, with a shield-like gorget'.13 Not only was this far too complicated for general use, it also confuses the modern reader by its reference to _Anas platyrhynchos_ – the Latin name for the mallard.
At a single stroke, Linnaeus did away with such unnecessary complication. With the publication of his masterwork _Systema Naturae_ , produced during the middle decades of the eighteenth century, he set in motion the modern science of taxonomy. At its heart was a new method of classification: the concept of binomial nomenclature.
As the phrase suggests, binomial nomenclature uses two words to do the job that had previously required whole phrases. The first name of the two is the genus (or grouping) to which the organism belongs; the second the specific (or species) name. Taken together, the binomial distinguishes that species from any other. Thus we have _Passer domesticus_ (house sparrow), _Troglodytes troglodytes_ (wren) and _Turdus merula_ (blackbird).xvi
Before I go on, however, one important misconception should be laid to rest. Linnaeus did not actually invent binomials. That honour goes to the Swiss botanist Caspar Bauhar, who lived and worked more than a century earlier, and who began pruning the over-ornate compound names into those with just two elements. However, Bauhar did not seek to adopt this as a universal system; Linnaeus did. Linnaeus's genius was to recognise that, if widely used, binomial nomenclature would revolutionise the study and classification of living things forever, as Anna Pavord points out: 'The binomial system worked... because it effectively mirrored the way that common names had evolved. Hoary plantain is, in effect, a binomial tag... In the English language the describing word comes before the generic one. In Latin it's the other way around.14
Linnaeus's simple but ingenious approach transformed the infant science of taxonomy, allowing all the world's organisms to be neatly classified in relationship to one another, and removing the room for error and confusion caused by over-complicated compound names.
Binomials are not simply used in academic or scientific circles. Even today birders, especially if they are amongst a multilingual group from several different countries, will often refer to a bird by its scientific name, in order to make it clear which particular species they are talking about.
I can still recall my first visit to Spain in the mid-1980s, when I was lucky enough to go to the Coto Doñana with the late Tono Valverde, the man who had done so much to save this extraordinary wetland, one of Europe's last great wildernesses. We travelled south from Seville in his dilapidated car on a warm spring day, finally reaching the edge of the vast reserve in the late afternoon. Birds were simply everywhere: herons and egrets, geese and flamingos, and many, many more. My Spanish was poor, and his English worse, but we soon found a workable means of communication by using binomials. Our conversation that day largely consisted of phrases such as ' _Gelochelidon nilotica_ ', ' _Glareola pratincola_ ' and ' _Sturnus unicolor_ ', allowing us to communicate easily through this universal language.xvii
*
Even today, scientific names continue to play a crucial role with regard to changing English bird names. At a time when many species still labour under a range of different local names, depending on where you are in the world – such as 'diver' and 'loon', 'skua' and jaeger', and 'bunting' and 'longspur', in Britain and North America respectively – scientific names help provide stability and continuity.
Or at least they did. However, now that the classification of all the world's birds is undergoing a major revolution, thanks to advances in our understanding of DNA, there has been a drive to change scientific names to reflect these new relationships; a move laudable from a scientific point of view, but likely to cause great confusion in the future (see Chapter 7).xviii
Notwithstanding this current complication, scientific ornithologists and ordinary birders the world over owe a massive debt to Linnaeus, for dragging the classification of birds and other living things into the modern age. As John Wright notes: 'It [binomial nomenclature] was by no means perfect... but it was good enough for the moment and, more importantly, became accepted by nearly everyone.'15
One other important consequence arose as a result of the new Linnaean system of classification: several scientific names were translated more or less directly from Latin into English, and became the standard name for the species. These included oriole, from _oriolus_ , which Lockwood suggests derives from the golden oriole's tuneful, whistling call, but may also be a nod to _aureolus_ , meaning 'golden-coloured'.
Another Latin-based name is phalarope. Two of the world's three species of phalaropes occur regularly in Britain: the red-necked, which breeds in Shetland and the Western Isles; and the grey, an autumn passage migrant. Their name derives (via French) from the Latin _phalaropus_ , which means 'coot-footed'. This refers to the lobes on phalaropes' toes that enable them to swim, and which resemble those on the feet of coots and moorhens.
As Lockwood points out, because phalaropes are so scarce and localised in most parts of Britain, neither species ever acquired an English folk name. But there is one notable exception. On Shetland, where it breeds, the red-necked phalarope is known in the local dialect as the 'peerie deuk', meaning 'little duck', from its tiny size and habit of swimming on shallow lochans when feeding. Intriguingly, the _Scottish National Dictionary_ defines the noun 'peerie' as 'a child's spinning-top', and this certainly fits the frantic feeding action of these tiny waders, which revolve like wound-up clockwork toys as they stir up tiny aquatic invertebrates.
*
On 10 January 1778, Carl Linnaeus, the man who began this revolution in the world of science and naming, died at his home in Hammarby, near Stockholm. He was seventy years old.
Linnaeus's final years had been blighted by illness, yet his scientific discoveries had also brought him fame and fortune. In 1761 he was ennobled by King Adolf Frederick (as Carl von Linné), creating a coat of arms divided into three, featuring what he considered to be the three kingdoms of nature – animal, vegetable and mineral – and after his death he was buried with great honour in Uppsala Cathedral. His continuing legacy, though, is the 4,400 species of animals and 7,700 species of plants to which he gave scientific names; including, of course, many species of birds.
## _5: A Correspondence Course_
In 1789, just four years before his death, Gilbert White, a country vicar in a rural parish in Hampshire, published _The Natural History of Selborne_ ,xxix which eventually became one of the best-selling books of all time.xx
Yet despite his enduring fame, when it comes to the story of the naming of our birds, White is something of a footnote. His correspondent Thomas Pennant, to whom many of the 'letters' in _Selborne_ are addressed, was far more important and influential in this regard.
Whether or not White actually sent his letters to Pennant (and his other correspondent, Daines Barrington), or simply used them as a literary device to impart information to the reader, is not especially relevant. What is significant – especially for our story – is that at the time the book was published, Pennant was far better known than White. He was one of the leading scientists of his day, and influenced, amongst others, that great man of letters Dr Samuel Johnson. So when he coined new names, or popularised existing ones, they were likely to be widely adopted and used by others.
Born in 1726, Thomas Pennant lived his whole life at the family estate in Flintshire, in his native Wales. Here, as a twelve-year-old boy, he was given a copy of Willughby's _Ornithology_ , an event to which he later ascribed his lifelong love of natural history. Pennant may have been true to his Welsh roots, but during a long and busy life he also travelled extensively around the British Isles, writing detailed notes on the plants, animals and landscapes he encountered.
His findings appeared in a series of highly influential books. The best known of these, _British Zoology_ , was published in several editions from the 1760s onwards, and soon became the definitive zoological work of its time.xxi This combination of scholarly rigour and wide readership meant that Pennant performed a crucial role in both developing new names and authorising existing ones. Indeed, he was so successful that we continue to use many of his chosen names today.
When it came to classifying birds, Pennant's approach broadly involved taking Linnaeus's classification of a group of related species into a genus or family, and then giving each member an English name within that grouping, to make these relationships more apparent. This was very helpful, as names such as 'sparrow' and 'wren' had until then been used interchangeably for several species from very different families.
Previously, for example, the reed bunting had often been called 'reed sparrow', which confusingly was also used for other small birds sharing the same habitat, including reed and sedge warblers. Pennant decided on the name reed bunting, along with 'common bunting' (later renamed corn bunting), and 'yellow bunting', the logical term for the yellowhammer, which nevertheless failed to catch on (as we saw in Chapter 2). The name of our hardiest species, the snow bunting, also emerged at about the same time.
The name 'bunting', which initially referred only to the corn bunting, goes back to the fourteenth century, and as a surname (meaning 'plump or thick-set person') is recorded as early as 1275. That original meaning also survives in the nursery rhyme 'Bye baby bunting', which first appears in print in 1784, but whose origin is almost certainly far older. So Pennant's role was more about organisation than innovation: it had long been known that these species were members of the bunting family but, as with so many other familiar species, their vernacular names had arisen by a series of accidents. The tidy-minded Pennant was not the first person, and will probably not be the last, to try to render bird names more logical.
During his career, he developed a number of compound names, combining a colour or shade with a part of the bird's body. These included white-fronted goose ('front' derives from the French for forehead, and refers to the white patch around the bird's bill), black-throated and red-throated divers, red-necked grebe, red-breasted merganser and red-backed shrike. And he either coined or popularised several other names derived from key plumage features, such as spotted flycatcher, ringed plover, long-tailed titmouse (later simplified to 'tit'), and long-eared and short-eared owls.
Pennant also adopted 'bearded titmouse', a direct translation from the French. The bearded tit, as we now call it, is an attractive bird found almost exclusively in reed beds, with a butterscotch and blue-grey plumage, long tail, and distinctive black markings on the sides of the male's bill. Yet these are more reminiscent of the fictional Chinese villain Fu Manchu's drooping moustaches than of any kind of beard. 'Tit' is also misleading: the species is entirely unrelated to the true tits, and indeed has now been placed in its very own family, Panuridae.
Avocet, bean goose, little egret, eider, linnet, night heron, oystercatcher, pochard, ruff, sanderling, tawny owl and wood sandpiper are just some of the many other species for which Pennant either invented the current name, or chose it from the various ones already in use.
Thanks to his influence, these are the names we still use today, even though perfectly acceptable alternatives (for example 'brown owl' instead of tawny owl, 'sea-pie' for oystercatcher) were available at the time. Pennant's choice of names did not always prevail: he preferred water ouzel instead of dipper, eared grebe (still used in North America) for black-necked grebe, golden-crested wren for goldcrest, goatsucker for nightjar and land rail for corncrake.xxii And in the early editions of _British Zoology_ he called the stone curlew the 'Norfolk plover', but later wrongly decided that this curious wader must be from a different family, and so gave it the wonderfully evocative name 'thick-kneed bustard', which also failed to catch on.xxiii But by and large, when Pennant named a bird, that name prevailed.
Popular and widely respected – he was described by one observer as an 'elegant scholar and refined gentleman' – Pennant died at home in 1798, aged seventy-two. In the centuries since, his fame has, as one early twentieth-century writer put it, 'suffered somewhat by the lapse of time'.16 But although he may no longer be a household name, like his friend and contemporary Gilbert White, Thomas Pennant's influence on the development of English bird names remains unmatched.
*
And what of Gilbert White? Perhaps I have been a little harsh on him; after all, he did famously sort out the confusion between three species of summer visitor to our shores: the chiffchaff, willow warbler and wood warbler.
To the modern birder, the very notion that these three superficially similar little birds might be impossible to distinguish from one another seems absurd. But we are forgetting two things: first, that during the late eighteenth century our knowledge of birds was both limited and piecemeal, as very few people took any interest at all in the natural world; and second, that the kind of sophisticated optical aids we now take for granted, such as binoculars, telescopes and digital cameras, were simply not available.
So as he wandered the highways and byways of his rural Hampshire parish, Gilbert White had to rely on his ability to observe bird behaviour at a distance, with his naked eyes. When it came to telling small, flighty warblers apart, this was fairly limited in its use and efficiency.
But White had another weapon up his sleeve, or rather, on the sides of his head: his ears. Today birders often use the sound of these birds, rather than their appearance, to tell them apart; White may not have been the first person to do so (surely the distinctive two-note sound of the chiffchaff would have aroused interest long before this?), but importantly he was the first to point out the key differences between the three species.
He set out his findings in Letter XIX of _The Natural History of Selborne_ , written on 17 August 1768, though presumably relying on evidence gained earlier that spring, when all three would have been singing: 'I have now, past dispute, made out three distinct species of the willow-wrens,... which _constantly_ and _invariably_ use distinct notes.' White had also noticed that
The yellowest bird [which we now know to be the wood warbler] is considerably the largest, and has its quill-feathers and secondary feathers tipped with white, which the others have not. This last haunts only the tops of trees in high beechen woods, and makes a sibilous grasshopper-like noise, now and then, at short intervals, shivering a little with its wings when it sings.17
As a description of the wood warbler's distinctive song, this could hardly be bettered. But if you are thinking of praising the good parson for his acute observations of the bird's plumage, you may have second thoughts when I tell you that he did have the advantage of having examined dead specimens, which he had presumably asked some local marksman to shoot – or perhaps even killed himself.
The other two species, both smaller and less distinctive than their scarcer relative, are the chiffchaff and willow warbler. Again, they are only superficially similar: as a long-distance migrant to and from southern Africa, the willow warbler needs longer wings, which give it a more elegant appearance. The marginally smaller, shorter-winged and more olive-plumaged chiffchaff is a short-distance migrant, with most of our breeding birds wintering in Spain, Portugal or North Africa.
For most birders, by far the easiest way to identify the chiffchaff is by the distinctive song that gives the species its onomatopoeic name, as White himself noted: 'The smallest uncrested willow wren, or chiffchaff... utters two sharp piercing notes, so loud in hollow woods, as to occasion an echo, and is usually first heard about the 20th of March.'
White does not appear to have got to grips with the third species – the one we now call the willow warbler – until some time after he identified the other two. At first this may seem rather odd, for the willow warbler is by far the commonest summer visitor to Britain, with well over two million pairs breeding here.xxiv Yet although its silvery song, descending the scale like water running down a slope, is very distinctive, the bird's habit of avoiding parks and gardens means it is far less well-known than its cousin the chiffchaff.
Incidentally, the name 'warbler', with which we are so familiar today, does not appear in print until – you've guessed it – the 1773 edition of Thomas Pennant's _British Zoology_. Although the new name eventually gained the upper hand over earlier epithets, the name wood-wren continued to be used for the wood warbler, and was stubbornly resistant to change. It was still preferred by William MacGillivray as late as 1839.
Gilbert White has many claims to fame, but two are especially pertinent to this story. First, there is his contribution to the pastime millions of us enjoy today. James Fisher called him 'the man who started us all birdwatching',18 and for me, this sums up his crucial contribution to the modern world. Before White, people had 'watched birds' so they could hunt and kill them for food, to observe their migratory journeys to try to predict the weather and the seasons, or as objects of superstition, folklore and worship. A few pioneers, such as William Turner, John Ray and Francis Willughby, had begun to carefully observe the habits and behaviour of birds in order to advance the cause of science. But Gilbert White brought a new and different viewpoint: clear, scientific inquiry, of course, but also a pure delight in the way birds are – an attitude that laid the foundations for the way we continue to watch and enjoy birds in the present day.
Gilbert White's other, more minor, distinction is that he is one of that small and select band of Britons who have had a bird named after them. Sadly, though, he never got the chance to see his eponymous species, an Asiatic relative of our own song and mistle thrushes that was named for him posthumously: White's thrush.
White's thrush breeds across a wide swathe of Asia, in the forests of central and eastern Siberia and the Himalayan foothills, and usually spends the winter in India or China. But in autumn, young birds occasionally go astray, heading in exactly the opposite direction from their usual migratory course, in a phenomenon known as 'reverse migration'. Most perish, but a tiny handful make it as far as Europe, which explains how one of these large and distinctive thrushes was shot near Christchurch (now Dorset, but at that time part of Hampshire) in January 1828, twenty-five years after Gilbert White had died.
This bird nearly escaped its fate, as this later account reveals: 'It attracted his attention, on disturbing it, in passing through a plantation, where it appeared to have established a haunt in a high furze brake, as it returned to it repeatedly before he could succeed in shooting it.'19
The eagle-eyed marksman was none other than James Edward Harris, 2nd Earl of Malmesbury. Recognising that the bird was unusual, he sent it to Thomas Eyton, who in his 1836 work _A History of the Rarer British Birds_ named the species 'in memory of one with whom everybody is familiar by name',20 his ornithological hero Gilbert White.
The choice of this species may not be particularly apt – the only connection being that White lived in the county where the bird was shot – but at least he does have a British bird named after him. It is deeply ironic that the name of his contemporary Thomas Pennant, who gave so many birds the names we still use today, can only be found in two now obsolete bird names that never made it to British shores: Pennant's parakeet (the common Australian species now known as crimson rosella), and the long-forgotten scientific name of the king penguin, which used to be known as _Aptenodytes pennanti_ (since changed to _Aptenodytes patagonicus_ ).xxv
But the naming of White's thrush did signal a new and growing trend that had begun in the eighteenth century, and would reach its zenith in the nineteenth: eponymous bird names, those named after a deserving (or occasionally undeserving) human being.
This is the theme I shall be exploring in the next two chapters. These stories – and a fascinating cast of characters – reflect a new era of competition: the race to give names to the last remaining regularly occurring British species that, until then, had remained anonymous and unknown.
#### Notes
1 The rather grandly titled _Inventory of all garments, jewels, silks, etc., in the queen's Garderobe of robes in the charge of Sir Thomas Gorges as surveyed by Sir Thomas, Lord Buckhurste, Lord Treasurer and others appointed by a commission under the great seal 4 July._
2 Lockwood, _op. cit._
3 Roy Porter, _The Enlightenment_ (Basingstoke, 1990).
4 Keith Thomas, _Man and the Natural World: Changing Attitudes in England 1500–1800_ (London, 1983).
5 For a more detailed analysis of this, see my earlier book _A Bird in the Bush: A Social History of Birdwatching_ (London, 2004).
6 In A. H. Evans (ed.), _Turner on Birds_ (Cambridge, 1903).
7 Thomas Moffett, _Health's improvement or, Rules comprizing and discovering the nature, method and manner of preparing all sorts of foods used in this nation_ (London, _c_. 1595).
8 James A. Jobling, _Helm Dictionary of Scientific Bird Names_ (London, 2010).
9 William Dampier, _A voyage to New Holland, etc., in the year 1699_ (London, 1703).
10 Ray Reedman, _Lapwings, Loons and Lousy Jacks_ (Exeter, 2016).
11 Anna Pavord, _The Naming of Names_ (London, 2005).
12 John Wright, op. cit..
13 Tim Birkhead, _The Wisdom of Birds_ (London, 2008).
14 Anna Pavord, op. cit.
15 John Wright, op. cit.
16 W. H. Mullens, 'Some Early British Ornithologists and Their Works', in _British Birds_ (1908–9).
17 Gilbert White, _The Natural History and Antiquities of Selborne_ (London, 1789).
18 In James Fisher, op. cit. See also my book _A Bird in the Bush_ (London, 2004).
19 T. C. Eyton, _A History of the Rarer British Birds_ (London, 1836).
20 ibid.
i Sixteen months later, Elisabeth gave birth to Isabella's younger sister Catherine; less than a year later, after miscarrying another daughter, she was dead. Isabella was just two years old.
ii In the words of the historian Anna Simoni, 'the Spanish assailed the unassailable; the Dutch defended the indefensible'. _Ostend Story_ (2003).
iii We may never know the true origin of a word that does not appear in the _OED_ until as late as 1859, in a description of desert-dwelling birds in the journal _Ibis_ written by the distinguished ornithologist Henry Baker Tristram (see Chapter 5): 'The upper plumage of every bird, whether Lark, Chat, Sylvian, or Sand-grouse... is of one uniform isabelline or sand colour.'
iv Alternative spellings include youldring, yowdring, yoldrin, yaldrin, yaldran, yeld(e)rin and yieldrin.
v It can, of course, be argued that many of the names of birds we use today are simply folk names that won the battle to be recognised as the 'official' name! For what are 'whitethroat' and 'blackcap', 'robin' and 'blackbird', if not folk names?
vi The term 'naturalist' is first recorded in 1581, in a work by the controversial Scottish Catholic John Hamilton. However, the first use of the word in its modern meaning is not until 1600, in Christopher Sutton's popular devotional work _Disce Mori: Learne to Dye_ , in which he writes of 'A lion, of whom the naturalist writeth, that he is of such courage...'
vii _Avium praecipuarum, quarum apud Plinium et Aristotelem mentio est, brevis et succincta historia_ , which roughly translates as: 'A Short and Succinct History of the Principal Birds Noticed by Pliny and Aristotle'.
viii Treecreeper is a relatively recent coinage, first noted in 1814, and not officially adopted by the BOU until as late as 1883. It soon caught on: one popular Victorian nature-writer, George Rooper, extolled 'the pretty lady-like tree-creeper [which] ran like a mouse up the tree.'
ix In Anna Pavord, _The Naming of Names_ (London, 2005). Ray's modesty was underscored by his deep religious beliefs, revealed in the title of his 1691 work, _The Wisdom of God Manifested in the Works of the Creation_.
x The full title is the rather unwieldy, _The ornithology of Francis Willughby of Middleton in the county of Warwick, Esq. In three books: wherein all the birds hitherto known, being reduced into a method sutable_ [sic] _to their natures, are accurately described: the descriptions illustrated by most elegant figures, nearly resembling the live birds, engraven in LXXVIII copper plates._
xi Shearwaters do exactly what their name suggests: glide low over the waves, their stiff wings almost touching the surface of the sea.
xii Physician, medical pioneer and co-founder of my old college, Gonville and Caius, Cambridge.
xiii Several other British birds are named after what they eat, including bee-eater (bees), mistle thrush (mistletoe berries) and linnets (flax seeds – _linum_ in Latin). But sparrowhawks rarely catch sparrows, goshawks don't usually hunt geese, and hen harriers don't eat poultry. Thanks to fish shortages, herring gulls now snatch ice creams from the hands of unwary holidaymakers, while oystercatchers don't often get the chance to feed on luxury shellfish. Hartlepool fishermen used to call this black-and-white wader 'mussel cracker', which is far more appropriate.
xiv Or, as they are more correctly known, 'scientific' names, because they frequently derive from Greek as well as Latin.
xv Professor Å. Gustafsson of the University of Lund, writing in 1979.
xvi Should a species of bird (or any other plant or animal) need to be further subdivided into different races or subspecies, then a third name (trinomial) is added. So the white wagtail, found over much of Europe and Asia, is called _Motacilla alba alba_ , while the pied wagtail, which we see here in Britain, is _Motacilla alba yarrelli._
xvii Gull-billed tern, collared pratincole and spotless starling respectively.
xviii For example, the black-headed gull has been transferred from the genus Larus to a new one, Chroicocephalus, and so is now known by the tongue-twister _Chroicocephalus ridibundus_. Meanwhile the blue, coal and crested tits (once in the genus Parus, along with the other British members of their family), are now in the genera Cyanistes, Periparus and Lophophanes respectively, while the marsh and willow tits have been moved to the genus Poecile. Only the great tit ( _Parus major_ ) remains unchanged.
xix In keeping with the pedantry of this age, its full title reads: _The Natural History and Antiquities of Selborne, in the County of Southampton. To which are added The Naturalists Calendar; Observations on Various Parts of Nature; and Poems._
xx Whether _Selborne_ can claim to have sold as many copies as _Quotations from Chairman Mao_ (better known as _The Little Red Book_ ), or various works by Tolkien, J. K. Rowling and Agatha Christie, is hard to tell, as reliable sales figures were not available back in the eighteenth and nineteenth centuries. But it has never been out of print, has been translated into many foreign languages, and in the two-and-a-half centuries since it was first published has appeared in almost 300 different editions.
xxi In 2008 a copy was sold at the London auction house Christie's for over £16,000.
xxii Some of these alternative names, such as dipper (1388), corncrake (1455) and nightjar (1630), were already in use; but goldcrest was not officially adopted until 1883, and black-necked grebe as recently as 1912. And of course 'Brown Owl' is still used today for the leader of a Brownie pack!
xxiii In some ways Pennant was ahead of his time: in most of the rest of the world the nine species in the same family as the stone curlew are known as 'thick-knees', after the bony protuberances on their legs. Oddly, even though this species is not closely related to the curlews, we still prefer the inaccurate and misleading name 'stone curlew'.
xxiv To put this figure into perspective, that means there are three times as many willow warblers in Britain as swallows, and the same number as the next two commonest migrants – chiffchaff and blackcap – combined.
xxv As some consolation, Pennant does have three species of mammal named after him, including the North American carnivore, and relative of the pine marten, the fisher _Martes pennanti_.
# TAMING NATURE
_The Organisation of Bird Names_
A named thing is a tamed thing.
Joanne Harris, _Runemarks_
## _1: A Man of Kent_
A thin layer of hoar frost coats every available surface, turning all to white. Ground, trees, bushes and sky merge into one, the sparse vegetation etched onto the landscape like a medieval engraving. In such intense winter cold, surely no small bird can survive – let alone a tiny warbler that depends on insects for its food?
So it is hardly surprising that the vast majority of this bird's relatives are far away to the south. Some, like the willow warbler, have flown all the way to Africa, where they now flit about on the parched savannah, feeding amongst elephants, lions and vast herds of wildebeest. Others, such as the blackcap, have stayed closer to home, on this side of the Sahara, and are foraging amongst the maquis bushes around the Mediterranean Sea. A few chiffchaffs, meanwhile, have remained in Britain, mostly heading to the south-west to take advantage of the milder winter climate in recent years.
But one kind of warbler has chosen a very different way of life. Instead of evolving to migrate each autumn, when the temperature drops and its food supply runs low, this species has chosen to remain right here on the Dorset heath where it was born – the landscape of Thomas Hardy's _Tess of the D'Urbervilles_.
Staying put is a big gamble. For this is one of our smallest and lightest birds, measuring just twelve centimetres from the tip of its bill to the end of its long, cocked tail, and weighing a mere ten grams – about the same as an old one pound coin.
It may be bone-chillingly cold, but according to the calendar, today is the equinox marking the start of spring – which means the start of the breeding season. With the passing of each day, there are a few more minutes of daylight; and in response to this, hormones are produced that encourage this warbler – indeed force it, for it has no real choice in the matter – to sing. So it hops up onto a sprig of gorse, whose custard-yellow flowers are just visible beneath a thick layer of frost, and bursts into song. A harsh, hesitant rattle of rapid-fire notes, rather like a speeded-up recording of a jammed machine-gun, floats over the heath, before evaporating in the chill March air.
This minuscule bird spends its whole life here, on this blasted heath. It is usually either perched on top of a gorse bush, or hidden inside the spiky foliage, searching for tiny insects, which it grabs and despatches with its small, pointed bill.
Given the bird's deep attachment to this one plant, and a plant itself so characteristic of its heathland home, you might imagine that it would be named after its habitat: heath warbler, perhaps, or gorse warbler. Indeed, one folk name is 'furze wren', after an old name for gorse. Instead, though, this species carries a misnomer so bizarre that when people hear it, they often assume they must be mistaken. For this bird is a Dartford warbler.
Yes, Dartford. A town that is famous for its tunnel and bridge, and for traffic snarl-ups during rush hours and on bank holiday weekends. Famous – amongst historians of rock music – as the childhood home of Mick Jagger and Keith Richards of the Rolling Stones. And famous, at least in the world of ornithology, as the place that gave its name to this tiny, and undeniably charming, little bird.i
Many birds are named after the habitat where they live: from marsh harrier to sand martin, and tree sparrow to wood pigeon. Some are named after buildings: house sparrow and barn owl, for example; others sport a name associated with farming, such as corncrake and corn bunting. And a few – less than a score of regularly occurring British birds, including the Scottish crossbill, Canada goose and Mediterranean gull – are named after a country or region. But of all the 600-plus species on the British Ornithologists' Union's official British List, only three are named after an English town or county: Kentish plover, Sandwich tern and Dartford warbler.
As you may have noticed, all three names originate in the same county – Kent. This is no coincidence, for the names of the plover, tern and warbler go back to one man: the eighteenth-century physician and amateur ornithologist Dr John Latham. It was he who first came across these three species, and gave them their fascinating – but utterly inappropriate – names.
*
John Latham was born in Eltham (now in south-east London, but then a village in Kent), on 27 June 1740. His father was a surgeon, while his mother was descended from the Sotheby family, the founders of the famous London auction house.
As the eldest son, it was inevitable that John Latham would follow his father into a medical career, and indeed he did. But like many professional gentlemen of this era, he used his ample leisure time to pursue his passion for nature, becoming a distinguished ornithologist, and doing much to extend our knowledge of British (and later Australian) birds.
Like all ornithologists at this time, Latham practised his science down the barrel of a gun. Optical aids such as telescopes were still in their infancy, and the first book that might enable observers to identify birds in the field – Thomas Bewick's _A History of British Birds_ – would not be published for another quarter of a century. So the only way to be absolutely sure of identifying any unusual bird was to shoot it.
Thus it was that early on a fine spring morning, in April 1773, John Latham left his home with his gun over his shoulder, to take a walk around Bexley Heath, near Dartford.ii
During John Latham's day, both Dartford and Bexley Heath were still very rural in character, with a large area of trees and scrub extending between the two towns. It was here that he came across a pair of birds he could not even recognise, let alone identify. So he did what he always did under such circumstances: lowered the barrel of his gun and discharged a volley of lead shot towards the unfortunate creatures. Being a practised marksman, he hit both his targets, and they fell lifeless to the ground.
He examined the corpses carefully. Superficially they resembled the common whitethroat: small and slender, with a thin bill and long tail. But the colour was like no bird he had ever seen. The male had deep magenta underparts, the shade of a fine red burgundy, a greyish-brown back and head, and a few tiny pale spots around the throat and bill. The female was more drab and browner in shade, but shared her mate's basic plumage pattern.
Excited but mystified by his find, Latham took the specimens home and showed them to his fellow bird collectors. These included Thomas Pennant, the man who, as we have seen, had already made such a huge contribution to the naming of Britain's birds.
Pennant agreed that this was indeed a new bird to science, and granted Latham the honour of naming it. Even though the birds had actually been shot on Bexley Heath, he decided for some unknown reason to call the species _Sylvia dartfordiensis_ :iii Dartford warbler.
*
Unlike its cousins, most of which have large breeding ranges stretching across Europe, western Asia and the Middle East, the Dartford warbler is confined to a small area of maritime western Europe. Its range runs from southern England, through western France, Spain and Portugal, to the north-western tip of Africa. With such a restricted distribution, and a declining population, the species has been categorised as 'Near Threatened' by the global conservation organisation BirdLife International.
Being on the very northern edge of its range here in Britain, the Dartford warbler is vulnerable to anything that might threaten its survival. In the two centuries or so since Latham's discovery, its primary habitat – lowland heath – has been largely destroyed. Today, less than one-fifth of England's original heathland remains, mostly in the southern counties of Dorset and Hampshire, with a few outlying patches in Devon, Surrey, Sussex and East Anglia.
But habitat loss is only one problem faced by this tenacious little bird. As a resident rather than a migrant, and with a diet of small insects, it is very susceptible to hard winters. This is particularly problematic during long spells of ice and snow, which make it hard to find food, and also weaken the birds by lowering their body temperature.
The cold winters of the first few decades of the twentieth century gradually reduced the population and range of Britain's Dartford warblers. The crunch came during the infamous 'Big Freeze' of 1962–3. For more than two months, from late December through to early March, Britain froze solid, with a thick layer of snow covering the ground, in conditions more severe than had been seen for over 200 years.
The winter of 1962–3 was horrendous enough for the human inhabitants of the British Isles; but for our birds it was, quite simply, a disaster. The ornithologist and broadcaster James Fisher summed up the enormity of the situation when, towards the end of the worst winter in living memory, he announced that 'It seems likely that at least half the wild birds living in this country before last Christmas are now dead.'
For the Dartford warbler, the situation was critical. Its population had already been heavily reduced by the previous year's cold winter, which had left just a hundred breeding pairs on the heaths of Dorset and the New Forest. But following the winter of 1962–63, numbers plummeted: in spring 1963 only a dozen pairs could be found, mostly on the coast around Poole Harbour, where temperatures had stayed marginally higher than elsewhere.
From this tiny and unpromising base, however, the fortunes of the Dartford warbler at last began to turn. Thanks to a long run of mild winters from the 1970s onwards, numbers steadily grew, to reach a peak by the mid-1990s of well over 3,000 breeding pairs. The two hard winters of 2009–10 and 2010–11 stalled the recovery a little, but the Dartford warbler is now thriving in a way that no-one could have predicted after those terrible events of the early 1960s.
As for the Dartford warbler's discoverer, John Latham, he lived a long and productive life, dying at his home in Hampshire on 4 February 1837, in his ninety-seventh year. And he certainly enjoyed the freedom he gained after his retirement from practising medicine. Having stopped working in 1796, he devoted the remaining four decades of his life to ornithology.
Like other pioneering ornithologists of his day, many of whom we shall encounter in the next chapter, he gave his name to a number of birds around the globe, such as Latham's snipe (sometimes known as Japanese snipe). Other species, including the glossy-black cockatoo, Australian brush-turkey, grosbeak starling and forest francolin, were also originally named after Latham, but have since been given new names.
Yet even he might be surprised to discover that the deeply unsuitable epithet he gave to the Dartford warbler has survived so long. And it's not the only one. Latham named two other British birds after his adopted county, Kent: the Sandwich tern, discovered by his fellow-ornithologist William Boys in 1784, and described and named by Latham three years later; and the Kentish plover, a small wader also discovered by Boys, when he shot three unfamiliar birds on the East Kent coast in May 1787.
Both names are, like that of the Dartford warbler, totally unsuited to the species that bear them. The Sandwich tern (named after the town, not the foodstuff) is found along the coasts of five continents, only being absent from Australasia and Antarctica; while the Kentish plover is also globally widespread. At least the Sandwich tern does regularly occur in Kent, whereas the Kentish plover no longer even breeds in Britain, having disappeared during the second half of the twentieth century. Yet their names survive: a reminder that the most suitable name for a bird is rarely the one by which it is known.
Latham spent many of his later years attempting to bring together the many new ornithological discoveries being made at that time, which he published in a massive ten-volume work, _A General History of Birds_ , which appeared in instalments between 1821 and 1828.1
However, his advanced age unfortunately meant that the work contained many basic mistakes, as the Revd Charles Swainson, a Victorian parson-ornithologist and the author of a seminal book on the folk names of birds,2 pointed out: 'His memory was not good; hence he has frequently described the same species by different names; and he placed too much faith in drawings, which led to the same error.'3
To be fair on Latham, he was doing his best to clarify a very confusing situation. The late eighteenth and early nineteenth centuries had seen a huge proliferation in the number and variety of bird specimens brought back to Britain from across the globe, thanks to the boom in exploration led by the expansion of the British Empire from roughly 1783 to 1815. 'The museums of Europe', Swainson also commented, 'became crowded with new birds, quite unknown to Linnaeus, without any one naturalist to describe them'.4
John Latham had set himself the mission of categorising and classifying all the world's birds; a daunting task for a man by then in his eighties. But although he did not fully succeed, he did at least try. And of all the new places being investigated at this time, and the new species of birds discovered, none were more exciting than the exotic specimens from that newly discovered land far away in the southern hemisphere: Australia.
Decades earlier, these extraordinary birds had captivated Latham, who by examining specimens had written the descriptions of many new species in _The Voyage of Governor Phillip to Botany Bay_ , by Arthur Phillip, the first governor of New South Wales, which was published in 1789. And although he never visited Australia himself – indeed, never actually set foot outside Britain – he was nevertheless instrumental in naming many of Australia's best-known birds, including the superb lyrebird, wedge-tailed eagle and the sulphur-crested cockatoo. His fascination with birds down under ultimately led to him being widely called 'the grandfather of Australian ornithology'.
Latham's fascination with Australia's birds was a consequence of a series of historical events, which make a fascinating diversion in our story of how birds got their English names. The coincidence of several factors brought it about: a rise in crime, the loss of a war, and the convenient and timely discovery of a new land on the other side of the globe.
Since the early 1700s, in an increasingly lawless Britain, it had made economic and political sense to send convicts abroad – a process known by the euphemistically benign word 'transportation'. During the middle years of the eighteenth century, more than a thousand people had been sent across the Atlantic Ocean to America, but after the War of Independence ended with the triumph of the rebel colonists in 1783, that avenue was closed. As a result, a new destination had to be found for these unfortunate prisoners. That place was just about as far away from Britain as you could get: Australia.
## _2: Flaming Galahs and Fairy-Wrens_
It had been a long, and at times unspeakably horrendous, voyage. The fleet of eleven ships had left Portsmouth on 13 May 1787, and sailed across the world's oceans for more than eight months, until the first vessel finally made landfall at Botany Bay on 18 January the following year. During the 15,000-mile journey, almost 1,500 crew and passengers had endured baking heat, freak storms, food and water rationing, an aborted mutiny, the company of rats, lice, bedbugs, fleas and cockroaches, and the deaths of almost fifty of their fellow men and women through sickness, violence and the occasional drowning.
The story of the First Fleet has become the stuff of legend. For these were no ordinary passengers, but convicts, being deported from England to this new and unknown land, in what would eventually become one of the greatest mass movements of people in the whole of history: the colonisation of Australia.
Along with the earlier settlement of North America by the Pilgrim Fathers, and the later expansion of the British Empire into Africa and Asia, this would extend the influence of English bird names around the world. But for the people being taken to the other side of the globe, the naming of the birds in this new land was probably the very last thing on their minds.
In _The Fatal Shore_ , his eloquent and moving account of the history of transportation, Robert Hughes points out that crew and convicts alike were being sent into the complete unknown:
Never had a colony been founded so far from its parent state, or in such ignorance of the land it occupied. There had been no reconnaissance. In 1770, Captain James Cook had made landfall on the unexplored east coast of his utterly enigmatic continent, stopped for a short while at a place named Botany Bay and gone north again. Since then, no ship had called: not a word, not an observation, for seventeen years, each one of which was exactly like the thousands that had preceded it...5
It is no wonder that, as the ships finally made landfall, the overriding emotion amongst the passengers was one of relief at having survived the voyage at all. But this was swiftly followed by bafflement: at the unforgiving sandstone landscape with its unusual greyish-green vegetation, the terrifying appearance of the indigenous peoples, and especially the truly bizarre wild creatures they encountered as they began to explore their new and utterly unfamiliar surroundings.
There were peculiar animals that jumped along on their hind legs, egg-laying mammals sporting a beak and webbed feet, and an astonishing array of impossibly colourful and noisy birds. As his ship _Lady Penrhyn_ was making her way up the narrow channel towards Port Jackson (now Sydney), the ship's surgeon Arthur Bowes Smyth made this excited entry in his journal: 'The singing of the various birds among the trees, and the flight of the numerous parraquets, lorrequets, cockatoos, and maccaws, made all around appear like an enchantment...'6
He may have been wrong about macaws (they are only found in the Americas) but, as Hughes points out, there were still plenty more birds for him to marvel at:
Several dozen kinds of parrot thronged the harbour bush: Galahs, bald-eyed Corellas, pink Leadbeater's [now Major Mitchell's] Cockatoos, black Funereal [now Yellow-tailed Black] Cockatoos, down through the rainbow-coloured lorikeets and rosellas to the tiny, seed-eating budgerigars which, when disturbed, flew up in green clouds so dense that they cast long rippling shadows on the ground.7
These birds can still be seen in the Sydney area, and across much of the rest of this vast nation, though sadly in far lower numbers than when the First Fleet arrived. But from the awestruck accounts of those early settlers, through later generations of Australians, right up to today's visitors from abroad, that sense that Australia's birds – and their names – are truly unique has never really gone away. I discovered this for myself more than two centuries later, in 2008, when I visited this strange and beguiling land.
*
The plains-wanderer may sound like the jolly swagman out of 'Waltzing Matilda', Australia's unofficial national anthem, but it is actually one of that country's rarest, least-known and most sought-after birds. About the size of a song thrush, and with the physique of a pot-bellied quail, its nearest relatives are the South American seedsnipes.
On that trip to Australia I had a rare opportunity to connect with this almost mythically elusive bird. Nervous with anticipation, a small group of us gathered under a clear, star-filled sky in a paddock in rural New South Wales. I say 'paddock' but, being Australia, it covered an area of more than 80 square miles, and its perimeter was comfortably longer than a marathon course.
Torches at the ready, we set off. It was a chilly night, and I was soon wishing I had worn warmer clothes, as it looked set to be a long one. Yet barely five minutes after we began our search, one of the guides caught a pale, moth-like creature in her spotlight beam. Fluttering on long, slender wings, it plummeted to the ground – surely never to be seen again.
The torches swept around like anti-aircraft lights, and then I heard a clear warning voice: 'Stephen... _don't_ step forward.'
As the torch beam reached me, I looked down, to find a small, brown creature crouching motionless, exactly where my foot was about to tread. It was a male plains-wanderer, staring right back at me with a mixture of fear and bafflement. Although at that point I had been watching birds for over 40 years, and seen almost 2,000 different species around the world, this truly was the most extraordinary, heart-stopping moment of my entire birding life.
The appreciative noises coming from the darkness around me suggested I was not alone. We stood and stared at this extraordinary creature like members of some minor religious sect until, a few moments later, it flew away into the darkness, never to be seen again.
This was my very first visit to Australia and, despite my long years of watching birds, on each of the other six continents, it was like starting birding all over again in some kind of weird parallel universe. That was because of the 200-plus different species of bird I saw on my whistle-stop tour, more than eighty per cent were completely new to me.
Before I go birding in a new place, I like to do my homework; looking up the birds I hope to see, and trying to learn the key plumage features that will help me identify them. But this time, the very act of doing so left me more confused than ever. What on earth were galahs and gerygones, currawongs and pardalotes, blue bonnets and bronzewings? What was a weebill, or a dollarbird? What was the difference between an apostlebird and a mistletoebird, apart from their vaguely festive names? And what – or perhaps who – was a Jacky winter or a Willie wagtail?
Some species – along with their names – were far more familiar. I was hoping to catch up with one of the largest of the all the world's birds, the emu, and one of the best known, the budgerigar. I knew that the brolga was a species of crane, and a cockatiel a kind of miniature cockatoo. I was well acquainted with bowerbirds and lyrebirds from David Attenborough's television programmes, and could guess that a varied sitella might look rather like a nuthatch (Sitta is the generic name for nuthatches) – as indeed it does.
But I had no idea that the bell miner, the noisy friarbird, and the eastern spinebill are all members of the vast honeyeater family, whose seventy-plus species covered page after page of my field guide. Gazing at their names, many of which appeared to be some combination of a colour and a part of the body – yellow-faced, white-eared, black-chinned, blue-faced, and brown-headed, together with scarlet, black, dusky, banded, striped and painted – I began to suffer from blurred vision, and wondered whether I would be able to cope when faced with this bewildering array of new birds.
*
The reason that Australian bird names seem so odd is that many of the birds themselves are confined to this particular part of the world. Australasia finally broke away from the much larger landmass of Gondwanaland about 100 million years ago; that's an awfully long time for its fauna and flora to have evolved along a very different path from the rest of the planet.
As a result, more than four-fifths of Australasian mammals, 90 per cent of reptiles, and 93 per cent of amphibian species are endemic – found only in Australasia, and nowhere else in the world. Because birds can fly, and therefore colonise new lands across the sea, a far lower proportion is endemic: even so, about 350 out of the 800 regularly occurring species of bird in Australasia – almost one in two – are only found here. Compare this to Britain, with just one endemic species (the Scottish crossbill), or Europe, with about a dozen.iv Yet despite the fact that many of Australia's birds cannot be seen anywhere else in the world, their names often seem strangely familiar to a British birder. So where did these names come from? On my week-long trek from Melbourne to Sydney, via the eucalypt forests, wetlands and mountains of Victoria and New South Wales, I came across the welcome swallow, Australian raven and black swan, all of which are indeed cousins of our own species.
But the woodswallows, fairy-wrens, robins, treecreepers and the Australian magpie are not at all closely related to their Eurasian counterparts. The magpie-lark, a black-and-white bird about the size of a blackbird, is neither a magpie nor a lark, but a monarch flycatcher – one of a family of small, insectivorous songbirds. Despite their familiar sounding names, many of these birds – including the fairy-wrens and robins – are more closely related to one another than they are to the groups of birds they superficially resemble.
The reason they have such unsuitable and misleading names is, of course, a legacy of those early settlers. Convicts – and many of the sailors – were forced to make a new life in this new land, but that did not mean that they forgot their former existence. And so when one homesick colonist came across a small, plump, perky bird with a red breast hopping across the ground in front of him, it was only natural that he should name it after a favourite bird from home.
This explains why a family of almost fifty species, taxonomically sandwiched between the birds-of-paradise of New Guinea and the picathartes (bald crows) of West Africa, is still known today as the 'Australasian robins'. Some, like the scarlet, rose, pink and flame robins, do indeed have that familiar red breast. But others, including the eastern and western yellow robins, the smart black-and-white hooded robin, and the skulking, greyish-coloured mangrove robin, sport a wide range of different colours. And none of them is even vaguely related to our own robin redbreast.
Not every Australian bird was named out of a nostalgic yearning for the English countryside and its native wildlife. Some were given names based on what they were called by Australia's indigenous population. But even though the aboriginal peoples had been on the continent for at least 30,000 – and perhaps as long as 60,000 – years before the western settlers first arrived, precious few Australian bird names derive from their languages.
Those that do include the gang-gang cockatoo, a greyish-brown and scarlet-headed bird confined to south-east Australia; and its noisy and colourful grey and shocking pink cousin, the galah, one of the most familiar of all Australian birds. This sociable, noisy species has given rise to the slang phrase 'flaming galah', still widely used as an insult to describe a simpleton or fool.
One of Australia's best-known birds also bears an aboriginal name. The laughing kookaburra is a dry-country species of kingfisher whose ringing call does indeed sound like hysterical human laughter. The name 'kookaburra' – clearly onomatopoeic in origin – has been traced back to several indigenous languages. Yet such was the prejudice against 'native' names that, despite being widely used for almost a century, kookaburra was only adopted as the species' official name in 1926. Before then it was known as the 'great brown kingfisher', a rather prosaic name originally coined by Latham.
Such an extrovert bird was also bound to attract its fair share of folk names – perhaps more than any other Australian bird. Many of the names listed by Ian Fraser and Jeannie Gray in their definitive work on the subject, _Australian Bird Names_ ,8 including 'Jacky', 'Jacko' and 'laughing John', derive from the English word 'jackass', referring to the kookaburra's donkey-like call. Others, such as 'alarm bird', 'breakfast bird' and 'clock bird', all acknowledge the impossibility of staying asleep when a kookaburra is calling outside your bedroom window, while 'woop woop pigeon' and 'ha ha duck' acknowledge the sound's similarity to human laughter.
The most famous of all aboriginal bird names is – thanks to its worldwide popularity as a cagebird – budgerigar.v The name is supposed to come from a phrase meaning 'good cockatoo'. 'Good' in this case refers to these birds' uncanny ability to find precious sources of water out in the bone-dry outback, a crucial aid to nomadic people in times of drought.
But other names which to our untrained ears may sound aboriginal, in fact have a very different origin. The word 'cockatoo', used for a family of birds closely associated with Australia, is actually from the Malay language – hence its early appearance in Arthur Bowes Smyth's 1788 journal. Likewise the names of Australia's tallest bird, the emu, and the heaviest, the southern cassowary, are not aboriginal either, but derive respectively from Portuguese and Malay, the latter coming into English via Dutch or French. In both cases, this is because members of the cockatoo and cassowary families had previously been discovered in south-east Asia, and so had already been given names.
Australian bird names are also distinguished by their frequent use of nicknames, and some over-the-top – yet often very appropriate – epithets. These include noisy (pitta, friarbird and miner), magnificent (riflebird), graceful (honeyeater), rainbow (lorikeet, bee-eater and pitta) and elegant (parrot). But for sheer hyperbole, it is hard to beat a quartet of fairy-wrens, tiny yet colourful birds with cocked tails, whose names get more and more elaborate as they go up the scale: from variegated, through lovely and splendid, to superb.
As for those originally based on nicknames, my favourites are Jacky winter, another member of the Australian robin family, and Willie wagtail, a species of fantail. Jacky winter may have arisen from the bird's habit – like its European counterpart the robin – of singing during the winter months. Willie wagtail, as Fraser and Gray point out, is the 'most archetypal of Australian bird names'. Yet it seems more likely that when the colonists came across a slender black-and-white bird with a long tail they simply named it after a bird familiar from back home, the pied wagtail ('willie wagtail' was already used as a folk name in Britain, especially in Scotland).9
Despite their informal and unscientific origins, a handful of other names coined by those early colonists have somehow managed to escape the tendency of modern ornithologists to 'tidy up' bird names, and are still used today. These include the Cape Barren goose, a peculiar-looking bird with a grey body and lime-green bill, named after an island off the north-east coast of Tasmania; the brush-turkey, a member of the megapode family, which incubates its eggs by burying them in a huge mound of earth and leaves; and the lyrebird, whose fan-like tail does indeed closely resemble that ancient stringed instrument.vi
*
Despite the awful start to their new life, those unfortunate convicts transported to Australia on the First Fleet, and the sailors who accompanied them, did ultimately manage to create a new home in this forbidding and quintessentially foreign land, many thousands of miles from their old lives.
In doing so, they began the process of globalisation that led to the world we know today, in which shrinking horizons have seen the sharing of knowledge and the homogenisation of culture and language. At the same time as these new names were springing up in Australia, successive revolutions were occurring in the way we understood the natural world, leading inevitably to slow but steady standardisation in the English names given to birds.
Not everyone was able to take advantage of this expansion in travel and knowledge. Back in Britain, the vast majority of people – especially those living in rural areas – led a virtually sedentary life, rarely travelling more than a few miles from where they were born. For them, the world was bounded by a few familiar places and people, and also by a few well-known and frequently encountered birds.
So despite the best efforts of pioneering ornithologists from Turner to Pennant – and of course Linnaeus – to standardise the names being used for birds, there was still a powerful pull in the opposite direction. Well into the nineteenth century, and sometimes beyond, ordinary folk still preferred to use folk names: those that had been used by their parents and grandparents before them. Many of the new names took a long time to reach rural communities; and even when they did, were unlikely to gain acceptance over the simpler and more familiar names used for centuries.
This tension between the new and the old names – and between the new science and the old traditions of the countryside – is demonstrated in the life story of one remarkable man: the poet and naturalist John Clare. Although he was courted by literary society, he still retained his rural roots – and the bird names he had learned as a boy – for the whole of his life. And even as ornithology was gaining reputation as a science, these old names were proving remarkably resistant to change.
## _3: The Nature Poet_
No other poet wrote about birds as often – or as well – as John Clare. This nineteenth-century farm labourer turned man of letters was, as the ornithologist and broadcaster James Fisher deftly put it: 'the finest poet of Britain's minor naturalists and the finest naturalist of Britain's major poets'.
Thanks to his field skills, observational talents and hard-won expertise, Clare's writings contain references to at least 120 (and possibly as many as 150) different species. These observations give us a profound insight into the dramatic and often devastating changes to the birds of our farmed countryside over the past two hundred years. Foremost amongst these is the loss of the bird Clare described as a ubiquitous 'summer noise among the meadow hay': the corncrake or, as Clare called it, the landrail.
Patronised by the London literati as a 'peasant poet', Clare, and in due course his poetry and prose, were intimately linked to the place where he grew up, and spent the majority of his life: the village of Helpston. Living on the edge of the flat, watery fens of East Anglia, but also close to the classic 'Middle England' landscape of Northamptonshire, the young Clare could explore fields and meadows, streams and rivers, woods and fens, and get to know their birdlife.
Through his rootedness in one place, during his twenties and thirties John Clare produced some of the finest nature poems ever written. However, for over a century these were neglected by literary critics and the general public alike, until from the 1950s onwards his reputation was restored and rehabilitated. Today he is widely hailed as one of the most influential of all writers on nature.
I became hooked on Clare's bird poetry more or less by accident, when studying English Literature at Cambridge back in the early 1980s. My Director of Studies at Gonville and Caius College, the famously enigmatic poet J. H. Prynne, learned of my interest in birds and enthusiasm for the poetry of John Clare, and suggested I meet John Barrell, who I later discovered was one of the world's greatest experts on Clare's writings.
Sitting in Professor Barrell's wood-panelled room in the forbidding surroundings of King's College, I nervously explained that I had noticed that the verse structure of Clare's bird poems somehow seemed to mimic the movements and behaviour of the bird itself. Encouraged by his positive response, I went on to write my undergraduate dissertation on this very subject.10
I am not alone in my love and admiration of John Clare; he has inspired many of today's cohort of 'new nature writers'. But at the time, not everyone approved of the way he wrote about the natural world. His contemporary John Keats complained that in his verse 'the description too much prevailed over the sentiment.' While that may occasionally be true, it is impossible to dispute Clare's intimate knowledge and understanding of nature, gained from day to day, season to season and year to year, and more importantly his skill in turning these marvellously detailed observations into poetry.vii
Yet despite Clare's undoubted influence and popularity today, for the new reader the poems can at first appear rather baffling. This is not just because of the style of writing, which, once you get used to his lack of punctuation and rather eccentric spelling, is actually very accessible – and full of delightful insights into bird behaviour – but also because in many cases the self-taught Clare chose to ignore the official name for the species, preferring to use the folk name he grew up with.
So in his poems we find the land rail and fern owl, butter bump and fire tail, water hen and peewit – now known respectively as the corncrake and nightjar, bittern and redstart, moorhen and lapwing.viii
At other times, Clare would use the correct epithet for the species, but ascribe it to the wrong family, as when he referred to the reed sparrow (reed bunting), reed wren (reed warbler) and grasshopper lark (grasshopper warbler). He would also often use names interchangeably for several different species, as in the opening lines of this sonnet, where 'black cap' refers to the great tit:
Under the twigs the black cap hangs in vain
With snow white patch streaked over either eye
The bird we now know as the blackcap – a member of the warbler family – Clare also called the March nightingale, because of its early return from its winter quarters and melodious song. However, while he may have titled his sonnet after the folk name, 'The March Nightingale', within the poem he preferred the official name:
The rocking clown leans oer the spinney rail
In admiration at the sunny sight
The while the Blackcap doth his ears assail
With such a rich and such an early song
He stops his own and thinks the nightingale
Hath of her monthly reckoning counted wrong
Clare's use of folk names is in some ways a throwback to an earlier age, before the standardisation brought about by ornithologists, when bird names were far more fluid and changeable.
But even Clare could not resist the tide forever. Changes were afoot, and nineteenth-century ornithologists were busily continuing the work of Thomas Pennant and his predecessors in standardising the names given to birds. In the longer term the messy assemblage of folk names so beloved of Clare, with several alternative names for each species, was simply not sustainable.
This also reflected bigger changes occurring at the time: the growing gulf between the rise of science, epitomised by a new generation of museum-based ornithologists who we shall meet in the next chapter, and men such as Clare, who based their knowledge on hard-won observations in the field, and were often contemptuous of the professionals. According to Clare scholar Eric Robinson, the poet did not have a very high opinion of revered ornithologists such as Pennant, as is evident in this comment on the plucking of geese: 'Mr Pennant says he saw the buisness of Geese pulling baere and that they pulled gosslings that were not above 6 weeks old I have no hesitation in saying that Mr Pennant is a Liar.'11
*
One reason why names had to be standardised was purely pragmatic, and due to the birth of a new publishing phenomenon: books about the natural world. Following the popularity of White's bestselling _The Natural History of Selborne_ (see Chapter 3), there was a growing demand for 'guidebooks' enabling ordinary people to identify the wild creatures they saw.
This gap in the market was soon filled by the engraver and political radical Thomas Bewick, with his pioneering _A History of British Birds_ , published in two volumes: _Land Birds_ (1797) and _Water Birds_ (1804). These reached a very wide audience, thanks to their delightful woodcut illustrations and clear, readable prose.
As was the custom of his day, Bewick referred to willow-wrens and throstles, titlarks and ringtails, pied and barred woodpeckers.ix Many of his readers would have known these, while some would have had their own preferred local versions. But Bewick also used a number of truly obscure names, including cravat goose, ash-coloured sandpiper and castaneous duck,x which would have been baffling to many of his readers. It was clear that from now on the names used for birds needed to be standardised, to enable the reader to know which species was being referred to, whether they lived in Penzance or Penarth, Inverness or the Isle of Wight. Someone needed to take on the thankless and time-consuming task of collating all the different names in use at the time, and making clear and lasting decisions on which should prevail. Such a person would have to be tough and uncompromising, knowledgeable and clear-thinking, assiduous and at times inspired, with the hard-won skills of a field observer combined with the scientific knowledge of the professional ornithologist. On top of all this, they would need the stamina to take on a workload that might defeat a lesser man.
The time was ripe for the entrance onto the scene of one of the least likely of all James Fisher's 'ornithological heroes'; a man who simply got on with the job in hand: 'In his efficient way, he swept up almost the last of our birds that were unknown because unrecognised, and usually unrecognised because undistinguished from some close relative.'12
The name of this often misunderstood and underrated man, who gave his name to one of the rarest and most elusive of all Britain's breeding birds? George Montagu.
## _4: The Military Man_
As I gazed across the field of golden barley, my eyes were momentarily dazzled by the reflection from the sun, high in the clear blue July sky. Then, in the distance, close to the hedgerow bordering the back of the field, I saw a movement. This time it wasn't just heat haze, but – at last – the bird I had come to see.
The female harrier rose up on long, slender wings just a few feet above the sea of barley, the crop waving to and fro in the gentle breeze. As she floated effortlessly in the thick summer air I could see the distinctive white rump at the base of her long, narrow tail, contrasting with the brownish hue of the rest of her plumage.
For a few moments, I took in her elegant shape and form before, seemingly out of nowhere, another bird appeared. This was the male: even slimmer and more aerodynamic than his mate, and sporting a pale, dove-grey plumage with black tips to his wings, as if he had just dipped them in a bottle of ink. As the male approached, she rose higher into the sky, and they began an aerial dance, twisting and turning in the warm summer air to cement their bonds of courtship.
Then I noticed that the male was carrying something in his talons – a vole, or perhaps a meadow pipit or skylark. He flew above his mate, stalled in mid-air, and dropped his prey; a fraction of a second later she stalled too, then twisted almost upside-down to grab her gift from the air, before flying down to her hidden nest.
I had witnessed one of the most intimate of all bird behaviours and, as the male powered away into the distance, I was left wondering if I had really seen it at all, so quickly had it happened. But I had – and I'm one of the few lucky ones, for although this elegant harrier breeds across a wide swathe of Europe, western Asia and north-west Africa, barely a dozen pairs of this beautiful creature return each spring to nest in the arable fields of southern Britain.
Yet despite this bird's rarity in the UK, it was in south Devon that, just over two centuries ago, one man identified it as a species new to science; a species that would eventually come to bear his name: Montagu's harrier.
*
George Montagu first came across what would become 'his' harrier on a hot August day in 1803. A local man had shot and killed a bird of prey and, unable to identify it, brought it to Montagu for him to inspect. As he dissected the bird, discovering that it had a freshly caught skylark in its stomach, Montagu became more and more excited. For although it superficially resembled a male hen harrier, this bird was noticeably smaller and more slender, and also had a longer tail, with reddish-brown markings on its pale grey wings.
Confident that it was a species hitherto unknown to science, Montagu gave the bird its original scientific and English names: _Falco cinerarius_ , the 'Ash-coloured Falcon'. Two decades after Montagu's death, the French ornithologist Louis Vieillot and his Dutch colleague Coenraad Temminck commemorated its discoverer by renaming it 'le busard Montagu', from which William MacGillivray coined the current English name: Montagu's harrier.
But George Montagu's contribution to our knowledge and understanding of Britain's birds went far beyond this tribute. His life story reveals much about the stifling formality of the society in which he lived; a society he ultimately chose to reject, so that he could pursue his lifelong passion for birds.
Montagu packed a lot into his six decades on this Earth. Born in 1753, he joined the army at the age of seventeen and was married a year later to Ann Courtenay, the high-born daughter of a nobleman. She gave birth to six healthy children – four sons and two daughters – while Montagu himself rose to become a lieutenant colonel in the county militia of Wiltshire.
But despite his outwardly successful and respectable life, all was not well with George Montagu. He had always been fascinated by the natural world, yet the demands of his military career, and his obligations towards his large and growing family, made him increasingly unhappy and frustrated. By the time he reached his thirty-sixth year he was suffering from what we might now call a mid-life crisis.
A letter survives which sheds some light on his state of mind: written in June 1789 to the Hampshire vicar and naturalist Gilbert White. White had just published his life's work, _The Natural History of Selborne_ , which would go on to become the bestselling nature book of all time. In the pages of his letter, the much younger Montagu pours out his heart, telling White that he has 'delighted in being an ornithologist from infancy, and, was I not bound by conjugal attachment, should like to ride my hobby into distant parts'.
Eventually, that's exactly what he did. But not before he faced crisis after crisis: a series of disasters that might have broken a lesser man. For during the following decade, Montagu's life began to fall apart in a quite spectacular manner.
First, in 1797, his unmarried elder brother James died suddenly, leaving George the family estates. There was just one condition: he must live in one of the houses on the estate with his wife. But by then, Montagu had already separated from Ann, and begun a secret affair with the wife of a London merchant, Mrs Eliza Dorville. The news of their clandestine relationship soon became public, causing outrage in polite Georgian society.
About this time Elizabeth, Lady Holland, a noted society hostess, encountered Montagu at a dinner party. She was clearly not impressed, as she waspishly confided to her journal:
Colonel Montagu I saw but once... and after dinner he gave the natural history of every bird that flies and every fish that swims. He is a man of bad temper, nor does it sound creditable to him that none of his officers speak to him, and they are on the eve of bringing him to Court-martial.
Soon after this inauspicious encounter, court-martial proceedings were indeed begun against Montagu. This was ostensibly because of insulting remarks he was supposed to have made to the wives of his fellow-officers, but he was surely not helped by his scandalous liaison with Eliza Dorville. On 15 October 1799 the tribunal found him guilty, and expelled him from the militia. His military career was over.
Worse was to come. Following the court-martial, Montagu became embroiled in a bitter dispute with his eldest son over the inheritance, which eventually led to the loss of most of the family estates. But with every cloud comes the proverbial silver lining. For George Montagu, it was the freedom to indulge in his twin passions: the love of his mistress and his burgeoning career as a naturalist.
He and Eliza fled westwards, settling at Knowle House, just outside the village of Kingsbridge on the South Devon coast. There, he pursued the study and classification of birds, and she – described by him as 'my friend in science' – provided illustrations of them. His masterwork, the two-volume _Ornithological Dictionary; or Alphabetical Synopsis of British Birds_ , was first published in 1802, with several revised editions after his death. A later ornithologist, Elliot Coues, described this as 'one of the most notable of treatises on British birds... which has held its place at a thousand elbows for three-quarters of a century'.
The reason the _Ornithological Dictionary_ is so important to our story is that – like all good dictionaries – it is both clear and comprehensive. There are long entries on each species, which in taut, closely spaced prose describe the bird and its appearance, plumage details, habitat, habits and other points of interest. But there are also dozens of one-line definitions, most of which give the alternative name for a species together with its official one, as in this series of entries:
CHURCH OWL – A name for the Barn Owl.
CHURN OWL – A name for the Nightjar.
CINEREOUS GODWIT – A name for the Greenshank.
CINEREOUS SHRIKE – A name for the Butcher-bird [red-backed shrike].
For anyone trying to fathom the confusing morass of alternative folk names of Britain's birds at the time, the book was a godsend. But it was far more than the dry reference work suggested by its title – for, having worked out which species was being referred to, the reader could then obtain a clear summary of the latest knowledge about that particular bird.
A flavour of Montagu's writing can be seen in the opening lines of the entry on the hoopoe, which then as now was a scarce but regular visitor to southern Britain:
The weight of this beautiful bird is about three ounces; length twelve inches; the bill is black, two inches and a half long, slender, and curved; irides hazel; the crown of the head is furnished with a crest composed of a double row of dull orange-coloured feathers, tipped with black, lengthening from the forehead backwards, the longest of which is above two inches.
This is precision writing at its best: the prose of a man who has looked really closely at the bird he is writing about. In referring to the hoopoe as 'beautiful' he even, rather uncharacteristically, allows himself a personal comment.
With its combination of forensic accuracy and extraordinary attention to detail, Montagu's _Ornithological Dictionary_ set the standard for the bird books that would follow during the Victorian era, such as the seminal multi-volume works by William Yarrell (1837–43) and William MacGillivray (1837–52).xi
But Montagu did not simply replicate and catalogue the work of others; he made many crucial discoveries of his own. As well as his eponymous harrier, near his Devon home he also discovered the cirl bunting, a species known from continental Europe but not recorded before in Britain.
The cirl bunting is a handsome yet rather curious-looking bird, which looks as if it has been assembled from different parts of other, more familiar species. Superficially similar to its cousin the yellowhammer, with a yellowish head, it also sports a streaky back like a reed bunting or dunnock, olive-green underparts like a greenfinch, and a black mask and throat like a house sparrow.
Montagu first came across cirl buntings near his home in the freezing winter of 1800, finding them 'not uncommon amongst flocks of yellow hammers and chaffinches.' Whether the species was there all along, and just needed someone of his skill and experience to notice it, is a moot point; some have suggested that it had only recently colonised England from across the Channel, though it seems far more likely that it had hitherto simply been overlooked. But as with so many other species, it was Montagu who cleared up any doubts as to its status as a British bird, unable to resist a dig at his fellow-ornithologists for their apparently poor observational skills: 'It is remarkable that so common a bird as the Cirl-Bunting seems to be in the west of England, should have so long escaped the notice of British naturalists.'
He himself had no doubt that the species had been present for some time. As he notes in the closing lines of his entry, even when the weather turned so cold during that bitter winter of 1800, the birds stayed put, suggesting that the species was not newly arrived from warmer climes, but had always been living there.
In the 200 years or so since Montagu made his momentous discovery, the cirl bunting's fortunes have waxed and waned. Until the middle of the twentieth century it could be found (albeit locally) across much of southern Britain, but after a rapid and precipitous decline the species retreated to the southern tip of Devon, close to where Montagu had first discovered it, and numbers fell to little more than a hundred pairs. The cirl bunting appeared to be on its way out as a British breeding bird.
Thanks to the efforts of the RSPB and local farmers, however, this curious yet attractive little bird has since made a dramatic comeback. Although, because of its highly sedentary nature and very specific habitat requirements, it is still largely confined to Devon, there are now almost 1,000 breeding pairs in the county. The species has also been successfully reintroduced to Cornwall, in what is thought to be the only example of the successful reintroduction of a songbird in the whole of Europe. I'm sure Montagu would have approved.xii
Sadly, George Montagu's later life was beset by tragedy. Three of his four sons were killed in the wars against France, and he never became reconciled with his surviving eldest son, leaving him out of his will. He and Eliza did have three children of their own – Henry, Isabella and Georgiana – and his two daughters with Ann also survived him. But it's not hard to imagine that one reason this proud, intense and reputedly difficult man may have thrown himself into his work was as one way of mitigating the terrible loss of his sons, and perhaps also assuaging his guilt at deserting them.
On 20 June 1815, just two days after the Duke of Wellington's famous victory over Napoleon at the Battle of Waterloo, the life of George Montagu came to a premature and painful end. A few days earlier, while building work was being carried out on his home in south Devon, he had inadvertently trodden on a rusty nail. Tetanus ensued, and he died in a high fever, aged sixty-two.
Later students of birds had good reason to thank George Montagu, for the _Ornithological Dictionary_ was the first systematic attempt to list all the birds found in Britain. It helped to kick-start the still young science of ornithology, and was widely used for at least a century after his death. And of course we still commemorate him – albeit often unwittingly – when a birder sights the rare and beautiful raptor and shouts excitedly to his companions: 'I've got a Monty's!'
For me, Montagu also represents a man who, despite all his many troubles, led a life well lived, eventually managing to fulfil the ambition to 'ride his hobby into distant parts' – albeit only as far as south Devon. Perhaps I also feel a personal connection with a man who, after a mid-life crisis, headed down to the West Country with the woman he loved, to begin a new and fulfilling life devoted to writing about birds.
*
The species named after George Montagu, Montagu's harrier, is one of a handful of regularly occurring British birds, and many more around the rest of the world, named after people. In the next chapter, I shall examine the golden age of eponymous bird names, during which many of the more obscure species that had not yet been given a common English name finally earned one. It was an era marked by mutual backscratching and backstabbing, as men (and a few women) competed with one another to have a new species of bird named after them – and by doing so, win everlasting fame.
#### Notes
1 John Latham, _A General History of Birds_ (Winchester, 1821–8). This built on his earlier work, _A General Synopsis of Birds_ (London 1781–1801), and its later summary, _Index Ornithologicus_ (London, 1790–1801).
2 Charles Swainson, _The Folk-Lore and Provincial Names of British Birds_ (London, 1885).
3 Charles Swainson quoted in W. H. Mullens and H. Kirke Swann, _A Bibliography of British Ornithology from the earliest times to the end of 1912_ (London, 1917).
4 Quoted in _A Bibliography of British Ornithology_.
5 Robert Hughes, _The Fatal Shore_ (London, 1996).
6 ibid.
7 ibid.
8 Ian Fraser and Jeannie Gray, _Australian Bird Names: A Complete Guide_ (Collingwood, 2013).
9 See Robin Jackson, _A Guide to Scots Bird Names_ (revised edition, Aboyne 2013), and also Francesca Greenoak, _British Birds: their Folklore, Names and Literature_ (London, 1997).
10 Stephen Moss, 'The Bird Poetry of John Clare' (unpublished dissertation, Cambridge, 1981).
11 Quoted in the Introduction to Robinson's 1982 anthology _John Clare's Birds_ , co-authored with the ornithologist Richard Fitter (Oxford, 1982).
12 James Fisher, op. cit.
i Even those who come from Dartford have never thought of the place with much affection. In 2010, the comedian Mark Steel returned to his home town to perform a live show for BBC Radio 4. He was not very flattering, but the locals were even less so: one audience member said that if Dartford were a three-course meal it would be 'McDonald's, KFC and a kebab'. Writing at the turn of the nineteenth century, William Cobbett made an equally barbed comment: 'After you leave Dartford, [the county of Kent] becomes excellent.'
ii Today, Dartford has long been joined to the urban sprawl of London – though officially, at least, it remains in the county of Kent. Bexleyheath, as it has now become, lies on the other side of the border, in Greater London, having been sucked into the metropolis following boundary changes in 1965.
iii Now _Sylvia undata._
iv Under current taxonomic rules there are thirteen: Balearic shearwater, Spanish imperial eagle, Caucasian snowcock, rock partridge, red-legged partridge, Marmora's warbler, Balearic warbler, crested tit, Corsican nuthatch, Scottish crossbill, parrot crossbill, citril finch and Corsican citril finch. However, as more and more species are 'split' (see Chapter 7) there may well be more – though nothing like as many as can be found in Australia.
v Also sometimes spelt as _betcherrygah_ , _betshiregah_ , _bougirigard_ and _budgeragar_.
vi The lyrebird also managed to acquire a wide range of alternative names, including 'pheasant', 'paradise-bird' and 'peacock-wren' – the latter aptly described by Fraser and Gray as 'surely one of the most creative or desperate of the many compound names coined in Australia'.
vii Clare was equally contemptuous of Keats's lack of first-hand knowledge of the natural world, pointing out that his more famous rival 'often described nature as she appeared to his fancies and not as he would have described her had he witnessed the things he describes...'
viii Clare's use of folk names is not some rustic affectation, as can be seen from the sonnet 'The Fern Owl's Nest', in which this alternative name for the nightjar is integral to the experience evoked by the poem:
The weary woodman rocking home beneath
His tightly banded faggot wonders oft
While crossing over the furze-crowded heath
To hear the fern owl's cry that whews aloft
In circling whirls and often by his head
Wizzes as quick as thought...
ix Willow warbler, song thrush, meadow pipit, hen harrier, great spotted and lesser spotted woodpeckers, respectively.
x For Canada goose, knot and ferruginous duck.
xi For more on William MacGillivray, the man described as 'Scotland's forgotten genius in the field of natural history', see Chapter 5.
xii If you're wondering about the meaning of the word 'cirl', John Latham coined it in 1783, as a direct translation of Linnaeus's scientific name _Emberiza cirlus_ , which we still use today. It comes from the Bolognese dialect of northern Italy, and may derive from an obsolete verb _zirlare_ , meaning 'to whistle like a thrush'. This would make cirl bunting another example of an onomatopoeic name hidden beneath layers of translation. It also means that neither of the two commonly used pronunciations – 'curl' and 'sirl' – is correct, as in Italian the combination of letters 'ci' is pronounced as 'ch'. So next time you come across the bird, confuse your companions by calling it a 'chirl bunting'.
# EPONYMS AND EXPLORATION
_Bird Names go Global_
Remember, they only name things after you when you're dead or really old.
Barbara Bush
## _1: The Museum Man_
As the drizzle continued to fall, soaking the rocks, grass and my clothes, I began to regret my earlier enthusiasm for our nocturnal expedition. It was a damp and uncomfortable August night, and a film crew and I were perched on slippery rocks at the top of Hirta, the largest island of the St Kilda archipelago. Our mission: to record the sounds of Leach's petrels returning to their nests.
For several hours, all we could see were mysterious shapes looming out of the murk, caught momentarily in the beams of our torches before disappearing into the darkness. These were Leach's petrels, though I could only be sure because of their extraordinary calls, which sounded like an amusement arcade machine suffering from radio interference: a constant outpouring of squeaks, clicks and yelps that, had I not known what was making them, would have chilled my blood.
Of all Britain's breeding birds, Leach's petrel is one of the hardest to see. That's not because it is especially rare – there are roughly fifty thousand pairs, far more than the UK population of coots, cormorants or grey herons – but because it chooses to nest on a few far-flung islands off north-west Scotland. Even here, in places such as North Rona, the Flannan Isles and St Kilda, this tiny seabird is almost impossible to find, as it only returns to its breeding colonies after dark, to avoid being attacked by predatory gulls.
When Leach's petrels have finished breeding, they head straight out over the open ocean. They then spend the autumn and winter on the high seas, rarely venturing close to shore unless forced to do so by strong gales. So it is perhaps not surprising that this species was not formally described and named until 1820, the year after the birth of Queen Victoria, and long after the vast majority of Britain's breeding birds had already been discovered.
The story of how Leach's petrel acquired its name embodies the changes occurring during this era, and touches on the life of one of the most eccentric men ever to be commemorated in the name of a British bird: Dr William Elford Leach.
Born in 1790, William Leach was for eight years the Assistant Keeper of Birds at what is now the Natural History Museum, though in later life he became better known as a specialist in insects and crustaceans.
A man of small and delicate build, Leach lived in two small rooms in the museum itself, decorated with an array of skulls and stuffed bats, which he dubbed the 'skullery and battery'. To his colleagues' amusement he kept fit by vaulting over the back of a stuffed zebra in the middle of his office. But beneath his unconventionality, Leach had a sharp and enquiring mind, and in between these gymnastic sessions he kept a keen eye out for new specimens to add to the museum's growing collection.
Attending a major auction of bird skins and eggs in May 1819, Leach found his attention caught by Lot 78, which came with an intriguing description: 'An undescribed petrel with a forked tail, taken at St Kilda in 1818; the only one known (with egg)'. Bidding was brisk, but Leach managed to purchase the petrel and its egg for £5 15 shillings, equivalent to about £420 at today's prices.i
A year after the auction, the Dutch ornithologist Coenraad Temminck (after whom Temminck's stint is named) visited Leach at the British Museum and examined the specimen. He named it in honour of his host: _Procellaria leachii_ – Leach's petrel. Although the species now has a different scientific name, _Oceanodroma leucorhoa_ (which roughly translates as 'white-rumped ocean-runner'), the original English name still stands.
Leach was undoubtedly flattered by having this newly discovered seabird named after him. But he may also have been slightly embarrassed, because he must have known the species was not a completely new discovery. In fact, it had already been found by another ornithologist: William Bullock – the man who sold Leach the specimen in the first place. Yet although Bullock had obtained the specimen of the petrel on one of his many collecting trips, and must surely have realised it was new to science, he had – either through carelessness or indifference – neglected to give it a name.
*
William Bullock was, like Leach, an eccentric and extraordinary man. Described by Barbara and Richard Mearns (authors of _Biographies for Birdwatchers_ ) as 'a naturalist, collector, traveller, antiquary, auctioneer, and showman',1 he ran a travelling museum that contained well over 30,000 exhibits, including more than 3,000 specimens of birds and their eggs, and attracted hordes of visitors. But in 1819 he decided to sell off this vast collection to fund a characteristically madcap scheme to make his fortune in Mexico. This was how Leach came to purchase the bird that still bears his name.
We have no idea whether Bullock was aware of the fact that he had a prior claim to the naming of this enigmatic seabird, but we do know that three years later, in 1822, he and his son (also called William) finally headed off to Mexico, where they hoped to make a killing by investing in silver mines. However, the scheme was not a success, and the elder William later returned to Britain via the United States, writing several books about his travels along the way. He died in Chelsea, in 1849, at the age of seventy-six.
William Bullock may have been denied his moment of fame by Leach, but he did live to see a bird named after him – and his son. While in Mexico, they had continued their obsession with shooting and collecting birds. Later, another William – Swainson – chose to name a new species of bright-orange-and-black bird discovered there after them, as Bullock's oriole. 'This, the most beautiful of the group yet discovered in Mexico', wrote Swainson,ii 'will record the name of those ornithologists who have thrown so much light on the birds of that country'.
However, despite its auspicious start Bullock's oriole has had a rather chequered history: for many years it was considered merely a well-marked race of the Baltimore oriole, but it has now been granted full specific status once again. (This is discussed in more detail in Chapter 7.)
Back in England, William Leach had also thrown himself into the nomenclatural fray, and was busily giving names to a number of new species. Unfortunately, most of these were later quietly dropped, as he had chosen some rather odd naming systems, such as the nine different genera of birds he christened with variations on the name Caroline (or the Latin 'Carolina'), including anagrams such as _Cirolana_ , _Conilera_ and _Rocinela_. As Barbara and Richard Mearns suggest, although some people took these to be a tribute to Queen Caroline of Brunswick, the wife of George IV, it is more likely that they referred to a mysterious woman with whom Leach was in love.
All this was taking its toll on Leach's already fragile mental and physical health. In 1821 he fell ill, having 'overworked himself to such a degree that his health and mind became affected', and retired to Italy, accompanied by his devoted sister.
But although gone, he was certainly not forgotten. As well as Leach's storm petrel, William Elford Leach's name featured, in one form or other, in the scientific names of well over a hundred species. More importantly, he was ultimately recognised for his efforts in putting museum science in Britain on a more organised and professional basis.
For decades, zoologists on the continent had pioneered the revision of Linnaeus's original method of classifying species into new and different groups, based on a range of different characters, rather than just one. This helped them work out the relationships between different species – along with larger groups such as families – with a far greater degree of accuracy than before. But British scientists, slavishly following the teachings of their Swedish master, had become inflexible in their thinking, refusing to budge from the original approach. As a result, British zoological studies had become stagnant and ossified.
Leach had always corresponded with French zoologists (something both difficult and unpopular at a time when the Napoleonic Wars between Britain and France were at their height), and soon realised that they were onto something. Despite opposition from his peers, through his determination, insight and hard work, he managed to drag the science of zoology in Britain into the modern era, and give it the status it deserved.
In 1836, the year of William Leach's untimely death from cholera, a parliamentary enquiry delivered its verdict on the management of the Natural History Museum. Leach, noted one observer 'was the first to make the English acquainted... with the progress that had been made in natural science on the Continent. Thus a new impetus was given to zoology'.iii
His career may have been cut short by illness but, by dragging the infant science of zoology into the modern world, William Leach had paved the way for the next generation, and in particular two far more famous men: Alfred Russel Wallace and Charles Darwin. It is fitting that he should be memorialised in the name of one of our scarcest and most elusive seabirds.
## _2: Eponymous Birds_
William Leach and George Montagu are just two of several thousand people commemorated in the vernacular or scientific names of the world's birds – or, as these names are often known, 'eponyms'.2 The heyday for this trend was during the eighteenth and nineteenth centuries, when new species of birds were being discovered at a tremendous rate.
This was fuelled by the expansion of the British Empire and its associated exploration of the globe, and especially by the rise of a new breed of intrepid gentleman-explorers, whose gung-ho attitudes would lay the foundation for much of our knowledge and understanding of the world's birds.
Appropriately, many of these men (and a handful of women) are still commemorated in the English names of birds. But even at a time when the fashion for eponyms was at its height, getting your name attached to a new species was not quite as easy as it might appear.
First, you had to travel to distant places with a shotgun over your shoulder, and enough supplies to enable you to spend long and arduous periods in the field. Then you had to find a bird that had never been seen before, shoot it, retrieve the lifeless corpse, and preserve this for long enough for someone else to examine it – ideally one of the museum-based ornithologists back home in Britain. They needed to verify that what you had found was indeed new to science, and not simply some aberrant form, or unknown plumage, of an already familiar species.
Finally, you had to persuade them to honour you by giving it your name – either in English or Latin, or preferably both. But this presented a further problem. The protocol was very clear: you were _not_ under any circumstances permitted to name a bird after yourself, but you _could_ name it after a fellow ornithologist, who would then, perhaps, return the favour by naming another new species after you.
That was the theory. Unfortunately, however, this cosy mutual arrangement did not always work. In 1826, Charles Payraudeau named Audouin's gull, which he had discovered on a visit to Corsica, after his 'excellent ami' Jean Victor Audouin. However, Audouin somehow neglected to return the favour, and so while his own name lives on in the field guides, the unfortunate Monsieur Payraudeau is consigned to ornithological obscurity. I like to imagine Payraudeau writing a series of increasingly desperate letters to Audouin, imploring him to fulfil his side of the bargain.iv
Charles Payraudeau was perhaps unfortunate – after all, many other ornithologists of his day did end up being commemorated in eponymous bird names. Yet looking down a list of the 250 or so different birds that occur regularly in Britain, it immediately strikes me how few are named after people. The reason is obvious: by the time it became the norm to do so, from the late eighteenth century onwards, the vast majority of common British birds had already been found, and so already had vernacular names.
Apart from Montagu's harrier and Leach's petrel, only one regular British breeding bird has an eponymous name: Cetti's warbler, named after an eighteenth-century Italian Jesuit priest, Francesco Cetti.
If we include passage migrants, wintering species and occasional breeders, six other species occurring in Britain are named after people: Bewick's swan, Cory's shearwater, Lady Amherst's pheasant, Temminck's stint, Richard's pipit and Savi's warbler. But when we consider vagrants – those rare birds that are occasional wanderers to our shores – then the picture changes dramatically, with another 42 species with eponymous names, making 51 in all.v Again, this makes sense: these mostly live in the more far-flung corners of the globe, so throughout this era were still being discovered and named.
When it comes to nationality, as you might expect, most of those who have given their names to our birds are British – eighteen in all – followed by eight Italians, seven Germans,vi five Frenchmen and three Americans, with one Swede and one Dutchman. Some are so obscure we know virtually nothing about them, while others are amongst the most famous naturalists of all time.vii
The vast majority of them – roughly three-quarters – were mainly active during the nineteenth century, though many were born in the eighteenth and did some of their most important work at this time. Spanning such a crucial period in British and world history, their names are not simply a narrative about ornithology and the naming of birds; they also conceal very human stories, which give us a detailed picture of the world in which they lived and worked.
When I look down the list of the forty-three individuals,viii however, one thing immediately strikes me: the gender imbalance. Only three are women.
They are Eleonora of Arborea, a fourteenth-century Sardinian princess, politician and military leader (Eleonora's falcon); the eighteenth-century museum curator Anna Blackburne (Blackburnian warbler); and the British aristocrat Sarah Amherst (Lady Amherst's pheasant). All three birds that bear their names are, like the women they honour, fascinating in their own particular way.
Eleonora's falcon is a slim but powerful raptor which has strayed northwards to Britain from its Mediterranean breeding grounds on only a handful of occasions, since the first was seen over Formby Point in Lancashire in August 1977. Uniquely amongst European birds of prey, Eleonora's falcons delay their breeding until late in the summer, so that they have chicks in the nest during September, the peak period for the autumn migration of songbirds. This guarantees a ready supply of fresh food for their youngsters, which having fledged then follow their parents all the way to Madagascar, where they spend the winter. The next spring, they return to the cliffs of islands such as Mallorca, where I have watched flocks of them during late April catching dragonflies in bright blue skies.
Despite its striking appearance – the species comes in two colour phases, with some birds similar in plumage to a hobby, others all dark – Eleonora's falcon was not described until 1839, from specimens shot by the Italian soldier and naturalist Alberto della Marmora (of whom more later) on the island of Sardinia. Della Marmora sent his specimens to his colleague in Turin, Giuseppe Gené, who decided to commemorate the location of the birds' discovery by naming it after Eleonora of Arborea, a Sardinian princess famed for leading her troops into battle. This may at first appear a rather odd choice, until we discover that she is not only still the island's greatest heroine, but also passed a law protecting the falcons, by preventing the young being taken from the nest – an act of benevolence far ahead of its time.
Anna Blackburne, after whom one of the most beautiful of all North American wood warblers is named, cannot claim quite such a heroic life. Indeed, as Barbara and Richard Mearns point out in their companion volume dealing with North American eponyms, _Audubon to Xantus_ , 'she is scarcely known at all'.3 But in her own quiet way she bucked the trend for women of her time, by becoming a professional naturalist in all but name.
Anna was born in 1726 near Warrington in Lancashire (now Cheshire), and after her mother's death spent much of her life looking after her rich industrialist father in his home at Orford Hall. Being both wealthy and unmarried, she could and did devote much of her spare time to studying nature, an interest she inherited from her father, who conveniently was a friend of Thomas Pennant.
To develop her knowledge of the natural world, Anna first learned Latin, and then began a lengthy correspondence with none other than Linnaeus himself. Inspired by the great man, she set up her own museum at Orford Hall, which eventually housed a major collection of birds, plants and insects. Some of these were sent by another of her correspondents, the German ornithologist and collector Peter Simon Pallas, who also had several species of bird named after him.ix
Meanwhile, Anna's younger brother Ashton had travelled to North America, where like many young men of this era he indulged his passion for shooting every living thing within range of his gun. Writing in 1784, after Ashton's death, Thomas Pennant was suitably impressed at his dedication and industry:
To the rich museum of _American_ birds, preserved by Mrs. ANNA BLACKBURN [ _sic_ ], of _Orford_ , near _Warrington_ , I am indebted for the opportunity of describing almost every one known in the provinces of _Jersey_ , _New York_ and _Connecticut_. They were sent over to the Lady by her brother, the late Mr. _Ashton Blackburn_ ; who added to the skill and zeal of a sportsman, the most pertinent remarks on the specimens he collected for his worthy and philosophical sister.4
Sadly, we do not know if the colourful, black and fiery orange specimen that Pennant named Blackburnian warbler was among those shot by Ashton, but, on the balance of probability, we can infer it was. This has led some to believe that the warbler was named after Ashton, and not his sister; but given Pennant's closeness to the family it is likely that in naming the bird he intended to commemorate both.x
The third member of this diverse trio of women after whom birds on the British List are named is the redoubtable Sarah, Lady Amherst. Born the Hon. Sarah Archer in July 1762,xi she was widowed with three children before she was forty. But less than a year after her first husband's death, she married again, to a man ten years younger than her: William, Lord Amherst.
Despite Sarah's relatively advanced age, she went on to bear him four children, making seven in all, before in 1823 Lord Amherst took up his post as Governor-General of India. Life there was far from easy, marked by war, mutiny and, for Lady Amherst, by then in her sixties, a dose of cholera that would have killed anyone with a weaker constitution.
After Sir Archibald Campbell, commander of the British forces, had made peace with the King of Burma, he presented Lord and Lady Amherst with two stunningly beautiful pheasants, which in 1828 they eventually brought back to England. A year later, the London taxidermist Benjamin Leadbeater named the species Lady Amherst's pheasant, 'as a tribute due to the distinguished lady to whom ornithologists are indebted' – even though all she had actually done was arrange for the birds to be transported back to England.
Sadly, the rest of her life was marred by tragedy. Having lost one son, Jeffrey, to fever in India, two of her remaining three sons also pre-deceased her, before her death in 1838, aged seventy-five.
The bird named after her has enjoyed mixed fortunes, too. Confined to a forested stretch of Asia from Myanmar in the west to southern China in the east, Lady Amherst's pheasant would not normally appear in any book on British birds. But from the late nineteenth century onwards, a number of these exotic gamebirds were bred and released in the grounds of stately homes such as Woburn Abbey in Bedfordshire.
For almost a century, the birds quietly got on with their lives in the woods and fields of Bedfordshire, near to where they had originally been released, with a few crossing the county border into Hertfordshire and Buckinghamshire. By the late 1960s, the population stood at between 100 and 200 pairs. But because they were regarded as little more than an escaped cagebird, few birders took any real interest in them.
Then, in 1971, all that changed, with the surprising (and, in hindsight, misguided) decision to elevate Lady Amherst's pheasant to the official British List, where it joined other originally feral species such as the Canada goose, mandarin duck, and its cousin the golden pheasant. This was done on the grounds that the population was thought to be self-sufficient, a decision that now appears to have been based on rather dubious evidence.
Almost as soon as the species gained official status, and birders finally began to seek it out to add to their lists, numbers began to fall. By the 1980s there were perhaps 200 individuals, and by 1990 as few as 60. I recall one day in early 1987 taking a walk around the woods near Ampthill in Bedfordshire where the few remaining pheasants were supposed to be. Having drawn a blank, I returned to where I had parked my car, only to notice a group of birds at the back of a field. Four magnificent male Lady Amherst's pheasants, their impossibly long tails barred with black-and-silver, were feeding unobtrusively along the edge of a wood. It was the only time I ever saw the species in Britain.
Soon afterwards, Lady Amherst's pheasant did make a brief comeback, but at the turn of the millennium the population was down to as few as 30 individuals. By then, its days as a British bird were numbered: genetic bottlenecks meant the species could not recover unless new birds were released, something no one was willing to do. At the time of writing the population is down to a single male, so the species is inevitably doomed to vanish from Britain.
As to why Lady Amherst's pheasant declined so rapidly, there are a number of reasons. According to the acknowledged authority on introduced species in Britain, Sir Christopher Lever,5 predation by foxes and the taking of their eggs and chicks by magpies may be one factor, as may human disturbance and loss of habitat. But ironically, two other introduced species may also be at least partly to blame. Goshawks, whose native population was augmented through the late twentieth century by birds released by falconers, would surely make short work of such a showy and colourful bird. Meanwhile another species introduced from Asia, Reeves's muntjac, has destroyed much of the woodland understorey where the pheasants find shelter and make their nests.
It's a matter for debate to whether the loss of such a bird is a cause for concern, considering that it should never really have been present in Britain in the first place. But when the species in question is one of just a handful of regularly occurring British birds named after people – and the only one named after a woman – it is, to my mind, rather sad.
*
So what of the only other regular British breeding bird to be named after a person: Cetti's warbler? It is one of five species of warbler whose names sound like the defensive line-up of an Italian football team: Bonelli, Cetti, Savi and Marmora, with Moltoni on the bench. Perhaps they'll play against a tight German midfield of Pallas, Radde and Ruppell. Or maybe they'll come up against the English forward line of Hume, Blyth and Sykes. This virtual soccer team comprises the men after whom no fewer than thirteen species of warbler on the British List are named (Pallas and Bonelli have bagged two species each).
Unlike the small, relentlessly active and difficult-to-identify birds that now bear their names, most of which are best told apart by their songs, these men were a pretty diverse bunch. Although by definition they were all amateur or professional ornithologists, for the most part they had other professions and callings, too. Francesco Cetti was a Jesuit priest and mathematician; Alberto della Marmora rose to become a general; Gustav Radde was an apothecary (equivalent to a modern-day pharmacist); Colonel William Henry Sykes was an army officer and later MP for Aberdeen; and Allan Octavian Hume – dubbed 'The Father of Indian Ornithology' – served as a colonial administrator in the Indian Raj.
As with the majority of people who have given their names to birds, all but three of the eleven were most active during the nineteenth century, the unofficial Age of Ornithological Discovery. Only Francesco Cetti (1726–78) and Peter Simon Pallas (1741–1811) lived earlier, while Professor Edgardo Moltoni was born at the tail end of the nineteenth century in 1896, and died in 1980.xii
The circumstances under which these eponymous ornithologists discovered their species were often quite random. On 22 September 1856, while exploring the remote Transbaikalia region of south-east Russia, the German explorer and naturalist Gustav Radde came across a bulky-looking leaf-warbler. Greenish-brown and with a distinctive pale stripe above its eye, it was hiding in the unlikely surroundings of a kitchen garden in a town with the tongue-twisting name of Kulussutajevsk.
Having finally managed to get reasonable views of the bird, Radde confidently declared it to be a new species: _Phylloscopus schwarzi_ , named after his friend and fellow-Prussian Ludwig Schwarz, the astronomer to the expedition. Two decades later, in 1881, another pioneering explorer of this region, the Yorkshireman steel manufacturer and inveterate traveller Henry Seebohm, gave the species its vernacular name, Radde's bush-warbler, later simplified to the one we use today.xiii
Savi's warbler – named after Paolo Savi, another academic who taught at the University of Pisa – was one of the last of Western Europe's breeding birds to be identified, less than 200 years ago. In 1824, when Savi was examining a small, nondescript bird he had shot some years earlier in Italy, he finally realised that he had discovered a species new to science.
Ironically, the first example had been originally found not in Italy but at Limpenhoe, a village along the Yare Valley in Norfolk, but was misidentified – by none other than Coenraad Temminck – as a Cetti's warbler. A decade or so later, the Norfolk bird was correctly re-identified as a Savi's, and the rather tatty specimen remains in Norwich's Castle Museum to this day.
I can still remember my excitement on seeing Cetti's and Savi's warblers at Stodmarsh in Kent back in the mid-1970s. They had recently colonised Britain, and were beginning to establish thriving populations in south-east England. To be honest, I hardly saw them at all, as both are so elusive that I barely glimpsed either species for longer than a couple of seconds.
I did, however, hear them and, though these two unstreaked, brown warblers may look superficially similar, it is hard to imagine two more strikingly different songs. Savi's warbler – like its cousin the grasshopper warbler – produces a low, insistent, buzzing sound, more like some kind of cricket than a bird, or like a fishing reel being rapidly unwound. This seems to melt into the evening soundscape, making it hard to pick out amongst the various chirping and buzzing insects, especially at a distance. The same definitely _cannot_ be said of Cetti's warbler, which has one of the loudest and most distinctive songs of any British bird. As writer and self-confessed 'bad birdwatcher' Simon Barnes has noted, there are several mnemonics that mimic its rhythm: including the unforgettable: 'Me – Cetti? If you don't like it... FUCK OFF!'
The Greta Garbo of birds, this small, grey and chestnut brown warbler spends its whole life hiding away in dense vegetation alongside water, and is hardly ever seen for more than a moment or two. But when you hear an explosion of notes emerge from a dense thicket of brambles, the identity of the singer is never in question.
These days this is a familiar sound throughout much of southern Britain, now that Cetti's warbler has firmly established itself as a breeding resident – unlike most other warblers, which migrate south in autumn, Cetti's stays put all year round. Walking around my local patch on the Somerset Levels, I hear its familiar song in every single month of the year. But until spring 2015, when a Savi's warbler unexpectedly turned up near my home, I hadn't heard one singing in Britain for almost forty years.
When I first saw Savi's warbler back in the 1970s I – and most other observers – assumed it would soon establish itself as a regular British breeding bird. But for some unknown reason, although it is a common breeder in the Netherlands and northern France, Savi's warbler remains only a sporadic visitor on this side of the Channel. Perhaps the one I heard in Somerset will be in the vanguard of a new invasion and, like its cousin Cetti's warbler, this elusive species will finally establish itself as a truly British bird, adding a second Italian eponym to the list of our regular breeding species.
## _3: Into the North_
When we think of the great polar explorers, the same names usually come to mind: Scott, Amundsen and Shackleton in the south, and Peary and Nansen in the north. James Clark Ross is not as well-known as any of these legendary men, and yet arguably he did more to pave the way for their achievements than any other early explorer.
Today, James Ross is commemorated in the name of one of the most beautiful and mysterious of all Arctic birds: Ross's gull. Ross's gull gives the lie to the widely held belief that gulls are ugly, boring, and all look the same. Unlike its larger, bulkier and more cantankerous relatives, it is a graceful, delicate creature, wafting buoyantly over the sea ice towards a passing ship like a visiting angel.
During the brief arctic summer, when Ross's gulls gather to breed on the rapidly thawing tundra, their normally snow-white breast acquires a delicate pinkish tinge, almost as if the bird is blushing at its newfound sexual potency. But few people have ever seen a Ross's gull in all its rosy glory. Indeed, given that this species lives in some of the remotest regions of the planet, very few people have seen one at all. James Fisher called Ross's gull 'one of the most mysterious birds in the world',6 and although more than sixty years have passed since he wrote those words, this enigmatic creature is still one of the most sought-after of all the Arctic birds that occasionally wander south to Britain.
When I was a teenage birder, back in the mid-1970s, this shadowy bird was barely on my radar. But a chance encounter in the summer of 1974 changed all that. A young Ross's gull turned up on the Dorset coast, near the holiday resort of Christchurch – a sighting described at the time as 'the most remarkable ornithological event of the year'.7
My school friend Daniel and I were camping nearby in the New Forest and, having heard about the bird's presence, cycled as fast as we could to where it had been seen. After failing to see the bird on the first day, we returned two days later. Finally we were rewarded, with a minute or so's sighting of this Arctic wanderer, as it drifted past us and eventually out of sight.
More than forty years on, I can still remember the sheer thrill of encountering this legendary bird in such bizarre circumstances: next to a beach at the height of the summer holidays, surrounded by families sunbathing and making sandcastles. There and then I resolved to find out more about this mysterious species – and about the man whose name it bears. So who exactly was James Clark Ross, and what connection did he have with his eponymous gull?
Like many of his fellow Victorian explorers, James Ross's life story reads like something out of the _Boy's Own Paper_. Born in 1800, he originally went to sea as a twelve-year-old ship's lad, on a vessel captained by his uncle, John Ross – himself a distinguished polar explorer and decorated veteran of the Napoleonic Wars.
During the next few years, the expedition's ships travelled back and forth through the Arctic seas, on an ultimately unsuccessful search for the legendary North-West Passage. For these early nineteenth-century explorers, the obsessive quest for this sea route from the Atlantic to the Pacific was the equivalent of later generations climbing Mount Everest or landing on the moon, and in its time perhaps even harder to achieve. For although the Vikings had sailed their longboats far into this land of ice-floes, bitter winds and violent seas, no-one had ever managed to find their way through to the other side – and on to the lucrative trade markets of Asia. It was a quest that would ultimately provide new insights into the geography and natural history of this remote and forbidding region, yet one that would also cost the lives of many brave men.
One young explorer determined to make his name by discovering the fabled North-West Passage – or die trying – was James Clark Ross. By the time he reached his early twenties, Ross had already risen to the rank of midshipman (an officer cadet, one of the junior ranks). More importantly for our story, he had also assumed the mantle of the expedition naturalist, and was keen to acquire interesting new specimens.
So when on a fine, cold day in June 1823, Ross spotted an unusual-looking gull flying alongside the ship close to the Melville Peninsula in the northern reaches of Arctic Canada, he was determined to get a closer look. As he approached, and realised that it was something different, he raised his musket and blasted the unfortunate seabird to kingdom come.
The ship's captain William Parry recorded the event for posterity in the expedition's journal:
Mr Ross had procured a specimen of gull having a black ring round its neck, and which in its present plumage, we could not find described. This bird was alone when killed but flying at no great distance from a flock of [Arctic] tern, which latter it somewhat resembles in size as well as in its red legs; but is on closer inspection easily distinguished by its beak and tail, as well as by a beautiful tint of most delicate rose-colour on its breast.8
When the expedition returned home later that year, having once again failed to discover the North-West Passage, Captain Parry presented the distinguished zoologist John Richardson with the bird and mammal specimens the crew had collected during the voyage. Richardson examined the mystery gull, and rightly concluded that it was indeed a species new to science. He named it the 'cuneate-tailed gull' (from the unusual wedge-shaped tail), and gave it the scientific name _Larus rossi_ ,xiv after its young discoverer.
And there things might have stood, were it not for what might be charitably called a mix-up or, less generously, an attempt by a fellow ornithologist, William MacGillivray, to grab all the credit for himself. Rather like William Leach a few years earlier, MacGillivray does not come out of the affair with much credit.
MacGillivray was not a man who you would think needed to resort to underhand tactics to cement his position. Later on in life, through his magisterial five-volume _History of British Birds_ , published over fifteen years from 1837-52, he would make huge advances in establishing the study of birds as a respectable and proper science. He also gave names to several new species, including the harrier that, following the example of the French ornithologists, he named after George Montagu.
Ironically, for such a key figure in the naming of birds, MacGillivray himself only has a single species named after him: MacGillivray's warbler, a scarce songbird that breeds in the forests of western USA and Canada, and spends the winter in Central America. Not only did MacGillivray never see the species that bears his name, he never even visited the continent where it lives. He owed the dubious honour to his long friendship with the legendary North American bird artist John James Audubon, whom he had helped to write the text to his monumental work _The Birds of America_.xv
Despite – or perhaps because of – his fame, MacGillivray was a troubled and difficult man, with a notoriously abrasive personality. This may have been because he had been born with the stigma of illegitimacy, so always saw himself as an outsider. He seemed to challenge himself at every opportunity: at just twelve years old he began his studies at Aberdeen University. Whether through genuine poverty or simple bloody-mindedness – or possibly a combination of the two – he would walk home to the Hebridean Isle of Harris at the end of each academic year: a distance of 180 miles.
Later on, he outdid even that feat of endurance. Deciding it would help his fledgling career as an ornithologist if he visited the British Museum's bird collections in London, he elected to walk there (via a circuitous route in order to see more of England), tramping more than 800 miles in all weathers.
In later life, his temper was legendary, as the American bird collector Elliot Coues later noted:
MacGillivray appears to have been of an irritable, highly sensitized temperament, fired with enthusiasm and ambition, yet contending... with poverty, ill-health, and a perhaps not well-founded, though not therefore the less acutely felt, sense of neglect; thus ceaselessly nerved to accomplish, yet as continually haunted with the dread of failure.9
Nor was MacGillivray very tolerant of other people's weaknesses, as Coues went on to explain:
He never hesitated to differ sharply with any one, or to express his own views pointedly... he scarcely disguised his contempt for triflers, blockheads, pedants, compilers, and theorizers.10
Whether MacGillivray thought John Richardson fell into any of these categories we cannot be sure; though given that Richardson was later knighted for his contributions to polar exploration and science, that is perhaps unlikely.
John Richardson had named Ross's specimen at a public meeting in Edinburgh (where both he and MacGillivray worked), so might have reasonably assumed that the species' name was now firmly established. But for some reason he neglected to confirm the new name in print until more than a year later, in an appendix to Parry's journal of Ross's voyage.
In the meantime, MacGillivray had also examined the specimen, and before Parry's journal appeared, published its name as 'Ross's rosy gull' ( _Larus roseus_ ). Because MacGillivray's chosen name was the first to appear in print, under the strict rules of scientific nomenclature it took priority, and he took the credit. So to Richardson's frustration his original (albeit rather cumbersome) name was relegated to the footnotes of ornithological history. Meanwhile, at the age of just twenty-four, James Ross had become the youngest person ever to be commemorated in the name of a British bird, an honour he still holds.
In later life, James Ross continued to make pioneering and hazardous expeditions to the High Arctic. His finest achievement, the discovery of Ross's gull notwithstanding, was reaching the Magnetic North Pole in June 1831, a discovery that allowed sailors to fix their position more easily, wherever they were in the world's oceans, and which undoubtedly helped to save many lives in the years that followed. He also explored the southern oceans, circumnavigating the whole of Antarctica, where his voyages are commemorated in place names such as the Ross Sea, Ross Island and the Ross Ice Shelf.
Another of Ross's fellow polar explorers, Edward Sabine, also had a species of gull named after him. Although Sabine was twelve years older than Ross, the two nevertheless became lifelong friends after they met on one of those early expeditions to search for the North-West Passage. During one of these, on 25 July 1818, Sabine and Ross sighted a series of rocky islands 20 miles offshore, and trekked off across the sea ice to investigate. On arrival, the two men noticed some unusual gulls breeding alongside Arctic terns, sporting forked tails, dark grey heads and black bills with canary-yellow tips. True to form, they took aim and shot them, later sending the skins back to London via a passing whaling ship. These ended up in the hands of Edward's elder brother Joseph, who presented them to the members of the Linnaean Society, and named them in honour of his brother: _Larus sabini_ – Sabine's gull.
Eventually Ross returned to Britain and was knighted for his achievements. Having reached his early forties, he finally settled down with his new wife Anne, their marriage producing four children. But the call of the Arctic proved too strong, and in the late 1840s he made his last voyage north. On his return, he continued to work as the leading authority on polar navigation, dying at his Buckinghamshire estate in 1862, the same age as the century.
*
A few years after Ross saw his gull for the very first time, another new species of gull was discovered on an Arctic voyage. Franklin's gull superficially resembles our own familiar black-headed gull, but has a darker grey back, a fully black (rather than brown) head, a white eye ring and a bright red bill and legs, making it appear altogether more becoming than its commoner relative – as if it were wearing make-up.
The gull was named after the man who arguably ranks at the very top of the hall of fame of polar explorers: Sir John Franklin. But unlike Ross, Franklin's distinguished career as a soldier, explorer and politician did not end in comfortable retirement back home in the English countryside. Instead he would suffer extraordinary hardship, tragedy, and a slow and painful death in the remote and frozen Arctic.
Even before he headed northwards, John Franklin's life was marked by extraordinary feats of endurance and suffering. Having joined the Royal Navy at the age of thirteen, he sailed to Australia on a voyage aiming to circumnavigate that vast and unknown land. But his vessel was wrecked, and he and the crew – most of them suffering from scurvy – found themselves marooned on a coral reef for several weeks before they were finally rescued.
Before he reached his twentieth birthday Franklin had fought at the Battle of Trafalgar, during which all but seven of the forty-seven men alongside him on deck were killed. Following this lucky escape, he went on his first voyage to the Arctic, an unsuccessful quest to reach the North Pole; he was then chosen to lead an expedition to try to discover the fabled North-West Passage. He was well aware that polar exploration had never been easy, but the trials and tribulations endured by Franklin and his men on this and later voyages almost beggar belief.
In August 1819, a reconnaissance party led by Franklin left their ship and headed off across the rapidly freezing tundra, where they underwent unimaginable hardships, eventually being forced to eat boiled leather and lichen to avoid starving to death. One member of the party, driven insane by hunger, even shot and killed a fellow crewman so he could eat his flesh.
Incredibly, Franklin, Richardson and a handful of others did somehow manage to survive their terrible ordeal. When they returned home the following autumn, they were given a heroes' welcome. You might imagine that these horrendous experiences would have put them off polar exploration forever, but over the next three decades Franklin continued to go off to search for the North-West Passage, continually being thwarted by the seemingly impenetrable barrier of the sea ice.
It was during one of these voyages, on 6 June 1827, that the bird that would be named after him was found: a male Franklin's gull, shot on the Saskatchewan River. In some ways it is surprising that this species had not been discovered earlier, for it is not a bird of the High Arctic like Sabine's and Ross's gulls. Franklin's gulls breed on the vast open prairies of Canada south and eastwards to Montana and Minnesota, and spend the winter along the Pacific coasts of Central and South America, migrating through much of the United States along the way.
In May 1845, almost two decades after he found his eponymous gull, Franklin and his 133-man crew set sail on what would be their final expedition, again heading north and west to chart the possible route through to the Pacific Ocean. This time the sea ice was so impenetrable that the two vessels became stuck – not just for one winter, but for two long years in a row.
Eventually, the fateful decision was taken to abandon the stranded ships and trek across the ice, in the hope of reaching land and safety. But this brave attempt was doomed to failure from the very start: already weakened by a combination of starvation, scurvy and the bone-chilling cold, the entire crew perished. Later a stone cairn was discovered, which revealed that Franklin had actually died on 11 June 1847, not long before his men had left the boat.
Today, a dramatic painting of the last days of this doomed expedition hangs in the National Maritime Museum in Greenwich, in east London. Painted by William Thomas Smith, it bears the dramatic title 'They forged the last link with their lives'. Taken from a letter from Sir John Richardson to the Prime Minister Lord Palmerston, this sentence commemorates the fact that, despite their deaths, the expedition had proved enormously valuable in surveying new territory.
These early voyages in turn paved the way for later exploration of both the Arctic and Antarctic, in which ornithologists continued to play a crucial part. The best known of these were both on the fateful expeditions with Captain Robert Falcon Scott. Edward 'Bill' Wilson perished with Scott on his last, doomed expedition to reach the South Pole, but not before he had produced a series of accurate and beautiful sketches of Antarctic birds. Apsley Cherry-Garrard did survive, but had the grim and thankless task of searching for – and ultimately discovering – the bodies of Scott and his men, frozen in their hut just 11 miles from safety.
Cherry-Garrard's legendary quest to collect the eggs of the emperor penguin was later documented in a book aptly titled _The Worst Journey in the World_ , in which he famously wrote that 'Polar exploration is at once the cleanest and most isolated way of having a bad time which has been devised.'11
The other important ornithological legacy of Captain Scott's ultimately disastrous expedition was the last, almost unbearably moving letter he wrote to his wife Kathleen. Knowing he was doomed to die, Scott sent her clear instructions on how to bring up their infant son, asking her to 'Make the boy interested in natural history if you can, it is better than games.' That young boy would grow up to be the best known naturalist, wildlife artist and conservationist of his, and arguably any other, era: Sir Peter Scott.
As for the legendary North-West Passage, the route through the ice was finally found during the first decade of the twentieth century by the Norwegian explorer Roald Amundsen – later conqueror of the South Pole – using charts made by Franklin and his crew on their final, ill-fated voyage. So Sir John Franklin and his brave men did not, it seems, die entirely in vain.
Yet even though our knowledge of the Arctic and its wildlife continued to expand, the gull discovered by James Ross remained a genuine enigma. As recently as 1938, well over a century after Ross's gull was first described, ornithologist Bernard Tucker could still write that 'Very few... have seen this gull alive.'12
During the intervening years, other polar explorers had occasionally come across this elusive bird. Their excitement when they did so is evident from this evocative entry from Fridtjof Nansen's diary for 3 August 1894:
Today my longing has at last been satisfied; I have shot Ross's gull. This rare and mysterious inhabitant of the unknown north, which is only occasionally seen, and of which no one knows whence it came and whither it goeth, which belongs exclusively to the world to which the imagination aspires, is what I have always longed to discover.13
A decade later, one mystery was finally solved, when the breeding grounds of Ross's gulls were finally discovered – completely by chance. The distinguished Russian ornithologist Sergei Aleksandrovich Buturlin was visiting Yakutia – a vast and remote region of north-eastern Siberia almost as big as India – when he found a colony of Ross's gulls nesting on the tundra outside the village of Pokhodsk.
Breeding colonies have since been discovered elsewhere in the Arctic, including north-east Greenland and the far north of Canada, not far from where Ross made his original find. The gulls probably spend the autumn and winter months somewhere in the North Atlantic, although even with the advances in tracking technology we have at our disposal today, we still do not know exactly where.14
## _4: Scotland's Forgotten Genius_
William MacGillivray is Scotland's forgotten genius in the field of natural history. There is no question of his pre-eminence as a naturalist, of his originality of mind, of his skill as a writer and above all his talents as an ornithologist.15
This tribute from William MacGillivray's biographer Robert Ralph identifies one aspect of the character of this pioneering nineteenth-century ornithologist. We have already seen aspects of his other, darker side, in the way he took the credit for the naming of Ross's gull, over the prior (but unpublished) claim of his colleague John Richardson, and in his ability to both offend and to find offence with others. But we have also witnessed his generosity of spirit, as shown in naming Montagu's harrier after his illustrious predecessor.
If MacGillivray is remembered nowadays at all, it is for what should have been the most influential and important ornithological work of the nineteenth century. The fact that he is largely forgotten, except by those few people who have actually read his writings, is partly due to his stubborn character and partly, as so often with the vagaries of fame, simply down to bad luck.
MacGillivray's _A History of British Birds_ is rarely referred to nowadays, and read even less frequently: the five stout, leather-bound volumes sit forgotten on library shelves, or linger unsold in the catalogues of antiquarian booksellers, gathering literal and metaphorical dust. After all, why would anyone bother to read a work now almost two centuries old, when so much has been discovered and written about our avifauna since it was published?
I first came across a set of MacGillivray's masterwork in a second-hand bookshop in Cambridge in the early 1980s, when I was writing my dissertation on the bird poems of John Clare. It was priced at a prohibitive £200, way beyond my student pocket, but thanks to the generosity of the bookseller I was able to make detailed notes on the books' contents without actually having to make a purchase.
Almost forty years later, I have finally acquired a set of my own, and have been entranced by the contents. The blend of forensic detail, together with extensive descriptions of each species' habits and behaviour, all wrapped up in that unmistakable musty smell of antiquarian books, take me straight back to this exciting era when so many discoveries about Britain's birds were being made. It also makes me appreciate the efforts of men such as MacGillivray and his English contemporary William Yarrell, who did so much to extend and consolidate our knowledge of Britain's birdlife.xvi
Like his predecessor George Montagu, MacGillivray was an obsessive completist, as the books' full title bears witness:
_A History of British Birds, Indigenous and Migratory:_
_Including_
_Their Organization, Habits and Relations;_
_Remarks on Classification and Nomenclature;_
_An Account of the Principal Organs of Birds, and_
_Observations Relative to Practical Ornithology._
Never let it be said of Victorian writers that they didn't provide enough information for their readers! So how was MacGillivray's work received? It is fair to say that reactions were somewhat mixed, as this comment, written eighty years after publication, reveals:
To MacGillivray has always belonged the enviable reputation of writing one of the most original histories of British birds we possess. The consensus of opinion accords his _History_ the merit of being original and accurate... but at the same time his peculiar methods of classification and nomenclature (most undoubtedly original) naturally aroused criticism and even condemnation.16
Talk about damning with faint praise – rarely can the word 'original' have been used so pejoratively. Yet the co-author of that verdict, William Mullens, was not unsympathetic to MacGillivray's work, considering it superior to that of any of his contemporaries. Mullens was also scathing about the critics who condemned MacGillivray for his eccentric ordering of families and obsession with detailed accounts of each species' anatomy, accusing them of having 'broken the heart of the greatest ornithologist this country has ever possessed', and almost preventing the completion of what he called 'one of the greatest books on British birds'.
It wasn't just MacGillivray's taxonomy that baffled his readers, but also the names he used for so many familiar species. A glance at the entries in the first volume, published in 1837, yields a truly baffling assortment, each invented by MacGillivray to impose some sense of order and logic on avian nomenclature. Fortunately for the modern reader, MacGillivray often added the more widely accepted name (usually the one we still use today) as an alternative:
The Mountain Finch, or Brambling
The Black-throated Grosbeak, or Hawfinch
The Red-fronted Thistlefinch, or Goldfinch
The Mountain Linnet, or Twite
In the second volume, published two years later in 1839, the names were even more eccentric. The thrushes appeared as black, ringed, chestnut-backed, red-sided and variegated – for blackbird, ring ouzel, fieldfare, redwing and the eponymous White's thrush respectively.
'Shore pipit' (for rock pipit) aside, the pipits and larks retained their more conventional names, but when it came to the pipits' cousins, the wagtails, MacGillivray went off-piste once again. Pied wagtail was the only instantly recognisable name, as the grey wagtail became 'grey-and-yellow', and the white wagtail (the continental race of pied) became 'grey-and-white'. For the yellow wagtail, and its continental relative the blue-headed wagtail, he invented an entirely new name – 'quaketail' – with the respective epithets 'green-headed' and 'blue-headed'.
I can see what MacGillivray was trying to do. A tidy-minded man, borderline obsessive-compulsive by nature, he was simply attempting to add some kind of rationality to bird names, where little or none had existed before. For instance, later in the same volume blackcap appeared as 'black-capped warbler', and whitethroat 'white-throated warbler' – both names are perfectly logical, yet nevertheless utterly inelegant.
MacGillivray created more evocative names for other members of the warbler family. He coined 'grasshopper chirper' for grasshopper warbler, and 'sedge and marsh reedlings' for sedge and reed warblers (the much rarer marsh warbler was yet to be discovered in Britain). But my favourite is 'Provence furzeling' – undeniably a more accurate name than the Dartford warbler, and redolent of the names still used for the species in Dutch, German and the Scandinavian languages, although not, oddly, in French.xvii
For the leaf warblers, he preferred 'yellow, willow and short-winged woodwrens' (wood and willow warblers and chiffchaff), while other new and rather bizarre names included 'long-tailed muffin' (long-tailed tit), 'hedge chanter' (dunnock), 'blue-throated redstart' (bluethroat) and 'white-rumped stonechat' (wheatear).xviii
It's easy to mock MacGillivray, especially as not a single one of the names he invented has stood the test of time; we might perhaps look on him more kindly if, as with so many of the new names coined by Thomas Pennant in the previous century, they had been adopted into general use. For who is to say that Pennant's 'tawny owl' is better than MacGillivray's 'tawny hooting-owl', or that the former's 'oystercatcher' is superior to the latter's 'pied oystercatcher'.
Sadly, for William MacGillivray, his plans to add rigour and logic foundered in the face of general usage and the mocking ridicule of his peers. In the meantime, Yarrell's shorter and more accessible work, also entitled _A History of British Birds_ (but lacking the long and convoluted subtitle), had become a popular bestseller. One of the main reasons for its wide appeal amongst the reading public was the inclusion of attractive woodcuts to illustrate each species.
Ironically, MacGillivray had also planned to have his work illustrated, and had painted many excellent plates himself, which had been praised by no less an authority than John James Audubon as equal to anything that great American artist had achieved himself. But stricken by poverty, as he was throughout his life, MacGillivray simply could not afford the cost of including these in the finished work.
Yarrell's chosen method of publication may also have helped promote his writing. Like many novelists of the period, including Dickens and Trollope, he issued his work not in one thick, heavy and expensive volume at a time, but periodically, in thirty-six affordable monthly parts.xix Yarrell also managed to complete his _History_ by 1843, whereas his rival took almost a decade longer to do so. This gave Yarrell a crucial head start in the market, from which MacGillivray never recovered.
MacGillivray's heartfelt opening words to the Preface of Volume IV, written in March 1852 from the Devon resort of Torquay, sought to excuse the long delay since the previous volume:
As the wounded bird seeks some quiet retreat, where, freed from the persecution of the pitiless fowler, it may pass the time of its anguish in forgetfulness of the outer world; so have I, assailed by disease, betaken myself to a sheltered nook, where, unannoyed by the piercing blasts of the North Sea, I had led to hope that my life might be protracted beyond the most dangerous season of the year...
This acid blend of bitterness and self-pity, lightened by a soupçon of black humour, is emblematic of MacGillivray's complex character. He was always an outsider, impatient and intolerant of others, and just as rigorous, it seems, at judging himself.
Yet the seaside cure seems to have worked – at least temporarily – for soon afterwards he returned north to his home in Aberdeen, where he was Professor of Natural History at the university. Here, on 31 July, he wrote the Preface to the fifth and final volume of his epic work. He once again outlined the case for the _History's_ importance, and added a telling comment on the poor reviews he expected to receive:
He who possesses the greatest contempt for public opinion is always the most anxious for general applause. I should, no doubt, be very well pleased to be commended; but I do not now anticipate great distress from the most virulent censure.
Ironically, MacGillivray never read any reviews of this final volume, whether praiseworthy or critical. For on 8 September 1852, barely a month after his final bitter sideswipe at the critics, he died, at the age of fifty-six. He was buried in Edinburgh's New Calton Cemetery, next to his late wife and two of his children, who had died in infancy. For half a century his grave remained unmarked, until some of his relatives and former students raised the funds for a huge and impressive granite monument that still stands today.
Ironically, after his death his fame grew, with the circulation of his posthumously published book _The Natural History of Dee Side and Braemar_ , privately produced with funding from none other than Queen Victoria. Towards the end of the Queen's long reign, the pioneering ornithologist Professor Alfred Newton bestowed upon MacGillivray the grand epitaph that 'after Willughby, MacGillivray was the greatest and most original ornithological genius... that this island has produced'.17
William MacGillivray may have lived a troubled life, and died in sad and difficult circumstances. But like the equally irascible George Montagu, he left us a masterwork that – if the critics had not become so hung up on his fanaticism for detail and eccentric names – might have changed the way we look at Britain's birds forever. Instead, it was Yarrell's _History_ , written in a less rigorous but undoubtedly more popular style, with more acceptable English names, that went on to influence birdwatchers and ornithologists for almost a century afterwards.xx
Had things been different, and MacGillivray's view prevailed, birders might even now be referring to furzelings and quaketails, reedlings and hedge chanters, blue-throated redstarts and long-tailed muffins. I have a tinge of regret this never came to pass, and that instead we are saddled with the far more familiar, yet perhaps less imaginative, names we use today.
## _5: Exploration and Empire_
As the nineteenth century rolled on, Britons continued to travel around the world, and discover more and more birds, for which new names had to be found – many of them prolonging the fashion for eponyms. Exploration and empire-building provided plenty of reading material for generations of schoolboys, with the dramatic episodes of polar exploration we have already witnessed – packed with adventure, suffering and derring-do – winning an eager audience through a plethora of classic Victorian books for boys.
But other travellers preferred to pursue their quarry at a slower and more gentlemanly pace, and in warmer, more equable climes. Typical of this latter breed was the Reverend Henry Baker Tristram (1822–1906). Tristram spent much of his working life as a country parson, though he eventually became Canon of Durham Cathedral. Yet despite his weighty clerical responsibilities, he still managed to pursue his abiding passion: the collection and study of the birds of North Africa and the Middle East.
He was so enthusiastic about his travels around the Middle East that, like the parody of an absentee clergyman Dr Vesey Stanhope in Anthony Trollope's _Barchester Chronicles_ , he appears to have spent considerably more time in the deserts of Palestine than the cloisters of Durham. What his long-suffering wife Eleanor (who bore him seven daughters and a son during their fifty-three-year marriage) thought of his peregrinations is not recorded.
Not for nothing was Henry Baker Tristram known as 'The Great Gun of Durham'.xxi But he did not simply aim and fire at any bird that moved, like some of his less fastidious contemporaries. As well as being a fine shot, he had a keen eye for the unusual. Moreover, at a time when few ornithologists ventured into these regions, he had the field more or less to himself. As a result eleven species of bird have, at one time or another, borne his name.xxii The best-known of these is Tristram's starling (often known as Tristram's grackle), a striking, glossy-black bird with orange wing-linings which can be seen over much of Israel, and whose loud, wolf-whistling call is very distinctive.
Canon Tristram lived a long and satisfying life, dying at the age of eighty-three in spring 1906. Before this, however, he had experienced a Damascene conversion. He acknowledged that amassing vast collections of bird skins and their eggs – in his case comprising well over 20,000 different specimens – was not how the study of birds should continue in this new century. Indeed, for the last two years of his life he served in the honorary role of Vice-President of the newly formed bird protection organisation the RSPB, just after King Edward VII had granted the Society its royal charter.
From this point onwards, the pendulum slowly began to swing, away from killing birds with shotguns and towards studying them using binoculars. Tristram was among the last of his kind: gentleman-naturalists and crack shots, with the time, money and inclination to travel to far-flung corners of the world, indulging their passion for killing birds in the name of furthering the science of ornithology.
But just before this, there had been one last hurrah for the old guard. During the final decades of Queen Victoria's reign, history provided the perfect opportunity for ambitious young men to contribute to the naming of the world's birds. It came about because of the most important (and arguably most controversial) institution of that complex and fascinating era: the British Empire.
*
On the last day of July 1912, amidst the searing summer heat, the city of Etawah in the Indian state of Uttar Pradesh came to a standstill. In the usually thriving and noisy bazaar, every shop and stall remained closed for the day, while the people mourned the loss of a man whose death, at the age of eighty-three, had just been announced. Yet this man was not some fêted maharajah – indeed, he was not Indian at all – but an officer of the ruling, and mostly despised, British Raj.
Allan Octavian Hume – the man mourned not just in Etawah but right across the vast nation of India – was by turns a civil servant, political reformer, co-founder of the Indian Congress Party, poet and naturalist.
So how did this modest, hardworking (and for his day surprisingly liberal) man earn such love and respect from the people he ruled? How on earth did he find time, in his busy political and administrative life, to put together the largest private collection of Indian birds ever created, and end up having no fewer than fourteen different species of bird named after him?xxiii And what does his life story tell us about the largest empire the world has ever seen?
At its height, the British Empire's colonies, dominions and protectorates covered almost one-quarter of the globe's land – more than 13 million square miles – and ruled over more than 450 million people, about one in five of the world's population.
From small beginnings, thanks to pioneering expeditions led by men such as John Cabot, Walter Ralegh and Francis Drake during the fifteenth and sixteenth centuries, the Empire had expanded hugely, to include large parts of the Americas, Africa, Asia and the Pacific. That oft-repeated cliché, 'the Empire on which the sun never sets' was not simply a jingoistic metaphor, but the literal truth: Britain's imperial possessions were so scattered around the world that there was always daylight falling on the Union Flag somewhere.
The vast size of the British Empire was not only its greatest strength, but also a fatal weakness. As other nations grew envious of Britain's powers, they wanted to diminish them and grab a share of the booty for themselves. Even though the British Empire continued to expand until the 1920s, the seeds of its downfall were already sown: the new world order decided at the Treaty of Versailles in 1919 would never allow one nation to be so globally dominant again.
During the second half of the nineteenth century, all this was yet to come. The Empire was still supreme, especially in India, where more than 300 million people lived under British rule, governed by men like Allan Octavian Hume.
Hume's career was certainly a colourful one. Having arrived in India in 1849, at the age of twenty, he had spent the next few years rising up the Indian Civil Service to become the chief officer of a district twice the size of Wales. Then, in 1857, the rebellion known as the Indian Mutiny began nearby. Showing great courage, Hume stormed a temple where the rebels were holed up, and later – despite having recently recovered from cholera – led a charge that forced them to retreat.
During the following decade, as the political situation eventually settled down, Hume finally had time to pursue his passionate interest in Indian birds. He began to amass a vast assemblage of skins, nests and eggs that eventually topped 100,000 specimens, the second largest private collection in the world after that owned by Walter Rothschild at Tring. In one single expedition, Hume collected no fewer than 1,200 skins of 250 different kinds of bird, of which eighteen species were new to India.
To house this extraordinary collection, he spent £20,000 (equivalent to more than a million pounds today) building a large extension on his home, in which row after row of beautifully crafted wooden cabinets housed his precious eggs and skins. He wrote several books on Indian birds, and started his own quarterly journal, entitled _Stray Feathers – a journal of ornithology for India and his dependencies_ , in which he wrote racy accounts of his collecting trips around the sub-continent.
In 1879, on what would be his final expedition, he noticed that the feathers in a ceremonial head-dress worn by a local official came from a species of gamebird he did not recognise. After sending hunters out to procure live examples, he named the bird Mrs Hume's pheasant (after his beloved wife Mary), the name still used today.xxiv
In 1882, at the age of fifty-three, Hume retired from the Indian Civil Service and returned to his home in Simla, to begin work on his life's masterwork: a book that would include every species of bird found in India. Soon afterwards, however, disaster struck. Having spent the winter of 1884 at his other residence on the lowland plains, Hume returned to discover – to his unimaginable horror – that all his research papers, weighing several hundredweight and containing more than twenty-five years of detailed notes and information, had been taken down to the local bazaar by his servants to be sold as waste paper. His dream was in tatters, and the world was deprived of what would have been the definitive work on Indian ornithology.
But Hume left another, arguably far more important, legacy. Having been so cruelly thwarted in his ornithological ambitions, he could have retired from the fray. Instead, he chose to devote the rest of his life to politics. Ornithology's loss was India's gain, because in 1885 Hume was instrumental in setting up the Indian National Congress. Guided by Mahatma Gandhi, who became the party's leader in 1921, Congress grew to be India's dominant political force, and spearheaded India's eventual independence from Britain in 1947. Without Hume's vision and hard work on behalf of the Indian people, it could be argued that independence might have come much later, or in a very different form.
In 1890 Hume took a trip back home to Britain, where on arrival he was informed of Mary's death back in Simla. Four years later, he decided to leave India for good, settling in the suburban district of Upper Norwood in south London, where he lived quietly until his death in 1912.
With Hume's passing, an era was over. The days of the British Empire were numbered, and the world was about to change dramatically with the onset of the First World War.
The way birds were named would change too, as the age of exploration drew to a close, and with it the tradition of new species being named after their discoverers, which had lasted for almost 200 years from the early eighteenth through to the late nineteenth centuries.
From now on, new bird names would be decided by committees using pen and paper, rather than by pioneering individuals carrying shotguns. The days when an amateur naturalist such as Allan Octavian Hume could push the boundaries of our knowledge of the world's birds, and give his name to so many species, were finally – and permanently – at an end.
#### Notes
1 Barbara and Richard Mearns, _Biographies for Birdwatchers: The Lives of Those Commemorated in Western Palearctic Bird Names_ (London, 1988).
2 For more about these men and women, see Bo Beolens and Michael Watkins, _The Eponym Dictionary of Birds_ (London, 2014).
3 Barbara and Richard Mearns, _Audubon to Xantus: The Lives of Those Commemorated_ _in North American Bird Names_ (London, 1992).
4 Thomas Pennant, _Arctic Zoology_ (London, 1784).
5 In Christopher Lever, _The Naturalized Animals of Britain and Ireland_ (London, 2009).
6 In James Fisher and Ronald Lockley, _Seabirds_ (London, 1954).
7 In _British Birds_ magazine, vol. LXXIX (1975).
8 William Parry, _Journal of a second voyage for the discovery of a North-West Passage_ (London, 1824); quoted from Michael Densley, _In Search of Ross's Gull_ (Leeds, 1999).
9 Elliot Coues, quoted in _A Bibliography of British Ornithology_ , op. cit.
10 ibid.
11 Apsley Cherry-Garrard, _The Worst Journey in the World_ (London, 1922).
12 In _The Handbook of British Birds_ , edited by Witherby et al (London, 1938–41).
13 Fridtjof Nansen, _Farthest North_ (London, 1897).
14 For more information on this mysterious seabird, I highly recommend Michael Densley, _In Search of Ross's Gull_ (Leeds, 1999).
15 From Robert Ralph, _William MacGillivray_ (London, 1993).
16 _A Bibliography of British Ornithology_ , op. cit.
17 Alfred Newton, quoted in _William MacGillivray_ , op. cit.
i To his delight he also acquired a great auk (also with an egg) for £16 – about £1,200 today. Less than half a century later, this statuesque flightless seabird would become globally extinct.
ii William Swainson is himself commemorated in the names of three North American birds – a hawk, a thrush and a warbler, as well as a host of tropical species, most of which have since been given new names.
iii The zoologist John Edward Gray, speaking to an 1836 parliamentary investigation into the management of the British Museum.
iv Actually he is not totally forgotten: Payraudeau's collection of bird specimens can still be seen at a small museum in La Chaize-le-Vicomte in the Vendée.
v In taxonomic order, these are: Steller's eider, Barrow's goldeneye, Fea's petrel, Scopoli's shearwater, Wilson's and Swinhoe's storm-petrels, Baillon's crake, Allen's gallinule, Macqueen's bustard, Baird's sandpiper, Wilson's phalarope, Wilson's snipe, Cabot's and Forster's terns, Sabine's, Bonaparte's, Ross's, Franklin's and Audouin's gulls, Pallas's sandgrouse, Tengmalm's owl, Eleonora's falcon, Pallas's, Hume's, Radde's, western and eastern Bonelli's, Marmora's, Ruppell's, Moltoni's subalpine, Pallas's grasshopper, Sykes and Blyth's reed warblers, White's, Swainson's and Naumann's thrushes, Moussier's redstart, Blyth's pipit, Cretzschmar's and Pallas's reed buntings, Blackburnian and Wilson's warblers.
vi Including some born in Prussia (part of which is in present-day Poland) – borders were fairly fluid at this time.
vii At one end of the fame scale we have Gilbert White (White's thrush), while at the other end there is 'Monsieur Richard of Lunéville' (the capital of Lorraine in eastern France). In October 1815 he 'collected' (i.e. shot) the bird that now bears his name, Richard's pipit. Yet today that is the only thing we know about him.
viii Forty-three, as opposed to fifty-one, because some people are commemorated in more than one species. Alexander Wilson (1766–1813) – a Scotsman who left to seek his fortune in the United States and became known as 'the father of American ornithology' – and Peter Simon Pallas each have four species named after them, while Edward Blyth and Franco Andrea Bonelli each have two.
ix Including Pallas's sandgrouse, Pallas's warbler, Pallas's grasshopper warbler and Pallas's reed bunting – all of which have been seen in Britain.
x Blackburnian warbler has only ever been recorded twice in Britain, both times in October on offshore islands: on Skomer in 1961 and on Fair Isle in 1988. I was the only birder on Fair Isle at the time who managed to miss the bird!
xi Sarah was the eldest daughter of Lord Archer, Baron of Umberslade (near Tanworth-in-Arden, Warwickshire).
xii This anomaly is explained by the fact that the bird bearing his name, Moltoni's subalpine warbler, was only recently elevated to full species status, having been separated from its very similar-looking cousin, the subalpine warbler.
xiii In France and Spain, however, Radde's warbler retains its link with the man originally honoured: _pouillot de Schwarz_ and _mosquitero de Schwarz_ respectively.
xiv Now in its own monotypic genus, _Rhodostethia rosea_.
xv Three species of bird are named after Audubon: a shearwater, a warbler and an oriole.
xvi In 1830, Yarrell was the first person to distinguish between the two species of 'wild swans', winter visitors from the north. He named the smaller of the pair Bewick's swan, after Thomas Bewick, who had died two years earlier. Later he popularised the name whooper swan for the larger species.
xvii _Provenceångare_ (Swedish), _Provencesanger_ (Danish and Norwegian), _Provencegrasmücke_ (German) and _Provenceaalse_ (Dutch). In French the Dartford warbler's name is _Fauvette pitchou_.
xviii A hundred and fifty years later, a cabal of late twentieth-century ornithologists attempted to standardise British bird names once again – and, just like MacGillivray, they failed to do so. See Chapter 7.
xix A hundred and thirty years later, from 1969–71, IPC magazines and John Gooders followed in Yarrell's footsteps with the ten-volume partwork _Birds of the World_ , a seminal influence on birders of my generation (see Prologue).
xx Yarrell's _History of British Birds_ was reprinted several times during the remainder of the nineteenth century, and formed the basis for a very popular single-volume work, Howard Saunders, _An Illustrated Manual of British Birds_ (London, 1889).
xxi He was also, more affectionately, dubbed the 'Sacred Ibis', after the symbol of the British Ornithologists' Union – hence the title of an excellent biography of Tristram by W. G. Hale, _Sacred Ibis: The Ornithology of Canon Henry Baker Tristram_ (Durham, 2016).
xxii Tristram's wheatear, serin, starling, warbler, bunting, scrubfowl, honeyeater, flowerpecker, storm-petrel, woodpecker and pygmy parrot. Only four (the starling, warbler, bunting and storm-petrel) still carry his eponym.
xxiii Hume's ground tit, wheatear, babbler, hawk-owl, lark, wren-babbler, blue-throated barbet, parakeet, leaf warbler, owl, swiftlet, whitethroat, treecreeper and reed warbler. Of these, only six (the wheatear, hawk-owl, lark, leaf warbler, owl and whitethroat) still bear his name.
xxiv Sadly, as with so many of that region's birds, it is now threatened by over-population, habitat loss and hunting.
# TWENTIETH-CENTURY FLOCKS
_The Names we use Today_
Names are not always what they seem.
Mark Twain
## _1: Redbreasts and Hedge Sparrows_
Max Nicholson – birdwatcher, scientist and pioneering conservationist – spanned the twentieth century like an ornithological Colossus. Born in 1904, the same year as Fats Waller, Salvador Dalí and Cary Grant, he lived to see the turn of the new millennium, before dying in 2003, in his ninety-ninth year. More than any other person, before or since, he witnessed – and to a great extent was also responsible for – the science of ornithology being dragged out of a bygone era and into the modern age.
In a long, active and quite remarkable life, Max Nicholson was instrumental in either founding or reforming many of today's leading conservation organisations, including the RSPB, BTO (British Trust for Ornithology), Natural England and, perhaps most crucially of all, the WWF (World Wide Fund for Nature). He also ran the Battle of the Atlantic shipping convoys during the Second World War and, as a senior civil servant, was at Winston Churchill's right hand at the historic Second World War conference with Stalin and Roosevelt at Yalta.i
Even before Nicholson's birth, however, what he memorably called 'the Victorian leprosy of collecting' was starting to give way to a more benevolent approach to bird study. This new, and very different, way of relating to the natural world would be achieved by looking through binoculars, rather than down the barrel of a gun. Developments in technology meant that optics were rapidly displacing firearms, and the time when an unusual bird needed to be shot to confirm its identity was finally coming to an end. The profession of taxidermy, so popular during the Victorian age, would soon become (quite literally) a dying art.
The turning point came in 1901, the year the old queen was finally laid to rest, with the publication of a book by Edmund Selous, simply entitled _Bird Watching_.1 Remarkable though it may now seem, this is the first recorded use of this phrase in the English language, at the start of a century that would end with birding – as it is now called – having become one of the most popular leisure activities in Britain.
Since Selous's book appeared, what we know about Britain's birds has increased exponentially. Much of this has been achieved through a dedicated cohort of 'amateur' (albeit highly skilled) birders who, even today, provide much of the raw data and field observations used by professional scientists, via surveys conducted by organisations such as the BTO. But from the turn of the last century, in Britain at least, amateurs and professionals alike turned their focus onto known, named species, rather than unknown, unnamed ones.
Changes to bird names remained the province of a small group of men who had the time, energy and inclination to sit on official committees. Top of the tree was the British Ornithologists' Union (BOU), an august body whose pronouncements on matters ornithological were handed down like tablets of stone to the masses below.
The BOU still makes the final decisions on the official 'British List', now standing at over 600 species,ii and also adjudicates on the names we call these different species – or perhaps, I should say, the names we are _supposed_ to use. For as we have already seen, bird names have long proved stubbornly resistant to what the mandarins of British ornithology have decreed that we should call them, even to the present day.
Take one of our commonest and most familiar birds, the robin. As late as 1952, the BOU insisted on calling this species by the official name 'redbreast', even though the name 'robin' had been widely used since at least the seventeenth century, and probably for far longer. Astonishingly, 'robin' was not formally adopted by the BOU as the official name for _Erithacus rubecula_ until the next checklist was published, in 1971.iii
During the course of the twentieth century, an interest in birds – and indeed all of nature – also became far more egalitarian. Once purely the preserve of a small and elite group of professional ornithologists, it was beginning to be enjoyed by people at every level of society.
Typical of the new breed of birdwatchers was the Cheshire-based ornithologist T. A. (Thomas) Coward. Although the elaborately moustachioed Coward was the middle-class son of a religious minister and businessman, he carefully cultivated a 'man-of-the-people' image, with his flat cap, pipe and bicycle. And despite their superficial differences in social class and appearance, both he and Max Nicholson (who came from the Anglo-Irish landed gentry) shared the same mission: to popularise the hobby and pastime of watching birds to the broadest possible audience.
As a working journalist, Coward did this initially as a contributor to the _Manchester Guardian_ 's 'Country Diary' column, which he wrote for many years until his death in 1933. But he is best known today for what could arguably described as the first modern 'field guide': a handy, portable aid to enable the small but growing cohort of birdwatchers to identify the species they were seeing.
_The Birds of the British Isles and Their Eggs_ , published in two stout volumes in 1919 (a third, containing background information on bird behaviour, was added in 1926), was ideal. The leather-bound, gilt-embossed books were small enough to fit into a coat or jacket pocket, but packed with enough detail to enable the quick and easy identification of unfamiliar birds. Indeed, they were so ahead of their time that they were still being used by post-war birdwatchers such as Bill Oddie and Ian Wallace well into the 1950s.
Being a forward-looking birdwatcher, you might expect Coward to have jettisoned the old names and embraced the new ones. And apart from a few exceptions, such as calling tits 'titmice', and giving equal weight to 'green plover' alongside lapwing, that's exactly what he did. Browsing its pages, we find that the names are more or less the same as in a modern field guide, and certainly far more familiar than those found in most Victorian bird books.
But there is one notable exception. On page 233 of Coward's first volume, sandwiched between the wheatear and the dipper, there is 'Hedge-Sparrow'. Younger readers may be puzzled by this name, though anyone over fifty years old will surely find it familiar. It refers to the species we now call the dunnock, the only member of the accentor family to occur regularly in Britain.
The name 'hedge sparrow' (with or without the hyphen) has a long and distinguished pedigree, having first been recorded by the Tudor priest and tutor John Palsgrave (as 'hedge sparowe') in 1530. By the early nineteenth century, however, professional ornithologists preferred the more taxonomically correct 'hedge accentor', to distinguish this species from the house and tree sparrows, which are in a completely different family. But despite persistent moves to have this adopted as the official English name, it never caught on, probably because the name 'accentor' is clearly one invented by scientists rather than by ordinary folk, and sounds somehow 'foreign'.iv
Yet still the hedge sparrow's clearly unsuitable name continued to be the subject of debate. Writing in 1895, the pioneering bird protectionist W. H. Hudson observed:
Most people know that a sparrow is a hard-billed bird of the finch family, and that the subject of this notice is not a sparrow, except in name... 'How absurd, then, to go on calling it a sparrow!' certain ornithologists have said from time to time, and have renamed it the hedge-accentor. But as Professor Newton has said... a name which has been part and parcel of our language for centuries, and which Shakespeare used,v 'is hardly likely to be dropped, even at the bidding of the wisest, so long as the English language lasts'. Now, as the English tongue promises to last a long time, it seems safest to retain the old and in one sense, incorrect name.2
The name 'hedge sparrow' did indeed prove remarkably persistent, as Hudson had predicted. Long after goldcrest had replaced 'golden-crested wren', and willow warbler had supplanted the equally inaccurate 'willow-wren', it remained the standard name for _Prunella modularis_. Even in Phyllis Barclay-Smith's slim volume _Garden Birds_ , published in 1945,3 she referred to the 'hedge-sparrow'; indeed, she did not mention the name dunnock at all.
But moves were afoot to rename this shy and retiring little bird. Ever the iconoclast, in his 1951 Collins New Naturalist volume _Birds and Men_4 Max Nicholson called for seven changes in the names of common birds. Despite his elevated status, and legendary skills of persuasion, no fewer than six of the seven, including the proposal to change song thrush to 'throstle', never got off the ground.vi
However, his final suggestion – to rename the hedge sparrow as the dunnock – was ultimately taken up. As he wrote, in his characteristically no-nonsense style: 'Dunnocks do no harm to us, but have in return been exposed to the undeserved insult and injury of being miscalled hedge-sparrows by people too stupid to see the absurdity of such a name, or too timid and conventional to revert to the older, briefer and better one.'
As Nicholson was at pains to stress, 'dunnock' has a longer pedigree than the name it supplanted, the first recorded reference coming almost half a century earlier than 'hedge-sparrow', in 1483. It even appeared in Emily Brontë's _Wuthering Heights_ , in which the character Hareton Earnshaw is, after his mother's death in childbirth and his grieving father's descent into alcoholism, 'cast out like an unfledged dunnock'. This is, of course, a reference to the cuckoo's habit of laying its egg in a dunnock's nest, whereby after hatching the cuckoo chick then ejects any dunnock offspring from its new home.
Nicholson also pointed out that the yoking of this harmless species with the house sparrow – in those days a major agricultural pest – had led to the inadvertent destruction of dunnocks' nests by members of 'sparrow clubs', who were paid a small bounty for every house sparrow they killed, or whose nest they destroyed.
His sheer persistence, together with the perennial confusion caused by the linking of this species with the unrelated tree and house sparrows, eventually won the day. By the time of the BOU's 1971 book-length publication _The Status of Birds in Britain and Ireland_ , the official name had finally been changed to dunnock.
As Max Nicholson confidently predicted, the time 'when the whole company of British bird-watchers will call a dunnock a dunnock' has now arrived. Well, almost. For even today, more than sixty years after he first made his proposal to change the name, some older observers still continue to refer to the little bird creeping around their rockery or garden lawn as the 'hedge sparrow'. A small but significant triumph, perhaps, for the forces of the common man (and woman) against the ornithological Establishment.
*
Hedge sparrow is just one of many examples of a persistent trend running through our bird names: naming a bird after where it lives. Or perhaps it is more accurate to say, where it is _meant_ to live. For even among many of the names we use today, many of these habitat-based epithets are at best questionable, and at worst downright misleading.
Take two closely related species, the marsh and willow tits. Few other British birds are so badly named, for the marsh tit's preferred habitat is not wetlands, but deciduous woodlands, parks and large rural gardens. Willow tits, on the other hand, prefer damp, marshy areas, often nesting close to streams and rivers, or alongside disused gravel pits.vii
The story of how these two species acquired their ill-fitting monikers is a salutary reminder of the ingrained 'messiness' of English bird names, however much some tidy-minded people might wish to sort out what we call our birds, so that each species has the name most appropriate to its sound, appearance, habitat or status.
## _2: Tit-Willow and Willow Tit_
On a tree by a river a little tom-tit
Sang 'Willow, tit-willow, tit-willow'
And I said to him, 'Dicky-bird, why do you sit
Singing 'Willow, tit-willow, tit-willow'
'Is it weakness of intellect, birdie?' I cried
'Or a rather tough worm in your little inside?'
With a shake of his poor little head, he replied
'Oh, willow, tit-willow, tit-willow!'
The opening lines of 'Tit-Willow', from Gilbert and Sullivan's 1885 comic opera _The Mikado_ , are still sung today in theatres and church halls up and down the country. When W. S. Gilbert wrote these lyrics, he may have simply chosen the name 'tit-willow' because of its euphonious beauty. But as birders have long been aware, when the phrase is inverted it turns into the name of one of our scarcest breeding birds: the willow tit.
Yet ironically, at the time these quintessentially English lyrics were written, the willow tit was not thought to be a British bird at all. It was an unfamiliar foreign species, whose range stretched from France and Scandinavia in the west, to Siberia and Japan (the setting for _The Mikado_ ) in the east. For the small band of Britons who were interested in birds, from casual bird-spotters to serious ornithologists, the willow tit might as well have been on another planet.
Then, in 1897 – twelve years after the first performance of _The Mikado_ at London's Savoy Theatre – two German ornithologists made a dramatic discovery; one that would send shock waves through the higher echelons of British ornithology. While examining a drawer containing the skins of marsh tits at the British Museum of Natural History, Ernst Hartert and Otto Kleinschmidt came across two unusual specimens, which they immediately realised were not what they seemed.
The two Germans noted subtle anomalies in plumage features, including a pale panel on the wing, a sooty (rather than glossy) black cap and a rather bull-necked appearance. Taken together, these led them to conclude that these birds were not marsh tits at all, but willow tits. A quick check of their provenance revealed that they had been shot in Hampstead in north London, and so they became the very first record of the willow tit in Britain.viii
A year later, the news was published in the journal the _Zoologist_ , under the intriguing title 'A hitherto overlooked British Bird'. And overlooked it most certainly had been. For unlike new additions to the British List today – which are mostly global wanderers from Asia or North America that are discovered on some remote, windswept offshore island – the willow tits were here all along. They had been hiding, as it were, in plain sight.
Imagine the embarrassment and shame felt by the members of Britain's ornithological establishment, who had hitherto assumed a lordly superiority over their continental rivals. Now that two Germans had made such a momentous discovery right under their noses, some of our most senior bird experts had to eat humble pie.
Yet to be fair on the embarrassed Brits, the observant Germans had managed to unravel one of the trickiest identification puzzles posed by any regularly occurring British birds. Willow and marsh tits are so similar that, even today, experienced birders using top-of-the-range optical equipment often find it difficult, even impossible, to tell them apart.
Although the two species do show subtle plumage differences, the most reliable method of knowing whether a particular bird is a marsh or willow tit is by listening for their distinctive calls. Whereas the willow tit makes a nasal sound, repeating three or four notes in rapid succession, the marsh tit delivers a short, explosive call, rather like a sneeze.
Even after the discovery of willow tits in Britain, the controversy over the species' true status continued. Sceptics claimed that it was merely a race of the marsh tit, or even that the so-called new specimens were merely the juvenile plumage of that species. But in the first issue of the long-running journal _British Birds_ , published in 1907, Walter Rothschild (the bird collector from the famous banking dynasty) comprehensively refuted these arguments, taking a swipe at his critics as he did so: 'Most of our older ornithologists have failed, or rather refused, to see the differences between the English Marsh and Willow Tits, and again, in this instance, the old proverb, "None so blind as those that will not see," has abundantly justified itself.'5
He went on to nail his argument in favour of the marsh and willow tits being separate species by pointing out a truism that forms the basis of modern systematics: that 'no two races or subspecies of the same bird [i.e. species] can live side by side; they must either inhabit different geographical areas or be found at different vertical heights.'6 But willow and marsh tits were indeed living alongside one another throughout much of Britain: conclusive proof that they were indeed two different species. And so the willow tit gained the distinction of being the last 'undiscovered breeding bird' to be added to the British List, from those 1897 specimens, at least until the separation of the common and Scottish crossbills in the 1980s – but more on that story later.
However, it turns out that over half a century earlier there had been a missed opportunity to discover the willow tit in Britain. All it would have taken was for a curious reader to have questioned an assertion by William Yarrell in a supplement to his popular three-volume work _A History of British Birds_ , published in 1845. For in the entry for marsh tit, Yarrell wrote: 'Colonel Montaguix says he has seen the Marsh Tit excavating the decayed part of such trees.'7
As Mark Cocker points out in _Birds Britannica_ ,8 this 'describes with great accuracy the willow tit's burrowing technique' – for, uniquely amongst our songbirds, the willow tit excavates its own nest-hole in the manner of a woodpecker. Had any of Yarrell's readers been familiar with the two species from travels in continental Europe, they would have immediately realised that the birds seen by Montagu must have been not marsh, but willow tits. As it was, a further fifty-two years would pass before the species was eventually discovered in Britain – by the Germans.x
## _3: Reed Warblers and Roasted Larks_
Warblers are one of those groups of birds that beginner birders find very tricky to identify – the classic 'little brown (or olive, yellow or green) jobs'. So it's perhaps not surprising that plumage features are rarely used to tell one species apart from another – with the exception, as already noted, of the distinctive blackcap and whitethroat. And while song is often used to identify them, only the chiffchaff and grasshopper warbler have sound-based names.
But when it comes to birds named after habitats, the warbler family has few rivals. Of the thirteen regular British breeding species, no fewer than six are named after the places where they live: reed, sedge, marsh, wood, willow and garden warblers.
Some of these names are perfectly correct: reed warblers do live in reed beds, from which their chuntering, repetitive song echoes from mid-April when they arrive back from Africa, through to the end of the breeding season in late June. Wood warblers are also aptly named: these lemon-green sprites are denizens of the oak woods of the north and west of Britain, where they sing their delightful song during the spring and early summer, high in the dense canopy of leaves. Likewise the marsh warbler: this now very rare British breeding bird does indeed live in wetlands, usually nesting in dense patches of nettles, willowherb and meadowsweet right alongside the water.xi
But the other three habitat-based names are woefully inaccurate. Sedge warblers don't nest in sedges, preferring small bushes in the heart of reed beds; garden warblers are rarely found in gardens, unlike their commoner cousin the blackcap; and willow warblers are not especially attracted to willows, but can be found in a wide range of woodland habitats.xii
Many birds named after their habitat, like the warblers, belong to families where several species look and sound the same as one another, so there are few distinguishing calls or plumage features that we can use to tell them apart. The pipits – small, brownish songbirds that look superficially similar, and sound even more so – are a classic example.
Britain is home to four species: meadow, tree, rock and water pipits.xiii Rock pipits do live on rocky coasts (the only European songbird to have colonised this constantly changing habitat, where they forage for insects along the tideline), while water pipits are indeed usually found in freshwater wetlands. Meadow and tree pipits are less easy to separate by habitat, but can also be told apart by their behaviour: whereas meadow pipits parachute all the way down to the ground after delivering their song, tree pipits tend to return to their original perch – as you might expect, on the branch of a tree.
The larks, too, are well named: woodlarks often breed in young forestry plantations with open heath nearby; shore larks (a scarce winter visitor to eastern Britain) are usually found along or near the coast; and what more appropriate name could there be for the skylark? No other bird spends quite so long simply hanging in the air on a fine summer's day, delivering that extraordinary outpouring of musical notes for what seems like hours on end.
And yet although 'skylark' sounds like a name that has been with us since time immemorial, it was actually coined fairly recently. In 1678 John Ray referred to 'the common or Skie-lark'; before then this quintessential sound of summer was simply known as the 'lark' (from a now long-forgotten Germanic word meaning 'songster').xiv
Larks were commonly kept as cagebirds – often cruelly blinded, as it was falsely believed this would make them sing more sweetly – and were also killed for the pot. The bestselling _Book of Household Management_ , written by the Victorian domestic goddess Mrs Beeton, included a recipe for lark pie, with the immortal instruction to 'roll the larks in flour, and stuff them.'xv
We no longer eat larks, but the ubiquity of this familiar rural species has led to its virtually unrivalled prominence in our language and culture (matched only by that other much-admired songster, the nightingale). Following Shelley's famous ode to a 'Blithe Spirit', a later poet, George Meredith, was also inspired by the skylark's song. His verse 'The Lark Ascending' is perhaps the best-known bird poem in the English language, and was later the inspiration for one of our best-loved pieces of music, by the twentieth-century composer Ralph Vaughan Williams.
He rises and begins to round,
He drops the silver chain of sound
Of many links without a break,
In chirrup, whistle, slur and shake,
All intervolv'd and spreading wide,
Like water-dimples down a tide
Where ripple ripple overcurls
And eddy into eddy whirls...
Many people who have never even seen or heard a skylark (which, given its rapid decline over the past fifty years is probably quite a few), may nevertheless refer to the bird in phrases such as 'up with the lark', 'sing like a lark' and 'larking about'.xvi The idea of 'having a lark' – meaning to have fun – is thought to derive from nineteenth-century naval slang, when sailors might mess about high in the rigging of a ship – just as a lark hangs like a dust-speck up in the sky.xvii
*
Given our long and close relationship with birds, it's hardly surprising that some of our most familiar species are named after an artificial 'habitat', such as an agricultural crop or human habitation. These include the corncrake and corn bunting, house sparrow and house martin, barn owl and barn swallow (the official name for our familiar swallow).
As I look down this list, I am immediately struck by the fact that most of these species are in trouble. Having chosen to make their lives alongside us, and having prospered for centuries, they are now facing serious – and in some cases possibly terminal – declines.
The corncrake and the corn bunting are both named after our main arable crops, corn being a synonym for any staple grain, including wheat, barley or oats. But they have both suffered massively from the post-war move from traditional, mixed farming to more intensive and industrialised modern agriculture.
As we saw in the writings of John Clare, in the early nineteenth century the corncrake – or as he called it, the land rail – was a familiar 'summer noise among the meadow hay'. Yet by the latter part of Queen Victoria's reign, the corncrake's days as a widespread British breeding bird were already numbered.
In the north, the Highland Clearances, and the resulting replacement of arable crops by sheep, led to a steep decline in the bird's population in the Scottish Highlands. Farther south, in lowland Scotland, Wales and England, mechanised mowing was beginning to displace scything by hand. This spelt disaster for this shy and elusive bird, which is very reluctant to abandon its eggs and chicks, even when under threat. Because of its exemplary parenting, not only were many young corncrakes killed but the adults perished too, caught up in the relentless blades of the mowing machines.
As mechanised harvesting spread north and west, so the corncrake began to disappear from much of the countryside. At the start of the Second World War, the species was still common in Ireland and the Northern and Western Isles of Scotland, but had vanished from large swathes of England and Wales. By the time of the first _BTO Atlas_ , for which fieldwork took place from 1968 to 1972, the corncrake was virtually absent from the British mainland, having retreated to the islands of the Outer Hebrides, where traditional farming methods still persisted.9
In recent years, thanks to concerted efforts from conservationists and farmers, the corncrake has made a modest comeback. Meanwhile, a scheme to re-introduce the species into lowland England, on the Nene Washes near Peterborough, has been a partial success. But the days when the corncrake was as familiar as the skylark are long gone, and are unlikely ever to return.xviii
Other birds that have thrown in their lot with humans are also in trouble. Barn owls and barn swallows both struggle to find suitable places to nest, as so many rural barns and other farm buildings have been converted into homes. And although house martins and house sparrows are still found across much of Britain and Ireland (in about 90% of all 10-km squares) both species have suffered major falls in abundance in recent years, with the biggest declines, ironically, in our towns and cities. This is down to a complex combination of factors, including loss of nest sites, a lack of food and, in the case of the house sparrow, possibly also the effects of air pollution on this very sedentary bird.
For the house martin, it is tempting to wonder where this charismatic little member of the swallow family would have bred before we built houses. I have seen them nesting in crevices in sea-cliffs on the coast of Wales, which must have been their original habitat. They also still nest in crevices in the crags of Malham Cove in North Yorkshire, where they have suffered in recent years from disturbance by climbers scaling the famous cliffs.
But for centuries now, house martins have preferred to build their cup-shaped nests, patiently constructed from hundreds of tiny balls of mud, on the sides of our homes. So it comes as no surprise that the folk names for this species reflect its deep connection with our own lives. They include 'eaves swallow' and 'house swallow', the latter the same as the German folk name _Hausschwalbe_ (far more appropriate than the official name _Mehlschwalbe_ , which rather oddly translates as 'flour swallow').
Another folk name is 'window swallow' – a direct translation of the French _hirondelle de fenêtre_ – which reflects the way house martins often look as if they are going to fly straight through our bedroom windows, before veering off at the very last moment.xix
Many centuries ago, this motley group of species chose to throw in their lot with us. They made their nests on our homes and in our outbuildings, and fed in the fields where we grew our crops. In the process, they became some of our most familiar and best-loved birds. Today, though, they face an uncertain future, with global climate change now adding to the problems already brought about by intensive agriculture, pollution and habitat loss. It does not feel too extreme to say that we have betrayed them.
## _4: Canada Geese and Crossbills_
As well as many birds being named after their habitat, some also feature a geographical region or place in their names. And as with habitat, in a tidy, organised world these would be entirely logical: Iceland gulls would breed mainly in Iceland (they don't), and Canada geese in Canada (they do, but having been introduced to Britain there are now an awful lot breeding here too). But given the haphazard way in which birds have acquired their names, you may not be surprised that the vast majority of geographical names are rarely accurate, or even particularly helpful.
Sometimes this is because the description covers such a vague area, as is the case with the Arctic tern, great northern diver and northern wheatear.xx Other names are more specific, but just as unhelpful.
The Mediterranean gull is a smart cousin of the more familiar black-headed variety, with a jet-black head and blood-red bill. Once its name was reasonably accurate: until about half a century ago this was a globally rare bird, found mainly in southern Europe and western Asia, with a significant part of the world population wintering around the Med. In those days the Mediterranean gull was so rare that in 1960 the eminent Dutch ornithologist Karel Voous described it as 'an unmistakable relict... probably in the course of becoming completely extinct'.10
Yet very soon after Voous made that doom-laden prediction, the species' fortunes began to turn. Like other members of its family, the Mediterranean gull is an opportunistic feeder, and from the 1960s onwards was able to take advantage of the exponential increase in surplus food created by our increasingly wasteful consumer society. Numbers started to increase, and its range began to expand to the north and west.
In 1968 a Mediterranean gull paired with a black-headed gull at a colony in Hampshire and successfully bred; following which several pure-bred pairs of the species became established, so that by the turn of the millennium there were at least 500 breeding pairs in Britain, including a huge colony in Poole Harbour. The global population of this handsome bird is now estimated at between 230,000 and 660,000 pairs,11 and today its future is well and truly secure.
The same cannot be said for probably the most globally threatened bird regularly seen in Britain, which also has a connection with the Mediterranean Sea: the Balearic shearwater. Like other members of its family, this svelte seabird hugs the waves on narrow, stiff wings. It is named after that trio of holiday islands in the western Mediterranean – Mallorca, Menorca and Ibiza – the only places on the planet where it breeds. Today, there are thought to be just 10,000 individuals of this enigmatic species in the world, some of which can be seen as they fly past our southern coasts each year on their post-breeding wanderings, in late summer and early autumn.
*
Surprisingly few British birds are named after countries. Indeed, the nation with the most birds named after it on the official British List (with three species) is Egypt – no doubt reflecting our long colonial connection with that North African nation.xxi The only other British bird named after a nation-state is the aforementioned Canada goose.xxii
Three other regularly occurring British birds are named after islands or regions. They are the Manx shearwater, named after its breeding colony on the Isle of Man; Slavonian grebe, named either after the region of that name in Croatia, or more likely from Scalovia (also spelt Sclavonia), a part of Prussia, now in modern-day Lithuania; and the Lapland bunting, which is found across a wide swathe of northern Europe, Asia and North America.
As for birds named after the United Kingdom's own nations, the Scottish crossbill, an enigmatic bird which (if it truly exists) is Britain's only truly endemic species, is found here and nowhere else in the world. As we saw in Chapter 3, crossbills sport a unique feature in the form of crossed mandibles, enabling them to extract the papery seeds from pinecones, a food source not readily available to any other species of bird.
Over time, ornithologists observing crossbills in the Caledonian pine forests of the Scottish Highlands began to suspect that there were two discrete populations there, each with a subtly different bill size and shape. This allowed each cohort of birds to exploit two different kinds of food: one feeding mostly on the seeds from larch and spruce cones, the other on a more mixed diet including the cones of Scots pine.
As early as 1975, the ornithologist Alan Knox tentatively suggested that these might represent two separate species, whose divergent diets had led to a permanent change in their bill size. This would, he reasoned, keep their populations ecologically separate, allowing them to live alongside one another in the same forests without interbreeding. Five years later, the British Ornithologists' Union boldly declared that there were indeed two species: the common (or red) crossbill, found across a wide swathe of the northern hemisphere, and the Scottish crossbill, found only in a small area of the Scottish Highlands and therefore an endemic British bird.xxiii
Having 'lost' the red grouse a few years earlier, when the gamebird previously considered to be Britain's only unique species was downgraded to a mere subspecies of the far more widespread willow grouse, the Scots were delighted to have a new endemic species. But as with all complex taxonomic decisions, not everyone agreed with the change. Some observers accused the BOU of jumping the gun, fuelled perhaps by a jingoistic desire to announce the discovery of a genuine British endemic.
That controversy has not fully died down, even though research by the RSPB has recently shown that the Scottish crossbills do indeed have larger bills, and also make a different sound – publicised gleefully in the press as the birds having a 'Scottish accent'. Taken together, these two factors would indeed allow the Scottish crossbills to maintain reproductive isolation from their cousins, even when the two different species are living and breeding cheek-by-jowl with one another in the same place.
Thus, more than a century after the last 'new' British breeding bird, the willow tit, was discovered hiding in our midst, it has been supplanted by the Scottish crossbill – as distinctively Caledonian as single malt whisky, sporrans and Andy Murray.
So could there be any more 'cryptic species' awaiting discovery, somewhere in Britain? Astonishingly, given how well we think we know our birds, there may well be, but it will take a revolution in genetics for us to find out (see Chapter 7).
## _5: Eiderdowns, Cranes and Kites_
I was once told a (possibly apocryphal) tale about a young boy being taken on a visit to the Farne Islands, the thriving seabird colony off the coast of Northumberland.
Just as the boat was leaving the harbour, the youngster spotted a small flock of eider ducks. Most were nut-brown females, with their delicately vermiculated plumage, looking as if someone had painstakingly drawn wavy lines across their wings and body, and etched even finer markings on their head. They were accompanied by a smaller group of males: boldly marked in slabs of black and white, with a strange greenish patch like a birthmark on the side of their necks, and a rosy flush across their breasts.
As the boy watched, to his delight the males began displaying to their mates, each throwing his head back onto his mantle like an over-enthusiastic gymnast. All the while, they were uttering one of the most bizarre sounds in the bird world: a call memorably described by Bill Oddie as sounding like a cross between a shocked old lady and the late Frankie Howerd.
The boy excitedly grabbed his father's arm as he struggled to remember the name of this striking bird. Overwhelmed with excitement, he finally managed to exclaim, 'Look daddy... it's... it's... a duvet duck!'
The great Dr Johnson would surely have been amused by the boy's etymological error. For in an essay in the _Idler_ magazine from January 1759, this pioneering lexicographer made the very first published reference to the word 'duvet': 'There are now to be sold... some Duvets for bed-coverings, of down.'
The name 'eider' (meaning 'down bird') comes originally from the Icelandic, and the bird's modern French name is _eider à duvet_. 'Duvet' also means 'down', and refers to the small, incredibly soft feathers that female wildfowl pluck from their breasts, to line their nests and keep the eggs snug and warm. Eider ducks produce large amounts of particularly soft and thermally efficient down, which has traditionally been harvested for human use.xxiv
The eider is a common and familiar breeding bird along the coasts of north-west Europe. So it is likely that our prehistoric ancestors, many of whom lived close to the sea, were the first people to begin harvesting the birds' down to keep themselves warm. But the first named individual to do so lived much later, in the seventh century ad, on the remote and chilly Holy Island in Northumberland.
Cut off from the mainland twice a day by the tides, Holy Island – also known as Lindisfarne – was at that time home to a small and isolated community of monks. Amongst them was a man named Cuthbert, who later rose to become the Bishop of Lindisfarne. After his death in AD 687 Cuthbert was canonised, and is now widely regarded as the patron saint of the north of England.
Holy Island was – and still is – home to a thriving population of eider ducks, which hide their nests away deep in the heather and bracken above the tideline to avoid being preyed on by gulls. Medieval legend has it that Cuthbert cultivated a special relationship with the eiders, harvesting their down and in turn looking after them by passing laws against the taking of their eggs – the first recorded instance of a nesting bird being given official protection, anywhere in the world.xxv
Today, St Cuthbert is still honoured in the local name for the eider, which is known as 'Cuthbert's duck', sometimes affectionately shortened to 'Cuddy's duck', in memory of this far-sighted and benevolent holy man.
But what of the eiderdown itself? The idea of making and marketing a quilted bed cover stuffed with duck down was first brought to Britain from Germany by an English diplomat, Paul Rycaut, in 1689 – almost exactly a thousand years after Cuthbert's death. Rycaut sent his friends large bags filled with eider down, instructing them that 'the coverlet must be quilted high and in large panes, or otherwise it will not be warme.'12
Some ideas perhaps arise too early for their own good. The British public, it seemed, was not yet ready for this strange European invention, either because it was too expensive, or more likely because they preferred the masochistic practice of sleeping beneath scratchy, flea-ridden woollen blankets.
Gradually, though, things began to change. In 1841, _The Times_ included an advert for an 'eiderdown quilt, or duvet', and by 1859, the novelist Wilkie Collins could write of 'a sweet little eider-down quilt, as light as roses'. By the 1950s the eiderdown had become the standard form of bed covering, placed on top of a layer of sheets and blankets to add an extra layer of warmth. But as a stern letter to _the Times_ reveals, not everyone approved of the use of this portmanteau word, especially when the filling came from a lesser form of wildfowl: 'I ask you... to lend your pen to scotching the unwarrantable term "eiderdown" when applied to the ordinary goose-down quilt.'xxvi
Yet even as this was being written, the days of the traditional eiderdown were numbered. After several false starts, it took the vision of one man, the legendary retail entrepreneur Terence Conran, to consign the eiderdown to the history-books. In 1964, when Conran opened his first Habitat store on London's trendy King's Road, one of the most popular items on sale was the 'continental quilt' – the product we now know as a duvet.xxvii
*
The eiderdown is just one of many examples of the way bird names have entered our language as similes and metaphors. We may refer to a greedy eater as a 'gannet', a mad person as 'cuckoo', or say someone is 'as bald as a coot'. But how often do we give a moment's thought to the origin of these words and phrases, or how they first came into our day-to-day speech?
Sometimes, of course, the link is pure coincidence: when someone is said to 'grouse' (meaning complain), there is no obvious link with the bird of that name; nor does the verb 'to quail' appear to have any connection with our smallest gamebird, unless it refers to its legendary shyness. The chess piece known as the rook has nothing to do with that member of the crow family, and the fungal disease thrush is likewise unrelated to the bird.
But there are many genuine connections between the meaning of a word and its origin as a bird's name. These include 'sniper', meaning a hidden marksman, a nod to the difficulty of shooting this fast-flying wader; and the colour 'teal', a deep, rich shade of bluish-green which comes from the patches on either side of the male teal's head.xxviii
The origin and meaning of another familiar word containing a bird's name, scarecrow, seems obvious – but delve a little deeper, and confusion reigns. When the word was first coined, around the middle of the sixteenth century, it referred to a young boy employed to frighten the birds away from the newly sown seed in a farmer's field, by throwing stones and making a loud noise. Perhaps because small boys tended to get bored and wander off, farmers soon began to use a substitute: a cross-shaped structure hung with clothes and a hat, which was supposed to resemble a human being.
But which species are we actually talking about here? For carrion crows – the commonest species across most of the UK – are frequently confused with rooks. Both are all black, though the rook does have a distinctive greyish-white patch around its beak, and a smaller, more angular head. They also sound subtly different, with the rook's call being less harsh – as the nature writer Dominic Couzens explains, it sounds like a crow that has been on an anger-management course.
One proverbial way of separating the two species is referred to in an old Norfolk rhyme:
When thass a rook, thass a crow,
And when thass crows, thass rooks.xxix
This relies on the fact that while crows are usually (though not always) solitary, rooks are more sociable birds, generally found in flocks. And while we tend to be suspicious of the all-black, rather sinister-looking crow, we take a more benevolent attitude towards rooks. This is perhaps because our relationship with them goes back many thousands of years, as the late Derek Goodwin, an ornithologist who made a special study of the crow family, pointed out: 'The rook is often thought of as one of the most characteristic birds of the British countryside. So it is, at least of the agricultural countryside... it is unlikely that there were any rooks in Britain, or indeed in Western Europe, before there were any farmers.'
The farmers themselves might take a less friendly approach, because if the premise behind the Norfolk rhyme is correct, the main threat to their precious crops would not have been the solitary crow but the gregarious rooks, whose large, noisy flocks can strip a newly planted field bare in an hour or two. This being the case, the scarecrow should really be called a scarerook; it is just another curious example of how words can mislead us.xxx
Whether it means rook or crow, it's obvious that the word scarecrow is named after a bird, and not the other way around. Likewise, we know that the Harrier jump jet, the first aircraft capable of vertical take-off and landing, was christened after the low-flying bird of prey; and that the imprint Puffin Books was named after the seabird, whose comical appearance and brightly coloured bill clearly appealed to children.xxxi But in other cases, it can be hard to work out which came first: was it the name of a bird or that of the object?
Take the kite and the crane. You might think that kites were named after the children's toy, while cranes were so called because of their resemblance to the mechanical version. Actually it's the other way around. Like so many of our oldest bird names, these have onomatopoeic origins. Kite – originally 'cyta' in Old English – has no counterparts in any other European language, and so we know it originated in Britain, some time between the sixth and eleventh centuries, as it must have developed after the Anglo-Saxon invasion but before the Norman Conquest. The name comes from an imitation of the bird's high-pitched, whistling call, though W. B. Lockwood suggested that it might have initially been applied to the mewing cry of the buzzard, and only later adopted for its scarcer cousin.
Likewise, 'crane' is also likely to have come about as a representation of the bird's deep, honking call. The name has counterparts in the various Old Germanic and Scandinavian languages, showing that, as with other ancient names such as goose, it almost certainly has a Proto-Indo-European origin.
With both kites and cranes, some have argued that the use of the same names for the birds and man-made objects is simply coincidence. But it is also widely believed that both toy kites and mechanical cranes were named after their resemblance to these particular species of bird.
Kites (the birds) really do look like their namesake: they hang in the air on long, fingered wings with effortless ease, twisting and turning with each new gust of wind, and using their forked tail as a rudder to control their position. So it would be reasonable to assume the name derives from the bird's aerobatic antics.
The evidence for and against this is mostly circumstantial. The _OED_ certainly favours a connection, suggesting that the name of the toy derived from 'its hovering in the air like the bird'. And in the first recorded use of the new meaning, in Samuel Butler's mock-heroic poem _Hudibras_ , published in 1664, the author conveniently provided a direct comparison between the two:
As a Boy, one night, did fly his Tarsel of a Kite, The strangest long-wing'd Hauk that flies.
With 'crane', the link between bird and object is far more clear-cut. Unlike 'kite', variations of the same word are used in many European languages for both the bird and the machine: _grue_ and _grue_ in French, _grúa_ and _grulla_ in Spanish, and _kraan_ and _kraanvogel_ (literally 'cranebird') in Dutch. Looking at a crane as it stands tall and stately, its head and bill curving forward at right angles to its long, straight neck, the connection between the two does seem irrefutable.
Another reason why we might query the links between these bird names and their man-made equivalents is that we tend to think of both kites and cranes (the birds, not the objects) as scarce and limited in their range. But this is an illusion, caused by the way we view the historical status of birds through the prism of our own current experience. If a bird is rare or common to us, we tend to assume that it has always been so, in what ecologists call 'shifting baseline syndrome'.
For cranes and kites, this assumption is quite misleading: a thousand years ago they would both have been far more widespread than they are today. Kites were the street-cleaners of medieval cities, snatching up any spilt food or unsavoury objects with alacrity, and earning a reputation as thieves and vagabonds. This explains Lear's angry insult to his deceitful daughter Goneril, whom he calls 'detested kite', and also the reference in _Coriolanus_ to 'the city of kites and crows'. Although referring to Rome, this must surely have come about because Shakespeare had seen kites scavenging on the streets of London.xxxii
As wetland specialists, cranes would have been less widespread than kites, but were still common enough to be caught and slaughtered by the dozen to supply vast medieval feasts. Indeed, until their watery habitats were destroyed in the late Middle Ages, these lanky waterbirds would have been found across much of eastern England. The evidence of their presence here can be seen from the many place names featuring the name, such as Cranfield, Cranbrook and Cranford, the last of which appears in the title of Elizabeth Gaskell's 1853 novel, which was televised in 2007.xxxiii
But over time, as the human population increased, persecution rose and wetlands were drained, both the kite and the crane began to decline in numbers. Red kites disappeared from the capital during the middle years of Queen Victoria's reign, eventually retreating to a few hidden valleys in central Wales, where they just managed to cling on, despite the attentions of egg collectors.
Cranes were not so fortunate: they vanished altogether as a British breeding bird during the Tudor period. Apart from the occasional wandering flock from the continent, they were absent for more than 400 years, until a small group returned to breed in Norfolk in the late 1970s. I can still remember my first, unforgettable sighting of cranes one chilly November afternoon in a remote corner of the Broads when, an hour or so before dusk, three huge birds flew past me uttering their haunting, honking calls. My first red kite is also sealed in my memory. Back in the mid-1970s, having spent three days combing the valleys of mid-Wales for these elusive birds, my mother and I finally struck lucky when a single kite drifted high overhead on a cloudless July day, its long wings glowing russet-orange against the deep blue sky.
Since then, both kites and cranes have made a dramatic comeback, and are far easier to see than they used to be. Cranes are still thriving in Norfolk, and have also been reintroduced onto the Somerset Levels near my home. Red kites are now a regular sight in many parts of England and Scotland as well as Wales – one occasionally drifts over my garden on sunny spring days, while I have seen dozens of these aerobatic raptors hanging in the air over the M40 motorway in rural Oxfordshire, and even soaring over Lord's Cricket Ground in the centre of London.xxxiv
But not every bird name in everyday language is as obvious as the kite and the crane, as two tales – that of a young inventor and our best-known fictional spy – reveal...
## _6: Hobbies and Spies_
The winter of 1946-7 was one of the coldest on record. Freezing temperatures persisted for days, weeks, then months, as the whole of Britain was blanketed with snow and ice. For a nation still reeling from the Second World War, and enduring the continuation of food rationing, it must have been a grim and miserable time.
But deep in the county of Kent, one young man was working hard to cheer the nation up, by promoting a way for the dads and lads of post-war Britain to have fun. A year earlier, Peter Adolph had come up with an idea he was sure would be a huge success: a kind of table football that involved each participant flicking the diminutive 'players' against the ball, to shoot, tackle, save or score a goal.
All he needed was a name: and one day, as he toiled to perfect the exact shape of his cardboard playing figures, he came up with one: 'Hobby'. But the officials at the Patent Office rejected his application outright. As one jobsworth pointed out to the crestfallen inventor, 'You might as well call a game "Game".'
Peter Adolph was back at square one. But then he had a clever idea. If he couldn't call his new game 'Hobby', surely he could use the scientific name of his favourite bird of prey, _Falco subbuteo_ – the hobby?xxxv And so, thanks to his ornithological expertise, the name of his product was born. A decade later, Subbuteo had become a fixture in homes up and down the country – and Peter Adolph was a millionaire.
So where does the bird's English name, hobby, come from? It first appears in the fifteenth century, spelt 'hoby', before it transmutes into the present-day spelling in 1642, at the height of the English Civil War. The name refers to the characteristic way this slender raptor hunts for its prey.
When hobbies return to Britain each spring, after a long and arduous flight from their African wintering grounds, they need a rapid boost of energy. They gather over large wetlands such as the Somerset Levels, Cotswold Water Park and the Stour Valley in East Kent, where they hunt dragonflies and other insects, providing a spectacular sight for watching birders.
After becoming more elusive for the rest of the breeding season, hobbies then reappear later in the summer, seeking out gatherings of swallows and house martins, and snatching their unwary youngsters out of the air with their razor-sharp talons.
Hunting both dragonflies and swallows requires a high level of agility, so the hobby sweeps back its wings and zigzags across the sky in hot pursuit of its victim. It is this jerky flight-action that gave the species its name – from the Old French verb 'hober', meaning 'to jump about'. However, unlike the connection between the scientific name of the species and the tabletop football game, the link with the word hobby – meaning pastime – is just a coincidence.
*
The influence of _Falco subbuteo_ on the games industry is one of the more unusual examples of how bird names have been incorporated into popular culture. One often overlooked way that bird names enter society is as first or Christian names. The names of two falcons, Peregrine and Merlin (sometimes spelt Merlyn) are, mainly among the upper classes, given to boys, as in the veteran journalist and political commentator Peregrine Worsthorne and the former Home Secretary Merlyn Rees.xxxvi
Girls called Sylvia share their name with a genus of warblers, while Phoebe is the common name for three species of American flycatchers (in both cases the names have a classical origin, meaning 'of the woods' and 'bright as the moon' respectively). Robin, however, doesn't count – as the bird was called after the boy's name, not the other way around!
Bird names are also surprisingly common as nicknames for sporting teams: no fewer than five football or national league clubs – Bristol City, Charlton Athletic, Cheltenham Town, Swindon Town and Wrexham – are known as the 'Robins' (because of the prominence of red in their playing strip), as is the rugby league team Hull Kingston Rovers. West Bromwich Albion (usually called the Baggies) are also known as the Throstles, a name that would presumably have pleased Max Nicholson; Norwich City, whose kit is yellow, are called the Canaries; while Newcastle United and Notts County, who both play in black-and-white, are known as the Magpies.xxxvii
Similarly, in Canada and the USA, a host of sporting teams are named after birds; most famously the major-league baseball teams the Baltimore Orioles, Toronto Blue Jays and St. Louis Cardinals. As with many British sports clubs, this is purely because of the colour of their strip, and not from any other affinity with the bird in question.
Sometimes the nickname has arisen as a result of the club's geographical setting. So Torquay United are known as the Gulls, and Brighton and Hove Albion the Seagulls, simply because they are based by the seaside.xxxviii But the origin of these bird-related nicknames isn't always obvious. Nowadays Crystal Palace's nickname is the Eagles, but until 1974 they were known as the Glazers. The name change happened when Malcolm Allison, one of the most flamboyant characters in football history, became their manager. Wishing to cultivate a more powerful image, he simply borrowed the nickname 'Eagles' from the Portuguese club Benfica.
No fewer than four amateur clubs in the English and Welsh leagues are known as the Linnets. This appears to be a curious choice, given that the linnet is not a very showy bird, and its only prominent colour is the pink patches that appear on the male's breast during spring and summer. When you discover that Burscough, King's Lynn, Runcorn Linnets and Barry Town don't play in pink, but in green, the name becomes even more baffling. The somewhat obscured reasoning behind this is that they are actually named after the species once known as the 'green linnet', a now long-forgotten folk name for the greenfinch, which is also commemorated in the title of a poem by William Wordsworth.
Birds also feature in the names of several music groups. In some cases, like the Eagles (and indeed the Byrds), there is no genuine ornithological link, but some bands were named deliberately after birds. Capercaillie is the name of a longstanding Scottish folk group, founded in the 1980s, and was chosen to celebrate that iconic highland grouse.
The band names Doves and Guillemots are no coincidence, either: founder members Jimi Goodwin (Doves) and Fyfe Dangerfield (Guillemots) are both keen birders – as are a surprising number of other members of rock bands, including Guy Garvey of Elbow and Martin Noble of British Sea Power, both of whom, however, chose non-ornithological names for their bands.
One of the most intriguing references to a bird name in popular culture is the goldeneye: a handsome species of diving duck that breeds in the Scottish Highlands and spends the winter on lakes and reservoirs in many parts of Britain.
Fans of James Bond will recognise Goldeneye as the name of author Ian Fleming's home in Jamaica – and, long after his death, the title of a 1995 film starring Pierce Brosnan as the eponymous hero. But sadly the villa was not named after this handsome duck, but rather Operation Goldeneye, a Second-World-War sabotage operation in which Fleming was involved (the operation, however, may well have been named after the bird).
There is, nevertheless, an avian connection with the Bond franchise. When Ian Fleming was looking for a name for his superspy hero, he wanted one that sounded 'as ordinary as possible... brief, unromantic, Anglo-Saxon and yet very masculine'. With the deadline fast approaching for the delivery of his first book _Casino Royale_ , he happened to glance at his bookshelf, and noticed a slim volume entitled _Birds of the West Indies_. On the spine was the name of the author: the renowned American ornithologist, James Bond.xxxix
That's not the end of the story. Several years later the real James Bond's wife wrote to Fleming complaining about the way her husband's name had become associated with the hard-drinking and womanising spy. Fleming sent a contrite reply:
Your husband has every reason to sue me... for practically every kind of libel in the book. In return I can only offer your James Bond unlimited use of the name Ian Fleming for any purposes he may think fit. Perhaps one day he will discover some particularly horrible species of bird, which he would like to christen in an insulting fashion...xl
Sadly, Bond never took Fleming up on his offer, so there is no Fleming's leaftosser skulking in the Amazonian rainforest, no Fleming's cisticola roaming the African plains, nor a Fleming's laughingthrush hiding halfway up a mountainside in Asia. But if there were it would hardly come as a surprise, even in these exotic locations. For there are an astonishing number of similarly bizarre official English names amongst what is, at the last count, the world's 10,700 or so species of birds. From Himalayan flameback to Indian pitta, malleefowl to magnificent frigatebird, sad flycatcher to joyful greenbul, and aberrant warbler to invisible rail, these birds and their wonderful names continue to inspire and delight us.
*
Now that we appear to have reached the point at which almost every species of bird in the world has been found and named, you might imagine that we are coming towards the end of our long and eventful story of the origins of English bird names.
Yet that assumption might turn out to be a little premature. For thanks to exciting new advances in science, we have recently discovered – to our astonishment, delight and perhaps some apprehension – that the birds we already know about, and have given names to, may simply represent the tip of a much larger iceberg. And so we come to the final chapter of this story.
#### Notes
1 Edmund Selous, _Bird Watching_ (London, 1901).
2 W. H. Hudson, _British Birds_ (London, 1895).
3 Phyllis Barclay-Smith, _Garden Birds_ (London & New York, 1945). Incidentally, this is the first mention I can find of the phrase 'garden birds' – which strikes me as surprisingly late in the day
4 E. M. Nicholson, _Birds and Men_ (London, 1951).
5 _British Birds_ , vol. I (1907).
6 ibid.
7 William Yarrell, _Supplement to the History of British Birds_ (London, 1845).
8 Mark Cocker and Richard Mabey, _Birds Britannica_ (London, 2005).
9 For more information on the corncrake's decline, and the status of other birds during the late nineteenth century, see Simon Holloway, _The Historical Atlas of Breeding Birds of Britain and Ireland 1875–1900_ (London, 1996).
10 K. H. Voous, _Atlas of European Birds_ (London, 1960).
11 BirdLife International estimate: <http://datazone.birdlife.org/species/factsheet/22694443>.
12 Quoted on the BBC News website, 25 December 2015, <http://www.bbc.co.uk/news/magazine-34848546>.
i One can only assume that Nicholson turned down a knighthood and a peerage; both of which were richly deserved but never bestowed.
ii With the addition of the 600th species, a Yelkouan shearwater off the coast of Devon (seen in 2008, but not finally accepted until 2016), the rapid growth of the British List has long confounded James Fisher's 1966 prediction that 'we are... unlikely to reach a list of more than 480 wild species by the year 2000, or more than 500 ever' _(The Shell Bird Book_ , London, _op. cit._ ).
iii Other old names also lasted longer than we might imagine. In the 1912 _A Hand-list of British Birds_ , compiled by Dr Ernst Hartert, our smallest species of bird was still known by the taxonomically misleading name 'golden-crested wren', even though the BOU had preferred goldcrest almost thirty years earlier.
iv As W. B. Lockwood acidly remarked, 'This half-Latin book name found little support... and quickly fell into disuse'. The name 'accentor' was coined by the German naturalist Johann Matthäus Bechstein in the early nineteenth century, and is derived from the Latin _cantor_ , meaning 'singer'.
v See _King Lear_ , Act 1 Scene IV, in which the Fool warns Lear against his scheming daughters:
For you know, nuncle,
The hedge-sparrow fed the cuckoo so long,
That it's had it head bit off by it young.
vi The other proposed changes were: great spotted and lesser spotted woodpeckers to 'pied' and 'barred' woodpeckers, great black-backed and lesser black-backed gulls to 'great blackback' and 'lesser blackback', and common gull to 'mew gull'.
vii It has been suggested – not entirely in jest – that the willow tit should now be renamed the 'marsh tit', and the marsh tit given the new name of 'oak tit'.
viii Soon afterwards, Lord Walter Rothschild was sent two similar birds shot in a wood in nearby Finchley; these too proved to be willow tits.
ix See Chapter 4 for more on Colonel Montagu.
x This also makes me wonder whether the bird given the name 'marsh titmouse' by John Ray, back in the late seventeenth century, was also a willow tit – especially as another name from that period was 'fen titmouse'. It's tempting to think that had the early ornithologists spent less time firing off their shotguns and more time studying bird behaviour and choice of habitat, the confusion between the two species might have been sorted out a couple of centuries earlier.
xi If we include European warblers that are rare or occasional visitors to Britain, several other names derive from watery habitats. These include river, paddyfield and aquatic warblers.
xii Other habitat-based names are more accurate: woodcocks inhabit woods, sand martins nest in sandbanks, and water rails and marsh harriers live in wetlands, as, indeed, do moorhens – the 'moor' part of the name doesn't refer to the heather-covered moors of the Brontë sisters, but is a corruption of the word 'marsh' or 'mere'. (The fact that a widespread folk name for marsh harrier is 'moor buzzard' confirms this, as does the official name for the area around my home, the Somerset Moors and Levels.)
xiii Despite their obvious behavioural differences and distinct plumage, rock and water pipits were not 'split' by taxonomists from one into two separate species until 1986.
xiv Folk names for this familiar countryside bird include 'heaven's hen', 'rising lark', 'sky-flapper' and the widespread 'laverock' – the last most often used in the north of England and Scotland.
xv A Roman recipe for larks' tongues outdoes even this:
Get 1,000 larks.
Remove their tongues and set aside.
Discard the larks.
Put the tongues in a pan with a little oil and sauté quickly.
Transfer to a hot platter.
Serves four.
xvi The phrase 'up with the lark', denoting an early riser, goes back to Tudor times, while 'as happy as a lark' appears a little later, during the eighteenth century. For more examples of the way bird names have influenced our day-to-day language, see Chapter 6, Part 5.
xvii A poignant example appears in Charles Dickens' novel (and David Lean's film) _Great Expectations_. After Pip has been transformed into a gentleman thanks to an unexpected bequest from the former convict Magwitch, his brother-in-law Joe visits him at his new home. The change in Pip's social status has created a barrier between them, which Joe attempts to overcome by telling Pip what a good time they'll have when he returns home: 'And when you're well enough to go out for a ride – what larks!' Yet sadly, as we – and they – realise, this will never come to pass.
xviii The other species named after the food we grow, the corn bunting, is not in quite so dire straits as its near namesake. But 'the fat bird of the barley', as it is affectionately known, has now vanished from large areas of lowland Britain. This includes my own home county of Somerset, where in 2012, for the first time since records began in the nineteenth century, not a single corn bunting was seen.
xix Note that, as with 'pigeon' and 'dove', 'swallow' and 'martin' are more or less interchangeable. For example, in North America the sand martin is known as the 'bank swallow'.
xx Northern wheatear is, like barn swallow, a relatively new name. It was deliberately coined in the latter years of the twentieth century to help sort out any confusion between the bird we simply call the 'wheatear' and its many other relatives, most of which live further south, in the Middle East or around the Mediterranean Sea (see Chapter 7).
xxi However, one of these, the Egyptian goose, was introduced here, and the other two, Egyptian vulture and Egyptian nightjar, are very rare vagrants with just two accepted records of each.
xxii Some might reasonably argue that 'American' is synonymous with the USA, but I believe that in the case of bird names it refers to the continent of North America, not the country. After recent taxonomic changes, 'American' is found in the name of eight species on the British List: American wigeon, bittern, coot, golden plover, herring gull, kestrel, robin and redstart. The recent addition of Chinese pond heron to the British List, thanks to a record in Kent in 2014, means there are now five 'British birds' named after nation-states.
xxiii To add to the confusion and complexity of the situation, there is now a third species present, with an even bigger bill, called the parrot crossbill.
xxiv Because eider ducks live at such cold northerly latitudes, and mainly hunt for food at sea, their down has a unique structure that traps warm air better than any other material – natural or synthetic. It is also produced in very small quantities: while tens of thousands of tonnes of goose down are sold worldwide every year, the entire annual global production of eider down could fit onto a single truck.
xxv Whether he did this out of a desire to protect this vulnerable wild creature, or from a less altruistic desire to keep all the down for himself, is not known.
xxvi _The Times_ , 26 April 1950. But the march of the eiderdown could not be stopped, even by such determined ornithological pedantry.
xxvii Many years later, Conran recalled that he had first come across it on a visit to Scandinavia: 'I had been in Sweden in the 1950s and was given a duvet to sleep under. I probably had a girl with me and I thought this was all part of the mood of the time – liberated sex and easy living.' A shrewd businessman, Conran understood that Britain's hard-pressed housewives would be less interested in the erotic possibilities of his new product, and more keen on the practical benefits. So he marketed the duvet as 'the ten-second bed': so much easier to make than the traditional version. It worked. Nowadays, with well over ten million duvets sold in Britain every year, virtually everybody sleeps beneath one.
xxviii The use of the name of the duck for this attractive colour appears to be very recent: the first _OED_ reference is as late as 1923.
xxix Roughly translated, this means, 'If you see one rook, it's a crow; if you see lots of crows, they're rooks'.
xxx Even Shakespeare was caught up in the confusion between the two species: _Macbeth_ features the memorable but puzzling line, 'Light thickens, and the Crow Makes Wing to th' Rookie Wood...'
xxxi And which followed in the footsteps of Penguin and Pelican Books.
xxxii The Bard's best-known line about kites is Autolycus's cautionary comment in _The Winter's Tale_ : 'When the kite builds, look to lesser linen.' This refers to the kite's unusual custom of stealing items of underwear to decorate its nest, earning it a deserved reputation as a kleptomaniac.
xxxiii Incidentally, the widespread belief that the place names beginning with 'Cran-' actually refer to herons is given short shrift by Eilert Ekwall, author of the authoritative _Concise Oxford Dictionary of English Place-names_ (Oxford, 1936; Fourth Edition 1959), who writes: 'There is no reason to assume any other meaning for the word than "crane", such as "heron". The two birds are always kept well apart in early records.' I agree with him.
xxxiv By what seems to be pure coincidence – but is perhaps related to the recent comebacks made by both species – an indie rock band from Edinburgh formed in 2012 call themselves Kite and the Crane. Appropriately, perhaps, they specialise in telling stories, using rich harmonies and soaring vocals – just like their avian namesakes.
xxxv _Subbuteo_ is Latin for 'small buzzard'. This is technically inaccurate as, despite their superficial similarity, falcons and buzzards are not related to one another. But it was nevertheless good enough for the founding father of the science of taxonomy, Linnaeus (see Chapter 3).
xxxvi The conservationist Sir Peter Scott, son of Scott of the Antarctic, named his son Falcon and one of his daughters Dafila (then the generic name for the pintail duck).
xxxvii Cardiff City are nicknamed the Bluebirds (after the celebrated North American songbird) because they play in blue. In 2012 their new owner, the Malaysian businessman Vincent Tan, controversially switched the colour of their strip to red – a lucky colour in the Far East. Less than three years later, after protests from fans, he was forced to change it back to blue, in line with the club's avian nickname.
xxxviii Sheffield Wednesday are known as the Owls, but this has no ornithological connection – it comes from the name of a local area of the city, Owlerton.
xxxix In an interview in 1962 for the _New Yorker_ magazine, Fleming claimed that when he wrote _Casino Royale_ , 'I wanted Bond to be an extremely dull, uninteresting man... when I was casting around for a name for my protagonist I thought, by God, [James Bond] is the dullest name I ever heard.'
xl Ian Fleming did eventually make peace with the real-life Bond and his wife, giving them a first edition of his 1964 novel _You Only Live Twice_ , with the inscription: 'To the real James Bond, from the thief of his identity'. A few months later, Fleming died, aged just fifty-six, from heart disease caused by his industrial consumption of cigarettes and alcohol. The cleaner-living James Bond survived another quarter of a century, dying in his home town of Philadelphia in 1989. Many years later, the signed copy of _You Only Live Twice_ sold at auction for $84,000 (£56,000).
# TOMORROW NEVER KNOWS
_The Future of Bird Names_
Names and attributes must be accommodated to the essence of things, and not the essence to the names, since things come first and names afterwards.
Galileo Galilei
## _1: Bird Names at a Crossroads_
As dawn was about to break over Kibbutz Lotan, just outside the Red Sea resort of Eilat, we heard the strange, metronomic call of the scops owl – our first species of the day. Sixteen hours later, our 139th and last species was the night heron, as we watched a flock passing high overhead in the rapidly darkening sky.
By then, we were exhausted. We'd driven almost 250 miles through this arid, sun-drenched land, and seen a dozen species of warbler, ten birds of prey, half-a-dozen kinds of wheatear and more than a score of different waders. We'd found exotic birds, including wrynecks, bee-eaters and hoopoes, and more familiar ones, such as swallows, house martins and great tits.
But on this day, 1 April 2014, they all ranked equally. This was because we were taking part in that most curious of ornithological pastimes: a twenty-four-hour bird race, in which different teams try to see or hear as many species of bird as possible during the course of a single day. Our three-man team consisted of me, David Lindo (aka The Urban Birder) and Tim Appleton (founder of the British Birdwatching Fair).
As an international birding occasion, Champions of the Flyway takes some beating. A dozen teams from all over the world had converged on the Red Sea resort of Eilat to take part in this marathon event. That was the fun part; the serious purpose was to raise awareness of the plight of migrant birds, and funds to help save them from the many threats they face on their travels.i
And what travels these are. The vast majority of migrant birds in the world breed in the temperate or Arctic latitudes of the northern hemisphere, fly south in autumn to spend the winter below the Equator, and then head back north in spring to raise a family once again. They do so for one simple reason: light. In summer, the farther north you go, the more hours of daylight there are; and this – together with warmer temperatures – produces a glut of insects on which these birds can feed their hungry offspring.
These migratory birds follow three major global routes: known as the Africa-Eurasia, East Asia-Australasia and the Americas flyways, which between them witness the global travels of billions of birds each spring and autumn.
Israel is slap-bang in the middle of the Africa-Eurasia flyway. Each year tens of millions of birds, of more than 300 different species, pass through the narrow strip of land that divides the Middle East from Africa, flooding down towards their winter quarters south of the Sahara in autumn, and heading back north to breed in the temperate regions of Europe and northern Asia in spring.
We have known about this biannual spectacle since the dawn of civilisation. In the Old Testament Book of Jeremiah, written in the sixth century BC, the prophet clearly refers to the spring arrival of birds in the Holy Land: 'Yea, the stork in the heaven knoweth her appointed times; and the turtle [dove] and the crane and the swallow observe the time of their coming.'
For decades, this migratory crossroads has attracted birders from all over the world, and this now-annual event has been no exception. At least a dozen nationalities were represented here: the Brits rubbing shoulders with Americans, Danes with Dutchmen, and, most tellingly of all, a joint Israeli-Palestinian team, who used their expert local knowledge to win the race, racking up an extraordinary tally of 169 species.
It was only afterwards that something struck me about the people taking part: not only did they all speak English, but throughout the contest they also used English bird names. This was even though their native languages included Hebrew, Arabic, Dutch, Finnish and German.
There were, it's true, a few concessions made by the Brits to our international colleagues: the use of 'northern wheatear' and 'barn swallow', for example, to distinguish our familiar species (known in the UK simply as the wheatear and swallow) from their more exotic relatives. But otherwise, the names corresponded to those you might hear back home in Britain, along with a handful of species whose English names reflect their limited Middle Eastern range, such as the Sinai rosefinch, Dead Sea sparrow and Palestine sunbird.ii
The use of English bird names amongst birders of different nationalities is not confined to Israel. Wherever in the world I have travelled to watch birds, I hear these names spoken. Sometimes, as in India, Sri Lanka, Kenya and Botswana, this is because these countries were once part of the British Empire, and so English is still widely used in everyday speech. But I have also heard English bird names in Spain and Argentina, Morocco and Mexico, Poland and Sweden – and spoken by Spanish, French, Dutch, German, Swedish and Finnish birders all over the world.
In one respect, this simply reflects the fact that birding originated as a pastime in Britain, before spreading around the globe. But this growth in the popularity of birdwatching has also gone hand-in-hand with a far more important phenomenon: the inexorable rise of the English language.
*
English is the dominant language of the Internet, Hollywood movies and pop music. It is used by Interpol and the international airline industry, and dominates the worlds of television and information technology. English appears on billboards and in viral videos, in adverts and in scientific papers – wherever and whenever the writer wants to reach the widest possible audience. And the endless, twenty-four-hour buzz of social media – the new religion of the twenty-first century – is predominantly conducted in English.iii
More than a third of a billion people now speak English as their first language: in Britain of course, but also in North America, South Africa, Australia and New Zealand. Yet this is dwarfed by the huge number of speakers who use English as their second (or even third or fourth) language – an estimated 600 million people around the world.iv When we add the tens of millions of people who have a working knowledge of English in their everyday lives, then English can justifiably claim to be the global _lingua franca_ for the twenty-first century and beyond.v
It is ironic that, at a time when the days of the British Empire are long gone, and when Britain is rapidly withdrawing from the world stage, the English language is not only still so dominant, but increasingly so – although this is largely down to the continuing global power of the USA.
Looking back one-and-a-half millennia, to when those Anglo-Saxon invaders first crossed the North Sea and brought their strange Germanic tongue to our shores, the notion that English would have eventually risen to be the world's main language would have seemed unthinkable. But as the saying goes, 'a language is a dialect that has met with success', and English certainly fits that bill.
And so – with a little help from the Vikings and Normans, Chaucer and Shakespeare, and generations of explorers, empire-builders, moviemakers, songwriters and computer geeks – we have now reached that stage. English is well and truly here to stay.
As we have seen, English isn't just the global _lingua franca_ – it is also the _lingua franca_ of birding across the world. But despite the overwhelming dominance of English in this field, it certainly doesn't mean that bird names are universally accepted wherever you go. For a start, there's the perennial problem of what Oscar Wilde (or George Bernard Shaw, opinion being divided as to who originated the phrase) called 'two nations separated by a common language': Britain and the USA.
Look through any North American field guide and you'll come across some strange and unfamiliar names: common loon, brant, parasitic jaeger, eared grebe, red phalarope, horned lark and Lapland longspur. For an unwary British birder visiting America for the first time, this can cause confusion; until, that is, you realise that these are actually very familiar species: great northern diver, brent goose, Arctic skua, black-necked grebe, grey phalarope, shore lark and Lapland bunting respectively. Despite decades of wrangling, the British and Americans simply won't agree on which names should take precedence.
From time to time, some authors have taken the plunge and attempted to sort out the situation. But this has only led to even greater confusion, as when the editors of the _Collins Bird Guide_ (to the birds of Britain and Europe)1 plumped for some rather strange, hybrid names, including 'great northern loon' and 'parasitic skua' (which by the time of the second edition had already reverted to Arctic skua).vi The fact that they tried to make a compromise and failed simply highlighted the problem: Brits will continue to refer to divers and skuas, and Yanks to loons and jaegers, for many years to come.
A similar, but more colonially sensitive, situation has arisen in Africa. Here the legacy of empires – the Dutch as well as the British – led to a schism between the English names used in East Africa and those, mainly derived from Afrikaans, used in South Africa.
Again, a quick comparative look through field guides for the two regions can be very confusing. Surely that strikingly blue, pheasant-like bird with the russet wings and yellow face is the same species as this one; yet in the East African guide it is called Ross's turaco, while in the South African book it appears as Ross's lourie. Of course they are the same species, as confirmed by both their shared eponym and their identical scientific name ( _Musophaga rossae_ ).vii The name 'lourie', from the Afrikaans, first surfaces in South Africa in the late eighteenth century, while in 1822 the visiting English ornithologist W. J. Burchell mentions the confusion already arising between the two alternative names, noting that 'In the aviary, I saw the Touracoo, called Loeri by the colonists.'
Likewise, in each guide we find plates featuring a series of plump, long-legged birds: one labelled as 'bustards' (East Africa), and the other as 'korhaans' (South Africa); while a group of strange waders resembling our own (misnamed) stone-curlew are shown as thick-knees in the East African guide and dikkops (meaning 'thick-head') in the South African one. Once again, they are the same family.
But gradually, as birding becomes more and more global, the Afrikaans-based South African names are beginning to give way to the more widely used (and English-based) East African ones. The latest edition of the main field guide to southern Africa lists turacos and thick-knees (with the South African names in brackets), but confusingly still keeps 'korhaan' for some of the smaller species of bustard.2
Keeping up with these changes in bird names, with decisions made unilaterally by authors in each region, presents a perennial problem not just for travelling birders, but also for the international ornithological community, who need to avoid confusion in scientific papers and when communicating with colleagues in different countries. So when, in 2006, two US ornithologists produced a slim volume entitled _Birds of the World: Recommended English Names_ ,3 many people heaved a sigh of relief.
Backed by the International Ornithological Congress – the United Nations of the bird world – Frank Gill and Minturn Wright (who in his day job is, appropriately, a lawyer) attempted to produce a consensus view on the world's bird names. The back-cover blurb summed up their aims and intentions: 'This book provides the first standardised English-language nomenclature for all living birds of the world... based on the rules and principles developed by leading ornithologists worldwide.'
It was a good and timely idea, and in the absence of something more definitive it is the best we have, at least in a volume conveniently small enough to be taken on your travels. But it still had to deal with minor issues (such as the US/UK difference between 'gray' and 'grey') as well as the thornier problems. These of course included the perennial issue of loons vs divers (they chose loons) and skuas vs jaegers, where they compromised, favouring skua for the two larger species (great and pomarine) and jaeger for the smaller duo (Arctic and long-tailed).
But the biggest problem faced by anyone who tries to standardise the English names of birds is that – just like the rest of our language – these are not fixed, but fluid. Most importantly, change doesn't happen because of top-down decrees, but through the normal processes of the evolution of the English language, from which, as we have seen, even some of our longest-established bird names are not immune.
The paradox of standardising the English names of birds is this: although in the short term it may make communication easier – especially between two different groups of English-speaking peoples, such as the Americans and British – it also runs the risk of losing the original reasons why the birds were named.
So although the eponymous bird names we celebrated in Chapter 5 may appear anachronistic, and although many other names we use are at best illogical and at worst downright misleading, if we change them, we lose the stories of their origin; we lose the complex connections between language, history and the real world; and we perhaps also lose a little of the contingency, contradiction and whimsy that make us human. And as we shall see, when the world's birdlife is facing greater threats than ever before, we cannot afford to forget these profound and ancient links between humanity and birds.
## _2: Titmice and Ring-Doves_
The English language is constantly shifting, but slowly and gradually: it is only after a change in meaning has occurred and been widely accepted that we realise it has happened.viii
Recent examples include 'decimated', which used to mean 'cut down by one-tenth', but is now widely used as a synonym for devastated; 'enormity', originally 'a great tragedy', but now simply 'a huge event'; and 'disinterested', now more or less synonymous with 'uninterested', rather than the original sense of unbiased. By adopting these new meanings, we effectively lose the original sense of each word. In many people's view, this risks diminishing the English language.
But as linguists have long pointed out, to try to resist such changes in spoken English is to fight against a tidal wave of actual usage. New words are being appropriated or invented all the time. Sometimes this is a deliberate act, as with the adoption of 'hygge' from Swedish, or the invention of 'Brexiteer' and 'chatbot'.ix Others are coined accidentally, as when Sarah Palin used the word 'refudiate' (instead of 'refuse' or 'repudiate'), or George W. Bush said 'misunderestimate' (whose meaning is, as usual with the former President's sayings, wonderfully vague).4
But whether a completely new word has been coined, or an established word has changed in meaning, the pedants must simply grin and bear it. Otherwise, I suppose, we would all be speaking the language of Chaucer or Shakespeare.
Bird names change over time in much the same way. We have already seen how during the twentieth century 'redbreast' became robin, 'golden-crested wren' was simplified to goldcrest and, much further back in time, 'gowk' was displaced by cuckoo and 'ouzel' by blackbird.
Leafing through a list of bird names in a 1923 issue of the British Ornithologists' Union journal _Ibis_ , we find almost a hundred names that are no longer in general use. Many of these changes are the result of minor tweaks, such as the removal of hyphens in compound names such as 'sky-lark', 'sheld-duck', 'bean-goose', 'oyster-catcher', 'black-grouse' and 'marsh-harrier'.x Other changes, though, are more radical. A modern birder might struggle to work out the identity of 'Richardson's and Buffon's skuas' (now Arctic and long-tailed), while 'buff-backed heron' (cattle egret) has also long fallen out of fashion.
Some species have simply gained a descriptive epithet: thus at some point 'heron' became grey heron, and 'kite' turned into red kite, to distinguish them from their rarer relatives, the purple heron and black kite. Names that were in the process of changing are shown in the 1923 list as alternatives: such as 'wild duck or mallard', and 'ring-dove or wood-pigeon'.
One of the most striking changes in usage is the switch in the name of some of our most familiar garden birds, from 'titmouse' to 'tit'. The use of the older name – which dates back to the fourteenth century, and as we have seen simply means 'small creature' – lasted longer than we might imagine. So even though the shorter and more convenient term 'tit' has been in widespread use since the 1700s, a book published as recently as 1975 was still called _The Titmice of the British Isles_.5
However, just four years later a Collins New Naturalist volume, written by the Oxford scientist Christopher Perrins, rejoiced in the rather saucy title _British Tits_.6 This was apparently confirmed only after a lengthy correspondence between the author and the publishers, who were afraid the ambiguity might offend the delicate sensibilities of their readers.
With all these changes to bird names taking place gradually, over almost a hundred years, they have been accepted into general usage and, over time, become the norm. But in the final decade of the twentieth century, a radical proposal was made to change a significant minority of English bird names. Unlike previous changes, however, its proponents aimed to make it happen not gradually and organically, but virtually overnight...
*
It was as if Moses had come down from the mountain, breathing fire and fury, and carrying the stone tablets on which were carved the Ten Commandments. But this time the pronouncements did not deal with such minor transgressions as murder, adultery or coveting thy neighbour's ass. Instead, they condemned what to many was a far more pressing and significant subject: a proposed series of radical changes to the English names of birds.
The speaker was Ian Wallace, a man widely regarded as the godfather of modern birding. With his flowing white locks and beard, he certainly looked the part of an Old Testament prophet. More importantly, he commanded both affection and respect from his audience – as well he might, for through the sheer force of his personality, expressed through his lively writings and quirky illustrations, he has influenced successive generations of birders from the 1950s to the present.xi
Never one to duck controversy, Wallace had seized an opportunity to speak in a debate before the great and the good of the bird world. This took place at a conference jointly organised by the British Trust for Ornithology and _Birding World_ magazine, held at Swanwick in Derbyshire in March 1993, and aptly named 'Pride and Prejudice'.
Wallace's words, combined with his animated and energetic delivery, suggested an imminent apocalypse of Biblical proportions. Hands waving, voice rising in volume and pitch with every sentence, he railed against a new proposal that wholesale changes should be made to the names of the birds of the Western Palearctic – the zoogeographical region comprising Britain, Europe, North Africa and the Middle East.
Following Wallace's oration his opponent, the highly respected ornithologist Tim Inskipp, did his best. He put forward some excellent arguments, founded mostly in the urgent need to remove confusion amongst birders and ornithologists from different nations around the world. But in the face of Wallace's whirlwind, he simply had no chance. When a show of hands for or against the changes was taken at the conclusion of the debate, it was overwhelmingly in favour of respecting the status quo.
Ironically, given the result of the debate, few birders and ornithologists could deny that many of our bird names were indeed completely illogical. Why blackcap and whitethroat, and not 'black-capped warbler' and 'white-throated warbler', for example (as MacGillivray had attempted to rename them nearly two centuries earlier)? Why, as we have already seen, were British and American names for the same species or family sometimes different? And most of all, how, in this post-imperial world, could we possibly defend our continued use of single names for 'our' swallow, cuckoo, wheatear, kingfisher, jay and wren, when each is just one of dozens of species in their respective families found across the globe?
These were the driving forces behind the call for change. That was why a few months earlier, in the June 1992 edition of the influential monthly magazine _British Birds_ , Tim Inskipp and Tim Sharrock had published a short but detailed paper setting out the reasons for the proposed changes, and a list of suggested new names.
These fell into three main categories. First, there were those birds for which the American and English names were significantly different. These had been discussed in detail three years earlier, at the 1990 International Ornithological Congress in New Zealand.
So while the names 'diver' and 'skua' remained, Inskipp and Sharrock's paper proposed that the descriptive English names should give way to the American versions. Thus white-billed diver (known in North America as the yellow-billed loon) became 'yellow-billed diver', while Arctic skua (known in North America as the parasitic jaeger) became 'parasitic skua'. Such messy compromises satisfied no one, and set the tone for the acrimonious opposition to the changes that would follow.xii
The next category of proposed changes was for those instances where two similar species had names distinguished only by a qualifying adjective – such as ringed plover and its smaller, scarcer relative the little ringed plover; and black tern and its rarer cousin, the white-winged black tern. This was deemed to be illogical, and so the latter were to be changed to 'little plover' and 'white-winged tern'. Again, this was completely logical, and yet somehow felt 'wrong'.
Other suggested changes were more arbitrary. The small seabirds known for generations as Leach's and storm petrels were renamed 'Leach's storm-petrel' and 'European storm-petrel' respectively, on the grounds that this distinguished them from the larger petrel species. Further modifications included changing stock and rock doves to 'stock pigeon' and 'rock pigeon', turning dunnock into 'hedge accentor', and replacing bearded tit with the simple word 'reedling' (but not, however, adopting William MacGillivray's splendid suggestion 'furzeling' for the equally ill-named Dartford warbler).
At this stage, alarm bells were beginning to sound in the minds of most readers. It was hard to imagine any birder in the field ever using the proposed new names for any of these species. And if a name is never actually used, except in print, can it really be considered acceptable?
Then there was the addition of gaps between words, in order, so it was said, to avoid confusion and be consistent. So skylark and woodlark became 'sky lark' and 'wood lark' (presumably to match shore lark) and corncrake turned into 'corn crake'. Although this made logical sense, and even, it could be argued, reflected the older, hyphenated versions of these names, they still looked very odd when set down in print.
The third and final major proposal was the addition of a distinguishing epithet to almost a hundred one-word names: these were birds where only a single species in their family is commonly found in Britain, and so for centuries they have been known by a single-word name. Examples included pheasant, crane, cuckoo, kingfisher, bee-eater, roller, swift, nightjar, swallow, nuthatch, wren, wheatear and starling.xiii
These are, without doubt, classic examples of British insularity, arrogance and jingoism. Each is just one member of a much larger family, containing many more species found around the world. But because we named our own familiar species first, before we were aware of any of its relatives, most of us continue to call them by their original, abbreviated name, without any qualifying adjectives.
Thus 'our' nuthatch is just one of two dozen species, which include the Kashmir, Chinese, Corsican and Algerian nuthatches, the eastern and western rock nuthatches and, supreme amongst them all, the beautiful nuthatch, whose bright blue plumage lives up to its name. Likewise, what we call the 'swallow' is one of about fifty species, the 'wren' one of eighty, and the 'kingfisher' one of almost a hundred.
Hence the proposed new names, with added descriptors to bring them into line with the rest of the world. By far the largest category of proposed new descriptors was the fairly meaningless adjective 'common', to be applied to thirty-one species, closely followed by the more helpful geographical distinctions 'Eurasian' (twenty-two species, such as wigeon and hobby) and 'European' (eleven species, including nightjar and bee-eater). 'Northern' was added to the names of eight species, but 'Western' to just one (the capercaillie).
Oddly, the committee in charge of these proposals then went off-piste. It was as though, bored with the addition of such dull and predictable words as 'common' and 'Eurasian', they rebelled. So amongst this rather mundane list there were a dozen marginally more imaginative labels: rock ptarmigan, pied avocet, red knot, black-legged kittiwake, Atlantic puffin, barn swallow, white-throated dipper, winter wren, wood nuthatch, black-billed magpie, spotted nutcracker and red-billed chough.
In some cases, this involved adopting a name widely used elsewhere in the world, particularly in places where two similar species occur. For instance, black-billed magpie was already in use in North America, to distinguish it from its close relative found there, the yellow-billed magpie. Similarly, the name black-legged kittiwake was already used to differentiate from its cousin, the red-legged kittiwake.
But ironically, what really riled some of the opponents to these changes was the inconsistency of the proposals. A hundred and fifty years earlier, William MacGillivray had at least had the courage to be truly radical: to sweep away the old and suggest an entirely new and systematic approach. Yet when it came to some of the least logical of our bird names, Inskipp and Sharrock held back. So they kept such oddities as fieldfare, redwing and black-cap, even though these do not provide any clue as to the family to which the birds belong.
The immediate reaction to Inskipp and Sharrock's proposals was a potent blend of bemusement and hostility. One correspondent to _British Birds_ called the whole debate 'a waste of time', and pointed out that – thanks to Linnaeus and his successors – we already had a perfectly good and universal way of telling species apart: 'Have Inskipp and Sharrock forgotten what the Latin system of scientific names is meant to be for?'7 However, not everyone was against the changes: another reader praised the authors for 'producing such a well-thought-out and comprehensive review [of] this perennial problem'.8
But perhaps the most pertinent response to the proposals – Ian Wallace's verbal tour de force notwithstanding – came in the pages of a scruffy parody of _British Birds_ (which is widely known as 'BB'). This was little more than a few cheaply printed pages held together by staples, which rejoiced in the title _Not BB_ , and ran to five quirky and often hilarious editions.
In a short feature entitled 'Changes to the English Language', the anonymous authors produced their own version of the proposed name changes.9 These included:
Black Vulture TO BECOME Afro-Caribbean Vulture
Garden Warbler TO BECOME Stately Home Warbler
House Sparrow TO BECOME Slum Weaver
Robin TO BECOME Northern Red-breasted Bush-Robin
Wren TO BECOME Jenny Wren
As the authors sardonically noted, in what would become the final nail in the coffin for the suggested changes:
The following names are so manifestly apt that they are TO REMAIN:
Barnacle Goose, Black-headed Gull, Dartford Warbler, Kentish Plover, Iceland Gull, Marsh Tit, Slavonian Grebe, Turtle Dove...
And yet, despite the widespread hostility towards the name-changes, a few of these new names – including barn swallow and northern wheatear – have gradually crept into use, especially when British birders travel abroad. They are also used in scientific papers, and in magazines such as _British Birds_ , though I still can't get used to the clumsy 'sky lark' to describe that wondrous aerial songster – just try saying 'skylark' and 'sky lark' out loud, and you'll soon see what I mean. Shelley would surely be turning in his grave.
But the vast majority of the new names will never be spoken. And that, surely, is the key reason why they never really caught on.
*
My real objection to every proposal to impose a definitive list of English names on us – from MacGillivray's doomed attempt in the mid-nineteenth century to the _British Birds_ affair – is not so much practical as philosophical. While I can understand the advantages of standardised names, and do occasionally use them myself when it will avoid confusion, the drive towards homogeneity goes against the very reason I am writing this book.
To me, the diversity of bird names is not an inconvenience but a wonder. The fact that we can choose to call _Prunella modularis_ the hedge sparrow, hedge accentor or dunnock, or _Erithacus rubecula_ the ruddock, redbreast or robin, or that there are more than thirty different folk names for the barn owl,xiv is for me not a cause of frustration, but something to celebrate. Like the rest of the English language – and especially the names we call ourselves and the places where we live – our bird names display a richness, complexity and downright quixotic quality that marks them out as part of our culture and heritage.
However, in a changing world, some bird names are now definitely considered unacceptable, as two stories from opposite sides of the Atlantic reveal.
## _3: Politics and Political Correctness_
In the year 2000, the American Ornithologists' Union (AOU) made one of its official pronouncements on bird names. They had decided to change the traditional name for the duck _Clangula hyemalis_ from 'oldsquaw' (a relatively late name, first appearing in print in 1834) to long-tailed duck – the name we have used in Britain since the naturalist George Edwards first coined it in the middle of the eighteenth century. Pamela Rasmussen, the distinguished US ornithologist based at Michigan State University, who was on the committee that made this decision, recalls that this proposal caused more controversy than any other name change, before or since.10
The AOU suggested that this was 'to concur with world-wide usage', even though it was abundantly clear that it was actually to avoid offending Native Americans, some of whom maintain that the word 'squaw' is unacceptable and demeaning. Ironically, though, this worthy intention may have been misplaced, as one anonymous contributor to a blog thread pointed out:
Did anyone stop to think that possibly an original person from the Penobscot Nation may have named the bird, 'Oldsquaw', and removing that name may dishonor the origin of it? I heard that when an old Native American woman dies it is believed her spirit goes into a bird... hence 'Oldsquaw'.11
But the AOU's decision was small beer when compared with the wholesale changes carried out by the Swedish Ornithological Society in 2015. As they were compiling a list of more than 10,700 Swedish names for the world's birds, they came across some that they realised, to their horror, could easily be considered racist.
These included a small African duck named the _hottentott_ , based on a derogatory term for the Khoikhoi people of south-western Africa;xv _kafferseglaren_ , which translates as 'kaffir-sailor', another deeply offensive South African insult, for the white-rumped swift; and _zigenarfågel_ ('gypsy-bird') for the peculiar South American bird known as the hoatzin – the only bird, incidentally, whose youngsters have claws on their wings to enable them to clamber around the forest foliage. Four other names included the word _neger_ , which translates as 'negro': these were changed to _svart_ , a less offensive term translating simply as 'black'.xvi
Around the same time, the author Robert Macfarlane published his book _Landmarks_ , full of rediscovered dialect words for features of the British landscape. As the journalist Patrick Barkham wryly observed in the _Guardian_ , 'I wonder how many racist words he has discovered and quietly not added to his word-hoard?'
Barkham also touched on the question of whether any current English bird names can be considered offensive. Here we must tread carefully, as offence is very much in the eye – or ear – of the beholder. Surely even the most devout Roman Catholic is unlikely to take exception to the use of the name cardinal for the bright red North American finch, or a lawyer be offended by the name prothonotary warbler (the descriptor refers to a high ranking court official).xvii
But prudish people can still be offended by the apparently rude names of some common and familiar birds: tit, shag and booby are the first that come to mind (meaning 'small', 'crested' and 'stupid' respectively). Other dubious names, some of which must be said out loud to be properly appreciated, include hoopoe, hawfinch, chough, bonxie (the Shetland dialect name for the great skua, still in common use amongst birders), great bustard and nutcracker.
Across the pond, a similar list might contain dickcissel, tufted titmouse, horned puffin and that wondrous orange and black South American bird, the cock-of-the-rock. But apart from providing amusement for bored bloggers, in most cases there is no genuine linguistic linkage between the bird name and the apparently rude meaning.
This is not the case, however, when it comes to the ultimate taboo in rude bird names – one that puts tit, shag and booby in the shade: windfucker. The origins of this now obsolete folk name for the kestrel reveal the fascinating history of what must surely be the best known swear word in the English language today.
Anyone who has ever watched a hunting kestrel as it hovers in the air, holding its head virtually motionless while its wings beat rapidly to keep in position, has marvelled at the bird's skill and technique. The Victorian poet Gerard Manley Hopkins was suitably impressed, and celebrated the kestrel's extraordinary abilities in his 1877 poem 'The Windhover':
I caught this morning morning's minion, kingdom
of daylight's dauphin, dapple-dawn-drawn Falcon, in his riding
Of the rolling level underneath him steady air, and striding
High there, how he rung upon the rein of a wimpling wing
In his ecstasy! Then off, off forth on swing,
As a skate's heel sweeps smooth on a bow-bend: the hurl and gliding
Rebuffed the big wind. My heart in hiding
Stirred for a bird – the achieve of, the mastery of the thing!
But as a Jesuit priest, who dedicated this poem 'To Christ our Lord', Hopkins would surely have been deeply shocked by 'windfucker', an earlier name for our commonest falcon.
The first – and indeed only – reference to this folk name for the kestrel dates back to 1599, when the Elizabethan pamphleteer, playwright and poet Thomas Nashe wrote of 'the kistrilles or windfuckers that filling themselues with winde, fly against the winde euermore'. Soon afterwards, in the early seventeenth century, it was adopted as what the _OED_ calls 'a term of opprobrium', in phrases such as Ben Jonson's 'did you euer heare such a wind-fucker, as this?'xviii Alternative versions included 'wind bibber' (Sussex), 'wind cuffer' (Orkney), 'wind fanner' (Surrey and Sussex) and the blunter 'fuckwind', which was still being used in the north of England as late as 1847.
On the surface, the origin of these names is clear: like 'windhover', they all refer to the way the kestrel's wings beat the air so that the hovering bird can stay in one place and keep its eyes fixed on its target below. But although we may be shocked at the use of the terms 'windfucker' and 'fuckwind', our ancestors might not have found them quite so offensive.
According to W. B. Lockwood, the word 'fuck' was originally a euphemism, meaning 'to beat'12 – rather like the word 'bonk' is used as a euphemism for the word 'fuck' today. However, we have only his word for this, as the term 'windfucker' is the only recorded use of that word before it took on the offensive meaning it has nowadays. We can, though, assume that it won't be considered as an alternative name for the kestrel, at least in the foreseeable future.
## _4: Splitting Species_
In the aftermath of Inskipp and Sharrock's controversial proposals, while we in Britain were focusing on proposed changes to English bird names, on the other side of the Atlantic a more momentous revolution was occurring in the world of scientific ornithology. This would not simply shake up the names we give to different birds: it would change the way we look at the relationships between the species themselves.
In 1990 three US ornithologists and biologists, Charles G. Sibley, Burt L. Monroe Jr. and Jon E. Ahlquist, produced two blockbuster books that between them would put a metaphorical bomb under what had been, until then, the stable and predictable world of bird taxonomy. Their proposals would completely disrupt the longstanding way in which we classify the relationships between different species of birds into larger groups and families.13
Until Sibley, Monroe and Ahlquist came along, to determine which species belonged to the same family, and to differentiate one species as separate from another, ornithologists had mainly relied on the long-established principles of external and internal morphology. These involved looking at the physical form and structure of different birds, and using these characteristics to establish how they were related.
But this approach could put the cart before the horse, because it made no real attempt to reflect what scientists call 'phylogeny' – the history of the evolutionary descent of different organisms. Just because two species look similar, or have similar physical characteristics, doesn't necessarily mean they're related: they could have evolved to look alike because their lifestyles are similar.
The new approach put forward by these three scientists began with phylogeny: it was an ambitious attempt to recreate the 'Tree of Life', to accurately represent the way the various species of bird had evolved, and therefore to show the relationships between them. In two huge volumes, each weighing several kilos, which between them contained more than 2,000 pages of text, Sibley, Monroe and Ahlquist completely overturned our established knowledge and understanding of the way we classify birds.
For the phylogeny they employed was based on a controversial new biochemical technique known as DNA–DNA hybridisation. This involved heating strands of DNA taken from one species until they separated, and then recombining the individual strands with similarly isolated strands taken from another species. When this new 'hybrid DNA' was in turn heated, the higher the temperature needed for the DNA to separate, the more closely the two species being compared were related, and vice versa.
This revolutionary approach was – so the authors and their supporters claimed – 'the Holy Grail of avian systematics'. They believed it would finally provide a definitive answer to the thorny problem of how closely two species – or on a larger scale, two orders or familiesxix – were actually related. Instead of looking at external features or internal structure, all you needed to do, they suggested, was to cook up samples of their DNA, and then do the maths.
The US team's suggested changes to classification were contentious, to say the least. They proposed wholesale changes in the family tree of the world's birds, involving some dramatic – and to some, frankly bizarre – new relationships. For example, they put those masters of the air, the albatrosses, into the same 'super-family' as the flightless penguins, and yoked together the apparently dissimilar families of grebes, shearwaters, waders, falcons and flamingos into another massive 'super-family' (Ciconiiformes).
Less controversially (at least at the time), they suggested that the bare faces and broad wings of storks and the New World vultures were not, as had previously been thought, the chance result of convergent evolution brought about by their scavenging habits, but had arisen because they were genetically close cousins.
Almost immediately, the backlash began. There was a widespread feeling amongst the wider ornithological community that although the overall aims might be laudable, the pioneering scientific methods being used were simply not accurate enough to support the conclusions. There was also a suggestion that Sibley was manipulating some of his experimental results to fit prior assumptions – either through carelessness or to get the answers he wanted – making it hard to know whether any particular set of results was valid or not.
During the quarter of a century since this bombshell first hit the world of ornithology, some of these radical conclusions have indeed been discredited.14 These include, ironically, the one proposal with which most people had agreed – the placing of New World vultures in the same group as storks. This was later retracted as having arisen from erroneous data, and most authorities now place the vultures back with the Old World hawks and eagles in their original order, Accipitriformes.
Right up to his death in 1998 Charles Sibley passionately continued to defend the new approach, pointing to the fact that many of their findings had been accepted by the wider scientific community.15 These included the once radical idea that Australasia's songbirds had – like that continent's marsupial mammals – evolved separately from those found elsewhere in the world, and so despite their superficial differences were in fact more closely related to one another than they were to their Old World counterparts (see Chapter 4). That one turned out to be spot-on.
Today, thanks to the pioneering work by the US evolutionary ornithologist Professor Richard O. Prum of Yale University, the big picture of avian phylogeny – at least at the level of orders and families – is becoming clearer, though there are still many gaps in our knowledge. In a 2015 paper in the journal _Nature_ ,16 Prum and his colleagues produced a series of what one of their peers called 'robust conclusions, representing a decisive technical advance on Sibley and Ahlquist'.17
To the ordinary birder, their findings are actually quite comforting, as they appear to reflect the way we see and understand relationships between birds in the field. For example, one grouping consists of those aerial acrobats the nightjars, swifts and hummingbirds, while two huge groupings are broadly separated into waterbirds (including all diving, wading and shorebirds) and landbirds.
The authors believe that these splits occurred in the wake of the last mass extinction, the Cretaceous-Palaeogene, some 66 million years ago. This was almost certainly the result of a massive meteorite hitting the planet, creating catastrophic changes to the world's climate and atmosphere, and killing off three-quarters of the plant and animal species on Earth – including the non-avian dinosaurs. This left the playing field clear for new lineages to evolve, including modern birds (or as we should perhaps now call them, avian dinosaurs).18
*
These days, however, the argument about the potential relatedness at the higher level of orders and families has faded into the background. A second revolution has been taking place in the world of ornithology, putting a potentially far more iconoclastic issue under the spotlight: the whole way we decide what defines a species. It has had a profound effect on the names we give to birds, as the result has been a huge increase in the number of different kinds of bird found around the globe.
When I was growing up in the 1960s and 1970s, the number of different species of bird in the world was thought to be roughly 8,600. This figure had been arrived at many years earlier, in 1951, by the German/American evolutionary biologist Ernst Mayr and his colleague, the US ornithologist Dean Amadon. At the time it was generally assumed that this total would increase only very slowly, as a handful of new species (or re-discovered species hitherto thought to be extinct) might still be found in the remote rainforests of South America, Africa or south-east Asia.
For several decades, that's exactly what happened. Then things began to change, with the number of species suddenly rising dramatically, to more than 9,700.19 As Charles Sibley later noted, only fifty or sixty of the thousand or so new species were previously unknown to science; the rest – around 95% – were the result of a new and radical way of classifying birds.
Before this time, the most widely accepted way of defining a species was Mayr's Biological Species Concept (BSC). This assumed, as we have seen earlier in the debate around the identification of the willow tit as a British breeding species, that if two different populations of birds overlap in range without interbreeding, they must be two separate species. But a problem arises when there are two populations that look or sound slightly different, but whose ranges do not overlap, which means that the hypothesis cannot be tested in the field.
Under the strict rules of the BSC, it had previously been assumed that these were _not_ separate species, but merely races or subspecies. To get round this problem, Sibley and Monroe made one crucial change. They promoted the 'allospecies concept', which gives birds that look or sound significantly different from their relatives the benefit of the doubt, and regards them as separate species – whether or not this can be confirmed in the field.
This may all sound like an arcane and meaningless distinction: the academic equivalent of medieval scholars debating how many angels can fit on the head of a pin. But in fact it was a critical paradigm shift: not just scientifically, but from a practical viewpoint too. For birders all over the world, it has made a huge difference, as it revealed the hidden presence of an extraordinary number of new species – perhaps as many as 1,100.
At this stage it is worth reminding ourselves of the old joke, 'a species is whatever a competent taxonomist says is a species'.xx Given the ambiguity, even the greatest experts in their field have been unable to agree on a working definition of when two discrete populations of birds can be defined as separate species, rather than just distinctive subspecies.
But whatever the arguments, the general move towards a looser definition of what makes a species has led to an epidemic of what ornithologists call 'splitting'. This is the process by which what were once considered distinctive races of a single species are 'promoted', thus creating two new species where only one had existed before.
This process has always been with us: when I started birding, back in the 1960s, pink-footed and bean geese were still considered conspecific – i.e. two different races of the same species. Soon afterwards, they were each elevated to specific status. Later on, the 'bean goose' was in turn split into two full species, known as tundra bean and taiga bean geese. Likewise rock and water pipits were lumped together as a single species until as recently as the late 1980s, despite their very obvious differences in appearance and ecology.
Today more and more species are being split: according to some authorities the familiar brent goose may comprise three different species: pale-bellied, dark-bellied and the North American version known as 'black brant'.
Further afield, there are now three different versions of the elegant raptor known as black-winged or black-shouldered kite (one in southern Europe and Africa, one in Australia and one, known as white-tailed kite, in the Americas); there are four different darters or 'snakebirds', where there were once two; and no fewer than six different versions of the giant cousin of our moorhen that used to be called 'purple swamphen', found from Spain and Portugal, through Africa and Asia, to New Zealand.
For keen 'world listers', desperate to see every species of bird on the planet, this unexpected development has been a mixed blessing. The good news is that there are suddenly all sorts of new birds to go out and see – or if they are lucky, to 'tick off' their list while sitting at home, as they have already seen them without hitherto realising that they were actually separate species. But for some, the new approach has proved deeply frustrating, because unless they have kept very detailed notes about the birds on their various travels, and exactly where they came across them, they have found it impossible to work out which of two or more new species they have actually seen.
The upshot of all this is that the current generally accepted total of different species in the world has now reached a figure of between 10,500 and 10,700, an increase of almost a quarter on Mayr and Amadon's original estimate. But is that the limit, apart from a handful of new discoveries to be made over the next few decades? Or might it be just the beginning of an explosion in the number of different species of bird in the world? I gained a fascinating insight into this issue in 2004, on a birding trip to Morocco.
*
The Dutch birder and ornithologist Arnoud van den Berg knows more about the birds of the Western Palearctic than virtually anyone. So when I accompanied him on a visit to Morocco I expected – and duly received – the benefit of his vast wealth of knowledge about that country's birds.
What I didn't expect was that, while we were watching one of the rarest birds in the world, a flock of northern bald ibises, Arnoud would casually point out a small, cormorant-like bird perching on the rocks below. I was even more surprised when he explained that this bird, the North African subspecies of the European shag, was potentially the most endangered bird we would see on the trip, with a global population of just a hundred pairs.
g
Another member of our group tactlessly pointed out that it was 'not a real species', which prompted a characteristically thoughtful response from Arnoud. He patiently explained that, given the difficulty in trying to define what is a species and what is not, we should perhaps take a different approach. In his view, we should simply focus on each distinctively different kind of bird – whether we consider it a subspecies or a full species – and do our best to conserve it. That way we would ensure that we retain avian biodiversity, and can leave future generations to argue whether or not a particular bird is indeed a different species from its cousin.xxi
Arnoud van den Berg is not the first naturalist to take this approach to classification and taxonomy. Back in the late nineteenth century, the Victorian ornithologist Richard Bowdler Sharpe, curator of the bird collection at the British Museum of Natural History, took a similarly pragmatic view. Indeed, he went even further: by ignoring the distinction between species and subspecies, he concluded that there were almost 19,000 species of bird in the world.xxii
Astonishingly, some ornithologists now believe that Bowdler Sharpe may have been right all along. In December 2016, a new study led by the highly respected American Museum of Natural History (AMNH) suggested that there could be as many as 18,000 different species of bird on the planet – close to Bowdler Sharpe's estimate, and not far off twice the current accepted total.20 The authors based their conclusions on the concept of 'hidden avian diversity'; the idea that there are many birds out there that look so similar to one another that they have previously been either ignored, or thought to be subspecies.
Considering that birds are probably the most studied group of wild creatures in the world, this seems hard to believe. But perhaps we shouldn't be so surprised: after all, as recently as 1999 scientists discovered that the small bat known as the pipistrelle – one of Britain's most widespread mammals – was in fact two totally separate species, each echolocating using different sound frequencies.
In some ways this is also the logical consequence of the new way of looking at what makes a species. As Charles Sibley remarked, '"Splitting" and "lumping" will continue as we try to make nature fit our concepts. Of course, we like to think that we are making our concepts fit Nature.'
Fundamentally, this comes down to the mismatch between the way we try to classify the natural world and the way it actually arose – something that, of course, we can never truly know or understand.
*
So where does this leave English bird names? Well, for a start, it would suggest that if there really are almost twice as many species out there, then we are going to need an awful lot more names. Perhaps fortunately, this is not quite true: in many cases the various subspecies already have a perfectly serviceable name, which can be adopted when the species is split.
Take the case of one of North America's most familiar and colourful birds, the Baltimore oriole. The Baltimore Orioles baseball team, based in Baltimore, Maryland, was named in the 1950s after the national bird of that state – a striking, orange-and-black songbird. But in 1973, to the horror of the team's many fans, the AOU decided to lump this species with its western counterpart, Bullock's oriole,xxiii into a single species, which was given the deeply unimaginative name 'northern oriole'. The reason given for the change was that, in the zone where both forms encountered one another, ornithologists discovered that they were interbreeding and perhaps producing fertile offspring – a classic sign, at least according to the more traditional BSC, that they must be the same species.
Fortunately for devoted followers of the Baltimore Orioles, the story has a happy ending. Later research by a team of Canadian ornithologists showed that while there was some interbreeding between the two forms, it was nowhere near as frequent as had previously been thought, so in 1995 they were separated once again, and given back their old (and to my mind far more interesting) names.xxiv
In other cases, when a species new to science is discovered, it is often named after the locality where it lives – especially when it has a very limited territory, as many of them do. Leafing through the final 'Special Volume' in _Handbook of the Birds of the World_ ,21 published as a supplement to the series in 2013, we find that of the eighty-four newly discovered species featured, the vast majority are named after the location where the bird was found.
These areas range from the geographically large, as with the New Zealand storm-petrel, to the tiny, such as the Rubeho forest-partridge, known only from a single, densely forested mountainside in northern Tanzania. Other species named after very localised place-names include the Delta Amacuro spinetail, from the Orinoco Delta in Venezuela, and the Acre antshrike, found only along a short section of Brazil's border with Peru.xxv Another recently bestowed bird name, Jocotoco antpitta, sounds as though it comes from a location, but 'Jocotoco' is actually the name given by the local indigenous people to the bird – a useful reminder that although we often consider a species to be 'newly discovered', it may in fact have been known to its human neighbours all along.
This new way of looking at the relationships between birds does have other, more serious implications for the names we give to birds. On a broader scale, falcons have recently been found to be far more closely related to parrots (and adjacent in the classification system to songbirds) than they are to other diurnal birds of prey such as hawks, buzzards and eagles.
So can we still refer to falcons as 'raptors', or indeed 'birds of prey'? And if we do continue to do so, why do we not include owls under the same term?xxvi After all, they are as closely – or distantly – related to the 'true' raptors as falcons are, and early ornithologists used to classify all three groups under the wonderfully evocative term 'rapacious birds'.
At a more detailed level, the latest DNA studies have revealed some surprising relationships – or perhaps that should be 'lack of relationships' – within one of our best-known bird families, the warblers.
We all know – or at least _think_ we know – what a warbler looks like: in Britain at least, they are (mainly) migratory, insectivorous songbirds, often rather drab in plumage and hard to see, so that we tend to tell them apart by their songs. But as I look through the latest version of the BOUs' British List (updated December 2016), I see to my surprise that Cetti's warbler is now separated from the rest of its tribe by the long-tailed tit.22 So does that mean that Cetti's warbler is not a warbler, or that the long-tailed tit is, or that neither species belongs to the warbler family?
Some authorities go much further, splitting Sylviidae, which used to include all the European warblers, into five separate families. These are the 'leaf warblers' (willow, chiffchaff etc.), 'bush warblers' (an African and Asian group, of which Cetti's is the only European member), 'grass warblers' (such as grasshopper and Savi's), 'marsh and tree warblers' (from the genera _Acrocephalus_ and _Hippolais_ , including reed, sedge and icterine) and finally the 'true warblers', of the genus _Sylvia_ , whose members include our familiar whitethroat and blackcap. As if that wasn't confusing enough, this last group may prove to not actually be warblers at all, but members of the babbler family.
If you are feeling a little baffled, then you had better get used to it, for this is the future of ornithology. In the meantime, we can either spend time arguing over whether 'warblers' actually exist, or take a more relaxed view, and continue to lump all these birds under that convenient (though perhaps scientifically inaccurate) linguistic term.
One result of all this change and upheaval, however, has been a very sensible proposal to standardise the sequence in field guides. When I was growing up the 'Wetmore Order' prevailed (established in 1930 by the US ornithologist Alexander Wetmore, and revised again in 1951 and 1960), with divers and grebes at the start and buntings and sparrows at the end. This had in turn displaced the sequence used in _The Handbook of British Birds_ , published at the start of the Second World War, which was more or less in the opposite order, placing crows at the start and gamebirds at the end.
Modern guides have now changed yet again, and although they keep the old sequence for the majority of birds, they now usually start with wildfowl (ducks, geese and swans) and gamebirds (pheasants, partridges, grouse etc.). This can make it hard for those from an earlier generation to find these species rapidly in the book when out in the field.
If taxonomy continues to change every year or two, publishers will inevitably struggle to keep up, so we face potential chaos. No wonder that several authors, including the expat Briton Richard Crossley (who lives in New Jersey and has developed a series of pioneering and innovative booksxxvii), have called for a new, fixed and consistent sequence to be used in all field guides. This would be established independently of any further taxonomic changes, and instead would be based on simple categories such as 'seabirds', 'waterbirds' and 'land birds'. At a time when even the scientific names of some birds are changing,xxviii anything that leads to greater stability and consistency is surely to be welcomed. And as we have seen, it broadly fits the latest conclusions regarding avian phylogeny.
Ironically this proposal is much the same as the system found in what is widely regarded as the very first portable bird book written in English, Thomas Bewick's _A History of British Birds_ , published in two volumes (Land Birds and Water Birds) in 1797 and 1804.23 This just shows that if we wait long enough, the way we look at birds really does turn full circle.
## _5: New Birds, New Names_
From time to time, a pioneering ornithologist actually does discover a bird that is completely unknown to science. So it was that in 1991, the British ornithologist Paul Salaman found a new species of vireo – a small, olive-and-yellow, insectivorous Neotropical bird superficially similar to a warbler – in the Chocó department of western Colombia.24
That it took so long for the bird to be discovered is hardly surprising, when you consider that the cloud-forest where it lives is not only remote, tricky to access and often blanketed in mist, but was also a hotspot in the long-running conflict between the Colombian government and the FARC guerrilla movement.
As we have noted, newly discovered (and newly split) species are often named after the region where they were found, and this species was indeed given the name Chocó vireo. Afterwards it might have sunk back into obscurity, were it not that its scientific name was then auctioned off to the highest bidder by Salaman, through the UK-based global conservation organisation BirdLife International. So it was that in 1996, for the surprisingly modest sum of $70,000, the Chocó vireo was granted the scientific name _Vireo masteri._25
The name commemorated Dr Bernard Master, a retired doctor and lifelong birder from Ohio, who on New Year's Day 2010, after searching for three weeks, finally got to see the species that bears his name. As a dedicated 'world lister', whose aim is to see every single one of the world's bird species, Master was naturally delighted at having finally caught up with his eponymous bird.
When it was first announced that the species' Latin name was being sold to the highest bidder, the response ranged from outrage to laughter. _Birdwatch_ magazine ran a competition to suggest other potential candidates for corporate sponsorship, which included such witty entries as 'Dulux roller' and 'Kellogg's corncrake', and the winner, the rather less wholesome 'Durex shag'.
Others were less amused at what they regarded as the thin end of the wedge. Would we eventually see birds being auctioned off to whichever multinational corporation could stump up the most money, allowing them to 'greenwash' their misdeeds by appearing to care about conservation? The influential World Wide Fund for Nature (WWF) was especially vocal in its opposition, but has since been proved wrong: many species names have been 'sold' in the past two decades, including a new species of barbet in Peru, which reportedly raised over $300,000 for conservation.
And the good news is that the Chocó vireo has also been discovered at several other locations, including across the border in north-west Ecuador – as predicted by Paul Salaman.26
But let us push the moral of this tale further: despite what purists may think, should we not perhaps be following this idea to its logical conclusion, and naming more and more of the world's endangered species after anyone – individual or company – prepared to donate enough hard cash to save them?xxix
Why shouldn't the global corporations who claim to want a sustainable future actually put their hands in their pockets and donate proper amounts of money to bird and wildlife conservation – and in exchange have a species named after them? Or should that be _re_ -named? For example, the Hood (or Española) mockingbird, found only on a single island in the Galápagos, already boasts the scientific name _Mimus macdonaldi_. However, this is not after the ubiquitous burger-retailer, but to commemorate the confederate soldier and naturalist Colonel Marshall McDonald [sic] who, oddly, specialised not in birds, but fish. Maybe we should approach his multinational near-namesake and ask them to make a suitable offer.
As we noted in Chapter 5, some of the people after whom birds were named were not necessarily the most deserving: Lady Amherst may, for example, have been a remarkable woman, but her ornithological credentials were pretty much non-existent. Likewise, Thekla Brehm, whose only claim to fame is that, when she died in 1857 at the tender age of twenty-four, her grieving father gave her name to a newly discovered species of lark, which had been shot in Spain by her elder brothers.
And then of course there are the deserving, among whom I would count the woman after whom one of the world's most obscure and elusive birds was named: Winifred Moreau. In an era when the couple's extensive ornithological research was automatically attributed for the most part to Reg Moreau, it is rather wonderful that her name lives on in her eponymous warbler, a testament to her contribution to the world of ornithology – and also to the strength of Reg and Winnie Moreau's marriage.
Fifty years after I first came across this bird in the pages of _Birds of the World_ , the search for Mrs Moreau's legacy led me to travel five thousand miles to the Uluguru Mountains in the heart of East Africa, and trek several thousand feet up a forest track, where I hoped to finally come upon the bird whose name provides the title of this book.
#### Notes
1 Lars Svensson, Killian Mullarney and Dan Zetterstrom, _Collins Bird Guide_ (2nd edition, London, 2009).
2 Ian Sinclair, Phil Hockey and Warwick Tarboton, _SASOL Birds of Southern Africa_ (3rd edition, Cape Town, 2002).
3 Frank Gill and Minturn Wright, _Birds of the World: Recommended English Names_ (London, 2006).
4 For more examples, see Simon Horobin, _How English Became English_ (Oxford, 2016).
5 John A. G. Barnes, _The Titmice of the British Isles_ (London, 1975).
6 Christopher Perrins, _British Tits_ (London, 1979).
7 Simon Dowell, _British Birds_ vol. 85 (1992) p. 620.
8 I. M. Lewis, _British Birds_ vol. 85 (1992) p. 620.
9 Dr Martin Williams, Simon Stirrup, Dave Hatton and Dan Duff. _Not BB_ is still available online at: <http://www.drmartinwilliams.com/not-bb/not-bb-iii.html>.
10 Author in conversation with Pamela Rasmussen.
11 <http://robins-chaos.blogspot.co.uk/2010/01/politically-incorrect-duck-or-long.html> (2010).
12 In _The Oxford Book of British Bird Names_ , op. cit.
13 Charles G. Sibley and Burt L. Monroe, _Distribution and Taxonomy of Birds_ _of the World_ (New Haven, 1990), and Charles G. Sibley and Jon E. Ahlquist, _Phylogeny and Classification of Birds: A Study in Molecular Evolution_ (New Haven, 1990).
14 For a very helpful discussion, see G. W. H. Davidson, 'Scientific Controversy over Avian Taxonomic Changes, based on DNA Hybridisation', _The Raffles Bulletin of Zoology_ 46 (2), 1998.
15 See, for example, his article 'On the Phylogeny and Classification of Living Birds', reproduced here: <http://digilander.libero.it/avifauna/classificazione/sequence5.htm>. Burt Monroe died in 1994, aged sixty-three; Jon Ahlquist (born 1944) is still alive.
16 'A comprehensive phylogeny of birds (Aves) using targeted next-generation DNA sequencing', by Prum et al. <http://www.nature.com/nature/journal/v526/n7574/full/nature15697.html>.
17 Professor Daniel Osorio, University of Sussex, _in litt._
18 For a really clear pictorial and diagrammatic guide to the relationships between birds on a species and family level, check out OneZoom <http://www.onezoom.org/>. More detailed findings can be seen at <http://birdtree.org/> (A Global Phylogeny of Birds). Alternatively, see the paper by Erich D. Jarvis _et al_ : 'Whole-genome analyses resolve early branches in the tree of life of modern bird', in _Science_ 346, 1320 (2014).
19 See <http://www.ornitaxa.com/SM/SMOrg/sibley3.html>.
20 See <http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0166307>.
21 _Handbook of the Birds of the World: Special Volume – New Species_ (Barcelona, 2013).
22 <https://www.bou.org.uk/wp-content/uploads/2016/12/British-List-12-Dec-2016.pdf>.
23 Thomas Bewick, _A History of British Birds_ (Newcastle, 1797 and 1804).
24 See _Threatened Birds of the World_ by BirdLife International (Barcelona and Cambridge, 2000).
25 For an entertaining account of its discovery and naming, see David Turner, _Was Beethoven a Birdwatcher?_ (Chichester, 2011).
26 D. M. Brinkhuizen and A. Solano-Ugalde, 'Range extension of Chocó Vireo _Vireo masteri_ in Ecuador, with a description of the species' song', _Cotinga_ 34: 73–77 (2012).
i The money raised by sponsorship – more than $60,000 in the first year, and far more since – has gone to support conservation projects in Israel, Turkey and Georgia.
ii At a brief stop for refreshments we also came across a flock of wolf-whistling Tristram's starlings, named after the Victorian bird collector Henry Baker Tristram (whom we met in Chapter 5) – a timely reminder of our previous colonial presence in the region.
iii English is also gradually creeping into areas once dominated by other languages. In India's Bollywood film industry, for example, Hindi and Urdu are giving way to the use of English, along with a hybrid of Hindi and English known disparagingly as 'Hinglish'.
iv More than the combined total of native speakers of Spanish, Arabic, Hindi and Russian, the next four languages in descending order of popularity.
v However, in terms of raw numbers English isn't actually the world's top language. Roughly 1.3 billion people – more than one in six of the entire world population – speak Mandarin Chinese. But we are entitled to take that figure with a large pinch of salt. For although 900 million native Mandarin speakers comfortably outrank those for whom English is their first language, when it comes to its status as a second language, English wins hands down.
vi They also preferred the American horned lark and red phalarope (but helpfully added 'shore lark' and 'grey phalarope' in brackets), yet left Lapland bunting, black-necked grebe and brent goose unchanged.
vii Incidentally, this bird is named not after the Arctic explorer James Clark Ross (featured in Chapter 5), but for the wife of the governor of St Helena, Lady Eliza Ross.
viii For example, some time during the early twentieth century the meaning of the word 'hopefully' changed from its original sense of 'in a hopeful manner', as in 'to travel hopefully is a better thing than to arrive' (a saying originally attributed to the Victorian author and explorer Robert Louis Stevenson) to the modern meaning, 'it is to be hoped'.
ix Just some of the Oxford Dictionaries' new words for 2016: <https://en.oxforddictionaries.com/word-of-the-year/word-of-the-year-2016>
x This reflects wider changes in the English language, in which hyphenated words have a tendency to be replaced by single words – as in 'bookshelf' and 'suitcase'; and the more recent trend towards 'portmanteau words', made from two abbreviated words yoked together, such as 'sitcom', 'Britpop' and the much-overused 'Brexit'.
xi One reviewer said of Wallace's captivating memoirs _Beguiled by Birds_ : 'It makes you want to bring Ian home, feed him, open a bottle of something – preferably Scottish in origin – wind him up and then sit back as he regales you with stories all night' (Gordon Hamlett, _Bird Watching_ magazine).
xii One that really riled the purists was the proposed change from red grouse to the potentially confusing 'willow ptarmigan' (and consequently changing ptarmigan to 'rock ptarmigan', as it is known in North America).
xiii There are many other birds for which there is at least one other species bearing the same name: shelduck, teal, wigeon, pintail, shoveler, pochard, scaup, eider, golden-eye, fulmar, capercaillie, quail, gannet, shag, cormorant, bittern, osprey, buzzard, sparrowhawk, goshawk, kestrel, quail, coot, moorhen, lapwing, dotterel, turnstone, oystercatcher, curlew, whimbrel, redshank, greenshank, knot, snipe, woodcock, avocet, kittiwake, guillemot, puffin, wryneck, skylark, stonechat, redstart, nightingale, robin, blackbird, whitethroat, chiffchaff, goldcrest, treecreeper, jackdaw, raven, chough, jay, magpie, chaffinch, linnet, greenfinch, goldfinch, bullfinch, crossbill and serin.
xiv Including church owl, white owl, screech owl (also, confusingly, the name of two dozen or more Neotropical species in the genus _Megascops_ ), screaming owl and hissing owl, respectively referring to the bird's home, pallid colour and strange, nocturnal call.
xv The official English name for this duck is still the 'hottentot teal'.
xvi Another potentially tricky name, 'negrofinch', has been quietly changed to 'nigrita', again to avoid giving offence.
xvii The prothonotary warbler itself has a bright yellow plumage, and is so named because papal officials in the Vatican wore uniforms of this colour. When US President Harry S. Truman was introduced to one such official, he is said to have asked, 'What the hell is a prothonotary?'
xviii In his 1609 comedy _Epicene, or the Silent Woman_ , to refer to Sir John Daw, a rather ridiculous character.
xix All species of bird (and indeed any other organism) belong to a particular family, which may contain anything from several hundred species to just one. In turn, each family is combined with others to create an order. So, for example, the house sparrow _Passer domesticus_ belongs to the family Passeridae, which in turn is part of the much larger order the Passeriformes.
xx In _Ever Since Darwin_ (1977), the US biologist Stephen Jay Gould referred to 'the fuzziness of all supposedly absolute taxonomic distinctions', while the Dutch author Kees van Deemter has gone even further: 'Species and subspecies are but a convenient fiction' ( _In Praise of Vagueness_ , 2010).
xxi As Bret M. Whitney and Mario Cohn-Haft point out, in the Special Volume of _Handbook of the Birds of the World_ (Barcelona, 2013), conservationists are now beginning to recognise subspecies and dubious species in their official plans, in order to preserve the building blocks of biodiversity before they disappear.
xxii In his journal _A hand-list of the genera and species of birds_ , Vol. 5 (London, 1909). Despite spending most of his life in the museum, Sharpe has several species named after him, including a rosefinch and a starling. He also had ten daughters.
xxiii Named after a Briton, William Bullock (see Chapter 5).
xxiv A similar situation occurred with myrtle and Audubon's warblers, which are considered by some authorities to be two separate species, and by others to be one, known (rather unimaginatively) as the yellow-rumped warbler.
xxv The volume also includes the Rubeho warbler, previously considered to be an isolated population of Mrs Moreau's warbler, and found in an adjacent mountain range, but now regarded as a separate species. Ironically, this means that Mrs Moreau's warbler is under even greater threat.
xxvi Although owls are often popularly referred to as raptors, the term is usually confined to day-flying birds of prey such as eagles, hawks, kites, harriers, buzzards and falcons.
xxvii Such as Richard Crossley and Dominic Couzens, _The Crossley Guide: Britain and Ireland_ (Princeton, 2014). The innovation lies in using a series of many different photos from different angles and perspectives, to create what he believes is a more realistic view of the bird.
xxviii For example, as has already been noted, blue tit has gone from the genus Parus into a new genus, Cyanistes, while black-headed gull, formerly in the genus Larus (along with most British gulls), now has the scientific name _Chroicocephalus ridibundus_.
xxix This is already starting to happen: the World Land Trust has recently launched its 'Name an Orchid' campaign, supported by Sir David Attenborough, in which people are invited to give the name of themselves or a loved-one to newly-discovered orchid species from Ecuador: <http://www.worldlandtrust.org/name-an-orchid>.
# EPILOGUE
_Winifred's Warbler_
If the names are lost, the knowledge also disappears.
Johann Christian Fabricius, _Philosophia Entomologica_ (1778)
Dawn breaks over the Uluguru Mountains in eastern Tanzania, as we rise and prepare for the day ahead. I am about to embark on the final leg of my quest to see a very special bird. Our local guides, Elia and James, are quietly confident; I only wish I could share their optimism.
The journey here has been a long and eventful one. A few days ago, I flew almost five thousand miles, with my companions Kevin and Graeme, from the UK to Dar-es-Salaam. There we met our leader Roy, who drove us in his battered Land Cruiser from the hot, fetid lowlands around the Tanzanian capital into the Eastern Arc Mountains, where we pitched camp by a fast-flowing stream.
Known as 'Africa's Galápagos', because millions of years of ecological isolation have led to a huge diversity of endemic plants and animals, this is the only place on the planet where the species we are searching for can be found. Now we are about to set off on a trek to one of the highest mountains in this remarkable range, a peak the locals call Kilangala. Here I hope to finally catch up with the bird in question: Mrs Moreau's warbler – now also known, rather less formally, as Winifred's warbler.
*
We make slow progress: not least because, as we are visiting this region for the first time, we keep coming across new and exciting birds, each with its own bizarre and delightful name. We see stripe-cheeked greenbul and white-chested alethe, African hill-babbler and yellow-bellied waxbill, white-tailed crested flycatcher and scarlet-chested sunbird – the last hovering momentarily by a flower, showing off its dazzling red breast as it sups the nectar.
As we trek higher up the hillside, the mist begins to clear and the temperature rises with the morning sun. A flock of swallows swoop around us, hawking for tiny insects in the sky. At first we ignore them, until Elia points out that these are not our familiar British swallows, as we had assumed, but the rare Angola swallow – another new bird for all of us. We pause to watch them, and take in their subtle differences in plumage and flight action.
But every time we delay, I feel increasingly anxious: will we be too late to find the bird we are searching for? And, as with any hill-climb, whenever I think we have got to the top, yet another rise in the land appears before us, until I despair of ever reaching our destination.
Eventually, after two long and exhausting hours, we arrive on the forested ridge just below the summit. Drenched in sweat, we take long gulps from our water bottles, whose contents are now unpleasantly lukewarm. But we can't afford to rest. With forest birds, the first few hours after sunrise are the key; later in the day they often fall silent, making them almost impossible to find amidst the dense foliage. As we enter the forest, it is indeed much quieter than the terraced farmland below, though we know it holds many birds found nowhere else on Earth: Loveridge's sunbird, Uluguru mountain greenbul and, of course, our target bird.
Fortunately, we have a secret weapon. Elia has brought sound equipment and a speaker, so that he can play the song of Mrs Moreau's warbler. Hopefully this will entice the bird to appear, and perhaps even sing back to what it will assume is a rival male intruding on its territory. So as we hike along the tree-lined ridge, every hundred metres or so we stop, play the call, and listen. Yet every time we do so... nothing.
Time and again we repeat this ritual: stop, play, listen... and move on. I am beginning to get nervous: what if we don't see the bird? Would it be worse if I hear it but don't manage to see it? And how will I feel if I do glimpse it, but get what birders call 'untickable views', frustratingly too brief to appreciate its key identification features? Even that would be better than nothing, I think grimly as we trudge towards the summit.
Then, just as I have almost given up hope, Elia pauses, and gestures urgently down below us and to the left. He thinks he has heard the warbler's call. We step off the path into the forest, and make our way gingerly through dense foliage, treading carefully on the uneven and treacherous ground. I look up momentarily as a bright blue butterfly floats past, and notice that we seem to be heading towards a small patch where two saplings have grown across one another, to form a distinctive, X-shaped cross.
And then I hear it for myself. The unmistakable notes of not just one, but two birds – a breeding pair, performing together in a synchronised duet. Elia plays the sound again, and the birds immediately respond. Then he excitedly grabs my arm: 'There... _there_!'
Fumbling with my binoculars, I lift them to my eyes and look towards where he is pointing. The sound is really loud now – a series of flute-like notes, so perfectly integrated I can't tell which are being sung by the male and which by the female. But still I see no movement.
Behind me, I can hear Graeme, Kevin and Roy, each exclaiming in turn as they manage to get a sighting of the bird. Cries of delight and relief as they congratulate one another make me even more frantic. My heavy breathing has made my lenses fog up, and I now face the very real prospect that everyone else will see the bird I have travelled so far to find – but that I shall miss out.
I take a deep breath and try to stop panicking. Then I hear Graeme's calm, reassuring, Scottish brogue: 'Look Stephen, _there_ – X marks the spot!'
I look again, at exactly the point where the two branches cross. And sitting right out in the open, so obvious I can't believe I have taken so long to see it, is a small, slender bird. Brownish buff, with a long, thin bill, and an orange chest, throat, head and neck, it looks rather like a robin whose red breast has extended upwards to cover its whole face and crown. As if to acknowledge me, it utters one more burst of song, and then melts back into the forest.
It has been a long time coming, but Mrs Moreau's warbler is finally, as birders say, 'in the bag'.
*
I think back to what Reg Moreau must have thought when he first laid eyes on this bird. How did he feel when he realised that it was a species unknown to science? When did it occur to him that he could name it after his beloved wife Winnie? And how did she react when he told her of his intention to do so?
Sadly, I have not been able to find any account of this momentous discovery in his writings. Maybe it was just too personal to put down on paper. Then again, that allows me to let my imagination run free: to visualise the moment when they returned home to Amani, and he revealed to Winnie his plans to name the bird after her. Did they have a celebratory gin and tonic on the verandah, as the sun went down over the Usambaras? I like to think so...
My reverie is broken, as the male warbler hops into view once again. This time, to my delight, his mate joins him. Once again, they begin to duet: a wonderfully tuneful performance – short, but very, very sweet. Then, after a few moments, they disappear back into the dense, dark-green foliage. The show is finally over.
We turn, and begin the long hike down the mountain and back to camp. I need some space and time to reflect on my encounter with the birds so, as my companions head along the path, I lag a little way behind.
How do I feel? Relieved, certainly. Happy, and fulfilled, after almost fifty years of waiting, that my long quest is finally over. The final piece in the jigsaw of the Moreaus' story has fallen into place for me. And what a bird! It was far more exciting than I could have imagined: its perky stance, subtle but attractive plumage, and delightful song, all made the experience quite unforgettable.
But I also feel a deep sense of sadness. As we climbed up the mountain earlier this morning, and the mist parted to reveal the whole landscape, it soon became clear just how much of the forest has already been cut down for farming. From the lower slopes, almost to the summit, the land has been cleared and planted with crops, to feed a rapidly growing local population.
I can hardly blame them for wanting to produce food for their families. But every tree that is felled marks another setback for the endemic birds of this region; birds like Mrs Moreau's warbler, which are found here, and nowhere else in the world. As I walk away from the duetting couple, with the last notes finally fading away into the forest, it strikes me as highly likely that this species will go extinct during my lifetime. For, as the latest report from BirdLife International reveals, the global population may now be as low as 500 individual birds.1
*
Today, we face a huge and disturbing paradox. Even as we are discovering more and more new species – either by finding them for the very first time, or by 'splitting' them because of differences in their DNA – many of the world's birds are heading towards the edge of extinction. What chance does Mrs Moreau's warbler – and its recently split cousin the Rubeho warbler, found in an adjacent mountain range – have of surviving in the modern world, where billions of human beings demand so much land and space?
It dawns on me that this is where my passion for bird names has finally reached the end of its long journey. It began when, as a ten-year-old boy, I first came across the name Mrs Moreau's warbler in _Birds of the World_. It developed through my lifelong passion for birds, and my growing love of the English language, both of which come together in the names we have given to birds down through the ages. And it ends with the realisation that my fascination with the history and origin of bird names – whether of common and familiar, or rare and unusual species – is ultimately because these names hold within them the incredible variety of birds around the globe, and the rich stories of their interactions over time with us.
From the familiar robin, chaffinch and blackbird (whose names turned out to be far less straightforward than we might have imagined) to the Uluguru violet-backed sunbird, Udzungwa forest partridge and Mrs Moreau's warbler, bird names are far more than just words. Every single one of them tells a story – a story that runs parallel with our own human narrative, expressed through our history, language and culture.
The naming of birds is, of course, a purely human pursuit; as we have seen, it helps us make sense of a complex and eternally diverse avian world. The birds themselves are entirely oblivious to what we decide to call them. And yet we insist on doing so, and indeed we go further, in celebrating our own world by the names we choose to bestow. In his own small way, that's what Reg Moreau was doing when he decided to immortalise his wife – and their love and devotion towards one another – in the name of this obscure little bird.
So what happens if Mrs Moreau's warbler disappears from the face of the earth? The extinction of any species is a tragedy, not just for the creature in question, but for us too; in John Donne's words, like any man's death, it diminishes us. But when we lose a species its name is also diminished, for who can hear of the most famous extinct bird without thinking of the proverbial phrase 'as dead as a dodo'? What was once a living, breathing creature is now simply a symbol of extinction and loss; no longer part of the wondrous diversity and complexity of the natural world.
To me, this shift in meaning is almost as important as the loss of the living bird. For, as this story has revealed, the names of birds are central to our own identity – a crucial part of what makes us human. As John Clare aptly lamented, 'O words are poor receipts for what time hath stole away'.2
The names we have given to birds down the ages reflect every aspect of our own lives: primitive superstitions, myths and legends, invasions and conquests, shifts in language, rigorous scientific observation, our love of sound, colour and pattern, and a sense of place. And, last, but certainly not least, some commemorate the extraordinary achievements of the men and women after whom they are named: including, of course, Reg and Winnie Moreau.
*
May their warbler continue to sing forever.
#### Notes
1 <http://www.birdlife.org/globally-threatened-bird-forums/2016/11/mrs-moreaus-warbler-bathmocercus-winifredae-request-for-information/>
2 From 'Remembrances'.
# ACKNOWLEDGEMENTS
This book arose from a lifelong fascination with birds and language, which began when my mother took me as an infant to feed the ducks near my home, and developed through her dedicated encouragement of my love of reading, books and language.
Many years later, these came together when I met Laura Hassan of Faber, who immediately saw the potential of the complex and fascinating story of how birds got their names. Laura has been fantastically supportive throughout the writing of the book, as has editor Katherine Ailes, whose perceptive comments and ability to see how the narrative should develop have been incredibly helpful. My dear friend Graham Coster copy-edited the book, making many helpful last-minute suggestions, while my agent Broo Doherty has, as ever, provided wise advice and great support. I would also like to thank the team at Faber, Fred Baty in editorial, Eleanor Crow in design, Kate Burton in publicity and John Grindrod in marketing.
Several experts have kindly taken time to read through excerpts and make corrections and suggestions. They are the linguists David Crystal and Simon Horobin, and ornithologists Jonathan Meyrav, Richard Prum, Paul Salaman, Arnoud van den Berg and Ian Wallace, along with the team behind the satirical magazine _Not BB_. Any errors that remain are of course my own.
The Appendices – lists of bird names under various categories, included for my more obsessive readers (you know who you are!) – were a perennial topic of conversation on boat trips along Peru's Manu River in May 2017. I would like to thank my companions Neil Glenn, Jo Wimpenny, Kyle Carlsen and Brian Egan for their very helpful suggestions (and some really silly ones). Nigel Redman also used his vast knowledge of the world's birds to add a number of names to this section. The expedition to Tanzania, featured in the Prologue and Epilogue, was organised by Zoe and Roy Hinde of Wild Things Safaris, and led by Roy and our excellent guide Ezra. I should also like to thank Colin Watkins and Nigel Simpson, both of whom provided very helpful and detailed advice to help us plan our trip; thanks also to Nigel for his generous gift of Martin Woodcock's evocative portrait of Mrs Moreau's Warbler.
I have also been inspired by a number of other people who have found the origin of bird names compelling. The late W. B. Lockwood, philologist and author of the slim but indispensable volume _The Oxford Book of Bird Names_ , has been a constant inspiration. Barbara and Richard Mearns, authors of two fine books on the origin of eponymous names, _Biographies for Birdwatchers_ and _Audubon to Xantus_ , have provided much useful biographical information on the people in Chapters 4 & 5. I also referred to the extraordinary two-volume work by Michel Desfayes, _A Thesaurus of Bird Names_.
Three of my greatest friends kindly read the whole book from cover to cover, providing really helpful comments throughout. They are my childhood birding companion Daniel Osorio, Graeme Mitchell, with whom I now regularly go birding in Somerset, and Kevin Cox. Kevin and his wife Donna also kindly offered me the use of the cottage in the grounds of their Devon home as a writing retreat.
Graeme and Kevin joined me on our fabulous trip in search of the elusive Mrs Moreau's warbler, the eponymous title of this book. I could not have wished for better companions on this, the journey of a lifetime.
STEPHEN MOSS
# APPENDIX
### _Positive and Negative Bird Names_
#### POSITIVE AND UPBEAT
beautiful nuthatch, jay, firetail, rosefinch etc.
elegant parrot, tern, honeyeater, tit, sunbird, trogon, pitta etc.
exclamatory paradise whydah
festive coquette
foxy cisticola
gorgeous bushshrike
joyful greenbul
handsome flycatcher, francolin
immaculate antbird
laughing dove, gull, kookaburra etc.
laughingthrushes
lovely cotinga, fairywren, sunbird
magnificent frigatebird, sunbird, riflebird, bird-of-paradise etc.
many-coloured rush-tyrant
marvellous spatuletail
paradise jacamar, kingfisher, drongo etc.
rainbow pitta
resplendent quetzal
royal sunangel, parrotfinch, penguin, albatross, flycatcher
sacred ibis, kingfisher
sociable lapwing
splendid fairywren, astrapia, starling, sunbird
superb fruit dove, lyrebird, pitta, parrot, starling, sunbird, bird-of-paradise etc.
#### NEGATIVE OR UNDERWHELMING
bentbills
drab water-tyrant
dull-blue flycatcher
dull-coloured grassquit
fearful owl
go-away-birds
inaccessible rail
intermediate egret
invisible rail
lachrymose mountain tanager
lazy cisticola
least tern, bittern etc.
medium ground finch, tree finch
middle-spotted woodpecker
modest tiger parrot
mourning dove, wheatear, warbler etc.
one-colored becard
plain antvireo, pigeon, flowerpecker etc.
sad flycatcher
screamers
shy albatross, heathwren
simple greenbul
snoring rail
sombre tit
solitary sandpiper, snipe, eagle, cacique etc.
spotless starling, crake
stout cisticola
tiny greenbul, tyrant-manakin, hawk, cisticola, sunbird etc.
unadorned flycatcher
unicolored antwren, tapaculo, jay, thrush, blackbird
uniform swiftlet, crake, finch
weebill
widowbirds
### _Long and short names_
#### LONG (MORE THAN TWENTY-FIVE LETTERS – HYPHENS AND SPACES NOT COUNTED)
26 LETTERS: chestnut-backed jewel-babbler, cinnamon-breasted tody-tyrant, Eastern wattled cuckooshrike, grey-headed canary-flycatcher, King of Saxony bird-of-paradise, Northern rough-winged swallow, plumbeous-crowned tyrannulet, purple-tailed imperial pigeon, Rüppell's long-tailed starling, Southern rough-winged swallow, Western wattled cuckooshrike
27 LETTERS: American three-toed woodpecker, amethyst-throated mountaingem, black-and-white tody-flycatcher, black-casqued wattled hornbill, chestnut-fronted helmet-shrike, Eurasian three-toed woodpecker, Northern beardless tyrannulet, Northern brown-throated weaver, Southern beardless tyrannulet, Southern brown-throated weaver
28 LETTERS: black-and-white casqued hornbill, chestnut-breasted chlorophonia, chestnut-crowned sparrow-weaver, cinnamon-bellied flowerpiercer, cinnamon-rumped foliage-gleaner, Donaldson Smith's sparrow-weaver, ochraceous-breasted flycatcher, red-bellied paradise flycatcher, slaty-backed nightingale-thrush, slender-billed scimitar babbler, yellow-casqued wattled hornbill
29 LETTERS: black-and-white shrike-flycatcher, black-and-yellow silky-flycatcher, lesser necklaced laughingthrush, white-bellied crested flycatcher, yellow-throated woodland warbler
30 LETTERS: buff-breasted paradise kingfisher, greater necklaced laughingthrush, Middendorff's grasshopper warbler, rufous-vented paradise flycatcher, Ruwenzori double-collared sunbird
31 LETTERS: Prigogine's double-collared sunbird
#### SHORT (FEWER THAN 6 LETTERS)
5 LETTERS: besra, cutia, galah, ifrit, kamao, kikau, kioea, maleo, malia, twite, veery
4 LETTERS: dodo, huia, liwi, kagu, kaka, nene, omao, ruff, rook, smew, sora, weka
3 LETTERS: emu, kea, moa, tui
2 LETTERS: ou
NB: only standalone names acceptable: coot, shag, wren etc. are always qualified (e.g. Eurasian wren)
### _Parts of the Body in Bird Names_
backed, banded, beaked, bearded, bellied, belted, billed, breasted, bridled, browed, capped, cheeked, chested, chinned, crested, collared, crowned, eared, eyed, faced, flanked, footed, fronted, headed, helmeted, hooded, horned, legged, lored, mandibled, mantled, masked, moustached, naped, necked, necklaced, plumed, ruffed, rumped, scarfed, shouldered, sided, spectacled, superciliaried, tailed, thighed, throated, tipped, toed, toothed, tufted, vented, webbed, whiskered, winged
### _Birds Named after States in the US_
Arizona woodpecker
California condor, gnatcatcher, gull, quail, scrub jay, thrasher, towhee
Carolina chickadee, wren, parakeet
Connecticut warbler
Florida scrub jay
Hawaiian akepa, amakihi, coot, creeper, crow, duck, elepaio, hawk, petrel
Kentucky warbler
Louisiana waterthrush
Mississippi kite
Tennessee warbler
Virginia rail
### _Birds Named after Man-Made Objects_
barn swallow, owl etc.
boat-billed heron
booted racket-tail
buttonquails
canvasback
fantails
gartered trogon
Gould's jewelfront
helmeted woodpecker, hornbill, myna etc.
helmet-shrikes
house martin, finch, wren etc.
ladder-tailed nightjar
lancebills
lyre-tailed nightjar
mitred parakeet
needletails
ovenbird
pennant-winged nightjar
pin-tailed snipe, pintail
razorbill, razor-billed curassow
riflebirds
saddlebacks
saddle-billed stork
saw-wings
scimitarbills
scissor-tailed flycatcher, nightjar
scythebills
sheathbills
shoebill
shovel-billed kookaburra
sickle-winged guan
spadebills
spoonbills
spoon-billed sandpiper
standardwing
sword-billed hummingbird
tambourine dove
trainbearers
trumpeter swan, finch, hornbill etc.
umbrellabirds
whipbirds
wire-tailed swallow
yellow-scarfed tanager
yellowhammer
### _Birds Named after Elements, Compounds and Minerals_
bronzed drongo
bronze mannikin
bronze-tailed peacock-pheasant
bronzewings
cobalt-winged parakeet
copper sunbird, seedeater, pheasant etc.
copper-rumped hummingbird
goldfinch
goldeneye
golden eagle, pheasant, weaver, sparrow, bush robin etc.
lead-coloured flycatcher
leaden antwren, honeyeater, flycatcher
metallic starling
metaltails
silver oriole, pheasant, teal
silver-beaked tanager
silverbills
silverbird
silvereye
silvery grebe, pigeon, kingfisher
steel-blue whydah, flycatcher
steely-vented hummingbird
### _Birds Named after Gems and Precious Stones_
amethyst sunbird, starling, brown dove, woodstar
beryl-spangled tanager
berylline hummingbird
Brazilian ruby
crimson topaz
diamond firetail, dove
emeralds
emerald cuckoo, dove, starling, tanager
emerald-spotted wood dove
fiery topaz
garnet pitta, robin
garnet-throated hummingbird
opal-crowned tanager, manakin
pearl kite
pearl-breasted swallow, conebill
pearly-eyed thrasher
ruby-cheeked sunbird
ruby-crowned kinglet, tanager
ruby-throated hummingbird, myzomela, bulbul
ruby-topaz hummingbird
sapphire-throated hummingbird
sapphire quail-dove, flycatcher
sapphires
Siberian rubythroat
### _Politically Incorrect Names_
dwarf bittern, tinamou, sparrowhawk, koel etc.
hottentot teal
kaffir rail
midget flowerpecker
negrito
negrofinch
oldsquaw
pygmy antwren, falcon, eagle etc.
### _Birds Named after Professions, Callings and Religious Orders_
adjutants
apostlebird
bearded mountaineer
bishops
blacksmith plover
capuchinbird
cardinals
coppersmith barbet
friarbirds
lanceolated monklet
millerbird
miners
monk parakeet
nunbirds, nunlets
prothonotary warbler
purple grenadier
secretarybird
tailorbirds
tinkerbirds
tyrants
weavers
### _Bird Names derived from Mythology, Ancient Civilisations etc._
Aztec rail, thrush
Calliope hummingbird
Cinderella waxbill
eastern, Say's and black Phoebe
Griffon vulture
Inca tern, flycatcher, dove, jay, wren
collared Inca
Lucifer hummingbird
Mayan antthrush
Mesopotamian crow
Montezuma quail, oropendola
Persian shearwater
Pharaoh eagle owl
Satanic nightjar
Stygian owl
### _Birds' Names Including Other Animals' Names (not including other birds)_
antbirds, antpeckers, antpittas, antpipits, antshrikes, ant-tanagers, antthrushes, antvireos, antwrens
bat hawk, falcon
bee-eaters
buffalo weavers
bullfinches
catbirds
cattle egret
cicadabirds
cowbirds
fish crow, fish eagles, fish owls
flycatchers
frogmouths
fox sparrow, kestrel
lizard cuckoos
mousebirds
oxpeckers
rhinoceros auklet
snail kite
snake eagles
spiderhunters
squirrel cuckoo
tiger herons
### _Birds Named after Royalty and Nobility (in order of rank)_
emperor penguin
imperial eagle, shag, snipe, pigeon, woodpecker, Amazon
monarchs
king vulture, eider, quail
kingbirds
kingfishers
Queen Carola's parotia
Prince Ruspoli's turaco
princess parrot
Princess Stephanie's astrapia
duchess lorikeet
Lord Derby's parakeet
Lord Howe woodhen, parakeet, gerygone
Lady Amherst's pheasant
### _Thirty-three Amazing Names_
bananaquit
bearded mountaineer
bokikokiko
bokmakierie
brownish twistwing
chuck-will's-widow
crinkle-collared manucode
fasciated tiger-heron
firewood-gatherer
forty-spotted pardalote
giant cowbird
glowing puffleg
hardhead
horned screamer
kinglet calyptura
lachrymose mountain tanager
leaf-love
Luzon bleeding-heart
marvellous spatuletail
oleaginous hemispingus
pink-legged graveteiro
Rock-loving cisticola
scaly-throated leaftosser
screaming piha
sharp-tailed streamcreeper
shining sunbeam
strange-tailed tyrant
teardrop white-eye
Upper Magdalena tapaculo
vermiculated screech owl
whip-poor-will
zigzag heron
zitting cisticola
# INDEX
1. Aberdeen, University of 1
2. accentor, hedge 1, 2, 3, 4
1. _see also_ dunnock
3. Accipitriformes 1
4. Adolf Frederick, King of Sweden 1
5. Adolph, Peter 1
6. Africa
1. and British Empire 1, 2, 3, 4
2. -Eurasian flyway 1
7. Afrikaans language 1
8. Age of Ornithological Discovery 1
9. Age of Reason/Enlightenment 1
10. agriculture
1. modern 1, 2, 3
2. origins of 1, 2, 3, 4
11. Ahlquist, Jon E. 1, 2
12. Albert VII, Archduke of Austria 1
13. Allison, Malcolm 1
14. Amadon, Dean 1
15. American Museum of Natural History (AMNH) 1
16. American Ornithologists' Union (AOU) 1, 2
17. Amherst, Lady Sarah (née Archer) 1, 2, 3, 4, 5, 6
18. Amherst, Lord William 1
19. 'Among the Ornithologists' (Harrold) 1
20. Amundsen, Roald 1
21. _Anas clypeata see_ shoveler
1. _A. platyrhynchos se_ e mallard
22. Anglo-Saxon
1. invasion 1, 2, 3, 4
2. language 1, 2, 3, 4, 5, 6n, 7, 8, 9, 10, 11, 12, 13
1. _see also_ English language, Old
23. _Anser anser see_ goose, greylag
24. Antarctica 1, 2
25. Anthony, David 1
26. _Anthus pratensis see_ pipit, meadow
27. antpitta, Jocotoco 1
28. antshrike
1. Acre 1
2. fulvous 1
29. Appleton, Tim 1
30. _Aptenodytes patagonicus see_ penguin, king
31. Archer, Sarah 1
1. _see also_ Amherst, Lady Sarah
32. Arctic 1, 2, 3, 4, 5, 6
1. _see also_ North-West Passage
33. Aristotle 1, 2
34. Audouin, Jean Victor 1
35. Audubon, John James 1, 2
36. _Audubon to Xantus_ (Mearns & Mearns) 1
37. auk 1
1. great 1n 2n
38. Australia 1, 2, 3
1. First Fleet 1, 2, 3
39. _Australian Bird Names_ (Fraser & Gray) 1
40. _Avium Praecipuarum_ (Turner) 1
41. avocet 1, 2n, 3
1. babbler 1
1. African hill- 1
2. fulvous 1
3. Hume's 1n
2. Ball, Philip 1n
3. Baltimore Orioles (baseball team) 1, 2
4. barbet 1n, 2
5. Barclay-Smith, Phyllis 1
6. Barkham, Patrick 1
7. Barnes, Simon 1
8. Barrell, Prof. John 1
9. Barrington, Daines 1
10. baseball teams 1, 2
11. Bass Rock, Scotland 1
12. bat, pipistrelle 1
13. Bauhar, Caspar 1
14. Bayeux Tapestry 1
15. bear, Himalayan brown ( _Ursus arctos isabellinus_ ) 1
16. Bechstein, Johann Matthäus 1n
17. bee-eater 1n, 2, 3
1. European 1
2. rainbow 1
18. Beeton, Mrs 1
19. _Beowulf_ 1, 2, 3
20. Berrick Salome, Oxfordshire 1
21. Bewick, Thomas 1, 2, 3, 4
22. Bexley Heath 1, 2
23. Bible (King James version, 1611) 1, 2
24. Biebrza Marshes, Poland 1
25. binoculars 1, 2, 3, 4
26. binomial nomenclature 1
27. biodiversity 1, 2
28. biomusicology 1n
29. bird, _OED_ definition 1
30. bird names
1. aboriginal 1, 2
2. auctioning 1
3. colour-based 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19
4. English 1, 2, 3
5. English, official 1, 2, 3
6. English _vs_ N. American 1, 2
7. eponymous 1, 2, 3, 4, 5
8. folk 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16
9. habitat-based 1, 2, 3n, 4, 5, 6, 7, 8
10. New World 1, 2, 3, 4
11. onomatopoeic 1, 2, 3, 4, 5, 6
12. in popular culture 1
13. rude/offensive/racist 1
14. scientific _see_ binomial nomenclature
31. bird of paradise 1, 2
32. bird song/calls, purpose 1
33. _Bird Watching_ (Selous) 1, 2
34. birders
1. confusion over bird names 1, 2, 3, 4
2. surveys 1
3. 'untickable views' 1
4. use of English names 1, 2, 3, 4
5. use of scientific names 1, 2
6. use of song for identification 1, 2, 3, 4
7. 'world listers' 1, 2
35. _Birding World_ 1
36. BirdLife International 1
37. _Birds and Men_ (Nicholson) 1
38. _Birds of America, The_ (Audubon) 1
39. _Birds Britannia_ (BBC4) 1
40. _Birds Britannica_ (Cocker) 1
41. _Birds of the British Isles and Their Eggs, The_ (Coward) 1
42. _Birds of the West Indies_ (Bond) 1
43. _Birds of the World_ 1, 2n, 3, 4
44. _Birds of the World: Recommended English Names_ (Gill & Wright) 1
45. _Birdwatch_ (magazine) 1
46. bittern 1, 2, 3, 4, 5n
47. blackbird ( _Turdus merula_ ) 1, 2, 3, 4, 5, 6n
1. Asiatic/New World 1
2. origin of name 1, 2, 3, 4
3. red-winged 1
4. song 1n, 2, 3
48. 'black cap' 1
1. _see also_ tit, great
49. Blackburne, Anna 1, 2
50. blackcap ( _Sylvia atricapilla_ ) 1, 2, 3, 4, 5, 6, 7, 8n, 9, 10, 11, 12, 13, 14, 15, 16
51. 'Blithe Spirit' (Shelley) 1
52. Bluebirds (Cardiff City nickname) 1n
53. bluethroat 1
54. Blyth, Edward 1n
55. Boadicea 1
56. _Bombycilla garrulus see_ waxwing
57. Bond, James (ornithologist) 1
58. Bonelli, Franco Andrea 1n
59. bonxie 1
1. _see also_ skua, great
60. _Book of Household Management_ (Beeton) 1
61. Borkowski, Marek 1
62. Botany Bay 1, 2, 3
63. bowerbird 1
64. Boys, William 1
65. brambling 1, 2
66. Brereton, John 1
67. Breton language 1
68. _British Birds_ (magazine) 1, 2, 3
69. British Empire 1, 2, 3, 4, 5, 6, 7, 8
70. British List 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
71. British Museum of Natural History 1, 2, 3, 4, 5
72. British Ornithologists' Union (BOU) 1n, 2, 3n, 4, 5, 6, 7
1. _see also_ British List
73. British Raj 1
74. _British Tits_ (Perrins) 1
75. British Trust for Ornithology (BTO) 1, 2, 3, 4
76. _British Zoology_ 1, 2, 3
77. brolga 1
78. Brontë, Emily 1
79. Brosnan, Pierce 1
80. _BTO Atlas_ 1
81. budgerigar 1, 2
82. bullfinch 1n
83. Bullock, William 1, 2
84. Bullock, William (son) 1
85. bunting
1. cirl 1
2. 'common' _see_ corn
3. corn 1, 2, 3, 4, 5n
4. Lapland 1, 2
5. 'longspur' (N. American name) 1
6. Pallas's reed 1n, 2n
7. reed 1, 2, 3, 4
8. snow 1, 2
9. Tristram's 1n
10. yellow 1, 2
1. _see also_ yellowhammer
86. Burchell, W. J. 1
87. bush-hen 1
88. Bush, George W. 1
89. bustard 1, 2, 3
1. great 1
2. 'thick-kneed' 1
1. _see also_ curlew, stone
90. butcher-bird (red-backed shrike) 1, 2
91. Butler, Samuel 1
92. butter bump 1
1. _see also_ bittern
93. Buturlin, Sergei Aleksandrovich 1
94. buzzard 1, 2, 3, 4
1. honey 1
95. 'Bye baby bunting' 1
1. Cabot, John 1
2. Caius, John 1
3. Caledonian pine forests 1
4. Cambridge, University of 1
5. Campbell, Sir Archibald 1
6. Canada, Arctic 1, 2
1. _see also_ North-West Passage
7. Canaries (Norwich City football club) 1
8. _Canterbury Tales, The_ (Chaucer) 1
9. capercaillie 1, 2, 3n
1. Western 1
10. Capercaillie (band) 1
1. Caroline of Brunswick, Queen 1
11. _Casino Royale_ (book) 1
12. Castle Museum, Norwich 1
13. Catholic Church 1, 2, 3
14. Celts 1n
15. Cetti, Francesco 1, 2
16. chaffinch 1n, 2, 3n, 4
1. origin of name 1
2. song accents 1
17. Champions of the Flyway 1
18. Charles I, King 1
19. Charles II, King 1
20. Charles V, Holy Roman Emperor 1
21. Chaucer, Geoffrey 1, 2, 3, 4, 5, 6, 7
22. Cherry-Garrard, Apsley 1
23. chickens 1, 2, 3
24. chiffchaff 1, 2, 3n, 4
1. migration 1, 2
2. 'shortwinged woodwrens' 1
3. song 1, 2, 3, 4, 5, 6
25. Chocó vireo ( _Vireo masteri_ ) 1, 2
26. chough 1, 2, 3n, 4, 5
27. Christchurch, Dorset 1, 2
28. _Chroicocephalus ridibundus see_ gull, black-headed
29. citrine (coloration) 1
30. Civil Service 1, 2
31. _Clangula hyemalis see_ duck, long-tailed
32. Clare, John 1, 2, 3, 4, 5, 6, 7
33. Cobbett, William 1
34. cockatoo 1, 2
1. gang-gang 1
35. cock-of-the-rock 1
36. _Coccothraustes coccothraustes see_ hawfinch
37. Cohn-Haft, Mario 1n
38. _Collins Bird Guide_ 1
39. Collins, Wilkie 1
40. Colombia 1
41. Columbidae 1
42. _Concise Oxford Dictionary of English Placenames_ 1n
43. Conran, Terence 1
44. conservation 1, 2, 3, 4n, 5, 6
45. Cook, Capt. James 1
46. coot 1, 2, 3, 4, 5
1. American 1
47. cormorant 1, 2n
48. corncrake (land rail) ( _Crex crex_ ) 1, 2n, 3, 4, 5, 6, 7, 8
49. Cornish language 1, 2
50. Cornwall 1, 2, 3
51. _Corpus Glossary_ 1
52. Corsica 1
53. Coto Doñana, Spain 1
54. Cotswold Water Park 1
55. Coues, Elliot 1, 2
56. 'Country Diary' ( _Manchester Guardian_ ) 1
57. Courtenay, Ann 1, 2, 3
58. courtship display 1
59. Couzens, Dominic 1, 2n
60. Coward, T. A. (Thomas) 1
61. cowbird 1
62. crane 1n, 2, 3, 4, 5
1. origin of name 1, 2
2. persecution/comeback 1, 2n
3. place names 1
63. Cretaceous-Palaeogene extinction 1
64. _Crex crex see_ corncrake
65. Cromwell, Oliver 1
66. crossbill 1, 2n
1. common 1, 2
2. parrot 1n, 2n
3. Scottish 1, 2, 3, 4
67. Crossley, Richard 1
68. crow 1, 2n, 3n, 4
1. carrion 1, 2, 3
2. family 1, 2
3. hooded 1
4. origin of name 1, 2, 3
69. Crystal, David 1, 2, 3
70. cuckoo
1. call 1, 2, 3, 4, 5, 6, 7
2. egg laying 1, 2
3. _Exeter Book_ verse 1
4. fairs 1
5. harbinger of spring 1, 2, 3, 4
6. metaphor 1
7. migration/tracking 1
8. origin of name 1, 2, 3, 4
71. curlew 1n
1. 'jack' 1n
1. _see also_ whimbrel stone 1, 2, 3
1. Dampier, William 1
2. Dangerfield, Fyfe 1
3. Danish language 1, 2
4. darters ('snakebirds') 1
5. Dartford, Kent 1, 2, 3
6. Darwin, Charles 1
7. 'Dawn Chorus' (Harrold) 1
8. Dee, Tim 1n
9. Desfayes, Michel 1
10. Devon 1, 2, 3
11. Dickens, Charles 1n
12. _Dictionary of the English Language_ (Johnson) 1
13. dikkop 1
14. Dilger, Mike 1
15. dinosaurs 1n, 2
16. dipper 1n, 2, 3, 4
17. diver
1. black-throated 1
2. great northern 1
3. red-throated 1
4. yellow/white-billed 1
18. DNA analysis 1
19. DNA–DNA hybridisation 1
20. domestication 1, 2
21. Donne, John 1, 2
22. Dorville, Eliza 1, 2, 3
23. dotterel 1n
24. dove 1, 2n
1. origin of name 1, 2
2. ring- 1
1. _see also_ pigeon, wood rock 1
3. turtle 1n, 2, 3, 4
25. Doves (band) 1
26. Drake, Sir Francis 1
1. duck
1. 'black sea' 1
1. _see also_ scoter
2. castaneous 1
2. classification 1
3. 'Cuthbert's' 1
1. _see also_ eider
4. domestication 1
5. ferruginous 1, 2
6. fulvous whistling 1
7. 'ha ha' 1
8. long-tailed ('oldsquaw') ( _Clangula hyemalis_ ) 1
9. mandarin 1
10. ruddy 1
1. _see also individual ducks_
27. dunlin 1
28. dunnock ( _Prunella modularis_ ) 1, 2, 3n, 4, 5, 6, 7, 8, 9, 10, 11
29. Dutch language 1, 2, 3, 4, 5
30. 'duvet' 1
1. eagle
1. classification 1, 2
2. golden 1
3. origin of name 1, 2
4. sea/white-tailed ('erne') 1n, 2, 3, 4, 5
5. Spanish imperial 1n
6. wedge-tailed 1
2. Eagles (band) 1
3. Eagles (Crystal Palace football club) 1
4. East Africa 1
5. East Anglia 1, 2, 3
6. Eastern Arc Mountains, Tanzania 1, 2
7. Edith the Fair 1
8. Edward VI, King 1
9. Edward VII, King 1
10. Edwards, George 1
11. egret
1. cattle 1
2. little 1
3. reddish 1
12. Egypt 1, 2
13. eider 1, 2, 3, 4, 5n
14. eiderdown 1, 2
15. Eilat, Israel 1
16. Ekwall, Eilert 1n
17. Elenora of Arborea 1
18. Eliot, T. S. 1
19. Elizabeth I, Queen 1, 2
20. emu 1, 2
21. English Civil War 1
22. English language
1. Middle 1, 2n, 3, 4, 5, 6
2. Modern 1, 2, 3, 4, 5n, 6, 7, 8
3. Old 1, 2, 3, 4, 5, 6n, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18
4. origins of 1
5. surnames 1
23. _Erithacus rubecula see_ robin
24. 'erne' 1
1. _see also_ eagle, sea
25. Etawah, Uttar Pradesh 1
26. evolution 1, 2
27. _Exeter Book_ 1
28. Exeter Cathedral 1
29. extinctions 1, 2, 3
30. Eyton, Thomas 1
1. Fair Isle 1n
2. _Falco subbuteo see_ hobby
3. falcon
1. 'ash-coloured' ( _Falco cinerarius_ ) 1
1. _see also_ harrier, Montagu's
2. classification 1n, 2, 3
3. Eleonora's 1n, 2
4. peregrine 1, 2, 3n, 4
1. _see also_ hobby; kestrel; merlin
4. falconry 1n, 2
5. Farne Islands 1
6. _Fatal Shore, The_ (Hughes) 1
7. _Fauvette pitchou see_ warbler, Dartford
8. Ferdinand of Aragon 1
9. Ferguson-Lees, James 1
10. ferruginous (coloration) 1
11. fieldfare 1, 2, 3, 4, 5, 6
12. finch
1. cardinal 1
2. Corsican citril 1n
3. Locust 1
4. origin of name 1
1. _see also individual finches_
13. First Fleet 1, 2, 3
14. Fisher, James 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12n
15. flameback, Himalayan 1
16. flamingo 1
17. Flannan Isles 1
18. Fleming, Ian 1
19. 'flop wing' 1
1. _see also_ lapwing
20. Florio, John 1n
21. flycatcher
1. monarch 1
2. Phoebe 1
3. pied 1
4. sad 1
5. spotted 1
6. white-tailed crested 1
7. yellowish 1
22. flyways, global 1
23. folk names _see_ bird names
24. football clubs 1
25. Formby Point, Lancashire 1
26. 'fowl'/'foule' 1, 2
27. Fowles, John 1
28. Franklin, Sir John 1, 2
29. Fraser, Ian 1
30. _Fratercula arctica see_ puffin
31. French language
1. Modern 1
2. Norman 1, 2, 3, 4, 5, 6
3. Old 1, 2, 3, 4
32. friarbird 1, 2
33. frigatebird, magnificent 1
34. 'fugol' 1
1. _see also_ 'fowl'
35. Fuller, Thomas 1
36. fulvous (coloration) 1
37. 'furzeling' 1
1. _see also_ warbler, Dartford
1. Gaelic, Scottish 1, 2, 3, 4
2. gallinule 1n
3. Gandhi, Mahatma 1
4. gannet 1n, 2, 3n
1. metaphor 1
2. origin of name 1, 2
5. _Garden Birds_ (Barclay-Smith) 1
6. Garvey, Guy 1
7. Gaskell, Elizabeth 1
8. Gené, Giuseppe 1
9. _General History of Birds, A_ (Latham) 1
10. Genesis, Book of 1, 2
11. Germanic language 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
1. Proto-/Old 1, 2, 3
2. West 1, 2
1. _see also_ Old Norse
12. Gessner, Conrad 1, 2
13. Gilbert, W. S. 1
14. Gill, Frank 1
15. Glanville, Eleanor 1
16. glaucous (coloration) 1
17. 'goatsucker' 1, 2
1. _see also_ nightjar
18. Godwinson, Gytha 1
19. godwit
1. bar-tailed 1
2. black-tailed 1
3. 'cinereous' (greenshank) 1
20. goldcrest 1, 2, 3, 4, 5, 6n
21. goldeneye 1n, 2, 3n
22. _Goldeneye_ (film) 1
23. Goldeneye, Operation 1
24. goldfinch 1, 2, 3, 4, 5, 6n
25. Gondwanaland 1
26. Gooders, John 1, 2
27. Goodwin, Derek 1
28. Goodwin, Jimi 1
29. goose
1. barnacle 1, 2
2. bean 1, 2, 3
3. brent 1, 2, 3
4. Canada 1, 2n, 3, 4, 5
5. Cape Barren 1
6. cravat 1
7. domestication 1
8. down 1n, 2
9. Egyptian 1n
10. greylag ( _Anser anser_ ) 1
11. migration 1
12. origin of name 1, 2, 3, 4
13. pink-footed 1, 2n, 3
14. white-fronted 1, 2
30. goshawk 1n, 2, 3n
31. Gould, Stephen Jay 1n
32. 'gowk' 1, 2, 3
1. _see also_ cuckoo
33. grackle 1
34. Granada, Siege of 1
35. Gray, Jeannie 1
36. Gray, John Edward 1n
37. _Great Expectations_ 1
38. Great Fen 1
39. grebe 1, 2
1. black-necked/eared 1, 2
2. red-necked 1
3. Slavonian 1, 2
40. Greek language 1, 2, 3
41. greenbul, joyful 1
42. Greenewalt, C. H. 1
43. greenfinch 1, 2, 3, 4, 5, 6n
44. Greenland 1
45. greenshank 1, 2n
46. greyish mourner 1
47. grosbeak 1
48. grouse
1. black 1, 2
2. red 1, 2, 3, 4n
3. willow 1
1. _see also_ capercaillie
49. grouse (verb) 1
50. _Guardian_ 1
51. guidebooks 1, 2, 3, 4, 5
52. guillemot 1, 2n
53. Guillemots (band) 1
54. guineafowl 1
55. gull
1. Audouin's 1, 2n
2. black-headed ( _Chroicocephalus ridibundus_ ) 1, 2n, 3, 4, 5n
3. common 1, 2n
4. Franklin's 1n, 2, 3
5. glaucous 1
6. great/lesser black-backed 1n
7. herring 1, 2n, 3n
8. Iceland 1, 2
9. ivory 1
10. Mediterranean 1, 2
11. origin of name 1, 2
12. Ross's ( _Rhodostethia rosea_ ) 1, 2, 3, 4, 5, 6, 7
13. Sabine's ( _Larus sabini_ ) 1, 2
56. Gulls, the (Torquay United football club) 1
57. 'gypsy-bird' (hoatzin) 1
1. habitat
1. artificial/human 1
2. -based names 1, 2n, 3, 4, 5, 6, 7, 8
3. loss 1, 2, 3n
2. Hamilton, John 1
3. Hamlett, Gordon 1
4. _Hand-list of British Birds, A_ (Hartert) 1n
5. _Handbook of British Birds, The_ (Witherby et al.) 1
6. Hardy, Thomas 1
7. Harold Godwinson, King 1
8. harrier
1. hen 1, 2, 3n, 4n
2. marsh 1, 2n, 3
3. Montagu's 1, 2, 3, 4, 5, 6, 7
9. Harrier (jump jet) 1
10. Harris, James Edward 1
11. Harris, Joanne 1, 2
12. Harris, Scotland 1
13. Harrold, A. F. 1, 2
14. Hartert, Ernst 1n, 2
15. hawfinch 1, 2n, 3
16. hawks, Old World 1
17. hawk-owl, Hume's 1n
18. heathland 1, 2
19. 'hedge chanter' 1, 2
1. _see also_ dunnock
20. Helpston 1
21. Henry II, King of France 1, 2
22. Henry VIII, King 1, 2
23. heron 1, 2n
1. 'buff-backed' (cattle egret) 1
2. Chinese pond 1n
3. grey 1, 2
4. night 1, 2
5. purple 1n, 2
6. 'spoon-billed' 1
1. _see also_ spoonbill
7. tiger 1
24. Highland Clearances 1
25. Highlands, Scottish 1, 2, 3
26. Hill, Christopher 1
27. Hindi language 1
28. Hirta, St Kilda 1
29. _History of British Birds, A_ (Berwick) 1, 2
30. _History of British Birds, A_ (MacGillivray) 1, 2, 3, 4, 5
31. _History of British Birds, A_ (Yarrell) 1, 2, 3, 4
32. _History of the Rarer British Birds, A_ (Eyton) 1
33. hoatzin ('gypsy-bird') 1
34. Hobbes, Thomas 1
35. hobby ( _Falco subbuteo_ ) 1, 2, 3, 4, 5, 6
36. Holland, Lady Elizabeth 1
37. Holme, Randle 1
38. Holme, Richard 1
39. Holy Island 1
40. honeyeater, graceful 1
41. hoopoe ( _Upupa epops_ ) 1, 2, 3, 4
42. Hopkins, Gerard Manley 1
43. Horobin, Prof. Simon 1
44. horse, domestication of 1
45. _Horse, the Wheel and Language, The_ (Anthony) 1
46. Howerd, Frankie 1
47. _Hudibras_ (Butler) 1
48. Hudson, W. H. 1
49. Hughes, Robert 1
50. Hume, Allan Octavian 1, 2, 3, 4
51. Hume, Mary 1, 2
52. hummingbird 1
53. hunting 1, 2, 3, 4, 5
54. hunter-gatherers 1, 2, 3, 4
1. _Ibis_ (journal) 1n, 2
2. Iceland 1, 2, 3
3. Icelandic language 1n, 2
4. Icteridae 1
5. icterine (coloration) 1
6. _Idler_ (magazine) 1
7. _Illustrated Manual of British Birds, An_ (Saunders) 1n
8. Indian Civil Service 1, 2
9. Indian Congress Party 1
10. Indian Mutiny 1
11. Indian National Congress 1
12. International Ornithological Congress 1, 2
13. Ireland 1, 2, 3
14. Isabella Clara Eugenia, Princess 1, 2
15. Isabella of Castile 1, 2
16. 'Isabelline' (adjective) 1, 2, 3
17. Isle of Man 1, 2
18. Israel 1
19. Italian language 1n, 2, 3n
1. jackdaw 1, 2n
2. Jacky winter 1, 2
3. Jacobs, Nancy J. 1
4. jaeger 1, 2
1. _see also_ skua
5. Jagger, Mick 1
6. James I and VI, King 1
7. James Bond 1
8. jay 1, 2, 3n
1. 'Bohemian' 1
1. _see also_ waxwing
9. Johns Revd C. A. 1
10. Johnson, Dr Samuel 1, 2
11. Jonson, Ben 1
12. Julius Caesar 1
13. Jutes 1
1. Keats, John 1, 2
2. Kilangala (mountain), Tanzania 1
3. _King Lear_ (Shakespeare) 1n
4. kingfisher 1, 2, 3, 4
1. origin of name 1
5. Kingsley, Charles 1n
6. kite 1, 2, 3, 4, 5n
1. black 1
2. black-winged 1
3. origin of name 1, 2
4. red 1, 2
5. white-tailed 1
7. kite (toy) 1, 2
8. kittiwake 1n, 2, 3, 4n
1. black-legged 1
2. red-legged 1
9. Kleinschmidt, Otto 1
10. knot 1, 2n, 3n, 4
11. Knowle House, Devon 1
12. Knox, Alan 1
13. Koepcke, Maria 1n
14. kookaburra 1
15. korhaan 1
16. Kulussutajevsk 1
1. _Lady Penrhyn_ (ship) 1
2. Lambert, Rob 1
3. _Land Birds_ (Bewick) 1, 2
4. land rail 1, 2, 3
1. _see also_ corncrake
5. _Landmarks_ (Macfarlane) 1
6. language, origin of 1, 2, 3
1. _see also individual languages_
7. lapwing 1, 2, 3, 4, 5, 6, 7, 8, 9, 10n
8. _Lapwings, Loons and Lousy Jacks_ (Reedman) 1
9. lark
1. 'grasshopper' 1
1. _see also_ warbler, grasshopper
2. horned 1
3. 4. Hume's 1n
5. magpie- 1
6. origin of name 1
7. pie (recipe) 1
8. 'scribble' 1
1. _see also_ yellowhammer
9. shore 1, 2, 3
10. use of word in English 1
1. _see also_ skylark; woodlark
10. 'The Lark Ascending' (Meredith) 1
11. _Larus sabini see_ gull, Sabine's
12. larynx (human) 1
13. Latham, John 1, 2, 3, 4, 5n
14. Latimer, Hugh 1
15. Latin language 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11n, 12n
1. _see also_ binomial nomenclature
16. Lawrence, D. H. 1
17. 'LBJ' (little brown job) 1
18. Leach, Dr William Elford 1, 2, 3, 4
19. Leadbeater, Benjamin 1
20. leaftosser, short-billed 1
21. Lever, Sir Christopher 1
22. Limpenhoe, Norfolk 1
23. Lincolnshire Fens 1
24. Lindisfarne 1
25. Lindo, David 1, 2
26. Linnaean Society 1
27. Linnaeus, Carl 1, 2, 3
28. linnet 1n, 2, 3, 4n
29. Linnets (football teams) 1
30. Lockwood, Prof. W. B. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11n, 12
31. longspur 1
1. Lapland 1
1. _see also_ bunting
32. loon
1. common 1
2. 'great northern' 1
3. yellow-billed 1
1. _see also_ diver
33. lorikeet, rainbow 1
34. lorrequet 1
35. lourie, Ross's ( _Musophaga rossae_ ) 1
36. Lowland Scots (dialect) 1
37. lyrebird 1, 2
1. macaw 1
2. McDonald, Colonel Marshall 1
3. Macfarlane, Robert 1
4. MacGillivray, William 1, 2, 3, 4, 5, 6, 7
5. Madagascar 1
6. 'mew' (mæw) 1
7. Magnetic North 1, 2
8. magpie 1, 2, 3
1. Australian 1
9. magpie-lark 1
10. Magpies (football teams) 1
11. Malham Cove, N. Yorkshire 1
12. mallard ( _Anas platyrhynchos_ ) 1, 2, 3
13. malleefowl 1
14. Mallorca 1
15. _Manchester Guardian_ 1
16. Mandarin Chinese language 1n
17. Marmora, Alberto della 1, 2
18. Marsden 'Cuckoo Day Festival' 1
19. martin
1. house 1, 2, 3
2. sand 1, 2n
20. Mary, Queen 1
21. Massie, Allan 1
22. Master, Dr Bernard 1
23. Mayr, Ernst 1
24. Mayr's Biological Species Concept (BSC) 1
25. meadowlark 1
26. Mearns, Barbara 1, 2, 3
27. Mearns, Richard 1, 2, 3
28. Mediterranean Sea 1, 2, 3
29. Melville Peninsula, Canada 1
30. Meredith, George 1
31. merganser, red-breasted 1
32. merlin 1
33. Mexico 1
34. Middle Ages 1, 2
35. Middle East 1, 2, 3, 4, 5, 6, 7
36. Middleton Hall, Warwickshire 1
37. _Midsummer Night's Dream, A_ 1
38. migration 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11
39. _Mikado, The_ (Gilbert and Sullivan) 1
40. Miles, John 1
41. _Mimus macdonaldi see_ mockingbird, Hood miner
1. bell 1
2. noisy 1
42. mockingbird, Hood (Española) ( _Mimus macdonaldi_ ) 1
43. Moltoni, Prof. Edgardo 1
44. Monroe Jr, Burt L. 1, 2, 3
45. Montagu, Ann (née Courtenay) 1, 2, 3
46. Montagu, George 1, 2, 3, 4, 5, 6, 7, 8, 9
47. Montagu, Georgina 1
48. Montagu, Henry 1
49. Montagu, Isabella 1
50. Montagu, James 1
51. moorhen 1, 2, 3, 4n, 5
52. Moreau, David 1, 2
53. Moreau, Prinia 1
54. Moreau, Reginald Ernest 1, 2, 3, 4
55. Moreau, Winifred 1, 2, 3, 4, 5
56. _Mori, Disce_ (Sutton) 1n
57. Morocco 1
58. Moss, Kay 1
59. _Motacilla alba yarrelli see_ wagtail, pied
1. _M. alba alba see_ wagtail, white
60. Mullens, William 1
61. _Music Instinct, The_ (Ball) 1n
62. _Musophaga rossae see_ lourie, Ross's
63. Mynott, Jeremy 1
1. _Naming Nature_ (Yoon) 1n
2. Nansen, Fridtjof 1
3. Napoleonic Wars 1, 2
4. Nashe, Thomas 1
5. National Maritime Museum, Greenwich 1
6. Natural England 1
7. _Natural History of Dee Side and Braemar, The_ (MacGillivray) 1
8. _Natural History of Selborne, The_ (White) 1, 2, 3, 4, 5
9. Natural History Museum _see_ British Museum of Natural History
10. nature, mankind's dominion over 1
11. _Nature_ (journal) 1
12. Nene Washes, Cambridgeshire 1
13. Netherlands 1
14. New Calton Cemetery, Edinburgh 1
15. New Forest, Hampshire 1, 2
16. New Guinea 1
17. New South Wales 1, 2, 3
18. New Testament 1
19. _New Yorker_ (magazine) 1n
20. New Zealand 1, 2, 3, 4
21. Newton, Prof. Alfred 1, 2
22. Nicholson, Max 1, 2, 3, 4
23. nightingale 1n, 2n
24. March 1
1. _see also_ blackcap
2. migration 1
3. song 1, 2, 3
25. nightjar 1, 2, 3, 4, 5, 6, 7
26. Noble, Martin 1
27. Norman Conquest 1, 2, 3
28. North Africa 1, 2, 3, 4, 5, 6, 7, 8
29. North America 1, 2, 3n, 4, 5, 6, 7, 8n, 9, 10n, 11, 12, 13
30. North Pole, Magnetic 1, 2
31. North Rona 1
32. North Sea 1, 2, 3, 4
33. North-West Passage 1, 2, 3, 4, 5, 6
34. Norwegian language 1, 2
35. _Not BB_ 1
36. nutcracker 1, 2
37. nuthatch 1, 2, 3n, 4, 5
1. _Observer's Book of Birds, The_ 1
2. Oddie, Bill 1, 2
3. 'oldsquaw' _see_ duck, long-tailed
4. Old Norse 1, 2, 3, 4, 5, 6
5. Old Testament 1, 2, 3, 4, 5
6. olivaceous (coloration) 1
7. Orford Hall, Warrington 1
8. Orinoco Delta 1
9. oriole
1. Audubon's 1n
2. Baltimore 1, 2
3. Bullock's 1, 2
4. golden 1
5. New World 1
6. origin of name 1
10. _Ornithological Dictionary; or Alphabetical Synopsis of British Birds_ 1, 2, 3
11. _Ornithology_ (Willughby & Ray) 1, 2, 3, 4
12. oropendolas, New World 1
13. Osorio, Prof. Daniel 1, 2
14. osprey 1, 2n
15. Ostend, Siege of 1–9
16. ouzel
1. 'black' 1, 2
1. _see also_ blackbird
2. ring 1, 2
3. 'water' 1n, 2
1. _see also_ dipper
17. owl
1. barn 1, 2, 3, 4
2. brown _see_ tawny
3. 'churn'/'fern'/'goat' _see_ nightjar
4. eagle 1n
5. fulvous 1
6. Hume's 1n
7. Koepcke's screech- 1n
8. long-eared 1
9. scops 1
10. short-eared 1
11. snowy 1
12. tawny 1, 2, 3
13. Tengmalm's 1n
18. Owls (Sheffield Wednesday nickname) 1n
19. _Oxford Book of English Verse_ 1
20. _Oxford English Dictionary_ ( _OED_ ) 1n, 2n, 3n, 4, 5n, 6n, 7, 8n, 9, 10n, 11
1. definition of bird 1
21. oystercatcher 1, 2n, 3, 4, 5, 6n
1. _Palearctic-African Bird Migration Systems, The_ (Moreau) 1
2. Palearctic, Western 1, 2
3. Palin, Sarah 1
4. Pallas, Peter Simon 1n, 2, 3
5. Palsgrave, John 1
6. Panuridae 1
7. parakeet 1, 2
1. Pennant's 1
1. _see also_ rosella, crimson
2. rose-ringed 1
8. Pareek, Aishwarya Shiva 1
9. Paris, Matthew 1
10. 'The Parlement of Foules' (Chaucer) 1, 2
11. parrot, elegant 1
12. Parry, Capt. William 1
13. partridge
1. red-legged 1n
2. Udzungwa forest 1
14. Parulidae 1
15. _Passer domesticus see_ sparrow, house
16. Patagonia 1
17. Pavord, Anna 1n, 2, 3
18. Payraudeau, Charles 1
19. peacock 1
20. peafowl 1
21. 'peerie deuk' 1
1. _see also_ phalarope, red-necked
22. peewit 1, 2, 3
1. _see also_ lapwing
23. Pelican Books 1
24. penguin
1. emperor 1
2. king ( _Aptenodytes patagonicus_ ) 1
25. Pennant, Thomas 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
26. Perrins, Christopher 1
27. petrel
1. European 1
2. Fea's 1n
3. Leach's storm- 1, 2, 3, 4, 5, 6
4. New Zealand storm- 1
5. origin of name 1
6. Swinhoe's storm- 1n
7. Tristram's storm- 1n
28. phalarope
1. grey 1, 2, 3
2. origin of name 1
3. red-necked 1
4. Wilson's 1n
29. pheasant 1, 2, 3, 4, 5
1. Lady Amherst's 1, 2, 3
30. Philip II, King of Spain 1
31. Phillip, Arthur 1
32. Phoebe (name) 1
33. _Phylloscopus schwarzi see_ warbler, Radde's
34. phylogeny 1
35. picathartes (bald crow) 1
36. pigeon
1. rock 1
2. wood 1, 2n, 3, 4, 5, 6, 7, 8, 9, 10, 11
37. Pilgrim Fathers 1
38. pipit
1. Blyth's 1n
2. buff-bellied 1
3. meadow 1, 2n, 3n, 4
4. Richard's 1, 2n
5. rock 1, 2, 3
6. tawny 1
7. tree 1
8. water 1, 2
39. pitta 1
1. Indian 1
40. place names, British 1, 2
41. plains-wanderer 1
42. Pliny the Elder 1, 2
43. plover
1. golden 1
2. grey 1
3. Kentish 1
4. little 1
5. 'Norfolk' 1
1. _see also_ curlew
6. ringed 1, 2
44. plumage 1, 2, 3
1. colour range 1
2. identification features 1, 2, 3, 4
45. pochard 1, 2n
46. Poole Harbour 1, 2
47. Porter, Roy 1
48. 'The Progress of Rhyme' (Clare) 1
49. Protestantism 1
50. Proto-Indo-European (PIE) language 1, 2, 3
51. Prum, Prof. Richard O. 1
52. _Prunella modularis see_ dunnock
53. Prynne, J. H. 1
54. ptarmigan 1
1. 'rock' 1n, 2
2. 'willow' 1n
1. _see also_ grouse, red
55. puffin ( _Fratercula arctica_ ) 1, 2, 3n, 4, 5
56. origin of name 1
57. Puffin Books 1
58. _Puffinus puffinus see_ shearwater, Manx
59. 'purple swamphen' 1
1. quail 1n, 2, 3n
2. quail (verb) 1
1. Radde, Gustav 1, 2
2. rail 1, 2
1. invisible 1
2. land 1, 2, 3, 4
1. _see also_ corncrake
3. water 1n
3. Raleigh, Walter 1
4. Ralph, Robert 1
5. raptors 1
6. Rasmussen, Pamela 1
7. raven 1, 2, 3n, 4, 5, 6, 7, 8, 9, 10, 11n
8. Ray, John 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
9. Red Sea 1
10. 'redbreast' _see_ robin
11. redshank 1n
12. redstart 1, 2, 3, 4, 5, 6, 7, 8n, 9, 10n
1. black 1, 2
2. 'blue-throated' (bluethroat) 1, 2
13. redwing 1, 2, 3, 4, 5
14. 'reedling' 1
1. _see also_ tit, bearded
15. Reedman, Ray 1
16. Rees, Merlyn 1
17. Renaissance 1
18. _Rhodostethia rosea see_ gull, Ross's
19. Richard I, King (Richard the Lionheart) 1
20. Richard of Lunéville, Monsieur 1n
21. Richards, Keith 1
22. Richardson, Sir John 1, 2, 3, 4
23. Ridley, Nicholas 1
24. riflebird, graceful 1
25. robin ( _Erithacus rubecula_ ) 1n, 2, 3n, 4
1. American 1
2. eastern/western yellow 1
3. hooded 1
4. mangrove 1
5. origin of name 1, 2, 3, 4
6. 'redbreast' 1, 2n, 3n, 4, 5, 6
7. 'ruddock' 1, 2
8. song 1, 2, 3
26. Robins (football teams) 1
27. Robinson, Eric 1
28. roller 1, 2
29. Roman Empire 1
30. rook 1, 2, 3, 4, 5
31. Rooper, George 1n
32. rosefinch, Sinai 1
33. rosella, crimson 1
34. Ross Island/Sea 1
35. Ross, James Clark 1, 2, 3, 4, 5
36. Ross, John 1
37. Ross, Lady Eliza 1n
38. Rothschild, Lord Walter 1, 2n 3
39. Rousseau, Jean-Jacques 1
40. Royal Navy 1
41. RSPB 1, 2, 3, 4, 5
42. Rubens, Peter Paul 1
43. 'ruddock' _see_ robin
44. Rudolf II, Holy Roman Emperor 1
45. rufescent (coloration) 1
46. ruff 1
47. rush-tyrant, many-coloured 1
48. Rycaut, Paul 1
1. Sabine, Edward 1
2. _Sacred Ibis: The Ornithology of Canon Henry Baker Tristram_ (Hale) 1n
3. St Albans monastery 1
4. St Cuthbert 1
5. St Guthlac of Crowland 1
6. St Kilda (archipelago) 1, 2
7. St Peter 1
8. Salaman, Paul 1, 2
9. Sample, Geoff 1, 2
10. sanderling 1
11. sandgrouse 1n
1. Pallas's 1n, 2n
12. sandpiper
1. 'ash-coloured' 1
2. Baird's 1n
3. buff-breasted 1
4. wood 1
13. sapsucker, yellow-bellied 1
14. Saskatchewan River 1
15. Saunders, Howard 1n
16. Savi, Paolo 1
17. Scandinavian languages 1, 2, 3n, 4, 5n, 6, 7, 8
1. _see also_ Old Norse
18. scarecrow 1, 2
19. scaup 1n
20. Schwarz, Ludwig 1
21. scoter 1
22. Scotland _see_ Highlands, Scottish; St Kilda; Western Isles
23. Scott, Capt. Robert Falcon 1, 2
24. Scott, Kathleen 1
25. Scott, Sir Peter 1, 2n
26. _Scottish National Dictionary_ 1
27. sea parrot 1
1. _see also_ puffin
28. 'sea-mew' 1, 2
1. _see also_ kittiwake
29. 'sea-pie' 1, 2
1. _see also_ oystercatcher
30. _Seafarer, The_ 1
31. Seagulls (Brighton and Hove Albion football club) 1
32. Seebohm, Henry 1
33. seedsnipe 1
34. Selous, Edmund 1
35. serin, Tristram's 1n
36. shag 1n, 2, 3, 4, 5
37. Shakespeare, William 1, 2, 3, 4, 5, 6n, 7, 8, 9
38. Sharpe, Richard Bowdler 1, 2
39. Sharrock, Tim 1, 2
40. shearwater 1
1. Audubon's 1n
2. Balearic 1n, 2
3. Cory's 1
4. Manx 1, 2, 3
5. Scopoli's 1n
6. Yelkouan 1n
41. shelduck 1, 2n
42. _Shell Bird Book, The_ 1n, 2n
43. Shelley, Percy Bysshe 1
44. Shetland 1, 2, 3
45. shooting 1, 2, 3, 4, 5
46. shoveler ( _Anas clypeata_ ) 1, 2, 3, 4n
47. shrike
1. brown 1n
2. 'cinereous' _see_ red-backed
3. Isabelline 1, 2
4. red-backed ('butcher-bird') 1, 2
48. Sibley, Charles G. 1, 2, 3, 4, 5, 6, 7
49. 'silk-tail' 1
1. _see also_ waxwing
50. Simoni, Anna 1n
51. sitella, varied 1
52. Skomer 1n
53. skua 1, 2, 3
1. Arctic 1, 2, 3
2. great 1
3. long-tailed 1
4. 'parasitic' 1, 2
1. _see also_ Arctic
54. skylark 1, 2, 3, 4, 5, 6, 7n, 8
1. origin of name 1
55. smew 1
56. Smith (surname) 1
57. Smith, William Thomas 1
58. Smyth, Arthur Bowes 1, 2
59. 'snakebirds' (darters) 1–8
60. sniper 1
61. snowcock, Caucasian 1n
62. Somerset (Moors and Levels) 1, 2n, 3, 4
63. South Africa 1, 2, 3, 4
64. South America 1, 2, 3, 4, 5
65. South Pole 1, 2
66. Spanish Empire 1
67. Spanish language 1, 2, 3n, 4, 5
68. sparrow 1, 2, 3, 4, 5n, 6
1. Dead Sea 1
2. 'hedge' 1, 2, 3, 4, 5, 6, 7, 8
1. _see also_ dunnock
3. house ( _Passer domesticus_ ) 1, 2, 3, 4, 5, 6, 7, 8n
4. 'reed' 1, 2
1. _see also_ bunting, reed
5. tree 1, 2
69. sparrowhawk 1n, 2n
70. species
1. defining 1
2. lumping 1, 2, 3
3. new 1
4. splitting 1, 2, 3
5. world total 1, 2
71. Spenser, Edmund 1
72. spinebill, eastern 1
73. spinetail, Delta Amacuro 1
74. spoonbill 1
75. Sri Lanka 1
76. starling 1, 2, 3, 4, 5, 6
1. Tristram's 1n, 2, 3n
77. _Status of Birds in Britain and Ireland, The_ (BOU) 1
78. Stedman, Capt. J. G. 1
79. Steel, Mark 1n
80. stint, Temminck's 1, 2
81. Stodmarsh, Kent 1
82. stonechat 1, 2n
1. 'white-rumped' 1
1. _see also_ wheatear
83. stork 1, 2, 3
84. Stour Valley 1
85. Stover, Matthew Woodring 1
86. _Stray Feathers – a journal of ornithology for India and his dependencies_ (Hume) 1
87. Subbuteo (game) 1
88. sunbird
1. Palestine 1
2. scarlet-chested 1
3. Uluguru violet-backed 1
89. Surflet, Richard 1
90. Sussex University 1
91. Sutton, Christopher 1
92. Swainson, Revd Charles 1
93. Swainson, William 1
94. swallow 1n, 2, 3, 4, 5, 6, 7, 8, 9
1. barn 1, 2, 3n, 4, 5, 6
2. 'hibernation' 1
3. 'house' 1
1. _see also_ martin, house
4. 'sea' 1
1. _see also_ tern, common
95. swan
1. Berwick's 1, 2, 3, 4, 5n
2. black 1
3. mute 1n, 2
4. origin of name 1, 2
5. whooper 1n, 2, 3, 4, 5n
96. Swedish language 1, 2
97. Swedish Ornithological Society 1
98. swift 1, 2
1. white-rumped 1
99. Sykes, Col William Henry 1
100. _Sylvia_ 1
1. _S. atricapilla see_ blackcap
2. _S. dartfordiensis_ 1
1. _see also_ warbler, Dartford
101. Sylvia (name) 1
102. Sylviidae 1
103. syrinx (vocal organ) 1
104. _Systema Naturae_ (Linnaeus) 1
1. Tan, Vincent 1n
2. Tasmania 1
3. Taylor, John 1
4. teal 1, 2n
1. 'hottentot' 1n
5. Temminck, Coenraad 1, 2, 3
6. Ten Commandments 1
7. tern 1n, 2, 3, 4n
1. Arctic 1, 2, 3
2. black 1
3. common 1
4. gull-billed 1n
5. origin of name 1
6. roseate 1
7. Sandwich 1, 2, 3
8. white-winged 1
8. territory, defending 1n, 2, 3
9. _Tess of the D'Urbervilles_ (Hardy) 1
10. thick-knees 1
1. _see also_ curlew, stone
11. Thomas, Keith 1
12. throstle 1
1. _see also_ thrush, song
13. Throstles (West Bromwich Albion football club) 1
14. thrush
1. MacGillivray's names 1
2. mistle 1n, 2
3. Naumann's 1n
4. song 1, 2, 3, 4, 5n, 6
5. Swainson's 1n, 2n
6. White's 1, 2n, 3
1. _see also_ blackbird; fieldfare; redwing
15. _Times, The_ 1, 2, 3n
16. tit
1. bearded 1, 2, 3
2. blue 1, 2n, 3, 4n, 5n
3. coal 1, 2, 3n
4. crested 1n, 2n
5. great 1, 2, 3n, 4
6. long-tailed 1, 2, 3
7. marsh 1, 2n, 3, 4, 5, 6
8. as rude word 1, 2
9. 'titmouse' 1, 2
10. willow 1n, 2, 3, 4
17. _Titmice of the British Isles, The_ 1
18. 'Tit-Willow' (Gilbert and Sullivan) 1
19. Toms, Mike 1
20. Treaty of Versailles 1
21. treecreeper 1, 2, 3, 4n
22. Tristram, Revd Henry Baker 1n, 2, 3n
23. _Troglodytes troglodytes see_ wren
24. Trollope, Anthony 1
25. Truman, Harry S. 1n
26. Tucker, Bernard 1
27. turaco 1
1. Ross's ( _Musophaga rossae_ ) 1
28. _Turdus merula see_ blackbird
29. Turner, William 1, 2, 3
30. turnstone 1n
1. Uluguru Mountains, Tanzania 1, 2, 3, 4, 5
2. Uppsala Cathedral 1
3. _Upupa epops see_ hoopoe
4. _Ursus arctos isabellinus see_ bear, brown
5. Usambara Mountains, Tanzania 1, 2
1. Valverde, Tono 1
2. van Deemter, Kees 1n
3. van den Berg, Arnoud 1
4. Vaughan Williams, Ralph 1
5. Victor, Jean 1
6. Victoria, Queen 1, 2, 3, 4, 5
7. Victorian period 1
8. Vieillot, Louis 1
9. Viking invasion 1n, 2, 3, 4
10. Voous, Karel 1
11. _Voyage of Governor Phillip to Botany Bay, The_ (Phillip) 1
12. vulture
1. classification 1, 2
2. Egyptian 1n
1. waders 1, 2, 3, 4, 5
2. wagtail
1. blue-headed 1
2. grey 1, 2, 3
3. pied ( _Motacilla alba yarrelli_ ) 1n, 2, 3n, 4
4. white ( _Motacilla alba alba_ ) 1n, 2
5. Willie 1, 2
6. yellow 1, 2, 3
3. Wales 1, 2, 3
4. Wallace, Alfred Russel 1
5. Wallace, Ian 1, 2, 3
6. Waltham Abbey, Essex 1, 2
7. War of Independence 1
8. warbler
1. aberrant 1
2. aquatic 1n
3. Balearic 1n
4. Blackburnian 1n, 2, 3
5. 'black-capped' 1, 2
1. _see also_ blackcap
6. blackpoll 1n
7. Bonelli's 1
8. Cetti's 1, 2, 3, 4
9. Dartford ( _Fauvette pitchou_ ) 1, 2, 3, 4, 5, 6n, 7
10. garden 1, 2, 3
11. 'grass'/'leaf' 1
12. grasshopper 1, 2, 3n, 4, 5, 6
13. Hume's leaf 1n
14. MacGillivray's 1
15. magnolia 1n
16. Marmora's 1n, 2
17. marsh 1, 2
18. 'marsh and tree' 1
19. melodious ( _Hippolais polyglotta_ ) 1
20. Moltoni's 1n
21. Mrs Moreau's (Winifred's) ( _Scepomycter winifredae_ ) 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
22. Old/New World 1, 2, 3
23. paddyfield 1n
24. Pallas's 1n
25. Radde's ( _Phylloscopus schwarzi_ ) 1, 2
26. reed 1, 2n, 3, 4, 5n, 6, 7, 8
27. river 1n
28. Savi's 1, 2, 3
29. sedge 1, 2n, 3, 4, 5, 6
30. Tristram's 1n
31. 'true' 1
32. 'white-throated' 1, 2
1. _see also_ whitethroat
33. willow 1, 2, 3, 4, 5n, 6, 7, 8, 9
34. Wilson's 1n
35. wood 1, 2, 3, 4, 5
36. wood- (New World) 1
37. yellow 1n
1. _see also_ blackcap; chiffchaff
9. _Waste Land, The_ (Eliot) 1
10. _Water Birds_ (Bewick) 1, 2
11. Watson, Chris 1n
12. waxbill, yellow-bellied 1
13. waxwing ( _Bombycilla garrulus_ ) 1
14. Welsh language 1, 2
15. West Africa 1
16. Western Isles, Scotland 1, 2, 3
17. 'Wetmore Order' 1
18. Wetmore, Alexander 1
19. wheatear 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
1. Hume's 1n
2. Isabelline 1, 2
3. northern 1, 2, 3
4. origin of name 1
5. Tristram's 1n
20. wheel, invention of 1
21. whimbrel 1n, 2, 3n, 4n
22. whinchat 1
23. White, Gilbert 1, 2, 3, 4, 5, 6, 7
24. whitethroat 1, 2, 3, 4, 5, 6, 7, 8, 9n, 10
1. Hume's 1n
2. lesser 1
25. Whitney, Bret M. 1n
26. wigeon 1, 2, 3n, 4
27. Wilde, Oscar 1
28. William I, King 1
29. Willughby, Francis 1, 2, 3, 4, 5, 6
30. Wilson, Alexander 1n
31. Wilson, Edward 'Bill' 1
32. 'wind-hover'/ 'windfucker' 1
1. _see also_ kestrel
33. 'The Windhover' (Hopkins) 1
34. winter, British
1. 1946–7 1
2. 1962–3 1
3. visitors 1
35. Woburn Abbey, Bedfordshire 1
36. woodcock 1n, 2n
37. woodlark 1, 2, 3
38. woodpecker 1, 2, 3, 4
1. great spotted (pied) 1n
2. green 1n, 2
3. lesser spotted (barred) 1n
39. woodswallow 1
40. Wordsworth, William 1
41. World Cup football (1970) 1
42. World Land Trust 1n
43. World War
1. First 1, 2
2. Second 1, 2, 3, 4, 5
44. _Worst Journey in the World, The_ (Cherry-Garrard) 1
45. Worsthorne, Peregrine 1
46. _Worthies of England_ (Fuller) 1
47. wren ( _Troglodytes troglodytes_ ) 1, 2, 3, 4, 5, 6, 7
1. fairy- 1, 2
2. fulvous 1
3. 'furze' 1, 2
1. _see also_ warbler, Dartford
4. 'golden-crested' 1, 2n, 3, 4
1. _see also_ goldcrest
5. Jenny 1n, 2
6. 'peacock-' 1n
1. _see also_ lyrebird
7. 'reed' 1
1. _see also_ warbler, reed
8. 'willow-' 1, 2, 3
1. _see also_ warbler, willow
9. 'wood-' 1
1. _see also_ warbler, wood
48. Wright, John 1
49. Wright, Minturn 1
50. wryneck 1, 2n
51. _Wuthering Heights_ (Brontë) 1
52. WWF (World Wide Fund for Nature) 1, 2
1. Yakutia, Siberia 1
2. Yale University 1
3. Yare Valley, Norfolk 1
4. Yarrell, William 1, 2, 3, 4, 5, 6, 7
5. yellowhammer 1n, 2, 3, 4, 5, 6, 7
6. yellowthroat 1
7. Yoon, Carol Kaesuk 1n
1. _Zoologist_ (journal) 1
# About the Author
Stephen Moss is a naturalist, broadcaster, television producer and author. In a distinguished career at the BBC Natural History Unit his credits included _Springwatch, Birds Britannia_ and _The Nature of Britain_. His books include _The Robin: A Biography, A Bird in the Bush, The Bumper Book of Nature, Wild Hares and Hummingbirds_ and _Wild Kingdom_. He is also Senior Lecturer in Nature and Travel Writing at Bath Spa University. Originally from London, he lives with his family on the Somerset Levels, and is President of the Somerset Wildlife Trust.
# Also by the Author
Selected titles by Stephen Moss:
A BIRD IN THE BUSH: A SOCIAL HISTORY OF BIRDWATCHING
WILD HARES AND HUMMINGBIRDS
WILD KINGDOM
THE ROBIN: A BIOGRAPHY
# Copyright
First published by Guardian Faber in 2018
Guardian Faber is an imprint of Faber & Faber Ltd
Bloomsbury House,
74–77 Great Russell Street
London WC1B 3DA
Guardian is a registered trademark of
Guardian News & Media Ltd,
Kings Place, 90 York Way,
London N1 9GU
This ebook edition first published in 2018
All rights reserved
© Stephen Moss, 2018
Cover design by Faber
Illustration © Neil Gower
The right of Stephen Moss to be identified as author of this work has been asserted in accordance with Section 77 of the Copyright, Designs and Patents Act 1988
Illustrations by Alexander Fussell and John Thompson, originally produced for William Yarrell's _A History of British Birds_ (1843)
This ebook is copyright material and must not be copied, reproduced, transferred, distributed, leased, licensed or publicly performed or used in any way except as specifically permitted in writing by the publishers, as allowed under the terms and conditions under which it was purchased or as strictly permitted by applicable copyright law. Any unauthorised distribution or use of this text may be a direct infringement of the author's and publisher's rights, and those responsible may be liable in law accordingly
ISBN 978–1–78335–092–6
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{
"redpajama_set_name": "RedPajamaBook"
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{"url":"https:\/\/en.wikipedia.org\/wiki\/Wikipedia_talk:WikiProject_Mathematics\/Archive_17","text":"# Wikipedia talk:WikiProject Mathematics\/Archive 17\n\n## 13-digit ISBNs\n\nAbove KSmrq, suggests the use of 13-digit ISBNs. However, since many (most?) sites (e.g. Amazon) can not handle 13-digit ISBNs, using them will make many of the \"Find this book\" links fail when clicking on the ISBN links. For example clicking on: ISBN 0-7167-0344-0, then clicking on \"Find this book\" link for the Amazon.com entry under the section \"Individual online booksellers\" finds this page, while doing the same thing for ISBN 978-0-7167-0344-0, gives this result So we might want to hold off for now on using 13-digit ISBNs. In the future I'm sure some enterprising bot will come along and convert all our ISBNs for us anyway\u00a0;-) \u2014 Paul August 16:03, 1 September 2006 (UTC)\n\nThe future arrives four months from today. Rich Farmbrough has a bot [User talk:Rich Farmbrough\/Archive\/2006Sep#The bot and ISBN-13 contemplating] an automatic change-over. In the linked discussion I mention a few other issues as well. I'm wondering if it would be too cumbersome to provide both ISBN forms (especially for print). Perhaps the MediaWiki ISBN magic could handle it for online use, like the handling of date formats; but, as always, implementation is not in our hands.\nMeanwhile, my feeling is that the ISBN-13 form is future-proof and international, and allows the intended book to be found, even if it doesn't find all the sellers the ISBN-10 form matches. Every ISBN has annoying limitations. A paperback and a hardback have different numbers, as do versions of classics provided by different publishers; and each edition has its own number, which is at times good and at other times an obstacle.\nRegardless of which ISBN you prefer, please do take a moment to provide one (and, ideally, check its validity).\nAnother way to assist readers in finding books is to check against online versions. Some texts can be found at Project Gutenberg, but mathematics is a minority there. Fortunately, we have alternatives.\nThese sites also include links to others. --KSmrqT 18:19, 1 September 2006 (UTC)\n\n## Good articles\n\nI've been going through the list of mathematics Good articles and I'm not sure that some of them really meet the grade. Riemann hypothesis is what I would consider to be the standard for a good article. My main concern is that the articles are either lacking in any history of the topic failing criteria (3a). Also it would be good to see some illustrations (6).\n\nMoreover, I think there is some need to discuss what makes a mathematics good article so we can establish a standard. Maths articles seem to be a bit of a special case as they are often highly technical, so they are likely to have problems with GA criteria 1a: it has compelling prose, and is readily comprehensible to non-specialist readers. We also seem to run into problems with 2b the citation of its sources is essential, and the use of inline citations is desirable, although not mandatory. Often inline citations are not really appropriate as the topic as a whole will be covered in cited textbooks.\n\nGenerally our number of GA's is very low with only 15 articles. Are there any other articles out there which people think are especially good? --Salix alba (talk) 10:37, 3 September 2006 (UTC)\n\nI think the article on knot theory is a good target to turn into a GA, and eventually maybe even an FA. It doesn't try to do too much, and what is there currently should be fairly easy to brush up. I note that the section on Conway notation and planar graph notation is incomplete, but shouldn't take too much time to complete. There are several obvious ways to add good illustrations (and illustrative examples) to the article. --C S (Talk) 11:10, 3 September 2006 (UTC)\nThere's also quite a bit of bickering going on at Grigori Perelman, but it seems to me that this article has recently undergone a great deal of attention and editing and if all disputes can be resolved, I expect it could become a GA. Perhaps even Poincar\u00e9 conjecture...but that will require a lot more work, and I've dropped the ball on that for which I apologize. But eventually I'll have a decent writeup of Perelman's proof (\"alpha\" version is at User:C S\/todo\/PC proof) and we can rewrite the article around that or whatever. --C S (Talk) 11:21, 3 September 2006 (UTC)\nYes I agree that knot theory could be a good target. Are people happy to defend the article, if so I think it should be nominated.\nGrigori Perelman and Poincar\u00e9 conjecture are probably too volitile at the moment GA 5 It is stable, i.e. it does not change significantly from day to day and is not the subject of ongoing edit wars. , that said it might be a good time to list if there are active contributors.\nI'd quite like to create a B+ rating, for articles which are nearly but not quite at the standard of GA, we do have a good number of articles listed on Mathematics 1.0 which would fit this category, for example Pi which is good but has been delisted from GA. --Salix alba (talk) 12:05, 3 September 2006 (UTC)\n\n## Rename \"Ordinal number\"? God forbid!\n\nUser:Salix alba wants to rename (move) Ordinal number which is (in my opinion) one of the most important articles in the general area of Set theory. There are more than FIVE HUNDRED articles which link to it by its current name. Now, admittedly the majority of them would just as well be linked to the article which he proposes to put in its place -- an article on \"first, second, third, fourth, fifth, etc.\", but there is still a large fraction of them which are important mathematics articles. Please resist this disruptive change by talking at Talk:Ordinal number and elsewhere. Notice that there is already a link at the beginning of \"Ordinal number\" to the section Names of numbers in English#Ordinal numbers which covers the material in which he is interested. JRSpriggs 02:53, 4 September 2006 (UTC)\n\nI agree, of course: \"ordinal number\" is correct. You can point him to this book, for example:\nHalmos, Paul (1974). Naive Set Theory. Springer. ISBN 0-387-900092\u00a0Invalid\u00a0ISBN-6. (reprint of 1960 classic)\nChapter 19 is entitled \"ordinal numbers\".---CH 21:17, 6 September 2006 (UTC)\n\n## McNugget number is up for AFD\n\nI've listed McNugget number for AFD. This is the second nom (first was by somebody else in October). AFD discussion page People may be interested in looking over the first discussion, which ended as \"no consensus\". --C S (Talk) 01:03, 5 September 2006 (UTC)\n\n## Multidimensional Gaussian integrals\n\nUser:EulerGamma recently removed a section about multidimensional generalizations from the Gaussian integral article for being \"complicated\" and lacking sources. The topic is real, but the lack of sources for the details is a valid complaint. Unfortunately, the original author seems to have been inactive for several months. I'm sure some people here are knowledgeable enough to check the content (I'm not); please have a look if you do. Fredrik Johansson 20:40, 6 September 2006 (UTC)\n\n## Leonhard Euler is up for FAC\n\nPlease see this page for the discussion. Borisblue 00:39, 7 September 2006 (UTC)\n\n## Mathematical Wikiers in Chinese\n\nDmharvy, here is your link. zh:Wikipedia talk:\u6570\u5b66\u5174\u8da3\u5c0f\u7ec4\u7ef4\u57fa\u4eba\u5217\u8868----Hillgentleman 03:41, 7 September 2006 (UTC)\n\n## User:WATARU\n\nNew user WATARU appears to me to be almost certainly User:WAREL. However he hasn't yet done any of the things that got him banned before. Let's keep an eye out, but not provoke. \"Don't start none, won't be none\", as Huey P Freeman would say. --Trovatore 20:30, 9 September 2006 (UTC)\n\nsee [1]. --Trovatore 18:33, 11 September 2006 (UTC)\n\nNow he's changed the Japanese link at division ring to something else. I don't read Japanese, so I don't know if it's appropriate or not, but given his history I'm not inclined to trust him. He may well be planning some shenanigans at ja.wiki and making edits here to prepare for them. (It goes without saying that he has long since used up his assumption of good faith.) Would someone with some competence in Japanese please look at this? --Trovatore 21:02, 12 September 2006 (UTC)\n\nAnd he is insisting on using the Big Omega function on square number, where it is pointless showing off. (See diffs: [2][3].) Given the number of complaints we get for being technical where we have to be, there is no excuse for this in an article that proves that the squares of odd numbers are odd. Septentrionalis 19:48, 13 September 2006 (UTC)\n\n## Articles tagged as too technical\n\nFor a list see Wikipedia:WikiProject_Mathematics\/Current_activity\/Lists#Articles_that_are_too_technical. I've noticed, as I'm sure others have, that sometimes well-meaning editors just go through mathematical articles tagging them as \"too technical\". For example simple module has been tagged; however, I don't really see why it was tagged other than it looks like \"gobbly-gook\" to someone who doesn't know what a ring or module is. I can't see how this article can really be improved in a significant way to be accessible to someone without such a background. Perhaps an example built from the ground up would help...but that would seem to be the equivalent of writing a wikibook on abstract algebra. In any case, I believe this article (and some others) have been tagged wrongly.\n\nThe unfortunate thing about all this is that it makes it hard to find the actual overly-technical articles that can be made much more accessible. As a first step to making articles more accessible, therefore, I suggest that some people take some time and untag as many articles as they can -- those that are very advanced topics or seem to have been made as accessible as possible. --C S (Talk) 02:30, 11 September 2006 (UTC)\n\nI added a sentence about graphical projection to Projection (linear algebra) and removed the tags. There wasn't anything in the talk page about why the tags were added. User:ST47 who added the \"technical\" tag was bot assisted. User:Srleffler added the original tag didn't leave any explanation. It seems Srleffler's attention was drawn to the article through graphical projection; they also left the same tag on projection (relational algebra) which Jon Awbrey summarily removed. Guess it's just another example of what you're talking about. (I know it's just one article. Sorry.) Lunch 23:15, 12 September 2006 (UTC)\ngot a bunch more. btw, it seems the current activity list hasn't been updated in a couple of weeks. did the bot run out of gas? Lunch 04:58, 24 September 2006 (UTC)\n\n## page move?\n\nthe article on Robert Berger, the mathematician, was linked to by several film-related articles mentioning the writer robert berger. i changed those to refer to Robert Berger (writer). might it be a good idea to move Robert Berger to Robert Berger (mathematician) and put a redirect in its place? how does one go about doing this? tia. Lunch 03:38, 11 September 2006 (UTC)\n\nThe easiest is to use the \"move\" tab at the top of the article to move Robert Berger to Robert Berger (mathematician). This will automagically leave a redirect in its place. --LambiamTalk 05:31, 11 September 2006 (UTC)\nBut looking at this stubby article, I think there is not enough info to merit having a separate article here, as was noted by others on its talk page. --LambiamTalk 06:16, 11 September 2006 (UTC)\nTwo points here: (1) Are you sure that these are two different people? Sometimes one person does work in two completely unrelated fields. For example, Dorthy Lamour (hope I remembered the right actress) Hedy Lamarr was both a film actress and the inventor of a method of encryption. (2) There is no point in moving the page unless you replace the redirect with a disambiguation page listing various people named \"Robert Berger\" and giving links to their pages. JRSpriggs 07:05, 11 September 2006 (UTC)\nTry Hedy Lamarr for the inventive star. --LambiamTalk 10:09, 11 September 2006 (UTC)\nMy thanks to Lambiam for the correction. JRSpriggs 05:26, 12 September 2006 (UTC)\nAccording to his entry at the IMDB, the writer Robert Berger was credited as \"Robert H. Berger M.D.\" for being a consultant for the movie Final Analysis. As that movie is about a psychiatrist, that Robert Berger is very likely too a shrink. Citations of (Berger, Robert. \"The undecidability of the domino problem\". Memoirs of the American Mathematical Society, 66, (1966), 1\u201372) all appear not to give a middle initial. --LambiamTalk 10:35, 11 September 2006 (UTC)\nThis Robert Berger seems not to have an entry in the Library of Congress, but he IS in the Harvard library catalog! He is given as Berger, Robert (born 1938), author of the AMS memoir on domino undecidability. They don't know his middle initial. I also looked up the memoir itself, and it includes no middle name, middle initial, thesis advisor, and no acknowledgments that I could find. There were four references, including one to a paper of Hao Wang. WP's entry for Wang says he was at Harvard from 1961 to 1967, so it's reasonable he could have been Berger's advisor. AMS MathSciNet does not seem to have any papers by this Robert Berger besides the domino memoir. EdJohnston 19:36, 11 September 2006 (UTC)\n\nthe harvard library catalog lists several holdings under the title \"the undecibility of the domino problem.\" one of them is the AMS publication. another one of them is a copy of his dissertation. the title page there probably has his advisor's name. i'll be visiting there at the beginning of november; if i get a chance, i'll look it up. (i'm also morbidly curious to see ted kaczynski's dissertation, too, so i might actually take the time.\u00a0:) UMI has him listed at harvard in 1965, too, but they don't have a copy of his dissertation (not even the abstract).\n\nwhat originally brought me to the article was just a haphazard meandering. i saw the article on the list of \"too technical\" articles and was curious why it was there. when i looked at the list of \"what links here,\" i noticed the three (four?) links to the movie writer\/producer. although a quick check through IMDB now makes me think there are at least three robert bergers of note: the mathematician; Robert H. Berger, M.D., the writer\/consultant for \"final analysis\"; and robert berger, the producer. this last fellow was making films as far back as 1962 so unless the mathematician robert berger was also a rookie film-maker during his harvard days, they're not the same person. (and incidentally, robert berger has produced almost three dozen movies; maybe there should be an article on him.) that doesn't rule out that the mathematician went out and got an M.D. and got into the film business, but i'd hazard a guess that didn't happen.\n\nanywho, all this attention seems way out of proportion, but i'm glad to see some other amateur sleuths out there too. \u00a0:) i s'pose my two bits is that i go back an un-wiki-link robert berger, the writer\/consultant of final analysis; make a stub on robert berger, the producer; and move robert berger, the mathematician. whaddya all think? too much?\n\nthanks. Lunch 20:21, 11 September 2006 (UTC)\n\n(oops, kaczynski did his PhD at michigan. he was an undergrad at harvard. oh well, maybe some other time.) Lunch 17:27, 22 September 2006 (UTC)\nOK with me. The Harvard library catalog shows many, many Robert Bergers. But this man is the most famous of the mathematical Robert Bergers. Google Scholar still shows 216 citations to the domino paper, so he is notable. EdJohnston 22:55, 11 September 2006 (UTC)\nThe plan sounds fine. Just be careful of the other mathematician named Robert W. Berger who wrote quite a few papers, mostly in German. His genealogy can be found here. I don't know how notable he was\/is.\nBy the way, this book review (a postscript file) asserts that the Robert Berger we have been discussing was indeed Hao Wang's student. Michael Kinyon 23:10, 11 September 2006 (UTC)\n\nthanks. (i think the link is [4] for the postscript or [5] for the pdf, but i think the pdf got chopped off.) to address lambiam's early point, should the robert berger article mention all three since separate articles would be too short? i started a stub for Robert Berger (producer); potentially it could be much longer (he was rather prolific), but isn't long now. i dunno how long the article on robert berger the aperiodic tiler could be, or how long the article on robert w. berger could be. Lunch 00:12, 12 September 2006 (UTC)\n\n## WP:BLP violation at Louis de Branges de Bourcia\n\nI've changed to a far better version while trying to incorporate some of the recent factual additions. But the previous version definitely had way too much speculation, ramblings, and just poor sourcing. Given the number of people (although maybe some of the IPs are really the same person), who have edited it into this state, I think it's wise if people keep an eye on this page. --C S (Talk) 06:04, 11 September 2006 (UTC)\n\nSome of the details are from Sabbagh's book, but I have not seen it recently enough to edit. Septentrionalis 20:32, 11 September 2006 (UTC)\n\n## What does it mean?\n\nI find that many math articles give definitions in a way that is 100% accurate but only 10% useful. (This is true of math writing beyond Wikipedia.) For example, until recently the definition of symmetric matrix simply stated that ${\\displaystyle A_{ij}=A_{ji}}$. That's all well and good\u2014it correctly defines the term\u2014but it does not answer the question \"what does it mean for a matrix to be symmetric?\". As best I can tell, the answer is \"it means the eigenvectors are orthogonal\", which I added. After all, this is what mathematicians think when they think \"symmetric\".\n\nI propose a concerted effort to get answers of this form into the definitions of math terms\u2014answers that allow readers to think like a mathematician rather than stare at syntax. Perhaps a template Template:what_does_it_mean? \u2014Ben FrantzDale 23:35, 11 September 2006 (UTC)\n\nAs for the statement you added, it wasn't quite correct, so I fixed it in the article. (It turns out to be exactly the symmetric matrices that have orthonormal bases of eigenvectors which makes your addition even more appropriate to this particular article.)\nAs for your suggestion, I agree with you in principle but not in practice. A mathematical definition is just that--a definition. While it may be equivalent to any number of conditions, some of which are intuitively more appealing than others, the definition is usually the more straightforward one. In this case \"symmetric\" means literally that the matrix entries exhibit some kind of symmetry, in this case with respect to the matrix transpose. That's why we have a whole article to follow; the article should explain \"what does it mean\". A good article probably does not need any additional template if it's doing its job correctly.\nHaving said this, thanks for your contribution and suggestion. We do need to make sure that the math articles fully explain the \"why\". VectorPosse 00:26, 12 September 2006 (UTC)\nwhat do you mean by \"mean\"?\u00a0;) that there is a complete set of orthonormal eigenvectors of a symmetric matrix (along with real eigenvalues) is usually called a theorem, and the symmetry of matrix entries is usually called the definition. of course, it is equivalent to do the reverse. (and there are several other definitions that result in equivalence.)\nbut the symmetry of matrix entries is by far the simplest definition, and the eigenvector\/value property is listed shortly thereafter in the article (and this is good practice). also, the symmetry of matrix entries does have significance: if two vectors are related by multiplication by a symmetric matrix, then changes in entry i wiggle entry j as much as entry j wiggles entry i. symmetry is also preserved under a congruence transform (as like with change of coordinates applied to a quadratic form - not to be confused with a similarity transform, a change of coords for a linear system). physicists love these sorts of things. (as do mathematicians, engineers, and a whole party of people.\u00a0:) but i'd stick this in a list of properties...\ni guess my point is that people usually go with the simplest possible definition and stick equivalent definitions under \"properties\" or \"lemmas\/theorems\". Lunch 00:43, 12 September 2006 (UTC) (oops. edit conflict.)\nmaybe i'd add that \"simplest\" doesn't always mean \"most intuitive\" or \"most informative about why this is useful\/interesting\/wheretheheckdidTHIScomefrom\". you're right in thinking that an article on such a subject deserves a bit of history\/motivation in the leading paragraph(s). or maybe i'm not thinking what you're thinking. Lunch 00:54, 12 September 2006 (UTC)\nLunch, I think we are on the same page when you say \u201c\u2018simplest\u2019 doesn't always mean \u2018most intuative\u2019...\". In the case of this example I'd argue that the obvious definition of symmetry, while important, is essentially intuition-free and so not very helpful for newbies. That's why I like the format \"X is defined as y but really a mathematician is thinking z.\" Overall I'd like to see a move towards systematically answering \u201cwheretheheckdidTHIScomefrom\u201d. \u2014Ben FrantzDale 02:40, 12 September 2006 (UTC)\nyou mean you didn't like my wiggling components analysis? \u00a0;) not to beat a dead horse, but as a mathematician who spends a lot of time doing linear algebra, i do think in components often enough. imho, the component-wise definition of a symmetric matrix is a good one and does have intuitive appeal. (i'd also add that linear algebra is almost always first introduced to students from a components point of view -- and with good reason.) Lunch 19:47, 12 September 2006 (UTC)\nVectorPosse, as for the template idea, to clarify I was thinking a cleanup-style template not an infobox\u2014something to tag an article with when it feels like it's skirting the \"mathematician's intuition\" definition. \u2014Ben FrantzDale 02:40, 12 September 2006 (UTC)\nOh, I see what you mean now. Well, I'm not sure that changes my opinion much. I'm rather new here myself so I don't know much about templates; nevertheless, I suspect there's already a common template to indicate that an article needs more explanation or clarification. I'll leave that to more experienced editors to decide. I still agree, of course, that any \"mathematical intuition\" should be explained in the article (but not in the definition). VectorPosse 04:44, 12 September 2006 (UTC)\nI'm not sure this is really necessary. Mathematical objects can have many properties, and one of them is not necessarily more important than others. We have a whole article to explain these properties and what is useful\/interesting about them, and the intro should summarise the article. JPD (talk) 08:08, 12 September 2006 (UTC)\nI agree that what does it mean? is really context-dependent. We would probably not say that Rn means \"cofunctor of an abelian variety\". A symmetric matrix may appear in several contexts without reference to spectral properties. pom 15:06, 12 September 2006 (UTC)\nA distance matrix is symmetric. This is an easily understood elementary property. Few mathematicians will think: \"Oh, I know what that means. It has an orthonormal basis of eigenvectors!' \u00a0--LambiamTalk 15:11, 12 September 2006 (UTC)\nGood point, and good example. I assume the eigenvector symmetry property isn't interesting in that case because the matrix isn't used as a transformation. For a distance matrix, it seams that symmetry is a trivial and not-too-interesting fact. The distance matrix page could do with some \"what does it mean\" love itself, actually; it says what one is and the fields in which they are used but not how they are used.\u2014Ben FrantzDale 18:09, 12 September 2006 (UTC)\nSymmetry isn't trivial or uninteresting in this case: it's one of the three key axioms defining a metric. \u2014David Eppstein 21:28, 12 September 2006 (UTC)\nI've been bold and added a Mathematical intuition project sub-page to try to address this issue. \u2014Ben FrantzDale 18:14, 12 September 2006 (UTC)\n\nI created extension (mathematics) as a new disambiguation page with more than 30 entries. I think it ought to get organized into sections and subsections. Could Wikipedia's many mathematicians please help? Michael Hardy 21:31, 12 September 2006 (UTC)\n\nI put them into some vague sections, people should feel free to subdivide further. Of course, most of these are algebra. -- Deville (Talk) 22:02, 12 September 2006 (UTC)\n\nDoes anyone else think it's a little weird that Extension problem is strictly about group extensions, while the stub Group extension mentions fields and other algebraic structures? Michael Kinyon 18:25, 15 September 2006 (UTC)\n\nYeah, I thought it was weird, so I changed Group extension to mention only groups and added a link at the bottom to Ring extension. This is a stub that could be greatly expanded. The article Extension problem actually has a lot of the material I would put in Group extension if it were up to me. Ah, if I only had the time... VectorPosse 19:03, 15 September 2006 (UTC)\nWhat problems would result from just switching the names around? Michael Kinyon 20:07, 15 September 2006 (UTC)\nI like it! If we did that, we would need to restore the few words I removed (probably with some editing), but I think this is a great idea. The page Extension problem ought to be a small-ish, more general page about any kind of extension problem. Then its links direct readers to the particulars of specific kinds of extensions. There is something in the page's discussion about calling it Extension (algebra) (which currently redirects to Group extension) and I think that would be necessary for this solution. Otherwise, one would have to include material on extension problems in all fields and that would be the same list that started this thread to begin with. VectorPosse 23:23, 15 September 2006 (UTC)\nIt seems fine to me. I am going on a Wikibreak for a bit more than a week starting tonight, and you have thought in more detail about what would be needed than I have. So my \"vote\" is: go for it! Anyone else have any thoughts about this? Michael Kinyon 03:47, 16 September 2006 (UTC)\n\n## Discussion at Euclidean space\n\nThere is a discussion occurring at Euclidean space concerning how best to write the introduction to be more accessible (see: Talk:Euclidean space#Obnoxious article and following). Interested parties may wish to join the discussion. Paul August 23:23, 12 September 2006 (UTC)\n\n## Peer review: Boy's surface\n\nBoy's surface (talk) is up for peer review. Please offer any insights (there, not here).\u2014msh210 21:36, 13 September 2006 (UTC)\n\nMartingale paradox has been put up for deletion: Wikipedia: Articles for deletion\/Martingale paradox. The author has spent a lot of effort on Usenet at promoting this material, e.g. [6] (see User:AntiochCollege for suspiciously similar material). --C S (Talk) 00:21, 15 September 2006 (UTC)\n\n## What happened\u00a0?\n\nI created a page for Pierre Rosenstiehl yesterday. It just disappeared today (even the traces of the changes I made). I am sure to have saved it after editing and the page is still in my watchlist... If it has been deleted, it would have been fair to post some message on my talk page. Otherwise, what did happen? pom 10:26, 15 September 2006 (UTC)\n\nHere's the entry from the deletion log:\nGo complain. --KSmrqT 12:13, 15 September 2006 (UTC)\nI put a message on Gustafson's Talk page asking him to consider restoring it and, if he still thinks Rosenstiehl is non-notable, putting the article up for AfD so that the rest of us can have some input. Michael Kinyon 12:42, 15 September 2006 (UTC)\nSpeedy deletion under A7: unremarkable people or groups\/vanity pages. An article about a real person, group of people, band, or club that does not assert the importance or significance of its subject. If the assertion is disputed or controversial, it should be taken to AfD instead. I think that was wrongly applied. Charles Matthews 13:31, 15 September 2006 (UTC)\nI think the \"proper\" method would be to take it to DRV. Or you could just recreate it with a {{hangon}} tag. But asking the deleting admin for reconsideration is always in order.\nThe page came back and I put a {{hangon}} tag. Actually, I am not sure it should be kept as I am not aware of the threshold considered by Wikipedia for notability. Whatever decision is taken does not care too much. However, deletion \/ restoration without a slightest explanation from an admin is an attitude which does not encourage editing at all. pom 16:05, 15 September 2006 (UTC)\nNotability is well known to be a difficult concept to apply in practice. A better question: who would consult Wikipedia as a reference about a given person (excluding family, friends, colleagues)? For a member of Oulipo, it is easy to see that many people might look here. It is an argument you could all there-are-no-minor-poets: of course almost all poets are 'minor', as almost all mathematicians fail to be 'major'. But if someone likes a poem and has only a name, then, yes, they might use a reference work to discover more. Charles Matthews 21:44, 15 September 2006 (UTC)\nOk, but from a practical point a view, what should I do if I want to start to write pages on living combinatorists? Should I consider there is limit on the number of bigraphies and that I should prioritize the additions. If so, what would be the order of magnitude of this limit? pom 22:34, 15 September 2006 (UTC)\nMr. Gustafson pulled the trigger on the article (and perhaps should have known better), but an anonymous user User:151.200.246.168 was the one who tagged the article for speedy deletion in the first place. In the span of just over two hours, they tagged 18 articles for speedy deletion. It wasn't quite vandalism; many of the articles were marginal at best, but didn't quite seem like candidates for speedy deletion either. Weird. Lunch 17:18, 15 September 2006 (UTC)\nWeird, indeed. In good faith, perhaps it is just someone who doesn't understand the speedy deletion criteria. In any case, I think this WikiProject can congratulate itself on how this was handled. (But will our backs hurt from patting them so hard?) Michael Kinyon 18:15, 15 September 2006 (UTC)\n\n## Etymology\n\nSome unusual updates have been made to the etymology at pentagon (disambiguation), heptagon and polygon. I'm no expert, but I never heard that these terms had a Sanskrit origin before, so I am rather doubtful about the accuracy of these updates. Any comments\u00a0? Gandalf61 10:31, 15 September 2006 (UTC)\n\nHere are some etymologies from the OED:\npentagon In A, ad. L. pentagon-us, a. Gr. pentagwn-oj pentagonal, five-cornered, f. penta- penta- + -gwn-oj from stem of gwnia angle. In B, ad. L. pentagon-um, Gr. pentagwnon, the neuter adj. used as sb. Cf. Fr. pentagone sb. (13th c. in Littr\u00e9), whence the Eng; form in -gone.\npenta- penta, before a vowel pent-, a. Gr. penta-, combining form of pente five, occurring in many words in Greek as a variant of the earlier pente-, and forming the initial element in various modern technical words adopted from Greek, or formed from Greek elements or on Greek analogies.\nI'm not convinced that those articles need any etymologies, much less ones that seem to have little support in standard references. It may be possible that the words came to Greek from Sanskrit, but without any documentation I think it is better to just remove the anonymous edits instead of correcting or expanding them. CMummert 10:49, 15 September 2006 (UTC)\nThe Greek did not \"come from\" Sanskrit any more than the Sanskrit came from Greek. I've removed these changes. --LambiamTalk 17:19, 15 September 2006 (UTC)\nI asked on wikitionary and got\nEr \u2013 no, it's wrong. All these related \u2018shape\u2019 nouns are from Greek. The Greek suffix was -\u03b3\u03c9\u03bd\u03bf\u03c2, literally \u2018angled\u2019, and in this case combined with \u03c0\u03b5\u03bd\u03c4\u03b1-, from \u03c0\u03ad\u03bd\u03c4\u03b5 \u2018five\u2019. The Sanskrit forms are cognate (i.e. both Sanskrit and Greek are descendants of Proto-Indo-European *penk\u02b7e \u2018five\u2019), but Sanskrit is not the immediate source of the English word. Very few words in English come from Sanskrit. -- Widsith\nSo now we know. --Salix alba (talk) 17:44, 15 September 2006 (UTC)\nThere was a habit of calling Proto-Indo-European \"Sanskrit\" a century ago, before the decipherment of Hittite and the present understanding of IE vowels. It should be suppressed where found. Septentrionalis 18:48, 15 September 2006 (UTC)\n\n## Polar coordinate system\n\nHi everyone! An article that I've been working on quite a bit, Polar coordinate system, has just become a good article. We've requested a peer review to find out how it can be improved to featured article status, and it's great so far. Any other comments would be appreciated. Thanks. \u2014 14:20, 16 September 2006 (UTC)\n\n## Subcategory for geometric graph theory?\n\nI've been working on a few pages lately that have the flavor of geometric graph theory \u2014 that is, about graphs that are either embedded in a geometric space themselves, or that arise from configurations in a geometric space. I'm wondering whether it would be appropriate to make a new category for them, as a subcategory of both geometry and graph theory.\n\nEvidence that organizing things this way is not just my own hobby horse: Pach's edited collection Towards a Theory of Geometric Graphs (to which I contributed a paper on geometric thickness, a subject that would fit here as well but one that I think someone else should add if it deserves adding).\n\nAnyway, this seems a widespread enough change that I felt I should open up the question for debate here rather than just going ahead and doing it. So, does anyone have an opinion on this possible reorganization? \u2014David Eppstein 21:30, 17 September 2006 (UTC)\n\nCategory:Geometric graph theory sounds good to me. --Salix alba (talk) 21:14, 17 September 2006 (UTC)\nI don't know if it will be so easy to make the distinction between Geometric Graph Theory and Topological Graph Theory. For instance: the usual crossing number is of topological nature, while the rectilinear crossing number is of geometric nature. Don't you think it could be better to (at least temporarily) merge the two in a Topological and Geometric Graph Theory subcategory? Of course, there are purely topological or geometric results (rotation system \/ Erd\u0151s\u2013Szekeres theorem) but most have several aspects. Graph drawings may rely on spectral analysis or poset related properties (like track drawing). The classification of theoretical results may also be problematic (e.g.: Schnyder's theorem is about planarity, poset dimension, decompositions into particular forests, and induce a straight line drawing on a linear grid). All of this does not mean I am against subcategories, but rather that I am afraid by the number of topics which will cross the boundaries of categories. pom 21:59, 17 September 2006 (UTC)\nTo me the distinction seems clear enough: topological graph theory concerns graphs embedded on 2-manifolds such as the Euclidean plane, with vertices as points and edges as curves, while geometric graph theory either considers similar type embeddings with edges as straight line segments or other restricted geometric curves (polygonal paths with few bends, or circular arcs, though I doubt there is much already in WP that mentions these), or graphs coming from other geometric constructions (intersection graphs, visibility graphs, arrangements, etc). But of course there is overlap between the two; fortunately WP allows entries to have multiple categories. Perhaps I shouldn't have included Crossing number (graph theory) above since it's about the topological version of the problem; it's a long article so it might make sense to have a separate article on Rectilinear crossing number or Geometric crossing number (two different names for the same thing). F\u00e1ry's theorem seems like a good example of an article that overlaps both categories since it states that a topological graph has a stricter geometric representation; Scheinerman's conjecture is also of that type. \u2014David Eppstein 03:20, 18 September 2006 (UTC)\nYou are fully right. pom 05:41, 18 September 2006 (UTC)\n\n## Can we put the Leonhard Euler FAC nomination on the project page?\n\nLeonhard Euler is nominated for Featured Article status. I know that the nominator of the article has already posted this 10 days ago on this talk page but I think it would also be worth putting the info more prominently on the welcome page of the project. There's not that much work left to do on it to push it up to the desired quality and it's clearly a goal that should be among the project's priorities. Pascal.Tesson 23:44, 17 September 2006 (UTC)\n\nIn related news I've put Ackermann function on FA review. I think it lacks in laymans explination and is not up to current FA standards. --Salix alba (talk) 00:03, 18 September 2006 (UTC)\nSpeaking of which the primitive recursive article is also in a very sad state. But I digress. Pascal.Tesson 06:17, 18 September 2006 (UTC)\n\n## the apes are in question\n\nI just contributed here calculating something. It would be nice if someone could verify what i wrote, because it seems the article contains a mistake. Nerdi 17:50, 18 September 2006 (UTC)\n\nI was going to move the link to the musical ensemble The Exponents from list of exponential topics to exponent (disambiguation) and add this (using the \"dablink\" template, since the various \"otheruses\" and \"alternateuses\" templates are an odious and execrable abomination abhored by all good people):\n\nBut the latter page does not exist. This caused me to notice that the list of exponential topics is quite incomplete as a list of Wikipedia articles already existing that belong there. Here's what needs to be done:\n\n\u2022 Enter \"exponent\" in the search bar and click \"search\", not \"go\".\n\u2022 Add to the list of exponential topics the mathematics articles that belong there.\n\u2022 Add to a new exponent (disambiguation) page the many \"exponent\" topics on non-mathematics topics, and also add the list of exponential topics to that page after a few of the most prominent mathematical senses of the word, with a note saying the list is fairly long.\n\nI'll be back later to participate in this, but maybe not till tomorrow. Michael Hardy 21:20, 18 September 2006 (UTC)\n\n## Ackermann function\n\nAckermann function is up for a featured article review. Detailed concerns may be found here. Please leave your comments and help us address and maintain this article's featured quality. Sandy 15:46, 19 September 2006 (UTC)\n\n## 256^(4.7*10^9) on prod\n\nIt's not in any math categories, so it won't show up on current activity; listing here. --Trovatore 21:15, 19 September 2006 (UTC)\n\nand current activity hasn't updated for a week; is something wrong? Septentrionalis 23:25, 19 September 2006 (UTC)\n\n## Tagging talk pages and assessing articles\n\nWikipedia Assessments within AWB. Click on the image to see it in better resolution\n\nHi. If you still have work to do tagging talk pages and assessing articles, my AWB plugin might be of interest to you.\n\nThe plugin has two main modes of operation:\n\n\u2022 Tagging talk pages, great for high-speed tagging\n\u2022 Assessments mode, for reviewing articles (pictured)\n\nAs of the current version, WikiProjects with simple \"generic\" templates are supported by the plugin without the need for any special programatic support by me. I've had a look at your project's template and you seem to qualify.\n\nHope that helps. If you have any questions or find any bugs please let me know on the plugin's talk page. --Kingboyk 14:01, 20 September 2006 (UTC)\n\n## Manifold Destiny (article)\n\nIt has been suggested to me that this page, dealing with a controversial New Yorker dirt-digging story about Perelman, needs semi-protection. I can't quite see that it fits the guidelines at Wikipedia:Semi-protection policy, although there have been some anons making edits there that could get WP into legal trouble. In any case this page is potentially something very troublesome. Charles Matthews 21:28, 21 September 2006 (UTC)\n\nI think you've been mislead by Lubos Motl's comment on your talk page [7]. Look through the history of the article. In particular, look at all the anon edits. I don't see what is trouble some about them; the worst I can see is a new user (not anon) that added an unsourced statement that Tian had never spoken to the New Yorker, but it was later removed by an anon.\nI think given that this article exists, the edits that have been made by new or anonymous users thus far, are in fact of decent quality, certainly better than some by registered users! So I don't see any valid reason someone could want the page semi-protected.\nWhether this page should exist is another issue. I didn't used to think so, but given the media coverage, it seems to me that this article is sufficiently notable. Some may not like what is going on, or that dirty underwear is being aired, but this kind of thing is par for the course on many topics. The mathematical community does not have any special protection on Wikipedia against this kind of stuff and shouldn't. Sure, the article could be potentially troublesome, but that is true of many controversial articles. We should deal with it like any other. Keep an eye on it and make sure people don't turn it into a version of their blog. --C S (Talk) 10:56, 22 September 2006 (UTC)\n\nThanks for filling me in. As I said, after I had been asked my conclusion was not to semi-protect. As you say, watching should be enough for the present. Charles Matthews 11:08, 22 September 2006 (UTC)\n\n## Order 3 groups are cyclic proof\n\nOrder 3 groups are cyclic proof is up for deletion. Chime in at the appropriate spot. Michael Kinyon 00:28, 23 September 2006 (UTC)\n\n## Gleason's theorem\n\nThis page was tagged as needing attention. It was a stub which simply stated the theorem in question. I have expanded it quite a bit, and removed the tag; I hope that my edits were sufficient to do so! My expansion has centred mainly on the application of the theorem, in quantum mechanics and the philosophy thereof. On the talk page, someone suggested sketching an outline of the proof of the theorem, which could be a worthwhile addition at some stage, but since most references to the theorem are centred on its uses and implications, this is probably not necessary at the moment (the proof is also hideously long and complicated, and not easy to summarise for an encyclopaedia article). Anyway, I wanted to find out the following. Now that the tag is gone, does your magical bot remove the page from the \"needing attention\" lists your project maintains? Or should I do that manually? I didn't want to just go ahead and do it, in case it interferes with the bot somehow...do let me know! Byrgenwulf 18:06, 23 September 2006 (UTC)\n\nthanks! the article certainly no longer is a stub (and the \"expert needed\" tag was probably misplaced). and don't worry, the 'bot will eventually pick up on the tags. Lunch 05:12, 24 September 2006 (UTC)\nMaybe this should be raised on the article's talk page, but I don't get the bit about P(y) being 1 for every lattice point y. Is 0 not also a lattice point? Doesn't this require y to be the sum of n (instead of any r) orthogonal atoms? And if true, isn't \"the probability is fixed\" a weak way of saying: the event is almost sure? --LambiamTalk 07:36, 24 September 2006 (UTC)\nThat's a typo...the \"=1\" shouldn't have been there (it's gone now). So: any y can be expressed as the union of some (not necessarily n) number of orthogonal atoms xi. The probability P(y) is the sum of the probabilities P(xi) (all r of them). \"The probability is fixed\" simply means \"uniquely determined\" in this context. Byrgenwulf 10:19, 24 September 2006 (UTC)\nRegarding the bot, there is some problem with the computer on which it runs. I'm on the other side of the planet and can't reach the computer remotely. I should be able to bring it up next week after I return to my office. Sorry about the problems (but it is nice to see that people are noticing). -- Jitse Niesen (talk) 14:45, 27 September 2006 (UTC)\n\n## Ear curve\n\nEar curve is up for deletion. Opine at its AfD page. Michael Kinyon 11:22, 24 September 2006 (UTC)\n\n## number needs attention\n\nThe section on real numbers is quite weak and maybe even misleading. Michael Hardy 02:15, 25 September 2006 (UTC)\n\nI do agree that it is weak. Did you have something particular in mind when you say it's misleading? VectorPosse 02:39, 25 September 2006 (UTC)\nI don't know what Michael Hardy had in mind; but the real numbers section conveys the impression that some of them (e.g., 0.1010010001...) are not constants. That is definitely misleading. JoergenB 13:31, 25 September 2006 (UTC)\nWhile we're at it, the \"Infinitary extensions\" subsection is very misleading, and the \"Transcendental numbers and reals\" subsection is worth a look (the first paragraph does not deal with transcendental numbers at all). -- Meni Rosenfeld (talk) 10:49, 25 September 2006 (UTC)\n\nIt was less misleading after my edit, just before I posted this comment. It was written so as to make it appear that a real number is by definition a decimal expansion. I suppose in some ways that's no worse than saying a real number is a Dedekind cut, or that it is an equivalence class of Cauchy sequences, or any of various other members of that same isomorphism class, but the prevalence of popular errors about the definition of rational and irrational numbers (thinking that those concepts are defined in terms of decimal expansions) makes me cringe at that way of introducing the idea. Michael Hardy 19:47, 25 September 2006 (UTC)\n\nTwo remarks:\n\u2022 the \"needs attention\" note is still in. It would help if you could copy-and-paste your detailed explanation to the talk page, so people can have a shot at fixing it.\n\u2022 defining real numbers as (equivalence classes) of decimal expansions, and rational numbers by properties of such expansions, is correct, if awkward.\n\u2022 I'm not aware of a definition of real numbers that's better than the one via decimal representations, but still has some connection with non-mathematical culture, and can easily be grasped by non-mathematicians. Dedekind cuts and Cauchy sequences, I'm afraid, are right out for that.\nRandomP 20:03, 25 September 2006 (UTC)\nSaying real numbers correspond to points on a continuous line certainly can be grasped by non-mathematicians. Michael Hardy 20:06, 26 September 2006 (UTC)\nFor most people, formally defining any kind of number is a strange ritual. Does a definition of positive integers in terms of sets or successors connect with non-mathematical culture? Mathematics itself was very slow to make numbers formal objects. But in terms of historical development, there is evidence that the Dedekind cut idea is the earliest of the four major approaches. (These are: cuts, decimals, sequences, field axioms.) Cuts are also technically simple, while decimals are a beast. However, the number article should mostly leave the formalities to specialized articles, and concentrate on the big picture, which is that real numbers \u2014 however defined \u2014 \"complete\" (fill in the gaps of) the line (rationals). Concretely, what's the diagonal of a unit square? What's the area of a unit circle? --KSmrqT 14:54, 26 September 2006 (UTC)\nThis may be OR, but I've found that a good way to explain real numbers is by a sequence of shrinking intervals \u2013 possibly of zero width, although that's not essential \u2013 [Li, Ri] with Li \u2264 Li+1 and Ri \u2265 Ri+1, and Ri \u2212 Li \u2192 0. The claim is that this determines a unique real number x that is contained in all intervals: Li \u2264 x \u2264 Ri. It is easy (for us) to see that this induces a Cauchy sequence as well as a Dedekind cut. There is no need to require the Li and Ri to be rationals when explaining the idea. The point is, rather, to formalize the notion that \"there are no gaps\", a closure property. I've found that for psychological reasons I can't explain the notion of an interval shrinking \"in the limit\" to zero is easier to grasp than limit in general, even for a monotonic sequence. --LambiamTalk 22:13, 26 September 2006 (UTC)\nYou and your students might appreciate Archimedes' proof that the area of a circle is the same as that of a right triangle with base equal to the circumference and height equal to the radius, found in \"Measurement of a circle\". (See Heath's translation, ISBN 978-0-486-42084-4.)\nWe want to be careful about the distinction between conveying intuition, which \"number\" should do, and establishing a workable definition, which \"real number\" should do. Working from the definition alone we need to be able to do arithmetic, comparisons, and proofs. That's one reason why Dedekind cuts are more appealing than decimal expansions for formal work. Compare with the modern definition of \"compact space\" in topology, where the \"finite subcover\" idea is less intuitive but more effective than \"closed and bounded\".\nBack to your original point: Mental models are important for teaching; they are also important for functioning in the real world, a theme that artificial intelligence research has explored under the names \"naive physics\" or \"qualitative reasoning\". (See Smith and AAAI for sample reading.) --KSmrqT 15:56, 27 September 2006 (UTC)\n\n## In which subject areas is the term basis function used?\n\nThere seems to be disagreement over whether the term basis function is used in functional analysis. I don't know enough about the subject to have an opinion. Could someone comment at Talk:Basis function? --Jtir 13:04, 25 September 2006 (UTC)\n\nThere is a problem with the weakness of the article. There must be several areas, eg wavelets, where this is a relevant concept. Charles Matthews 13:12, 25 September 2006 (UTC)\nCorrect. I looked at the what links here list and found wavelets, plus articles in chemistry, physics, engineering, and business that link to Basis function (I've put a culled, classified, and alphabetized list of linked articles at Talk:Basis_function). A wikipedia search finds many other examples of the term being used. It is starting to seem to me that making the article a dab would be preferable to trying explain all possible uses of the term. I don't have enough WP experience, though, to know what the implications are. --Jtir 21:26, 25 September 2006 (UTC)\nA dab page makes mainly sense if we have separate articles on different meanings of the words \"basis function\". In mathematical use, isn't there a common meaning: an element of some basis of a vector space whose elements are functions? The main problem of the article may be that it starts with the words \"In functional analysis\" instead of \"In mathematics\". --LambiamTalk 22:38, 25 September 2006 (UTC)\nan intro sentence might be, \"In mathematics -- particularly analysis -- a basis function is an element of the basis for a function space. The use of the term is analogous to basis vector for a vector space.\" (NB: some of those words are dab pages so the links are Analysis (mathematics) and Basis (linear algebra).) Lunch 00:56, 26 September 2006 (UTC)\nWith this formulation, couldn't all the technical content of the article be removed? Basically the article is saying that basis function is a synonym for basis vector in some usages. If so, the article could become a redirect to basis (?) which could parenthetically note the same thing. --Jtir 16:10, 26 September 2006 (UTC)\n\nI don't think a simple redirect to Basis is a good idea. When dealing with bases in function spaces, a Hamel basis (which is what that page focuses on) is usually not the tool of choice. Instead one typically deals with a Schauder basis or, in the more specific Hilbert space setting, an orthonormal basis. Sometimes the word is stretched a bit, such as in the context of Riesz basis (which I think is really just a frame). Michael Kinyon 16:20, 26 September 2006 (UTC)\n\n(I'm gonna CC the conversation here to the basis function talk page. There's some good stuff here that hasn't been mentioned there, and vice versa (along with some repeats). Come on over and join in!) Lunch 19:04, 26 September 2006 (UTC)\n\n## Should \"Recursively presented group\" redirect to \"Presentation of a group\"\n\nBoth finite and recursively presented groups are defined on the page \"Presentation of a group\". At present \"Finitely presented group\" redirects to \"Presentation of a group\" but \"Recursively presented group\" is just a fairly minimal stub. It would make more sense to me if it too redirected to \"Presentation of a group\". Bernard Hurley 20:38, 25 September 2006 (UTC)\n\nYes, it would. I think I created the article, and wasn't aware of that (probably because I thinko'd and created it under the wrong title - sorry, I was just upset we didn't have those articles when rereading Rotman).\nRandomP 20:53, 25 September 2006 (UTC)\nWell, merge and redirect. Charles Matthews 09:36, 26 September 2006 (UTC)\n\n## 'Determinants' is a featured article on the French Wikipedia\n\nAre we allowed to steal from the other language Wikipedias? See [10] for a rather nifty treatment of Determinants. It's 111K (vs the 55K of our own English article) and has some nice color illustrations. The language used is not 100% familiar to someone whose linear algebra is several years in the past, but maybe this is the latest thing.\n\nHere are the opening sentences:\n\n\"First introduced in algebra to determine the number of solutions of a system of linear equations, the determinant reveals itself as a very powerful tool in numerous domains (study of endomorphisms, search for eigenvalues, differential calculus). It is in this manner that we define the determinant of a system of equations, the determinant of an endomorphism or the determinant of a system of vectors.\n\"For many operations, the determinant can be defined by a collection of properties (axioms) that we summarize by the term \"alternating n-linear form\". This definition allows us to make a complete theoretical study and to enlarge further its field of application. But the determinant may also be conceived as a generalization to n-dimensional space of the notion of oriented surfaces and volume. This aspect, often neglected, is a practical and illuminating approach to the properties of the determinant.\" EdJohnston 23:59, 25 September 2006 (UTC)\nI believe that the other language wikipedias use the same license which we have here. So you can use their content freely provide that you give them credit for it and offer it to others under the same condition. In other words, go ahead and copy any of their text and translate it into English. But make sure that you attribute it to them in your edit summary -- specify that it was the French wikipedia and name the article, so that anyone can look in their revision history to see who put the material into it in the first place. JRSpriggs 05:50, 26 September 2006 (UTC)\nYes, you can translate and use here freely. Charles Matthews 09:38, 26 September 2006 (UTC)\n\nActually, translation is not just permitted (and as far as I can see often not accompanied by credits), but encouraged. Read e.g. Wikipedia:Translations into English. JoergenB 10:13, 26 September 2006 (UTC)\n\nYou ought to give credit, though. Anything else is risky under the GFDL. Remember that the original authors still hold copyright, even though they've licensed it to you. If you don't comply with the terms of the license (which requires attribution) you could be infringing. --Trovatore 06:43, 27 September 2006 (UTC)\nWell, this sounds reasonable; and 'there are some nice templates', which make it very easy to inform the reader of sources from sister Wikipedia, and which you may place under the heading references. It might be a good idea to use them whenever material is translated, which is seemingly not done now. Not only the determinants article lack such information. JoergenB 13:51, 29 September 2006 (UTC)\nRegretfully, I'll have to qualify the statement there are some nice templates. After having been around at the template pages a little, getting more and more confused, but finally finding some adequate explanation, I'l have to rephrase it there are two nice templates (namely Template:German and Template:Polnish). I accidently started by looking at a list of recently translated articles from German to English, and then assumed that I knew the pattern... However, there may be other such templates without proper categorisation (and of course they should not be too hard to create, I suppose, if we want to encourage translators to give more credit).\nThat discussion perhaps should move to another page, though. JoergenB 16:43, 29 September 2006 (UTC)\n\n## Something has gone wrong with LaTeX interpreter\n\nBeing realtively new to wikipedia I'm not sure where to post this so it's going here. Something has gone wrog with the LaTeX interpreter on wikipedia so that maths pages are full of lots of raw LaTeX. Bernard Hurley 23:27, 26 September 2006 (UTC)\n\nIt seems that the server of formula PNGs (http:\/\/upload.wikimedia.org\/) is unreachable. As a consequence, PNG formulas only appear in their HTML version. pom 23:52, 26 September 2006 (UTC)\nSome images also seem to be broken. I suspect this is a tempory problem which will be fixed in a few hours. Its happened before. --Salix alba (talk) 00:29, 27 September 2006 (UTC)\nHave you tried control-shift-R? For several days now, I have occassionally been seeing the formulas unconverted. But they always become correct after control-shift-R. JRSpriggs 06:13, 27 September 2006 (UTC)\nThat is curious because the LaTeX interpreter is on the server. I can't test this because the problem seems to have gone away, but thanks for the suggestion. Bernard Hurley 08:29, 27 September 2006 (UTC)\nIf you visted the page while the PNGs are not accesible, the HTML version could be stored in the cache and so appear even after the problem is gone. Purging the cache when everything is working again would fix this problem. JPD (talk) 10:31, 27 September 2006 (UTC)\n\n## Spurious dashes\n\nHmm. Gleason's theorem, at least, still has issues with spurious dashes, though. Does this happen to anyone else? RandomP 02:11, 27 September 2006 (UTC)\n\nYes, I noticed it an hour or so ago in Character theory. Michael Kinyon 06:37, 27 September 2006 (UTC)\n\nThis is a bug in the LaTeX interpreter on wikipedia. The LaTeX interpreter seems to add a dash to the end of formulas containing certain letters and ending in certain other characters. So in the following paragraph \"B(m,n)\" gets an extra dash:\n\nLet ${\\displaystyle n,m\\in \\mathbb {Z} }$ where ${\\displaystyle m}$ is odd, ${\\displaystyle n>10^{78}}$ and ${\\displaystyle m>1}$, and let ${\\displaystyle B(m,n)}$ be the free m-generator Burnside group, then every non-cyclic subgroup of ${\\displaystyle B(m,n)}$ is SQ-universal in the class of groups of exponent ${\\displaystyle n}$.\n\nIf I change it to \"B(x,y)\" it is OK:\n\nLet ${\\displaystyle x,y\\in \\mathbb {Z} }$ where ${\\displaystyle x}$ is odd, ${\\displaystyle y>10^{78}}$ and ${\\displaystyle x>1}$, and let ${\\displaystyle B(x,y)}$ be the free m-generator Burnside group, then every non-cyclic subgroup of ${\\displaystyle B(x,y)}$ is SQ-universal in the class of groups of exponent ${\\displaystyle y}$.\n\nA fix seems to be to add a LaTeX space at the end of the formula but in this case the formula gets displayed with a larger font! So using \"B(m,n)\\ \" we get:\n\nLet ${\\displaystyle n,m\\in \\mathbb {Z} }$ where ${\\displaystyle m}$ is odd, ${\\displaystyle n>10^{78}}$ and ${\\displaystyle m>1}$, and let ${\\displaystyle B(m,n)\\ }$ be the free m-generator Burnside group, then every non-cyclic subgroup of ${\\displaystyle B(m,n)\\ }$ is SQ-universal in the class of groups of exponent ${\\displaystyle n}$.\n\nIncidentally you can get the larger font by including a LaTeX space so:\n\n\u2022 \"a\" gets interpreted as ${\\displaystyle a}$\n\u2022 \"a\\ \" gets interpreted as ${\\displaystyle a\\ }$\n\nBernard Hurley 08:23, 27 September 2006 (UTC)\n\nAnd a thinner space by using \\,: \"[$a\\$]\" gives \"[${\\displaystyle a\\ }$]\", while \"[$a\\,$]\" gives \"[${\\displaystyle a\\,}$]\". --LambiamTalk 10:27, 27 September 2006 (UTC)\nThere is a bugzilla bug on this. Vote for it to encourage a quick fix. I'd recommend against short term fixes in articles. --Salix alba (talk) 08:27, 27 September 2006 (UTC)\nGuys it's totally about the caching. When wikipedia sees an equation it's seen before it re-uses the old image. Sometimes they change the image renderer so you get a version from an old renderer. e.g.: ${\\displaystyle B(m,n)}$ ${\\displaystyle B(x,y)}$ ${\\displaystyle B(asixhux,sdkcjnjzz)}$. So it seems ${\\displaystyle B(m,n)}$ is from the old renderer. ${\\displaystyle B(mmmnn,nnn)}$ Here's another example: ${\\displaystyle \\oint }$. Oh look it's broken. That one's cached. But with the new renderer: ${\\displaystyle Babcasisxajnsx\\oint }$. I think that last one got fixed when they switched over to dvipng. When you do experiements like this you should always insert random text to trick the caching. 'course I could be wrong about all this, it's just a theory. Dmharvey 12:45, 27 September 2006 (UTC)\nThat seems to make sense. It would be nice if there were some mechanism to force the re-caching of a formula, but I suppose that could be open to abuse, it would also break any pages that rely on an incorrect old rendition. Bernard Hurley 13:02, 27 September 2006 (UTC)\nI did a checkout on phase3 (is this the correct tree?) and was able to reproduce the bug with a fresh mw installation. I found a problem in render.ml, and after fixing it, the problem went away (it was necessary to clear the math table, of course). However, this can't explain why the bug does nolonger occur for new formulas. For details, see bugzilla.--gwaihir 16:25, 27 September 2006 (UTC)\nGood stuff. Have you tried running with the preference set to MathML if possible. This has the effect of rendering simple maths as html and for these I'm getting the same problem without a image being used anywhere, so $B(m, n)$ produces the html <span class=\"texhtml\"><i>B<\/i>(<i>m<\/i>,<i>n<\/i>)-<\/span>. --Salix alba (talk) 17:48, 27 September 2006 (UTC)\nAh the cache issue explaines the difference of apprearance, in some equations\n${\\displaystyle a_{n}x^{n}+a_{n-1}x^{n-1}+\\cdots +a_{2}x^{2}+a_{1}x+a_{0}}$.\n${\\displaystyle ((\\ldots (a_{n}x+a_{n-1})x+...+a_{2})x+a_{1})x+a_{0}\\,}$.\nto me the first looks good, but the letters in the second seem rather blury. I guess the first is cached using an old renderer, but the second is generated using a new renderer. --Salix alba (talk) 08:43, 29 September 2006 (UTC)\n\n## Good articles and inline cites\n\nOn Wikipedia talk:Good article candidates they have been reworking the criteria, which now currently require the use of inline cites. This resulted in all 11 of our mathematic GA receiving a message warning that the articles may be up for review. Lots of other articles also received the same messages and the physists especially have visiforously protested against the change. Theres now an atempt to reach a consensus on the issue. This particularly affect maths articles as we tend not to use inline cites for the main mathematical content, in Wikipedia:Featured article review\/Eigenvalue, eigenvector and eigenspace use of manitory inline cites was contested.\n\nPeople might like the add their views at on the issue at Wikipedia talk:Good article candidates. --Salix alba (talk) 18:03, 27 September 2006 (UTC)\n\nThere is also discussion on Wikipedia talk:Citing sources. This is very relevant as the proposed GA standards would make it difficult for math articles to receive GA status. And there is also discussion on Wikipedia talk:WikiProject Physics. CMummert 03:23, 28 September 2006 (UTC)\n\n## Category:Math wars\n\nI nominated this for deletion. The discussion is at Wikipedia:Categories for deletion#Category:Math wars. Comments are welcome. Oleg Alexandrov (talk) 01:51, 28 September 2006 (UTC)\n\n## Intro line to analysis\n\nIn Areas of mathematics, I think it is misleading to say that analysis is primarily related to rates of change. Many aspects to the theory do not arise in this way. I think it would be better to say that analysis is the study of inequalities, because this is the theme that runs through every branch, at least it seems to me. To quote Krantz (from a book review of 'A Companion to Analysis: A Second First and First Second Course in Analysis') \"Analysis is dirty, rotten, hard work. It is estimates and more estimates. And what are those estimates good for? Additional estimates, of course. We do hard estimates of integrals in order to obtain estimates for operators. We obtain estimates for operators in order to say something about estimates for solutions of partial differential equations. And so it goes.\" Any comments? I tried to change it initially myself, but instantly got reverted.\u00a0:) I should have started here I suppose, thanks to Oleg for pointing this out to me. Thenub314 03:29, 28 September 2006 (UTC)\n\nWhatever it is, it should match the Mathematical analysis article. (I personally have no really clear \"intrinsic\" concept of analysis - I know what would be considered analysis amongst the things I know, but if confronted with some mathematics that was totally unknown to me, I might be in doubt as to whether to consider it analysis.\nRight now, the areas of mathematics article claims\n\"Analysis is primarily concerned with change. Rates of change, accumulated change, multiple things changing relative to (or independently of) one another, etc.\"\nwhich appears to me to be based on real analysis in one variable. Mathematical analysis has:\n\"Analysis is a branch of mathematics that depends upon the concepts of limits and convergence.\"\nwhich is what I would consider more appropriate for describing topology. Of course, one approach would be to define analysis historically, as that branch of mathematics that begins with the study of \"nice\" real functions, integration, and differentiation.\nRandomP 03:58, 28 September 2006 (UTC)\nJordan would have probably have considered the \"Analysis is... limits and convergence\" definition to be correct, but at that time topology did not exist as a separate area of study. It's a matter of historical perspective. The contents of undergraduate analysis courses seem to have been fixed for about the last 50 years, but apart from that it would seem quite difficult to define.\nBernard Hurley 09:31, 28 September 2006 (UTC)\nPart of the problem is that \"analysis\" is actually at least two fields: functional analysis and something which I might term \"higher calculus\". The former does often concern itself with limits and convergence, but in function spaces rather than spaces like ${\\displaystyle \\mathbb {R} ^{n}}$. The latter considers individual functions on ${\\displaystyle \\mathbb {R} ^{n}}$ using calculus-like ideas such as the derivative (of course, in many variables). Then there is the mysterious realm of PDE's, which bleeds into differential geometry, while perhaps the entire field is haunted by the ghost of operator theory. Perhaps the best one-sentence summary is that \"Analysis is the field of mathematics which studies functions or spaces of functions using techniques related to the notion of limits and convergence.\" If I wanted another sentence, I would write, \"Although all of its techniques, such as differentiation, integration, Fourier analysis, and so on, have seen vast generalizations (for example, p-adic analysis, generalized measure theory, and harmonic analysis on an arbitrary locally compact topological abelian group), it is over the connected, locally compact, and complete metric spaces ${\\displaystyle \\mathbb {R} ,\\mathbb {C} }$ that they wield the greatest power and demand the most extensive use.\" This sentence disposes of the vast confusion that arises when you try to generalize about \"analysis\", since it is now so big. On the other hand, I'm not an analyist, and it seems that by doing this I might be doing the moral equivalent of saying \"Algebraic geometry, although generalized to work over arbitrary commutative rings and to answer questions of number theory and even algebra itself, is essentially the study of complex algebraic varieties.\" Ryan Reich 13:33, 28 September 2006 (UTC)\nFunny that this came up; a few days ago, a professor of mine remarked \"It's not an exaggeration to say that analysis is the study of estimates\". I think there might be some merit to incorporating that word in the definition. Fredrik Johansson 13:36, 28 September 2006 (UTC)\n\n## History of Analysis article\n\nI am not a historian, so I probably should not really comment, but is the history section under Mathematical Analysis article seems a bit too good to be true. I did some reading up on the MacTutor math history site. It doesn't seem to indicate that calculus was known in india in the 12th-14th century. Is this really true? In terms of verifiability all I found in any of the articles was a link to some physics prof's web site. Does anyone know more? Thenub314 00:30, 30 September 2006 (UTC)\n\n## Citation issues\n\nLately, there have been many discussions of how to cite science and math articles at WP:GA and WP:CITE. In particular, there are editors out there in Wikipedia-land which would like to see every line in Wikipedia-articles cited. That would include, for example, line-by-line citations for mathematical proofs which I think would be ridiculous. There is currently a proposal at WP:CITE to include an important modification to the guidelines that would state that elementary facts should not\/may not be cited. I tried to qualify this with a statement of what things I think (and maybe others think) should be cited in science and math articles and what things should not (and why). Please read, comment, and modify this work here. --ScienceApologist 05:53, 29 September 2006 (UTC)\n\nThere is little point giving citations for 'well-known' facts, anyway. Putting a huge effort into that is not going to solve the issue of references for genuinely recherch\u00e9 facts, which are those for which it is valuable to give pointers. Charles Matthews 07:00, 29 September 2006 (UTC)\nI would go further. Peppering an article with extra citations is harmful, not helpful, for readers.\nExtremists at Wikipedia insist otherwise.\nOne distorting force is the use of inline citations to address the reliability of our articles, which I believe is a fundamental mistake. Editorial debates belong on a talk page, not an article page. A reader should be able to have confidence that the Wikipedia quality control process has done its job, so that they can safely focus on absorbing content from the article.\nExcess citations make it impossible to assess salience. If we cite both for \"1+1=2\" and \"the Riemann hypothesis is true\", a reader has no indication that the former is trival and uncontroversial while the latter would be a major claim. Nor will many editors wish to verify dozens and dozens of such citations, so garbage can easily creep in.\nIf only common sense were more common \u2026 --KSmrqT 17:49, 29 September 2006 (UTC)\nagree with above, c.f. 0.999, where every trivial thing, it seems, is cited. Mct mht 00:45, 30 September 2006 (UTC)\nI encourage people with views like those above to follow and contribute to the discussions at WP:CITE and WP:GA. Discussing this here won't help to convince the editors who recently revised the GA guidelines. \"Consensus\" was reached on the changes because nobody from the sciences was contributing to those discussions. CMummert 17:57, 29 September 2006 (UTC)\n\n## GA status for Addition?\n\nTo my eyes Addition seems to be a good quality article. It might be an idea to put it forward to Wikipedia:Good article candidates, if anyone willing to defend it. BTW it is well cited with both inline and overal cites. --Salix alba (talk) 16:30, 30 September 2006 (UTC)\n\n## Citation guidelines proposal\n\nI know you've been having similar concerns about citations and Good Articles here as we have over at Wikipedia:WikiProject Physics. I have a proposal to deal with this debacle. Let's establish, by consensus within the project, a set of guidelines for referencing physics and mathematics articles in Wikipedia. Then, at least, we will have a set of clear guidelines and an established consensus to refer to if we start having problems with WP:GA and WP:FA. I think if we write a reasonable set of guidelines, which respect WP:V and WP:CITE, we'll get little argument from the vast majority of the people over there.\n\nI have already written a proposal, available here: Wikipedia:WikiProject Physics\/Citation guidelines proposal. It definitely has a whiff of the first draft about it (some sentences seem pretty tortured), but I'm confident we can bang it into something that is clear and concise. I've tried to write the guidelines in such a way that they don't apply just to physics, although the examples are (by necessity) taken from articles I'm familiar with. I'm hoping that we can get the editors from both WikiProjects (physics and mathematics) to form some kind of a consensus for referencing our articles, which would give it increased legitimacy: we could incorporate the guidelines into both our projects.\n\nTo keep the discussion (semi-)unified, please comment at Wikipedia talk:WikiProject Physics or Wikipedia talk:WikiProject Physics\/Citation guidelines proposal. \u2013Joke 16:59, 30 September 2006 (UTC)\n\nI urge the participants here to go over the proposal and help reach concensus. I expect it will carry more weight if it is a joint proposal of two active WikiProjects. \u00a0--LambiamTalk 23:16, 30 September 2006 (UTC)\n\n## History of mathematical notation - peer review\n\nHistory of mathematical notation is seeking peer review. --Salix alba (talk) 19:02, 30 September 2006 (UTC)","date":"2017-08-19 14:26:04","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 38, \"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\": 0.6494812369346619, \"perplexity\": 1309.0056151376884}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"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-2017-34\/segments\/1502886105451.99\/warc\/CC-MAIN-20170819124333-20170819144333-00304.warc.gz\"}"}
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\section{Introduction}
\label{sec:intro}
The deployment of multiple antennas in a wireless communication system offers key advantages to its performance in terms of power gain, channel robustness, diversity, and spatial multiplexing. The use of multiple antennas in rich scattering environments provides effective utilization of the scarcely available spectrum resources. As a result, multiple-input-multiple-output (MIMO) technology has gained much interest in the research community.
Installing extra antennas to a MIMO system can introduce substantial enhancements in both link reliability and data throughput of the system \cite{LTEAdvanced}. Specifically, the use of very large antenna arrays (in the order of multiples of a hundred antennas)
has been found to be beneficial to overcome problems encountered in traditional MIMO settings. Such systems, known as \emph{massive} MIMO or \emph{large-scale} MIMO \cite{scaling_up},\cite{how_many_antennas}, also have the potential to scale down the transmission power because of the use of small active antennas with very low power. Moreover, in massive MIMO systems fast fading is averaged out and intracell interference almost vanishes. Thus, using large antenna arrays can play a key role in exploiting the true potential of traditional MIMO systems while at the same time overcoming several challenges. Massive MIMO is therefore considered as an emerging key technology that can meet the growing demands of current wireless systems. For interested readers, some other advantages of adding more antennas to the base station (BS) have been discussed in \cite{training_MU_MIMO, 6736761, non_cooperative_BS_antennas}.
In order to benefit from the advantages of massive MIMO systems, we need to determine the channel impulse response (CIR) for each transmit-receive link. In a typical massive MIMO system, a BS is equipped with a large antenna array and communicates with several users resulting in a large number of channels that need to be estimated. This results in a substantial increase in complexity which causes performance limitations at the BS.
Obviously, obtaining CIR requires training data (pilots) to be sent by the users. It is known that the number of pilot symbols required is proportional to the total number of users \cite{training_MU_MIMO}. Therefore, as the number of users increases, there is a higher chance that the pilot sequences in the neighboring cells interfere with each other. This pilot contamination problem is a major limiting factor for the massive MIMO systems \cite{5898372, 5947131}. However, pilot contamination could be reduced if the reserved number of pilot tones is reduced. Therefore, in a multi-user scenario there is a need to reduce the number of pilots without affecting the CIR quality. Hence the development of efficient channel estimation techniques for massive MIMO that are computationally less complex and require less number of pilots is a challenge that needs to be thoroughly addressed
Massive MIMO channel estimation is similar to the MIMO channel estimation. Existing literature includes several methods proposed for channel estimation in MIMO systems \cite{medles1, semi_blind_Swamy, semi_blind_MIMO_OFDM,6288608,5670986}. However, it is difficult to directly adopt these approaches for a number of reasons. For example,
\begin{enumerate}
\item There is a need to reduce the number of pilots.
\item All received (thousands of) signals in a massive MIMO system can not be processed efficiently at one central processor. Therefore, there is a need for methods/algorithms which are
\begin{inparaenum}[\itshape a\upshape)]
\item distributed;
\item computationally efficient; and
\item require little communication overhead.
\end{inparaenum}
\item The antenna arrays could spread over a large space making it quite different in its model than a regular compact MIMO receiver.
\end{enumerate}
Recent works have indicated increased interest in the problem of massive MIMO channel estimation (see for example \cite{2014arXiv1401.5703S, 6415397, 5947131, 5898372, 6555020}). Most of these algorithms make use of the channel statistics. However, these statistics are usually not known and therefore some kind of assumption is made about the distribution of channel taps. Moreover, some of the techniques involve computationally expensive operations like inversion of channel covariance matrices which is not reasonable for the massive MIMO scenario.
It is well known that many wireless channels have impulse response that is sparse in the sense that they have very few significant paths. For example, see \cite{387095, barbotin2011estimating, semi_blind_MIMO_OFDM, 771349, wireless_sparse3, 4202180, 4042341, Rappaport2002wireless, 5670986, comp_channel_sensing} and the references therein. We would also like to add that in massive MIMO since a large number of antennas has to be placed usually it is difficult due to several space, structural, aesthetic constraints that antennas are positioned far from each other. Antenna separation is bound to decrease as we increase the number of antennas. This means that for antennas that are close to each other the times of arrival will be similar however the amplitudes and phases of the paths will be different, implying common support. The ideas that we will put forward in this paper take advantage of the above-mentioned two properties.
In this paper, we propose a set of algorithms for channel estimation in massive MIMO. Specifically, we consider a base-station equipped with a large number of antennas serving several single-antenna user-equipments (UE). Our approach makes use of the fact that the wireless channels between a UE and base-station antennas are expected to be sparse and that neighboring antennas observe channels with similar support (i.e., sparsity pattern) but not necessarily the same fading along the active taps. The antennas share information with their neighbors to reach a decision on the most probable support. Decisions are made in a distributed manner with low complexity and communication overhead. In summary, the set of algorithms we propose in this paper has the following distinctive features:
\begin{enumerate}
\item It utilizes the sparsity of the CIR and the fact that channel supports for neighboring antennas are approximately the same.
\item It is Bayesian in nature. It utilizes the sparsity of CIR and acknowledges the Gaussianity of the additive noise but is agnostic to the distribution of the active taps of the CIR.
\item It has a distributed nature requiring limited communication between neighboring antennas. One version of the algorithm requires only integer communication between antennas.
\item It has a data-aided extension that identifies reliable carriers and uses them to further reduce the number of pilots and enhance the CIR estimate.
\end{enumerate}
The distributed Bayesian algorithm we develop in this paper is based on the Support Agnostic Bayesian Matching Pursuit algorithm (SABMP) developed by the authors in \cite{sabmp}.
The remainder of this paper is organized as follows. In Section \ref{sec:sysmodel}, we present the system model and formulate the channel estimation problem. In Section \ref{sec:sabmp}, we present a simple channel recovery method and propose enhancements to it that are required for the development of our coordinated recovery algorithms proposed in Section \ref{sec:coordinated}. A data-aided version of this algorithm is presented in Section \ref{sec:data-aided}. Simulation results are discussed in Section \ref{sec:results} and Section \ref{sec:conclusions} concludes the paper.
\subsection{Notation}
We denote vectors with small-case bold-face letters (e.g., ${\bf x}$), matrices with upper-case, bold-face letters (e.g., ${\bf X}$), and reserve calligraphic notation for symbols in frequency domain (e.g., $\bm{{\cal X}}$). We use ${\bf x}_i$ to denote the $i^{th}$ column of matrix ${\bf X}$ and $x(j)$ to denote the $j^{th}$ entry of vector ${\bf x}$. We also use ${\bf X}_{{\cal S}}$ to denote the sub-matrix formed by the columns $\{{\bf x}_i : i \in {\cal S} \}$, indexed by the set ${\cal S}$. We use $\widehat{{\bf x}}$, ${\bf x}^*$, ${\bf x}^{\sf T}$, and ${\bf x}^{\sf H}$ to respectively denote the estimate, conjugate, transpose, and conjugate transpose of the vector ${\bf x}$. Finally we use $\mathrm{diag}({\bf x})$ to transform the vector ${\bf x}$ into a diagonal matrix with the entries of ${\bf x}$ spread along the diagonal.
\section{System Model and Problem Formulation}
\label{sec:sysmodel}
\subsection{Transmission Model}
We consider a MIMO-OFDM system in which the BS is equipped with a large two-dimensional antenna array consisting of $R=M \times G$ antennas distributed across $M$ rows and $G$ columns.\footnote{Depending on the value of $M$ and $G$, the antennas could have a linear or a rectangular configuration. Further, we would like to stress that while we confine our attention to rectangular configurations for convenience, our approach applies to any one-, two- or three-dimensional configuration of antennas as explained at the end of Sec. \ref{sec:data-aided}.} The base station serves a number of single-antenna terminals. Orthogonal frequency division multiplexing (OFDM) is adopted as the signaling mechanism. In an OFDM system, serially incoming bits are divided into $N$ parallel streams and mapped into a $Q$-ary QAM alphabet $\{ {\cal A}_1, {\cal A}_2, \dots, {\cal A}_{Q}\}$. This results in an $N$-dimensional data vector $\bm{{\cal X}} = \left[ {\cal X}(1), {\cal X}(2), \dots,{\cal X}(N)\right]^{\sf T}$. The equivalent time-domain signal ${\bf x}$ is obtained by taking the inverse Fourier transform of the data vector, i.e.,
\begin{align}
{\bf x}&={\bf F}^{\sf H} \bm{{\cal X}},
\end{align}
where ${\bf F}$ is an $N \times N$ unitary discrete Fourier transform (DFT) matrix whose $(k,l)$ entry is given by
\begin{align}
f_{k,l}&=\frac{1}{\sqrt{N}} \exp{(-\jmath\frac{2\pi}{N}kl)}.
\end{align}
A cyclic prefix is inserted at the beginning of each symbol and the resulting signal is transmitted.
\subsection{Channel Model}\label{sec:channelmodel}
It is known that most wireless channels can be modeled as discrete multipath channels with large delay spread and very few significant paths as scatterers are sparsely distributed in space (see Fig. \ref{fig:scatterers}). This makes the CIR sparse \cite{wireless_sparse1,wireless_sparse2,Rappaport2002wireless}. Thus, for each transmit-receive link, we need only estimate a few significant multipath channel gains, which has the potential to reduce the pilot overhead substantially. We explicitly mention the sparsity property as property 1.
\begin{center}
\fbox
\parbox{0.9\columnwidth}
\begin{center}
\textbf{Property 1:} \emph{The channel impulse response is sparse.}
\end{center}
}
\end{center}
\begin{figure}[htbp]
\centerline{ \includegraphics[width=0.9\columnwidth] {channels}
} \caption{In most wireless channels scatterers are sparsely distributed and the resulting channel impulse response is sparse.}
\label{fig:scatterers}
\end{figure}
Let ${\bf h}^{r} \in \mathbb{C}^L$ denote the CIR which models the channel between a typical single antenna user and the receive antenna $r=(m,g)$ where $m\in\{1,2,\dots, M\}$ and $g\in\{1,2,\dots, G\}$ as shown in Fig. \ref{fig:2Dantennagrid}.
\begin{figure}[htbp]
\centering
\includegraphics[scale=0.75]{Fig2}
\caption{2-D antenna grid of size $M\times G$. An arbitrarily selected antenna is highlighted in red along with its neighboring antennas in blue. In this context, the red antenna is the central antenna $r$ and $r_U, r_R, r_D,$ and $r_L$ are its 4-neighbors.}
\label{fig:2Dantennagrid}
\end{figure}
We assume that ${\bf h}^r$ is sparse and is modeled as \cite{6280684}
\begin{align}
{\bf h}^r = {\bf h}_A^r \odot {\bf h}_B^r,
\end{align}
where $\odot$ indicates element-by-element multiplication. The vector ${\bf h}_A^r$ consists of elements that are drawn from some distribution\footnote{We put no restriction on the distribution of ${\bf h}_A^r$ which could be Gaussian or not. The distribution might even be unknown and the coefficients of ${\bf h}_A^r$ need not be iid. The implementation in this paper is agnostic to the distribution of channel coefficients.} and ${\bf h}_B^r$ is a Bernoulli random vector with independent entries that are distributed as \cite{6280684}
\begin{align}\label{eq:priorshB}
\mathrm{P}(h_B^r(i) = j) &=\begin{cases}
\lambda_i, & \text{for $j=1$}.\\
1-\lambda_i, & \text{for $j=0$}.
\end{cases}
\end{align}
In other words, the entries of ${\bf h}_B^r$ form a collection of independent (and possibly non-identically distributed) Bernoulli random variables. Thus, ${\bf h}^r$ is an $L$-tap discrete-time sparse channel, where no assumption whatsoever is made about the distribution of its non-zero complex-valued coefficients.
The received signal at the $r$th antenna is best described in the frequency domain and is given by
\begin{align}\label{eq:received_signal_freq_domain}
{\bm{\Yc}}^r &= \mathrm{diag}(\bm{{\cal X}}) {\bm{\Hc}}^r + {\bm{\Wc}}^r,
\end{align}
where ${\bm{\Yc}}^r$ is obtained from the time-domain received signal by removing the cyclic prefix and pre-multiplying by the Fourier matrix ${\bf F}$. The noise ${\bm{\Wc}}^r \sim {\cal C}{\cal N}({\bf 0}, \sigma_w^2{\bf I})$ is the frequency-domain noise vector of dimension $N\times 1$ and ${\bm{\Hc}}^r$ is the $N \times 1$ channel frequency response vector i.e.,
\begin{align}\label{eq:Hr}
{\bm{\Hc}}^r &={\bf F}\begin{bmatrix}{\bf h}^r\\
{\bf 0}_{N-L \times 1}
\end{bmatrix}=\underbar{{\bf F}}{\bf h}^r
\end{align}
where $\underbar{{\bf F}}$ is the truncated Fourier matrix of size $N \times L$ formed by selecting the first $L$ columns of ${\bf F}$. Using (\ref{eq:Hr}), we can rewrite (\ref{eq:received_signal_freq_domain}) as
\begin{align}\label{eq:received_signal_freq_domain_final}
{\bm{\Yc}}^r &= \mathrm{diag}(\bm{{\cal X}}) \underbar{{\bf F}}{\bf h}^r + {\bm{\Wc}}^r= {\bf A}{\bf h}^r + {\bm{\Wc}}^r,
\end{align}
where ${\bf A} \triangleq \mathrm{diag}(\bm{{\cal X}})\underbar{{\bf F}}$ is an $N \times L$ matrix.
\subsection{Spatial Channel Model}\label{sec:SpatialChannelModel}
The large number of antennas in massive MIMO can be arranged in different configurations. For example, \begin{inparaenum}[\itshape a\upshape)]
\item linear,
\item planar (rectangular), and
\item cylindrical (circular).
\end{inparaenum} Our algorithm is capable of working on any configuration as will be explained later in the paper. However, for convenience, we adopt the uniform rectangular array.
In massive MIMO it is reasonable to assume that antenna elements in the same vicinity will observe almost same echoes from different scatterers and therefore the corresponding channels will have common support. For the wireless system under study, the signal bandwidth, operating frequencies and antenna separation controls the supports commonality across the large arrays.
Specifically, the time difference of arrival $\Delta \tau$ of a wavefront to two antennas separated by a distance $d$ satisfies $\Delta \tau \le \frac{d}{C}$, where $C$ is the speed of light. The authors in \cite{barbotin2011estimating} suggest that two channel taps are resolvable if the time difference of arrival is larger than $\frac{1}{10BW}$ where $BW$ is the signal bandwidth. Thus, let $d_{\mathrm{max}}$ be the distance between the farthest antennas of an array ($d_{\mathrm{max}}$ is a function of the antenna spacing $d$ and the number of antennas in the array), then it follows easily from the above that the antenna array might exhibit one of the two possible scenarios:
\subsubsection{The array is spatially invariant with respect to the CIR support if $\frac{d_{\mathrm{max}}}{C} \le \frac{1}{10BW}$}
Here, all ${\bf h}^r$'s will have same sparsity pattern, which means the amplitudes of the channel taps might be different but the positions of the most significant taps (MST) will not change. This assumption is motivated by the fact that for closely spaced antenna elements, the times of arrival are quite close, though the paths amplitudes and phases could be different. Therefore, different antenna elements will experience almost the same echoes from the different scatterers. In other words, the support of the channels will not change as we move from one antenna to another throughout the array but the tap strengths might be different as evident from Fig. \ref{fig:SIA}. We call such arrays space-invariant arrays (SIA).
\begin{figure}[htbp]
\centering
\includegraphics[width=0.75\columnwidth]{SIA}
\caption{CIRs of a $5\times 5$ section of a space-invariant antenna array. Plots show the tap strengths on y-axis with respect to the tap locations on x-axis for antennas in this space-invariant antenna array section. Note that the support is invariant across the array but the taps strengths fade differently across the array.}
\label{fig:SIA}
\end{figure}
\subsubsection{The array is spatially variant with respect to the CIR support if $\frac{d_{\mathrm{max}}}{C} > \frac{1}{10BW}$}\label{sec:SpaceVariant}
In this case, the channel support varies across the array. Note that such variation takes place slowly and, therefore, it is safe to assume that the following property is always valid
\begin{center}
\fbox
\parbox{0.9\columnwidth}
\begin{center}
\textbf{Property 2:} \emph{Any central antenna and its 4-neighbors have \underline{approximately} common support.}
\end{center}
}
\end{center}
See Fig. \ref{fig:SVA} for an example where the neighboring antennas have approximately the same support. We call such arrays space-variant arrays (SVA).
\begin{figure}[htbp]
\centering
\includegraphics[width=0.75\columnwidth]{SVA}
\caption{CIRs of a $5\times 5$ section of a space-variant antenna array. Plots show the tap strengths on y-axis with respect to the tap locations on x-axis for antennas in this space-variant antenna array section. Note that neighboring antennas have \emph{approximately} the same support.}
\label{fig:SVA}
\end{figure}
In Table \ref{table:system_parameters} we classify antenna arrays of three different dimensions as either SIA or SVA. Specifically, the table illustrates the relationship between the maximum resolvable distance ($d_{\mathrm{max}}$) and the dimensions of the arrays for three different communications standards. For instance, in the 3GPP LTE standard, the distance between two antenna elements on the far ends of a $10 \times 10$ array is $9d<d_{\mathrm{max}}$, thus the array is SIA. Whereas, for a $50 \times 50$ array, the distance is $49d > d_{\mathrm{max}}$ causing the array to be SVA. Note that in this table the distance between two adjacent antennas is assumed to be $d = \lambda/2$ where $\lambda$ is the signal wavelength.
While most of the available research in MIMO channel estimation deals with the space-invariant case (for example, \cite{semi_blind_MIMO_OFDM, barbotin2011estimating, blind_sparse, blind_sparse2}), very limited research has been conducted for the space-variant scenario. Similarly, the literature related to the estimation of space-variant sparse channels in massive MIMO is limited (e.g., see \cite{library977759} and the references therein). The approach we pursue in this paper is capable of dealing with both the space-variant and space-invariant cases.
\begin{table*}[htbp]
\centering
\renewcommand{\arraystretch}{1.3}
{\small
\begin{tabular}{c | c |p{2.1cm} |c |c |c |c |c}
\hline\hlin
\textbf{Standard} & \textbf{Bandwidth} ($BW$)& \centering{\textbf{Center frequency } ($f_c$)} & ${d}_{\mathrm{max}}=\frac{C}{10 BW}$ & $d=\frac{\lambda}{2}$ & $10 \times 10$ array & $50 \times 50$ array & $100 \times 100$ array \\
\hline \hline
CDMA2000 & $1.25$ MHz & \centering{$1$ GHz} & $24$ m & $0.150$ m & SIA & SIA & SIA \\
\hline
3GPP LTE & $20$ MHz & \centering{$2.6$ GHz} & $1.5$ m & $0.058$ m & SIA & SVA & SVA \\
\hline
UWB & $500$ MHz & \centering{$3$ GHz} & $0.06$ m & $0.050$ m & SVA & SVA & SVA \\
\hline
\end{tabular}}
\caption{Wireless Systems Parameters
\label{table:system_parameters}
\end{table*}
\subsection{Pilots}
Pilots are needed for channel estimation where the transmitter reserves $K$ subcarriers for pilots and uses the remaining $N-K$ carriers for data transmission. Let ${\cal P}$ denote the set of indices of pilot carriers. Using (\ref{eq:received_signal_freq_domain_final}), the received pilots at receive antenna $r$ are then given by
\begin{align}\label{eq:probmodel}
{\bm{\Yc}}^r({\cal P}) = {\bf A}({\cal P}) {\bf h}^r + {\bm{\Wc}}^r({\cal P})
\end{align}
where ${\bm{\Yc}}^r({\cal P})$ and ${\bm{\Wc}}^r({\cal P})$ are $K \times 1$ vectors formed, respectively, by selecting the $K$ entries of ${\bm{\Yc}}^r$ and ${\bm{\Wc}}^r$ indexed by ${\cal P}$. Similarly, ${\bf A}({\cal P})$ is a $K \times L$ matrix formed by selecting the rows of ${\bf A}$ indexed by ${\cal P}$. Solving (\ref{eq:probmodel}) for ${\bf h}^r$ obviously requires that we at least have more pilots than the channel delay spread (i.e., $K\ge L$), which impacts the spectral efficiency of the system. Here, however, we use the sparse nature of the channel and the fact that adjacent antennas have almost the same support (i.e., properties 1 and 2) to substantially reduce the number of pilots needed as promised by the compressed sensing theory \cite{donoho,candes}.
Several pilot placement schemes have been suggested for OFDM channel estimation. It is best to allocate the pilots uniformly in conventional OFDM channel estimation (which does not make use of sparsity) \cite{5715843,5947173,1284837,1318954}. However, when the channel is sparse a random assignment of pilots has been observed to be optimal \cite{det_pilots_sparse,random_pilots_sparse2}.
With this model, we are now ready to tackle the problem of channel estimation. We do that in three steps spread over three sections
\begin{enumerate}
\item Bayesian channel estimation at each antenna,
\item Distributed channel estimation, and
\item Data-aided channel estimation.
\end{enumerate}
\section{Sparsity-aware Distribution Agnostic Bayesian Channel Estimation}\label{sec:sabmp}
Consider the linear regression model presented in (\ref{eq:probmodel}). For notational convenience, we will drop the superscript $r$ and the symbol ${\cal P}$ unless these are required for clarity. Hence (\ref{eq:probmodel}) becomes
\begin{align}\label{eq:sigmodel}
{\bm{\Yc}} = {\bf A} {\bf h} + {\bm{\Wc}},
\end{align}
where ${\bm{\Yc}}$ and ${\bm{\Wc}}$ are vectors of dimension $K \times 1$, ${\bf h}$ is a vector of dimension $L \times 1$ and ${\bf A}$ is a matrix of dimension $K \times L$. Here we are interested in performing Bayesian estimation of the wireless CIR ${\bf h}$. Bayesian approaches assume a prior distribution, however, given the dynamic nature of wireless channels, it is usually impossible to characterize the distribution. Moreover, such an assumption is usually not suitable as it does not reflect the reality and might result in performance degradation. Additionally, even if the distribution is known, it is very difficult to estimate the distribution parameters (e.g., mean and variance for Gaussian), especially when the channel statistics are not i.i.d. In that respect, the use of distribution agnostic Bayesian sparse signal recovery (SABMP) developed by the authors in \cite{sabmp, wosspa_sabmp} is quite attractive, as it provides Bayesian estimates even when the prior is non-Gaussian or unknown.
\subsection{Simple Channel Estimation using SABMP}\label{sec:simple_chann_est}
The set of channel estimation algorithms that we propose in this paper (Sec. \ref{sec:coordinated} and \ref{sec:data-aided}) use a modified version of the \texttt{SABMP} algorithm proposed by the authors in \cite{sabmp, wosspa_sabmp}. The modifications to \texttt{SABMP} required for the development of our distributed and data-aided channel estimation methods are proposed in Sec. \ref{sec:covar} and \ref{sec:FindingMarginals}. However, before presenting the modifications we consider it essential to quickly go through the steps followed by the \texttt{SABMP} algorithm. In that respect, we briefly describe a straightforward approach for sparse channel estimation using \texttt{SABMP}. In this approach, all channels ${\bf h}^r$ are estimated independently using the \texttt{SABMP} algorithm. Since no collaboration takes place among antennas in this approach, it is not possible to take advantage of property 2 mentioned earlier.
To estimate the $L \times 1$ sparse channel ${\bf h}$, from the $K \times 1$ observations vector ${\bm{\Yc}}$ related by the linear regression model given in (\ref{eq:sigmodel}), \texttt{SABMP} pursues an MMSE estimate of ${\bf h}$ given ${\bm{\Yc}}$ which is formally defined by
\begin{equation}\label{eq:app:xmmse}
\widehat{{\bf h}}_{\rm{MMSE}} \triangleq \mathbb{E}[{\bf h}|{\bm{\Yc}}] = \sum_{{\cal S}} p({\cal S}|{\bm{\Yc}})\mathbb{E}[{\bf h}|{\bm{\Yc}},{\cal S}].
\end{equation}
Here the sum is executed over all possible $2^{L}$ support sets of ${\bf h}$. However, computing this sum is a challenging task when the channel delay spread $(L)$ is large because the number of possible support sets can be extremely large and the computational complexity will become unrealistic. To have a computationally feasible solution, this sum can be approximated by considering only those support sets which include the most significant taps with high probability. These few support sets correspond to the sets with significant posteriors $p({\cal S}|{\bm{\Yc}})$. Let ${\cal S}_d$ be the set of supports for which the posteriors are significant. Hence, (\ref{eq:app:xmmse}) can be approximated by\footnote{Note that $\sum_{{\cal S}\in{\cal S}_d} p({\cal S}|{\bf y}) < 1$ since ${\cal S}_d \subset {\cal S}$. This would render the estimate in (\ref{eq:app:xammse}) biased. To ensure an unbiased estimate, we normalize $p({\cal S}|{\bf y})$ so that $\sum_{{\cal S}\in{\cal S}_d} p({\cal S}|{\bf y}) = 1$.}
\begin{equation}\label{eq:app:xammse}
\widehat{{\bf h}}_{\rm{AMMSE}} = \sum_{{\cal S}\in{\cal S}_d} p({\cal S}|{\bm{\Yc}})\mathbb{E}[{\bf h}|{\bm{\Yc}},{\cal S}].
\end{equation}
We could determine ${\cal S}_d$ and $\widehat{{\bf h}}_{\rm AMMSE}$ in a greedy manner using the dominant support selection metric defined as the log posterior
\begin{align}\label{eq:app:dssm1}
\nu({\cal S}) &\triangleq \ln p({\cal S} | {\bm{\Yc}}) = \ln p({\bm{\Yc}}|{\cal S}) p({\cal S}).
\end{align}
The greedy algorithm of \texttt{SABMP} starts by first finding the best support of size 1. This requires evaluating $\nu({\cal S})$ for ${\cal S}=\{ 1 \}, \dots, \{ L \}$, i.e., a total of $\binom{L}{1}$ search points. Let ${\cal S}_1 = \{ \alpha_1 \}$ be the optimal support. Now, the optimal support of size 2 is found. Ideally, this involves a search over a space of size $\binom{L}{2}$. To reduce the search space, however, the greedy approach looks for the tap location $\alpha_2 \neq \alpha_1$ such that ${\cal S}_2=\{ \alpha_1, \alpha_2 \}$ maximizes $\nu({\cal S}_2)$. This involves $\binom{L-1}{1}$ search points (as opposed to the optimal search over $\binom{L}{2}$ points). The process continues in this manner by forming ${\cal S}_3 = \{ \alpha_1, \alpha_2, \alpha_3 \}$ and so on. Therefore, ${\cal S}_d$, the set of dominant support sets is composed of support sets that are incremental in nature and is given by\footnote{In (\ref{eq:app:Sd}), $T_{\mathrm{max}}$ refers to the maximum number of non-zero elements in the sparse ${\bf h}$. $T_{\mathrm{max}}$ is selected to be slightly larger than the expected number of active taps in the estimated CIR using the de Moivre-Laplace theorem. For details, readers are referred to \cite{sabmp}.\label{ftn:Tmax}}
\begin{align}\label{eq:app:Sd}
{\cal S}_d &= \left\{ {\cal S}_1, {\cal S}_2, \dots, {\cal S}_{T_{\mathrm{max}}} \right\},\nonumber\\
{\cal S}_d &= \left\{ \{ \alpha_1\}, \{ \alpha_1, \alpha_2\}, \{ \alpha_1, \alpha_2, \alpha_3\}, \dots, \{ \alpha_1, \alpha_2, \dots, \alpha_{T_{\mathrm{max}}}\} \right\}.
\end{align}
The development of the \texttt{SABMP} algorithm in \cite{sabmp} assumes that the taps of ${\bf h}$ are activated with equal probability $\lambda$ (i.e., i.i.d. Bernoulli with probability $\lambda$). However, here we consider the case where some taps are more probable than others (based on the available information), and hence it is desirable to assign those taps a higher probability. This requires us to assume an independent and non-identically distributed Bernoulli behavior for the unknown sparse vector and therefore the prior is given by
\begin{align}\label{eq:app:pS}
p({\cal S}) &= \prod_{i\in {\cal S}} \lambda_i \prod_{j\in\{1,\dots,L\}\backslash {\cal S} } (1-\lambda_j),
\end{align}
where, $\lambda_i$ is the probability that the $i$th tap of ${\bf h}$ is active. Moreover, the likelihood is approximated as
\begin{align}\label{eq:app:pyS}
p({\bm{\Yc}}|{\cal S}) &= \exp \left( -\frac{1}{2\sigma_w^2} \left\|{\bf P}_{\cal S}^{\bot} {\bm{\Yc}}\right\|_2^2 \right),
\end{align}
where, ${\bf P}_{\cal S}^\bot ={\bf I} - {\bf P}_{\cal S} ={\bf I} - {\bf A}_{\cal S}\left( {\bf A}_{\cal S}^{\sf H} {\bf A}_{\cal S}\right)^{-1} {\bf A}_{\cal S}^{\sf H}$ is the projection matrix and ${\bf A}_{\cal S}$ is a matrix formed by selecting columns of ${\bf A}$ indexed by support ${\cal S}$. Substituting (\ref{eq:app:pS}) and (\ref{eq:app:pyS}) in (\ref{eq:app:dssm1}) yields
\begin{align}\label{eq:app:dssm2}
\nu({\cal S}) \triangleq \ln p({\cal S} | {\bm{\Yc}}) &= ( -\frac{1}{2\sigma_w^2}) \left\|{\bf P}_{\cal S}^{\bot} {\bm{\Yc}}\right\|_2^2 + \sum_{i\in {\cal S}}\ln \lambda_i \nonumber\\
&+ \sum_{j \in \{1,\cdots,L\}\backslash{\cal S}}\ln(1-\lambda_j)
\end{align}
Now the only term that is left to be evaluated in (\ref{eq:app:xammse}) is $\mathbb{E}[{\bf h}|{\bm{\Yc}},{\cal S}]$. Note that it is difficult or even impossible to evaluate this quantity because the distribution of the active taps of ${\bf h}$ is unknown. Therefore, we replace it by the best linear unbiased (BLUE) estimate given by
\begin{align}\label{eq:BLUE}
\mathbb{E}[{\bf h}|{\bm{\Yc}},{\cal S}] \leftarrow \left( {\bf A}_{\cal S}^{\sf H} {\bf A}_{\cal S}\right)^{-1} {\bf A}_{\cal S}^{\sf H} {\bm{\Yc}}.
\end{align}
This provides us all the required quantities to evaluate $\widehat{{\bf h}}_{\rm AMMSE}$. Note that all parameters including $\sigma_w^2, \bm{\lambda}=\{\lambda_i\}_{i=1}^L$ and the possible size of support $T_{\rm max}$ need not be known and are estimated by the algorithm. Specifically, $\lambda_i$'s are initialized as
\begin{align*}
\lambda_{i} &= \frac{1}{L}\left| \left\{ j : \left| {\bf a}_j^{\sf H} {\bm{\Yc}} \right| \ge \frac{1}{2} \left\| {\bf a}^{\sf H} {\bm{\Yc}} \right\|_\infty \right\} \right|,
\end{align*}
where ${\bf a}_j$ is the $j$th column of the matrix ${\bf A}$. Moreover, $\sigma_w^2$ is initialized simply as a scaled version of the variance of the received signal i.e., $\sigma_{\bm{\Yc}}^2$. Finally, $T_{\mathrm{max}}$ is selected to be slightly larger than the expected number of active taps in the estimated CIR using the de Moivre-Laplace theorem. Note that our algorithm is robust to these initial estimates and can find right support even if these parameters are initialized away from their true values. For more details the interested readers are referred to \cite{sabmp}.
By following this greedy approach, each antenna node estimates the corresponding approximate sparse CIR (\ref{eq:app:xammse}) in a distribution agnostic manner. A detailed statement of the greedy algorithm is presented in Table \ref{alg:greedy}.
\begin{table}
\begin{algorithmic}[1]
\Procedure{Greedy} {${\bf A}, {\bm{\Yc}}, \bm{\lambda}, \sigma_w^2, T_{\rm max}$}
\State \textbf{initialize} $J \gets \{1, 2, \hdots, L\},\, i \gets 1$
\State \textbf{initialize empty sets} ${\cal S}_{max},\, {\cal S}_d,\, p({\cal S}_d|{\bm{\Yc}}),\, \mathbb{E}[{\bf h}|{\bm{\Yc}},{\cal S}_d]$
\State $J_i \gets J$
\While{$i \le T_{\rm max}$}
\State $\Omega \gets \{ {\cal S}_{max} \cup \{ \alpha_1 \},
{\cal S}_{max} \cup \{ \alpha_2 \},
\cdots,
{\cal S}_{max} \cup \{ \alpha_{|J_i|} \} \mid \alpha_k \in J_i\}$
\State \textbf{compute }$\{ \nu({\cal S}_k) \mid {\cal S}_k \in \Omega \}$
\State \textbf{find} ${\cal S}_\star \in \Omega$ \textbf{such that} $\nu({\cal S}_\star) \ge \max_j \nu({\cal S}_j)$
\State ${\cal S}_d \gets \{{\cal S}_d, {\cal S}_\star\}$
\State \textbf{compute} $p({\cal S}_\star|{\bm{\Yc}}), \mathbb{E}[{\bf h}|{\bm{\Yc}},{\cal S}_\star]$
\State $p({\cal S}_d|{\bm{\Yc}}) \gets \{p({\cal S}_d|{\bm{\Yc}}), p({\cal S}_\star|{\bm{\Yc}})\}$
\State $\mathbb{E}[{\bf h}|{\bm{\Yc}},{\cal S}_d] \gets \{\mathbb{E}[{\bf h}|{\bm{\Yc}},{\cal S}_d] , \mathbb{E}[{\bf h}|{\bm{\Yc}},{\cal S}_\star\}$
\State ${\cal S}_{max}\gets {\cal S}_\star$
\State $J_{i+1} \gets L~\backslash~{\cal S}_\star$
\State $i \gets i+1$
\EndWhile\label{euclidendwhile}
\State\label{alg:greedy:returnline} \textbf{return} ${\cal S}_d, p({\cal S}_d|{\bm{\Yc}}), \mathbb{E}[{\bf h}|{\bm{\Yc}},{\cal S}_d]$
\EndProcedure
\end{algorithmic}
\caption{Support Agnostic Bayesian Matching Pursuit Algorithm (SABMP)}\label{alg:greedy}
\end{table}
In addition to the non-iid generalization above, we develop in the following two necessary modifications to the \texttt{SABMP} algorithm. Specifically, we modify \texttt{SABMP} to output \begin{inparaenum}[\itshape a\upshape)]
\item the channel estimation error covariance matrix and
\item the marginal probabilities of the detected MSTs
\end{inparaenum} that are needed for the distributed and data-aided versions of the channel estimation algorithms proposed in Sec. \ref{sec:coordinated} and \ref{sec:data-aided} respectively.
\subsection{Error Covariance and Estimation Error}\label{sec:covar}
The channel estimation error and the covariance could be computed as follows.
Let,
\begin{align}\label{eq:channelerror}
\widetilde{{\bf h}} = \widehat{{\bf h}}_{\rm AMMSE} - {\bf h}
\end{align}
be the error vector and ${\bf R}_{\widetilde{{\bf h}}} \triangleq {\rm cov}[\widetilde{{\bf h}}|{\bm{\Yc}}]$ represent the error covariance matrix. The trace of ${\bf R}_{\widetilde{{\bf h}}}$ i.e., $\Tr[{\bf R}_{\widetilde{{\bf h}}}]$ gives the MMSE estimation error. In order to evaluate ${\bf R}_{\widetilde{{\bf h}}}$, let us define the error vector $\widetilde{{\bf h}}_{\cal S} = \widehat{{\bf h}}_{\cal S} - {\bf h}$ for a given support ${\cal S}$, where $\widehat{{\bf h}}_{\cal S} = \mathbb{E}[{\bf h}|{\bm{\Yc}},{\cal S}]$. Let the corresponding error covariance matrix be ${\bf R}_{\widetilde{{\bf h}}|{\cal S}} \triangleq {\rm cov}[\widetilde{{\bf h}}|{\bm{\Yc}},{\cal S}]$. Then ${\bf R}_{\widetilde{{\bf h}}}$ could be expressed in terms of ${\bf R}_{\widetilde{{\bf h}}|{\cal S}}$ by summing it over the dominant support set ${\cal S}_d$ and is given by
\begin{align}\label{eq:Rh}
{\bf R}_{\widetilde{{\bf h}}} &= \sum_{{\cal S}\in{\cal S}_d} p({\cal S}|{\bm{\Yc}}) \,\, {\bf R}_{\widetilde{{\bf h}}|{\cal S}}.
\end{align}
Since we replace $\mathbb{E}[{\bf h}|{\bm{\Yc}},{\cal S}]$ with a BLUE estimate, the conditional error covariance matrix will be ${\bf R}_{\widetilde{{\bf h}}|{\cal S}} = ({\bf A}_{\cal S}^{\sf H} {\bf C}^{-1} {\bf A}_{\cal S})^{-1}$ \cite{poor1994introduction} (where ${\bf C} = \sigma_w^2 {\bf I}$ is the noise covariance matrix). Combining this fact with (\ref{eq:Rh}) yields
\begin{align}\label{eq:error_covariance_matrix}
{\bf R}_{\widetilde{{\bf h}}} &= \sigma_w^2\sum_{{\cal S}\in{\cal S}_d} p({\cal S}|{\bm{\Yc}}) \,\, ({\bf A}_{\cal S}^{\sf H} {\bf A}_{\cal S})^{-1}.
\end{align}
Note that the calculation of covariance matrix involves a matrix inversion term which is a computationally expensive task. However, we would like to highlight that these inverses are available as part of intermediate calculations in the \texttt{SABMP} algorithm and hence do not pose any additional computational burden. The error covariance matrix and the estimation error play a vital role in the development of the data-aided approach presented in Sec. \ref{sec:data-aided}.
\subsection{Finding Marginals}\label{sec:FindingMarginals}
The marginal probabilities are not directly available at the output of \texttt{SABMP} and could be computed from the posteriors $p({\cal S}|{\bm{\Yc}}), \forall {\cal S}\in {\cal S}_d$ in a simple manner described below.
Let $T^r = \{ \alpha_1^r, \alpha_2^r, \dots, \alpha_{T_{\rm max}}^r\}$ be the set of MST locations of channel ${\bf h}^r$ as detected by \texttt{SABMP} algorithm. Then the marginal probabilities of $\alpha_i^r$, $\forall i\in \{1,2,\dots,T_{\rm max} \}$ could be computed as
\begin{align}\label{eq:palphat}
p(\alpha_i^r|{\bm{\Yc}}) &= \sum_{\alpha_i^r \cap {\cal S} \ne \varnothing } p({\cal S}|{\bm{\Yc}}),
\end{align}
where the sum is evaluated over all ${\cal S} \in {\cal S}_L^r$ satisfying the condition ${\alpha_i^r \cap {\cal S} \ne \varnothing }$ where ${\cal S}_L^r$ contains all $2^L - 1$ support sets that could be created for ${\bf h}^r$ (recall that ${\bf h}^r$ is a vector of length $L$). Among these $2^{L}-1$ support sets there are only $2^{T_{\rm max}}-1$ support sets which involve purely the ${T_{\rm max}}$ detected non-zero locations. Let us denote the set of these $2^{T_{\rm max}}-1$ support sets by ${\cal S}_{T_{\rm max}}^r$. We assert that only the support sets present in ${\cal S}_{T_{\rm max}}^r$ have significant posteriors $p({\cal S}|{\bm{\Yc}})$ as compared to the others which have very small values. This follows from the findings of \cite{sabmp} (see Fig. 7 therein). Thus, we can evaluate the sum in (\ref{eq:palphat}) over ${\cal S} \in {\cal S}_{T_{\rm max}}^r$ to find the marginal probability of each detected non-zero location. However, the \texttt{SABMP} algorithm returns ${\cal S}_d^r$, which unlike ${\cal S}_{T_{\rm max}}^r$ contains only ${T_{\rm max}}$ support sets (see (\ref{eq:app:Sd})). Therefore, we modify the \texttt{SABMP} algorithm so that it outputs $p({\cal S}|{\bm{\Yc}})$ for all supports in ${\cal S}_{T_{\rm max}}^r$. For illustration purpose Table \ref{tab:Sd_vs_Sp} provides an example of the support sets ${\cal S}_d^r$ and ${\cal S}_{T_{\rm max}}^r$ when ${T_{\rm max}}=3$. Please note that for convenience of notation, we shall from now on use $\lambda(\alpha^r_i)$ as a shorthand notation for $p(\alpha_i^r|{\bm{\Yc}})$.
\begin{table}
\centering
\begin{tabular}{|c|c|c|}
\hline
& ${\cal S}_d^r$ & ${\cal S}_{T_{\rm max}}^r$\\
\hline
1 & $\{\alpha_1\}$ & $\{\alpha_1\}$ \\
2 & $\{\alpha_1, \alpha_2\}$ & $\{\alpha_2\}$ \\
3 & $\{\alpha_1, \alpha_2, \alpha_3\}$ & $\{\alpha_3\}$ \\
4 & & $\{\alpha_1, \alpha_2\}$ \\
5 & & $\{\alpha_1, \alpha_3\}$ \\
6 & & $\{\alpha_2, \alpha_3\}$ \\
7 & & $\{\alpha_1, \alpha_2, \alpha_3\}$\\
\hline
\end{tabular}
\caption{Sets ${\cal S}_d^r$ and ${\cal S}_{T_{\rm max}}^r$ for ${T_{\rm max}}=3$.}
\label{tab:Sd_vs_Sp}
\end{table}
Since ${\cal S}_{T_{\rm max}}^r$ has more support sets, this modification obviously results in increased computational complexity. However, utilizing the available intermediate information in \texttt{SABMP} helps to compute the marginalized posterior probabilities $p({\cal S}|{\bm{\Yc}})$ in an efficient manner. Specifically, note that for the example of Table \ref{tab:Sd_vs_Sp}, we require posteriors of $\{\alpha_2\}, \{\alpha_3\}, \{\alpha_1,\alpha_3\},$ and $\{\alpha_2, \alpha_3\}$ in addition to those returned by \texttt{SABMP}. However, it follows from the explanation given in Sec. \ref{sec:simple_chann_est}, that the posteriors for $\{\alpha_2\}, \{\alpha_3\}$, and $\{\alpha_1,\alpha_3\}$ are already available to the algorithm by virtue of the intermediate computations. Therefore, the only missing computation which has to be performed additionally is that of $\{\alpha_2, \alpha_3\}$. The same reasoning applies for larger support sizes. Therefore, we state that the increase in computational complexity is not significant.
For ease of reference, we name the modified version of \texttt{SABMP} as \texttt{RS1}. It has the following additional features:
\begin{itemize}
\item it considers the non-zero taps to follow independent and non-identically distributed Bernoulli behavior (as highlighted in Sec. \ref{sec:simple_chann_est})
\item it returns the error covariance of our estimate which is needed for the data-aided part (Sec. \ref{sec:covar})
\item it outputs the belief/probability that a given tap is active (Sec. \ref{sec:FindingMarginals}).
\end{itemize}
This algorithm along with \texttt{SABMP} will be used in the discussion that follows to develop the coordinated channel recovery algorithms.
\section{Coordinated Channel Estimation}\label{sec:coordinated}
In the coordinated channel recovery method, the receive antennas collaborate with each other to take advantage of property 2 and estimate the MST locations jointly. In order to realize this coordinated method, we assume baseband processing at each receive antenna with an additional processor on each baseband card to implement the collaboration strategy described in this section. At the heart of the collaboration strategy followed by the proposed method is the following simple information-sharing step.
\begin{center}
\fbox
\parbox{0.9\columnwidth}
\begin{center}
\textbf{Sharing Step: } Each antenna acting as a central antenna $r_C$ receives information from its direct 4-neighbors ${\cal N} = \{ r_U, r_D, r_R, r_L\}$.\footnotemark
\end{center}
}
\end{center}
\footnotetext{For the elements lying at the edges of the array the number of neighbors are different. We use ${\cal N}$ to denote the set of neighbors irrespective of the position of $r$ and therefore $2 \le |{\cal N}| \le 4 $.}
It is obvious that repetitive application of this sharing step would result in information diffusion throughout the antenna grid. For example, in the first iteration $r_C$ receives information from just the first tier of antennas (i.e., the neighboring 4 antennas). In just two iterations $r_C$ receives information from $12$ antennas (i.e., the first and second antenna tiers). Therefore, with the help of this simple step each antenna is able to incorporate information from its neighbors to enhance its decision about the MSTs of its channel. This ultimately helps in estimating the channels accurately. There are two advantages of this stepwise collaboration mechanism:
\begin{enumerate}
\item It gives us the flexibility to control the number of collaborators for each receiver which is essential for the space-variant case (Sec. \ref{sec:SpaceVariant}).
\item The collaboration mechanism is computationally efficient as the antennas do not all collaborate with each other at the same time. This also reduces the communication overhead.
\end{enumerate}
We now present two algorithms for channel estimation based on this simple stepwise information sharing strategy.
\subsection{Algorithm 1: Marginal-based Channel Estimation using Pilots}\label{sec:alg1}
We seek to solve the problem mentioned in (\ref{eq:probmodel}). The proposed algorithm starts by estimating the sparse channels ${\bf h}^r$ at each receive antenna $r$ using the \texttt{RS1} algorithm. We initialize the algorithm by assuming that all taps of ${\bf h}^r$ have equal active probability $\lambda_{\mathrm{init}}$, i.e.,
\begin{align}
\mathrm{P}\left(h_{B}^r(l)= 1\right)= \lambda_{\mathrm{init}}, \quad l \in \left\{1,2, \cdots, L\right\}, \forall r.
\end{align}
Let $T^{r} = \{ \alpha_1^{r}, \alpha_2^{r}, \cdots, \alpha_{T_{\mathrm{max}}}^{r}\}$ be the set of MSTs of channel ${\bf h}^r$ as detected by \texttt{RS1}. Here $\alpha_i^r$ is the location of the $i$th detected tap of receiver $r$. Note that since $\lambda_{\mathrm{init}}$ is same throughout the array, the number of detected MSTs, i.e., $T_{\mathrm{max}}$, will also be same for all receivers in the array.\footnote{The value of $T_{\mathrm{max}}$ is selected to be slightly larger than the expected number of active MSTs in the estimated CIR using the de Moivre-Laplace theorem which relies on $\lambda_{\mathrm{init}}$.} In other words, the cardinality $|T^r| = T_{\mathrm{max}}, \forall r$. However, the actual tap locations might differ from one antenna to another. Therefore, it is not necessary that $|T^{r_1} \cap T^{r_2}| = T_{\rm {max}}, \text{ for } r_1 \ne r_2$. Along with the MSTs, \texttt{RS1} also returns the marginals $\lambda(\alpha^r_t) \triangleq \mathrm{P}\left(h_{B}^r(\alpha_t^r)=1\right), t\in\{ 1,2,\cdots, T_{\mathrm{max}}\}$. At this point, we invoke the sharing step mentioned previously and share the marginals. Hence, each antenna, acting as central antenna $r_C$, collects these marginals from its 4-neighbors and computes the average marginal for each tap given by
\begin{align}\label{eq:newmarginals}
\lambda(\alpha_i^{r_C}) &= \begin{cases}
\sum\limits_{j \in {\cal N}^+} \lambda(\alpha_i^j) \Big/ |{\cal N}^+|, & \text{if $\alpha_i^{r_C} \in \mathop\bigcup\limits_{j \in {\cal N}^+}T^j$}\\
\lambda_{\mathrm{small}}, & \text{otherwise}
\end{cases},
\end{align}
where ${\cal N}^+ ={\cal N} \cup r_C$, $i\in \{1,2,\cdots,L\}$ and $\lambda(\alpha_i^{r_C})$ can be seen as the updated marginal of the $i$th tap detected at $r_C$. Here, $\lambda_{\mathrm{small}}$ is an arbitrarily small value assigned to those taps which have not been detected by any of the receivers in ${\cal N}^+$; it is highly probable that these taps have almost zero gains. Note that (\ref{eq:newmarginals}) is performed at each antenna as each antenna is the center of some neighbors. Moreover, each antenna repeats these sharing and averaging steps $D$ times where $D$ is selected based on whether the array under consideration is classified as SIA or SVA. This repetition allows each antenna to utilize the observations of distant antenna tiers to bolster its support estimates. Note that when $D=1$, information from only the immediate neighbors is taken into consideration and for $D=2$, information belonging to the neighbors of neighbors is also incorporated in the computations. Therefore, in this fashion, higher values of $D$ make it possible to extend the scope of information sharing to distant antennas.
In the space-invariant array case, since the MST locations do not vary across the array, contribution from as many antennas as possible will always strengthen our belief in these locations. Therefore, we may select $D=\max(M,G)$ which equals to the largest dimension of the antenna array. This particular choice of $D$ ensures that each antenna receives information from every other antenna in the array. However, one might not need to select such high value of $D$ and a smaller number of iterations might be sufficient based on the problem parameters such as the observation size ($K$) and the sparsity ($n$) of the channels. In fact we could establish a loose lower bound on $D$ in the noise free case as a function of these quantities using lemma 1 in \cite{1453780}. According to this lemma, if observations from $q$ antennas are used to recover $n$-sparse channel vectors using $K$ pilots then for a unique solution the following relationship holds
\begin{align}\label{eq:nbound}
n \le \lceil (K+q)/2\rceil -1,
\end{align}
where $\lceil \cdot \rceil$ denotes the ceiling operation. This yields the lower bound on $q$ which is given by
\begin{align}\label{eq:qbound}
q > 2n - K.
\end{align}
Furthermore, it could be easily deduced that the total number of antennas that take part in the $D$th sharing and averaging step is $2D(D+1)+1$. Thus relating this number with $q$ above we conclude that, to guarantee a unique solution in the noise free case, $D$ must satisfy
\begin{align}\label{eq:lowerboundonD_pre}
2D(D+1)+1 > 2n - K,
\end{align}
which simplifies to
\begin{align}\label{eq:lowerboundonD}
D > \sqrt{n-\frac{K}{2}-\frac{1}{4}} - \frac{1}{2}.
\end{align}
For a detailed account of the lemma and its requirements please refer to \cite{1453780}.
In the space-variant case, depending upon how fast the MST locations (support) change across the array, we might or might not gain from sharing. Specifically, if the change in support is fast, using higher values of $D$ would degrade the estimates. On the other hand, if the support changes very slowly, we expect that the neighborhood around a given antenna would behave \emph{approximately} as SIA. Therefore, we select a value for $D$ such that collaboration among antennas in that neighborhood would improve the estimates. In fact, from the discussion in Sec. \ref{sec:SpatialChannelModel}, we can determine the value of $D$ which will ensure that all sharing and averaging takes place among antennas having same support. Specifically, the number of tiers (also $D$) selected for sharing could be represented in terms of the distance between two adjacent antennas $d$ and the signal bandwidth $BW$. Note that the distance between the farthest antennas in tier 1 (i.e., when $D=1$) is $2d$. Similarly, for tier 2 this distance is $4d$. In general, the distance is directly related to $D$ and is given by $2Dd$. Now to ensure space-invariance for antennas up to tier $D$, it follows that we should require $2Dd \le \frac{C}{10BW}$ (see Sec. \ref{sec:SpatialChannelModel}). Therefore,
\begin{align*}
D \le& \frac{C}{20\cdot d\cdot BW},
\end{align*}
or
\begin{align}\label{eq:Dval_for_spaceinvariance}
D =& \left\lfloor \frac{C}{20\cdot d\cdot BW} \right\rfloor,
\end{align}
where $\lfloor \cdot \rfloor$ denotes the floor operation. The value of $D$ in (\ref{eq:Dval_for_spaceinvariance}) will ensure that sharing happens among antennas whose support is approximately the same. That said the number of pilots should also be large enough such that (\ref{eq:lowerboundonD}) is also satisfied.
At the end of $D$ iterations each antenna has a new set of marginals which are used as new priors with \texttt{SABMP} to get the final sparse CIR estimate. This final estimate is more accurate as the antennas have shared their information to strengthen their beliefs about the locations of the active taps. We call this algorithm the \emph{Marginal-based Algorithm}. A graphical description of the algorithm is given in Fig. \ref{fig:AlgorithmSteps} and a summary of the steps followed by the algorithm is presented in Algorithm \ref{alg:Alg1}.
\begin{algorithm}
\caption{Marginal-based Channel Estimation using Pilots\label{alg:Alg1}}
\begin{enumerate}
\item Initialize $\mathrm{P}\left(h^r_{B}(l)= 1\right)= \lambda_{\mathrm{init}}, \quad l \in \left\{1,2, \cdots, L\right\}, \forall r$.
\item Run \texttt{RS1} at each antenna to estimate its $\bm\lambda$. (Fig. \ref{fig:AlgStep1})
\item\label{en:step1} Each antenna, acting as central antenna, receives marginals from its neighbors. (Fig. \ref{fig:AlgStep2})
\item\label{en:step2} Each antenna computes average marginals $(\bm\lambda')$. (see (\ref{eq:newmarginals}) and Fig. \ref{fig:AlgStep3})
\item Repeat steps \ref{en:step1}-\ref{en:step2} above, $D$ times. (Fig. \ref{fig:AlgStep4})
\item All antennas re-estimate channels using these marginals as new priors with \texttt{SABMP} algorithm.
\end{enumerate}
\end{algorithm}
\begin{figure*}
\centering
\begin{subfigure}[b]{0.35\textwidth}
\includegraphics[width=\textwidth]{gridExplanationAlgorithm}
\caption{Step 1: Each antenna finds MSTs and their corresponding marginals $\bm{\lambda}$.}
\label{fig:AlgStep1}
\end{subfigure}\qua
~
\begin{subfigure}[b]{0.35\textwidth}
\includegraphics[width=\textwidth]{gridExplanationAlgorithm2}
\caption{Step 2: Each antenna receives marginals from its 4-neighbors (highlighted red).}
\label{fig:AlgStep2}
\end{subfigure}\qua
\begin{subfigure}[b]{0.35\textwidth}
\includegraphics[width=\textwidth]{gridExplanationAlgorithm3}
\caption{Step 3: Each antenna computes the mean of the received marginals $\bm{\lambda'}$.}
\label{fig:AlgStep3}
\end{subfigure}\qua
\begin{subfigure}[b]{0.35\textwidth}
\includegraphics[width=\textwidth]{gridExplanationAlgorithm4}
\caption{Step 4: Repeat steps 2 \& 3. Information from green antennas comes in.}
\label{fig:AlgStep4}
\end{subfigure}
\caption{Description of the steps followed by Algorithm 1 when $D=2$. Although these steps are followed by all antennas in parallel, the process is highlighted only for the blue antenna.}\label{fig:AlgorithmSteps}
\end{figure*}
\subsection{Algorithm 2: Reduced Communication and Computational Cost -- Integer-based Channel Estimation}
We would like to point out that sharing the marginals vectors among the receiver antennas puts a high communication overhead on the massive-MIMO system. This is because the marginals are floating point numbers and communicating these numbers requires complex signalling. This increases the communication overhead between antennas. However, if just integers are shared, the communication cost could be reduced significantly. We therefore, propose a variant of Algorithm 1 which uses integers for communication among receiver antennas. Since we are not interested in sharing marginals, we do not calculate these and rely on the original \texttt{SABMP} algorithm. Therefore, this algorithm has an additional advantage of low computational complexity as marginals are not calculated.
The algorithm starts by estimating channels at each receiver using \texttt{SABMP} and depends completely on the amplitudes of the estimated MSTs. Following the same reasoning given in the previous section, let $T^{r} = \{ \alpha_1^{r}, \alpha_2^{r}, \cdots, \alpha_{T_{\mathrm{max}}}^{r}\}$ be the set of MSTs of channel ${\bf h}^r$ as detected by \texttt{SABMP} and ${\bf h}^r(T^{r})$ be the corresponding amplitudes. Based on these amplitudes, we define an integer metric for each tap which we call \emph{score} and denote it by $\psi$. For a given $T^{r}$, the highest score is assigned to the channel tap with maximum amplitude in absolute sense. Similarly, the tap (in $T^r$) with minimum amplitude gets the least score. Specifically, since there are $T_{\mathrm{max}}$ MSTs, we assign a score of $T_{\mathrm{max}}$ to the channel tap with maximum amplitude, a score of $T_{\mathrm{max}}-1$ to the second highest tap and so on until a score of $1$ is assigned to the tap with the smallest amplitude among these $T_{\mathrm{max}}$ taps. Therefore, if $|h^r(\alpha_1^r)| > |h^r(\alpha_2^r)| > \cdots > |h^r(\alpha_{T_{\mathrm{max}}}^r)|$ then $\psi(\alpha_1^r)=T_{\mathrm{max}}, \psi(\alpha_2^r)=T_{\mathrm{max}}-1, \cdots, \psi(\alpha_{T_{\mathrm{max}}}^r)=1$ where $\psi(\alpha_i^r)$ is the score of the $i$th detected tap of receiver $r$. All other $L-T_{\mathrm{max}}$ tap locations are assigned a score of zero.
Once each receiver has assigned scores to its detected MSTs, we are ready to invoke the sharing step. Thus, each antenna acting as a central antenna $r_C$ collects these scores from its 4-neighbors and computes the average score for each tap given by
\begin{align}\label{eq:newscores}
\psi(\alpha_i^{r_C}) &= \begin{cases}
\ceil[\Bigg]{\sum\limits_{j \in {\cal N}^+} \psi(\alpha_i^j) \Big/ |{\cal N}^+|}, & \text{if $\alpha_i \in \mathop\bigcup\limits_{j \in {\cal N}^+}T^j$}\\
0, & \text{otherwise}
\end{cases}
\end{align}
where $i\in \{1,2,\cdots,L\}$. The sharing and averaging process is repeated $D$ times and all the related discussion in marginal-based algorithm applies to this algorithm as well. The averaging step of (\ref{eq:newscores}) is similar to the averaging step of Algorithm 1 given in (\ref{eq:newmarginals}) except that we round up the averaging result to the nearest largest integer. This ensures that the resulting score is always an integer. However, note that the rounding operation is not required in the last step as no sharing takes place after that. Therefore, to avoid unnecessary computation and the resulting information loss, the round up operation is not performed on the average scores. At the end of the $D$ sharing and averaging steps each node computes a belief metric given by
\begin{align}\label{eq:newbeliefs}
b(\alpha_i^r) &= \psi(\alpha_i^r)/T_{\mathrm{max}},
\end{align}
where $b(\alpha_i^r)$ is the estimated belief that the $i$th tap of receiver $r$ is active. Each node uses the beliefs as the Bernoulli priors to re-estimate the channels using \texttt{RS1}. We call this algorithm the \emph{Integer-based Algorithm}. The steps followed by this algorithm are summarized in Algorithm \ref{alg:Alg2}. This algorithm has the following advantages over the marginal-based algorithm:
\begin{enumerate}
\item Reduced communication cost since it totally avoids communicating floating point numbers and,
\item Lower computational complexity since it does not compute marginal probabilities.
\end{enumerate}
\begin{algorithm}
\caption{Integer-based Channel Estimation using Pilots \label{alg:Alg2}}
\begin{enumerate}
\item Run \texttt{SABMP} at each antenna
\item\label{enIB:step1} Each antenna receives scores from its neighbors
\item\label{enIB:step2} Each antenna computes average scores (\ref{eq:newscores})
\item Repeat steps \ref{enIB:step1}-\ref{enIB:step2} above, $D$ times
\item Each antenna computes a belief vector (\ref{eq:newbeliefs})
\item All antennas re-estimate channels using these belief vectors in place of Bernoulli priors with \texttt{SABMP} algorithm
\end{enumerate}
\end{algorithm}
We now move on to suggest another level of refinement for the marginal probability/scores vectors by selecting reliable data carriers to perform channel estimation.
\section{Data-aided Channel Estimation}\label{sec:data-aided}
By virtue of the channel sparsity property we can perform channel estimation using a small number of pilots $K$ compared to the channel length $L$ as discussed in the last two sections. We can enhance the channel estimate by increasing the number of pilots. Alternatively, we take a data-aided approach as it is more spectrally efficient. Here, the pilot-based channel estimate is used for data detection which along with the pilots is used to enhance the channel estimate further. Note however that we do not need to use all the detected data for channel estimation thanks to the channel sparsity; a few additional observations would enhance the channel estimate significantly. Therefore, we can be selective and use only the samples which are reliable. So each antenna could independently determine which carrier is reliable by assigning a reliability measure $\mathfrak{R}(i), \; i\in \{ 1, \cdots, N\} \backslash {\cal P}$ to each of the $N-|{\cal P}|$ data carriers. That said, we recognize that there are two sources of error in data detection that play important role in determining the reliable data carriers, namely,
\begin{inparaenum}[\itshape a\upshape)] \item noise, and \item error in channel estimation\end{inparaenum}. Their combined distortion effect could be expressed by substituting the estimated channel $\widehat{{\bf h}}_{\rm AMMSE}$ from (\ref{eq:channelerror}) into the system model (\ref{eq:sigmodel}) as follows
\begin{align}
{\bm{\Yc}} &= {\bf A}({\bf h}+\widetilde{{\bf h}})+{\bm{\Wc}}\nonumber= {\bf A}{\bf h}+{\bm{\Zc}},
\end{align}
where, ${\bm{\Zc}} = {\bf A}\widetilde{{\bf h}}+{\bm{\Wc}}$ is the combined distortion which is assumed to be Gaussian with zero mean and covariance ${\bf R}_{\cal Z}$, where ${\bf R}_{\cal Z}$ is represented in terms of the error covariance ${\bf R}_{\widetilde{h}}$, calculated in (\ref{eq:error_covariance_matrix}), as
\begin{align}
{\bf R}_{\cal Z} &= \mathbb{E}[{\bm{\Zc}} {\bm{\Zc}}^{\sf H}]= \mathbb{E}[({\bf A}\widetilde{{\bf h}} + {\bm{\Wc}})({\bf A}\widetilde{{\bf h}} + {\bm{\Wc}})^{\sf H}]\nonumber\\
&= \mathbb{E}[{\bf A}\widetilde{{\bf h}}\widetilde{{\bf h}}^{\sf H}{\bf A}^{\sf H} + {\bm{\Wc}}\Wbc^{\sf H}]= {\bf A} {\bf R}_{\widetilde{h}} {\bf A}^{\sf H} + \sigma_w^2{\bf I}.
\end{align}
Here we have assumed that noise ${\bm{\Wc}}$ and error $\widetilde{{\bf h}}$ are uncorrelated. Note that ${\bm{\Zc}}$ includes the effect of both the channel estimation error and the noise and plays the central role in the calculation of reliability measure. Specifically, we use the reliability criterion proposed in \cite{6292976} which takes into consideration the fact that for some carrier $i$, the distortion ${\cal Z}(i)$ might be strong enough to take the estimated data symbol $\widehat{{\cal X}}(i)$ out of its correct decision region, while for some other carriers the distortion is not strong enough and the data is decoded correctly. All those data carriers $i$ which satisfy this condition $\langle \widehat{{\cal X}}(i) \rangle = {\cal X}(i)$, where $\langle \cdot \rangle$ represents rounding to the nearest constellation point, are termed \emph{reliable} carriers and the following metric is used to compute the reliability of carrier $i$
\begin{align}\label{eq:reliability}
\mathfrak{R}(i) & = \frac{p({\cal Z}(i) = {\cal X}(i) - \langle \widehat{{\cal X}}(i) \rangle)}{\sum_{k=0,{\cal A}_k \ne \langle \widehat{{\cal X}}(i) \rangle}^{Q-1}p({\cal Z}(i) = {\cal X}(i) - {\cal A}_k)}
\end{align}
where $p(\cdot)$ represents the pdf of ${\bm{\Zc}}$. The numerator in (\ref{eq:reliability}) is the probability that ${\cal Z}(i)$ does not take ${\cal X}(i)$ beyond its correct decision region and the denominator sums the probabilities of all possible incorrect decisions that ${\cal Z}(i)$ can cause (i.e., ${\cal Z}(i)$ takes $\langle\widehat{\cal X}(i)\rangle$ to a QAM constellation point ${\cal A}_k$ different from ${\cal X}(i)$). The idea of reliability calculation is shown graphically in Fig. \ref{fig:GeomRel}. In this figure, although both $\widehat{{\cal X}}(1)$ and $\widehat{{\cal X}}(2)$ are equiprobable to be decoded as ${\cal X}$ (numerator of (\ref{eq:reliability}) will have same value), $\widehat{{\cal X}}(2)$ is less likely to be decoded as any other constellation point (denominator of (\ref{eq:reliability}) for $\widehat{{\cal X}}(2)$ will be smaller) and thus more reliable. Thus, it is obvious that the higher the value of $\mathfrak{R}$ the higher the probability of staying in correct decision region and hence the higher the reliability of the carrier. Note that our reliability calculations of (\ref{eq:reliability}) require the error covariance which is already available at the output of the \texttt{RS1} algorithm as mentioned in Sec. \ref{sec:covar}. Here we would like to point out that the general approach of using reliable data carriers for enhanced channel estimation is not new and techniques employing reliable carriers exist \cite{5419090, 4359544, 1275673}. Specifically, the reliability measure $\mathfrak{R}$ in (\ref{eq:reliability}) is similar to log-likelihood ratios (LLRs) commonly used in joint channel estimation and data detection methods similar to turbo-equalizers (for example, see \cite{1267050} and the references therein).
\begin{figure}[htbp]
\centering
\input{ReliabilityGeometrical.tikz}
\caption{Geometrical representation of the reliability measurement. $\widehat{{\cal X}}(2)$ is more reliable than $\widehat{{\cal X}}(1)$ as it is less probable to be confused with other constellation points.}\label{fig:GeomRel}
\end{figure}
\begin{algorithm}
\caption{Channel Estimation using Pilots + Reliable Carriers\label{alg:Alg3}}
\begin{enumerate}
\item Run \texttt{Algorithm 1} or \texttt{2} to get CIR estimates
\item Each antenna $r$ uses its estimated channel to find top $U$ reliable carriers ${\cal R}^r$ and sends ${\cal R}^r$ to its central antenna
\item Each antenna finds the intersection of received reliable carriers ${\cal R} = \bigcap_{r\in {\cal N}^+} {\cal R}^r$ and sends it back to its neighbors
\item Each antenna sends back data corresponding to ${\cal R}$ to its central antenna
\item Each antenna further refines the reliable carriers by selecting only those with same data. Call this list ${\cal R}^\star$.
\item Each antenna uses the carriers ${\cal R}^\star$ and the pilots to perform \texttt{SABMP} recovery
\end{enumerate}
\end{algorithm}
Using (\ref{eq:reliability}) each antenna determines the reliability of all data carriers and then select the $U$ carriers with highest reliability values. Let ${\cal R}^r$ denote the index set of these $U$ reliable carriers for antenna $r$. One possible approach could be that each receiver uses these reliable carriers to enhance the CIR estimate by using Algorithm 1 or 2. However, the antennas can collaborate to enhance the reliability even further. First, each antenna $r_C$ acting as a central antenna collects the indices of the reliable carriers from its 4-neighbors and returns the indices of the reliable carriers common to all antennas, i.e., ${\cal R} = \bigcap_{r\in {\cal N}^+} {\cal R}^r$.
The central antenna can go one step further and ask its neighbors to share their equalized data on the common carriers. The central antenna in turn prunes the set ${\cal R}$ further and only retains those carriers ${\cal R}^\star$ on which there is agreement among the neighbors on the value of the transmitted data. The central antenna can now use the enlarged set of pilots plus reliable carriers ${\cal P} \cup {\cal R}^\star$ to revisit the channel estimation problem starting from the system of equations
\begin{align}\label{eq:reliabilityprobmodel}
{\bm{\Yc}}^r({\cal P} \cup {{\cal R}^\star}) = {\bf A}({\cal P} \cup {\cal R}^\star) {\bf h}^r + {\bm{\Wc}}^r({\cal P} \cup {\cal R}^\star)
\end{align}
and estimate channel ${\bf h}^r$. The resulting algorithm is presented in Algorithm \ref{alg:Alg3}.
At this stage we would like to point out that implementation of all three algorithms is independent of the antenna array configuration. This is due to the fact that each antenna only deals with its direct neighbors. Therefore, as far as the antennas have the knowledge of their neighbors these algorithms can be implemented on any one-, two-, or three- dimensional arrays with arbitrary topology. Furthermore, the development of algorithms assumed single-antenna UE's; however, the techniques could be easily extended to multiple-antenna UE's such as the LTE UE's which are often equipped with two or four closely located antennas. Since the antennas are closely located, their channels will exhibit the approximately common support property. Therefore, the algorithms explained above could be used to exploit correlation among the channel tap locations to further improve the channel estimates. The only difference is that the effective number of collaborating antennas in each tier will scale with the number of antennas on UE. For example, if there are two transmit antennas on a UE, the number of collaborating antennas in each tier will double.
\section{Simulation Results}\label{sec:results}
\subsection{System Setup}
In this section we will present extensive simulation results to demonstrate the performance of our proposed channel estimation algorithms. Specifically, we consider a MIMO-OFDM system with the simulation parameters given in Table \ref{tab:simparams}.
\begin{table}[htbp]
\centering
{\small
\begin{tabular}{| l | c |}
\hline\hlin
\textbf{Parameters} & \textbf{Value}\\
\hline
Uniform Rectangular Array $(M \times G)$ & $20 \times 20$\\
\hline
Number of carriers $(N)$& $512$\\
\hline
Number of pilots $(K)$ & $8, 16$\\
\hline
QAM modulation order $(Q)$ & $4,16$\\
\hline
Channel length $(L)$& $32, 64$\\
\hline
Channel sparsity $(n)$& $\approx 3, 5, 7$\\
\hline
Collaboration parameter $(D)$& $3$\\
\hline
\end{tabular}}
\caption{System Parameters for simulation
\label{tab:simparams}
\end{table}
For simulations, sparse Rayleigh channels are generated where the channel statistics are assumed to be unknown at the receivers. For space-invariant arrays the active tap locations remain fixed across the array. However, for the space-variant case the active tap locations vary slowly across the array. Specifically, we use the IlmProp channel modeling tool \cite{Del_Galdo_ARS_03, IlmProp} for channel generation. It is important to note that there is a general lack of channel models for massive MIMO scenarios and currently IlmProp seems to be one of the best options available to the research community for channel generation. Please refer to Appendix \ref{apd:channelmodel} for relevant discussion. The channels are generated We generate the channels by placing point-like scatterers and the transmitter randomly in the environment and make sure that the line-of-sight is obstructed. Moreover, the number of scatterers is set according to the desired sparsity i.e., $n$. Since the resulting CIR contains many small non-zero components along with the dominant ones, we discard the small ones and keep just the top $n$ components. Further, the center frequency and signal bandwidth are chosen to be $2.6$ GHz and $20$ MHz respectively as specified in the 3GPP-LTE standard. Moreover, to generate the SIA and SVA behavior the distance between antennas was adjusted accordingly.
\subsection{Methods for Performance Comparisons}
The channel vectors ${\bf h}^r$ are estimated using
\begin{inparaenum}[\itshape a\upshape)]
\item least-squares method with known true MST locations (oracle-LS),
\item block-sparse recovery method (BR),
\item proposed Marginal-based channel estimation using pilots (MB-P),
\item proposed Integer-based channel estimation using pilots (IB-P), and
\item proposed Marginal- or Integer-based channel estimation using pilots and reliable carriers (MB-R / IB-R),
\end{inparaenum}
The first two methods are used to benchmark the performance of our algorithm. Oracle-LS knows the channel support at each antenna and hence the only burden is tap estimation using the available pilots. The block sparse recovery method (BR) works in the space-invariant case and uses the fact that the channel support is the same across the array. It casts the problem as several block sparse problems where each receiver collects all observations from its neighbors to estimate the channels. We use the block sparse Bayesian learning algorithm (BSBL) proposed in \cite{BSBL} for block sparse vector estimation as it has been shown to be superior to other methods.
\subsection{Evaluation Criteria}
To evaluate channel estimation performance we use:
\begin{enumerate}
\item Normalized mean-squared error (NMSE) between true and estimated channel vectors.
\begin{equation}
\text{NMSE} = 10 \log_{10} \left( \frac{1}{\Theta} \sum_{\theta=1}^\Theta \frac{\left\| \widehat{{\bf h}}_\theta - {\bf h}_\theta \right\|^2}{\left\| {\bf h}_\theta \right\|^2} \right),
\end{equation}
where $\Theta$ is the number of trials. ${\bf h}_\theta$ and $\widehat{{\bf h}}_\theta$ are the original and estimated CIR at the $\theta$th iteration respectively.
\item Bit-error-rate (BER) between the transmitted data and the recovered data at receivers using the estimated channels. We use zero-forcing equalization to recover the data passed through the channels.
In all of the experiments we average the NMSE and BER over $\Theta=100$ trials.
\end{enumerate}
\subsection{Experiments}
\subsubsection{Experiment 1 - How many pilots?}
In this experiment, we are interested in finding the required number of pilots for successful recovery of channels of length $L=64$. The graphs in Fig. \ref{fig:sim:Exp1} show the channel recovery success rate vs varying number of pilots for both SIA and SVA.
\begin{figure*}[htbp]
\centering
\input{Exp1Results.tikz}
\caption{Experiment 1: How many pilots are needed to successfully recover the CIR?}\label{fig:sim:Exp1}
\end{figure*}
Note that both pilot-based and data-aided versions of MB and IB algorithms were simulated. Here, success rate is defined as the ratio of the number of successful trials to the number of total trials, where a trial was considered successful when the NMSE was better than $-10$ dB. The SNR was fixed at $10$ dB and the number of pilots $K$ was varied from $2$ to $42$ while $\Theta=100$ trials were conducted for each value of $K$. Channel sparsity was assumed to be $n=3$ and QAM signals of order $Q=4$ were passed through the channels. It is evident from the graphs that for SIA just $6$ pilots are needed by both MB-R and IB-R to achieve a success rate $> 50\%$ and only $12$ pilots to achieve a $100\%$ success rate. This is a small fraction of the channel length $L=64$ (i.e., $9.37\%$ and $18.75\%$ respectively).
\subsubsection{Experiment 2 - Comparison between MB and IB}
In this experiment, we compare the performance of the proposed MB and IB channel estimation algorithms. Channels of length $L=64$ and sparsity $n=3$ were estimated using $K=16$ pilots. The top row of Fig. \ref{fig:sim:Exp2} shows the NMSE in estimated CIRs while the bottom row shows the BER of the recovered data using these CIR estimates. Moreover, as apparent from the labels, this experiment was run using three different choices of parameters, namely, ($Q=4$, SIA), ($Q=4$, SVA), and ($Q=16$, SIA) respectively. The figure shows that incorporating reliable carriers results in significant performance gains. The figure also shows that the algorithms perform equally well for both SIA and SVA configurations and, hence are robust to how accurate property 2 is.
\begin{figure*}[htbp]
\centering
\input{Exp2Results.tikz}
\caption{Experiment 2: Performance comparison between marginal-based (MB) and integer-based (IB) algorithms.}\label{fig:sim:Exp2}
\end{figure*}
An important observation is that there is little advantage of MB algorithms over IB algorithms in this setting. However in some scenarios the improvement could be better as it uses more (and accurate) information to estimate the channels. For example, the difference between the performance of MB and IB algorithms is more evident when the number of pilots is further reduced, as could be seen in the success rate curves of Fig. \ref{fig:sim:Exp1}. Also note that the gain of MB algorithms is much more noticeable for their pilot-based versions. However, this advantage is at the expense of a relatively high computational and communication cost as depicted in Table \ref{tab:sim:PBvsIB_runtime} which compares the runtime for the two algorithms.
\begin{table}[htbp]
\centering
{\small
\begin{tabular}{|c|c|}
\hline\hline
\textbf{Algorithm} & \textbf{Run time (sec)}\\
\hline
MB-P & 0.3897\\
\hline
IB-P & 0.3095\\
\hline
\end{tabular}}
\caption{Run time comparison of the MB-P and IB-P.
\label{tab:sim:PBvsIB_runtime}
\end{table}
\subsubsection{Experiment 3 - Comparison with BR and oracle-LS algorithms
In this experiment, we benchmark the performance of the proposed algorithms against BR and oracle-LS. Here we use QAM signals of order $Q=4$ and $Q=16$ and pass them through a channel of length $L=32$ and sparsity $n=3$. We use $K=8$ pilots and confine our attention to the space-invariant case as this is an essential requirement for block sparsity algorithm to work.
\begin{figure*}[htbp]
\centering
\input{Exp3Results.tikz}
\caption{Experiment 3: Performance comparison between the proposed and the BR and oracle-LS algorithms.}\label{fig:sim:allcomparison}
\end{figure*}
It is obvious from the graphs of Fig. \ref{fig:sim:allcomparison} that the proposed MB-R algorithm has the best performance among all algorithms.
The gain in the performance of the proposed algorithm over others is more prominent for higher values of SNR. Specifically, note that there is a difference of nearly two orders of magnitude in the BER of MB-R and BR when SNR $=35$ dB and $Q=4$
\subsubsection{Experiment 4 - Effect of sparsity rate}
In this experiment, we study the performance of the proposed algorithms under different sparsity rates. Channels of length $L=64$ were generated having $n=3,5$ and $7$ non-zero taps corresponding to sparsity rate of $4.7\% - 11\%$. In this experiment, QAM signals of order $Q=4$ were passed through the channels and $K=16$ number of pilots were used. Fig. \ref{fig:sim:EffectOfSparsityRate} shows the NMSE performance of the proposed IB-R algorithm. It is evident from the graphs that the performance of the proposed algorithm degrades gracefully as CIR gets denser. A similar performance is achieved for the MB-R algorithm. The degradation in reconstruction accuracy with increased number of non-zeros is a common trend in all sparse recovery algorithms
\begin{figure}[htbp]
\centering
\input{Exp4Results.tikz}
\caption{Experiment 4: Effect of channel sparsity on its recovery.}\label{fig:sim:EffectOfSparsityRate}
\end{figure}
\subsubsection{Experiment 5 - Effect of $D$}\label{sec:Exp5}
In this experiment, we study the effect of the number of collaborating antennas on channel estimation for both the SIA and SVA cases. In this experiment we compute the BER in the recovered data for various values of SNR and parameter $D$. Fig. \ref{fig:sim:ChoiceOfDSIA} and \ref{fig:sim:ChoiceOfDSVA} show the BER vs SNR graphs for SIA and SVA respectively. Specifically, we plot for $D=1,2,3,4$ and $5$. Note that when $D=1$, information from only the direct 4 neighbors is taken into consideration and for $D=2$ information belonging to the neighbors of neighbors is also incorporated in the computations.
In the SIA case (Fig. \ref{fig:sim:ChoiceOfDSIA}) QAM signals of order $Q=4$ were passed through channels of length $L=32$ and sparsity $n=3$ and the corresponding CIRs were estimated using IB-P with the help of $K=8$ pilots. Fig. \ref{fig:sim:ChoiceOfDSIA} shows that sharing improves the BER performance. Specifically, as we increase the scope of sharing from the first tier of neighbors ($D=1$) to the fifth tier ($D=5$), we observe a drop in BER. However, the improvement in BER is not significant beyond $D=3$. This behavior is dependent on several factors such as the length of channel ($L$) to be estimated, number of pilots ($K$) and the channel sparsity ($n$). For instance, in this example, if the number of pilots is reduced (i.e., $K<8$) the estimated CIRs will be more erroneous. However, the effect of this error is compensated by using more neighbors to average the marginals/scores (i.e., higher $D$). Therefore, in this reduced number of pilots scenario we might observe significant improvement beyond $D=3$ as well.
In the SVA case (Fig. \ref{fig:sim:ChoiceOfDSVA}) QAM signals of order $Q=4$ were passed through channels of length $L=64$ and sparsity $n=3$ and the corresponding CIRs were estimated using IB-P with the help of $K=16$ pilots. Fig. \ref{fig:sim:ChoiceOfDSVA} shows that we do not gain anything for higher values of $D$. This is due to the space-variant nature of the impulse response that adding more information does not help in the improvement of estimation accuracy. Therefore, setting $D=1$ or $2$ might be sufficient in this scenario.
\begin{figure}[htbp]
\centering
\includegraphics[width=1.15\columnwidth]{piters_SIA}
\caption{Experiment 5: Information sharing among antennas belonging to different neighbor levels ($D = 1 - 5$) adds to CIR estimation accuracy in the SIA case.}\label{fig:sim:ChoiceOfDSIA}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=1.15\columnwidth]{piters_SVA}
\caption{Experiment 5: In the SVA case information sharing does not help in the improvement of CIR estimation accuracy.}\label{fig:sim:ChoiceOfDSVA}
\end{figure}
\section{Conclusion and Future Work}\label{sec:conclusions}
Massive MIMO systems provide substantial performance gains as compared to the traditional MIMO systems. However, these gains come with a huge requirement of estimating a large number of channels. In this paper we have shown that these channels could be estimated in a collaborative manner where the antennas collaborate with their neighboring antennas. Three algorithms based on this collaborative method have been presented. The algorithms show good performance under different scenarios and that too while using a relatively small number of pilots.
\appendices
\section{Channel Models}\label{apd:channelmodel}
Numerous good/accurate models exist for MIMO wireless channels. However, almost all consider the uniform linear array configuration of the antennas \cite{chan_mod_bolcskei, SCM_3GPP}. It should be noted that from a practical point of view, assuming a uniform linear array (ULA) becomes unfeasible for the purpose of modeling a large scale antenna array. Therefore, the use of two- or three-dimensional antenna arrays can be more appropriate in massive MIMO systems. There have been a few attempts in developing channel models for the two- and three-dimensional antenna array configurations. For example, the IlmProp tool available online \cite{IlmProp} allows to generate CIR for 2D antenna array configuration. A similar proposal was put forward in \cite{boon_2D_antennaarray, 3D_chann_proposal}, to extend the spatial channel model (SCM) standard \cite{SCM_3GPP}. Similar models have also been considered in Winner II \cite{WinnerII} and Winner+ \cite{Winner_plus} initiatives. However, these models come with their limitations. For example, in IlmProp the parameters have been estimated using much lower spatial resolution. Indeed, there is a general lack of channel models for the massive MIMO scenarios and there is a need of more work in this direction.
\bibliographystyle{IEEEtran}
|
{
"redpajama_set_name": "RedPajamaArXiv"
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| 3,651
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{"url":"https:\/\/www.physicsforums.com\/threads\/dirac-equation-for-the-conjugated-field.362835\/","text":"Dirac equation for the conjugated field\n\n1. Dec 12, 2009\n\nphsopher\n\nThis is probably a stupid question, but when I apply the Euler-Lagrange equation to the Lagrangian density of the Dirac field I get for the conjugate field\n\n$$\\bar{\\psi} (-i \\partial_\\mu \\gamma^{\\mu} -m) = 0$$ (derivative acts to the left).\n\nBut when I take a hermitian conjugate of the Dirac equation for the field I get an extra $$\\gamma^0$$:\n\n$$0 = \\left[ (i \\partial_\\mu \\gamma^{\\mu} -m)\\psi \\right]^\\dagger = \\psi^\\dagger (-i \\partial_\\mu (\\gamma^{\\mu})^\\dagger -m) = \\psi^\\dagger (-i \\partial_\\mu \\gamma^0 \\gamma^{\\mu} \\gamma^0 -m) = \\psi^\\dagger \\gamma^0(-i \\partial_\\mu \\gamma^{\\mu} \\gamma^0 -m) = \\bar{\\psi} (-i \\partial_\\mu \\gamma^{\\mu} \\gamma^0 -m)$$.\n\nWhat am I missing?\n\n2. Dec 12, 2009\n\ndextercioby\n\nTaking the normal hermitean conjugate of the <original> Dirac eqn will not give you the <conjugated> equation. You need an extra $\\gamma_0$.\n\nActually, you took out the $\\gamma_0$ from the paranthesis without it being there next to the <m>. That's wrong.\n\n3. Dec 13, 2009\n\nphsopher\n\nAh yes, of course. Then I can multiply with $$\\gamma^0$$ from the right and get the same equation as from Euler-Lagrange. I'm such an idiot. Thanks a lot.","date":"2018-03-21 15:22:17","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 0.9407432675361633, \"perplexity\": 660.6532781064305}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-13\/segments\/1521257647660.83\/warc\/CC-MAIN-20180321141313-20180321161313-00738.warc.gz\"}"}
| null | null |
Brand new 1:64 scale diecast model of Mack R Model With 28' Pop Trailer Briggs Transportation die cast model by First Gear.
Opening hood with complete engine detail.
Pivoting fifth-wheel for trailer attachments.
|
{
"redpajama_set_name": "RedPajamaC4"
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| 5,459
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Justia Forms Business Contracts Leatt Corp Leatt Corp. Amended and Restated 2011 Equity Incentive Plan
Leatt Corp. Amended and Restated 2011 Equity Incentive Plan as amended
EX-4.2 2 exhibit4-2.htm EX-4.2 Leatt Corp.: Exhibit 4.2 - Filed by newsfilecorp.com
LEATT CORPORATION
AMENDED AND RESTATED 2011 EQUITY INCENTIVE PLAN
Purposes of the Plan. Leatt Corporation, a Nevada corporation (the "Company") hereby establishes the LEATT CORPORATION AMENDED AND RESTATED 2011 EQUITY INCENTIVE PLAN (the "Plan").The purposes of this Plan are to attract and retain the best available personnel for positions of substantial responsibility, to provide additional incentive to Employees, Directors and Consultants, and to promote the long-term growth and profitability of the Company. The Plan permits the grant of Incentive Stock Options, Nonstatutory Stock Options, Restricted Stock, Restricted Stock Units, Stock Appreciation Rights, Performance Units and Performance Shares as the Administrator may determine.
Definitions. The following definitions will apply to the terms in the Plan:
"Administrator" means the Board or any of its Committees as will be administering the Plan, in accordance with Section 4.
"Applicable Laws" means the requirements relating to the administration of equity-based awards under U.S. state corporate laws, U.S. federal and state securities laws, the Code, any stock exchange or quotation system on which the Common Stock is listed or quoted and the applicable laws of any foreign country or jurisdiction where Awards are, or will be, granted under the Plan.
"Award" means, individually or collectively, a grant under the Plan of Options, SARs, Restricted Stock, Restricted Stock Units, Performance Units or Performance Shares.
"Award Agreement" means the written or electronic agreement setting forth the terms and provisions applicable to each Award granted under the Plan. The Award Agreement is subject to the terms and conditions of the Plan.
"Board" means the Board of Directors of the Company.
"Change in Control" means the occurrence of any of the following events:
(i) Any "person" (as such term is used in Sections 13(d) and 14(d) of the Exchange Act) becomes the "beneficial owner" (as defined in Rule 13d-3 of the Exchange Act), directly or indirectly, of securities of the Company representing fifty percent (50%) or more of the total voting power represented by the Company's then outstanding voting securities; provided however, that for purposes of this subsection (i) any acquisition of securities directly from the Company shall not constitute a Change in Control; or
(ii) The consummation of the sale or disposition by the Company of all or substantially all of the Company's assets;
(iii) A change in the composition of the Board occurring within a two-year period, as a result of which fewer than a majority of the directors are Incumbent Directors. "Incumbent Directors" means directors who either (A) are Directors as of the effective date of the Plan, or (B) are elected, or nominated for election, to the Board with the affirmative votes of at least a majority of the Incumbent Directors at the time of such election or nomination (but will not include an individual whose election or nomination is in connection with an actual or threatened proxy contest relating to the election of directors to the Company); or
(iv) The consummation of a merger or consolidation of the Company with any other corporation, other than a merger or consolidation which would result in the voting securities of the Company outstanding immediately prior thereto continuing to represent (either by remaining outstanding or by being converted into voting securities of the surviving entity or its parent) at least fifty percent (50%) of the total voting power represented by the voting securities of the Company or such surviving entity or its parent outstanding immediately after such merger or consolidation.
For avoidance of doubt, a transaction will not constitute a Change in Control if: (i) its sole purpose is the change the state of the Company's incorporation, or (ii) its sole purpose is to create a holding company that will be owned in substantially the same proportions by the persons who held the Company's securities immediately before such transaction.
"Code" means the Internal Revenue Code of 1986, as amended. Any reference in the Plan to a section of the Code will be a reference to any successor or amended section of the Code.
"Committee" means a committee of Directors or of other individuals satisfying Applicable Laws appointed by the Board in accordance with Section 4 hereof.
"Common Stock" means the common stock of the Company.
"Company" means Leatt Corporation, a Nevada corporation, or any successor thereto.
"Consultant" means any person, including an advisor, engaged by the Company or a Parent or Subsidiary to render services to such entity.
"Director" means a member of the Board.
"Disability" means total and permanent disability as determined by the Administrator in its discretion in accordance with uniform and non-discriminatory standards adopted by the Administrator from time to time.
"Employee" means any person, including Officers and Directors, employed by the Company or any Parent or Subsidiary of the Company. Neither service as a Director nor payment of a director's fee by the Company will be sufficient to constitute "employment" by the Company.
"Exchange Act" means the Securities Exchange Act of 1934, as amended.
"Fair Market Value" means, as of any date, the value of Common Stock determined as follows:
(i) If the Common Stock is listed on any established stock exchange or a national market system, including without limitation any division or subdivision of the Nasdaq Stock Market, its Fair Market Value will be the closing sales price for such stock (or the closing bid, if no sales were reported) as quoted on such exchange or system on the day of determination, as reported in The Wall Street Journal or such other source as the Administrator deems reliable;
(ii) If the Common Stock is regularly quoted by a recognized securities dealer but selling prices are not reported, including without limitation quotation through the over the counter bulletin board ("OTCBB") quotation service administered by the Financial Industry Regulatory Authority ("FINRA"), the Fair Market Value of a Share will be the mean between the high bid and low asked prices for the Common Stock on the day of determination, as reported in The Wall Street Journal or such other source as the Administrator deems reliable; or
(iii) In the absence of an established market for the Common Stock, the Fair Market Value will be determined in good faith by the Administrator, and to the extent Section 15 applies (a) with respect to ISOs, the Fair Market Value shall be determined in a manner consistent with Code section 422 or (b) with respect to NSOs or SARs, the Fair Market Value shall be determined in a manner consistent with Code section 409A.
"Fiscal Year" means the fiscal year of the Company.
"Grant Date" means, for all purposes, the date on which the Administrator determines to grant an Award, or such other later date as is determined by the Administrator, provided that the Administrator cannot grant an Award prior to the date the material terms of the Award are established. Notice of the Administrator's determination to grant an Award will be provided to each Participant within a reasonable time after the Grant Date.
"Incentive Stock Option" or "ISO" means an Option that by its terms qualifies and is otherwise intended to qualify as an incentive stock option within the meaning of Section 422 of the Code and the regulations promulgated thereunder.
"Nonstatutory Stock Option" or "NSO" means an Option that by its terms does not qualify or is not intended to qualify as an ISO.
"Officer" means a person who is an officer of the Company within the meaning of Section 16 of the Exchange Act and the rules and regulations promulgated thereunder.
"Option" means a stock option granted pursuant to the Plan.
"Optioned Shares" means the Common Stock subject to an Option.
"Optionee" means the holder of an outstanding Option.
"Parent" means a "parent corporation," whether now or hereafter existing, as defined in Section 424(e) of the Code.
"Participant" means the holder of an outstanding Award.
"Performance Share" means an Award denominated in Shares which may vest in whole or in part upon attainment of performance goals or other vesting criteria as the Administrator may determine pursuant to Section 10.
"Performance Unit" means an Award which may vest in whole or in part upon attainment of performance goals or other vesting criteria as the Administrator may determine and which may be settled for cash, Shares or other securities or a combination of the foregoing pursuant to Section 10.
"Period of Restriction" means the period during which Shares of Restricted Stock are subject to forfeiture or restrictions on transfer pursuant to Section 7.
"Plan" means this 2011 Equity Incentive Plan.
"Restricted Stock" means Shares awarded to a Participant which are subject to forfeiture and restrictions on transferability in accordance with Section 7.
"Restricted Stock Unit" means the right to receive one Share at the end of a specified period of time, which right is subject to forfeiture in accordance with Section 8 of the Plan.
"Rule 16b-3" means Rule 16b-3 of the Exchange Act or any successor to Rule 16b-3.
"Section" means a paragraph or section of this Plan.
"Section 16(b)" means Section 16(b) of the Exchange Act.
"Service" shall mean service as an Employee, Director or Consultant.
"Service Provider" means an Employee, Director or Consultant.
"Share" means a share of the Common Stock, as adjusted in accordance with Section 13.
"Stock Appreciation Right" or "SAR" means the right to receive payment from the Company in an amount no greater than the excess of the Fair Market Value of a Share at the date the SAR is exercised over a specified price fixed by the Administrator in the Award Agreement, which shall not be less than the Fair Market Value of a Share on the Grant Date. In the case of a SAR which is granted in connection with an Option, the specified price shall be the Option exercise price.
"Subsidiary" means a "subsidiary corporation," whether now or hereafter existing, as defined in Section 424(f) of the Code.
"Ten Percent Owner" means any Service Provider who is, on the grant date of an ISO, the owner of Shares (determined with application of ownership attribution rules of Code Section 424(d)) possessing more than 10% of the total combined voting power of all classes of stock of the Company or any of its Subsidiaries.
Stock Subject to the Plan.
Stock Subject to the Plan. Subject to the provisions of Section 13, the maximum aggregate number of Shares that may be issued under the Plan is Nine Hundred and Twenty Thousand (920,000) Shares. The Shares may be authorized but unissued, or reacquired Common Stock.
Lapsed Awards. If an Award expires or becomes unexercisable without having been exercised in full or, with respect to Restricted Stock, Restricted Stock Units, Performance Shares or Performance Units, is forfeited in whole or in part to the Company, the unpurchased Shares (or for Awards other than Options and SARs, the forfeited or unissued Shares) which were subject to the Award will become available for future grant or sale under the Plan (unless the Plan has terminated). With respect to SARs, only Shares actually issued pursuant to a SAR will cease to be available under the Plan; all remaining Shares subject to the SARs will remain available for future grant or sale under the Plan (unless the Plan has terminated). Shares that have actually been issued under the Plan under any Award will not be returned to the Plan and will not become available for future distribution under the Plan; provided, however, that if Shares issued pursuant to Awards of Restricted Stock, Restricted Stock Units, Performance Shares or Performance Units are forfeited to the Company, such Shares will become available for future grant under the Plan. Shares withheld by the Company to pay the exercise price of an Award or to satisfy tax withholding obligations with respect to an Award will become available for future grant or sale under the Plan. To the extent an Award under the Plan is paid out in cash rather than Shares, such cash payment will not result in reducing the number of Shares available for issuance under the Plan.
Share Reserve. The Company, during the term of this Plan, will at all times reserve and keep available such number of Shares as will be sufficient to satisfy the requirements of the Plan.
Administration of the Plan.
Procedure. The Plan shall be administered by the Board or a Committee (or Committees) appointed by the Board, which Committee shall be constituted to comply with Applicable Laws. If and so long as the Common Stock is registered under Section 12(b) or 12(g) of the Exchange Act, the Board shall consider in selecting the Administrator and the membership of any committee acting as Administrator the requirements regarding: (i) "nonemployee directors" within the meaning of Rule 16b-3 under the Exchange Act; (ii) "independent directors" as described in the listing requirements for any stock exchange on which Shares are listed; and (iii) Section 15(b)(i) of the Plan, if the Company pays salaries for which it claims deductions that are subject to the Code section 162(m) limitation on its U.S. tax returns. The Board may delegate the responsibility for administering the Plan with respect to designated classes of eligible Participants to different committees consisting of two or more members of the Board, subject to such limitations as the Board or the Administrator deems appropriate. Committee members shall serve for such term as the Board may determine, subject to removal by the Board at any time.
Powers of the Administrator. Subject to the provisions of the Plan and the approval of any relevant authorities, and in the case of a Committee, subject to the specific duties delegated by the Board to such Committee, the Administrator will have the authority, in its discretion:
i. to determine the Fair Market Value;
ii. to select the Service Providers to whom Awards may be granted hereunder;
iii. to determine the number of Shares to be covered by each Award granted hereunder;
iv. to approve forms of agreement for use under the Plan;
v. to determine the terms and conditions, not inconsistent with the terms of the Plan, of any Award granted hereunder. Such terms and conditions include, but are not limited to, the exercise price, the time or times when Awards may be exercised (which may be based on continued employment, continued service or performance criteria), any vesting acceleration (whether by reason of a Change of Control or otherwise) or waiver of forfeiture restrictions, and any restriction or limitation regarding any Award or the Shares relating thereto, based in each case on such factors as the Administrator, in its sole discretion, will determine;
vi. to construe and interpret the terms of the Plan and Awards granted pursuant to the Plan, including the right to construe disputed or doubtful Plan and Award provisions;
vii. to prescribe, amend and rescind rules and regulations relating to the Plan;
viii. to modify or amend each Award (subject to Section 19(c)) to the extent any modification or amendment is consistent with the terms of the Plan. The Administrator shall have the discretion to extend the exercise period of Options generally provided the exercise period is not extended beyond the earlier of the original term of the Option or 10 years from the original grant date, or specifically (1) if the exercise period of an Option is extended (but to no more than 10 years from the original grant date) at a time when the exercise price equals or exceeds the fair market value of the Optioned Shares or (2) an Option cannot be exercised because such exercise would violate Applicable Laws, provided that the exercise period is not extended more than 30 days after the exercise of the Option would no longer violate Applicable Laws.
ix. to allow Participants to satisfy withholding tax obligations in such manner as prescribed in Section 14;
x. to authorize any person to execute on behalf of the Company any instrument required to effect the grant of an Award previously granted by the Administrator;
xi. to delay issuance of Shares or suspend Participant's right to exercise an Award as deemed necessary to comply with Applicable Laws; and
xii. to make all other determinations deemed necessary or advisable for administering the Plan.
Effect of Administrator's Decision. The Administrator's decisions, determinations and interpretations will be final and binding on all Participants and any other holders of Awards. Any decision or action taken or to be taken by the Administrator, arising out of or in connection with the construction, administration, interpretation and effect of the Plan and of its rules and regulations, shall, to the maximum extent permitted by Applicable Laws, be within its absolute discretion (except as otherwise specifically provided in the Plan) and shall be final, binding and conclusive upon the Company, all Participants and any person claiming under or through any Participant.
Eligibility. NSOs, Restricted Stock, Restricted Stock Units, SARs, Performance Units and Performance Shares may be granted to Service Providers. ISOs may be granted as specified in Section 15(a).
Stock Options.
a. Grant of Options. Subject to the terms and conditions of the Plan, the Administrator, at any time and from time to time, may grant Options to Service Providers in such amounts as the Administrator will determine in its sole discretion. For purposes of the foregoing sentence, Service Providers shall include prospective employees or consultants to whom Options are granted in connection with written offers of employment or engagement of services, respectively, with the Company; provided that no Option granted to a prospective employee or consultant may be exercised prior to the commencement of employment or services with the Company. The Administrator may grant NSOs, ISOs, or any combination of the two. ISOs shall be granted in accordance with Section 15(a) of the Plan.
b. Option Award Agreement. Each Option shall be evidenced by an Award Agreement that shall specify the type of Option granted, the Option price, the exercise date, the term of the Option, the number of Shares to which the Option pertains, and such other terms and conditions (which need not be identical among Participants) as the Administrator shall determine in its sole discretion. If the Award Agreement does not specify that the Option is to be treated as an ISO, the Option shall be deemed a NSO.
c. Exercise Price. The per Share exercise price for the Shares to be issued pursuant to exercise of an Option will be no less than the Fair Market Value per Share on the Grant Date.
d. Term of Options. The term of each Option will be stated in the Award Agreement. Unless terminated sooner in accordance with the remaining provisions of this Section 6, each Option shall expire either ten (10) years after the Grant Date, or after a shorter term as may be fixed by the Board. Each Award Agreement shall set forth the extent to which the Option may be exercised following termination of Service. Each Award Agreement shall provide the holder with the right to exercise the Option following the Service Provider's termination of Service during the Option term, to the extent the Option was exercisable for vested Shares upon termination of Service, for at least thirty (30) days if termination of Service is due to any reason other than cause (as defined for this purpose by applicable law, the terms of the Award Agreement or a contract of employment), death or Disability, and for at least six (6) months after termination of Service if due to death or Disability (but in no event later than the expiration of the Option term). If Service is terminated for cause, the Award Agreement may provide that the right to exercise the Option terminates immediately on the effective date of termination of Service. To the extent the Option was not exercisable for vested Shares upon termination of Service, the Option shall terminate on the date of termination of Service. Subject to the foregoing, such provisions shall be determined in the sole discretion of the Administrator, need not be uniform among all Options issued pursuant to the Plan, and may reflect distinctions based on the reasons for termination of Service.
e. Time and Form of Payment.
i. Exercise Date. Each Award Agreement shall specify how and when Shares covered by an Option may be purchased. The Award Agreement may specify waiting periods, the dates on which Options become exercisable or "vested" and, subject to the termination provisions of this section, exercise periods. The Administrator may accelerate the exercisability of any Option or portion thereof.
ii. Exercise of Option. Any Option granted hereunder will be exercisable according to the terms of the Plan and at such times and under such conditions as determined by the Administrator and set forth in the Award Agreement. An Option may not be exercised for a fraction of a Share. An Option will be deemed exercised when the Company receives: (1) notice of exercise (in such form as the Administrator specify from time to time) from the person entitled to exercise the Option, and (2) full payment for the Shares with respect to which the Option is exercised (together with all applicable withholding taxes). Full payment may consist of any consideration and method of payment authorized by the Administrator and permitted by the Award Agreement and the Plan (together with all applicable withholding taxes). Shares issued upon exercise of an Option will be issued in the name of the Optionee or, if requested by the Optionee, in the name of the Optionee and his or her spouse. Until the Shares are issued (as evidenced by the appropriate entry on the books of the Company or of a duly authorized transfer agent of the Company), no right to vote or receive dividends or any other rights as a stockholder will exist with respect to the Optioned Shares, notwithstanding the exercise of the Option. The Company will issue (or cause to be issued) such Shares promptly after the Option is exercised. No adjustment will be made for a dividend or other right for which the record date is prior to the date the Shares are issued, except as provided in Section 13.
iii. Payment. The Administrator will determine the acceptable form of consideration for exercising an Option, including the method of payment. Such consideration may consist entirely of:
(1) cash;
(2) check;
(3) to the extent not prohibited by Section 402 of the Sarbanes-Oxley Act of 2002, a promissory note;
(4) other Shares, provided Shares have a Fair Market Value on the date of surrender equal to the aggregate exercise price of the Shares as to which said Option will be exercised;
(5) to the extent not prohibited by Section 402 of the Sarbanes-Oxley Act of 2002, in accordance with any broker-assisted cashless exercise procedures approved by the Company and as in effect from time to time;
(6) by asking the Company to withhold Shares from the total Shares to be delivered upon exercise equal to the number of Shares having a value equal to the aggregate Exercise Price of the Shares being acquired;
(7) any combination of the foregoing methods of payment; or
(8) such other consideration and method of payment for the issuance of Shares to the extent permitted by Applicable Laws.
f. Forfeiture of Options. All unexercised Options shall be forfeited to the Company in accordance with the terms and conditions set forth in the Award Agreement and again will become available for grant under the Plan.
Restricted Stock.
a. Grant of Restricted Stock. Subject to the terms and conditions of the Plan, the Administrator, at any time and from time to time, may grant Shares of Restricted Stock to Service Providers in such amounts as the Administrator will determine in its sole discretion.
b. Restricted Stock Award Agreement. Each Award of Restricted Stock will be evidenced by an Award Agreement that will specify the Period of Restriction, the number of Shares granted, and such other terms and conditions (which need not be identical among Participants) as the Administrator will determine in its sole discretion. Unless the Administrator determines otherwise, the Company as escrow agent will hold Shares of Restricted Stock until the restrictions on such Shares have lapsed.
c. Vesting Conditions and Other Terms.
i. Vesting Conditions. The Administrator, in its sole discretion, may impose such conditions on the vesting of Shares of Restricted Stock as it may deem advisable or appropriate, including but not limited to, achievement of Company-wide, business unit, or individual goals (including, but not limited to, continued employment or service), or any other basis determined by the Administrator in its discretion. The Administrator, in its discretion, may accelerate the time at which any restrictions will lapse or be removed. The Administrator may, in its discretion, also provide for such complete or partial exceptions to an employment or service restriction as it deems equitable.
ii. Voting Rights. During the Period of Restriction, Service Providers holding Shares of Restricted Stock granted hereunder may exercise full voting rights with respect to those Shares, unless the Administrator determines otherwise.
iii. Dividends and Other Distributions. During the Period of Restriction, Service Providers holding Shares of Restricted Stock will be entitled to receive all dividends and other distributions paid with respect to such Shares, unless the Administrator determines otherwise. If any such dividends or distributions are paid in Shares, the Shares will be subject to the same restrictions on transferability and forfeitability as the Shares of Restricted Stock with respect to which they were paid.
iv. Transferability. Except as provided in this Section, Shares of Restricted Stock may not be sold, transferred, pledged, assigned, or otherwise alienated or hypothecated until the end of the applicable Period of Restriction.
d. Removal of Restrictions. All restrictions imposed on Shares of Restricted Stock shall lapse and the Period of Restriction shall end upon the satisfaction of the vesting conditions imposed by the Administrator. Vested Shares of Restricted Stock will be released from escrow as soon as practicable after the last day of the Period of Restriction or at such other time as the Administrator may determine, but in no event later than the 15th day of the third month following the end of the year in which vesting occurred.
e. Forfeiture of Restricted Stock. On the date set forth in the Award Agreement, the Shares of Restricted Stock for which restrictions have not lapsed will be forfeited and revert to the Company and again will become available for grant under the Plan.
Restricted Stock Units.
a. Grant of Restricted Stock Units. Subject to the terms and conditions of the Plan, the Administrator, at any time and from time to time, may grant Restricted Stock Units to Service Providers in such amounts as the Administrator will determine in its sole discretion.
b. Restricted Stock Units Award Agreement. Each Award of Restricted Stock Units will be evidenced by an Award Agreement that will specify the number of Restricted Stock Units granted, vesting criteria, form of payout, and such other terms and conditions (which need not be identical among Participants) as the Administrator will determine in its sole discretion.
c. Vesting Conditions. The Administrator shall set vesting criteria in its discretion, which, depending on the extent to which the criteria are met, will determine the number of Restricted Stock Units that will be paid out to the Participant. The Administrator may set vesting criteria based upon the achievement of Company-wide, business unit, or individual goals (including, but not limited to, continued employment or service), or any other basis determined by the Administrator in its discretion. At any time after the grant of Restricted Stock Units, the Administrator, in its sole discretion, may reduce or waive any vesting criteria that must be met to receive a payout.
d. Time and Form of Payment. Upon satisfaction of the applicable vesting conditions, payment of vested Restricted Stock Units shall occur in the manner and at the time provided in the Award Agreement, but in no event later than the 15th day of the third month following the end of the year in which vesting occurred. Except as otherwise provided in the Award Agreement, Restricted Stock Units may be paid in cash, Shares, or a combination thereof at the sole discretion of the Administrator. Restricted Stock Units that are fully paid in cash will not reduce the number of Shares available for issuance under the Plan.
e. Forfeiture of Restricted Stock Units. All unvested Restricted Stock Units shall be forfeited to the Company on the date set forth in the Award Agreement and again will become available for grant under the Plan.
Stock Appreciation Rights.
a. Grant of SARs. Subject to the terms and conditions of the Plan, the Administrator, at any time and from time to time, may grant SARs to Service Providers in such amounts as the Administrator will determine in its sole discretion.
b. Award Agreement. Each SAR grant will be evidenced by an Award Agreement that will specify the exercise price, the number of Shares underlying the SAR grant, the term of the SAR, the conditions of exercise, and such other terms and conditions (which need not be identical among Participants) as the Administrator will determine in its sole discretion.
c. Exercise Price and Other Terms. The per Share exercise price for the exercise of an SAR will be no less than the Fair Market Value per Share on the Grant Date.
d. Term of SARs. The term of each SAR will be stated in the Award Agreement. Unless terminated sooner in accordance with the remaining provisions of this Section 9, each SAR shall expire either ten (10) years after the Grant Date, or after a shorter term as may be fixed by the Board. Each Award Agreement shall set forth the extent to which the SAR may be exercised following termination of Service. Each Award Agreement shall provide the holder with the right to exercise the SAR following the Service Provider's termination of Service during the SAR term, to the extent the SAR was vested upon termination of Service, for at least thirty (30) days if termination of Service is due to any reason other than cause (as defined for this purpose by applicable law, the terms of the Award Agreement or a contract of employment), death or Disability, and for at least six (6) months after termination of Service if due to death or Disability (but in no event later than the expiration of the SAR term). If Service is terminated for cause, the Award Agreement may provide that the right to exercise the SAR terminates immediately on the effective date of termination of Service. To the extent the SAR was not vested upon termination of Service, the SAR shall terminate on the date of termination of Service. Subject to the foregoing, such provisions shall be determined in the sole discretion of the Administrator, need not be uniform among all SARs issued pursuant to the Plan, and may reflect distinctions based on the reasons for termination of Service.
e. Time and Form of Payment of SAR Amount. Upon exercise of a SAR, a Participant will be entitled to receive payment from the Company in an amount no greater than: (i) the difference between the Fair Market Value of a Share on the date of exercise over the exercise price; times (ii) the number of Shares with respect to which the SAR is exercised. An Award Agreement may provide for a SAR to be paid in cash, Shares of equivalent value, or a combination thereof.
f. Forfeiture of SARs. All unexercised SARs shall be forfeited to the Company in accordance with the terms and conditions set forth in the Award Agreement and again will become available for grant under the Plan.
Performance Units and Performance Shares.
a. Grant of Performance Units and Performance Shares. Performance Units or Performance Shares may be granted to Service Providers at any time and from time to time, as will be determined by the Administrator, in its sole discretion. The Administrator will have complete discretion in determining the number of Performance Units and Performance Shares granted to each Participant.
b. Award Agreement. Each Award of Performance Units and Shares will be evidenced by an Award Agreement that will specify the initial value, the Performance Period, the number of Performance Units or Performance Shares granted, and such other terms and conditions (which need not be identical among Participants) as the Administrator will determine in its sole discretion.
c. Value of Performance Units and Performance Shares. Each Performance Unit will have an initial value that is established by the Administrator on or before the Grant Date. Each Performance Share will have an initial value equal to the Fair Market Value of a Share on the Grant Date.
d. Vesting Conditions and Performance Period. The Administrator will set performance objectives or other vesting provisions (including, without limitation, continued status as a Service Provider) in its discretion which, depending on the extent to which they are met, will determine the number or value of Performance Units or Performance Shares that will be paid out to the Service Providers. The time period during which the performance objectives or other vesting provisions must be met will be called the "Performance Period." The Administrator may set performance objectives based upon the achievement of Company-wide, divisional, or individual goals or any other basis determined by the Administrator in its discretion.
e. Time and Form of Payment. After the applicable Performance Period has ended, the holder of Performance Units or Performance Shares will be entitled to receive a payout of the number of vested Performance Units or Performance Shares by the Participant over the Performance Period, to be determined as a function of the extent to which the corresponding performance objectives or other vesting provisions have been achieved. Vested Performance Units or Performance Shares will be paid as soon as practicable after the expiration of the applicable Performance Period, but in no event later than the 15th day of the third month following the end of the year the applicable Performance Period expired. An Award Agreement may provide for the satisfaction of Performance Unit or Performance Share Awards in cash or Shares (which have an aggregate Fair Market Value equal to the value of the vested Performance Units or Performance Shares at the close of the applicable Performance Period) or in a combination thereof.
f. Forfeiture of Performance Units and Performance Shares. All unvested Performance Units or Performance Shares will be forfeited to the Company on the date set forth in the Award Agreement, and again will become available for grant under the Plan.
Leaves of Absence/Transfer Between Locations. Unless the Administrator provides otherwise or as required by Applicable Laws, vesting of Awards will be suspended during any unpaid leave of absence. An Employee will not cease to be an Employee in the case of (i) any leave of absence approved by the Company or (ii) transfers between locations of the Company or between the Company, its Parent, or any Subsidiary.
Transferability of Awards. Unless determined otherwise by the Administrator, an Award may not be sold, pledged, assigned, hypothecated, transferred, or disposed of in any manner other than by will or by the laws of descent or distribution and may be exercised, during the lifetime of the Participant, only by the Participant. If the Administrator makes an Award transferable, such Award will contain such additional terms and conditions as the Administrator deems appropriate, and transfers will be permitted only to a revocable trust or to one or more family members or a trust established for the benefit of the Participant and/or one or more family members to the extent permitted by Rule 701 of the Securities Act.
Adjustments; Dissolution or Liquidation; Merger or Change in Control.
a. Adjustments. In the event that any dividend or other distribution (whether in the form of cash, Shares, other securities, or other property), recapitalization, stock split, reverse stock split, reorganization, merger, consolidation, split-up, spin-off, combination, repurchase, or exchange of Shares or other securities of the Company, or other change in the corporate structure of the Company affecting the Shares occurs, the Administrator, in order to prevent diminution or enlargement of the benefits or potential benefits intended to be made available under the Plan, shall appropriately adjust the number and class of Shares that may be delivered under the Plan and/or the number, class, and price of Shares covered by each outstanding Award.
b. Dissolution or Liquidation. In the event of the proposed dissolution or liquidation of the Company, the Administrator will notify each Participant as soon as practicable prior to the effective date of such proposed transaction. To the extent it has not been previously exercised, an Award will terminate immediately prior to the consummation of such proposed action.
c. Change in Control. In the event of a merger or Change in Control, any or all outstanding Awards may be assumed by the successor corporation, which assumption shall be binding on all Participants. In the alternative, the successor corporation may substitute equivalent Awards (after taking into account the existing provisions of the Awards). The successor corporation may also issue, in place of outstanding Shares of the Company held by the Participant, substantially similar shares or other property subject to vesting requirements and repurchase restrictions no less favorable to the Participant than those in effect prior to the merger or Change in Control.
In the event that the successor corporation does not assume or substitute for the Award, unless the Administrator provides otherwise, the Participant will fully vest in and have the right to exercise all of his or her outstanding Options and SARs, including Shares as to which such Awards would not otherwise be vested or exercisable, all restrictions on Restricted Stock and Restricted Stock Units will lapse, and, with respect to Performance Shares and Performance Units, all Performance Goals or other vesting criteria will be deemed achieved at target levels and all other terms and conditions met. In addition, if an Option or SAR is not assumed or substituted in the event of a Change in Control, the Administrator will notify the Participant in writing or electronically that the Option or SAR will be exercisable for a period of time determined by the Administrator in its sole discretion, and the Option or SAR will terminate upon the expiration of such period.
For the purposes of this Section 13(c), an Award will be considered assumed if, following the Change in Control, the Award confers the right to purchase or receive, for each Share subject to the Award immediately prior to the Change in Control, the consideration (whether stock, cash, or other securities or property) or, in the case of a SAR upon the exercise of which the Administrator determines to pay cash or a Performance Share or Performance Unit which the Administrator can determine to pay in cash, the fair market value of the consideration received in the merger or Change in Control by holders of Common Stock for each Share held on the effective date of the transaction (and if holders were offered a choice of consideration, the type of consideration chosen by the holders of a majority of the outstanding Shares); provided, however, that if such consideration received in the Change in Control is not solely common stock of the successor corporation or its Parent, the Administrator may, with the consent of the successor corporation, provide for the consideration to be received upon the exercise of an Option or SAR or upon the payout of a Restricted Stock Unit, Performance Share or Performance Unit, for each Share subject to such Award (or in the case of Restricted Stock Units and Performance Units, the number of implied shares determined by dividing the value of the Restricted Stock Units and Performance Units, as applicable, by the per share consideration received by holders of Common Stock in the Change in Control), to be solely common stock of the successor corporation or its Parent equal in fair market value to the per share consideration received by holders of Common Stock in the Change in Control.
Notwithstanding anything in this Section 13(c) to the contrary, an Award that vests, is earned or paid-out upon the satisfaction of one or more performance goals will not be considered assumed if the Company or its successor modifies any of such performance goals without the Participant's consent; provided, however, a modification to such performance goals only to reflect the successor corporation's post-Change in Control corporate structure will not be deemed to invalidate an otherwise valid Award assumption.
Tax Withholding.
a. Withholding Requirements. Prior to the delivery of any Shares or cash pursuant to an Award (or exercise thereof), the Company will have the power and the right to deduct or withhold, or require a Participant to remit to the Company, an amount sufficient to satisfy federal, state, local, foreign or other taxes required by Applicable Laws to be withheld with respect to such Award (or exercise thereof).
b. Withholding Arrangements. The Administrator, in its sole discretion and pursuant to such procedures as it may specify from time to time, may permit a Participant to satisfy such tax withholding obligation, in whole or in part by (without limitation) (i) paying cash, (ii) electing to have the Company withhold otherwise deliverable Shares having a Fair Market Value equal to the amount required to be withheld, or (iii) delivering to the Company already-owned Shares having a Fair Market Value equal to the amount required to be withheld. The amount of the withholding requirement will be deemed to include any amount which the Administrator agrees may be withheld at the time the election is made. The Fair Market Value of the Shares to be withheld or delivered will be determined as of the date that the taxes are required to be withheld.
Provisions Applicable In the Event the Company or the Service Provider is Subject to U.S. Taxation.
Grant of Incentive Stock Options. If the Administrator grants Options to Employees subject to U.S. taxation, the Administrator may grant such Employee an ISO and the following terms shall also apply:
i. Maximum Amount. Subject to the provisions of Section 13, to the extent consistent with Section 422 of the Code, not more than an aggregate of Nine Hundred and Twenty Thousand (920,000) Shares may be issued as ISOs under the Plan.
ii. General Rule. Only Employees shall be eligible for the grant of ISOs.
iii. Continuous Employment. The Optionee must remain in the continuous employ of the Company or its Subsidiaries from the date the ISO is granted until not more than three months before the date on which it is exercised. A leave of absence approved by the Company may exceed ninety (90) days if reemployment upon expiration of such leave is guaranteed by statute or contract. If reemployment upon expiration of a leave of absence approved by the Company is not so guaranteed, then three (3) months following the ninety-first (91st) day of such leave any ISO held by the Optionee will cease to be treated as an ISO.
iv. Award Agreement.
(1) The Administrator shall designate Options granted as ISOs in the Award Agreement. Notwithstanding such designation, to the extent that the aggregate Fair Market Value of the Shares with respect to which ISOs are exercisable for the first time by the Optionee during any calendar year (under all plans of the Company and any Parent or Subsidiary) exceeds one hundred thousand dollars ($100,000), Options will not qualify as an ISO. For purposes of this section, ISOs will be taken into account in the order in which they were granted. The Fair Market Value of the Shares will be determined as of the time the Option with respect to such Shares is granted.
(2) The Award Agreement shall specify the term of the ISO. The term shall not exceed ten (10) years from the Grant Date or five (5) years from the Grant Date for Ten Percent Owners.
(3) The Award Agreement shall specify an exercise price of not less than the Fair Market Value per Share on the Grant Date or one hundred ten percent (110%) of the Fair Market Value per Share on the Grant Date for Ten Percent Owners.
(4) The Award Agreement shall specify that an ISO is not transferable except by will, beneficiary designation or the laws of descent and distribution.
v. Form of Payment. The consideration to be paid for the Shares to be issued upon exercise of an ISO, including the method of payment, shall be determined by the Administrator at the time of grant in accordance with Section 6(e)(iii).
vi. "Disability", for purposes of an ISO, means total and permanent disability as defined in Section 22(e)(3) of the Code.
vii. Notice. In the event of any disposition of the Shares acquired pursuant to the exercise of an ISO within two years from the Grant Date or one year from the exercise date, the Optionee will notify the Company thereof in writing within thirty (30) days after such disposition. In addition, the Optionee shall provide the Company with such information as the Company shall reasonably request in connection with determining the amount and character of Optionee's income, the Company's deduction, and the Company's obligation to withhold taxes or other amounts incurred by reason of a disqualifying disposition, including the amount thereof.
Performance-based Compensation. If the Company pays salaries for which it claims deductions that are subject to the Code Section 162(m) limitation on its U.S. tax returns, then the following terms shall be applied in a manner consistent with the requirements of, and only to the extent required for compliance with, the exclusion from the limitation on deductibility of compensation under Code Section 162(m):
i. Outside Directors. The Board shall consider in selecting the Administrator and the membership of any committee acting as Administrator the provisions regarding "outside directors" within the meaning of Code Section 162(m).
ii. Maximum Amount.
(1) Subject to the provisions of Section 13, the maximum number of Shares that can be awarded to any individual Participant in the aggregate in any one fiscal year of the Company is Seventy-Eight Thousand (78,000) Shares;
(2) For Awards denominated in Shares and satisfied in cash, the maximum Award to any individual Participant in the aggregate in any one fiscal year of the Company is the Fair Market Value of Seventy-Eight Thousand (78,000) Shares on the Grant Date; and
(3) The maximum amount payable pursuant to any cash Awards to any individual Participant in the aggregate in any one fiscal year of the Company is the Fair Market Value of Seventy-Eight Thousand (78,000) Shares on the Grant Date.
iii. Performance Criteria. All performance criteria must be objective and be established in writing prior to the beginning of the performance period or at later time as permitted by Code Section 162(m). Performance criteria may include alternative and multiple performance goals and may be based on one or more business and/or financial criteria. In establishing the performance goals, the Committee in its discretion may include one or any combination of the following criteria in either absolute or relative terms, for the Company or any Subsidiary:
Increased revenue;
Net income measures (including but not limited to income after capital costs and income before or after taxes);
Stock price measures (including but not limited to growth measures and total stockholder return);
Market share;
Earnings per Share (actual or targeted growth);
Earnings before interest, taxes, depreciation, and amortization ("EBITDA");
Cash flow measures (including but not limited to net cash flow and net cash flow before financing activities);
Return measures (including but not limited to return on equity, return on average assets, return on capital, risk-adjusted return on capital, return on investors' capital and return on average equity);
Operating measures (including operating income, funds from operations, cash from operations, after-tax operating income, sales volumes, production volumes, and production efficiency);
Expense measures (including but not limited to overhead cost and general and administrative expense);
Margins;
Stockholder value;
Total stockholder return;
Proceeds from dispositions;
Production volumes;
Total market value; and
Corporate values measures (including but not limited to ethics compliance, environmental, and safety).
Stock Options and SARs Exempt from Code section 409A. If the Administrator grants Options or SARs to Employees subject to U.S. taxation the Administrator may not modify or amend the Options or SARs to the extent that the modification or amendment adds a feature allowing for additional deferral within the meaning of Code section 409A.
No Effect on Employment or Service. Neither the Plan nor any Award will confer upon any Participant any right with respect to continuing the Participant's relationship as a Service Provider with the Company or any Parent or Subsidiary of the Company, nor will they interfere in any way with the Participant's right or the Company's or its Parent's or Subsidiary's right to terminate such relationship at any time, with or without cause, to the extent permitted by Applicable Laws.
Effective Date. The Plan's effective date is the date on which it is adopted by the Board, so long as it is approved by the Company's stockholders at any time within twelve (12) months of such adoption. Upon approval of the Plan by the stockholders of the Company, all Awards issued pursuant to the Plan on or after the Effective Date shall be fully effective as if the stockholders of the Company had approved the Plan on the Effective Date. If the stockholders fail to approve the Plan within one year after the Effective Date, any Awards made hereunder shall be null and void and of no effect.
Term of Plan. The Plan will terminate 10 years following the earlier of (i) the date it was adopted by the Board or (ii) the date it became effective upon approval by stockholders of the Company, unless sooner terminated by the Board pursuant to Section 19.
Amendment and Termination of the Plan.
a. Amendment and Termination. The Board may at any time amend, alter, suspend or terminate the Plan.
b. Stockholder Approval. The Company will obtain stockholder approval of any Plan amendment to the extent necessary and desirable to comply with Applicable Laws.
c. Effect of Amendment or Termination. No amendment, alteration, suspension or termination of the Plan will impair the rights of any Participant, unless mutually agreed otherwise between the Participant and the Administrator, which agreement must be in writing and signed by the Participant and the Company. Termination of the Plan will not affect the Administrator's ability to exercise the powers granted to it hereunder with respect to Awards granted under the Plan prior to the date of such termination.
Conditions Upon Issuance of Shares.
a. Legal Compliance. The Administrator may delay or suspend the issuance and delivery of Shares, suspend the exercise of Options or SARs, or suspend the Plan as necessary to comply Applicable Laws. Shares will not be issued pursuant to the exercise of an Award unless the exercise of such Award and the issuance and delivery of such Shares will comply with Applicable Laws and will be further subject to the approval of counsel for the Company with respect to such compliance.
b. Investment Representations. As a condition to the exercise of an Award, the Company may require the person exercising such Award to represent and warrant at the time of any such exercise that the Shares are being purchased only for investment and without any present intention to sell or distribute such Shares if, in the opinion of counsel for the Company, such a representation is required.
Inability to Obtain Authority. The inability of the Company to obtain authority from any regulatory body having jurisdiction, which authority is deemed by the Company's counsel to be necessary to the lawful issuance and sale of any Shares hereunder, will relieve the Company of any liability in respect of the failure to issue or sell such Shares as to which such requisite authority will not have been obtained.
Repricing Prohibited; Exchange And Buyout of Awards. The repricing of Options or SARs is prohibited without prior stockholder approval. The Administrator may authorize the Company, with prior stockholder approval and the consent of the respective Participants, to issue new Option or SAR Awards in exchange for the surrender and cancellation of any or all outstanding Awards. The Administrator may at any time repurchase Options with payment in cash, Shares or other consideration, based on such terms and conditions as the Administrator and the Participant shall agree.
Substitution and Assumption of Awards. The Administrator may make Awards under the Plan by assumption, substitution or replacement of performance shares, phantom shares, stock awards, stock options, stock appreciation rights or similar awards granted by another entity (including an Parent or Subsidiary), if such assumption, substitution or replacement is connection with an asset acquisition, stock acquisition, merger, consolidation or similar transaction involving the Company (and/or its Parent or Subsidiary) and such other entity (and/or its affiliate). The Administrator may also make Awards under the Plan by assumption, substitution or replacement of a similar type of award granted by the Company prior to the adoption and approval of the Plan. Notwithstanding any provision of the Plan (other than the maximum number of shares of Common Stock that may be issued under the Plan), the terms of such assumed, substituted or replaced Awards shall be as the Administrator, in its discretion, determines is appropriate.
Governing Law. The Plan and all Agreements shall be construed in accordance with and governed by the laws of the State of Nevada.
Adopted by the Board of Directors on November 4, 2015
AMENDMENT NO. 1 TO THE LEATT CORPORATION
This AMENDMENT NO. 1 TO THE LEATT CORPORATION AMENDED AND RESTATED 2011 EQUITY INCENTIVE PLAN (this "Amendment") is effective as of the date adopted by the Board of Directors of the Company (the "Board") below. Capitalized terms used, but not otherwise defined, herein have the meanings ascribed to such terms in the Plan (as defined below).
The Board has previously adopted, and the stockholders of the Company have previously ratified and approved, the Leatt Corporation Amended and Restated 2011 Equity Incentive Plan, allowing the Company to grant equity incentives in the form of options, stock appreciation rights, restricted stock, restricted stock units, performance shares, performance units and other share-based awards to employees, directors, and consultants (the "Plan").
In accordance with Section 19 of the Plan, the Board, as Administrator of the Plan, desires to increase the amount of the Company's common stock, par value $0.001 available for issuance under the Plan from 920,000 shares to 1,120,000 shares as follows:
Amendment to Section 3 (Stock Subject to the Plan). Section 3 of the Plan is deleted in its entirety and in lieu thereof the following provision is inserted:
Stock Subject to the Plan. Subject to the provisions of Section 13, the maximum aggregate number of Shares that may be issued under the Plan is Eleven Hundred and Twenty Thousand (1,120,000) Shares. The Shares may be authorized but unissued, or reacquired Common Stock.
Full Force and Effect. In all other respects, the Plan shall remain in full force and effect.
Adopted by the Board of Directors on September 10, 2018
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,682
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Q: Safely require gems in Ruby Is there a way to require a ruby gem safely so as to not raise an exception if the gem is not found?
I am looking a solution close to this:
if require 'hirb'
# do some hirb related stuff
else
# do other stuff
end
I want this to make sure no unnecessary gems are failing my deploys to production.
A: It would probably be done like this:
begin
require 'hirb'
rescue LoadError => e
puts "could not find hirb"
end
A: The best way to do this is to use bundler, that way you can be sure your gems really will be installed.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,537
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Q: Tensor product of modules Let $R$ be a polynomial ring over $\mathbb{C}$. Let $R_1=R/I$ for some ideal $I \subset R$. Let $M_1, M_2$ be $R_1$-modules. So, they are $R$-modules as well. Is it true that $M_1 \otimes_{R_1} M_2 \cong M_1 \otimes_{R} M_2$?
A: Yes. You can argue, vaguely, that $M_1 \otimes_R M_2$ "only depends on the action of $R$", and this is defined through its quotient $R/I$. More precisely, $M_1 \otimes_R M_2$ is the quotient of the abelian group tensor product $M_1 \otimes M_2$ by the module generated by all relations $rm_1 \otimes m_2 - m_1 \otimes rm_2$, for $r \in R$ and $m_i \in M_i$. And if $\bar{r}$ is the image of $r$ in $R/I$, these relations are the same as the one $\bar{r} m_1 \otimes m_2 - m_1 \otimes \bar{r} m_2$ for $M_1 \otimes_{R_1} M_2$.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,358
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NuWatt Energy is a company that specializes in solar energy services and is headquartered in Austin, Texas, and Woburn, Massachusetts. NuWatt Energy markets, manufactures, and installs residential and commercial solar panels in Texas, Massachusetts, California, Vermont, and New Hampshire. NuWatt Energy specializes in turn-key solar electric (photovoltaic) solutions for residential, commercial, and institutional clients in New England.
NuWatt also develops solutions within the solar energy sector. In 2017, NuWatt worked with the town of Lexington, Massachusetts to retrofit a "solar noise barrier." The barrier generates 825 megawatt hours (MWh) per year of electricity, and is the first initiative of its kind in the United States.
History
NuWatt Energy was founded in 2009 by Dr. Aiman Alawa, based on his structural engineering expertise and an examination of the future of global energy. NuWatt enables small businesses to benefit from renewable energy while supporting green jobs.
Innovation
NuWatt has been at the forefront of innovating the rise of solar PV noise barriers (PVNB), which serve both as noise barriers and power generation facilities for local communities and State Departments of Transportation.
It is Privately funded, PVNBs are built using private funds, which ensures expansion without federal or state budget impact. The energy produced is then sold at a discount to DOTs or communities.
See also
Efficient energy use
List of energy storage projects
Solar power
References
American companies established in 2009
Energy companies established in 2009
Solar energy companies of the United States
Energy in Massachusetts
Technology companies based in Massachusetts
Companies based in Massachusetts
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 2,012
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{"url":"http:\/\/mathhelpforum.com\/calculus\/97707-squareroot-1-log-1-a-print.html","text":"# squareroot (1 + ...) , log (1 + ...)\n\n\u2022 August 11th 2009, 01:24 PM\nSebastian de Vries\nsquareroot (1 + ...) , log (1 + ...)\nSince I suspect that both cases have a similar apoach I combine these two.\nAccording to exercise-exams of my university $\\sqrt{1+(1\/x^2)}$ is aproximately equal to $1 + (1\/2x^2)$ and $log{(1 + (1\/4x^2))}$ aproximately equal to $1\/(4x^2)$\n\nWhy?\n\u2022 August 11th 2009, 01:51 PM\nMatt Westwood\nI presume you double-posted by accident?\n\u2022 August 11th 2009, 04:21 PM\nSebastian de Vries\nYes, of course. (Wondering)\nA double post is the dumbest thing you can do when you want an answer, it's annoying.\n\nI edited my post right after posting the thread, I realy don't know how it's possible.\nAs far as I'm concerned this thread (not the other one) can be deleted.\nThanks for the answer in the other thread and apology to the mods for the unnessecary work.","date":"2015-01-29 22:47:41","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\": 0, \"codecogs_latex\": 4, \"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\": 0.7804011106491089, \"perplexity\": 3266.471893511712}, \"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-2015-06\/segments\/1422115863063.84\/warc\/CC-MAIN-20150124161103-00212-ip-10-180-212-252.ec2.internal.warc.gz\"}"}
| null | null |
{"url":"http:\/\/www.ebroadcast.com.au\/lookup\/encyclopedia\/re\/Rest_mass.html","text":"Contents\n\nRest mass\n\nIn special relativity, rest mass is an obsolete term used to describe what is today simply referred to as mass. Relativistic mass[?] mr (formerly called \"mass\", but now called energy) increases with velocity (v), and the rest mass m is the inertial mass at v=0, which corresponds to the classical notion of mass.\n\nWhen an object is travelling at a velocity v relatively to an inertial reference frame, then the relativistic mass increases according to:\n\n$m_r = \\gamma m$\n\n$\\gamma = {1 \\over {\\sqrt{1 - v^2\/c^2}}}$\n\nIt should be noted that the rest mass is one of the invariant scalar quantities of special relativity.\n\nThe terms rest mass and relativistic mass can be still be found in elementary textbooks and popularisations of relativity, but their use is depreciated in graduate-level physics.","date":"2013-05-23 14:33:26","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\": 0.630318284034729, \"perplexity\": 623.2458166339931}, \"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-2013-20\/segments\/1368703334458\/warc\/CC-MAIN-20130516112214-00052-ip-10-60-113-184.ec2.internal.warc.gz\"}"}
| null | null |
Q: Lubuntu 16.04 - no internet on my new ISP "Gateway/Cable modem/coax" connection It is Lubuntu mini.iso with no desktop environment, only fvwm, so please, I can use only the terminal. I don't have any network manager installed.
There are two different routers:
On the SIM router the wired internet connection works for the Lubuntu device.
As soon as I change the Ethernet cable from this SIM router, to the other provider (router) which has only tv cable/coax connection through only a cable modem (Gateway), there is no more internet connection.
With the same network cable/router port (from the tv cable/coax/Gateway ISP) the internet works on Windows 7 and 10.
From what I could see for the two different routers/connections the IPs are assigned dynamically.
Tell me please what logs do you need from the working and not-working connections, in order to see the differences...
Thank you.
Working internet connection (SIM router):
~$ ifconfig -a
eth0 Link encap:Ethernet HWaddr 68:f7:28:21:c8:85
inet addr:192.168.8.100 Bcast:192.168.8.255 Mask:255.255.255.0
UP BROADCAST RUNNING MULTICAST MTU:1500 Metric:1
RX packets:6773 errors:0 dropped:0 overruns:0 frame:0
TX packets:6029 errors:0 dropped:0 overruns:0 carrier:0
collisions:0 txqueuelen:1000
RX bytes:5842607 (5.8 MB) TX bytes:1038943 (1.0 MB)
lo Link encap:Local Loopback
inet addr:127.0.0.1 Mask:255.0.0.0
UP LOOPBACK RUNNING MTU:65536 Metric:1
RX packets:7609 errors:0 dropped:0 overruns:0 frame:0
TX packets:7609 errors:0 dropped:0 overruns:0 carrier:0
collisions:0 txqueuelen:1
RX bytes:766404 (766.4 KB) TX bytes:766404 (766.4 KB)
wlan0 Link encap:Ethernet HWaddr ac:b5:7d:f2:ef:2f
BROADCAST MULTICAST MTU:1500 Metric:1
RX packets:0 errors:0 dropped:0 overruns:0 frame:0
TX packets:0 errors:0 dropped:0 overruns:0 carrier:0
collisions:0 txqueuelen:1000
RX bytes:0 (0.0 B) TX bytes:0 (0.0 B)
Non interet working connection (immediately after I only change the ethernet cable from the SIM router to the other)
~$ ifconfig -a
eth0 Link encap:Ethernet HWaddr 68:f7:28:21:c8:85
inet addr:192.168.8.100 Bcast:192.168.8.255 Mask:255.255.255.0
UP BROADCAST RUNNING MULTICAST MTU:1500 Metric:1
RX packets:15580 errors:0 dropped:0 overruns:0 frame:0
TX packets:14234 errors:0 dropped:0 overruns:0 carrier:0
collisions:0 txqueuelen:1000
RX bytes:11441631 (11.4 MB) TX bytes:2455373 (2.4 MB)
lo Link encap:Local Loopback
inet addr:127.0.0.1 Mask:255.0.0.0
UP LOOPBACK RUNNING MTU:65536 Metric:1
RX packets:8093 errors:0 dropped:0 overruns:0 frame:0
TX packets:8093 errors:0 dropped:0 overruns:0 carrier:0
collisions:0 txqueuelen:1
RX bytes:839056 (839.0 KB) TX bytes:839056 (839.0 KB)
wlan0 Link encap:Ethernet HWaddr ac:b5:7d:f2:ef:2f
BROADCAST MULTICAST MTU:1500 Metric:1
RX packets:0 errors:0 dropped:0 overruns:0 frame:0
TX packets:0 errors:0 dropped:0 overruns:0 carrier:0
collisions:0 txqueuelen:1000
RX bytes:0 (0.0 B) TX bytes:0 (0.0 B)
I am not interested in the WiFi connection.
No configurations have been changed to any of the two routers or any machines Linux/Windows 7/Windows 10.
Only change the same cable for all the 3 difference devices from a router to another. Windows 10 and Windows 7 have internet connection on both routers.
Linux has internet only on the SIM router.
For the non working internet connection to the link led from the router where is the Linux Ethernet cabled plugged, only the orange
led one blinks, but for the other internet working devices connected via Ethernet cable to the same router, blinks also the green led on the router.
Working internet connection:
~$ ping google.com
PING google.com (195.249.145.114) 56(84) bytes of data.
64 bytes from cache.google.com (195.249.145.114): icmp_seq=1 ttl=58 time=26.0 ms
64 bytes from cache.google.com (195.249.145.114): icmp_seq=2 ttl=58 time=35.6 ms
64 bytes from cache.google.com (195.249.145.114): icmp_seq=3 ttl=58 time=44.4 ms
64 bytes from cache.google.com (195.249.145.114): icmp_seq=4 ttl=58 time=43.2 ms
64 bytes from cache.google.com (195.249.145.114): icmp_seq=5 ttl=58 time=43.2 ms
64 bytes from cache.google.com (195.249.145.114): icmp_seq=6 ttl=58 time=39.4 ms
^C
--- google.com ping statistics ---
6 packets transmitted, 6 received, 0% packet loss, time 5008ms
rtt min/avg/max/mdev = 26.059/38.670/44.463/6.378 ms
~$ ping 8.8.8.8
PING 8.8.8.8 (8.8.8.8) 56(84) bytes of data.
64 bytes from 8.8.8.8: icmp_seq=1 ttl=57 time=48.4 ms
64 bytes from 8.8.8.8: icmp_seq=2 ttl=57 time=57.6 ms
64 bytes from 8.8.8.8: icmp_seq=3 ttl=57 time=56.1 ms
^C
--- 8.8.8.8 ping statistics ---
3 packets transmitted, 3 received, 0% packet loss, time 2004ms
rtt min/avg/max/mdev = 48.494/54.102/57.670/4.023 ms
Non working internet connection for ping google.com after pending/waiting a few minutes, it looks like this:
~$ ping google.com
ping: unknown host google.com
~$ ping 8.8.8.8
PING 8.8.8.8 (8.8.8.8) 56(84) bytes of data.
From 192.168.8.100 icmp_seq=1 Destination Host Unreachable
From 192.168.8.100 icmp_seq=2 Destination Host Unreachable
From 192.168.8.100 icmp_seq=3 Destination Host Unreachable
From 192.168.8.100 icmp_seq=4 Destination Host Unreachable
From 192.168.8.100 icmp_seq=5 Destination Host Unreachable
From 192.168.8.100 icmp_seq=6 Destination Host Unreachable
From 192.168.8.100 icmp_seq=7 Destination Host Unreachable
From 192.168.8.100 icmp_seq=8 Destination Host Unreachable
^C
--- 8.8.8.8 ping statistics ---
10 packets transmitted, 0 received, +8 errors, 100% packet loss, time 9026ms
pipe 3
I run dhcpcd eth0 with the working internet cabled in, and after dhcpcd loaded everything, I plugged the cable in the non-working internet router, and the internet worked, maybe because dhcpcd was running.
I restarted the device to see if it will still work but when Linux loaded after restart this appeared in the console:
Failed to start LSB: IPV4 DHCP client with IPV4ALL support.
See 'systemctl status dhcpcd.service' for details
16.780656 usb 1-1.4.3: device descriptor read/64, error -110
A start job is running to Raise network interfaces ( ** min **s / 5min 7s)
The last part "A start job is running..." where I have to wait 5 min and 7s it seams that appears every time that the internet doesn't work. It has never appeared when the internet worked, but it always appears when internet doesn't work...
After starting Linux with the non-working internet in, I also tried dhcpcd eth0 but...
eth0: waiting for carrier
timed out
dhcpcd exited
And again...
I have just (nothing less or more) only moved the cable from the non-internet connection router to the internet working connection router
dhcpcd eth0
Loaded all the things successfully (with the internet connection plugged). Is it right that would not be a good idea to paste in here the output of "dhcpcd eth0" ?
Unplug the working internet connection and plug in to the non-internet connection, but surprise, now I am posting from the non-working internet connection which it seems that now is working because of the dhcpcd eth0
But, if I will restart like last time, I think it will happen again like before.
As soon as "waiting for carrier" without touching anything else, I unplugged only the head on the network cable from the Linux machine, insert it in the Windows 7 machine, disable the wireless at all from the Windows machine, enable LAN, and internet is working... This could mean that the router/cable/port of the router work properly?
I will try also to reset the router...
Could it be any other configurations of Linux, considering that the internet works after I use dhcpcd eth0 with the internet-working router plugged in, and after that, plugging in the non-internet router and still has internet connection... (until restart)
The only way the internet was working IN THE NON-INTERNET WORKING ROUTER (the same network cable/the same router port, everything the same like when it doesn't work) was dhcpcd eth0 WHEN THE NETWORK CABLE WAS PLUGGED IN THE WORKING INTERNET ROUTER and after that only moving the cable from the working-internet-router to the non-working-internet router.
But if I will restart the machine, no internet until I will not repeat the before steps.
I've also reset for a few times the router, restarted it (unplug from the electricity for at least 30 second) tried to reconnect the Linux machine to the non-internet-router with another cable router and also to another port router, but still not working, until I will not do the step from the beginning.
The same Ethernet cable and router port from the non-internet router provide internet instantly when is plugged into another Windows 7 and Windows 10 machines, without to change anything in Windows or in the fresh/new/from scratch reset router.
uname -a
Linux WindowsXP 4.4.0-78-generic #99-Ubuntu SMP Thu Apr 27 15:29:09 UTC 2017 x86_64 x86_64 x86_64 GNU/Linux
lsb_release -a
No LSB modules are available.
Distributor ID: Ubuntu
Description: Ubuntu 16.04.2 LTS
Release: 16.04
Codename: xenial
Here is /var/log/dmesg
https://paste.debian.net/hidden/8226e135/
All the following commands are outputed with the internet working(connected to the usually non-internet router, until I will not do the steps from the beginning of this replay)
lshw -C network *-network DISABLED
description: Wireless interface
product: QCA9565 / AR9565 Wireless Network Adapter
vendor: Qualcomm Atheros
physical id: 0
bus info: pci@0000:02:00.0
logical name: wlan0
version: 01
serial: ac:b5:7d:f2:ef:2f
width: 64 bits
clock: 33MHz
capabilities: pm msi pciexpress bus_master cap_list rom ethernet physical wireless
configuration: broadcast=yes driver=ath9k driverversion=4.4.0-78-generic firmware=N/A latency=0 link=no multicast=yes wireless=IEEE 802.11bgn
resources: irq:18 memory:90500000-9057ffff memory:90580000-9058ffff
*-network
description: Ethernet interface
product: RTL8111/8168/8411 PCI Express Gigabit Ethernet Controller
vendor: Realtek Semiconductor Co., Ltd.
physical id: 0
bus info: pci@0000:03:00.0
logical name: eth0
version: 10
serial: 68:f7:28:21:c8:85
size: 100Mbit/s
capacity: 1Gbit/s
width: 64 bits
clock: 33MHz
capabilities: pm msi pciexpress msix vpd bus_master cap_list ethernet physical tp mii 10bt 10bt-fd 100bt 100bt-fd 1000bt 1000bt-fd autonegotiation
configuration: autonegotiation=on broadcast=yes driver=r8169 driverversion=2.3LK-NAPI duplex=full firmware=rtl8168g-3_0.0.1 04/23/13 ip=192.168.110.11 latency=0 link=yes multicast=yes port=MII speed=100Mbit/s
resources: irq:88 ioport:1000(size=256) memory:90404000-90404fff memory:90400000-90403fff
more /etc/resolv.conf
# Dynamic resolv.conf(5) file for glibc resolver(3) generated by resolvconf(8)
# DO NOT EDIT THIS FILE BY HAND -- YOUR CHANGES WILL BE OVERWRITTEN
nameserver 208.67.222.222
nameserver 208.67.220.220
nameserver 192.168.8.1
route
Kernel IP routing table
Destination Gateway Genmask Flags Metric Ref Use Iface
default 192.168.110.1 0.0.0.0 UG 202 0 0 eth0
192.168.110.0 * 255.255.255.0 U 202 0 0 eth0
The var/log/kern.log it was to big to paste anywhere, and I don't know why I could not upload the file to any file hosting site, or on Google drive, or elsewhere. The file has only 2.1M. I've also tried to copy it/rename it, but it is not possible to upload it anywhere. even here on the forum as an File Attachment. Maybe because of the file content.
Considering the before post, when I was connected on the internet through the regularly/usually non-working-internet router, but this time, before restart the internet was working on it (the reason for the working internet connection you can see at the beginning of the before post)
I only restarted the machine, without to touch anything else, router, cable or anything else.
Immediately after restart:
~$ ping google.com
ping: unknown host google.com
~$ ping 8.8.8.8
connect: Network is unreachable
~$ sudo dhcpcd eth0
[sudo] password for globalisation:
eth0: waiting for carrier
timed out
dhcpcd exited
From the above post/replay it seams that 192.168.110.0 is the Gateway, It seems that it is right, because when I write in the browser it asks for the username and password of the modem/password
ping 192.168.110.1
connect: Network is unreachable
globalisation@WindowsXP:~$ ping 192.168.110.0
connect: Network is unreachable
globalisation@WindowsXP:~$ ping 192.168.8.1
connect: Network is unreachable
globalisation@WindowsXP:~$ ping 8.8.8.8
connect: Network is unreachable
globalisation@WindowsXP:~$ ping google.com
ping: unknown host google.com
:~$ dmesg |grep eth[0-9]
[ 2.981426] r8169 0000:03:00.0 eth0: RTL8168g/8111g at 0xffffc90000768000, 68:f7:28:21:c8:85, XID 10900800 IRQ 88
[ 2.981432] r8169 0000:03:00.0 eth0: jumbo features [frames: 9200 bytes, tx checksumming: ko]
[ 15.883064] r8169 0000:03:00.0 eth0: link down
[ 15.883066] r8169 0000:03:00.0 eth0: link down
:~$ sudo lshw -C network
*-network DISABLED
description: Wireless interface
product: QCA9565 / AR9565 Wireless Network Adapter
vendor: Qualcomm Atheros
physical id: 0
bus info: pci@0000:02:00.0
logical name: wlan0
version: 01
serial: ac:b5:7d:f2:ef:2f
width: 64 bits
clock: 33MHz
capabilities: pm msi pciexpress bus_master cap_list rom ethernet physical wireless
configuration: broadcast=yes driver=ath9k driverversion=4.4.0-78-generic firmware=N/A latency=0 link=no multicast=yes wireless=IEEE 802.11bgn
resources: irq:18 memory:90500000-9057ffff memory:90580000-9058ffff
*-network
description: Ethernet interface
product: RTL8111/8168/8411 PCI Express Gigabit Ethernet Controller
vendor: Realtek Semiconductor Co., Ltd.
physical id: 0
bus info: pci@0000:03:00.0
logical name: eth0
version: 10
serial: 68:f7:28:21:c8:85
size: 10Mbit/s
capacity: 1Gbit/s
width: 64 bits
clock: 33MHz
capabilities: pm msi pciexpress msix vpd bus_master cap_list ethernet physical tp mii 10bt 10bt-fd 100bt 100bt-fd 1000bt 1000bt-fd autonegotiation
configuration: autonegotiation=on broadcast=yes driver=r8169 driverversion=2.3LK-NAPI duplex=half firmware=rtl8168g-3_0.0.1 04/23/13 latency=0 link=no multicast=yes port=MII speed=10Mbit/s
resources: irq:88 ioport:1000(size=256) memory:90404000-90404fff memory:90400000-90403fff
:~$ ifconfig -a
eth0 Link encap:Ethernet HWaddr 68:f7:28:21:c8:85
UP BROADCAST MULTICAST MTU:1500 Metric:1
RX packets:0 errors:0 dropped:0 overruns:0 frame:0
TX packets:0 errors:0 dropped:0 overruns:0 carrier:0
collisions:0 txqueuelen:1000
RX bytes:0 (0.0 B) TX bytes:0 (0.0 B)
lo Link encap:Local Loopback
inet addr:127.0.0.1 Mask:255.0.0.0
UP LOOPBACK RUNNING MTU:65536 Metric:1
RX packets:1761 errors:0 dropped:0 overruns:0 frame:0
TX packets:1761 errors:0 dropped:0 overruns:0 carrier:0
collisions:0 txqueuelen:1
RX bytes:130352 (130.3 KB) TX bytes:130352 (130.3 KB)
wlan0 Link encap:Ethernet HWaddr ac:b5:7d:f2:ef:2f
BROADCAST MULTICAST MTU:1500 Metric:1
RX packets:0 errors:0 dropped:0 overruns:0 frame:0
TX packets:0 errors:0 dropped:0 overruns:0 carrier:0
collisions:0 txqueuelen:1000
RX bytes:0 (0.0 B) TX bytes:0 (0.0 B)
:~$ more /etc/resolv.conf
# Dynamic resolv.conf(5) file for glibc resolver(3) generated by resolvconf(8)
# DO NOT EDIT THIS FILE BY HAND -- YOUR CHANGES WILL BE OVERWRITTEN
nameserver 208.67.222.222
nameserver 208.67.220.220
:~$ route
Kernel IP routing table
Destination Gateway Genmask Flags Metric Ref Use Iface
Internet working connection on the regularly non-internet working router:
:~$ ethtool eth0
Settings for eth0:
Supported ports: [ TP MII ]
Supported link modes: 10baseT/Half 10baseT/Full
100baseT/Half 100baseT/Full
1000baseT/Half 1000baseT/Full
Supported pause frame use: No
Supports auto-negotiation: Yes
Advertised link modes: 10baseT/Half 10baseT/Full
100baseT/Half 100baseT/Full
Advertised pause frame use: Symmetric Receive-only
Advertised auto-negotiation: Yes
Link partner advertised link modes: 10baseT/Half 10baseT/Full
100baseT/Half 100baseT/Full
1000baseT/Full
Link partner advertised pause frame use: No
Link partner advertised auto-negotiation: Yes
Speed: 100Mb/s
Duplex: Full
Port: MII
PHYAD: 0
Transceiver: internal
Auto-negotiation: on
Cannot get wake-on-lan settings: Operation not permitted
Current message level: 0x00000033 (51)
drv probe ifdown ifup
Link detected: yes
No internet(because of restart) for the same above non-internet working router. There is no link detected, but this same cable/machine/port had internet before restart...
:~$ ethtool eth0
Settings for eth0:
Supported ports: [ TP MII ]
Supported link modes: 10baseT/Half 10baseT/Full
100baseT/Half 100baseT/Full
1000baseT/Half 1000baseT/Full
Supported pause frame use: No
Supports auto-negotiation: Yes
Advertised link modes: 10baseT/Half 10baseT/Full
100baseT/Half 100baseT/Full
Advertised pause frame use: Symmetric Receive-only
Advertised auto-negotiation: Yes
Link partner advertised link modes: 10baseT/Half 10baseT/Full
100baseT/Half 100baseT/Full
1000baseT/Full
Link partner advertised pause frame use: No
Link partner advertised auto-negotiation: Yes
Speed: 100Mb/s
Duplex: Full
Port: MII
PHYAD: 0
Transceiver: internal
Auto-negotiation: on
Cannot get wake-on-lan settings: Operation not permitted
Current message level: 0x00000033 (51)
drv probe ifdown ifup
Link detected: yes
The always internet working router( completely another provider, another connections, another router)
:~$ ethtool eth0
Settings for eth0:
Supported ports: [ TP MII ]
Supported link modes: 10baseT/Half 10baseT/Full
100baseT/Half 100baseT/Full
1000baseT/Half 1000baseT/Full
Supported pause frame use: No
Supports auto-negotiation: Yes
Advertised link modes: 10baseT/Half 10baseT/Full
100baseT/Half 100baseT/Full
1000baseT/Full
Advertised pause frame use: Symmetric Receive-only
Advertised auto-negotiation: Yes
Link partner advertised link modes: 10baseT/Half 10baseT/Full
100baseT/Half 100baseT/Full
Link partner advertised pause frame use: Symmetric
Link partner advertised auto-negotiation: Yes
Speed: 100Mb/s
Duplex: Full
Port: MII
PHYAD: 0
Transceiver: internal
Auto-negotiation: on
Cannot get wake-on-lan settings: Operation not permitted
Current message level: 0x00000033 (51)
drv probe ifdown ifup
Link detected: yes
Regards.
A: It seams that these two commands resolve the problem:
sudo ethtool -s eth0 autoneg off speed 100 duplex full
sudo dhcpcd eth0
Only that now, still have to find out how they could run on the system start, because after every restart the two commands should be used.
It seams that in adding them in /etc/rc.local doesn't have any effect.
Still at he boot there are these 3 issues:
[FAILED] Failed to start LSB: IPV4 DHCP client with IPV4LL support
[17.289877] usb 1-1.4.3: device descriptor read/64, error -110
A start job is running for Raise network interfaces (**min **s / 5min 8s)
And there is waiting time until the 5min 8s passes.
This last error it appears always only when the internet connection is not recognized, but never when the internet connection works from the beginning immediately after restart, without to use any commands in order to start the internet.
A: Running dhcpcd directly conflicts with the network configuration system. (In your case, the network configuration is in /etc/network/interfaces or /etc/network/interfaces.d). Usually you should only need to invoke dhcpcd for testing purposes.
To avoid conflict with the Debian network configuration system, use ifdown eth0 before you test dhcpcd.
The Debian network configuration system ifupdown is not expected to respond to the link status (cable unplugged and replugged). If you want this feature, NetworkManager can provide it. NetworkManager also works in a way that would avoid blocking the whole boot process when the network fails to come up. Otherwise, to switch networks, you must
*
*ifdown eth0 - do this first to avoid causing an address conflict with another computer on the new network!
*switch cables
*ifup eth0
Finally, from your answer it sounds like you have discovered a workaround for an unusual hardware problem (points at r8169 = cheap RealTek chips), which you want to run before ifupdown starts a DHCP client for you. Edit the /etc/network/interfaces file which contains your network configuration. Immediately after the line
iface eth0 inet dhcp
add
pre-up ethtool -s eth0 autoneg off speed 100 duplex full
The leading space / indent is customary, but it is not required.
To assure yourself that these commands are actually run, you can use logger. Note I've modified the line with ethtool as well as adding lines before and after.
pre-up logger -- running ethtool...
pre-up ethtool -s eth0 autoneg off speed 100 duplex full 2>&1 | logger
pre-up logger -- finished running ethtool
The log lines should appear in the system journal (e.g. journalctl -b), and inside them any output from ethtool, followed by the messages from dhclient. If there is no output from ethtool, then it believes it has succeeded. I believe the journal messages will also be forwarded to your persistent system log files (/var/log/messages or /var/log/syslog ?).
To avoid the five minute delay while testing, you could start by commenting out the auto eth0 line (add a # at the start to disable it). Then the computer will boot without trying to raise the eth0 interface, and you can run ifup manually to see if it works. Although it is sadly possible that it works this way, but not during boot, because of some nasty problem of timing.
Google search suggests that NetworkManager will not implement pre-up. Unfortunately, if you wanted to switch to NetworkManager, you would have to go into systemd and create a service. Probably need to order the service to run before NetworkManager.
A: It seems that the final result, in order to have internet access automatically after reboot/shutdown/sleep with no more manual commands:
/etc/network/interfaces
# This file describes the network interfaces available on your system
# and how to activate them. For more information, see interfaces(5).
source /etc/network/interfaces.d/*
# The loopback network interface
auto lo
iface lo inet loopback
# The primary network interface
iface eth0 inet dhcp
pre-up ethtool -s eth0 autoneg off speed 100 duplex full
allow-hotplug eth0
/etc/rc.local
#!/bin/sh -e
#
# rc.local
#
# This script is executed at the end of each multiuser runlevel.
# Make sure that the script will "exit 0" on success or any other
# value on error.
#
# In order to enable or disable this script just change the execution
# bits.
#
# By default this script does nothing.
echo 70 > /sys/class/backlight/intel_backlight/brightness
rfkill block bluetooth
rfkill block wifi
ethtool -s eth0 autoneg off speed 100 duplex full
ip link set eth0 up
exit 0
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,030
|
Q: Understanding notation of norm with three vertical lines We usually denote a norm with the notation $||\cdot ||$. However I've seen someone write $|||\cdot|||$. Is this the same as $||\cdot||$?
For instance, I have readed a note regarding functional analysis. A lemma states:
Lemma: Let $A$ be a Banach algebra with identity $I$. Then there is a norm $|||\cdot|||$ on $A$, equivalent to the original norm, such that $(A,|||\cdot|||)$ is a unital Banach algebra with $|||I|||=1$.
I am just curious. Could we just write the lemma as e.g.
Lemma: Let $A$ be a Banach algebra with identity $I$. Then there is a norm $||\cdot||$ on $A$, equivalent to the original norm, such that $(A,||\cdot||)$ is a unital Banach algebra with $||I||=1$.
A: So first, when we say something like let "$A$ be a Banach Algebra", we are implicitly assuming that there is a norm $||\cdot||$ defined on $A$ (after all, a Banach Algebra, by definition, is a type of normed space). The lemma which you have stated is saying something different. It might be the case that even though $|| \cdot ||$ is a norm, that $ ||I|| \neq 1$ with the norm $A$ has been endowed with. However, what the lemma is saying is that there exists a different norm, which they are denoting by $|||\cdot|||$, on $A$ such that $||| I ||| = 1$. They are stating this different norm is equivalent to the original norm $|| \cdot ||$ that $A$ was endowed with. The two norms are equivalent but they are not necessarily the same. Two norms $p$ and $q$ on a vector space $X$ over $\mathbb{R}$ (or $\mathbb{C}$) are equivalent if there exists real numbers $c, C >0$ such that $$cq(x) \leq p(x) \leq Cq(x) \ \text{for all} \ x \in X$$
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,499
|
Some western Iowa residents say they're concerned about wind projects being planned nearby. Energy developer Invenergy is planning almost 170 wind turbines in Sac and Ida counties.
According to Invenergy, the planned Sac and Ida county wind farms could produce enough energy to power about 90,000 homes each year. The projects' permits are still being reviewed in both counties. Invenergy's goal is to have them up and running between the end of 2019 and the end of 2020.
But some rural residents say they're concerned about noise and the view of large turbines on the horizon. Mason Fleenor, an Ida County farmer who lives near an Ida Grove wind farm, said the turbines are loud.
According to the Iowa Wind Energy Association's website, the state has more than 4,000 wind turbines and more than 7,000 megawatts of installed wind capacity. Nearly 40 percent of the state's electricity is produced from wind. But Janna Swanson, a Clay County farmer and a board member for the Coalition for Rural Property Rights, said wind farms bring a lot of disadvantages to rural residents.
Invenergy developed another wind farm in Ida County that's been operating for about two years. After phase I was complete, about 500 residents petitioned the county to have wind turbines be at least one mile away from a home. Ida County Board of Supervisors Chairman Rhett Leonard said the county's Planning and Zoning board reviewed an ordinance about wind energy and changed it from a 1,250-foot distance to a 1,500-foot distance away from a home. The 1,500-foot distance is equivalent to less than one-third of a mile.
The county's code also says wind turbines must be placed so that shadow flickering caused by sunlight shining through rotating blades does not exceed 30 hours per year on an occupied building.
Leonard said he has heard from residents both for and against large wind farms. One benefit to the county in having the them, he said, is revenue from the turbine property tax benefits projects to rebuild roads and deteriorating infrastructure.
"We were to the point where our roads and infrastructures were deteriorating so quickly and it's such a high-priced item to replace and maintain that we were getting to the point where we weren't sure what we were going to do with these," Leonard said.
The county estimates it receives more than $2.4 million per year "when taxed at the full 30 percent allowed by the law" Leonard said.
Invenergy spokeswoman Beth Conley said in an email that wind farm developments undergo thorough review processes with public comment. The company works with landowners across all of its projects in the state, and participating landowners receive yearly payments, she said.
"Wind farms are built where landowners voluntarily participate and have signed easements, or leases, for a wind turbine to be placed on their property," Conley said.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,884
|
MINNEAPOLIS (AP) _ The United States won the Curtis Cup for the first time in eight years, beating Britain-Ireland 10-8 in the two-day competition at The Minikahda Club in Minneapolis.
Singles victories by Kellee Booth and Brenda Corrie-Kuehn with four matches to go gave the U.S. team an insurmountable lead in the biennial women's amateur golf event.
Members of the U.S. team embraced each other after Corrie-Kuehn beat Becky Morgan with a par putt at the 17th hole.
Britain-Ireland won three of the last four matches Sunday, but the rally came after the United States had clinched the Cup.
Booth and Corrie-Kuehn teamed up for a pair of foursome victories and each won two individual matches, making them the fifth and sixth players to win four times in a single Curtis Cup.
The win completed a reversal for Booth, who was on the U.S. team that lost the Cup in 1996.
The Americans, who came into the final day needing four points to win, took a 7 1/2-4 1/2 lead in the morning after winning two of the three foursome matches.
The United States won despite a minimal contribution from Jenny Chuasiriporn. The U.S. Open runner-up and future Duke teammate Beth Bauer were beaten in both of their foursome matches, while Chuasiriporn earned a half-point in her singles match against Rebecca Hudson Saturday.
The win was only the second for the Americans since 1986. They lead the series 21-6-3.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,101
|
package com.github.phudekar.downloader.exceptions;
public class UnexpectedResponseException extends Throwable {
private final String location;
private final int responseCode;
public UnexpectedResponseException(int responseCode, String location) {
super("Could not connect. status code : " + responseCode);
this.responseCode = responseCode;
this.location = location;
}
public int getResponseCode(){
return this.responseCode;
}
public String getLocation() {
return location;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,851
|
\subsection{The Distance Problem at mmWave-THz Bands}
\hl{Comment the main aspects relative to the distance problem, wireless comms oriented.}
\cite{akyildiz2018combating}\textcolor{red}{Odysseas}
\\
\\
\\
\hl{Describe the existence of prototypes for RIS, wireless comms oriented.}\textcolor{red}{Odysseas and Sergi}
\\
\\
\\
\hl{Briefly describe how those can help to address the distance problem.}\textcolor{red}{Odysseas}
\\
\\
\\
\hl{Mention that there is not much work specific for mmWave-THz frequencies. We are different and aim to bridge this gap.}\textcolor{red}{Odysseas}
\cite{tsilipakostoward, abadal2020programmable}
\cite{pitchappa2020frequency}
\\
\\
\\
\hl{Mention that graphene-based metasurfaces are the excellent candidate at mmWave-THz bands (especially THz) }\textcolor{red}{Ana}
\cite{Correas2017}
\cite{han2020complete}
\cite{cheng2019recent}
\subsection{Multi-band Coding Formulation}
The realization of the promising paradigm of widely tunable MS integrating graphene-based unit cells needs careful consideration of multiple aspects related to, among others, graphene technology and metasurface design. We discuss them next.
\subsection*{Technological implementation}
Although there are a rising number of methods for making various forms of graphene, the volume production for some of those methods remains limited. The focus will have to be on material quality and manufacturing cost. The main challenge lies in maintaining the quality of graphene, i.e. large $\tau$, in large areas. Defects on the graphene layer will significantly affect the electrical conductivity.
Therefore, choosing the most accurate method for producing graphene and good control over the graphene layer deposition is very important.
Various fabrication approaches have been utilized to develop graphene. The more efficient among them, Chemical Vapor Deposition (CVD), has been shown to provide high quality, single layer, uniform graphene in large scale samples~\cite{Choi2017116X}.
The contact of graphene with other materials, for example, dielectric substrates or metallic electrodes, modifies the doping level and chemical potential of graphene. In this case, encapsulating graphene in insulating materials such as hexagonal Boron Nitride (hBN) helps minimizing the influence of contacting materials and, hence, maintaining a high $\tau$ \cite{wang2019graphene}.
For the implementation of graphene-based MSs, it is usual to deploy a patterned scheme applied in a uniform sheet, as the patches discussed here. For the patterning, post-process techniques in pre-synthesized graphene are available, i.e., electron-beam, helium ion beam, lithography, nanostructure-assisted, or nanoimprint technology~\cite{Wei20201655}.
\subsection*{Graphene tuning mechanisms}
As mentioned before, the properties of graphene depend strongly on aspects such as relaxation time and chemical potential, the latter of which can be tuned by imposing an external stimulus. Thus one can modify graphene's conductivity by altering the chemical doping, by thermal tuning, external electrostatic or magnetic field, and optical pumping or self-action. Many of the works investigating the physical aspects of graphene's modulation capabilities focus on the global modification usually involving uniform graphene sheets. The most prevalent technique for locally manipulating the optical properties of graphene is controlling the biasing voltage, often enhanced with the use of an electrolytic medium. In this case, care must be taken in controlling the biasing voltage at each unit cell individually as required by the MS coding. This non-uniform modification can be achieved by properly placed transparent electrodes or by alternative techniques, for example, uneven ground planes or homogeneous dielectric spacers~\cite{DeAbajo2014133}, which in any case are non-trivial at the granularity of the proposed unit cells. In all cases, we note that the tuning speed of graphene (in the order of picoseconds) is enough to support not only slow-paced indoor applications, but also the most demanding outdoor applications, even vehicular ones, whose reconfiguration speeds may be in the the order of hundreds of microseconds \cite{akyildiz2018combating}.
\subsection*{Multi-band metasurface design}
Assuming that the MS operation in all three bands is concurrent, e.g., without any time-multiplexing scheme to decouple them, means that setting the graphene chemical potentials $(\mu_1,\mu_2)$ to optimize the four states for one frequency band will unavoidably simultaneously affect the graphene response in the other bands. In line with the vision outlined in this paper, an ideal unit cell would provide a large number of states with high amplitude and all possible combinations of equally spaced phases in the three target frequencies, i.e. $\varphi=\varphi_{0}+\{0,\tfrac{\pi}{2},\pi,\tfrac{3\pi}{2}\}$ at $f_{1},f_{2},f_{3}$. This would allow controlling the response at the three bands simultaneously and independently. However, such a unit cell would probably require heavy optimization to strike a delicate balance among multiple resonances.
In less ideal unit cells, the complex reflection coefficients in the other bands will in principle form a sub-optimal set of states deteriorating the performance. It can be intuitively understood that optimization of the unit-cell geometry can potentially lead to adequately broadband behavior; alternatively, the four states can be assigned to four $(\mu_1,\mu_2)$ sets that lead to a tolerable performance, \textit{on average} across the operation bands. In Fig.~\ref{fig:coupled_response}, we present two-beam splitting of a normally incident plane wave by a circular-aperture MS (diameter 1.4~mm) of 35~$\upmu$m-wide cells. Each panel in this figure corresponds to an operating frequency band ($\{a,b,c\}=\{0.65,0.85,1.05\}$~THz) for which the $(\mu_1,\mu_2)$ for the four states have been optimally selected. Evidently, the scattering patterns present two well-defined lobes at the desired directions, $(\theta,\varphi)=(30^\circ,0)$ and $(45^\circ,90^\circ)$. In each case, the response at the bands for which the MS has not been tuned, which is not shown for space constraints, is degraded. Specifically, the scattering directivity in the desired lobes was decreased by 1-2~dB (worst-case) whereas parasitic side lobes were raised by 6-7~dB.
\begin{figure}[!t]
\centering
\includegraphics[width=\columnwidth]{figures/Fig_CoupledResponse.pdf}
\caption{Scattering patterns for two-beam splitting of a normally incident plane wave by a circular-aperture MS ($\{a,b,c\}=\{0.65,0.85,1.05\}$~THz).}
\label{fig:coupled_response}
\end{figure}
\subsection*{Near-field and link-budget considerations}
The Huygens-Fresnel principle introduced previously can be used to accurately estimate the far-field as a superposition of elementary scattering patterns from each cell.
Even though this Huygens-Fresnel framework is strictly valid in the far-field, i.e., in a context similar to geometric-optics/ray-tracing, its applicability for performance evaluation extends well into the `transition' (or Fresnel) region, bounded by the near-field and the Fraunhofer ($r_{FF}$) regions. This means that a RIS can be envisioned as a part of the antenna system of a transceiver base station, providing digital beamforming that enhances path loss to virtual-line-of-sight mobile stations. Moreover, in real-world applications relevant to 6G THz communications, such as pico/femto-cells or short-range indoor channels, all incoming sources will practically be in the far-field of the MS, even for cm-sized apertures for which the Fraunhofer region is a few meters.
Finally, note that this multi-band RIS is capable of operating at frequency bands of almost an octave distance. In this sense, the path loss will exhibit a 6~dB deterioration switching from the lowest to the highest frequency, which could be considered for a short-range link budget. Nevertheless, this difference can be partially compensated by an increased directivity of the scattering pattern assuming that: (i) the RIS aperture is fixed, (ii) the channel is appreciably unobstructed so that a low exponent $\lambda^n$, $n\rightarrow2$, can be assumed, and (iii) the MS reflection coefficients are approximately similar across the frequency bands; the latter can be attained by engineering the unit-cell and careful selection of its states, and/or using materials with sufficiently broadband response, such as graphene.
\section*{Introduction}
\label{sec:intro}
\input{intro.tex}
\section*{Graphene-based Multi-Wideband Metasurface}
\label{sec:unit}
\input{unitCell.tex}
\section*{Implementation Considerations}
\label{sec:challenges}
\input{challenges.tex}
\section*{Conclusions}
\label{sec:conc}
\input{conclusions.tex}
\bibliographystyle{IEEEtran}
\subsection*{Capacity Analysis}
\section{Discussion}
\hl{Metasurface: how frequency modulation affects directivity and other metrics.}\textcolor{red}{Hamidreza and Alex}
\subsection*{Graphene Modeling}
\subsection*{Unit Cell Design}
Fig.~\ref{fig:unit cells} shows the proposed unit cell whose lateral size is $c=35~\upmu$m (around $\lambda/10$), small enough to allow for fine resolution of the reflection phase profile, characteristic of MSs over standard reflectarrays. From top to bottom, the unit cell consists of a multi-layer structure with four parasitic graphene patches ($a=15~\upmu$m, chemical potential $\mu_2$) on top of an alumina layer with $0.1~\upmu$m thickness. Synthesis of graphene/alumina composite materials enhances strength, toughness, and wear-resistance with a low-cost process \cite{Kim2014}. This composite is stacked on high-density polyethylene (HDPE) substrate, due to its particularly low losses in the terahertz band \cite{polym12092094}. HDPE permittivity is $\epsilon_r=2.37$ and its thickness is selected $h_1=15~\upmu$m. The combination of graphene and HDPE leads to high electrical conductivity with good thermal and mechanical properties \cite{polym12092094}. Another layer of graphene/alumina composite ($b=20~\upmu$m, the chemical potential $\mu_1$) is sandwiched between HDPE and silicon ($h_2=10~\upmu$m) to allow for supporting multiple resonances and, thereby, being able to operate in three disjoint bands.
\begin{figure}[!t]
\vspace{-0.4cm}
\centering
\includegraphics[width=0.5\textwidth]{figures/unitcell3.png}
\caption{Schematic representation of the unit cell layout. This unit cell is composed of two main dielectric layers (i.e., HDPE and silicon), four parasitic graphene patches on top and one graphene sheet sandwiched between the dielectrics. The unit cell is back plated with a thin layer of gold to be fully reflective.}
\label{fig:unit cells}
\vspace{-0.1cm}
\end{figure}
Due to the native oxidation of silicon, a thin layer of silicon dioxide is grown in the outer layer in a short time. To analyze a realistic design we have considered a layer of silicon dioxide with $0.3~\upmu$m thickness on top of the silicon layer and the layout is back plated with gold. Regarding the fabrication feasibility of the proposed low-profile structure, there are advanced silicon substrate thinning techniques that can be used to achieve an ultra-thinning down to 4~$\upmu$m without damage occurred due to thinning processes.
As mentioned above, the frequency and reflection phase of a graphene-based unit cell can be controlled via changes in its biasing voltage. This is modeled through the frequency-dependent surface conductivity of graphene $\sigma(f)$, which in the THz band is given by
\begin{equation}
\sigma\left(f\right)=\frac{e^{2}\mu}{\pi\hbar^2}\frac{i}{2\pi f+i\tau^{-1}},
\label{eq:sigma_graphene}
\end{equation}
where $e$ and $\hbar$ are constants, while $\mu$ and $\tau$ are variables that correspond to the chemical potential and the relaxation time of the graphene layer, respectively \cite{wang2019graphene}. When a graphene sheet forms one plate of a capacitor, any applied static voltage will alter the carrier density at its surface thus modulating its EM material properties, i.e., its complex-valued electric conductivity; the chemical potential, expressed in eV units, is typically used as the controllable variable, linked to the externally applied tuning voltage as $\mu\propto\sqrt{V_\mathrm{bias}}$. On the one hand, the amplitude response depends on the losses within graphene, $\mathrm{Re}\{\sigma\}$, which in turn depend on the quality of the graphene sheets as modeled by the relaxation time value $\tau$. For the purpose of this work, the relaxation time of graphene is assumed to be $\tau=1$~ps, which has been considered in multiple works and is realizable with state-of-the-art fabrication and encapsulation techniques \cite{Banszerus2015}. On the other hand, to modify the resonance frequency and the phase of the response, the key tuning variable is the chemical potential value $\mu$ or, in our case, two distinct values $\mu_{1}$ and $\mu_{2}$ on top and bottom patches as shown next.
All full-wave EM simulations at the unit-cell level were conducted in CST Microwave Studio (frequency domain solver).
\subsection*{Unit Cell States} \label{cell_states}
We explore the chemical potential design space by sweeping the values of $\mu_{1}$ and $\mu_{2}$ in the three desired operational frequencies. The inset figures in Fig.~\ref{fig:apf} show a trade-off between reflection amplitude and reflection phase versus chemical potential values at 0.65 THz as an example. Hence, by controlling the bias voltage we can tune the reflection characteristics.
\begin{figure*}[!ht]
\centering
\vspace{-0.5cm}
\includegraphics[width=0.65\textwidth]{figures/APF.png}
\caption{Reflection characteristics of the selected states at respective operation frequency. The insets represent the design space of different combinations of chemical potentials at 0.65 THz. The horizontal axis represents the chemical potential of sandwiched graphene patch (see Fig. 3) and the vertical axis represents the chemical potential of the graphene patches on top. Four states corresponding to the highest possible amplitude and spaced with $\pi/2$ phase are selected (black circles in the inset and blue circles in the outset figure). Reflection amplitude and phase at 0.85 and 1.05 THz versus chemical potentials have similar design spaces. Selected states are plotted with red stars (0.85 THz) and yellow diamonds (1.05 THz) in the outset figure.}
\label{fig:apf}
\end{figure*}
The results show the response of the MS for a continuous range of chemical potentials. However, to design a digital reconfigurable MS, we need to discretize the design space and select a set of chemical potentials to obtain a finite set of addressable states. Firstly, the number of unit cell states (per frequency band) has to be selected. Here, the number of states will determine the phase difference between consecutive unit cell states. As shown in our previous work \cite{9109701}, four states (with $\pi/2$ phase difference between them) can offer high-quality beam-steering performance, given that the unit cell is fairly subwavelength as in our case here. In the insets of Fig.~\ref{fig:apf}, black circles indicate the four selected states at 0.65 THz. These states are selected manually with two goals i) $\pi/2$ phase shift to each other and maximum reflection amplitude. The blue symbols in Fig.~\ref{fig:apf} represent the selected states at 0.65 THz, whereas the red and yellow symbols represent the other two frequencies.
We note that the selected states may need to change when the incidence angle changes. Assuming that a completely different set of states is needed to cover the neighborhoods of 0, 30, and 60 degrees \cite{9109701}, the minimum \emph{total} number of required states is 4 states per frequency band $\times$ 3 frequency bands $\times$ 3 incidence angle neighborhoods $=36$ states. Moreover, if the metasurface is brought in the near-field of transmitting/receiving antennas, the reflection phase/amplitude may vary and depend on the distance \cite{Danufane2021}. This behavior should be taken into account with additional states being foreseen to cover operation in such short distances.
\subsection*{Metasurface coding for beam steering} \label{sec:MScoding}
Having selected the four sets of chemical potentials $(\mu_1,\mu_2)$ that define the cell states, we retrieve the corresponding complex reflection coefficients (phase and amplitude) for each band, from charts similar to those in the insets of Fig.~\ref{fig:apf}, as calculated from full-wave EM simulations. In this work, we assume that the operation between the three bands is decoupled, i.e., the four $(\mu_1,\mu_2)$ sets that define the four states are different for each band. In this case, achieving independent control over the three bands is only possible via space multiplexing (large RIS panels divided into subpanels devoted to a particular frequency) or time multiplexing (slots reserved to each particular frequency) leveraging the fact that the response of graphene to the electrical tuning of its chemical potential is practically instantaneous. As discussed above, the phase-stepping is very close to $\pi/2$ along with the states of all three bands (horizontal axis of Fig.~\ref{fig:apf}), with a worst-case reflection coefficient of $-4$~dB; these metrics ensure qualitative agreement with theory and can be further improved with cell redesign.
The Huygens-Fresnel principle (HFP) \cite{9109701} allows us to evaluate the scattering pattern of the MS for a given configuration as the aggregated response of its unit cells to a given incidence (frequency, polarization, wavefront shape, and direction). For linearly polarized incidence waves, the scattered field can be expressed as
\begin{equation}
\begin{split}
E(\theta, \varphi) = \sum_{m=1}^{M} \sum_{n=1}^{N} A_{mn} p_{mn}(\theta_{mn}, \varphi_{mn})\\ R_{mn} p_{mn}(\theta, \varphi)
e^{jk_0\zeta_{mn}(\theta, \varphi)}
\end{split}
\label{eq:Huygens}
\end{equation}
where $\varphi$ and $\theta$ are the azimuth and elevation angles, $A_{mn}$ corresponds to the complex amplitude of the wave incident for the $mn$-th unit cell, $p_{mn}$ denotes the scattering pattern of the $mn$-th unit cell, and $R_{mn}$ is the complex reflection coefficient for the $mn$-th unit cell.
Finally, $\zeta_{mn}(\theta, \varphi)$ is the relative phase shift of the unit cells with respect to the radiation pattern coordinates and their geometric arrangement on the MS. Tuning of the two voltages applied at each $mn$-cell alters graphene's local $\mu_{1,2}$ and consequently the cell's response, $R_{mn}$; the collective effect of the tuning of all $R_{mn}$ is a tuning of the scattering pattern, $E(\theta, \varphi)$. The underlying approximation of the HFP is that unit cells are uncoupled, so that their $R_{mn}$ can be tuned independently, which is valid for the cell sizes considered here. Finally, we note that HFP is an analytical method implemented on MATLAB using parameters calculated from 3D geometry and/or extracted from full-wave unit-cell level EM simulations.
The ideal MS coding or configuration, that is, the unit cell amplitudes and phases required to provide a given functionality, can also be analytically calculated in the same framework \cite{9109701}. This procedure can be summarized as forward-propagating the source `rays' while back-propagating the desired radiation pattern (prescribing a functionality) onto the MS, and then taking the complex-valued ratio of the propagated wavefronts on each cell; these ratios denote the required reflection coefficient $R_{mn}$ profile to realize the functionality.
Then, the unit cell states closest to the ideal ones are selected. As an example, normally incident plane waves can be steered away from the specular (normal) reflection direction by configuring the MS cells in stripes of progressively increasing reflection phase, according to diffraction grating theory. More elaborate beamforming patterns can be engineered by more complicated configuration patterns; for instance, beam-splitting in two different directions can be accomplished by superimposing the \emph{stripes} that correspond to single-beam steering thus generating a pattern of rectangular \emph{supercells}.
In Fig.~\ref{fig:scatpats}, we present a set of scattering patterns that demonstrate the potential of the proposed multi-wideband multi-functional RIS, assuming $40\times40$ 35~$\upmu$m-wide cells. This figure contains a selection of configurations and functionalities interesting in 6G contexts, demonstrated in three different frequencies, namely:
\begin{itemize}
\item \textbf{Three-beam splitting} at 0.85 THz, Fig.~\ref{fig:scatpats}(a), shows a case where a normally incident plane wave is split in three beams, each one steered to a different direction that can be controlled independently of others: $(\theta,\varphi)=(15^\circ,315^\circ)$, $(30^\circ,90^\circ)$, and $(45^\circ,135^\circ)$. Fig.~\ref{fig:scatpats}(a) depicts the scattering pattern both in spherical (3D) and in cylindrical (2.5D) coordinates, in the main view and top-left inset, respectively, together with the unit cell states with color-coded squares in the left-hand insets. This functionality could be useful for multicasting in multi-band 6G networks.
\item \textbf{Random scattering} of normally incident radiation at 0.65~THz, Fig.~\ref{fig:scatpats}(b). This is accomplished by totally random (uniform distribution) phase states across the MS; note that to eliminate back-scattering (or monostatic radar cross-section) the phase-states must be clustered in supercells of $\lambda/2$ size, as smaller randomly-coded cells emulate a metallic reflector and produce a specular reflection. This functionality is useful for physical-layer security methods, blocking certain users.
\item \textbf{Collimation of a spherical wavefront towards two far-field directions} $(\theta,\varphi)=(30^\circ,0)$ and $(45^\circ,90^\circ)$ at 1.05~THz, Fig.~\ref{fig:scatpats}(c). In this case, the illuminating point-source is outside the near-field border, at 3~mm above the MS axis, so that emitted `rays' impinge almost perpendicularly on the MS. This functionality is useful for the building of antennas where an integrated RIS acts as an intelligent reflector.
\end{itemize}
\begin{figure}[!t]
\centering
\vspace{-.1cm}
\includegraphics[width=\columnwidth]{figures/Fig_ScatPats.pdf}
\caption{Scattering patterns for normal illumination of the multi-band multi-functional RIS; scattering heatmaps are normalized to theoretical maximum directivity. (a) 3D scattering pattern of a normally incident 0.85~THz plane-wave, split/steered in 3 predetermined directions: $(\theta,\varphi)=(15^\circ,315^\circ)$, $(30^\circ,90^\circ)$, and $(45^\circ,135^\circ)$; left-hand insets depict the cylindrical-coordinate representation of the same pattern, the orthogonal color-coded view of the MS cells configuration, and the color-mapping of the phase-states. (b) MS configuration and corresponding cylindrical scattering pattern for diffused scattering, and (c) MS configuration and corresponding cylindrical scattering pattern for collimation/steering of a spherical wavefront towards two distinct directions.}
\label{fig:scatpats}
\vspace{-.5cm}
\end{figure}
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{
"redpajama_set_name": "RedPajamaArXiv"
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Presentation A 60 year old man presents with bilateral palpable scrotal masses. He had a vasectomy performed 25 years ago. A scrotal ultrasound was performed.
Caption: Transverse image of the right scrotum.
Description: The right epididymal head appears enlarged and mildly lobular.
Caption: Right oblique image of the scrotum.
Description: Numerous dilated fluid-filled spaces are seen in the region of the epididymal body and tail. The visualized right testis appears normal.
Caption: Transverse image of the right epididymal body.
Description: The rounded fluid filled structures appear to be arising from the epididymus, seen anteriorly in the center on this image. Note the enhanced through transmission beneath the largest cyst.
Caption: Transverse image of the right epididymal tail.
Description: More fluid filled structures are seen in the epididymal tail.
Caption: Color Doppler image of the right epididymis.
Description: No abnormal area of increased vascularity are seen in either the epididymis or testis.
Caption: Transverse left epididymal scan.
Description: The left epididymus also exhibits multiple cystic structures in the epididymal head and body.
Caption: Transverse scan in the upper part of scrotal sac.
Description: The left epididymal body is thickened and has adjacent fluid filled areas. Some of these areas show internal septations.
Caption: Color Doppler of the left epididymal cystic area.
Description: Sparse vascularity is noted around the periphery of the cystic area in the epididymal tail.
Caption: Midline transverse scan of the scrotum.
Description: This image shows the large cystic lesions associated with the epididymi bilaterally.
This case is presented to show the ultrasound appearance of a spermatocele.
A spermatocele is a benign clear fluid-filled cyst that may arise in any part of the epididymis and contains non-viable sperm. A spermatocele is usually asymptomatic but sometimes produces palpable masses that may cause discomfort to the patient. It is more common in older men.
No definitive casuative factor for spermatocele has been established. The sperm passes through the epididymis where it undergoes maturation. One theory that has been postulated in the pathogenesis of a spermatocele is that with advancing age, the seminiferous epithelium degenerates and the germ cells that are shed accumulate and block the efferent ducts, resulting in sterile cystic collections within the epididymis.
1. Spermatoceles may be single or multiple, unilateral or bilateral.
2. Commonly located in the epididymal head, however they can also be seen in the body and tail.
3. They appear as well-defined hypoechoic or cystic structures that show significant posterior enhancement.
4. Internal septations may be present in the cysts.
1. To establish a diagnosis and distinguish from other intrascrotal lesions such as a varicocele, hydrocele or a simple epididymal cyst.
2. Ultrasound guided therapeutic aspiration of a multilocular spermatocele may be performed to obtain relief of symptoms.
Yagi H, Igawa M, Shiina H, Shigeno K, Yoneda T, Wada Y. Multilocular spermatocele: a case report. Int Urol Nephrol. 2001;32(3):413-6.
Itoh M, Li XQ, Miyamoto K, Takeuchi Y. Degeneration of the seminiferous epithelium with ageing is a cause of spermatoceles? Int J Androl. 1999 Apr;22(2):91-6.
This patient refused any form of treatment and follow up scrotal ultrasound revealed no change in the spermatoceles.
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{
"redpajama_set_name": "RedPajamaC4"
}
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Q: Online resources for premade WPF styles/control templates? When it comes to UI design I generally don't care to do custom styling (appearance-wise), I'm not much of an artist and I hate trying to come up with interesting color schemes and the like. Are there any online repositories of premade WPF styles and/or control templates? I could have sworn I saw one before but I can't find it now, it might be gone.
Just curious if you guys know of anything like this. It'd be great to be able to drop in a premade style into my application to spruce it up a bit.
A: Here's another.
A: Here's one.
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,119
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package forum {
class thread(args: threadArgs) {
import scala.xml.NodeSeq
import scala.xml.NodeSeq.Empty
import javax.jcr.Node
import utils._
import utils.RichJCR._
import args._
def javascript(function: String, args: String*) =
args.mkString("javascript:" + function + "('", "', '", "')")
def versions(node: Node) = {
<span id={ node.uuid + "showversionlink" }>
<a href={ javascript("showVersions", node.uuid) }>old versions</a>
</span>
<span style="display:none" id={ node.uuid + "hideversionlink" }><p>
<a href={ javascript("hideVersions", node.uuid) }>hide versions</a>
</p></span>
<div style="display:none" id={ node.uuid + "versions" }>
{
for (version <- node.versions;
if version.predecessors.length > 0;
if version.successors.length > 0) yield {
<hr />
<p>version created at: { version.created }</p>
<p> { version.frozenNode("subject") } </p>
<p> { version.frozenNode("body") } </p>
}
}
<hr />
</div>
}
def detail(node: Node): NodeSeq = {
val threadPath = node.path.split("/").take(4).mkString("/")
<p>
{
emptyUnless(node.hasNode("logo")) {
<img src={ node.getNode("logo").path + "/jcr:content" } />
}
}
</p>
<p>
<span id={ node.uuid + "body" }>
<a name={ node.uuid } />
<p> { node("subject") } </p>
<p> { node("body") } </p>
</span>
</p>
<p>
<span id={ node.uuid + "showeditlink" }>
<a href={ javascript("showEdit", node.uuid) }> edit</a>
</span>
<span style="display:none" id={ node.uuid + "editform" }>
<form action={ node.path } method="POST">
<p><input type="text" name="subject" value={ node("subject") } /></p>
<textarea rows="3" name="body">{ node("body") }</textarea>
<p>
<input type="submit" value="update" />
<a href={ javascript("cancelEdit", node.uuid) }>cancel</a>
</p>
<input name=":redirect" value={ threadPath + ".thread.html#" + node.uuid } type="hidden" />
</form>
</span>
</p>
<p>created: { node.baseVersion.created }</p>
<div>
{
emptyUnless(node.versions.getSize > 2) {
<p> { versions(node) } </p>
}
}
</div>
<p><span id={ node.uuid + "showformlink" }>
<a href={ javascript("showComment", node.uuid) }>add comment</a>
</span></p>
<div style="display:none" id={ node.uuid + "form" }>
<form action={{ node.path } + "/*" } method="POST">
<p><input type="text" name="subject" /></p>
<textarea rows="2" name="body"></textarea>
<p>
<input type="submit" value="submit comment" />
<a href={ javascript("cancelComment", node.uuid) }>cancel</a>
</p>
<input name=":redirect" value={ threadPath + ".thread.html#" + node.uuid } type="hidden" />
</form>
</div>
<ul>
{
node.nodes.filter(_.getPrimaryNodeType.getName != "nt:file") map {node => {
<li> { detail(node) } </li>}
}
}
</ul>
}
println {
<html>
<head>
<link rel="stylesheet" href="/apps/forum/static/blue.css" />
<script type="text/javascript"><!--
function showComment(id) {
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function cancelComment(id) {
document.getElementById(id + "showformlink").style.display = 'block';
document.getElementById(id + "form").style.display = 'none';
}
function showEdit(id) {
document.getElementById(id + "showeditlink").style.display = 'none';
document.getElementById(id + "body").style.display = 'none';
document.getElementById(id + "editform").style.display = 'block';
}
function cancelEdit(id) {
document.getElementById(id + "showeditlink").style.display = 'block';
document.getElementById(id + "body").style.display = 'block';
document.getElementById(id + "editform").style.display = 'none';
}
function showVersions(id) {
document.getElementById(id + "showversionlink").style.display = 'none';
document.getElementById(id + "hideversionlink").style.display = 'block';
document.getElementById(id + "versions").style.display = 'block';
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document.getElementById(id + "showversionlink").style.display = 'block';
document.getElementById(id + "hideversionlink").style.display = 'none';
document.getElementById(id + "versions").style.display = 'none';
}
--></script>
</head>
<body>
<div id="Header">
<a href="/content/forum.html">< back to thread overview</a>
</div>
{ SearchBox.render(request) }
<div id="Content">
{ detail(currentNode) }
</div>
</body>
</html>
}
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"redpajama_set_name": "RedPajamaGithub"
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| 7,443
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{"url":"https:\/\/black.readthedocs.io\/en\/stable\/guides\/using_black_with_other_tools.html","text":"# Using Black with other tools\u00b6\n\n## Black compatible configurations\u00b6\n\nAll of Black\u2019s changes are harmless (or at least, they should be), but a few do conflict against other tools. It is not uncommon to be using other tools alongside Black like linters and type checkers. Some of them need a bit of tweaking to resolve the conflicts. Listed below are Black compatible configurations in various formats for the common tools out there.\n\nPlease note that Black only supports the TOML file format for its configuration (e.g. pyproject.toml). The provided examples are to only configure their corresponding tools, using their supported file formats.\n\nCompatible configuration files can be found here.\n\n### isort\u00b6\n\nisort helps to sort and format imports in your Python code. Black also formats imports, but in a different way from isort\u2019s defaults which leads to conflicting changes.\n\n#### Profile\u00b6\n\nSince version 5.0.0, isort supports profiles to allow easy interoperability with common code styles. You can set the black profile in any of the config files supported by isort. Below, an example for pyproject.toml:\n\n[tool.isort]\nprofile = \"black\"\n\n\n#### Custom Configuration\u00b6\n\nIf you\u2019re using an isort version that is older than 5.0.0 or you have some custom configuration for Black, you can tweak your isort configuration to make it compatible with Black. Below, an example for .isort.cfg:\n\nmulti_line_output = 3\ninclude_trailing_comma = True\nforce_grid_wrap = 0\nuse_parentheses = True\nline_length = 88\n\n\n#### Why those options above?\u00b6\n\nBlack wraps imports that surpass line-length by moving identifiers into their own indented line. If that still doesn\u2019t fit the bill, it will put all of them in separate lines and put a trailing comma. A more detailed explanation of this behaviour can be found here.\n\nisort\u2019s default mode of wrapping imports that extend past the line_length limit is \u201cGrid\u201d.\n\nfrom third_party import (lib1, lib2, lib3,\nlib4, lib5, ...)\n\n\nThis style is incompatible with Black, but isort can be configured to use a different wrapping mode called \u201cVertical Hanging Indent\u201d which looks like this:\n\nfrom third_party import (\nlib1,\nlib2,\nlib3,\nlib4,\n)\n\n\nThis style is Black compatible and can be achieved by multi-line-output = 3. Also, as mentioned above, when wrapping long imports Black puts a trailing comma and uses parentheses. isort should follow the same behaviour and passing the options include_trailing_comma = True and use_parentheses = True configures that.\n\nThe option force_grid_wrap = 0 is just to tell isort to only wrap imports that surpass the line_length limit.\n\nFinally, isort should be told to wrap imports when they surpass Black\u2019s default limit of 88 characters via line_length = 88 as well as ensure_newline_before_comments = True to ensure spacing import sections with comments works the same as with Black.\n\nPlease note ensure_newline_before_comments = True only works since isort >= 5 but does not break older versions so you can keep it if you are running previous versions.\n\n#### Formats\u00b6\n\n.isort.cfg\n[settings]\nprofile = black\n\nsetup.cfg\n[isort]\nprofile = black\n\npyproject.toml\n[tool.isort]\nprofile = 'black'\n\n.editorconfig\n[*.py]\nprofile = black\n\n\n### Flake8\u00b6\n\nFlake8 is a code linter. It warns you of syntax errors, possible bugs, stylistic errors, etc. For the most part, Flake8 follows PEP 8 when warning about stylistic errors. There are a few deviations that cause incompatibilities with Black.\n\n#### Configuration\u00b6\n\nmax-line-length = 88\nextend-ignore = E203\n\n\n#### Why those options above?\u00b6\n\nIn some cases, as determined by PEP 8, Black will enforce an equal amount of whitespace around slice operators. Due to this, Flake8 will raise E203 whitespace before ':' warnings. Since this warning is not PEP 8 compliant, Flake8 should be configured to ignore it via extend-ignore = E203.\n\nWhen breaking a line, Black will break it before a binary operator. This is compliant with PEP 8 as of April 2016. There\u2019s a disabled-by-default warning in Flake8 which goes against this PEP 8 recommendation called W503 line break before binary operator. It should not be enabled in your configuration.\n\nAlso, as like with isort, flake8 should be configured to allow lines up to the length limit of 88, Black\u2019s default. This explains max-line-length = 88.\n\n#### Formats\u00b6\n\n.flake8\n[flake8]\nmax-line-length = 88\nextend-ignore = E203\n\nsetup.cfg\n[flake8]\nmax-line-length = 88\nextend-ignore = E203\n\ntox.ini\n[flake8]\nmax-line-length = 88\nextend-ignore = E203\n\n\n### Pylint\u00b6\n\nPylint is also a code linter like Flake8. It has the same checks as flake8 and more. In particular, it has more formatting checks regarding style conventions like variable naming. With so many checks, Pylint is bound to have some mixed feelings about Black\u2019s formatting style.\n\n#### Configuration\u00b6\n\ndisable = C0330, C0326\nmax-line-length = 88\n\n\n#### Why those options above?\u00b6\n\nWhen Black is folding very long expressions, the closing brackets will be dedented.\n\nImportantClass.important_method(\nexc, limit, lookup_lines, capture_locals, callback\n)\n\n\nAlthough this style is PEP 8 compliant, Pylint will raise C0330: Wrong hanging indentation before block (add 4 spaces) warnings. Since Black isn\u2019t configurable on this style, Pylint should be told to ignore these warnings via disable = C0330.\n\nAlso, since Black deals with whitespace around operators and brackets, Pylint\u2019s warning C0326: Bad whitespace should be disabled using disable = C0326.\n\nAnd as usual, Pylint should be configured to only complain about lines that surpass 88 characters via max-line-length = 88.\n\n#### Formats\u00b6\n\npylintrc\n[MESSAGES CONTROL]\ndisable = C0330, C0326\n\n[format]\nmax-line-length = 88\n\nsetup.cfg\n[pylint]\nmax-line-length = 88\n\n[pylint.messages_control]\ndisable = C0330, C0326\n\npyproject.toml\n[tool.pylint.messages_control]\ndisable = \"C0330, C0326\"\n\n[tool.pylint.format]\nmax-line-length = \"88\"","date":"2021-09-17 16:49:09","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\": 0.29677683115005493, \"perplexity\": 7578.4536895381425}, \"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-2021-39\/segments\/1631780055684.76\/warc\/CC-MAIN-20210917151054-20210917181054-00675.warc.gz\"}"}
| null | null |
The Age of Adaline 2015 download from Movies Paper. Download The Age of Adaline 2015 in a single click. Click the download button to download The Age of Adaline 2015 Full Movie in HD 720p.
The Age of Adaline 2015 is a Hollywood movie in the English language. Lee Toland Krieger is the director of this movie. Sidney Kimmel, Gary Lucchesi and Tom Rosenberg are the producers of this movie. The writer of the movie is J. Mills Goodloe. The Age of Adaline 2015 movie cast includes Blake Lively, Michiel Huisman, Harrison Ford and other stars. Music of the movie is by Rob Simonsen. Download The Age of Adaline 2015 Movie HD 720p. The Cinematography of The Movie is by David Lanzenberg. The budget for this movie is $25 million. The release date for the movie is April 24, 2015.
One afternoon in San Francisco, Adaline archer purchases pretend IDs at associate living accommodations before returning home to feed her dog. She then goes to figure and opens a box of film reels, that highlight historical events and times. A voiceover ties along the film reels explaining the story of Adaline's life up to now. She was born on New Years Day 1908, then later married and gave birth to a female offspring, solely to become a widow once her husband died during a tragic accident. Years later, in 1937, Adaline crashed her car when she swerved into a ravine during a snowstorm and died in the freezing lake nearby, but a lightning strike suddenly revived her. From that moment on, Adaline has stayed physically twenty nine years previous.
The Age of Adaline 2015 Movie Download. One night, 2 suspicious FBI agents arrange to force her onto associate heavier-than-air craft for study, however she escapes captivity and realizes that she is going to have to be compelled to pay the remainder of her life on the run. Ever since, she has modified her identification and address per the age, whereas her female offspring Flemming ages ordinarily, to the purpose wherever she seems older than Adaline.
The Age of Adaline 2015 Movie Download. On national holiday Eve within the gift year, she attends a celebration wherever she meets Ellis Jones, introducing herself as her current alias, Jennifer. He asks to examine her once more however she refuses, knowing she will ne'er fall infatuated as a result of she will ne'er have a traditional future with somebody. The next day at work, Ellis arrives and once more asks Adaline to travel on a date with him. Finally she accepts.
In a flashback, Adaline is shown propulsion up during a cab to a park wherever a person is waiting, holding associate ring. Scared, she asks the cab driver to stay going. Back within the gift day, Adaline's dog falls unwell and he or she begins to ignore Ellis' calls. He shows up at her living accommodations however she pushes him away, solely to possess a amendment of heart whereas rummaging through some previous pictures and realizing she does not need to live an equivalent year 100 times while not having a natural life.
The Age of Adaline 2015 Movie Download. They later resolve their argument and Ellis asks Adaline to attend the party at his parents' house celebrating his parents' fortieth day, and he or she says affirmative. Upon their arrival, Ellis introduces her to his father, William, WHO acknowledges her instantly and calls her Adaline. She seems to acknowledge him, too, but lies, telling him that Adaline was her mother, WHO has since died. A flashback shows however they met and came to be infatuated, presently revealing that he was the person with the ring she stood up that afternoon. One night, Ellis tells Adaline he's falling infatuated together with her and he or she is unsure of the way to react.
The following day, Adaline talks with William outside and he notices a scar on her manus, and becomes agitated. Another flashback reveals that Adaline cut her hand whereas they were hiking decades past and he had seamed it up himself. He realizes that she is actually Adaline and confronts her. She becomes upset, speech she wont to be "normal" and does not grasp what modified her. He begs her to not run, for Ellis' sake, however she says she does not acumen to remain. She flees and returns to the house, writing a note to Ellis whereas he showers, then she packs her things and leaves. Moments later, Ellis finds the letter and confronts his father, WHO refuses to clarify.
While driving home, Adaline thinks of all the days she has run and suddenly contains a amendment of heart regarding the means she lives her life. She stops and calls her female offspring to inform her she goes to prevent running. As she turns the automobile around, a motortruck plows into her during a hit-and-run accident, deed her to die. Freezing and helpless, Adaline dies again. An motorcar arrives and he or she is revived by the electricity of the electronic device. Later within the hospital, she wakes up to Ellis, and also the 2 profess their love for each other. Adaline then tells him of her 107 years of life.
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{
"redpajama_set_name": "RedPajamaC4"
}
| 1,026
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namespace ui {
// static
Compositor* Compositor::Create(gfx::AcceleratedWidget widget) {
return NULL;
}
} // namespace ui
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 655
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{"url":"http:\/\/mathhelpforum.com\/calculus\/163214-another-integral-print.html","text":"# Another integral.\n\n\u2022 November 14th 2010, 07:57 AM\nKristina\nAnother integral.\nOkay,\nCan I write like this:\nintegral x^-4=x^-3\/-3 ?\nAnd one more question, from where is 8\/9? In this exercise\nhttp:\/\/www.part.lt\/img\/c1e69784c5724...e583256f86.jpg\n\u2022 November 14th 2010, 10:54 AM\nTheCoffeeMachine\nQuote:\n\nOriginally Posted by Kristina\nCan I write like this:\nintegral x^-4=x^-3\/-3 ?\n\nYes, $\\displaystyle \\int{\\frac{1}{x^4}\\;{dx} = -\\frac{1}{3x^3}+k$.\nQuote:\n\nAnd one more question, from where is 8\/9? In this exercise\nAfter differentiating, solve for [LaTeX ERROR: Convert failed] and multiply both sides by $8x^2$:\n\n$\\displaystyle u = 3x^3-1 \\Rightarrow \\frac{du}{dx} = 9x^2 \\Rightarrow dx = \\frac{1}{9x^2}\\;{du} \\Leftrightarrow 8x^2dx = \\frac{8}{9}\\;{du}.$\n\u2022 November 15th 2010, 03:33 AM\nKristina\nSo,\n1)-1\/3x^3 fluxcion will be -3x^-3=9x^-4 I'm not right? (Wondering)\n2)HOwever I have integration simalar like this example and I tried to use it. But I find problem\nIntegration (x^2-2) ln(x)dx\nu=x^2-2\ndu=2x\ndu\/dx=2x\ndx=du\/2x\nln(x)\/2x*du\nWhat i do wrong?\n\u2022 November 15th 2010, 05:06 AM\nharish21\nQuote:\n\nOriginally Posted by Kristina\nSo,\n1)-1\/3x^3 fluxcion will be -3x^-3=9x^-4 I'm not right? (Wondering)\n\nWhat are you doing here? trying to integrate? $\\displaystyle \\dfrac{-1}{3x^3} \\neq -3x^{_3}$\n\nIt is written as $\\displaystyle \\dfrac{-1}{3x^3} = \\dfrac{-x^{-3}}{3}$\n\nQuote:\n\nOriginally Posted by Kristina\n2)HOwever I have integration simalar like this example and I tried to use it. But I find problem\nIntegration (x^2-2) ln(x)dx\nu=x^2-2\ndu=2x\ndu\/dx=2x\ndx=du\/2x\nln(x)\/2x*du\nWhat i do wrong?\n\nIs your question $\\displaystyle \\int (x^2-2)\\;ln(x)\\;dx$???\n\nThen start with $\\displaystyle \\int (x^2-2)\\;ln(x)\\;dx = \\int x^2\\;ln(x)\\;dx - \\int 2\\;ln(x)\\;dx$\n\nUse integration by parts for the first term!\n\u2022 November 16th 2010, 10:34 AM\nKristina\nOkay,\nNow I do not understand nothing... (Headbang)\nintegration u*du dx=(x^3\/3-2x*1\/x (CAN SOMEONE EXPLAIN HOW TO WRITE FORMULAS?)\n\u2022 November 16th 2010, 12:01 PM\nmr fantastic\nQuote:\n\nOriginally Posted by Kristina\nOkay,\nNow I do not understand nothing... (Headbang)\nintegration u*du dx=(x^3\/3-2x*1\/x (CAN SOMEONE EXPLAIN HOW TO WRITE FORMULAS?)\n\nIf you posted more clearly and did not confuse your threads with new questions (which I have had to move several times now) you might have a better chance of understanding the help people are giving you.\n\nClick on the link in my signature for how to format equations.","date":"2014-07-25 02:02:46","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\": 0, \"codecogs_latex\": 7, \"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\": 0.8518618941307068, \"perplexity\": 5214.048344485546}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"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\/1405997892648.47\/warc\/CC-MAIN-20140722025812-00249-ip-10-33-131-23.ec2.internal.warc.gz\"}"}
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\section{Introduction}
Wall pressure fluctuations underneath a turbulent boundary layer (TBL) induce fluid-structural coupling, fatigue-induced component failure \cite{hambric2004vibrations} and vibro-acoustic noise transmission in many applications including air and ground transport, or wind turbine blades \cite{bull1996,Avallone2018,Tang2019,roger2005back}. In each case, the efficiency of the aero-vibro-acoustic coupling mechanism depends on the amplitude of the pressure fluctuations, on their spatial coherence, and on the velocity at which the pressure footprint of the TBL vortical structures is advected. Reliable engineering models are still needed for those characteristics, in particular when the TBL is subjected to a streamwise pressure gradient.
Considerable research effort has been devoted to the measurement and simulation of the above quantities for relatively canonical flows such as flat plate boundary layers in null \cite{blake1970} or moderate adverse pressure gradients \cite{keith1991}, or over airfoils \cite{moreau2005}. High fidelity simulations such as LES\cite{cohen2018} or DNS\cite{choi1990} can provide an unprecedented amount of detailed data about the unsteady flow field but remain too costly for the design stage in many engineering applications. Wind tunnel experiments\cite{farabee1991,van2018} share the same problem.
This explains the sustained interest in simplified, empirical or semi-empirical models, relating the wall pressure fluctuations, their spectrum and coherence lengths, to the main characteristics of a TBL\cite{bull1996}. Following the work of Kraichnan \cite{Kraichnan1956} and Panton \& Linebarger \cite{panton1974}, Blake \cite{Blake_1986} derived models based on the integration of the Poisson equation in the wavenumber-frequency domain. This method requires the vertical distribution of the mean longitudinal velocity and the wall-normal Reynolds stresses, and leads to high dimensional integrals, which can be difficult to compute\cite{grasso2019}.
A different approach was followed by Amiet \cite{Amiet1976} and Chase \& Howe \cite{Howe_1998} for the prediction of airfoil trailing-edge noise. It makes use of empirical or semi-empirical relations to relate the wall pressure spectra to statistical parameters of the boundary layer such as its thickness, external velocity or friction velocity. This procedure is attractive because the TBL parameters can be computed from low-CPU Reynolds Averaged Navier Stokes (RANS) simulations combined with more or less complex correlations.
Several improvements have thus been brought along this line over the last couple of decades. The empirical model of Chase-Howe was modified by Goody \cite{goody2004} to account for Reynolds number effects and improve the high-frequency behaviour. This model accurately predicts wall pressure spectra in the absence of pressure gradients, as demonstrated by Hwang \cite{Hwang2007}. Extensions of this last approach to account for the presence of pressure gradients were proposed by Kamruzzaman \cite{kamruzzaman2015}, Rozenberg \cite{rozenberg2012}, Hu \cite{Hu2018}, or Lee \cite{lee2018}.
Nevertheless, these models have proved valid only within specific ranges of the pressure gradient. Although rooted in the physics of the boundary layer, their derivation remains somewhat ad-hoc and usually requires the data-driven calibration of several coefficients\cite{catlett2016}. This process is essentially a regression problem, in which the definition of the parametric function is driven by physical considerations, scaling laws and modeller's ingenuity. Furthermore, for boundary layers in non-equilibrium the local turbulence dynamics and associated wall pressure fluctuations cannot be characterized exclusively using local parameters such as Clauser's parameter $\beta$ and the friction Reynolds number $Re_\tau$, but requires additional parameters to account for the boundary layer history\cite{Bobke2017} .
Given those difficulties, it becomes tempting to alleviate the problem of elucidating the complex dependencies between the TBL physical parameters, and to consider machine learning techniques instead, with their fast-growing arsenal of tools to solve regression problems\cite{Brunton2020}. Methods and tools from genetic programming\cite{Banzhaf1997} and deep learning\cite{Goodfellow} allow for deriving complex nonlinear functions from data, ultimately shifting the modelling challenge from the definition of the parametric function to the definition of an appropriate set of input parameters.
In this line, the present authors have recently proposed Genetic Programming to model pressure wall spectra \cite{dominique2021}. Genetic Programming is a collection of evolutionary techniques that automatically derive and calibrate analytic expressions linking input and output variables. While providing encouraging results (also reported in Section~\ref{sec:3E} of this article), the procedure requires a significant amount of time to converge when the parameter space formed by the number of operators and input functions becomes large\cite{cranmer2020}. On the other hand, limiting the set of candidate gene structures requires a large amount of knowledge about the regression task at hand or a tedious trial-and-error procedure.
Deep learning is considered in this work as an alternative. Deep learning is a subset of machine learning concerned with training large Artificial Neural Networks (ANN). These are recursive and distributed architectures of simple computational units, called neurons. ANNs are known as \emph{universal function approximators} because of their ability to approximate \emph{any} function\cite{Hornik1989}. This paper presents a wall pressure spectrum model based on ANNs. The model was calibrated (or \emph{trained}, in the machine learning terminology) on a wide range of conditions using experimental and numerical data from various authors. The model's predictive capabilities are compared with classic semi-empirical models, and model uncertainties are derived using an ensemble approach\cite{Abdar2021,Gawlikowski2021}. The ANN model and the uncertainty predictor have been made publicly available at \url{https://github.com/DominiqueVKI/VKI_researchWPS}.
The remaining of this paper is structured as follows. Section \ref{sec:2} presents the modelling problem, introducing the boundary layer parameters of existing wall pressure spectra models, described in Section \ref{sec:3}. Section \ref{sec:4} presents the numerical and experimental test cases that were used to calibrate the model in a first stage, and to validate it afterwards. Section \ref{sec:5} presents the methodology used to extract the boundary layer parameters, to design the ANN architecture to compute its uncertainty. Results are presented and discussed in Section \ref{sec:6} where we compare the prediction capabilities of the models on the present dataset. Finally, Section \ref{sec:7} collects conclusions and perspectives.
\section{Boundary layer wall pressure spectrum: definitions and parameters}
\label{sec:2}
The configuration of interest is sketched in Fig. \ref{fig:2dbl}. A two-dimensional boundary layer of thickness $\delta(x)$ develops as a flow with uniform velocity $U_\infty$ runs over a plate (or an airfoil). The fluid density and kinematic viscosity are denoted as $\rho$ and $\nu$, respectively. Because our focus is on turbulent flows, all velocities are to be interpreted as \emph{mean} velocities. Moreover, we consider conditions with negligible compressibility effects (i.e. with $Ma<0.3$, with $Ma=U_\infty/c_0$ the Mach number and $c_0$ the speed of sound).
\begin{figure}[h]
\includegraphics[width=0.45\textwidth]{figures/2dbl.pdf
\caption{Boundary layer developing over a large flat plate}
\label{fig:2dbl}
\end{figure}
The velocity of the flow at the edge of the boundary layer is the equilibrium velocity $u_e$. The boundary layer thickness is customarily defined as the distance from the wall at which the stream-wise velocity is $u(x,\delta)=0.99 U_\infty=u_e(x)$. This is the definition considered in this work for all the experimental datasets. An alternative definition is based on the pseudo-velocity\cite{deuse2019} $u^*(x,\delta)=0.99 u^*(x,\infty) = u_e(x)$, defined as
\begin{equation}
\label{pseudo}
u^*(x,y)=-\int^{y}_{0}\Omega_z(x,\xi) d\xi\,\,
\end{equation} where $\xi$ is a dummy integration variable along $y$ and $\Omega_z$ is the third component of the (mean) vorticity $\Omega=\nabla \times \mathbf{u}$, with $\mathbf{u}=(u,v)$ the (mean) velocity vector.
This quantity becomes constant when vorticity vanishes outside the boundary layer. The definition of $\delta$ based on the psuedo-velocity was used in this work for all the numerical datasets, as it was found to be more accurate in regions where the outer flow does not have uniform velocity.
Within the boundary layer region ($y<\delta$), the vertical pressure gradient is $\partial_y p =0$, while the acceleration of the flow in the outer region ($y>\delta$) can produce an Adverse Pressure Gradient (APG, $\partial_x p>0$), or a Favorable Pressure Gradient (FPG, $\partial_x p<0$). In the absence of flow acceleration, i.e. $\partial_x u_e=0$, one has a Zero Pressure Gradient (ZPG) condition.
The point-wise power spectral density of the pressure fluctuations can be obtained from the Fourier transform of the auto-correlation function:
\begin{equation}
\label{eq:Phipp}
\Phi_{pp}(x,\omega) = \int_{-\infty}^{+\infty} R(x,\tau) \, \textrm e^{\scriptsize -\textrm i\omega \tau} \, \textrm d\tau,
\end{equation}
where $\textrm i=\sqrt{-1}$, $\omega=2 \pi f$ is the angular pulsation and
\begin{equation}
\label{R}
R(x,\tau) = \int_0^\infty p'(x,t) \, p'(x,t+\tau) \, \textrm d t
\end{equation}
is the auto-correlation of the pressure fluctuations $p'$ at the wall ($y=0$), here assumed to be invariant along the spanwise direction $z$.
Empirical models aim at linking the space dependency of $\Phi_{pp}(x,\omega)$ to local boundary layer quantities. These include integral parameters measuring the boundary layer thickness and the expected averaged velocity profile, as well as others accounting for the influence of the pressure gradient.
Besides $\delta$, the boundary layer thickness can be measured also in terms of the displacement thickness $\delta^*$ and momentum thickness $\theta$:
\begin{subequations}
\label{eq:thickness}
\begin{gather}
\delta^*(x) = \int_0^{\delta}\left( 1 - \frac{u(x,y)}{u_e} \right) dy\,,\\
\theta(x) = \int_0^{\delta} \frac{u(x,y)}{u_e}\left( 1 - \frac{u(x,y)}{u_e} \right) dy\,.
\end{gather}
\end{subequations}
These account, respectively, for the mean flux and the momentum flux deficits due to the boundary layer's velocity profile $u(y)$. In a ZPG, the velocity profile is excepted to obey a universal form\cite{coles1956}:
\begin{equation}
\label{eq:universal_vel}
u^+(y^+) =
\begin{cases}
\mathcal{F}_w (y^+) + \frac{\Pi}{\kappa} \mathcal{W}\left(\eta\right) & \text{if } y < \delta \\
u_e/u_\tau & \text{if } y \geq \delta
\end{cases}
\end{equation} where $u^+=u/u_\tau$, with $u_\tau=\sqrt{\tau_w/\rho}$ the friction velocity and $\tau_w$ the wall shear stress, $y^+=y/\delta_v$, with $\delta_v=\nu/u_\tau$ the viscous length scale and $\eta=y/\delta$ a dimensionless coordinate which is better suited than $y^+$ far from the wall. The first term is the \emph{law of the wall}; the second term is a corrective term introduced by Coles\cite{coles1956} and is known as \emph{wake function}. The coefficient $\Pi$ is the wake strength parameter and $\kappa$ is the von Karman constant. We return on the definition of the velocity profiles wall function $\mathcal{F}_w (y^+)$ and the wake function $\mathcal{W}$ in Section \ref{sec:5p1}. For the moment, it suffices noticing that, as shown by Nagib and Chauhan \cite{nagib}, the impact of a pressure gradient can be accounted for while still keeping the ansatz in \eqref{eq:universal_vel} and modifying the functions $\mathcal{F}_w$ and $\kappa$ according to the strength and the sign of the pressure gradient.
From the set of parameters introduced thus far, empirical models for the wall pressure spectra search for a relation of the form:
\begin{equation}
\label{Model_Phi}
\Phi_{pp}(\omega)=f\left(\delta,\delta^*,\theta, u_e, \rho, \nu, \tau_w, \partial_x p, c_0, \Pi \right) \,,
\end{equation}
having omitted the functional dependency on the space coordinate $x$, which is included in the boundary layer parameters. It is worth noticing that the parameters $\delta^*$ and $\theta$ could be deduced from the model of the velocity profile in \eqref{eq:universal_vel} and thus one could postulate a relation of the form $\delta^*,\theta=g(\delta,u_e,\rho,\nu, \tau_w,\partial_x p, c_0)$. Nevertheless, in this work we use a model for $u(y)$ (as in equation \ref{eq:universal_vel}) only for the computation of $\tau_w$ and $\Pi$ and consider $\delta^*$ and $\theta$ as independent inputs to the correlation to be derived.
Because only three fundamental units are involved, the Buckingham-Pi theorem suggests that such a relation can be conveniently written in dimensionless form as
\begin{equation}
\frac{\Phi_{pp} u_e}{\delta^* \tau_w^2}\Biggl(\frac{\omega \delta^*}{u_e}\Biggr)=f\left(\frac{\delta}{\delta^*}, \frac{\theta}{\delta^*}, \frac{c_0}{u_e}, \frac{\rho u_e^2}{\tau_w}, \frac{\nu \delta^*}{u_e}, \frac{\delta^*}{\tau_w}\frac{dp}{dx}, \Pi\right) \,,
\end{equation} having used $\delta^*$, $\tau_w$ and $u_e$ as repeated variables in the scaling. Adjusting the derived dimensionless number to those more commonly encountered in the literature of wall pressure spectra, the dimensionless function to be derived is thus
\begin{equation} \label{eq:Buckingham}
\tilde{\Phi}_{pp}(\tilde{\omega}) = \tilde{f}\left(\Delta, H, Ma, \Pi, C_f, R_T, \beta \right),
\end{equation} where $\tilde{\Phi}_{pp}=\Phi_{pp}/(\delta^* \tau_w^2/u_e)$ is the dimensionless power spectral density, $\tilde{\omega}=\omega/(u_e/\delta^*)$ is the dimensionless frequency, $\Delta=\delta/\delta^*$ is the Zagarola-Smits's parameter \cite{zagarola}, $H=\delta^*/\theta$ is the boundary layer's shape factor, $Ma=u_e/c_0$ is the Mach number, $C_f=\tau_w/(\rho u_e^2)$ is the friction coefficient, $R_T=(\delta^*/u_e)/(\nu/u_\tau^2)$ is the time scale ratio, also known as the Reynolds number effect of the pressure spectrum \cite{goody2004}, and $\beta = (\theta/\rho u_\tau^2) \partial_x p $ is the Clauser parameter.
Finally, it is worth noticing that the scaling laws in \eqref{eq:Buckingham}, and particularly the definition of the dimensionless frequency and the wall pressure spectra, are known to be reasonable for the low and the mid-frequency range ($\omega \nu / u_\tau^2 < 100$) but not for the higher frequency range \cite{blake1986}, in which a scaling based on the inner variables, i.e. $\hat{\omega}=\omega \nu/(u_\tau^2)$, is more appropriate\cite{bull1996}. Nevertheless, the lack of universal scaling for all frequencies is compensated by the Reynolds number $R_T$, which is the ratio of the two leading time scales in the spectrum.
\section{Empirical models}
\label{sec:3}
We now introduce various forms of Eq.~\eqref{eq:Buckingham} from five models available in the literature. These are used to benchmark the ANN model derived in this work.
\subsection{Goody's ZPG model based on self-similarity}
The empirical wall pressure spectra proposed by Goody\cite{goody2004} was built based on the principle of scaling and self-similarity. The primary precept is that the wall pressure spectra are proportional to different scales at low and high frequencies. The low-frequency scales with outer flow variables ($\delta^*$ and $u_e$) and the high-frequency scales with the viscosity and inner variables ($\nu$ and $u_\tau$). Such scaling is similar to the two-layer scaling of turbulent boundary layer velocity profile using a viscous sublayer and a logarithmic layer \cite{coles1956}.
Based on this principle, Goody further extended the model of Chase and Howe \cite{Howe_1998} to account for this outer-to-inner scaling ratio by including the Reynolds number, with the following expression:
\begin{equation*}
\label{eq:Goody}
\frac{\Phi_{pp}(\omega) u_e}{\tau_w^2 \delta} = \frac{ C_2(\omega \delta / u_e)^2 }{\left( (\omega \delta / u_e)^{0.75} + C_1\right)^{3.7} + \left( C_3(\omega \delta / u_e) \right)^7},
\end{equation*} where $C_1 = 0.5$, $C_2=3.0$, and $C_3 = 1.1 R_{T,\delta}^{-0.57}$ describes the effect of the Reynolds number, which these author define with respect to $\delta$ and not $\delta^*$, i.e. $R_{T,\delta}=(\delta/u_e)/(\nu/u_\tau^2)$. This model, tuned using the constants listed above, describes accurately wall pressure spectra beneath a turbulent boundary layer in the absence of pressure gradients, as demonstrated by Hwang \cite{hwang2009comparison}.
\subsection{Rozenberg's extension for APG}
Rozenberg \emph{et al.}\cite{rozenberg2012} extended Goody's model by including the effects of adverse pressure gradients. The authors assumed these could be incorporated into the model using the Clauser parameter $\beta$ and the boundary layer wake parameter $\Pi$. This assumption originated from an analogy with the scaling of turbulent boundary layer velocity profiles, which requires a wake correction.
Rozenberg's model has three main extensions of Goody's model. First, the slope of the mid-frequency range is allowed to be a function of the pressure gradient. Second, the global level of pressure fluctuation is allowed to increase with $\beta$ and $\Pi$. Third, the reference length used for the scaling of the model is changed to the displacement thickness.
The resulting model is:
\begin{equation*}
\label{eq:Rozenberg}
\frac{\Phi_{pp}(\omega) u_e}{\tau_w^2 \delta^*} = \frac{\Bigg[ 2.82\Delta^2(6.13\Delta^{-0.75}+F_1)^{A_1} \Bigg] \left[4.2 \frac{\Pi}{\Delta} + 1\right] \tilde{\omega}^2}{[4.76 \tilde{\omega}^{0.75} + F_1]^{A_1} + (C_3\tilde{\omega})^{7}},
\end{equation*} where $F_1 = 4.76\left({1.4}/{\Delta}\right)^{0.75}[0.375A_1-1]$, $A_1 = 3.7 + 1.5\beta$ and $C_3 = 8.8R_{T,\delta}^{-0.57}$. This formulation falls back on Goody's for ZPG. However, this model was only calibrated on APG flows data and is therefore unable to predict FPG.
\subsection{Kamruzzaman's extension for FPG}
Kamruzzaman \emph{et al.}\cite{kamruzzaman2015} extended Rozenberg's model to a larger database, including APG and FPG conditions.
Differently from the previous works, the authors reduce the set of input parameters by linking Clauser parameter to the wake coefficient via the relation $\Pi = 0.8 \left( \beta + 0.5 \right)^{0.75}$.
The amplitude of the model predictions was slightly increased for both APG and FPG. The dependency of the Reynolds number suggested by Goody was also changed to best fit their database. As a result, the prediction of this model does not agree with Goody's model for ZPG. In opposition to Rozenberg's model, the variations of the slope in the mid-frequency range were not observed in Kamruzzaman's study. The resulting model is:
\begin{equation*}
\label{eq:Kamruzzaman}
\frac{\Phi_{pp}(\omega) u_e}{\tau_w^2 \delta^*} = \frac{B_2 \tilde{\omega}^2}{[\tilde{\omega}^{1.64} + 0.27]^{2.47} + [B_3 \tilde{\omega} ]^7},
\end{equation*} where $ B_2 = 0.45(1.75(\Pi^2\beta^2)^{0.5(H/1.31)^{0.3}}+15) $ and $ B_3 = (1.15R_T)^{-2/7}$.
Due to the empirical relationship between the wake coefficient and the Clauser parameters, the model dependency on $\Pi$ is enslaved to its dependency on $\beta$. The shape factor $H$ is also used for the first time as a modelling tool to account for history effects of the boundary layer developments. The use of the shape factor to model history effects in wall pressure spectra was later used in 2018 by Hu \emph{et al.} \cite{Hu2018} who proposed another empirical wall pressure spectral model with no direct influence of the pressure gradient.
\subsection{Lee's combined model}
Lee \emph{et al.}\cite{lee2018} performed an extensive comparison between all the models previously discussed, and highlighted that although each of these performed well on their training dataset, the match with other data proved to be limited. Lee \emph{et al} proposed a combination of the previous models with the following modifications: \textit{i}) the transition from mid to high frequency ranges is made dependent on the pressure gradient, \textit{ii}) a correction of the spectrum amplitude at low frequencies for ZPG and \textit{iii}) a larger increase of the amplitude for large APGs.
\begin{equation*}
\label{eq:Lee}
\frac{\Phi_{pp}(\omega) u_e}{\tau_w^2 \delta^*} = \frac{B_2 \tilde{\omega}^2}{[4.76\tilde{\omega}^{0.75}+d^*]^e + [C_3\tilde{\omega}]^{h^*}},
\end{equation*} where $B_2 = max(a,(0.25\beta - 0.52 )a)$, $a=2.82\Delta^2(6.13\Delta^{-0.75}+d)^{3.7+1.5\beta}$, $d^*=max(1.0,1.5d)$, $d=4.76(1.4/\Delta)^{0.75}[0375e-1]$, $e=3.7 + 1.5\beta$, $C_3 = 8.8 R_{T,\delta}^{-0.57}$ and finally $h^* = min(3,0.139+3.1043\beta)+7$.
Lee's model is close to the Goody model for ZPG flows with slight discrepancies at low frequencies. Despite its complexity and being non-continuous because of the minimum/maximum operators, this is nowadays known as one of the most accurate empirical models for FPG and APG boundary layers. However, as it will be shown in \textsc{Sec.} \ref{sec:6}, this model still faces difficulties in the presence of strong APG.
\subsection{Dominique's model obtained from Gene Expression Programming}
\label{sec:3E}
Dominique \emph{et al.}\cite{dominique2021} presented a data-driven methodology for empirical models of wall pressure spectra model. Unlike the models discussed thus far, this is not derived as an extension of previous models but is derived \textit{ex novo} using Gene Expression Programming \cite{ferreira2006} (GEP).
The method was originally implemented using a single experimental dataset of flat plate boundary layer \cite{salze2014}, but it is extended in this work to the larger dataset described in Sec \ref{sec:4}. This approach returns the following empirical model:
\begin{equation*}
\label{eq:Dominique}
\frac{\Phi_{pp}(\omega) u_e}{\tau_w^2 \delta^*} = \frac{\left( 5.41+ C_f(\beta_C + 1 )^{5.41}\right) \tilde{\omega}}{\tilde{\omega}^2 + \tilde{\omega} + (\beta+1)Ma + (\tilde{\omega}+3.6)\frac{\tilde{\omega}^{4.76}}{C_f R_T^{5.83}}},
\end{equation*}
The GEP approach returns a model which is structurally close to the other models described in this section in that it assumes a rational function in $\tilde{\omega}$. However, some differences are evident at low frequencies and high frequencies. At low frequencies, the GEP model yields an asymptotic dependence as $\tilde{\omega}^{1}$, instead of the $\tilde{\omega}^{2}$ predicted by the Kraichnan-Phillips theorem~\cite{Kraichnan1956}.
At high frequencies, the GEP model yields a decay as $\tilde{\omega}^{-4.76}$ instead of the $\tilde{\omega}^{-5}$ theoretically predicted by Blake\cite{blake1986}.
While these discrepancies might appear to violate physical models, we recall that these theoretical results are not always confirmed experimentally. For example, the theoretical slope at low frequencies has been rarely observed in wind tunnel measurements. One of the very few experimental confirmations, brought by Farabee and Casarella~\cite{farabee1991}, required low-speed measurements performed in a particularly quiet wind tunnel and dedicated noise cancellation techniques~\cite{goody2004}.
The GEP somehow accounts for these effects and appears to have better predictive performances on realistic scenarios. The differences at high frequencies ($\tilde{\omega}^{-4.76}$ instead of $\tilde{\omega}^{-5}$) is somewhat minor if one considers that the GEP model is the result of a pure regression, unaware of physics-based insights.
\section{Experimental and numerical datasets used for training and validation}\label{sec:4}
The dataset analyzed in this work include the experimental database by Salze \emph{et al}\cite{salze2014} on the boundary layer over a flat plate and three high-fidelity numerical simulations of the flow over a controlled diffusion (CD) airfoil, by Deuse \emph{et al}\cite{deuse2019}, Hao \emph{et al}\cite{wu2018} and Christophe \emph{et al}\cite{christophe2009}.
The flat plate experimental dataset from Salze \emph{et al}\cite{salze2014} includes ZPG and moderate APG and FPG conditions. The first should be suited for Goody's model whereas the others should be within the range of validity of Lee's or Kamruzzaman's models.
The CD airfoil is a benchmark problem for aero-acoustic noise prediction. This airfoil is representative of engine cooling axial fans for automotive or HVAC systems. It has a 4\% relative thickness and a leading-edge camber angle of $12^{\circ}$. In nominal operating conditions, the airfoil has a laminar boundary layer on the pressure side and a short separation bubble at the leading edge, on the suction side, followed by a reattachment and transition to a turbulent boundary layer.
In contrast with the flat plate case, the CD configuration provides a variety of conditions along the airfoil chord, as the development of the TBL continuously adapts to the local Reynolds number and pressure gradient. The TBL data have thus been extracted at a number of streamwise locations starting from the trailing edge (the relevant location for the prediction of trailing edge noise) and further upstream of it. A brief descrition of each dataset, as well as the precise locations at which each profiles are extracted are reported in the following subsections.
\subsection{Salze's flat plate TBL subjected to pressure gradients}
Salze\cite{salze2014} performed an experimental investigation of the wall pressure spectrum under moderately favourable and adverse pressure gradients in the closed test section wind tunnel of the University of Lyon. They used a rotating antenna of pinhole microphones to measure the wall pressure fluctuations and a sloped opposite wall to induce pressure gradients. All wall pressure spectra are obtained at the location of the antenna in the wind tunnel, changing the flow velocity.
The measurements of ZPG boundary layers were found close to the expected Goody model. However, only moderate effects of the pressure gradient were observed. At low frequency, the amplitude of the spectra was slightly decreased for FPG and increased for APG. As it can be observed in Fig. \ref{fig:salze_data}, the pressure gradients also affect the inertial slope in the mid-frequency region. At high frequency, the spectrum decays as $\omega^{-5}$ for all pressure gradients.
\begin{figure}[h]
\includegraphics[width=0.5\textwidth]{figures/SalzeDataV2.pdf
\caption{Experimental wall pressure spectra from the flat plate TBL database of Salze\cite{salze2014}.}
\label{fig:salze_data}
\end{figure}
\subsection{Deuse's CD airfoil TBL at 8 degrees of angle of incidence}
Deuse\cite{deuse2019} performed a DNS simulation of the CD airfoil using the finite difference solver named HipSTAR. The airfoil is embedded in an infinite uniform flow at a free-stream Mach $Ma=0.2$ with an angle of attack of $\alpha_e = 8^{\circ}$ and a Reynolds number $Re = 10^5$. The contour of the time-averaged static pressure field for this test case is shown in Fig.~\ref{fig:cd_comp}(a), together with the location at which the boundary layer profiles and the wall pressure spectra are sampled in this work.
The first point, from the trailing edge, is located at 2\% of the chord upstream of the airfoil trailing edge. From there, the profiles are taken at steps of 5\% chord lengths until 60\% of the chord. Points situated further upstream are not suited for the modelling of attached TBL because the flow separates.
The velocity profiles and their associated wall pressure spectra at three selected locations (labelled with $1,2,3$ in Fig.~\ref{fig:cd_comp}(a)) are shown in Fig.~\ref{fig:cd_comp}(c) and Fig.~\ref{fig:cd_comp}(d) respectively. The spectrum with the largest overall amplitude corresponds to the closest location to the trailing edge, and the other spectra decrease monotonously as the extraction location moves upstream. The fast-growing adverse pressure gradient explains the sharp increase of the pressure fluctuations near the trailing edge. The spectral levels also appear to be considerably larger than those obtained by Salze (Fig.~\ref{fig:salze_data}) for more moderate adverse pressure gradients. Finally, the inertial range is not observed in this simulation because of the low Reynolds number.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{figures/Cd_comp.pdf}
\caption{Comparison of the numerical wall pressure spectra and velocity profile extracted perpendicular to the airfoil from (a) Deuse and (b) Hao. The turbulent boundary layer velocity profiles are expressed in wall units where the dotted points correspond to the numerical points and the lines are the fit to the model in Eq.\ref{eq:universal_vel} using the methodology in Sec. \ref{sec:5}}
\label{fig:cd_comp}
\end{figure*}
\subsection{Hao's CD airfoil TBL at 4 degrees of effective angle of incidence}
Hao\cite{wu2018} has also used HipSTAR to perform a simulation similar to that of Deuse at $Ma=0.25$, and a Reynolds number of $1.5\times 10^5$. The sampling locations for the wall pressure fluctuations and velocity profiles are the same as in the previous section for Deuse's dataset but include three extra points until 75\% of the chord. The sampling locations are shown in Fig.~\ref{fig:cd_comp}(b) together with the time-averaged static pressure field. Three selected averaged velocity and wall pressure profiles (labelled as $4,5,6$ in Fig.~\ref{fig:cd_comp}(b)) are shown in Fig.~\ref{fig:cd_comp}(c) and Fig.~\ref{fig:cd_comp}(d).
Differently from Deuse\cite{deuse2019}, Hao simulated the finite dimensions of the wind tunnel jet in which the airfoil was placed in the experiments \cite{Padois2016}. Since a lifting airfoil placed in a jet induces a deflection of the stream, the practical angle of incidence of the airfoil is reduced compared to the infinite uniform flow case. An effective angle of incidence can however be estimated, which was found to be $\alpha_e = 4^{\circ}$ in this case. As a result, the aerodynamic loading around the airfoil is reduced compared to that of Deuse\cite{deuse2019}. This explains the lower spectral levels obtained by Hao\cite{wu2018} as visible in Fig.~\ref{fig:cd_comp}(d). Another noticeable difference is the high-frequency hump observed for large pressure gradients. The precise mechanism leading to this high-frequency hump remains unclear, but we conjecture that it may be caused by the nearby turbulence over the pressure side or by the near wake.
\subsection{Christophe's CD airfoil TBL at effective angles of incidence varying between 2 and 6 degrees}
Christophe\cite{christophe2009} performed six LES simulations on the CD airfoil at a free-stream Mach number of 0.05, and effective angles of incidence $\alpha_e $ varying between $2^{\circ}$ and $6^{\circ}$.
The velocity and the wall pressure spectra profiles are extracted from 5\% until 65\%, with steps of 5\%, of the chord upstream the airfoil trailing edge. This dataset, initially developed for uncertainty quantification, yields different boundary layer transitions and histories, affected by the presence and extent of the laminar separation bubble at the suction side. The wall pressure spectra were deemed reliable up to a frequency of about 4\,kHz, beyond which numerical dissipation caused an artificial attenuation of the pressure levels.
\begin{table*}[htbp]
\begin{center}
\includegraphics[width=\textwidth]{figures/table.pdf}
\end{center}
\caption{Range of turbulent boundary layers parameters covered by the four datasets. The points represent the mean value while the vertical lines represent respectively the 25\% and 75\% percentiles.\vspace*{-3ex}}
\label{fig:inputrange}
\end{table*}
\subsection{A Note on the full dataset}\label{Note}
The range of dimensionless numbers covered by the datasets is illustrated in Tab.~\ref{fig:inputrange} (Table \ref{tab:input} in appendix provides the exact values and Section \ref{sec:5} describes the methodology used in their computation).
The number of boundary layer profiles (and corresponding wall pressure spectra) for each test case is denoted as $N$ in column 2. Because these datasets have vastly different sizes, their relative contribution to the training of the ANN is weighted by the weights $W$ in column 3 (see section \ref{sec:subAAN}). This avoids over-representing some flow conditions or geometries that are more densely sampled. For the same reason, since the frequency resolution is not the same for all profiles, all wall pressure spectra have been re-sampled over 500 points logarithmically spaced over the frequency axis.
The combination of experimental and numerical data covers a wide range of operating conditions. For example, the experimental data cover higher Reynolds numbers $R_T$ cases than the numerical simulations of the CD airfoil, but the latter offer a broader range of pressure gradients, with associated Clauser parameter $\beta$ and wake parameter $\Pi$.
Figure.~\ref{fig:cd_comp}(d) reveals that fairly similar non-dimensional wall pressure spectra can be obtained in largely different flow conditions, at least over the analyzed frequency range. For example, the wall spectrum obtained by Hao close to the trailing edge (position ~4) of the airfoil closely matches the one sampled by Deuse at 60\% of chordwise distance from the trailing edge (position ~3), up to $\omega \delta^*/U_e=2$. Yet, the corresponding boundary layer profiles (Fig.~\ref{fig:cd_comp}(c)) are significantly different. Conversely, the available dataset doesn't contain samples with identical boundary layer parameters leading to different wall pressure spectra. This would make the problem of identifying models in the form of \eqref{Model_Phi} ill-posed, since the modelling would require additional parameters.
It is nevertheless possible to give a qualitative overview of the complexity of the function to be derived by mapping the full input space sampled in the dataset onto a plane. This exercise in high-dimensional cartography can be carried out using a dimensionality reduction approach known as t-SNE (t-Distributed Stochastic Neighbor Embedding\cite{vanDerMaaten2008}). Like many other nonlinear dimensionality reduction techniques (see Maaten \emph{et al}\cite{Maaten2009DimensionalityRA} for a review), the t-SNE maps a large dimensional dataset into a lower dimensional one while preserving some nonlinear metrics of similarity. In other words, points that are close or far away in the initial space (according to a certain metrics) will also be close or far away (according to another metrics) in the reduced space provided by the t-SNE embedding.
In the t-SNE, the similarity metrics is taken using Gaussian (in the higher dimensional space) and t-Student distributions (in the low dimensional space). More specifically, given $\boldsymbol{x}'_i$ and $\boldsymbol{x}'_j$ two of the $n_p$ vectors in the original space $\mathbb{R}^{n_x}$, the degree of similarity under a Gaussian distribution is
\begin{equation}
\label{d_ij_EQ}
q_{i,j}=q(\boldsymbol{x}'_i,\boldsymbol{x}'_j)=\frac{\exp(-||\boldsymbol{x}_i-\boldsymbol{x}'_j||^2/(2\sigma^2_i))}{\sum_{k\neq j} \exp(-||\boldsymbol{x}'_i-\boldsymbol{x}'_j||^2/(2\sigma^2_i))}\,.
\end{equation}\, where $\sigma_i$ is a user defined parameter and assuming that both inputs have been normalized in the range $[0,1]$. Hence two samples which are similar (close to each other in $\mathbb{R}^{n_x}$) leads to large $q_{i,j}$ while samples that are largely different (far away in $\mathbb{R}^{n_x}$) leads to $q_{i,j}\approx 0$. The normalization in \eqref{d_ij_EQ} allows to interpret $q_{i,j}$ as a distribution. A similar metric $d_{i,j}(\boldsymbol{z}_i,\boldsymbol{z}_j)$ is defined in the reduced space (albeit with a different distribution) and the mapping can be done in such a way that the pair-wise distances are preserved as much as possible.
We use this approach to visualize the input dataset. For a given wall pressure spectra, we map the input space for the wall pressure spectra model, i.e. $\boldsymbol{x}'=[\Delta, H, Ma, \Pi,C_f,R_T, \beta]\in\mathbb{R}^{7}$, into a two-dimensional plane $\boldsymbol{z}=({z}_1,{z}_2)$. The result is shown in Fig. \ref{fig:tsne}, with different markers denoting the datasets previously introduced.
The figure shows that Deuse's and Hao's datasets are close (according to a metric like \eqref{d_ij_EQ}) in the input space while Salze's is different from all the others. Christophe's dataset covers the largest span and has conditions similar to Deuse's (at high $z_1$). This figure should be analyzed together with Fig. \ref{fig:tsne2}, collecting all the 117 available power spectral densities, labelled by the dataset with the same markers as in Figure \ref{fig:tsne}.
\begin{figure}[h]
\centering
\includegraphics[width=0.5\textwidth]{figures/tsne.pdf}
\caption{T-SNE manifold 2D map of the 7D inputs of Tab. \ref{fig:inputrange}. \textcolor{color11}{\ding{53}} for Christophe, \textcolor{color12}{\ding{108}} for Salze, \textcolor{color13}{\ding{117}} for Deuse and \textcolor{color14}{\ding{116}} for Hao.}
\label{fig:tsne}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.5\textwidth]{figures/tsne-output.pdf}
\caption{Wall pressure spectra \textcolor{color11}{\ding{53}} for Christophe, \textcolor{color12}{\ding{108}} for Salze, \textcolor{color13}{\ding{117}} for Deuse and \textcolor{color14}{\ding{116}} for Hao.}
\label{fig:tsne2}
\end{figure}
The plot confirms the expectation that the wall pressure spectra in Hao and Deuse are relatively similar, given their similarity in the input space. However, their spectral amplitudes vary significantly, potentially indicating strong sensitivity to the inputs. The amplitude of the wall pressure spectra in Salze falls somewhere in between the one of Hao and Deuse despite the largely different input space. However, Salze's wall pressure spectra differ significantly in shape since the inertial range is clearly observable contrary to the spectra obtained on the CD airfoil.
While the available dataset does not allow for assessing the well-posedness of the regression problem, it is clear that the function at hand is particularly complicated. As we could not find similar turbulent boundary layers with different wall pressure spectra, we have no evidence that the development history of the non-equilibrium wall pressure spectra used in this study is not already accounted for by the set of parameters from \eqref{eq:Buckingham}.
\section{Methodology}
\label{sec:5}
\subsection{Computation of the boundary layer parameters}\label{sec:5p1}
The boundary layer parameters $\Pi$ and $\tau_w$ are obtained from a nonlinear regression of the boundary layer velocity profile $u(y)$ by the model laid out in Eq.~(\ref{eq:universal_vel}). The method is inspired by Clauser's method \cite{clauser1956turbulent} and the work of Kendal \cite{kendall2008} and Rodríguez‑López \cite{rodriguez2015} for the determination of the wall shear stress by fitting the data to a theoretical velocity profile. We extend this approach with an iterative approach, illustrated in Fig.~\ref{fig:bl_model}, to account for the presence of APG or FPG. While Clauser's method adjusts the friction velocity $u_\tau$ to match the universal logarithmic law in the overlap region of the boundary layer, we adjust both $u_\tau$ and the wake parameter $\Pi$ to fit the extracted data with a complete velocity profile accounting for the presence of a pressure gradient.
Following Eq.~(\ref{eq:universal_vel}) in Section \ref{sec:2}, the law of the wall $\mathcal{F}_w(y^+)$ used in this work is the one proposed by Musker \cite{musker}:
\begin{equation}\label{eq:musker}
\mathcal{F}_w(y^+) = \int_0^{y^+}\frac{s \xi^2 + \kappa }{\kappa \, s\, \xi^3+ s\,\xi^2+\kappa} d \xi,
\end{equation}
with $\xi$ a dummy integration variable, $\kappa$ the von Karman constant and $s$ the free parameters of this formulation. For a ZPG, setting $s=0.001093$ and $\kappa=0.41$ results in a velocity profile matching the classic logarithmic profile $1/\kappa \,ln(y^+)+B$ with $B=5.0$. This formulation has the main advantage of not requiring a piece-wise definition of the law of the wall, which is also usually problematic in the buffer region ($5\leq y^+ \leq 30$).
To account for the pressure gradient effect, we follow Nickels' approach\cite{nickels} and modify the von Karman constant as:
\begin{equation}
\label{eq:nickel_kappa}
\frac{\kappa}{\kappa_0} = \frac{1}{\sqrt{1 + p_x^+ y_c^+}} \,
\end{equation} where $\kappa_0 = 0.41$ is the von Karman constant for ZPG, and $y_c^+$ is the point at which the logarithm layer intersects the linear prediction in the viscous sub-layer. This quantity is also a function of the pressure gradient, and can be computed as the smallest positive root of the cubic:
\begin{equation}
\label{eq:nickel_yc}
p_x^+(y_c^+)^3 + (y_c^+)^2 - 144 = 0,
\end{equation} where $p^+_x= (\delta_\nu/\theta)\beta$ is the dimensionless pressure gradient.
Combining \eqref{eq:nickel_yc}-\eqref{eq:nickel_kappa}, one sees that $\kappa > \kappa_0$ for APG and $\kappa < \kappa_0 $ for FPG. This is consistent with what observed in this study and with the discussion on the non-universality of the von Karman coefficient by Nagib \emph{et al} \cite{nagib}. These authors have also introduced an empirical relation to link $\kappa$ and $B$ in the logarithmic region ($y^+\geq 30$) in the presence of a pressure gradient:
\begin{equation}
\label{eq:nagib}
B = \frac{1.6[\exp(0.1663B)-1]}{\kappa} \,.
\end{equation}
The alteration of the von Karman constant permits a better fit in the logarithmic region when a pressure gradient is present, and also influences indirectly the wake parameter $\Pi$.
The wake function $\mathcal{W}(\eta)$ in Eq.~(\ref{eq:universal_vel}) used in this work is the one proposed by Chauhan \cite{chauhan2007}:
{\fontsize{8}{8}\selectfont %
\begin{align}
\label{Chauhan}
\mathcal{W}(\eta) &=\frac{1 -\exp\left(-0.25\left(5a_2 + 6a_3 +7a_4)\eta^4 + a_2\eta^5 + a_3 \eta^6 + a_4\eta^7\right)\right)}{1-\exp(-0.25(a_2+2a_3+3a_4))} \nonumber \\
& \times \left( 1-\frac{1}{2\Pi} \ln(\eta) \right),
\end{align}}
where $a_2 = 132.84$, $a_3 = -166.2$, $a_4 = 71.91$ ensure a continuous slope at the edge of the wake region. The wake parameter can be identified by noticing that the normalizing condition imposes $\mathcal{W}(1)=2$ and thus at the edge of the boundary layer one has
\begin{equation}
\label{eq:wake}
\frac{u_e}{u_\tau} = \frac{1}{\kappa} \ln \left(\frac{\delta u_\tau}{\nu}\right) + B+ \frac{2 \Pi}{\kappa}\,.
\end{equation}
Combining the equations introduced thus far (cf. Figure \ref{fig:bl_model}), it is possible to set up a regression problem in which the friction velocity $u_\tau$ and the wake parameter $\Pi$ are the ones that allows minimizing the loss function
\begin{equation}
{J}(u_\tau,\Pi)=||u^+(y^+)-\tilde{u}^+({y}^+)||_2
\end{equation}
where $||\bullet||_2$ is the $l_2$ error, $u^+(y^+)$ is the available data and $\tilde{u}^+({y}^+)$ is the model prediction.
\begin{figure}[h!]
\includegraphics[width=0.4\textwidth]{figures/optimisation_loop.pdf
\caption{Computation loop for the error used within the optimisation of the friction velocity $u_\tau$ and wake parameter $\Pi$.}
\label{fig:bl_model}
\end{figure}
\begin{figure}[h]
\includegraphics[width=0.45\textwidth]{figures/bl_example_V2.pdf
\caption{Example of boundary layer velocity fitting with pressure gradient correction}
\label{fig:p_corr}
\end{figure}
An example of boundary layer velocity profile in presence of pressure gradient is shown in Fig. \ref{fig:p_corr}, considering a test case with a strong pressure gradient ($\beta=5.93$) from Hao at 2\% chord upstream the the airfoil trailing edge (position no~4 in Fig. \ref{fig:cd_comp}). The proposed optimization loop leads to $u_\tau=2.37$ m/s and $\Pi=2.26$, producing an excellent match with the available data. For completeness, the figure also reports the prediction of a classic piece-wise linear-logarithmic model and Musker's model in \eqref{eq:musker} with $\kappa = 0.25$ and $B=-0.6$. The classic piece-wise linear-logarithmic model is unable to match the correct slope in the logarithmic region while the Musker law modified with the pressure gradient correction from \eqref{eq:nagib} and \eqref{eq:nickel_kappa} provides an excellent match with the data.
\subsection{The Artifical Neural Network (ANN)} \label{sec:subAAN}
ANNs are distributed architecture of simple computational units\cite{Goodfellow}, called \emph{neurons}, connected in layers as shown in Fig. \ref{fig:NN_architecture}. These architectures provide a parametric function $\boldsymbol{y}=f(\boldsymbol{x},\boldsymbol{w},\boldsymbol{b})$ mapping an input $\boldsymbol{x}\in\mathbb{R}^{n_x}$ to an output $\boldsymbol{y}\in\mathbb{R}^{n_y}$ according to a set of parameters called \emph{weights} $\boldsymbol{w}\in\mathbb{R}^{n_w}$ and \emph{biases} $\boldsymbol{b}\in\mathbb{R}^{n_b}$. In the simplest configuration used in this work, namely a fully connected feed-forward network, this function is defined in a recursive way, such that the output of each layer is the input of the following. Specifically, given $\boldsymbol{y}^{(l)}\in\mathbb{R}^{n_l}$ the output of a layer containing $n_l$ neurons, the output of the following layer is
\begin{equation}
\label{ANN_EQ}
\boldsymbol{y}^{(l+1)}=\sigma^{(l+1)}(\boldsymbol{z}^{(l+1)}) \,,
\end{equation} with
\begin{equation}
\label{ANN_EQ_2}
\boldsymbol{z}^{(l+1)}=\boldsymbol{W}^{(l)}\boldsymbol{y}^{(l)}+\boldsymbol{b}^{(l+1)}\,,
\end{equation} and $\sigma^{(l+1)}()$ the \emph{activation} function at layer $(l+1)$, $\boldsymbol{W}^{(l)}\in\mathbb{R}^{n_l\times n_{l+1}}$ the matrix of \emph{weights} connecting the $n_l$ neurons in layer $l$ to the $n_{l+1}$ neurons in layer $l+1$ and $\boldsymbol{b}^{l+1}\in\mathbb{R}^{n_{l+1}}$ the vector of \emph{biases} at layer $l+1$. The input and outputs of the ANN are, respectively, the input of the first layer and the output of the last one.
\begin{figure}[h!]
\includegraphics[width=0.4\textwidth]{figures/NN_architecture.pdf
\caption{Example of Neural network architecture and activation function.}
\label{fig:NN_architecture}
\end{figure}
ANNs are known as universal function approximators because they can approximate any function if a sufficiently large number of neurons and layers is used \cite{Cybenko1989}. In this work, the ANN is called to approximate eq. \eqref{eq:Buckingham}, hence the input vector is $\mathbf{x}=[\tilde{\omega},\Delta, H, Ma, \Pi, C_f, R_T, \beta ]\in\mathbb{R}^{8}$ and the output is the scalar $y=10\log_{10}(\Phi_{pp} u_e / \tau_w^2 \delta^*)\in\mathbb{R}$ which expresses the dimensionless wall pressure spectra in Decibel per Hertz. The training of the ANN consists in determining the set of weights $\boldsymbol{w}=(\boldsymbol{W}^{(1)},\cdots \boldsymbol{W}^{(n_L)})$ and biases $\boldsymbol{b}=(\boldsymbol{b}^{(1)},\cdots \boldsymbol{b}^{(n_L)})$, with $n_L$ the total number of layers, such that a loss function measuring the error in the prediction over the available data is minimized. In this work, the loss function is taken as the weighted mean square error on the logarithmic values (lMSE):
\begin{equation} \label{eq:MSE}
lMSE(\boldsymbol{w},\boldsymbol{b}) = \frac{1}{N}\sum_{i=1}^N W_i \left(y_i - y_{i}^{A}(\boldsymbol{w},\boldsymbol{b}) \right)^2
\end{equation} where $ y_{i}^{A}(\boldsymbol{w},\boldsymbol{b})$ is the prediction of the ANN for the $i-th$ sample, $y_i$ is the $i-th$ available data point and $W_i$ is the weight (shown also in table \ref{fig:inputrange}) of each data point. These weights are chosen to give equal importance to the four available datasets despite their widely different sizes. In the machine learning literature, this is a common approach when dealing with datasets in which some classes or some portions of the sample space is more present than others in the training data\cite{cui2019}.
The training of the network was carried out using Nadam optimizer \cite{dozat2016} with a learning rate of $10^{-4}$. This is a variant of the stochastic gradient descent which uses inertia during the descent and the classic back-propagation algorithm\cite{David} to compute the gradient of the loss function with respect to the ANN parameters. Following a mini-batch approach, the gradient of the loss function is computed on a batch of $n_B=32$ randomly chosen training sample.
During the training, the dataset is split into training data ($80\%$) and validation data ($20\%$). The lMSE in the first drives the optimization (training), while the lMSE in the second allows for identifying the stopping point using the early stopping approach. This stops the training when the validation error stops decreasing within a user-defined tolerance. In addition, six profiles were removed from the dataset and used as testing data to assess the ANN's ability to generalize beyond the training data. The performances of the ANN on these profiles is illustrated in Sec. \ref{sec:6}.
Fig.~\ref{fig:NN_training} shows a typical learning curve for the ANNs used in this work. These have four layers between the input and outputs. The first is a normalization layer that scales the inputs using their mean and standard deviation, as this helps to stabilize and accelerate the learning process \cite{ba2016}. The remaining layers are fully connected with $n_N=10$ neurons and Selu activation functions, as recommended with normalized inputs \cite{klambauer2017}.
\begin{figure}[h]
\includegraphics[width=0.45\textwidth]{figures/overfiting.pdf
\caption{Evolution of training and validation error.}
\label{fig:NN_training}
\end{figure}
The proposed architecture resulted as the most appropriate compromise between model complexity and accuracy. Considering layers of equal size ($n^{(1)}=n^{(2)}\dots=n_N$), we performed a sensitivity study testing different combinations of $n_N$ and $n_L$. For each combination, an ANN was trained and a performance measure was defined as
\begin{equation}
\label{Perf}
P(n_N,n_L)=lMSE+0.01 ||(\boldsymbol{w},\boldsymbol{b})||_F
\end{equation} where $||(\boldsymbol{w},\boldsymbol{b})||_F$ is the Frobenious norm of the weights and biases in the network. This term acts as a regularization by penalizing solutions with large weights. The analysis shows that different architectures lead to comparable performances on the available data as long as they include sufficient parameters. However, increasing the number of layers and nodes increases the computational cost and the chances of overfitting the data, causing poor generalization. Similar performances could be obtained by ANNs with either $n_L=2$ with $n_N=15$ to $24$ or $n_L=3$ layers with $n_N=10$ to $20$ (\textsc{Fig.}~\ref{fig:NN_geometry}). Among these, the option with fewer number of weight and biases was favoured, leading to $n_L=3$ and $n_N=10$.
\begin{figure}[h]
\includegraphics[width=0.45\textwidth]{figures/NNgeometry.pdf
\caption{Performance indicator \eqref{Perf} for different network architecture composed of $nL$ layers and $nN$ nodes.}
\label{fig:NN_geometry}
\end{figure}
\subsection{The ANN uncertainties}
We estimate the uncertainties of the ANN model using an ensemble learning approach\cite{Abdar2021,Gawlikowski2021}. The focus is placed on the model (epistemic\cite{Huellermeier2021}) uncertainties, as these allow quantifying the performances of the ANN in relation to the available data. The epistemic uncertainties are linked to the uncertain determination of the ANN's weight and biases. We are here interested in their impact on the model predictions, especially in regions of the input space that have not been sampled adequately. Besides providing confidence intervals, these allow unveiling regions of the input space that lack training data.
The ensemble approach implemented in this work (see also Lee \emph{et al}\cite{Lee2015} and Lakshminarayanan \emph{et al}\cite{Lakshminarayanan2016}) consists in training an ensemble of $n_E=100$ versions of the same ANN. These networks are trained on the same dataset of $111$ profiles (having removed 6 for testing purposes), but differ because of the different (random) initialization of weight and biases and because of the inherent stochasticity of the training process. Each network is trained with a different (random) splitting between training and validation data ($80\%$ and $20\%$ respectively) and different (random) batch selection during the computation of the loss function gradients.
For a given input vector $\boldsymbol{x}=[\tilde{\omega},\Delta, H, Ma, \Pi, C_f, R_T, \beta ]\in\mathbb{R}^{8}$, given $y^{A,j}$ the prediction of the $j^{th}$ ANN, the ensemble prediction and variance are given as
\begin{equation}\label{eq:ANN_epistemic}
\mu_{e}(\boldsymbol{x})=\mathbb{E}_{\sim E}\{y^{A,j}\}\quad \mbox{and}\quad
\sigma^2_{e}(\boldsymbol{x})=\mathbb{E}_{\sim E}\{(y^{A,j}-\mu_e)^2\}\,,
\end{equation} where $\mathbb{E}_{\sim E}$ is the expectation operator over the ensemble of networks. These contributions are function of the inputs and one would expect $\sigma_{e}(\boldsymbol{x})$ to be large for $\boldsymbol{x}$ in poorly sampled regions.
As a simple model for the aleatory uncertainty, we consider a symmetric distribution centered in the ensemble mean. Therefore, assuming that the minimization of the lMSE in \eqref{eq:MSE} during the training ensures a symmetric distribution of the errors, the aleatory contribution to the uncertainty is constant for the entire input space and can be estimated as
\begin{equation}\label{eq:ANN_aleatory}
\sigma^2_{a} = \mathbb{E}_{\sim E} \{ lMSE_j\}\,.
\end{equation}
This estimate is optimistically biased because it assumes that the training data is rich enough to have converged statistics and assumes that the same statistics hold outside the training range. Nevertheless, this simple estimate fits the purpose of providing a first glance at the ANN predictive capabilities and succeeds in providing confidence intervals that accommodate the training, the validation, and the test data.
The standard deviation due to the aleatory contribution was found to be equal to $0.54$ dB/Hz. This is the irreducible uncertainty of the data (due to measurements or numerical uncertainties) which would, in principle, remain unvaried even if the training of the network is enriched with more profiles.
Assuming that these contributions are independent, the standard deviation in the prediction is thus $\sigma_E=\sqrt{\sigma^2_{e}+\sigma^2_{a}}$. Taking a 95\% confidence interval for the prediction, the uncertainty intervals can be taken as $\mu_E \pm 1.96 \sigma_E$.
\begin{figure*}
\centering
\begin{subfigure}[b]{0.33\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/perf_goody.pdf}
\caption{Goody lMSE = 90.12 dB/Hz}
\label{fig:perf_goody}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.33\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/perf_kamruzzaman.pdf}
\caption{Kamruzzaman lMSE = 160.98 dB/Hz}
\label{fig:perf_kamruzzaman}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.33\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/perf_rozengerg.pdf}
\caption{Rozenberg lMSE = 76.14 dB/Hz}
\label{fig:perf_Rozenberg}
\end{subfigure}
\vskip\baselineskip
\begin{subfigure}[b]{0.33\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/perf_lee.pdf}
\caption{Lee lMSE = 74.89 dB/Hz}
\label{fig:perf_lee}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.33\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/perf_dominique.pdf}
\caption{Dominique lMSE = 59.46 dB/Hz}
\label{fig:perf_dominique}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.33\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/perf_ANN.pdf}
\caption{ANN lMSE = 0.88 dB/Hz}
\label{fig:perf_ANN}
\end{subfigure}
\caption{Global performance of the empirical models discussed in Sec \ref{sec:3}, on the dataset presented in Sec. \ref{sec:4}. The markers are \textcolor{color1}{\textbullet} for ZPG, \textcolor{color2}{\textbullet} for FPG and \textcolor{color3}{\textbullet} for APG, regardless of the dataset. For plotting purposes, only one point every ten is shown. }
\label{fig:models_perf}
\end{figure*}
\begin{figure*}[p]
\centering
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/salze.pdf}
\caption{Salze's case with ZPG at $u_e = 45m/s$ ($\beta=0.0$, $R_T=13.11$, $C_f=2.64\times10^{-3}$, $\Pi=0.47$, $H=1.32$, $\Delta=7.8$, Ma = 0.133) }
\label{fig:NN Salze}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/deuse.pdf}
\caption{Deuse's case with APG at $x_c = 0.02$ ($\beta=15.59$, $R_T=2.58$, $C_f=1.02\times10^{-3}$, $\Pi=2.26$, $H=2.12$, $\Delta=3.1$, Ma= 0.211)}
\label{fig:NN_deuse}
\end{subfigure}
\vskip\baselineskip
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/hao.pdf}
\caption{Hao's case with APG at $x_c = 0.02$ ($\beta=5.93$, $R_T=1.85$, $C_f=1.34\times10^{-3}$, $\Pi=2.26$, $H=2.15$, $\Delta=2.8$, Ma = 0.267)}
\label{fig:NN_Hao}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/christophe2.pdf}
\caption{Christophe's case with APG at $x_c = 0.05$ ($\beta=6.32$, $R_T=X3.06$, $C_f=2.02\times10^{-3}$, $\Pi=1.28$, $H=1.82$, $\Delta=3.7$, Ma = 0.049)}
\label{fig:NN ch2}
\end{subfigure}
\vskip\baselineskip
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/christophe3.pdf}
\caption{Christophe's case with APG at $x_c = 0.2$ ($\beta=3.79$, $R_T=3.47$, $C_f=3.13 \times 10^{-3}$, $\Pi=0.53$, $H=1.56$, $\Delta=5.0$, Ma = 0.053)}
\label{fig:NN ch1}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/christophe1.pdf}
\caption{Christophe's case with APG at $x_c = 0.10$ ($\beta=3.37$, $R_T=2.33$, $C_f=2.52\times10^{-3}$, $\Pi=1.13$, $H=1.77$, $\Delta=3.8$, Ma= 0.051)}
\label{fig:NN ch3}
\end{subfigure}
\caption{Prediction of the ANN ensemble $\mu_{e}(\boldsymbol{x})$ form \eqref{eq:ANN_epistemic}, with confidence intervals $\mu\pm1.96\sigma(\boldsymbol{x})$, and comparison with selected semi-empirical models from Sec. \ref{sec:3} on the testing data.}
\label{fig:NN_predictions}
\end{figure*}
\section{Results and Discussion}
\label{sec:6}
\subsection{Model Accuracy}\label{sec:6p1}
We provide a global performance score to all models discussed in Section \ref{sec:3}, and the new ANN model, on the available data. This score is defined as the average lMSE from \eqref{eq:MSE} over whole dataset. The global performance of all models is illustrated in Figure \ref{fig:models_perf}. A figure plots the prediction versus the available data for each model, and the caption reports the average lMSE error. The markers in the figure distinguish datasets in ZPG, APG and FPG.
Because of its inability to account for pressure gradients, Goody's model performs poorly in terms of global performance (average $lMSE=90.12$ dB/Hz) despite the excellent predictions for the ZPG conditions. More surprisingly, Kamruzzaman's model yields the largest errors (average $lMSE=160.98$ dB/Hz) despite being designed to account for pressure gradients, and having been shown to match well with Kamruzzaman's own experimental data \cite{kamruzzaman2015}. This discrepancy was also pointed out by Lee \emph{et al}\cite{lee2018} for both the CD airfoil and flat plate boundary layers.
Rosenberg's (average $lMSE=76.14$ dB/Hz) and Lee's (average $lMSE=74.89$ dB/Hz) models yield similar global performances, gaining $20\%$ global lMSE over Goody's model. Much of the gain is observed in cases in APG. Their similar performance could also be partially attributed to the dataset used in this work: both of these models were calibrated to give similar predictions on the CD airfoil, but Lee's model was extended to account for various other cases that are not featured in the dataset collected in this work.
A further tangible improvement is obtained by Dominique's GEP model (average $lMSE=59.46$ dB/Hz), which performs better than the previous ones in both APG and FPG. Although this model has the same architecture as the previous models (a rational function in $\tilde{\omega}$), it accounts differently for pressure gradients (through $\beta$), introduces a function of $C_f$ and is independent of $\Pi$. The performance in APG conditions, however, remains limited.
Finally, the ANN's model shows a ten-fold gain in performance (average $lMSE=0.88$ dB/Hz) over the full range of conditions in the dataset. This shows the strength of the ANN as universal function approximator and offers some insights on the wall-pressure modeling problem: the success of the ANN model is reasonably explained by the fact that its functional relation is far more complex than the rational function assumed by all the other models. On the one hand, one might thus wonder whether the balance between model complexity and prediction performances should move the modeller's ingenuity beyond rational functionals. On the other hand, the comparatively poorer performances of the previous models, especially if compared to their performances on the data from which they were derived, shows that the dataset used in this work might not be sufficiently large and diverse to develop the most general ANN-based wall pressure spectra model. Nevertheless, the methodology described in this work could be easily extended, and the promising predictive capabilities of the ANN could be improved with more data.
We now move to the generalization capabilities of the ANN by testing it on the six profiles that were left out of the training. These were selected from all the available datasets and include one case in ZPG from Salze and five cases in APG, as these proved to be the most difficult to predict. The results are shown in Figure \ref{fig:NN_predictions}, with the caption recalling the dataset from which they are sampled and the input parameters. The prediction of all models is shown in the figure together with the ensemble mean prediction and confidence intervals for the ANNs.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{figures/uncertainty.pdf}
\caption{Neural network uncertainties and average sensitivity via ensemble learning. For each combination of boundary layer parameters used as input, the contour map show the global measure of confidence $\mathcal{C}(\boldsymbol{x'})$ from \eqref{eq:ANN_confidence}, varying only two parameters and keeping the others at their average value. The red dots are the training points from the database. The diagonal shows the average sensitivity $\mathcal{S}_{\boldsymbol{x}'_i}$ from \eqref{S_omega} with respect to the input parameter $\boldsymbol{x}_i$, as a function of the dimensionless frequency $\tilde{\omega}$. The line with $\mathcal{S}_{\boldsymbol{x}'_i}=0$ is highlighted in black for plotting purposes. }
\label{fig:NN_accuracy}
\end{figure*}
For the first profile in \ref{fig:NN Salze}, in ZPG, all models yield good predictions except for Kamruzzaman's model. This model misses the roll-off high-frequency point, presumably because of its different dependency on the Reynolds number. The ANN provides an excellent fit within its training frequency range, but departs from the data and from Goody's model at both high and low frequencies. As expected, Rozenberg's closely match Goody's model in this condition, but both models slightly overpredict Salze's data at low frequencies. The models of Lee and Dominique also match closely Goody's model prediction while providing a better fit at low frequencies.
Large uncertainty band are observed for the ANN for some conditions, e.g. for $\omega \delta^*/u_e >25$ in ZPG. They can be explained by the lack of training data. In these cases, the ANN predictions are biased by the few profiles that include data in this frequency range. These are the profiles from Hao (see also Fig. \ref{fig:NN_Hao}), which are in APG conditions and feature a steeper decay.
The extrapolation problem is less severe for Dominique's GEP model because its rational function form keeps the expected trends outside the training range. The trade-off between model complexity and model performance is a classic problem in the machine learning literature\cite{Abu} and its resolution for wall pressure spectra modelling requires a larger dataset. This suggests a possible hybrid approach where the rational function remains imposed, and the ANN is used to predict the necessary coefficients.
The complexity of the ANN model pays off in the APG conditions, as clearly shown by the test case from Deuse's (Fig. \ref{fig:NN_deuse}) and Hao's datasets (Fig. \ref{fig:NN_Hao}). These are the test cases for which data is available at the highest frequencies and exhibit wall pressure spectra that rational functions don't seem to describe well. This is particularly true for the test case in Hao's datasets (Fig. \ref{fig:NN_Hao}), characterized by a hump at high frequencies. Although the prediction of Lee's model shows a remarkable agreement with the data, the ANN is the only model capable of predicting such a trend. The same conclusions hold for the test case in Deuse's dataset (Fig. \ref{fig:NN_deuse}), which also feature a gentle change in the slope of the power spectra at $\tilde{\omega}\approx 10$. In this case, Lee's model is less accurate, and the ANN approach largely outperforms all empirical models.
Due to the lack of high-frequency data in Christophe's strong APG database, the ANN is asked to extrapolate over a significant frequency range to predict those spectra. Relatively large uncertainty bands are indeed observed above $\tilde{\omega} \simeq 4$. Such extrapolation seems to be in reasonable agreement with the empirical models for the test case in \ref{fig:NN ch1} but not for the ones in Figs \ref{fig:NN ch2} and \ref{fig:NN ch3}. On the other hand, the accuracy at the lowest frequency is remarkable and unmatched by the any of the semi-empirical models.
Finally, most of the points in the test set fall within the ANN confidence intervals. This validates the ensemble methodology for uncertainty quantification.
\subsection{Uncertainties and Sensitivities}\label{sec:6p2}
We analyze the confidence intervals and the sensitivity of the ANN ensemble over the full range of available data. The goal was to identify regions where additional data would be mostly beneficial to improve the model and to analyze the sensitivity of the confidence bounds to the input parameters.
Let $\sigma_E(\boldsymbol{x'},\tilde{\omega})$ be the standard deviation of the ANN ensemble from \eqref{eq:ANN_epistemic}, as a function of the dimensionless frequency $\tilde{\omega}$, for a given input in $\boldsymbol{x'}=[\Delta, Ma, \Pi, C_f, R_T, \beta ]$. Let $[\omega_1,\omega_2]$ be the largest range of available dimensionless frequencies, which in this work is $[0.01,0.35]$. For every input $\boldsymbol{x'}$, we define a global measure of confidence as
\begin{equation}\label{eq:ANN_confidence}
\mathcal{C}(\boldsymbol{x'})=\frac{1}{\tilde{\omega}_2-\tilde{\omega}_1}\int^{\tilde{\omega}_2}_{\tilde{\omega}_1} \sigma_E(\boldsymbol{x'},\tilde{\omega}) d\tilde{\omega}\,.
\end{equation}
The contour-plot of the confidence measure $\mathcal{C}$ is shown in Figure \ref{fig:NN_accuracy} in the form of a scatter matrix on the planes defined by pairs of inputs (e.g. $\Delta$ versus $Ma$, $\delta$ versus $\Pi$, etc.). The contour plot shows the value of $\mathcal{C}$ together with the available data, shown as a scatter plot.
The diagonal of the scatter matrix shows the averaged sensitivity to each input parameter as a function of the dimensionless frequency $\tilde{\omega}$. In other words, given $y=10\log_{10}(\tilde{\Phi}_{pp}(\tilde{\omega},\boldsymbol{x}))$ the output of the ANNs for an input $\boldsymbol{x}'$ and a frequency $\tilde{\omega}$, the global (averaged) ensemble sensitivity is defined as
\begin{equation}
\label{S_omega}
\mathcal{S}_{\boldsymbol{x}'_i} (\tilde{\omega})=\mathbb{E}_{\sim E, \boldsymbol{X}} \{ \partial_{\boldsymbol{x}'_i} y\}\,,
\end{equation} where the expectation is computed over both the ensemble ($E$) and the full dataset ($\boldsymbol{X}$) and the partial derivative $\partial_{\boldsymbol{x}'_i} y$ with respect to the input $\boldsymbol{x}'_i$ is computed analytically via back-propagation.
The contour plots confirm high uncertainty in regions lacking data. The region of highest uncertainties are observed at large $R_T$, large $\beta$ and low $\Delta$. Of these three parameters, the uncertainties grow more significantly with decreasing $\Delta$. The figures in the diagonal show that this last parameter has a strong impact at low frequency, while others, such as $Ma$ and $R_T$, mostly play at high frequencies. It is worth noticing that the averaged value of the derivative with respect to $R_T$ is $\partial_{R_T} y \approx 3.5$ at high frequencies, in agreement with the expected dependency of the Reynolds number in Goody's model which should averaged at 4.0. Moreover, the averaged sensitivity $\mathcal{S}_{\boldsymbol{x}'_i} (\tilde{\omega})$ is always positive for ${\boldsymbol{x}'_i}=\Pi,\,R_T, \,\beta$ as it is the case in other empirical models, while the sensitivity changes sign for ${\boldsymbol{x}'_i}=C_f, Ma$.
Some of the regions of high uncertainties (e.g. large $R_T\sim 20$ \emph{and} $\beta\sim 10$) are unexplored territory for which data would be particularly valuable in improving the model performances. Others are less interesting because physically inaccessible due to the correlation between inputs. For example, the plane $\Pi-\Delta$ reveals a strong correlation between these two variables, and an apparent impossiblity for the boundary layer to exhibit high values for both simultaneously. The corresponding region of high uncertainty is thus not a concern. Strongly correlated variables were used in this work as a way to increase the predictive capabilities of the model. This is the case for example of $\Delta$ and $H$, which contains the parameters $\delta$, $\delta^*$ and $\theta$. These are almost linearly correlated in the training data used in this work, as shown by the scatter plot in Figure \ref{fig:NN_correlation} together with the contour of $\mathcal{C}$. Therefore, the regions of high uncertainties in this plot do not seem physically accessible. At least within the range of parameters investigated, one might consider removing one of those two parameters in future modeling efforts.
\begin{figure}[h]
\includegraphics[width=0.45\textwidth]{figures/correlation.pdf
\caption{Correlation between the variables $\delta$, $\delta^*$ and $\theta$. The red dots are the training points from the database. The contour plots display display the global measure of confidence $\mathcal{C}(\boldsymbol{x'})$ from \eqref{eq:ANN_confidence} for $\delta^*/\delta$ and $\theta/\delta$, while the others parameters are fixed to their mean averaged value.}
\label{fig:NN_correlation}
\end{figure}
\section{Conclusion}
\label{sec:7}
Existing semi-empirical models fail to predict wall pressure spectra for strong adverse pressure gradients, and data-driven models such as neural networks offer an attractive alternative. The artificial neural network is trained on a dataset constituted of three high-fidelity numerical simulations on the CD-airfoil and one set of experimental measurements on a flat plate boundary layer. The methodology used to extract the relevant boundary layer parameters required by the model was extended to account for large pressure gradients encountered in industrial applications and the CD-airfoil simulations. The neural network constructed in this work provided a significant improvement over the other existing models in those conditions. Finally, we investigated the uncertainties of the neural network predictions and the sensitivity to its inputs using an ensemble approach. Those allow for identifying regions where new training data would be most beneficial to the model's accuracy.
In contrast with semi-empirical models assuming a functional dependence (in this instance, a ratio of polynomials), which can filter out outliers to some extent, data-driven methods rely entirely on the quality of their inputs. Therefore, an elaborate methodology has been necessary for computing the parameters of boundary layers subjected to an adverse pressure gradient. This method is based on the theoretical developments of Nickles \cite{nickels} and Nagib \cite{nagib}, about the non-universality of the von Karman coefficient. This procedure showed that the pressure gradient strongly impacts the von Karman coefficients of the log law. We accounted for these changes while computing the boundary layer parameters for three high-fidelity numerical simulations on the CD-airfoil and one set of experimental measurements on a flat plate boundary layer.
Methods for data-driven regression such as genetic programming and neural network allow for deriving complex nonlinear functions from data but require an appropriate set of input parameters. A dimensional analysis using the Buckingham-Pi theorem yielded a set of eight dimensionless numbers to model the wall pressure spectra. We verified that this set of parameters was sufficient to model the present dataset using a manifold mapping approach for dimensionality reduction. We observed that the available dataset doesn't contain samples with identical boundary layer parameters leading to different wall pressure spectra. While exploring the dataset, we observed a clear correlation between the Zagarola-Smits's parameter $\Delta$ and the shape factor $H$. If this correlation proved true on extended data, this suggests that both those parameters are unnecessary, and we could consider removing one.
The database investigated in this work provides a thorough investigation for the CD airfoil under a broad range of inflow parameters, including many strong adverse pressure gradient boundary layers. Among all the empirical models presented in this work, Lee's and Rozenberg's model provided the best match to the data. However, even those model tends to underpredict the exact amplitude of the spectra by 5 to 10 dB/Hz.
Dominique's data-driven model using GEP slightly improves over Lee's predictions in some cases, but this improvement is not consistent for all turbulent boundary layers. In addition, it does not capture the correct roll-off frequency for the transition toward the high-frequency spectral decay. The imperfections of the GEP model are the result of the convergence difficulties of Genetic programming approaches when the parameter space formed by the number of operators and input functions becomes too large.
The neural network constructed in this work provided a significant improvement over the other models, with accurate predictions globally within $\pm 1 dB/Hz $ accuracy for the different flow cases. However, the model tends to feature a steeper decay at high frequencies due to the limited training data.
An analysis of the uncertainties through ensemble learning highlighted the lack of data for large pressure gradients and medium to large Reynolds numbers or low Zagarola-Smits's parameter. This caused an increase in the model uncertainty for such flow configurations and indicated this area of parameters as a promising region for additional data in the future. In addition, different airfoil geometries should be added to the dataset as it is currently mainly composed of simulations over the CD airfoil.
The authors conjecture that this approach could be extended to a broader range of boundary layer parameters without additional modifications in the long term. This would make a great tool to propose models that perform well on a larger dataset, including different physical assumptions (e.g. pressure gradient, compressibility effects, etc.) without requiring additional effort on the methodology front.
\begin{acknowledgments}
The French global automotive Valeo partly supported this research in the context of a thesis for the investigation of noise induced by HVAC systems. The authors thank Dr. Salze and Pr. Bailly from Ecole Centrale de Lyon, Ing. Deuse and Pr. Sandberg from the University of Melbourne, Dr. Hao and Pr. Moreau from Université de Sherbrooke and Dr. Christophe from von Karman Institute for providing the experimental and numerical data.
\end{acknowledgments}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 6,113
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INSTI® HIV-1 / HIV-2
INSTI® Multiplex
INSTI® HIV Self Test (Box)
INSTI® HIV Self Test (Pouch)
Pastors' Wives HIV Initiative Produce Record Results
First Ladies Health Initiative Executive Director Tracey Alston credits the high volume of HIV tests taken in Chicago and Northwest Indiana this year in part to a partnership with bioLytical Laboratories that is expanding. At the Chicago Health Day, bioLytical Laboratories donated its INSTI™ 60-second rapid HIV testing kits to the Health Day and agreed that for each test administered during the day, it would donate a free test kit to the Desmond Tutu HIV Foundation in South Africa. In all, more than 1,200 HIV tests were administered.
CHICAGO (December 2015) — Amid renewed focus on raising awareness on HIV/AIDS this month, African-American pastors' wives across the country report making major inroads in encouraging people to get life-saving screenings for the virus.
The First Ladies Health Initiative succeeded in getting more than 1,200 community residents to get free screenings for HIV in a single day at 70 churches in Chicago and Northwest Indiana at its Health Day this year — a record for faith-based organizations.
Since launching in Chicago in 2008, the nonprofit initiative's Walgreens-sponsored First Ladies Health Days, held at their churches annually, have enabled more than 200,000 individuals to get an array of free health screenings, including for HIV/AIDS.
Because of its influence, the First Ladies Health Initiative has been selected by the Centers for Disease Control and Prevention to participate in a national HIV campaign.
"I think the philanthropic partnership component helped encourage more people to get tested," Alston said. "People felt compelled to do this, not only to be proactive about their health, but to also help others across the globe in South Africa."
That partnership's reach now extends to California. For every HIV test administered during the First Ladies Health Days in Los Angeles and Orange County, CA. next year, bioLytical Laboratories will donate a free test to the foundation. It has committed to a minimum of 1,000 tests.
"We're hopeful that the opportunity to help others while also helping themselves will be a strong motivator for Californians to get HIV tested as it was at our Health Day in Chicago and Northwest Indiana," Alston said.
The First Ladies Health Initiative has grown to comprise 152 pastors' wives, also known as First Ladies, across denominations at chapters in Northwest Indiana, Los Angeles and Orange County, CA, Cincinnati, OH, in addition to Chicago.
United for healthier communities, the initiative targets illnesses that disproportionately affect African-Americans and Hispanics and encourages them to be proactive about their health.
"We've come a long ways since we began in 2008," Alston said. "We recognized then that one of the most urgent health needs was and still is the devastating spread of HIV/AIDS in communities of color."
That concern led Walgreens to initiate an HIV/AIDS Task Force, including the AIDS Foundation of Chicago that culminated in the formation of the First Ladies Health Days.
"What began as a pilot program combating HIV/AIDS has now expanded to also include a focus on other major illnesses that rob lives in our communities, including cardiovascular disease, diabetes, hypertension, hepatitis C, cancer and other illnesses," Alston said.
But HIV remains a major priority, given the continuing risk it poses to communities of color. At some point in their lifetimes, an estimated 1 in 16 African-American men and one in 32 African-American women will be diagnosed with HIV infection, according to the Centers for Disease Control and Prevention.
Meanwhile, Hispanics accounted for 23 percent of the estimated new diagnoses in 2013, and less than half of Hispanics living with HIV are receiving medicines to treat their infection, according to the CDC, putting their lives at greater risk.
Read full article at: http://chicagodefender.com/2015/12/10/pastors-wives-hiv-initiative-produce-record-results/
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|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,452
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Sand dunes are ridges or hills of sand found at the top of a beach, above the usual maximum reach of the waves.
The image below shows Harlech Beach, North Wales. Its large tidal range supports the development of sand dunes. The tidal range is the difference between the high tide and the following low tide.
How does wind transport sand?
1% of the movement of sand is caused by suspension. This is when sand is picked up and carried within the wind. 95% of sand movement results from saltation. This is when grains of sand bounce along the beach as they are picked up and dropped by the wind. Finally, 4% of transportation is by creep. This is when sand grains collide and push each along other grains.
The video below shows a combination of these processes of transportation.
As the wind blows up the beach it will transport material. Larger pieces of sediment will rest against an obstacle forming a ridge while smaller particles will settle on the other side of it. On the side facing the wind, the material begins to reach a crest. This is because the pile of material becomes steep and unstable and begins to collapse. When this happens smaller particles fall down the other side. Once there is a stable angle (30-34 degrees) the sand stops slipping. This cycle repeats. As the sand becomes an obstacle itself more dunes may form in front of it. The stronger the wind the higher the dunes.
How do sand dunes change as you move inland?
How do sand dunes change with distance from the beach?
Moving inland sand dunes become taller. Embryo dunes (youngest sand dunes) are only a few metres high whereas mature dunes are up to 15m high. This is because marram grass and other vegetation colonise the sand dune and hold it together with long roots, stopping the migration of the sand dune. Sand dunes closer to the beach are more yellow in colour whereas further away they are grey due to humus and bacteria from plants and animals being added. Each sand dune is separated by a trough (dip). This is known as a slack. They are formed by the removal of sediment from the sheltered lee side of the dune and the windward side of the next dune. Slacks can be eroded so much that they reach the water table resulting in the formation of salty dunes. The video below illustrates how vegetation in a sand dune ecosystem changes as you move inland (known as vegetation succession).
The image below shows vegetation succession on sand dunes at Harlech, North Wales.
The video below shows the extent of roots and illustrates the way vegetation helps stabilise sand dunes.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,013
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Just had a brochure through the post from Borders College, listing out their evening class programme for September 2007. There is a range of classes at their places in the Borders, from the Know Your Motorcycle course which I did earlier in the year to Belly dancing.
Another one that looks really interesting is mountain bike maintenance at Glentress – I know quite a few folks that go down there to use the trails.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,032
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Q: How to return and display a count result json response api in flutter I created an api that counts total category in my database table using laravel, and am trying to consume it in flutter and display the total count value. Please how can i achieve that
My Code:
//Laravel Query that count
public function count_categories(){
$count_category = DB::table('category_menu')
->count();
return response()->json($count_category);
}
//Flutter Code
//API URL
static const COUNT_CATEGORY = "/count_category";
Future<List<Category>> countCategory() async {
String countCategory = COUNT_CATEGORY;
Map<String, String> headers = {'Accept': 'application/json'};
var response = await http.get(countCategory, headers: headers);
if (response.statusCode == 200) {
var body = jsonDecode(response.body);
print(response.body);
}
}
//The response.body is printing the count value correctly, how can i display the response body result in a class widget?
A: If I correctly understand what you are trying to do here is serialize the json response and then pass it onto a Widget.
Although more details on your objective would be useful, here it goes
There are many ways of doing it but the most basic approach is to use the json_serializable or the online converter.
For some reference refer to this answer here.
Once you convert your json response to a Dart Class (which might look something like)
class SampleModel {
String name;
int count;
SampleModel({this.name, this.count});
// You use this function to make a instance
// of class SampleModel and use it inside an UI Widget
SampleModel.fromJson(Map<String, dynamic> json) {
name = json['name'];
count = json['count'];
}
Map<String, dynamic> toJson() {
final Map<String, dynamic> data = new Map<String, dynamic>();
data['name'] = this.name;
data['count'] = this.count;
return data;
}
}
Now you can either use a FutureBuilder, setState() or a proper statemanagement solution to load this into a Widget.
class SampleWidget extends StatelessWidget {
SampleWidget({this.model})
@override
Widget build(BuildContext context) {
return Container(
child: Text(model.count),
);
}
}
A: First of all, from what I see, from the API you are sending the amount (number), therefore when making the GET request from flutter you will receive a number in the body.
All good so far, but you must modify the data type of the function, you currently have defined:
Future<List<Category>> countCategory() async {...}
You must modify it for this if you want to return the value.
Future<int> countCategory() async {...}
Now, it can also be of type void, but you must save the value in a variable and then use this variable in the corresponding widget.
Future<void> countCategory() async {
...
setState(() {
count = response.body; // Assuming the variable is globally defined.
})
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,522
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/*****************************************************************************/
/**
* @file xtime_l.c
*
* This file contains low level functions to get/set time from the Global Timer
* register in the ARM Cortex A9 MP core.
*
* <pre>
* MODIFICATION HISTORY:
*
* Ver Who Date Changes
* ----- ------ -------- ---------------------------------------------------
* 1.00a rp/sdm 11/03/09 Initial release.
* 3.07a sgd 07/05/12 Upadted get/set time functions to make use Global Timer
* </pre>
*
* @note None.
*
******************************************************************************/
/***************************** Include Files *********************************/
#include "xtime_l.h"
#include "xpseudo_asm.h"
#include "xil_types.h"
#include "xil_assert.h"
#include "xil_io.h"
/***************** Macros (Inline Functions) Definitions *********************/
/**************************** Type Definitions *******************************/
/************************** Constant Definitions *****************************/
/************************** Variable Definitions *****************************/
/************************** Function Prototypes ******************************/
/****************************************************************************
*
* Set the time in the Global Timer Counter Register.
*
* @param Value to be written to the Global Timer Counter Register.
*
* @return None.
*
* @note In multiprocessor environment reference time will reset/lost for
* all processors, when this function called by any one processor.
*
****************************************************************************/
void XTime_SetTime(XTime Xtime)
{
/* Disable Global Timer */
Xil_Out32(GLOBAL_TMR_BASEADDR + GTIMER_CONTROL_OFFSET, 0x0);
/* Updating Global Timer Counter Register */
Xil_Out32(GLOBAL_TMR_BASEADDR + GTIMER_COUNTER_LOWER_OFFSET, (u32)Xtime);
Xil_Out32(GLOBAL_TMR_BASEADDR + GTIMER_COUNTER_UPPER_OFFSET,
(u32)(Xtime>>32));
/* Enable Global Timer */
Xil_Out32(GLOBAL_TMR_BASEADDR + GTIMER_CONTROL_OFFSET, 0x1);
}
/****************************************************************************
*
* Get the time from the Global Timer Counter Register.
*
* @param Pointer to the location to be updated with the time.
*
* @return None.
*
* @note None.
*
****************************************************************************/
void XTime_GetTime(XTime *Xtime)
{
u32 low;
u32 high;
/* Reading Global Timer Counter Register */
do
{
high = Xil_In32(GLOBAL_TMR_BASEADDR + GTIMER_COUNTER_UPPER_OFFSET);
low = Xil_In32(GLOBAL_TMR_BASEADDR + GTIMER_COUNTER_LOWER_OFFSET);
} while(Xil_In32(GLOBAL_TMR_BASEADDR + GTIMER_COUNTER_UPPER_OFFSET) != high);
*Xtime = (((XTime) high) << 32) | (XTime) low;
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,353
|
Totalcontent | heard that one before?
Home » blog » heard that one before?
Picked up this CD by the Black Keys the other week, and the rather self-conscious cover art seemed curiously familiar. That's probably because the much fêted record sleeve design company Hipgnosis had come up with the idea some 30 years earlier. It was said to have been rejected by regular clients Pink Floyd, but XTC (remember them?) were more than happy with the crumbs from the table for their second album 'Go' (released October 1978).
While the Black Keys' effort has a certain laconic charm – I particularly like the deadpan line on the back cover "These are the names of the songs on this album/These are the guys in the band" – XTC's takes the idea much further in the copy, the voice becoming increasingly involved, and almost getting into an argument with itself.
Anticipating post-modernism, it debunks the whole notion of the record sleeve as a sales tool, the music industry, and capitalism in a deliciously tongue-in-cheek way. When you consider that the Bee Gees' 'Saturday Night Fever', Springsteen's 'Darkness on the Edge of Town' and the eponymous 'Dire Straits' were the big sellers back then, you realise quite how far ahead of its time Hipgnosis' design idea was. And actually, it happens to fit perfectly with XTC's witty, English, self-deprecating style. They were a super-talented band who never quite fulfilled their potential, mainly because of front man Andy Partridge's paralysing stage fright.
Which brings me neatly on to Hard Fi's 'Once Upon A Time In The West', designed by London graphics house Intro, whose output is generally far more original. Another self-referential piece, once again this pokes fun at the consumerist machine, though this time in a more brutally Modernist typographic style. Actually, it bears a striking resemblance to Intro's own monograph 'Display Copy Only', with its pared-down, black on yellow colour scheme.
What does all this tell us? That there's nothing new under the sun. That plagiarism abounds. That some ideas are worth revisiting (with a twist). That talent borrows, genius steals. Perhaps a bit of each. Certainly appropriation is routine in the design industry, as the amusing Dopplegänger Design blog so eloquently proves. Click through for several revealing hours of spot the difference.
Pingback: Totalcontent | the first take.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,640
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$ to go to the end of the line.
bdw to delete the last word on the line (this is the email address or account name).
A to enter insert mode.
:wq to save the file and exit vim.
newaliases to update the aliases database.
That seems like a lot but I timed myself and I can easily accomplish all of these steps in under 30 seconds. What could be easier? I doubt this task could be accomplished in under 30 seconds with a GUI. If you are not fluent in the Unix shell you are probably getting quite angry at me right now. "But I don't know those commands" you say. This is where the term "easy to use" breaks down. The average computer user is not looking for easy to use. They are looking for easy to discover. The normal computer user does not care if a task takes a little longer than the optimal way. All a normal computer user cares about is the ability to easily easily re-discover the steps necessary to accomplish the task the next time they need to do it. These users don't want to learn the skills necessary to optimally control their computer. Instead of talking about computer UIs with the term "easy to use" I think it's time we start talking about "easy to do" and "easy to discover".
Posted byDan Siemon 2004-06-01 2004-06-01 Posted inMusingsTags: ComputersLeave a comment on Easy to what?
The continuing saga of the The Alexis de Tocqueville Institute FUD machine is quite amazing. It scares the crap out of me that the world is influenced by people like this. Check out the discussion in Tim Labert's Blog. Groklaw is also getting in on this.
I just got back from the National Capital Race Weekend in Ottawa. Ran the half marathon (21.1 Km). Finished with a time of 2:16:08. This is the first time I have ran a marathon so I am pretty happy with that time. Surprisingly, it didn't hurt as much as I thought it would. Pictures from the weekend are in the Photo album.
I finally got Moe's pictures from the Cozumel trip online. That's another ~150 pictures added to the Cozumel album. Now I just need to get Allison's ~380 uploaded and that will be all of the Cozumel photos. Hmm…. I should probably put up the pictures from the Dominican trip in 2001 too.
Well, I have spent way more time than I should have playing with the CSS to make this site look the way I want. Hopefully it works in IE too. If you want to see an amazing example of just how much CSS can change a website without any XHTML changes check this out.
It's odd. Having my life completely saturated with Internet technologies both in the ISP work and University I should have taken time to investigate CSS before now. Wow, this is pretty darn cool.
So I decided it was time I do something useful with my website and setup some blogging software. I must say that so far I am quite impressed with WordPress.
Posted byDan Siemon 2004-05-27 2004-05-27 Posted inGeneral2 Comments on Blog!
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\section{Introduction}
High-dimensional sparse regression is central to prediction with massive datasets and many predictors. Amazon, Facebook, Google, and Walmart employ data analytics to analyze customer behavior and to maximize customer value given purchase histories. For example, Amazon has an "anticipatory shipping" patent to pre-ship products to reduce waiting times based on purchase order history, search history, and shopping cart activities\footnote{https://www.marketingweek.com/2014/01/22/amazon-has-seen-the-future-of-predictability/}.
Netflix and Spotify rely heavily on extensive user databases to provide recommendations, make bids for rights of television dramas, and suggest movies to license\footnote{https://blog.kissmetrics.com/how-Netflix-uses-analytics/}.
Uber calculates fares automatically and dynamically using GPS and street data, the rider's history, traditional routes, and other factors. Its carpooling service helps reduce average cumulative trip miles by 40\%, and the process takes millions of cars off the roads.
Alpha-norm ($\ell_\alpha$-norm) regularization provides an attractive high-dimensional predictor selection method to handle massive datasets. Proximal algorithms can find sparse solutions as they are scalable and use Majorization-Minimization (MM) and Iterative Shrinkage-Thresholding (IST) for convergence. Our algorithm design uses coordinate-descent (\cite*{marjanovic2013exact,marjanovic2014sparsity}) together with a closed-form proximal operator of $\ell_\alpha$ penalty (\cite*{polson2015proximal}). Convergence results are provided in \cite*{attouch2013convergence} which describes the $\ell_\alpha$ objective and derives its necessary Kurdyka-{\L}ojasiewicz (KL) condition. \cite*{zeng2014cyclic} improves this by eliminating the requirement that the columns of the design matrix be normalized, and introduces a step-size parameter to enhance the algorithm. \cite*{bredies2015minimization} provide convergence results for MM-based algorithms.
Our approach uses the closed-form proximal operator for $\ell_\alpha$ sparsity, where $0<\alpha<1$. Under $\ell_0$ sparsity, which penalizes the number of non-zero coefficients directly, Single Best Replacement (SBR) provides a fast, scalable alternative to direct posterior sampling using spike-and-slab priors, see \cite*{polson2017bayesian}. \cite*{mazumder2012sparsenet} discuss the limitations of the lasso $\ell_1$-norm versus $\ell_\alpha$-norm. The key property of $\ell_\alpha$ regularization is that it ``jumps" to a sparse solution, which we exploit in our application.
Sparsity in marketing and economics arises due to large internet-related economic transactional data where machine-learning tools are required for estimation and forecasting. Market-demand forecasts are necessary for predicting future sales, inventory planning, and understanding the effects of potential marketing strategies. Two approaches are used to avoid in-sample overfitting: imposing an informative structure and regularization and model selection through penalizations on parameter proliferation. Model selection is especially useful with well-defined subpopulations or segments. Nonparametric approaches are used to estimate incremental effects, see \cite*{varian1982nonparametric}. Quantitative variables of interests include lift of non-targeted promotions (see \cite*{briesch2010nonparametric}), targeted promotions (see \cite*{hartmann2011identifying}), and joint lift of price and promotions predictions (see \cite*{fong2010private} and \cite*{mullainathan2017machine}).
Scanner-panel data typically includes many discrete categorical variables, such as product and store attributes, over a long time span and dummy variables to control for the individual effects. \cite*{andrews2011comparison} show that a homogeneous demand model with random in-store heterogeneity can achieve similar accuracy to model with various heterogeneity specification. Tailored methods have been developed to deal with discrete quantities and quantity discounts for packaged goods (see \cite*{allenby2004choice}) and endogenous dummy regressors, see \cite*{angrist2001estimation}. In our empirical study, we find that $\ell_\alpha$ regularization provides better out-of-sample performance compared to linear and traditional shrinkage methods.
The rest of the paper is outlined as follows. Section \ref{alpha-regularization} shows the performance of the alpha norm in simulation studies. We demonstrate how the $\ell_\alpha$ can achieve better prediction and reduce bias in coefficient estimation in a general setting. Section \ref{simulation-study} shows the alpha norm can also produce more accurate prediction (smaller RMSE) in a simulated log-linear model and demand-estimation context. Section \ref{empirical-analysis} applies our methodology to scanner-panel data of sales for salty snacks within one grocery store chain. Our model can achieve smaller out-of-sample RMSE and substantially shrink the number of predictors. Section \ref{discussion} concludes with directions for future research.
\section{Sparse $\ell_\alpha$ Regularization }\label{alpha-regularization}
Consider an ultra high-dimensional regression model for an output, $y $, with many predictors $X=(x_1, x_2, \ldots, x_p) \in \mathds{R}^{N\times p}$. The columns of $X$ are standardized such that $\Vert x_i\Vert_2 = 1$ for all $i \in \{1, 2, ... , p\}$. Hence, our model assumes:
\begin{equation}
y=X\beta+\epsilon,
\end{equation}
where $y\in \mathds{R}^{N\times 1}$ are the response observations, $\beta \in \mathds{R}^{p\times 1}$ is the sparse coefficient of interest, $X\in \mathds{R}^{N\times p}$ is the design matrix, and $\epsilon \sim N(0,\sigma^2I_N)$ is the noise.
The $\ell_\alpha$ regularization is equivalent to an optimization problem which minimizes the following objective function with penalty parameter $\lambda>0$,
\begin{equation} \label{Jbeta}
J(\beta):=\frac{1}{2}||y-X\beta||_2^2+\lambda||\beta||_\alpha^\alpha, \text{ where } ||\beta||_\alpha:=\left(\sum_{i=1}^p|\beta_i^\alpha\right)^{\frac{1}{\alpha}}.
\end{equation}
For a given $\lambda$, we calculate a regularization path for $\hat \beta_\lambda$.
Define the proximal operator, $\tau_\lambda(z)$, by $\tau_\lambda(z) = \text{argmin}_\beta \frac{1}{2}(z-\beta)^2+\lambda |\beta|^\alpha$.
Then all its solutions are given by
\begin{equation*}
\tau_\lambda(z) =
\begin{cases}
0, &\text{if } |z|<h_{\lambda,\alpha}\\
\{0,\text{sgn}(z)b_{\lambda,\alpha}\}, &\text{if } |z|=h_{\lambda,\alpha}\\
\text{sgn}(z)\bar\beta, &\text{if } |z|>h_{\lambda,\alpha}
\end{cases}
\end{equation*}
where $b_{\lambda,\alpha}:=[2\lambda(1-\alpha)]^{\frac{1}{2-\alpha}}$ and $h_{\lambda,\alpha}:=b_{\lambda,\alpha}+\lambda \alpha b_{\lambda,\alpha}^{\alpha-1}$, and $\bar\beta>0$ satisfies $\bar\beta+\lambda \alpha\bar\beta^{\alpha-1} = |z|$. We derive two solutions, and set $\bar\beta\in(b_{\lambda,\alpha},|z|) $ to be the larger one in implementation.
Our algorithm is based on the result of \cite*{marjanovic2012optimization} who provide a solution to $\ell_\alpha$ minimization. First, as $\alpha\rightarrow 0^+$, the limit of $\tau_\lambda(z)$ is hard thresholding with the threshold value $h_{\lambda,0} = \sqrt{2\lambda}$, whereas as $\alpha\rightarrow 1^-$, the limit of $\tau_\lambda(z)$ is soft thresholding with the threshold value $\lambda$. For comparison, $\tau_\lambda(z)$ for different $\alpha$ are plotted in Figure \ref{tau}. The value of $\tau_\lambda(z)$ at the point $z = h_{\lambda,\alpha}$ is not unique with solution set $\{0, \text{sgn}(b_{\lambda,\alpha})\}$.
To show an improvement in the objective function, we use Lemma 1 of \cite*{marjanovic2014sparsity}:
\begin{equation}
\label{descent}
J(\beta_{-i}+\tau_\lambda(z_i)e_i)\leq J(\beta), \quad \text{for any } \beta.
\end{equation}
Define $z_i = z(\beta_{-i})$ as $x_i^T(y-X\beta_{-i})$ is the adjusted gradient for the $i$-th coordinate and $\beta_{-i} := \beta - \beta_ie_i$ where $e_i$ has a 1 in the $i$-th position and 0's in the rest. Similar to the soft-thresholding rule of lasso, $\tau_\lambda(z)$ function maps the gradient $z$ to 0 when it's smaller than the threshold $h_{\lambda,\alpha}$. Therefore, $\ell_\alpha$ regularization selects a null model (all coefficient estimates are 0) if max$|(x_i,y)|< h_{\lambda,\alpha}$ (we assume $x$ and $y$ are centered and $\Vert x \Vert_2=1$).
Coordinate-descent can then be used to iteratively minimize the objective function. In each iteration, the algorithm minimizes along one coordinate and solves the scalar optimization problem with $z_i$ replaced by the adjusted gradient. If $\alpha=1$, this is equivalent to the R package {\tt glmnet} of \cite*{friedman2010regularization} . At each step of the coordinate-descent, we use the closed-form proximal operator for $\ell_\alpha$.
Unlike soft-thresholding, $\tau_\lambda(z)$ jumps immediately from 0 to $b_{\lambda,\alpha}$ when $z$ arrives to $h_{\lambda,\alpha}$, which results in discontinuity of the coefficient-regularization path. When $\lambda$ is small, $h_{\lambda,\alpha} > \lambda$, and when $\lambda$ is large, $h_{\lambda,\alpha} < \lambda$, thus the estimates are sparser than those given by lasso when we choose a small $\lambda$ (increase the shrinkage of $\hat\beta$ when true $\beta=0$). On the other hand, they are more robust for $\lambda$'s that are too large (reduce the shrinkage of $\hat\beta$ when true $\beta\neq 0$). In Figure \ref{tau2}, we show $\tau_\lambda(z)$ when $\alpha=0.5$ and $\lambda = 1,10$. The larger $\alpha$ is, the quicker $h_{\lambda,\alpha}$ changes with $\lambda$.
Appendix \ref{appendix} provides full details of the algorithm.
\section{Applications}\label{simulation-study}
\subsection{Linear Regression Simulation }\label{linear-model-benchmark}
To illustrate the $\ell_\alpha$-norm estimator, we simulate data from the model:
\[y = X\beta + \epsilon, \text{ where } \epsilon \sim N(0, \sigma^2I_N), \beta \in \mathds{R}^p, \sigma^2=1,\]
and we minimize the objective function over a regularization path $\lambda>0$:
\begin{equation}
J(\beta)=\Vert y-X\beta\Vert_2^2+\lambda \Vert\beta\Vert_\alpha^\alpha.
\end{equation}
The design matrices $X = [X_1, X_2, ..., X_p]$ are drawn from a multivariate normal distribution where each $X_i$ has variance $10^2$ and mean 0. We also introduce correlation among $X_i$'s, Cor$(X_i, X_j) = 0.1^{|i-j|/3}$. The noise variance $\sigma^2=1$ and $\beta$ is the coefficient vector. For three different values of $\alpha\in (0.1, 0.5, 0.9)$, we show how coefficient estimates given by $\ell_\alpha$ change with regularization parameter $\lambda$. We set three different data dimensions---$p=50, 100, \mbox{ and } 500$---whereas the true model dimension is fixed as small as 5. Specifically, $\beta = [5,5,..,5,0,0,...0]$ so that only the first five coefficients are non-zero. The number of observations is $N = 600$.
Figure \ref{proxiS1} shows simulation results. The dataset dimension $p$'s are 50, 100, and 200 from top to bottom. The left column illustrates the case of non-zero coefficient $\beta_1 = 5$. As $\lambda$ increases, the estimate $\hat\beta_1$ is penalized from 5 to 0. Also note the jumps in the $\ell_\alpha$-paths, especially when $\alpha=0.1$, which is expected and due to the discontinuity nature of the $\tau$ function, as discussed previously. As the columns $X_1, X_2, ..., X_5$ are positively correlated, the drop of one $\hat\beta$ increases the estimated value of others. Figure \ref{proxiS1} shows the path of lasso ($\alpha = 1$) looks smoother and shrinks quickly. The behavior of $\ell_{0.9}$ regularization is similar to the lasso.
The larger $\alpha$ is, the quicker $\hat\beta_1$ shrinks. This finding suggests the $\ell_\alpha$ estimator is less biased than the lasso estimator ($\alpha = 1$) regarding the true non-zeros, and we can further reduce estimation bias by choosing a smaller $\alpha$. Figure \ref{betahat} shows that when $\hat\beta$ drops to 0 as $\lambda$ increases, the regularization paths of other $\hat\beta$'s are affected and immediately make jumps.
For a suitable range of $\lambda$, $\ell_\alpha$ regularization introduces more sparsity than lasso. The right column in Figure \ref{proxiS1} shows how the number of non-zero $\hat\beta_i$'s change with $\lambda$. We point out that, the path of $\ell_{0.1}$ drops most quickly when $\log(\lambda)$ is less than 0. Though the regularized models with this range of $\lambda$ are still redundant since number of non-zero $\hat\beta_i$'s is greater than the true value, $\ell_{\alpha}$ gives a more sparse model than lasso.
Once the true model is achieved, $\ell_\alpha$ tends to stay on it though the regularization parameter $\lambda$ keeps increasing, especially when $\alpha$ is small. In other words, when $\lambda$ is small and there are many redundant variables in the model, the regularization of $\ell_\alpha$ is stronger than lasso; when $\lambda$ is large and we are close to the true model, the regularization of $\ell_\alpha$ is weaker than lasso. In Figure \ref{proxiS1}, we see that when $\log(\lambda)$ is close to 0, the performances of $\ell_{0.1}$ and lasso are similar. This is where the relative strength of regularization gets reversed. The plots summarize the estimated regularization paths and indicate again that $\ell_\alpha$ regularization is less biased and more robust for variable selection. $\ell_\alpha$ regularization possesses better performances with a wider range of $\lambda$.
We also show the performance of $\ell_\alpha$ regularization under two different correlation strengths: Cor$(X_i, X_j) = \rho^{|i-j|/3}$, where $\rho=0.1, 0.6$, representing low and high correlation, respectively. In this example, the regularization parameter $\lambda$ is chosen using five-fold cross-validation. The sizes of the datasets are $N^{train} = 600, N^{test}=600$, and the total number of runs is 100. In Table \ref{bias-compare}, we list the average prediction RMSE (out of sample), and the bias/variance of coefficient estimates (in sample) of $\ell_\alpha$ with other linear methods, including lasso, OLS, and the elastic net. For the elastic net, we always use the penalty $\frac{1}{2}\Vert\beta\Vert_1 + \frac{1}{4}\Vert\beta\Vert_2^2$. The dataset dimensions are $p=50,100$,and $500$ as before. Lasso is a benchmark, and normalize results by their counterparts under lasso. The RMSE is simply calculated as
\begin{equation*}
\text{RMSE}(Y, \hat Y)=\sqrt{ \frac{1}{n}\sum_{i=1}^n (y_i -\hat y_i)^2}
\end{equation*}
where \((y_1, y_2, \ldots, y_n)\) are observed values and \( (\hat y_1,\hat y_2, \ldots,\hat y_n)\) are the predicted values.
We find that $\ell_\alpha$ gives more accurate predictions in all dimensional and correlation settings. By comparing the average RMSE, we see that small $\alpha$ produces small prediction errors. $\ell_\alpha$ also produces less bias than lasso and elastic net in almost all cases, though the estimator variances get larger for those non-zero estimates. For $\beta_6=0$, we also emphasize that in the high-correlation setting, $\ell_\alpha$ estimates $\beta_6$ with great accuracy. Both $\ell_{0.1}$ and $\ell_{0.5}$ give the correct $\hat\beta_6 = 0$ with zero estimator variance. $\ell_\alpha$ performs extremely well when dealing with those redundant variables. Imagining $\ell_\alpha$ encourages sparser models than the lasso and elastic net is therefore not hard. This advantage of $\ell_\alpha$ regularization tends to stay even under a high-correlation and high-dimensional design setting.
\subsection{Predictive Performance }\label{empirical-application}
The purpose of this section is to compare the prediction performance of our $\ell_\alpha$ regularization and other commonly used machine-learning methods. We follow the literature of market-demand estimation, particularly \cite*{bajari2015machine, bajari2015demand}. Marketing researchers use measures such price elasticities and lift estimates to assess the effectiveness of promotions. $\ell_\alpha$ provides less biased estimates with a prediction error that is only slightly higher for those black-box nonlinear methods. We compare the performance of the alpha norm with other widely used machine-learning methods: OLS, generalized linear model boosting (GLMBoosting), random forests, support vector machines (SVMs), lasso, ridge, and the elastic net.
GLMBoosting improves the predictive ability by iteratively reweighting the estimated regression and classification functions. Reducing the bias is profitable but may sacrifice some prediction accuracy as the trade-off. Random forests are the most popular machine-learning method. During the training process, the model constructs multiple decision trees and outputs the class that is the mode of the classes (classification) or mean prediction (regression) of the individual trees. This method is often time-consuming and overfits the data without proper trimming. A Support Vector Machine(SVM) map points into different categories in space, and choose the hyperplane that maximizes the distance between the nearest data points on each side.
Adding machine-learning methods checks the robustness of our results. Random forest can achieve better prediction in some cases, whereas $\ell_\alpha$ is easier to interpret. From a computational perspective, the $\ell_\alpha$ estimator takes a far shorter time in tuning and run-time relative to non-parametric tree methods.
\subsubsection{Discrete Choice Model}\label{set-up}
The goal is to analyze customers' preferences among products, where our products
are collectively exhaustive, mutually exclusive, and finite. Suppose we have $J$ products and observe their sales in $M$ markets, and each market has $N_m$ customers. Assuming each customer only chooses one product based on higher expected utility, the probability of
customer $n$ in market $m$ choosing product $j$ is defined as
\[P_{mnj}=\text{P}(\text{customer $n$ in market $m$ choose product $j$}).\]
\noindent Since each customer only chooses one product in each market, \(\sum_j P_{mnj}=1, 0 \leq P_{mnj} \leq 1, j=1,\ldots,J\). The choice of product $j$ by person $n$ in
market m, denoted by \(y_{mnj}\), is given by
\begin{equation*}
y_{mnj}=\left\{
\begin{array}{rl}
1, & U_{mnj}>U_{mnk}, k\neq j\\
0, & otherwise.
\end{array}
\right.
\end{equation*}
\noindent Here \(U_{mnj}\) is the utility of product $j$ for customer $n$ in
market $m$:
\begin{equation}
U_{mnj}=\beta_{0}+X_{mj}\beta+\epsilon_{mnj},
\end{equation}
\noindent where \(X_{mj}\) is a vector of characteristics of product $j$ in market $m$. The parameter \(\beta\) is a set of parameters giving the effects of variables on probabilities in the market, and \(\epsilon_{mnj}\) captures the individual difference for product $j$ in market $m$, specifically for person $n$. The choice probability is given by
\begin{align*}
P_{mnj}= &\text{P}(y_{mnj}=1)=\text{P}(\cap_{k\neq j} \{ U_{mnk}<U_{mnj}\})\\
= & \text{P}(\cap_{k\neq j} \{ \epsilon_{mnk}-\epsilon_{mnj}<X_{mj}\beta_{m}-X_{mk}\beta_{m}\}).
\end{align*}
In our simulation, we primarily explore the case of one product. Then we have the choice of choosing the product and not choosing the product (option 1 vs option 0):
\begin{equation}
\left\{
\begin{array}{l}
U_{mn1} = \beta_0 + X_{m}\beta+\epsilon_{mn}\\
U_{mn0}=0\\
\epsilon_{mn} \sim \mbox{Logis}(0,1).
\end{array}
\right.
\end{equation}
\noindent Here $U_{mn1}$ is the utility of this product for customer $n$ in market $m$. \(\beta\) describes how characteristics influence the
expected utility, and is also a vector
of length K, and it independently comes from a multivariate normal distribution. $\beta_0+X_{m}\beta$ is the systematic utility of this product in market $m$, and $\epsilon_{mn}$ is the random utility for customer $n$ in market $m$. Here $U_{mn0}$ is the utility of not choosing the product, and we set it to be 0. Here \(X_{m}\) is a vector of $K$ product characteristics.
Then \(P_{mn1}\) have a type 1 extreme distribution as
\begin{equation}
P_{mn1}=\frac{\exp(\beta_0+X_{m}\beta)}{1+\exp(\beta_0+X_{m}\beta)}.
\end{equation}
For large enough $N$, the sample probability of
product 1 being chosen in market $m$ converges to this extreme distribution value. \cite*{bajari2015demand} construct a data generating process for each of the characteristics \(X_{m}\). We simulate them independently and identically from multivariate log-normal distribution:
\begin{equation*}
X_{m} \sim \mbox{logNormal}(0,\Sigma_{m}), \mbox{diag}(\Sigma_{m}) \sim \mbox{Unif}(0.5,1.5;K).
\end{equation*}
\noindent Next we add confounding variables. Given $K_c$ confounding variables here, and that they are weakly correlated with true variables, we are actually constructing
\begin{equation*}
(X_{m}, X^c_{m}) \sim \mbox{logNormal}(0,\Sigma ), \Sigma=\left[
\begin{array}{cc}
\Sigma_{m} & C_1\\
C_1^T & \Sigma_{m}^c
\end{array}\right],
\end{equation*}
\noindent where $X^c_{m}$ is the matrix of confounding variables, $C_1$ is the covariance matrix of $X_{m}$ and $X^c_{m}$, and we control their correlation to be weak. $\Sigma^c_{m}$ is the covariance matrix of $X_{m}$. To control the correlation among true variables and confounding variables, we construct the big covariance matrix $\Sigma$ as
\begin{equation}\label{Sigma_Construct}
[\Sigma]_{i,j}=\rho^{\frac{|i-j|}{3}}.
\end{equation}
\noindent Here $\rho$ is a pre-set correlation parameter, $0<\rho<1$. Combining $X_{m}$ and $X^c_{m}$ gives us the total characteristics of the product in market $m$: $X^t_{m}=(X_{m}, X^c_{m}) $:
Then we add categorical variables
into the datasets. Specifically, we create
binary variables by setting a cutoff value $T$ and let
\begin{equation*}
\tilde{X}_{mk}=\left\{
\begin{array}{ll}
1,& X_{mk}>T\\
0, & otherwise.
\end{array}
\right.
\end{equation*}
\noindent The sample share of the product in market $m$ is defined as
\begin{equation}
S_{m}=\frac{1}{N_m}\sum_{n=1}^{N_m} \mathds{1}_{\{U_{mn1}>0\}}.
\end{equation}
\noindent Here $N_m$ is the number of customers in market $m$. To make the case simpler, we assume the number of customers in each market is the same. If we set $N_m$ large enough, $S_m$ follows the extreme value distribution and converges to the real share. Following our empirical analysis in Section \ref{empirical-analysis}, instead of using sample shares as a response, we model the log of the quantity(unit) sold in each market:
\begin{equation*}
\mbox{log}(Q_m)=\mbox{log}(\sum_{n=1}^{N_m} \mathds{1}_{\{U_{mn1}>0 \}}),
\end{equation*}
\noindent where \(\mbox{log}(Q_m)=(\mbox{log}(Q_1), \mbox{log}(Q_2),\ldots,\mbox{log}(Q_M))'\) is the vector of the log of units of this product sold in each market.
\(X_{m}=(X^t_{1},X^t_{2},\ldots,X^t_{M})'\) is this product's characteristics in each market.
\subsubsection{Empirical results: Model comparison}\label{comparision-of-model}
To estimate and perform model comparison, we partition the data into two
parts: training \(\mbox{DGP}^{(1)}\) and testing \(\mbox{DGP}^{(2)}\). And we control \(\mbox{DGP}^{(1)}\) and \(\mbox{DGP}^{(2)}\) to be the same size. For \(\mbox{DGP}^{(1)}\), we use this part of the data to estimate the best tuning
parameters for SVM, GLMBoosting, and random forests. We find the best $\hat \lambda$ and best $\hat \alpha$ for the alpha norm via a five-fold cross-validation, where $\alpha $ is chosen from $\{0.1,0.5,0.9\}$. We also use five-fold cross-validation to choose the best $\hat \lambda$ for lasso, ridge, and the elastic net. We plug in the tuning parameters to construct all models \(\{f_i\}\).
For \(\mbox{DGP}^{(2)}\), we obtain \(\hat{y_i}=\text{predict}(f_i,\mbox{DGP}^{(2)})\) for each method, and we estimate their corresponding out-of-sample RMSE.
As our panel data in Section \ref{empirical-analysis} only have two continuous predictors, we include two continuous predictors in the true-predictor set. We try two different true-predictor cases. In the first instance, we include two continuous predictors and two binary predictors as true predictors. We estimate the RMSE of all models in different situations and calculate their RMSE ratios with the alpha norm as the benchmark. Table \ref{ratio_2_2} displays the results. Similarly, in the second case, we use two continuous and 20 binary predictors as true predictors, and present the results in Table \ref{ratio_2_20}. To further explain our data-generating process, for the rest of the redundant predictors, we control their correlation with the true predictors via $\rho$ in equation \ref{Sigma_Construct}. In the low-correlation case, we let $\rho=0.1$,whereas in the high-correlation case, we make $\rho=0.6$. The confounding predictors are 50\% numerical and 50\% categorical. We want to show our $\ell_\alpha$ method outperforms other linear methods in a high-correlation and high-dimensional case, by altering correlation $\rho$, market size $M$, and the number of predictors in total $K$.
From the results in Table \ref{ratio_2_2} and Table \ref{ratio_2_20}, we see that most of the entries are larger than 1, and ratios tend to be greater in the high-correlation case, or in the high-dimensional case, where $K$ is equal to or greater than $M$. In most cases, our $\ell_\alpha$ method outperforms the other methods except the random forest. The alpha norm can provide proper model selection and marginal effect estimates, however, whereas the random forest is difficult to interpret and lacks flexibility when we want to define subgroups.
\section{Store-Level Market Demand Data}\label{empirical-analysis}
\subsection{Sales Prediction}\label{sales_prediction}
The marketing example is about grocery store sales for salted snacks. Our dataset uses scanner-panel data on grocery stores from IRI Marketing Research. A unit of observation is product $j$, uniquely defined by a UPC (Universal Product Code), in store $m$ in week $t$. The number of observations is
15,339, which includes 100 unique products.
Let \(q_{jmt}\) be the
number of bags of salty snack j sold in store $m$ in week $t$. If
\(q_{jmt} = 0\), the possible situation can be zero sale, being out of stock
or missing observation. The price \(p_{jmt}\) is defined
as the quantity-weighted average of prices for product $j$ in store $m$ in
week $t$. Therefore, if \(q_{jmt}\) = 0, the weight is also set to zero. The general regression model is of the form
\begin{equation*}
Y_{jmt}=f(X_{\mbox{price},jmt}, X_{\mbox{product},jmt},X_{\mbox{promotion},jmt},X_{\mbox{week},t})+\epsilon_{jmt}.
\end{equation*}
\noindent Table \ref{meaning-variables} provides detailed summary information on our predictors. In our model, we do not use the variable iri\_key, which is a unique label for each store, because it is severely unbalanced in the data. The weeks are transformed into a combination of year and week. Specifically, we use six years and 52 weeks to present the 300 weeks in our dataset to deal with the unbalanced data issue.
For model validation, again we randomly separate the dataset into two parts: training $\mbox{DGP}^{(1)}$ and testing $\mbox{DGP}^{(2)}$. Then we use $\mbox{DGP}^{(1)}$ to estimate the best tuning parameters and construct the model and apply the models to $\mbox{DGP}^{(2)}$ to evaluate the out-of-sample RMSE.
In total our dataset contains 15 predictors with three continuous variables: quantity (number of units sold), price, and equivalized volume (or all commodity volume), which represent the store revenue from all products sold. Quantity ranges from 1 to 1,505 with mean 16.36 and standard deviation 40.37. Price ranges from 0.2 to 9.99 with mean 2.11 and standard deviation 1.003. Equivalized volume ranges from 0.0469 to 3 with mean 0.5139 and standard deviation 0.266.
Table \ref{market_rmse} provides out-of-sample RMSE ratios for the $\ell_\alpha$ estimator as the benchmark and $R^2$ of all models . Out-of-sample $R_{\mbox{OOS}}^2$ is simply calculated as
\begin{equation}\label{eq:OOS_Rsquare}
R^2_{\mbox{OOS}} = 1- \frac{\sum (\hat{y_i}-y_i)^2}{\sum (y_i -\bar y)^2},
\end{equation}
\noindent where $\hat {y_i}$ and $y_i$ are the predicted and observed values, respectively. The $\ell_\alpha$ regularization beats all other linear methods, with a smaller RMSE and larger $R^2$. The final solution selects 158 predictors.
To better understand the process of regularization, we plot the trend of changes in RMSE, and the number of predictors of the $\ell_\alpha$ and ridge regression when we change lambda in Figure \ref{fig:15000}. (Notice we do not draw this plot with lasso from {\tt glmnet}; instead, we use the alpha norm with \(\alpha=1\).)
The top predictors of those regularized methods can be found from the penalization path in Figure \ref{fig:15000}. The top predictors we can extract from the path include price, equivalized volume, promotion, brands (e.g., Lays and Ruffles), flavors (e.g., original, classical, regular, and sour cream and onion), the cooking method (e.g.,kettle-cooked), and fat content (e.g., reduced fat). Among these top predictors, many strands of literature have discussed the effect of price, promotion, brand, and equivalized volume.
Our method, however, provides an even closer look at the incremental effect of a particular brand, flavor, cooking method, and fat content, which can help grocery stores develop detailed strategies to improve their inventory. For example, a larger proportion of Lays snacks stock leads to higher sales.
\subsection{Promotion Lift Estimates}\label{lift estimate}
Section \ref{sales_prediction} shows promotion is always selected as a top predictor. Now we estimate the lift of a promotion. The products in our dataset are tagged with a promotion label if their price deduction is greater than 5\%. We want to see the incremental effect on sales generated by promotion (see \cite*{bajari2015demand}).
First, we split the data into a training set $\mbox{DGP}^{(1)}$ and testing set $\mbox{DGP}^{(2)}$. $\mbox{DGP}^{(1)}$ contains all the records with no promotion, and $\mbox{DGP}^{(2)}$ contains all the records with promotion. We find that we have 11,348 non-promotion records, which account for 74\% of our data. We use $\mbox{DGP}^{(1)}$ to train all models and provide predictions for $\mbox{DGP}^{(2)}$. Then we calculate the lift factor:
\begin{equation*}
\mbox{Lift factor}=\frac{\mbox{Actual sales}}{\mbox{Baseline sales}}=\frac{y}{\hat y}.
\end{equation*}
We use the predictions from the non-promotion model as the baseline sales and compare them to the actual sales. The incremental effect on the lift factor is defined by
\begin{equation*}
\Delta Q =\mbox{Lift factor}\cdot \mbox{Baseline sales}- \mbox{Baseline sales}=(\mbox{Lift factor}-1)\cdot \mbox{Baseline sales}.
\end{equation*}
When we predict the sales using a log-linear model of the form:
\begin{align*}
&\log(Q)=\alpha-\eta \log(P)+\sum_{i=1}^p \beta_i X_i,\\
&\log(Q')=\alpha-\eta \log((1-\gamma)P)+\sum_{i=1}^p \beta_i X_i +\beta_{\mbox{prom}} \mbox{Prom}.
\end{align*}
\noindent Here, $Q$ is the quantity of products in the absence of promotion, and $X_i$ is all the predictors. $P$ is the price of the product, $Q'$ is the quantity of products when promotion is considered, and Prom is the dummy indicator for promotion. $\beta_{\mbox{prom}} $ is the corresponding coefficient and $\gamma$ is the discount applied to price in the promotion, and in our model, $\gamma>0.05$.
The lift factor is calculated from the comparison of the two models as
\begin{align*}
\mbox{log(Lift)}&=log(Q')-log(Q)=-\eta log(1-\gamma)+\beta_{\mbox{prom}}
\mbox{Prom},\\
\mbox{Lift factor}&=\frac{Q'}{Q}=\mbox{exp}(-\eta log(1-\gamma)+\beta_{\mbox{prom}} . {\mbox{Prom}} ).
\end{align*}
\noindent The average of realized lift factors can predict the future lift from a promotional event. Figure \ref{fig:dist_log_Lift} plots the distribution of log(lift) calculated under different models. The distributions generated from different models are similar, and the mean of each distribution is positive as expected.
Figure \ref{fig:dist_log_Lift} shows that a large number of models have negative estimates for log(Lift), though the average effect of promotion is positive and statistically significant. We want to investigate what may play a significant role in deciding the magnitude and sign of the promotional effect. Finally, models of log(lift) are helpful in determining promotional strategies concerning an individual product.
Our lift estimates based on the $\ell_\alpha$ estimation are less biased than linear regression methods. We use resampling bootstrap to get the distribution of $\beta_\text{Prom}$ from OLS estimate, and we hope it will approximate to the true distribution since OLS gives unbiased estimates. For each resampling model, we randomly select half of the observations and use them to fit the OLS. We plot the distribution of $\beta_\text{Prom}$ in Figure \ref{fig:ols_pr_boot}. The plot of log(lift) estimated by the alpha norm is very close to the mean of OLS bootstrapping. And it is expected to be less biased than rf and svm. All methods give positive average lifts, with positive and negative increments in the estimate. The large variance in the estimation suggests the negative lifts come from variance rather than bias, see \cite{bajari2015demand}. Further improvements in lift estimates occur if we use the variables selected in lasso or the alpha norm.
\section{Discussion}\label{discussion}
$\ell_\alpha$ regularization provides a useful tool for predictor selection in marketing and economics. Scanner-panel data usually have thousands of binary dummy variables, and many of the predictors do not predict sales; hence, variable selection is needed. Our $\ell_\alpha$ regularization can jump to a sparse solution. Post-lasso variables can increase the fit of the model in high-dimensional sparse cases. For the applications we use here, $\ell_\alpha$ regularization finds a better solution to in-sample overfitting and is more adaptive than lasso, as the degree of the norm can be chosen from 0 to 1, thus applying to both sparse and extreme sparse models.
Our empirical analysis shows $\ell_\alpha$ regularization does improve predictions versus traditional linear regression and machine-learning black-box techniques. Predicted sales and selected variables can be used for inventory planning and can predict the outcome of potential marketing strategies. Also, the alpha norm can be particularly useful when the practitioners want to study the significant predictors for a particular subpopulation. Our alpha norm can efficiently shrink the predictor size and pick significant predictors according to the features of the response.
In contrast to machine-learning approaches, the $\ell_\alpha$ regularization benefits from interpretability of the marginal effects. The impact of a particular product or flavor can be assessed. $\ell_\alpha$ regularization is more efficient in estimating the model than nonlinear methods while providing relatively similar performance. A further extension could be to apply our methodology to demand and supply-demand estimation as in \cite*{brian2015bayesian}, when targeting a specific incremental effect, such as the lift of promotion. Similar approaches can be applied to estimate lift of in-store displays, which usually have multiple levels.
\clearpage
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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If you had to get one, how would you know which one you'd like more?
Well I did order the 88 Shimmer Palette off ebay when I purchased my 120. And it took a little longer to arrive. But now, it's time for an initial comparison of both popular palettes!
Size of Case: Similar palette dimensions, but the 88 palette has all its eyeshadows on 1 side, whereas the 120 has 2 separate palettes within, so it's thicker and heavier.
Case quality: 120 wins. My 88 neutral arrived with the latch broken off, and the 88 shimmer is hard to open. They are also thinner and less sturdy than the 120. The only con for the 120 is that each of the 2 palettes are detached from the main case itself. Some people may like being able to bring the palettes out and about, but I find it too prone to eyeshadow incidents, so I actually glued my 2 palettes to the case.
Eyeshadow Volume: 120 wins. Besides the fact that there are 32 more pans, each pan is also at least 150% the size/volume of an 88 pan. As circles go, it's probably close to double the amount of shadow. Odds are, the 120 will last you a good while longer than the 88.
Color spectrum: I have to say the 88 wins, color-wise, and the 120 wins texture-wise. Despite having less colors, there are no repeats in the 88 shimmer palette. You have all the reds, pinks, browns, blues, greens, yellow, deeps, and highlights you need. The 120 has a some duplicates (or colors close enough to be so). It does offer a variety of matte and shimmer, which is great, but the fatal flaw is it's lack of soft nudes and really rich, deep tones (coffee, navy, indigo) for contouring and definition.
Texture/Pay-off: 88 Shimmer vs 120's non-matte Shades are like MAC's Lustre shadows vs Frosts. Frosts tend to be more opaque, with a satin-y finish. Lustres tend to be a little more translucent, but also more reflective and shimmery. It depends which sort of finish you like. Pay-off wise, the 88 is far more consistent than the 120 though. If that's important for you, get the 88.
Usable Wet? Yup for both! But I find the 88 Palette works better with water or mixing medium than the 120. Pretty much the whole palette can be converted to wicked chrome-finish liners because of the intensity of these shades when wet. Below, I picked 2 almost-identical teals, but as you can see, the level of sheen differs greatly. Only a very small handful of colors in the 120 palette can give you that level of sheen.
Longevity: Both of these wear well with a primer beneath, and I'm not about to go out and about without any on, so I am not a good judge on this.
Blendability: The 88 Shimmer palette is easier to work with as far as blending different shades goes. HOWEVER, unless you're going to use everything wet, you will get a stronger contrast with the more opaque colors in the 120 palette. It takes a little more work, but you will get a more dramatic effect without having to wet any except the hardest of the matte shades.
Verdict: If you are looking for the option to spice up your existing collection of shadows without forking out a ton of money on dramatic colors, get the 88 Shimmer. If you're just starting out and need something that will be the centerpiece and mainstay of your budding eyeshadow collection, get the 120.
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{
"redpajama_set_name": "RedPajamaC4"
}
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Q: WinSCP .NET Assembly impersonation issue: The stub received bad data I have an .NET console app using WinSCP assembly version 5.19.5. The app is under impersonation in order to access files on NAS so that I set ExecutableProcessUserName and ExecutableProcessPassword for the Session object.
Dim sessionOptions As New SessionOptions
With sessionOptions
.Protocol = Protocol.Sftp
.HostName = _SFTPServer
.UserName = _SFTPLoginID
.Password = _SFTPLoginPwd
.PortNumber = _SFTPPort
End With
sessionOptions.SshHostKeyFingerprint =
session.ScanFingerprint(sessionOptions, "SHA-256")
Using session As New Session
session.ExecutableProcessUserName = _ServiceAccountUsername
session.ExecutableProcessPassword =
ConvertToSecureString(_ServiceAccountPassword)
session.Open(sessionOptions)
It fails at session.Open with error:
The stub received bad data
Any ideas? Thanks
A: It seems that you need to specify the domain:
Stub received bad data?
session.ExecutableProcessUserName = "user@domain";
Side note: Setting SessionOptions.SshHostKeyFingerprint to the value returned by session.ScanFingerprint is just an ineffective equivalent of SshHostKeyPolicy.GiveUpSecurityAndAcceptAny.
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
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Q: Help understanding a theorem about group homology $H_0$ I'm self-studying homological algebra. I have problems on understanding a theorem about $H_0$. First, I don't know where the bottom row of the commutative diagram comes from, see the red line in figure 1. Second, what does $\mathcal{G}A$ mean? Is it $\mathcal{G}[A]$, a group ring produced by $A$?
Also, what does $NA$ mean in figure 2? $N$ even is not a algebraic structure, is it? (These figures are all from Rotman's Advance Modern Algebra Book II)
Figure 1:
Figure 2:
A: The bottom row comes from the fact (see below) that $\mathcal{G}A$ is a submodule of $A$, so of course you have a short exact sequence $0\to \mathcal{G}A\to A\to A/\mathcal{G}A\to 0$. More generally, whenever $N$ is a submodule of $M$, you have a short exact sequence $0\to N\to M\to M/N\to 0$
$\mathcal{G}$ fits into the short exact sequence $0\to\mathcal{G}\to \mathbb{Z}G\to \mathbb{Z}\to 0$ so it is the kernel of the augmentation map $\mathbb{Z}G\to \mathbb{Z}$. In other words, it is the augmentation ideal, and so if $A$ is a $G$-module, $\mathcal{G}A$ is just the submodule of $A$ defined by this ideal, that is the submodule generated by $\{xa, x\in\mathcal{G},a\in A\}$.
Similarly, $N$ is defined as an element of the group ring (specifically, $N=\displaystyle\sum_{g\in G}g$), and it is central, therefore $x\mapsto Nx, A\to A$ is a $G$-morphism, and its image is a sub-$G$-module : it clearly deserves the name $NA$ (in the same way as you would write $f(A)$ or $fA$ for the image of a morphism $f$)
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
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\section{Introduction}
Conservation laws have been used as a mathematical tool in a variety of
situations in order to provide a simplified description of complex
physical phenomena which nevertheless keeps the essential features
of the processes to be described, and the general theory of hyperbolic conservation
laws aims to provide a unified set of techniques needed to understand
the mathematical properties of such equations. However, in some cases,
the general theory fails to provide a firm mathematical description
for a particular case because some of the assumptions needed
in the theory are not in place.
In the present contribution we focus on such an example,
a hyperbolic conservation law appearing in ideal magnetohydrodynamics.
For this conservation law, solutions cannot be found using the classical
techniques of conservation laws, and a new approach is needed.
Magnetohydrodynamics (MHD) is the study of how electric currents in a moving
conductive fluid interact with the magnetic field created by the moving fluid itself.
The MHD equations are a combination of the Navier-Stokes equations of fluid mechanics
and Maxwell's equations of electromagnetism, and the equations are generally coupled
in such a way that they must be solved simultaneously. The ideal MHD equations are
based on a combination of the Euler equations of fluid mechanics
(i.e. for an inviscid and incompressible fluid)
and a simplified form of Maxwell's equations.
The resulting system is highly complex and one needs to rely on numerical
approximation of solutions in order to understand the dynamics of the system.
As even the numerical study of the full system is very challenging, it can be
convenient to introduce some simplifying assumptions -- valid in some limiting cases --
in order to get a better idea of the qualitative properties of the system,
and in order to provide some test cases against which numerical codes
for the full MHD system can be tested.
The emergence of coherent structures in turbulent plasmas has
been long observed both in numerical simulations and experiments.
Moreover, the tendency of the magnetic field to organize into
low-dimensional structures such as two-dimensional
magnetic pancakes and one-dimensional magnetic ropes is well known.
As a consequence, in certain cases it makes sense to use simplified
one or two dimensional model equations.
Such simplified equations will be easier to solve, but nevertheless preserve some
of the important features observed in MHD systems.
In \cite{brio}, a simplified model system for
ideal MHD was built using such phenomenological considerations. The system is written as
\begin{equation}\label{Brio System}
\begin{split}
\partial_t u+\partial_x\big(\Sfrac{u^2+v^2}{2}\big)=0,\\
\partial_t v+\partial_x\big(v(u-1)\big)=0.
\end{split}
\end{equation}
The quantities $u$ and $v$ are the velocity components of the fluid whose
dynamics is determined by MHD forces, and the system represents the conservation of the velocities.
Velocity conservation in this form holds only in idealized situations in the case of smooth solutions,
and
the limitation of this assumption manifests itself in the non-solvability of the system
even for the simplest piece-wise constant initial data, i.e.
for certain dispositions of the Riemann initial data
\begin{equation}
\label{riemann}
u|_{t=0}=\begin{cases}
U_L, &x\leq 0\\
U_R, &x> 0
\end{cases}, \qquad v|_{t=0}=\begin{cases}
V_L, &x\leq 0\\
V_R, &x> 0
\end{cases}.
\end{equation}
From a mathematical point of view, the characteristic fields of this system are neither
genuinely nonlinear nor linearly degenerate in certain regions in the $(u,v)$-plane (see \cite{HL}).
In this case the standard theory of hyperbolic conservation laws which can be found in e.g. \cite{Daf}
does not apply and one cannot find a classical Riemann solution admissible in the sense of
Lax \cite{Lax} or Liu \cite{Liu}.
In order to deal with the problem of non-existence of solutions to the
Riemann problem for certain conservation laws, the concept of singular solutions
incorporating $\delta$-distributions along shock trajectories was introduced in \cite{Korchinski}.
The idea was pursued further in \cite{KK,HL}, and by now, the literature on the subject is rather
extensive. Some authors have defined theories of distribution products in order to incorporate
the $\delta$-distributions into the notion of weak solutions \cite{DMLM,huangWang,Sarr}.
In other works, the need to multiply $\delta$-distributions has been avoided either
by working with integrated equations \cite{Huang,KaTe}, or by making an appropriate definition
of singular solutions \cite{DSH}. In order to find admissibility conditions for such
singular solutions, some authors have used the weak asymptotic method \cite{DSH,DOS,MN_arma,Omelyanov}.
With the aim of dealing with the nonlinearity featured by the system \eqref{Brio System},
the weak asymptotic method was also extended to include complex-valued approximations \cite{KM}.
The authors of \cite{KM} were able to provide singular solutions of \eqref{Brio System}
even in cases which could not be resolved earlier. However, even if \cite{KM} provides some
admissibility conditions, the authors of \cite{KM} did not succeed to prove uniqueness.
Existence of singular solutions to \eqref{Brio System} was also proved in \cite{Sarr} using
the theory of distribution products, but uniqueness could not be obtained.
Therefore, it was natural to ask whether the Brio system should be solved
in the framework of $\delta$-distributions
as conjectured in \cite{HL} where the system was first considered
from the viewpoint of the conservation laws theory.
The authors of \cite{HL} compared \eqref{Brio System} with the triangular system
\begin{equation}
\label{triang}
\begin{split}
\partial_t u+\partial_x\big(\Sfrac{u^2}{2}\big)=0,\\
\partial_t v+\partial_x\big(v(u-1)\big)=0.
\end{split}
\end{equation}
which differs from \eqref{Brio System} in the quadratic term $v^2$. However, the system \eqref{triang} is linear with respect to $v$ and it naturally admits $\delta$-type solutions (obtained e.g. via the vanishing viscosity approximation). To this end, let us remark that most of the systems admitting $\delta$-shock wave solutions are linear with respect to one of the unknown functions \cite{DMLM, DSH, HL, huangWang, KK}. There are also a number of systems which can be solved only by introducing the $\delta$-solution concept and which are non-linear with respect to both of the variables such as the chromatography system \cite{Sun} or the Chaplygin gas system \cite{N1}. However, in all such systems, it was possible to control the nonlinear operation over an approximation of the $\delta$-distribution. This is not the case with \eqref{Brio System} since the term $u^2+v^2$ will necessarily tend to infinity for any real approximation of the $\delta$-function.
This problem can be dealt with by introducing complex-valued approximations of the $\delta$-distribution.
Using this approach, a somewhat general theory can be developed as follows.
Consider the system
\begin{equation}
\label{gensystem}
\begin{split}
\partial_t u + \partial_x f(u,v) =& 0, \\
\partial_t v + \partial_x g(u,v) =& 0,
\end{split}
\end{equation}
The following definition gives the notion of $\delta$-shock solution to system \eqref{gensystem}.
\begin{df}
\label{def-gen}
The pair of distributions
\begin{equation}
\label{delta-sol}
u=U+\alpha(x,t)\delta(\Gamma) \ \ \text{and} \ \ v=V+\beta(x,t)\delta(\Gamma)
\end{equation} are called a generalized $\delta$-shock wave solution of system
\eqref{gensystem} with the initial data
$U_0(x)$ and $V_0(x)$
if the integral identities
\begin{align}
\label{m1-g1}
\nonumber
&\int_{I\!\!R_+} \! \! \int_{I\!\!R} \left( U\partial_t\varphi +f(U,V)
\partial_x\varphi\right)~dxdt \\
&\qquad + \sum\limits_{i\in I}\int_{\gamma_i}\alpha_i(x,t)
\Sfrac{\partial \varphi(x,t)}{\partial {\bf l}} \, + \int_{I\!\!R} U_0(x)\varphi(x,0)~dx = 0,
\end{align}
\begin{align}
\label{m2-g1}
\nonumber
&\int_{I\!\!R_+} \! \! \int_{I\!\!R} \left(V\partial_t\varphi+ g(U,V)\partial_x\varphi\right)~dxdt\\
&\qquad + \sum\limits_{i\in I}\int_{\gamma_i}\beta_i(x,t)
\Sfrac{\partial \varphi(x,t)}{\partial {\bf l}} \, + \int_{I\!\!R} V_0(x)
\varphi(x,0)~dx = 0,
\end{align}
hold for all test functions $
\varphi\in {\mathcal D}(I\!\!R\times I\!\!R_+)$.
\end{df}
This definition may be interpreted as an extension of the
classical notion of weak solutions.
The definition is consistent with the concept of measure solutions
as put forward in \cite{DMLM, huangWang} in the sense that the two singular parts
of the solution coincide, while the regular parts differ on a set of Lebesgue measure zero.
However, Definition \ref{def-gen} can be applied to any hyperbolic system of equations
while the solution concept from \cite{DMLM} only works in the special situation
when the $\delta$-distribution is attached to an unknown which appears linearly in the flux $f$ or $g$,
or when nonlinear operations on $\delta$ can somehow be controlled in another way.
Definition \ref{def-gen} is quite general, allowing a combination of initial
steps and delta distributions; but its effectiveness is already
demonstrated by considering the Riemann problem with a single jump.
Indeed, for this configuration it can be shown that a $\delta$-shock
wave solution exists for any $2\times 2$ system of conservation laws.
Consider the Riemann problem for \eqref{gensystem} with initial data
$u(x,0) = U_0(x)$ and $v(x,0) = V_0(x)$,
where
\begin{equation}
\label{rieman-data}
U_0(x)=\begin{cases} u_1, &x<0,\\
u_2, & x>0,
\end{cases} \quad
V_0(x)=\begin{cases} v_1, &x<0,\\
v_2, & x>0.
\end{cases}
\end{equation}
Then, the following theorem holds:
\begin{thm}\label{thm-cnl}
{\bf a)}
If $u_1\neq u_2$ then the pair of distributions
\begin{eqnarray}\label{sol-a1}
u(x,t) & = & U_0(x-ct), \\
\label{sol-a2}
v(x,t) & = & V_0(x-ct) + \beta(t) \delta(x-ct),
\end{eqnarray}
where
\begin{equation}
\label{RH-def1}
c=\frac{[f(U,V)]}{[U]}=\frac{f(u_2,v_2)-f(u_1,v_1)}{u_2-u_1}, \ \mbox{ and } \
\beta(t)=(c[V]-[g(U,V)])t
\end{equation}
represents the $\delta$-shock wave solution of \eqref{gensystem}
with initial data $U_0(x)$ and $V_0(x)$
in the sense of Definition \ref{def-gen} with $\alpha(t)=0$.
\vskip 0.1in
\noindent
{\bf b)} If $v_1\neq v_2$ then the pair of distributions
\begin{eqnarray}\label{sol-b1}
u(x,t) & = & U_0(x-ct) + \alpha(t) \delta(x-ct), \\
\label{sol-b2}
v(x,t) & = & V_0(x-ct),
\end{eqnarray}
where
\begin{equation}
\label{RH-def2}
c=\frac{[g(U,V)]}{[V]}=\frac{g(u_2,v_2)-g(u_1,v_1)}{v_2-v_1}, \ \
\alpha(t)=(c[U]-[f(U,V)])
\end{equation}
represents the $\delta$-shock solution of \eqref{gensystem} with
initial data $U_0(x)$ and $V_0(x)$ in the sense of Definition
\ref{def-gen} with $\beta(t)=0$.
\end{thm}
\begin{proof}
We will prove only the first part of the theorem as the
second part can be proved analogously. We immediately see that
$u$ and $v$ given by \eqref{sol-a1} and \eqref{sol-a2}
satisfy \eqref{m1-g1} since $c$ is given exactly by the
Rankine-Hugoniot condition derived from that system. By
substituting $u$ and $v$ into \eqref{m2-g1},
we get after standard transformations:
\begin{align*}
\int_{I\!\!R_+}\left(-c[V]+[g(u,V)]\right)\varphi(ct,t)~dt
-\int_{I\!\!R_+}\alpha'(t)\varphi(ct,t)~dt = 0.
\end{align*}
From here and since $\alpha(0)=0$, the conclusion follows
immediately.
\end{proof}
As the solution framework of Definition \ref{def-gen} is very weak,
one might expect non-uniqueness issues to arise.
This is indeed the case, and the proof of the following proposition
is an easy exercise.
\begin{prop}\label{non-unique}
System \eqref{gensystem} with the zero initial data:
$u|_{t=0}=v|_{t=0}=0$ admits $\delta$-shock solutions of the form:
\begin{align*}
u(x,t)=0, \ \ v(x,t)=\beta\delta(x-c_1t)-\beta\delta(x-c_2 t),
\end{align*}for arbitrary constants $\beta$, $c_1$ and $c_2$.
\end{prop}
As already alluded to, a different formal approach for solving \eqref{Brio System}
was used by \cite{Sarr}. However, just as in \cite{KM} the definition of singular solutions
used in \cite{Sarr} is so weak that uniqueness cannot be obtained.
Another problem left open in \cite{KM, Sarr} is the physical meaning of the $\delta$-distribution
appearing as the part of the solution.
Considering systems such as the Chaplygin gas system or \eqref{triang},
the use of the $\delta$-distribution in the solution can be justified by
invoking extreme concentration effects if we assume that $v$ represents density.
However, in the case of the Brio system, $u$ and $v$ are velocities and unbounded
velocities cannot be explained in any reasonable physical way.
In the present contribution, we shall try to explain necessity of $\delta$-type
solutions for \eqref{Brio System} following considerations from \cite{KT} where it was argued
(in a quite different setting) that the wrong variables are conserved.
In other words, the presence of a $\delta$-distribution in a weak solution actually signifies
the inadequacy of the corresponding conservation law in the case of weak solutions.
Similar consideration were recently put forward in the case of singular solutions in
the shallow-water system \cite{KMT}.
Starting from this point, we are able to formulate uniqueness requirement for the Riemann problem
for \eqref{Brio System}. First, we shall rewrite the system using the energy $q=(u^2+v^2)/2$
as one of the conserved quantities (which is actually an entropy function corresponding to \eqref{Brio System}).
Thus, we obtain a strictly hyperbolic and genuinely nonlinear system which admits a Lax admissible solution
for any Riemann problem. Such a solution is unique and it will give a unique $\delta$-type solution
to the original system. The $\delta$-distribution will necessarily appear
due to the nonlinear transformation that we apply.
The paper is organized as follows: In Section 2, we shall rewrite \eqref{Brio System}
in the new variables $q$ and $u$, and exhibit the admissible shock and rarefaction waves.
In Section 3, we shall introduce the admissibility concept for solutions of the
original system \eqref{Brio System},
and prove existence and uniqueness of a solution to the Riemann problem
in the framework of that definition.
\section{Energy-velocity conservation}
As mentioned above, conservation of velocity is not necessarily a physically well defined balance law,
and it might be preferable to specify conservation of energy for example.
Actually, in some cases, conservation of velocity does give an appropriate balance law,
such as for example in the case of shallow-water flows \cite{GKK}.
In the present situation, it appears natural to replace at least one of the
equations of velocity conservation. As will be seen momentarily,
such a system will be strictly
hyperbolic with genuinely nonlinear characteristic fields,
so that the system will be more amenable to standard method of hyperbolic conservation laws.
To introduce the new conservation law, we define an energy function
\begin{equation}\label{Entropy Function}
q(u,v)=\frac{u^2+v^2}{2},
\end{equation}
and note that this function is a mathematical entropy for the system \eqref{Brio System}.
Then we use the transformation
\begin{equation*}
(u,v)\rightarrow \big(u,\Sfrac{u^2+v^2}{2} \big),
\end{equation*}
to transform \eqref{Brio System} into the system
\begin{equation}\label{Transformed system}
\begin{split}
\partial_tu + \partial_xq=0,\\
\partial_tq + \partial_x \big((2u-1)q+\Sfrac{u^2}{2}-\Sfrac{2u^3}{3} \big)=0.
\end{split}
\end{equation}
System \eqref{Brio System} and the transformed system \eqref{Transformed system}
are equivalent for differentiable solutions.
However, as will be evident momentarily, the nonlinear transformation changes the character of the system,
and while \eqref{Brio System} is not always genuinely nonlinear,
the new system \eqref{Transformed system} is always strictly hyperbolic and genuinely nonlinear.
In the following, we analyze \eqref{Transformed system}, and find the elementary
waves for the solution of \eqref{Transformed system}.
The flux function of the new system is given by
\begin{equation*}
F=
\begin{pmatrix}
q\\ (2u-1)q+\frac{u^2}{2}-\frac{2u^3}{3}
\end{pmatrix}
\end{equation*}
with flux Jacobian
\begin{equation*}
DF=
\begin{pmatrix}
0 & 1\\ 2q+u-2u^2 & 2 u-1
\end{pmatrix}.
\end{equation*}
The characteristic velocities are given by
\begin{equation}\label{eigenvalues}
\lambda_{-,+} = \frac{2u-1 \mp \sqrt{8q-4u^2+1}}{2}.
\end{equation}
A direct consequence of \eqref{Entropy Function} gives the
relation $2q \geq u^2 \geq 0$ which implies that the quantity under the square
root is non-negative. Thus, $8q-4u^2+1>0$ and the eigenvalues are real and
distinct so that the system is strictly hyperbolic.
The right eigenvectors in this case are given by
\begin{equation}
\begin{split}
r_- =
\begin{pmatrix}
1 \\ u-\frac{1}{2} - \sqrt{2q-u^2+\frac{1}{4}}
\end{pmatrix},\\
r_+ =
\begin{pmatrix}
1 \\ u-\frac{1}{2} + \sqrt{2q-u^2+\frac{1}{4}}
\end{pmatrix}.
\end{split}
\end{equation}
It can be verified easily that these eigenvectors are linearly independent
and span the $(u,q)$-plane.
The associated characteristic fields
\begin{equation}
\nabla\lambda_-\cdot r_- = 2+\frac{1}{\sqrt{8q-4u^2+1}} \ ,
\end{equation}
\begin{equation}
\nabla\lambda_+\cdot r_+ = 2-\frac{1}{\sqrt{8q-4u^2+1}} \ ,
\end{equation}
are genuinely nonlinear and admit both shock and rarefaction waves.
For a shock profile connecting a constant left state $(u,q)=(u_L,q_L)$ to a constant right
state $(u,q)=(u_R,q_R)$, the Rankine-Hugoniot jump conditions for \eqref{Transformed system} are
\begin{align}\label{Rankine-Hugoniot}
c(u_L-u_R)&=~(q_L-q_R),\\
\label{Rankine-Hugoniot1}
c(q_L-q_R)&=\big( (2u_L-1)q_L+\Sfrac{u_L^2}{2}-\Sfrac{2u_L^3}{3} -
(2u_R-1)q_R-\Sfrac{u_R^2}{2}+\Sfrac{2u_R^3}{3} \big),
\end{align}
where $c$ is the shock speed. We want the speed
in \eqref{Rankine-Hugoniot}, \eqref{Rankine-Hugoniot1}
to satisfy the Lax admissibility condition
\begin{equation}
\label{Lax}
\lambda_\mp(u_L,q_L)\geq c \geq \lambda_\mp(u_R,q_R).
\end{equation}
To determine the set of all states that can be connected to a
fixed left state $(u_L,q_l)$, we eliminate the shock speed, $c$, from the above equations
to obtain the shock curves
\begin{small}
\begin{align*}
&(q_R)_{1,2}=\frac{2q_L-(u_L-u_R)(2u_R-1)}{2} \ \ \pm \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\
&\frac{\sqrt{[-2q_L+(u_L-u_R)(2u_R-1)]^2+4\left[(u_L-u_R)\left((2u_L-1)q_L+\frac{u_L^2}{2}-
\frac{u_R^2}{2}-\frac{2u_L^3}{3}+\frac{2u_R^3}{3}\right)-q_L^2\right]}}{2}.
\end{align*}
\end{small}
After basic algebraic manipulations, we obtain
\begin{align}\label{SW}
\nonumber
(q_R)_{1,2}= q_L&-\frac{1}{2}(u_L-u_R)(2u_R-1)\\
&\pm \mid u_L-u_R\mid\sqrt{2q_L + \Sfrac{1}{4} + \Sfrac{1}{2} (u_L-u_R)-\Sfrac{1}{3} \big( 2u_L^2 +2u_Lu_R-u_R^2\big)}
\end{align}
From here and \eqref{Lax}, by considering $(u_R,q_R)$ in a small neighborhood of $(u_L,q_L)$, we conclude that the shock wave of the first family (SW1), the shock wave of the second family (SW2), the rarefaction wave of the first family (RW1) and the rarefaction wave of the second family (RW2) are given as follows:
\begin{figure}
\begin{center}
{\includegraphics[scale=0.5]{ffigure1a.eps}}~~
{\includegraphics[scale=0.5]{ffigure1b.eps}}
(a)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(b)
\end{center}
\caption{\small (a) Shock wave curves of the first and the second families at the left state $(u_L,q_L)=(1,5)$.
(SW1) is indicated by the upper curve, while (SW2) is the lower curve.
The blue dotted curve shows the limiting curve $q=u^2/2$.
(b) Rarefaction wave curves of the first and the second families at the left state $(u_L,q_L)=(1,5)$.
The (RW1) is indicated by the lower curve while (RW2) is the upper curve.}
\label{Fig1}
\end{figure}
\begin{flalign}\label{SW1}
& \mathrm{(SW1)} \ \
\nonumber
& q_R=q_L -\frac{1}{2}\big(u_L-u_R\big)\big(2u_R-1\big)
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \
\\
& & + \mid u_L-u_R\mid \Big( 2q_L + \Sfrac{1}{2}(u_L-u_R)-\Sfrac{1}{3} \big(2u_L^2 +2u_Lu_R-u_R^2 \big)
+ \Sfrac{1}{4} \Big)^{\frac{1}{2}}, \ \
\end{flalign}
for $u_R<u_L$. To verify that this indeed is the shock wave of the first
family, we obtain from \eqref{Rankine-Hugoniot} and \eqref{Lax} that
\begin{equation*}
\lambda_-(u_L,q_L)\geq c=\frac{2u_R-1-\sqrt{8q_L+1+\frac{4u_R^2}{3}-\frac{8u_Lu_R}{3}-\frac{8u_L^2}{3}-2u_R+2u_L}}{2}.
\end{equation*}
Taking into account the form of $\lambda_-$, we conclude from the above equation that
\begin{equation*}
2(u_L-u_R)\geq \sqrt{8q_L+1-4u_L^2} -\sqrt{8q_L+1+\frac{4u_R^2}{3}-\frac{8u_Lu_R}{3}-\frac{8u_L^2}{3}-2u_R+2u_L}.
\end{equation*}
Further simplification leads to
\begin{equation*}
2 \geq \frac{-\frac{4}{3}(u_L-u_R)-2}{\sqrt{8q_L+1-4u_L^2} +\sqrt{8q_L+1+\frac{4u_R^2}{3}-\frac{8u_Lu_R}{3}-\frac{8u_L^2}{3}-2u_R+2u_L}},
\end{equation*}
which is obviously correct. In a similar way, the second part of the Lax condition,
\begin{equation*}
\lambda_-(u_R,q_R)\leq c,
\end{equation*}
can be verified. Moreover, it is trivial to verify the additional inequality $\lambda_+(u_R,q_R) \geq c$,
so that we have three characteristic curves entering the shock trajectory, and one characteristic curve
leaving the shock.
\begin{flalign}
& \mathrm{(SW2)} \ \
\nonumber
& q_R=q_L-\frac{1}{2}\big(u_L-u_R\big)\big(2u_R-1\big)
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \ \\
& & - \mid u_L-u_R\mid \Big( 2q_L + \Sfrac{1}{2} (u_L-u_R) - \Sfrac{1}{3}\big( 2u_L^2 +2u_Lu_R-u_R^2 \big) +
\Sfrac{1}{4} \Big)^{\frac{1}{2}}, \ \
\end{flalign}
for $u_R<u_L$. We will skip the proof since it is the same as in the case of (SW1).
Next, we have the rarefaction curves. \\
\noindent
(RW1), \ Using the method from \cite[Theorem 7.6.5]{Daf}, this wave can be written as
\begin{equation}
\label{RW1}
\frac{dq}{du}=\frac{2u-1-\sqrt{8q-4u^2+1}}{2}=\lambda_-(u,q), \ \ \ \ \ q(u_L)=q_L,
\end{equation}
for $u_R>u_L$.
Clearly, for $u_R<u_L$ we cannot have (RW1) since in that domain,
states are connected by (SW1) (see (SW1) above).
In order to prove that \eqref{RW1} indeed provides RW1,
we need to show that
\begin{equation}
\label{check-1}
\lambda_-(u_L,q_L)<\lambda_-(u_R,q_R) \ \ {\rm if} \ \ u_R>u_L.
\end{equation}
Introducing the change of variables $\tilde{q}=8q-4u^2+1$ in \eqref{RW1}, we can rewrite it in the form
$$
\frac{d\tilde{q}}{du}=-4(1+\sqrt{\tilde{q}})<0.
$$
From here, we see that $\tilde{q}$ is decreasing with respect to $u$ and thus,
for $u_L<u_R$, we must have
$$
8q_L-4u_L^2+1=\tilde{q}_L>\tilde{q}_R=8q_R-4u_R^2+1.
$$
This, together with $u_L<u_R$ immediately implies \eqref{check-1}. \\
\noindent
(RW2) Using again \cite[Theorem 7.6.5]{Daf}), we have
\begin{equation}
\label{RW2}
\frac{dq}{du}=\frac{2u-1+\sqrt{8q-4u^2+1}}{2}=\lambda_+(u,q), \ \ q(u_L)=q_L,
\end{equation}
for $u_R>u_L$.
It can be shown that \eqref{RW2} gives the rarefaction wave (RW2) in the same way
explained above for (RW1).
The wave fan issuing from the left state $(u_L,q_L)$
and the inverse wave fan issuing from the right state $(u_R,q_R)$
are given in Figure \ref{Fig2}(a) and Figure \ref{Fig2}(b), respectively.
\begin{figure}
\begin{center}
{\includegraphics[scale=0.5]{ffigure2a.eps}}~~
{\includegraphics[scale=0.5]{ffigure2b.eps}}
(a)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(b)
\end{center}
\caption{\small Shock and rarefaction wave curves of the first and the second families:
(a) shows SW1 (dashed) and RW1 (solid) at the left state $(u_L,q_L)=(1,5)$.
(b) shows inverse SW2 (dashed, red) and inverse RW2 (solid, red)
at the right state $(u_R,q_R)=(0.7,7)$.}
\label{Fig2}
\end{figure}
We next aim to prove existence of solution for arbitrary Riemann initial data
without necessarily assuming a small enough initial jump.
The only essential hypothesis is that both left and right states are above the critical curve $q_{crit}= u^2/2$:
\begin{equation}
\label{assumpt}
q_L\geq u_L^2/2, \ \ \ \ q_R \geq u_R^2/2.
\end{equation}
This assumptions is of course natural given the change of variables $q=\frac{u^2+v^2}{2}$.
Nevertheless, this condition makes complicates our task
since is also needs to be shown that the Lax admissible solution
to a Riemann problem remains in the area $q\geq u^2/2$.
To this end, the following lemma will be useful.
\begin{lem}
\label{L1}
The function $q_{crit}(u)=\frac{u^2}{2}$ satisfies \eqref{RW2}.
\end{lem}
\begin{proof}
The proof is obvious and we omit it.
\end{proof}
The above lemma is important since, according to the uniqueness of solutions
to the Cauchy problem for ordinary differential equations,
it shows that if the left and right states $(u_L,q_L)$ and $(u_R,q_R)$
are above the curve $q_{crit}(u)=\frac{u^2}{2}$,
then the simple waves (SW1, SW2, RW1, RW2) connecting the states will remain above it
which means that we can use the solution to \eqref{Transformed system}
to obtain a solutions of \eqref{Brio System}
since the square root giving the function $v=\sqrt{2q-u^2}$ will be well defined.
Concerning the Riemann problem, we have the following theorem.
\begin{thm}
\label{transf-thm}
Given a left state $(u_L,q_L)$ and a right state $(u_R,q_R)$, so that both are above the critical curve
$q_{crit}(u)=\frac{u^2}{2}$ i.e. we have $q_L\geq u_L^2/2$ and $q_R\geq u_R^2/2$,
the states $(u_L,q_L)$ and $(u_R,q_R)$ can be connected
Lax admissible shocks and rarefaction waves via a middle state belonging to the domain $q>u^2/2$.
\end{thm}
\begin{figure}
\begin{center}
{\includegraphics[scale=0.5]{Figure3.eps}}
\end{center}
\caption{\small Admissible connections between a given left state $(u_L,q_L)$ and a right state
can be classified into four regions in the phase plane.}
\label{Fig3}
\end{figure}
\begin{proof}
In order to find a connection between $(u_L,q_L)$ and $(u_R,q_R)$, we first draw the waves of the
first family (SW1 and RW1) through $(u_L,q_L)$ and waves of the second family (SW2 and RW2) through $(u_R,q_R)$.
The point of intersection will be the middle state through which we connect $(u_L,q_L)$ and $(u_R,q_R)$
(see Figure \ref{Fig4} for different dispositions of $(u_L,q_L)$ and $(u_R,q_R)$).
In this case, the intersection point will be unique which can be seen by considering
the four possible dispositions of the states $(u_L,q_L)$ and $(u_R,q_R)$ shown in Figure \ref{Fig4}:
\begin{itemize}
\ite
For right states in region $I$: RW1 followed by RW2;
\ite
For right states in region $II$: SW1 followed by RW2;
\ite
For right states in region $III$: RW1 followed by SW2;
\ite
For right states in region $IV$: SW1 followed by SW2;
\end{itemize}
Properties of the curves of the first and second families are provided in a)-d) above. The growth properties give also existence as we shall show in detail in the sequel of the proof.
Firstly, we remark that SW1 and RW1 emanating from $(u_L,q_L)$ cover the entire $q\geq u^2/2$ domain (see Figure \ref{Fig2}(a)). In other words, we have for the curve $q_R$ defining the SW1 by \eqref{SW1}:
\begin{equation*}
\lim\limits_{u_R\to - \infty} q(u_R)=\infty,
\end{equation*}
implying that the SW1 will take all $q$-values for $q_R> q_L$. More precisely, for every $q_R>q_L$ there exists $u_R<u_L$ such that $q_R(u_R)=q_R$ where $q_R$ is given by \eqref{SW1}.
\begin{figure}
\begin{center}
{\includegraphics[scale=0.45]{ffigure4a.eps}}~~
{\includegraphics[scale=0.45]{ffigure4b.eps}}\\
(a)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(b)
{\includegraphics[scale=0.45]{ffigure4c.eps}}~~
{\includegraphics[scale=0.45]{ffigure4d.eps}}\\
(c)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(d)
\end{center}
\caption{\small Shock and rarefaction wave curves of the first and the second families.
At the left state $L=(u_L,q_L)$, the curves SW1 (dashed), SW2 (dashed), RW1 (solid),
and RW2 (solid) are drawn in black. The inverse curves at the right state $R=(u_R,q_R)$
are indicated in red: SW1 (dashed), SW2 (dashed), RW1 (solid) and RW2 (solid).
Panel (a) shows the situation for region I,
Panel (b) shows the situation for region II,
Panel (c) shows the situation for region III and
Panel (d) shows the situation for region IV.
}
\label{Fig4}
\end{figure}
As for the RW1, it holds for $q$ given by \eqref{RW1} that
\begin{equation*}
\frac{dq}{du}-u \leq -1 \implies \frac{dq}{du}\leq u-1,
\end{equation*}
which means that the RW1 curve emanating from any $(u_L,q_L)$ for which $q_L>u_L^2/2$ will intersect the curve $q_{crit}=\frac{u^2}{2}$ (since $\frac{d q_{crit}}{d u}=u>u-1 \geq \frac{dq}{du}$) at some $u_R>u_L$ as shown in Figure \ref{Fig1}, b).
Now, we turn to the waves of the second family. Let us fix the right state $(u_R,q_R)$. We need to compute the inverse waves (i.e. for the given right state, we need to compute curves consisting of appropriate left states (see Figure \ref{Fig2}(b)). The inverse rarefaction curve of the second family is given by the equation \eqref{RW2}, but we need to take values for $u_R<u_L$ (opposite to the ones given in \eqref{RW2}). As for the inverse SW2, we compute from \eqref{Rankine-Hugoniot} and \eqref{Rankine-Hugoniot1} the value $q_L$:
\begin{align}\label{SW2-inv}
\nonumber
q_L= q_R&-\frac{1}{2} \big(u_L-u_R\big)\big(2u_L-1\big)\\
& + \frac{(u_L-u_R)}{2}\sqrt{8q_R+1+\Sfrac{4u_L^2}{3}-\Sfrac{8u_L u_R}{3}-\Sfrac{8u_R^2}{3}-2u_L+2u_R},
\end{align}
for $u_R<u_L$.
Clearly, the RW2 cannot intersect the critical line $q_{crit}=\frac{u^2}{2}$ since $q_{crit}$ satisfy \eqref{RW2} (see Lemma \ref{L1}) and the intersection would contradict uniqueness of solution to the Cauchy problem for \eqref{RW2}. However, a solution to \eqref{RW2} with the initial conditions $q(u_R)=q_R>u_R^2/2$ will converge toward the line $q_{crit}=u^2/2$ since for $q$ given by \eqref{RW2} we have
\begin{equation*}
\frac{dq}{du}-u \geq 0 \ \ {\rm and} \ \ \frac{dq}{du}\Big|_{(u,u^2/2)}-u=0,
\end{equation*}
implying that $q$ will decrease toward $q_{crit}=u^2/2$ and that they will merge as $u_L\to -\infty$ (see Figure \ref{Fig2}(b)). As for the inverse SW2 given by \eqref{SW2-inv}, we see that
\begin{equation*}
\lim\limits_{u_L\to \infty} q(u_L)=\infty,
\end{equation*}
which eventually imply that the 1-wave family emanating from $(u_L,q_L)$ must intersect with the inverse 2-wave family emanating from $(u_R,q_R)$ somewhere in the domain $q>u^2/2$ (see Figure \ref{Fig4}
for several dispositions of the left and right states).
Finally, we remark that according to the previous analysis, it follows that the intersection between curves of the first and the second family is unique. \end{proof}
\section{Admissibility conditions for $\delta$-shock wave solution to the original Brio System}
Our starting point is that the system original Brio system \eqref{Brio System}
is based on conservation of quantities which are not necessarily physically conserved,
and that the transformed system \eqref{Transformed system} is a closer representation
of the physical phenomenon to be described. The second principle is that $\delta$-distribution
represents actually a defect in the model and thus,
it should be necessarily present as a part of non-regular solutions to \eqref{Brio System}.
Moreover, the regular part of a solution to \eqref{Brio System}
should be an admissible solution to \eqref{Transformed system}.
Having these requirements in mind, we are able to introduce admissibility conditions
for a $\delta$-type solution to \eqref{Brio System}.
Let us first recall the characteristic speeds for \eqref{Brio System}. Following the \cite{HL}, we see immediately that
\begin{equation}
\label{lambdaB}
\lambda_1(u,v)=u-1/2-\sqrt{v^2+1/4}, \ \ \lambda_2(u,v)=u-1/2+\sqrt{v^2+1/4}.
\end{equation} The shock speed for \eqref{Brio System} for the shock determined by the left state $(U_L,V_L)$ and the right state $(U_R,V_R)$ is given by
\begin{equation}
\label{shockB}
s=\frac{U_L+U_R}{2}+\frac{V_L^2-V_R^2}{2(U_L-U_R)}.
\end{equation}
Now, we can formulate admissibility conditions for $\delta$-type solution to \eqref{Brio System} in the sense of Definition \ref{def-gen}. We shall require that the real part of $\delta$-type solution to \eqref{Brio System} satisfy the energy-velocity conservation system \eqref{Transformed system} and that the number of $\delta$-distributions appearing as part of the solution to \eqref{Brio System} is minimal.
\begin{df}
\label{admissibility} We say that the pair of distributions
$u=U+\alpha(x,t)\delta(\Gamma)$ and $v=V+\beta(x,t)\delta(\Gamma)$ satisfying Definition \ref{def-gen} with $f(u,v)=\frac{u^2+v^2}{2}$ and $g(u,v)=v(u-1)$ is an admissible $\delta$-type solution to \eqref{Brio System}, \eqref{riemann} if
\begin{itemize}
\item The regular parts of the distributions $u$ and $v$ are such that the functions $U$ and $q=(U^2+V^2)/2$ represent Lax-admissible solutions to \eqref{Transformed system} with the initial data
\begin{equation}
\label{Triemann}
u|_{t=0}=U_0, \ \ q|_{t=0}= q_0=(U_0^2+V_0^2)/2.
\end{equation}
\item For every $t\geq 0$, the support of the $\delta$-distributions appearing in $u$ and $v$ is of minimal cardinality.
\end{itemize}
\end{df} To be more precise, the second requirement in the last definition means that the admissible solution will have ``less" $\delta$-distributions as summands in the $\delta$-type solution than any other $\delta$-type solution to \eqref{Brio System}, \eqref{riemann}. We have the following theorem:
\begin{thm}
There exists a unique admissible $\delta$-type solution to \eqref{Brio System}, \eqref{riemann}.
\end{thm}
\begin{proof}
We divide the proof into two cases:
In the first case, we consider initial data such that both left and right states
of the function $V_0$ have the same sign. In the second case, we consider the initial data where left and right states
of the function $V_0$ have the opposite sign.
In the first case, we first solve \eqref{Transformed system} with the initial data $U_0$ and $q_0=(U_0^2+V_0^2)/2$.
According to Theorem \ref{transf-thm}, there exists a unique Lax admissible solution
to the problem denoted by $(U,q)$. Using this solution, we define $V=\sqrt{2q-U^2}$
if the sign of $V_0$ is positive and $V=-\sqrt{2q-U^2}$ if the sign of $V_0$ is negative.
To compute $\alpha$ and $\beta$ in \eqref{delta-sol}, we compute the Rankine-Hugoniot deficit
if it exists at all. According to Theorem \ref{transf-thm}
there are four possibilities.
\begin{figure}
\begin{center}
{\includegraphics[scale=0.8]{Figure5.eps}}
\end{center}
\caption{\small Admissible connection between rarefaction wave curves of the first and second families}
\label{Fig5}
\end{figure}
\begin{itemize}
\item Region $I$:
The states $(U_L,q_L)$ and $(U_R,q_R)$ are connected by a combination of RW1 and RW2
via the state $(U_M,q_M)$. In this situation, we do not have any Rankine-Hugoniot deficit
since the solution $(u,q)$ to \eqref{Transformed system} is continuous.
Thus, we simply write $(u,v)=(u,\sqrt{2q-u^2})$
and this is the solution to \eqref{Brio System}, \eqref{riemann}.
The solution is plotted in Figure \ref{Fig5}.
As for the uniqueness, we know that the function $u$ is unique
since it is the Lax admissible solution to \eqref{Transformed system}
with the initial data \eqref{Triemann}.
The function $v$ is determined by the unique functions $u$ and $q$ via
$$
v=\pm \sqrt{2q-u^2}.
$$
Thus, $v$ could change sign so that we connect $V_L$ by $V_{M1}$
and then skip to $-V_{M1}$ on $v=-\sqrt{2q-u^2}$ and then connect it by $-V_{M2}$.
From here we connect to $V_{M2}$ located on the original curve $v=\sqrt{2q-u^2}$
and then connect $V_{M2}$ to $V_M$. Finally, we connect $V_M$ with $V_R$.
The procedure is illustrated in Figure \ref{Fig6}.
However, since we imposed the requirement that the solutions have a minimal number
of $\delta$-distributions and we cannot connect the states $(U_{M1},V_{M1})$
and $(U_{M1},-V_{M1})$ using the $\delta$-shock since such a choice would yield a solutions
with a higher number of singular parts than the previously described solution.
Thus the shock connecting the states $(U_{M1},V_{M1})$ and $(U_{M1},-V_{M1})$
cannot be singular, (i.e. there can be no Rankine-Hugoniot deficit),
and therefore the speed $s$ of the shock must satisfy the Rankine-Hugoniot condition
$$
s=U_{M1}.
$$
On the other hand, the characteristic speeds of $(U_{M1},V_{M1})$ and $(U_{M1},-V_{M1})$
are $\lambda_1(U_{M1},V_{M1})=\lambda_1(U_{M1},-V_{M1}) \neq s$, and since these are equal,
the shock connection between $(U_{M1},V_{M1})$ and $(U_{M1},-V_{M1})$
is impossible with Rankine-Hugoniot condition satisfied.
Similarly, the same requirement makes it impossible to connect $(U_{M2},V_{M2})$ and $(U_{M2},-V_{M2})$
by a $\delta$-shock. In this case, the shock speed satisfies the Rankine-Hugoniot condition
$$
s=U_{M2}.
$$
Furthermore, we have equality of speeds $\lambda_2(U_{M2},V_{M2})=\lambda_2(U_{M2},-V_{M2})$,
but we have the contrasting inequality $\lambda_2(U_{M2},V_{M2})=\lambda_2(U_{M2},-V_{M2})\neq s$
implying that a shock connection between $(U_{M2},V_{M2})$ and $(U_{M2},-V_{M2})$ is not possible
if the Rankine-Hugoniot condition is satisfied.
The same procedure leads to the conclusion that a $\delta$-shock connection
between $(U_{M},V_{M})$ and $(U_{M},-V_{M})$ is impossible with Rankine-Hugoniot condition satisfied.
Hence, the only possible connection of $(U_L,V_L)$ and $(U_R,V_R)$ is
by the combination RW1 and RW2 via the state $(U_M,V_M)$.
Consequently, we remark that RW1 and RW2 corresponding to \eqref{Transformed system}
are transformed via $(u,q)\mapsto (u,\sqrt{2q-u^2})$ into RW1 and RW2 corresponding
to \eqref{Brio System}
(since $q$ is the entropy function for \eqref{Brio System},
and RW1 and RW2 are smooth solutions to \eqref{Transformed system}).
\begin{figure}
\begin{center}
{\includegraphics[scale=0.7]{Figure6.eps}}
\end{center}
\caption{\small Nonadmissible connection between rarefaction wave curves of the first and the second families}
\label{Fig6}
\end{figure}
\item Region $II$:
The states $(U_L,q_L)$ and $(U_R,q_R)$ are connected by the combination SW1 and RW2 via the state $(U_M,q_M)$.
Unlike the previous case, we have a shock wave in \eqref{Transformed system},
and we will necessarily have a Rankine-Hugoniot deficit in the original system \eqref{Brio System}.
We thus define
\begin{equation}
\label{S1R2}
(u,v)=(u,\sqrt{2q-u^2})+(0,\beta(t) \delta(x-ct)),
\end{equation}
where $c$ is the speed of the SW1 connecting the states $(U_L,q_L)$ and $(U_M,q_M)$
in \eqref{Transformed system}. The speed $c$ is given by \eqref{RH-def1}
as well as the corresponding Rankine-Hugoniot deficit $\beta(t)$:
\begin{equation}
\label{RHdef}
c=\frac{\frac{U_L^2+V_L^2}{2}-\frac{U_R^2+V_R^2}{2}}{U_L-U_R}, \ \ \beta(t)=(c(V_L-V_R)-(V_L(U_L-1)-V_R(U_R-1))t.
\end{equation}
Concerning the other possible solutions, as in the previous item,
we can only split the curve connecting $(U_L,V_L)$ and $(U_M,V_M)$ into several new curves
e.g. by connecting the states $(U_L,V_L)$ and $(U_{M1},V_{M1})$,
then the (opposite with respect to $v$) states $(U_{M1},V_{M1})$ and $(U_{M1},-V_{M1})$,
then $(U_{M1},-V_{M1})$ and $(U_{M2},-V_{M2})$,
then $(U_{M2},-V_{M2})$ and $(U_{M2},V_{M2})$ etc. until we reach $(U_M,V_M)$.
The states $(U_{M1},V_{M1})$ and $(U_{M1},-V_{M1})$ can be connected only by the shock satisfying
the Rankine-Hugoniot conditions
(due to the minimality condition on $\delta$-shocks, we cannot have a Rankine-Hugoniot deficit).
Since we cannot have the Rankine-Hugoniot deficit, as in the previous item,
we must connect the various states with shock waves satisfying the Rankine-Hugoniot conditions,
and at the same time being equal to the speed $c$
(the speed of the SW1 connecting the states $(U_L,q_L)$ and $(U_M,q_M)$ in \eqref{Transformed system}).
This is obviously never fulfilled i.e. the only solution in this case is \eqref{S1R2}.
\item Region $III$:
The states $(U_L,q_L)$ and $(U_R,q_R)$ are connected by the combination RW1 and SW2 via the state $(U_M,q_M)$.
The analysis for the existence and uniqueness proceeds along the same lines as the first two cases.
The admissible (and thus unique) $\delta$-type solution in this case has the form:
\begin{equation}
\label{R1S2}
(u,v)=\big(u,\sqrt{2q-u^2}\big)+\big(0,\beta(t) \delta(x-ct)\big),
\end{equation}
where $c$ in this case represents the speed of the SW2 connecting
the states $(U_R,q_R)$ and $(U_M,q_M)$ in \eqref{Transformed system}.
The speed $c$ and the corresponding Rankine-Hugoniot deficit $\beta(t)$ are given in \eqref{RH-def1}
and explicitly expressed as in \eqref{RHdef}. The solution structure is represented by
\begin{equation*}
(U_L,V_L)\xrightarrow{RW1}(U_M,V_M)\xrightarrow{SW2}(U_R,V_R),
\end{equation*}
where the $\delta$-shock propagates at the speed $c=\lambda_1(U_M,V_M)=\lambda_1(U_M,-V_M)$.
Notice that it is possible to generate infinitely many non-admissible
(in the sense of Definition \ref{admissibility}) solutions
(in the sense of Definition \ref{def-gen}) by partitioning the rarefaction wave
of the first family that connects the states $(U_L,V_L)$ and $(U_M,V_M)$.
The solution is constructed by connecting $(U_L, V_L)$ and $(U_{M1},V_{M1})$
by RW1 and then passing over to $(U_{M1},-V_{M1})$ by a shock
which satisfies the Rankine-Hugoniot conditions $s=U_{M1}$.
The procedure is advanced to connect all the finite possible states $(U_{Mk},V_{Mk})$ and $(U_{Mk},-V_{Mk})$
by a shock satisfying both the Rankine-Hugoniot conditions $s=U_{Mk}$, where $k\in\mathbb{Z}_+$
and the speed of the shock of the second family connecting the states $(U_M,-V_M)$ and $(U_R,V_R)$.
This process is carried out prior to the state $(U_M,V_M)$
and the shocks connecting pairs of states cannot be admissible in the sense of Definition 3.1
due to the minimality condition.
Consequently, the only solution admissible in this sense is \eqref{R1S2}.
\item Region $IV$:
The states $(U_L,q_L)$ and $(U_R,q_R)$ are connected by the combination SW1 and SW2 via the state $(U_M,q_M)$.
The presence of shocks in this case will necessarily introduce Rankine-Hugoniot
deficit in \eqref{Brio System}. The solution is constructed by solving
\eqref{Transformed system} for the solution $(u,q)$ and then go back to
\eqref{Brio System} to obtain the admissible $\delta$-type solution
\begin{equation}\label{S1andS2}
(u,v)=\big(u,\sqrt{2q-u^2}\big)+\big(0,\beta_1(t)\delta(x-c_1t)\big)+\big(0,\beta_2(t)\delta(x-c_2t)\big),
\end{equation}
where $c_1$ and $c_2$ given by the expressions
\begin{equation}
c_1=\frac{\frac{U_L^2+V_L^2}{2} - \frac{U_M^2+V_M^2}{2}}{U_L-U_M} \ \ \ \ \text{and} \ \ \ \ \ c_2=\frac{\frac{U_M^2+V_M^2}{2} - \frac{U_R^2+V_R^2}{2}}{-U_M-U_R},
\end{equation}
are the speeds of the shocks SW1 and SW2 respectively. The Rankine-Hugoniot
deficits $\beta_1(t)$ and $\beta_2(t)$ are expressed as in \eqref{RHdef} for
the appropriate states. The analysis for uniqueness of \eqref{S1andS2} is
similar to the above cases except that all the elementary waves involved in this
case are shocks.
\end{itemize}
Now, assume that $V_L>0$ and $V_R<0$. It was shown in \cite{HL} that in this case,
the Riemann problem \eqref{Brio System}, \eqref{riemann} does not admit a Lax admissible solution,
even for initial data with small variation.
In order to get an admissible $\delta$-type solution, as before,
we solve \eqref{Transformed system} with $(U_0,q_0)$ as the initial data.
The obtained solution connects $(U_L,q_L)$ with $(U_R,q_R)$ by Lax admissible waves through a
middle state $(U_M,q_M)$. Next, we go back to the original system \eqref{Brio System}
by connecting $(U_L,V_L)$ with $(U_M,\sqrt{2q_M-U_M^2})$ by an elementary wave
containing the corresponding Rankine-Hugoniot deficit corrected by the $\delta$-shock wave.
Then, we connect $(U_M,\sqrt{2q_M-U_M^2})$ with $(U_M,-\sqrt{2q_M-U_M^2})$
by the shock wave whose speed will obviously be $U_M$.
Finally, we connect $(U_M,-\sqrt{2q_M-U_M^2})$ with $(U_R,V_R)$
by an elementary wave containing corresponding Rankine-Hugoniot deficit
corrected by the $\delta$-shock wave.
Let us first show it is possible to apply the described procedure.
We again need to split considerations into four possibilities depending
on how the states $(U_L,q_L)$ and $(U_R,q_R)$ are connected.
\begin{figure}
\begin{center}
{\includegraphics[scale=0.8]{Figure7.eps}}
\end{center}
\caption{\small Admissible connection between rarefaction wave curves of the first and second families
in the case when the left state has $V_L < 0$ and the right state has $V_R > 0$. In this case, a shock
connecting the states $(-V_M,U_M)$ and $(V_M,U_M)$ has to be fitted between the rarefaction curves.
It is shown in the part of the proof pertaining to region I that this shock has the required speed.
}
\label{Fig7}
\end{figure}
\begin{itemize}
\item Region $I$:
The states $(U_L,q_L)$ and $(U_R,q_R)$ are connected by RW1 and RW2 via the middle state $(U_M,q_M)$.
It is clear that we can connect $(U_L,V_L)$ with $(U_M,\sqrt{2q_M-U_M^2})$ using RW1
(it is the same for both equations since RW1 and RW2 are smooth solutions to \eqref{Transformed system}).
Also, we can connect $(U_M,-\sqrt{2q_M-U_M^2})$ with $(U_R,V_R)$ using RW2.
We need to prove that the shock wave connecting $(U_M,\sqrt{2q_M-U_M^2})$
and $(U_M,-\sqrt{2q_M-U_M^2})$ has a speed which is
between $\lambda_1(U_M,\sqrt{2q_M-U_M^2})$ and $\lambda_2(U_M,-\sqrt{2q_M-U_M^2})$.
In other words, we need to check
\begin{align*}
U_M - \frac{1}{2} - \sqrt{V_M^2 + \Sfrac{1}{4}} \leq U_M \leq U_M - \frac{1}{2} + \sqrt{V_M^2 + \Sfrac{1}{4}}
\end{align*}
which is obviously correct. This configuration is depicted in Figure \ref{Fig7}.
\item Region $II$:
The states $(U_L,q_L)$ and $(U_R,q_R)$ are connected by SW1 and SW2 via the middle state $(U_M,q_M)$.
As in the previous item, we connect $(U_L,V_L)$ with $(U_M,\sqrt{2q_M-U_M^2})$ this time using the SW1 from \eqref{Transformed system} which will induce the Rankine-Hugoniot deficit in \eqref{Brio System}. Then, we skip from $(U_M,\sqrt{2q_M-U_M^2})$ to $(U_M,-\sqrt{2q_M-U_M^2})$ using the standard shock wave (the one satisfying the Rankine-Hugoniot conditions), and finally we go from $(U_M,-\sqrt{2q_M-U_M^2})$ to $(U_R,V_R)$ using the SW2 from \eqref{Transformed system} and corrected with an appropriate $\delta$-shock. More precisely, the admissible $\delta$-type solution will have the form:
\begin{equation}
\label{S1S2}
\begin{split}
u(t,x)&=U_L+(U_M-U_L)(H(x-c_1 t)-H(x-ct))\\&+(-U_M-U_L)(H(x-c t)-H(x-c_2t))+(U_R-U_L)H(x-c_2t)\\
v(t,x)&=V_L+(V_M-V_L) (H(x-c_1 t)-H(x-ct))\\&+(V_M-V_L)(H(x-c t)-H(x-c_2t))+(V_R-V_L)H(x-c_2t)\\&+\beta_1(t) \delta(x-c_1t)+\beta_2(t) \delta(x-c_2t),
\end{split}
\end{equation}where $c_1$ is the speed of the SW1 connecting $(U_L,q_L)$ with $(U_M,q_M)$ in \eqref{Transformed system}, $c_2$ is the speed of the SW2 connecting $(U_M,q_M)$ with $(U_R,q_R)$ in \eqref{Transformed system}, while $c$ is the speed of the shock connecting $(U_M,-\sqrt{2q_M-U_M^2})$ with $(U_M,\sqrt{2q_M-U_M^2})$ and it is given by the Rankine-Hugoniot conditions from \eqref{Brio System}. The deficits $\beta_1$ and $\beta_2$ are given by Theorem \ref{thm-cnl} (see \eqref{RHdef} for the analogical situation).
However, we still need to prove that \eqref{S1S2} is well defined, i.e. that
\begin{align*}
&c_1 \leq c \leq c_2 \ \ \implies \\
& \frac{2 U_M-1-\sqrt{8 q_L+1+\frac{4 U_M^2}{3}-\frac{8 U_L U_M}{3}-\frac{8 U_L^2}{3}-2 U_M + 2 U_L}}{2} \leq U_M \\&
\leq \frac{2U_M-1+\sqrt{8q_R+1+\frac{4U_M^2}{3}-\frac{8 U_M U_R}{3}-\frac{8 U_R^2}{3}-2U_M+2U_R}}{2}
\end{align*}
which is also clearly true.
\item Region $III$:
The states $(U_L,q_L)$ and $(U_R,q_R)$ are connected by RW1 and SW2 via the middle state $(U_M,q_M)$.
This case, as well as the following one, is handled by combining the previous two cases.
\item Region $IV$:
The states $(U_L,q_L)$ and $(U_R,q_R)$ are connected by SW1 and RW2 via the middle state $(U_M,q_M)$.
\end{itemize}
Uniqueness is obtained by arguing as in the first part of the proof.
\end{proof}
\section*{Acknowledgements}
This research was supported by the Research Council of Norway under grant no. 213747/F20
and grant no. 239033/F20.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,277
|
IC 2463 ist eine Galaxie vom Hubble-Typ S im Sternbild Löwe an der Ekliptik, die schätzungsweise 417 Millionen Lichtjahre von der Milchstraße entfernt ist.
Das Objekt wurde am 14. April 1896 von Stéphane Javelle entdeckt.
Einzelnachweise
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,601
|
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